6337:
5639:
91:, it allows us to make better posterior estimates as more observations become available. Thus unlike non-Bayesian approach where parameters of interest are assumed to be deterministic, but unknown constants, the Bayesian estimator seeks to estimate a parameter that is itself a random variable. Furthermore, Bayesian estimation can also deal with situations where the sequence of observations are not necessarily independent. Thus Bayesian estimation provides yet another alternative to the MVUE. This is useful when the MVUE does not exist or cannot be found.
6332:{\displaystyle {\begin{aligned}C_{e}&=\operatorname {E} \{({\hat {x}}-x)({\hat {x}}-x)^{T}\}\\&=\operatorname {E} \{({\hat {x}}-x)(W(y-{\bar {y}})-(x-{\bar {x}}))^{T}\}\\&=\underbrace {\operatorname {E} \{({\hat {x}}-x)(y-{\bar {y}})^{T}\}} _{0}W^{T}-\operatorname {E} \{({\hat {x}}-x)(x-{\bar {x}})^{T}\}\\&=-\operatorname {E} \{(W(y-{\bar {y}})-(x-{\bar {x}}))(x-{\bar {x}})^{T}\}\\&=\operatorname {E} \{(x-{\bar {x}})(x-{\bar {x}})^{T}\}-W\operatorname {E} \{(y-{\bar {y}})(x-{\bar {x}})^{T}\}\\&=C_{X}-WC_{YX}.\end{aligned}}}
20762:
4738:
20403:
62:. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated. Since the posterior mean is cumbersome to calculate, the form of the MMSE estimator is usually constrained to be within a certain class of functions. Linear MMSE estimators are a popular choice since they are easy to use, easy to calculate, and very versatile. It has given rise to many popular estimators such as the
22644:
13741:
4349:
20757:{\displaystyle {\begin{aligned}{\hat {x}}&=\left(1^{T}{\frac {1}{\sigma _{Z}^{2}}}I1+{\frac {1}{\sigma _{X}^{2}}}\right)^{-1}1^{T}{\frac {1}{\sigma _{Z}^{2}}}Iy\\&={\frac {1}{\sigma _{Z}^{2}}}\left({\frac {N}{\sigma _{Z}^{2}}}+{\frac {1}{\sigma _{X}^{2}}}\right)^{-1}1^{T}y\\&={\frac {\sigma _{X}^{2}}{\sigma _{X}^{2}+\sigma _{Z}^{2}/N}}{\bar {y}},\end{aligned}}}
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20185:
841:
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4733:{\displaystyle {\begin{aligned}\operatorname {E} \{({\hat {x}}-x)(y-{\bar {y}})^{T}\}&=\operatorname {E} \{(W(y-{\bar {y}})-(x-{\bar {x}}))(y-{\bar {y}})^{T}\}\\&=W\operatorname {E} \{(y-{\bar {y}})(y-{\bar {y}})^{T}\}-\operatorname {E} \{(x-{\bar {x}})(y-{\bar {y}})^{T}\}\\&=WC_{Y}-C_{XY}.\end{aligned}}}
22899:
16418:
82:
setting with quadratic cost function. The basic idea behind the
Bayesian approach to estimation stems from practical situations where we often have some prior information about the parameter to be estimated. For instance, we may have prior information about the range that the parameter can assume; or
10280:
In many real-time applications, observational data is not available in a single batch. Instead the observations are made in a sequence. One possible approach is to use the sequential observations to update an old estimate as additional data becomes available, leading to finer estimates. One crucial
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14005:, then after receiving another set of measurements, we should subtract out from these measurements that part that could be anticipated from the result of the first measurements. In other words, the updating must be based on that part of the new data which is orthogonal to the old data.
12510:
87:(MVUE) where absolutely nothing is assumed to be known about the parameter in advance and which does not account for such situations. In the Bayesian approach, such prior information is captured by the prior probability density function of the parameters; and based directly on
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is now a random variable, it is possible to form a meaningful estimate (namely its mean) even with no measurements. Every new measurement simply provides additional information which may modify our original estimate. Another feature of this estimate is that for
22639:{\displaystyle {\begin{aligned}w_{1}&={\frac {1/\sigma _{Z_{1}}^{2}}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}},\\w_{2}&={\frac {1/\sigma _{Z_{2}}^{2}}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}}.\end{aligned}}}
24251:
1948:
of the measurements. For random vectors, since the MSE for estimation of a random vector is the sum of the MSEs of the coordinates, finding the MMSE estimator of a random vector decomposes into finding the MMSE estimators of the coordinates of X separately:
21142:
13736:{\displaystyle {\begin{aligned}{\hat {x}}_{k}&={\hat {x}}_{k-1}+C_{e_{k-1}{\tilde {Y}}_{k}}C_{{\tilde {Y}}_{k}}^{-1}(y_{k}-{\bar {y}}_{k})\\&={\hat {x}}_{k-1}+C_{e_{k-1}}A^{T}(AC_{e_{k-1}}A^{T}+C_{Z})^{-1}(y_{k}-A{\hat {x}}_{k-1})\end{aligned}}}
23444:
921:
330:
16175:
scalar measurements. The use of scalar update formula avoids matrix inversion in the implementation of the covariance update equations, thus improving the numerical robustness against roundoff errors. The update can be implemented iteratively as:
2790:
are jointly
Gaussian, it is not necessary to make this assumption, so long as the assumed distribution has well defined first and second moments. The form of the linear estimator does not depend on the type of the assumed underlying distribution.
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One possibility is to abandon the full optimality requirements and seek a technique minimizing the MSE within a particular class of estimators, such as the class of linear estimators. Thus, we postulate that the conditional expectation of
23943:
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2396:; but this method still requires the evaluation of expectation. While these numerical methods have been fruitful, a closed form expression for the MMSE estimator is nevertheless possible if we are willing to make some compromises.
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5486:{\displaystyle {\begin{aligned}C_{\hat {X}}&=\operatorname {E} \{({\hat {x}}-{\bar {x}})({\hat {x}}-{\bar {x}})^{T}\}\\&=W\operatorname {E} \{(y-{\bar {y}})(y-{\bar {y}})^{T}\}W^{T}\\&=WC_{Y}W^{T}.\\\end{aligned}}}
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83:
we may have an old estimate of the parameter that we want to modify when a new observation is made available; or the statistics of an actual random signal such as speech. This is in contrast to the non-Bayesian approach like
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20180:{\displaystyle {\begin{aligned}&\operatorname {E} \{y\}=0,\\&C_{Y}=\operatorname {E} \{yy^{T}\}=\sigma _{X}^{2}11^{T}+\sigma _{Z}^{2}I,\\&C_{XY}=\operatorname {E} \{xy^{T}\}=\sigma _{X}^{2}1^{T}.\end{aligned}}}
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836:{\displaystyle \operatorname {tr} \left\{\operatorname {E} \{ee^{T}\}\right\}=\operatorname {E} \left\{\operatorname {tr} \{ee^{T}\}\right\}=\operatorname {E} \{e^{T}e\}=\sum _{i=1}^{n}\operatorname {E} \{e_{i}^{2}\}.}
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Suppose that a musician is playing an instrument and that the sound is received by two microphones, each of them located at two different places. Let the attenuation of sound due to distance at each microphone be
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In many cases, it is not possible to determine the analytical expression of the MMSE estimator. Two basic numerical approaches to obtain the MMSE estimate depends on either finding the conditional expectation
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10688:{\displaystyle {\begin{aligned}p(x_{k}|y_{1},\ldots ,y_{k})&\propto p(y_{k}|x,y_{1},\ldots ,y_{k-1})p(x_{k}|y_{1},\ldots ,y_{k-1})\\&=p(y_{k}|x_{k})p(x_{k}|y_{1},\ldots ,y_{k-1}).\end{aligned}}}
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Note that except for the mean and variance of the error, the error distribution is unspecified. How should the two polls be combined to obtain the voting prediction for the given candidate?
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or finding the minima of MSE. Direct numerical evaluation of the conditional expectation is computationally expensive since it often requires multidimensional integration usually done via
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fraction of votes. Since some error is always present due to finite sampling and the particular polling methodology adopted, the first pollster declares their estimate to have an error
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The repetition of these three steps as more data becomes available leads to an iterative estimation algorithm. The generalization of this idea to non-stationary cases gives rise to the
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21299:{\displaystyle \operatorname {LMMSE} =\sigma _{X}^{2}-\sigma _{\hat {X}}^{2}={\Big (}{\frac {\sigma _{Z}^{2}}{\sigma _{X}^{2}+\sigma _{Z}^{2}/N}}{\Big )}{\frac {\sigma _{X}^{2}}{N}}.}
8813:
8142:
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499:{\displaystyle \operatorname {MSE} =\operatorname {tr} \left\{\operatorname {E} \{({\hat {x}}-x)({\hat {x}}-x)^{T}\}\right\}=\operatorname {E} \{({\hat {x}}-x)^{T}({\hat {x}}-x)\},}
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A few weeks before the election, two independent public opinion polls were conducted by two different pollsters. The first poll revealed that the candidate is likely to get
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The repeated use of the above two equations as more observations become available lead to recursive estimation techniques. The expressions can be more compactly written as
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22894:{\displaystyle \sigma _{\hat {X}}^{2}={\frac {1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}}\sigma _{X}^{2},}
9834:
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17119:
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12708:, which is also referred to as innovation or residual. It is more convenient to represent the linear MMSE in terms of the prediction error, whose mean and covariance are
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also been
Gaussian, then the estimator would have been optimal. Notice, that the form of the estimator will remain unchanged, regardless of the apriori distribution of
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Consider a variation of the above example: Two candidates are standing for an election. Let the fraction of votes that a candidate will receive on an election day be
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Here, since the denominator term is constant, the poll with lower error is given higher weight in order to predict the election outcome. Lastly, the variance of
18051:
16413:{\displaystyle w_{k+1}^{(\ell )}={\frac {C_{e_{k}}^{(\ell )}A_{k+1}^{(\ell )T}}{C_{Z_{k+1}}^{(\ell )}+A_{k+1}^{(\ell )}C_{e_{k}}^{(\ell )}(A_{k+1}^{(\ell )T})}}}
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2611:. The linear MMSE estimator is the estimator achieving minimum MSE among all estimators of such form. That is, it solves the following optimization problem:
7534:
6852:{\displaystyle {\hat {x}}={\frac {\sigma _{XY}}{\sigma _{Y}^{2}}}(y-{\bar {y}})+{\bar {x}}=\rho {\frac {\sigma _{X}}{\sigma _{Y}}}(y-{\bar {y}})+{\bar {x}},}
22185:
22035:
21934:
21126:{\displaystyle \sigma _{\hat {X}}^{2}=C_{XY}C_{Y}^{-1}C_{YX}={\Big (}{\frac {\sigma _{X}^{2}}{\sigma _{X}^{2}+\sigma _{Z}^{2}/N}}{\Big )}\sigma _{X}^{2}.}
20367:{\displaystyle {\begin{aligned}{\hat {x}}&=C_{XY}C_{Y}^{-1}y\\&=\sigma _{X}^{2}1^{T}(\sigma _{X}^{2}11^{T}+\sigma _{Z}^{2}I)^{-1}y.\end{aligned}}}
941:
15988:
1723:
1383:{\displaystyle \operatorname {E} \{{\hat {x}}_{\operatorname {MMSE} }(y)\}=\operatorname {E} \{\operatorname {E} \{x\mid y\}\}=\operatorname {E} \{x\}.}
20848:
2340:. This can be directly shown using the Bayes theorem. As a consequence, to find the MMSE estimator, it is sufficient to find the linear MMSE estimator.
21329:, we see that the MMSE estimator of a scalar with uniform aprior distribution can be approximated by the arithmetic average of all the observed data
16784:{\displaystyle {\hat {x}}_{k+1}^{(\ell )}={\hat {x}}_{k}^{(\ell -1)}+w_{k+1}^{(\ell )}(y_{k+1}^{(\ell )}-A_{k+1}^{(\ell )}{\hat {x}}_{k}^{(\ell -1)})}
11272:
14588:
14146:
23188:
23155:{\displaystyle \mathrm {LMMSE} =\sigma _{X}^{2}-\sigma _{\hat {X}}^{2}={\frac {1}{1/\sigma _{Z_{1}}^{2}+1/\sigma _{Z_{2}}^{2}+1/\sigma _{X}^{2}}}.}
1084:
9668:
23951:
15553:, depends on our confidence in the new data sample, as measured by the noise variance, versus that in the previous data. The initial values of
11787:
3775:
4206:
2690:
One advantage of such linear MMSE estimator is that it is not necessary to explicitly calculate the posterior probability density function of
24576:
19388:. The matrix equation can be solved by well known methods such as Gauss elimination method. A shorter, non-numerical example can be found in
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14395:
12505:{\displaystyle {\begin{aligned}C_{Y_{k}|X_{k}}&=AC_{X_{k}|Y_{1},\ldots ,Y_{k-1}}A^{T}+C_{Z}=AC_{e_{k-1}}A^{T}+C_{Z}.\end{aligned}}}
10281:
difference between batch estimation and sequential estimation is that sequential estimation requires an additional Markov assumption.
17:
12218:
7351:{\displaystyle \rho ^{2}={\frac {\sigma _{X}^{2}-\sigma _{e}^{2}}{\sigma _{X}^{2}}}={\frac {\sigma _{\hat {X}}^{2}}{\sigma _{X}^{2}}}}
13327:{\displaystyle {\begin{aligned}C_{X_{k}Y_{k}|Y_{1},\ldots ,Y_{k-1}}&=C_{e_{k-1}{\tilde {Y}}_{k}}=C_{e_{k-1}}A^{T}.\end{aligned}}}
21335:
18578:
15674:
15408:
4108:
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1656:
5177:
1941:{\displaystyle {\mathcal {V}}=\{g(y)\mid g:\mathbb {R} ^{m}\rightarrow \mathbb {R} ,\operatorname {E} \{g(y)^{2}\}<+\infty \}}
16129:
as independent scalar measurements, rather than vector measurement. This allows us to reduce computation time by processing the
9424:
7217:{\displaystyle \left({\frac {{\hat {x}}-{\bar {x}}}{\sigma _{X}}}\right)=\rho \left({\frac {y-{\bar {y}}}{\sigma _{Y}}}\right)}
18856:
17516:
9859:
24581:
21414:
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7105:
The above two equations allows us to interpret the correlation coefficient either as normalized slope of linear regression
16912:
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2066:
24277:
6986:{\displaystyle \sigma _{e}^{2}=\sigma _{X}^{2}-{\frac {\sigma _{XY}^{2}}{\sigma _{Y}^{2}}}=(1-\rho ^{2})\sigma _{X}^{2},}
6429:
3004:
1512:{\displaystyle {\sqrt {n}}({\hat {x}}_{\operatorname {MMSE} }-x)\xrightarrow {d} {\mathcal {N}}\left(0,I^{-1}(x)\right),}
84:
24361:
8948:
is positive definite, the estimate still exists. Lastly, this technique can handle cases where the noise is correlated.
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2463:
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14274:
12546:
191:
known random vector variable (the measurement or observation), both of them not necessarily of the same dimension. An
24537:
24480:
24453:
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19666:
14509:
7435:
24246:{\displaystyle w_{i}={\frac {a_{i}/\sigma _{Z_{i}}^{2}}{\sum _{j}a_{j}^{2}/\sigma _{Z_{j}}^{2}+1/\sigma _{X}^{2}}}.}
11708:
11578:
10822:
10204:
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In terms of the terminology developed in the previous sections, for this problem we have the observation vector
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23439:{\displaystyle w_{i}={\frac {1/\sigma _{Z_{i}}^{2}}{\sum _{j=1}^{N}1/\sigma _{Z_{j}}^{2}+1/\sigma _{X}^{2}}}}
14389:
is often referred to as the Kalman gain factor. The alternative formulation of the above algorithm will give
4970:
916:{\displaystyle {\hat {x}}_{\operatorname {MMSE} }(y)=\operatorname {argmin} _{\hat {x}}\operatorname {MSE} .}
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9575:
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How should the recorded music from these two microphones be combined, after being synced with each other?
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In the context of linear MMSE estimator, the formula for the estimate will have the same form as before:
10307:
10012:
9156:
7482:. In this case, no new information is gleaned from the measurement which can decrease the uncertainty in
4769:
2897:
2393:
8148:
6345:
2392:. Another computational approach is to directly seek the minima of the MSE using techniques such as the
515:
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4886:
4296:
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12901:
23565:{\displaystyle \mathrm {LMMSE} ={\frac {1}{\sum _{j=1}^{N}1/\sigma _{Z_{j}}^{2}+1/\sigma _{X}^{2}}}.}
15640:
14924:+1)-th observation, the direct use of above recursive equations give the expression for the estimate
14891:
9143:{\displaystyle C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}A^{T}C_{Z}^{-1},}
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In the
Bayesian framework, such recursive estimation is easily facilitated using Bayes' rule. Given
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275:
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However, the estimator is suboptimal since it is constrained to be linear. Had the random variable
19111:
15825:{\displaystyle \nabla _{\hat {x}}\mathrm {E} \{{\tilde {y}}^{2}\}=-2\mathrm {E} \{{\tilde {y}}a\}.}
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8687:
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197:
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denote the sound produced by the musician, which is a random variable with zero mean and variance
15344:{\displaystyle w_{k+1}={\frac {C_{e_{k}}a_{k+1}}{\sigma _{k+1}^{2}+a_{k+1}^{T}C_{e_{k}}a_{k+1}}}.}
9798:
24283:
19389:
19150:
19076:
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17084:
15357:
13932:{\displaystyle C_{e_{k}}=C_{e_{k-1}}-C_{e_{k-1}}A^{T}(AC_{e_{k-1}}A^{T}+C_{Z})^{-1}AC_{e_{k-1}}.}
12943:
8911:
7505:
1586:
1577:
1397:
22652:
19334:{\displaystyle \left\Vert e\right\Vert _{\min }^{2}=\operatorname {E} -WC_{YX}=15-WC_{YX}=.2857}
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15556:
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932:
When the means and variances are finite, the MMSE estimator is uniquely defined and is given by:
24303:
23938:{\displaystyle {\begin{aligned}y_{1}&=a_{1}x+z_{1}\\y_{2}&=a_{2}x+z_{2}.\end{aligned}}}
18532:{\displaystyle C_{Y}=\left&E&E\\E&E&E\\E&E&E\end{array}}\right]=\left.}
17188:
17058:
16562:{\displaystyle C_{e_{k+1}}^{(\ell )}=(I-w_{k+1}^{(\ell )}A_{k+1}^{(\ell )})C_{e_{k}}^{(\ell )}}
16132:
15961:
15948:{\displaystyle {\hat {x}}_{k+1}={\hat {x}}_{k}+\eta _{k}\mathrm {E} \{{\tilde {y}}_{k}a_{k}\},}
7791:
318:
168:
122:
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14727:
As an important special case, an easy to use recursive expression can be derived when at each
12151:{\displaystyle C_{X_{k}|Y_{1},\ldots ,Y_{k-1}}=C_{X_{k-1}|Y_{1},\ldots ,Y_{k-1}}=C_{e_{k-1}},}
7985:
3171:
Thus, the expression for linear MMSE estimator, its mean, and its auto-covariance is given by
18056:
17214:
15671:
cannot be directly observed, these methods try to minimize the mean squared prediction error
15523:
15156:
15123:
10137:
is identical to the ordinary least square estimate. When apriori information is available as
8878:
7854:
7365:
2214:{\displaystyle \operatorname {E} \{({\hat {x}}_{\operatorname {MMSE} }-x){\hat {x}}^{T}\}=0.}
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From the point of view of linear algebra, for sequential estimation, if we have an estimate
13124:
11427:-1)-th time step. This structure allows us to formulate a recursive approach to estimation.
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10353:
9548:
8822:
7805:
7746:
6399:
5519:
4859:
3500:
Lastly, the error covariance and minimum mean square error achievable by such estimator is
3124:
23634:, which are assumed to be known constants. Similarly, let the noise at each microphone be
21566:
15110:{\displaystyle {\hat {x}}_{k+1}={\hat {x}}_{k}+w_{k+1}(y_{k+1}-a_{k+1}^{T}{\hat {x}}_{k})}
14719:. The three update steps outlined above indeed form the update step of the Kalman filter.
2680:{\displaystyle \min _{W,b}\operatorname {MSE} \qquad {\text{s.t.}}\qquad {\hat {x}}=Wy+b.}
2274:
1805:
1526:
8:
24465:
24394:
18899:
18468:
18141:
17991:
8672:{\displaystyle C_{e}=C_{X}-C_{\hat {X}}=C_{X}-C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}AC_{X}.}
8502:{\displaystyle {\hat {x}}=C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}(y-A{\bar {x}})+{\bar {x}}.}
24423:
24267:
24262:
23769:
23168:
21595:
21497:
21477:
21312:
20380:
19909:
19889:
19869:
19799:
19721:
19648:
19628:
19555:
19443:
19423:
19403:
19371:
16158:
16112:
15985:
is the scalar step size and the expectation is approximated by the instantaneous value
15654:
15612:
11558:
11538:
11249:
10981:
10287:
10184:
10058:
9992:
9839:
9528:
8849:
8721:
8697:
8207:
7927:
7907:
7836:
7799:
7773:
7722:
7690:
7670:
7622:{\displaystyle {\hat {x}}={\frac {\sigma _{XY}}{\sigma _{Y}}}(y-{\bar {y}})+{\bar {x}}}
7485:
7085:
7065:
6664:
6644:
5499:
4746:
3752:
3732:
3480:
3460:
3151:
3104:
3084:
2817:
2797:
2773:
2753:
2733:
2713:
2693:
2556:
2536:
2516:
2443:
2423:
2403:
2323:
2303:
2250:
2230:
1592:
1559:
1553:
1017:
575:
555:
535:
314:
255:
235:
148:
102:
79:
55:
51:
22310:{\displaystyle {\hat {x}}=w_{1}(y_{1}-{\bar {x}})+w_{2}(y_{2}-{\bar {x}})+{\bar {x}},}
22118:{\displaystyle \operatorname {E} \{y_{1}\}=\operatorname {E} \{y_{2}\}={\bar {x}}=1/2}
21615:
19672:
19579:
16109:
is a diagonal matrix. In such cases, it is advantageous to consider the components of
15609:
are taken to be the mean and covariance of the aprior probability density function of
24552:
24533:
24514:
24495:
24476:
24449:
24430:
24407:
24381:
24298:
22022:{\displaystyle {\begin{aligned}y_{1}&=x+z_{1}\\y_{2}&=x+z_{2}.\end{aligned}}}
17678:
are real
Gaussian random variables with zero mean and its covariance matrix given by
17419:
problem as an example. Let a linear combination of observed scalar random variables
17416:
14731:-th time instant the underlying linear observation process yields a scalar such that
8682:
The significant difference between the estimation problem treated above and those of
7716:
322:
47:
35:
19460:
disturbed by white
Gaussian noise. We can describe the process by a linear equation
15635:
This important special case has also given rise to many other iterative methods (or
1003:{\displaystyle {\hat {x}}_{\operatorname {MMSE} }(y)=\operatorname {E} \{x\mid y\}.}
11266:
changes with time, we will make a further stationarity assumption about the prior:
9784:
7740:
6342:
The first term in the third line is zero due to the orthogonality principle. Since
2268:
16067:{\displaystyle \mathrm {E} \{a_{k}{\tilde {y}}_{k}\}\approx a_{k}{\tilde {y}}_{k}}
2063:. More succinctly put, the cross-correlation between the minimum estimation error
1791:{\displaystyle \operatorname {E} \{({\hat {x}}_{\operatorname {MMSE} }-x)g(y)\}=0}
1270:
The MMSE estimator is unbiased (under the regularity assumptions mentioned above):
20938:{\displaystyle {\bar {y}}={\frac {1^{T}y}{N}}={\frac {\sum _{i=1}^{N}y_{i}}{N}}.}
15636:
7830:
50:(MSE), which is a common measure of estimator quality, of the fitted values of a
16082:
In many practical applications, the observation noise is uncorrelated. That is,
11409:{\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})=p(x_{k-1}|y_{1},\ldots ,y_{k-1}).}
7869:
Let us further model the underlying process of observation as a linear process:
14705:{\displaystyle {\hat {x}}_{k}={\hat {x}}_{k-1}+W_{k}(y_{k}-A{\hat {x}}_{k-1}),}
14263:{\displaystyle {\hat {x}}_{k}={\hat {x}}_{k-1}+W_{k}(y_{k}-A{\hat {x}}_{k-1}),}
8956:
An alternative form of expression can be obtained by using the matrix identity
510:
12161:
as per by the properties of MMSE estimators and the stationarity assumption.
24570:
24365:
24293:
24288:
24272:
23290:{\displaystyle {\hat {x}}=\sum _{i=1}^{N}w_{i}(y_{i}-{\bar {x}})+{\bar {x}},}
14716:
11986:
and its covariance matrix is given by the previous error covariance matrix,
8683:
7858:
5633:
Lastly, the covariance of linear MMSE estimation error will then be given by
88:
67:
63:
59:
58:
setting, the term MMSE more specifically refers to estimation with quadratic
8846:
is positive definite. Physically the reason for this property is that since
7857:
process. In such stationary cases, these estimators are also referred to as
6582:
Thus the minimum mean square error achievable by such a linear estimator is
1171:{\displaystyle C_{e}=\operatorname {E} \{({\hat {x}}-x)({\hat {x}}-x)^{T}\}}
16077:
9836:, corresponding to infinite variance of the apriori information concerning
846:
The MMSE estimator is then defined as the estimator achieving minimal MSE:
12164:
Similarly, for the linear observation process, the mean of the likelihood
9773:{\displaystyle {\hat {x}}=C_{e}A^{T}C_{Z}^{-1}(y-A{\bar {x}})+{\bar {x}}.}
24011:{\displaystyle \operatorname {E} \{y_{1}\}=\operatorname {E} \{y_{2}\}=0}
15730:. For instance, in the case of scalar observations, we have the gradient
11975:{\displaystyle {\bar {x}}_{k}=\mathrm {E} =\mathrm {E} ={\hat {x}}_{k-1}}
8738:). The estimate for the linear observation process exists so long as the
3827:{\displaystyle \operatorname {E} \{{\hat {x}}\}=\operatorname {E} \{x\}.}
2573:
is a vector. This can be seen as the first order Taylor approximation of
21514:, so long as the mean and variance of these distributions are the same.
19341:. Note that it is not necessary to obtain an explicit matrix inverse of
16074:. As we can see, these methods bypass the need for covariance matrices.
4286:{\displaystyle \operatorname {E} \{({\hat {x}}-x)(y-{\bar {y}})^{T}\}=0}
15832:
Thus, the update equation for the least mean square filter is given by
7864:
31:
22125:. Thus, we can obtain the LMMSE estimate as the linear combination of
3619:{\displaystyle C_{e}=C_{X}-C_{\hat {X}}=C_{X}-C_{XY}C_{Y}^{-1}C_{YX},}
14135:{\displaystyle W_{k}=C_{e_{k-1}}A^{T}(AC_{e_{k-1}}A^{T}+C_{Z})^{-1},}
11528:{\displaystyle {\hat {x}}=C_{XY}C_{Y}^{-1}(y-{\bar {y}})+{\bar {x}}.}
5135:{\displaystyle {\hat {x}}=C_{XY}C_{Y}^{-1}(y-{\bar {y}})+{\bar {x}}.}
3272:{\displaystyle {\hat {x}}=C_{XY}C_{Y}^{-1}(y-{\bar {y}})+{\bar {x}},}
2044:{\displaystyle \operatorname {E} \{(g_{i}^{*}(y)-x_{i})g_{j}(y)\}=0,}
1014:
In other words, the MMSE estimator is the conditional expectation of
192:
21563:
Thus the fraction of votes the other candidate will receive will be
11123:{\displaystyle p(y_{k}|x_{k},y_{1},\ldots ,y_{k-1})=p(y_{k}|x_{k}).}
10899:-th time step. Here we have assumed the conditional independence of
9408:{\displaystyle W=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}A^{T}C_{Z}^{-1},}
7051:{\displaystyle \rho ={\frac {\sigma _{XY}}{\sigma _{X}\sigma _{Y}}}}
3668:{\displaystyle \operatorname {LMMSE} =\operatorname {tr} \{C_{e}\}.}
1455:
15647:, that directly solves the original MSE optimization problem using
6623:{\displaystyle \operatorname {LMMSE} =\operatorname {tr} \{C_{e}\}}
649:{\displaystyle \operatorname {E} \left\{({\hat {x}}-x)^{2}\right\}}
3769:. It is required that the MMSE estimator be unbiased. This means,
20377:
We can simplify the expression by using the alternative form for
17840:{\displaystyle \operatorname {cov} (Z)=\operatorname {E} =\left,}
14498:{\displaystyle C_{e_{k}}^{-1}=C_{e_{k-1}}^{-1}+A^{T}C_{Z}^{-1}A,}
2710:. Such linear estimator only depends on the first two moments of
656:. Note that MSE can equivalently be defined in other ways, since
24490:
Luenberger, D.G. (1969). "Chapter 4, Least-squares estimation".
24467:
Fundamentals of
Statistical Signal Processing: Estimation Theory
12300:{\displaystyle {\bar {y}}_{k}=A{\bar {x}}_{k}=A{\hat {x}}_{k-1}}
1034:
given the known observed value of the measurements. Also, since
21401:{\displaystyle {\hat {x}}={\frac {1}{N}}\sum _{i=1}^{N}y_{i},}
15520:
Here, no matrix inversion is required. Also, the gain factor,
9954:
is identical to the weighed linear least square estimate with
9783:
In this form the above expression can be easily compared with
21829:
Similarly, the second pollster declares their estimate to be
14830:-by-1 known column vector whose values can change with time,
8018:. Here the required mean and the covariance matrices will be
21612:
as a random variable with a uniform prior distribution over
18760:{\displaystyle C_{YX}=\left\\E\\E\end{array}}\right]=\left.}
15723:{\displaystyle \mathrm {E} \{{\tilde {y}}^{T}{\tilde {y}}\}}
15510:{\displaystyle C_{e_{k+1}}=(I-w_{k+1}a_{k+1}^{T})C_{e_{k}}.}
4193:{\displaystyle {\hat {x}}-x=W(y-{\bar {y}})-(x-{\bar {x}}).}
3332:{\displaystyle \operatorname {E} \{{\hat {x}}\}={\bar {x}},}
1707:{\displaystyle {\hat {x}}_{\operatorname {MMSE} }=g^{*}(y),}
1609:
is a scalar, an estimator constrained to be of certain form
1400:
and it converges in distribution to the normal distribution:
8714:) need not be at least as large as the number of unknowns,
5223:{\displaystyle \operatorname {E} \{{\hat {x}}\}={\bar {x}}}
4743:
When equated to zero, we obtain the desired expression for
2271:, then the MMSE estimator is linear, i.e., it has the form
17486:
be used to estimate another future scalar random variable
11246:. With the lack of dynamical information on how the state
11176:-th observation is then the mean of the posterior density
5034:
Thus the full expression for the linear MMSE estimator is
78:
The term MMSE more specifically refers to estimation in a
24511:
Mathematical
Methods and Algorithms for Signal Processing
12842:
Hence, in the estimate update formula, we should replace
9515:{\displaystyle C_{e}=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}.}
8204:
Thus the expression for the linear MMSE estimator matrix
19063:{\displaystyle C_{Y}^{-1}C_{YX}=\left\left=\left=W^{T}.}
17586:{\displaystyle {\hat {z}}_{4}=\sum _{i=1}^{3}w_{i}z_{i}}
16078:
Special Case: vector observation with uncorrelated noise
9947:{\displaystyle W=(A^{T}C_{Z}^{-1}A)^{-1}A^{T}C_{Z}^{-1}}
8875:, there need be no measurement error. Thus, we may have
46:) estimator is an estimation method which minimizes the
21464:{\displaystyle \sigma _{\hat {X}}^{2}=\sigma _{X}^{2},}
17395:{\displaystyle {\hat {x}}_{k+1}^{(m)}={\hat {x}}_{k+1}}
11575:
will need to be replaced by those of the prior density
3685:
Let us have the optimal linear MMSE estimator given as
592:
is a scalar variable, the MSE expression simplifies to
23824:
We can model the sound received by each microphone as
19013:
18977:
18726:
18603:
27:
Estimation method that minimizes the mean square error
24103:
24027:
23954:
23833:
23792:
23772:
23733:
23694:
23667:
23640:
23613:
23586:
23452:
23307:
23191:
23171:
22987:
22949:
22910:
22687:
22655:
22329:
22188:
22158:
22131:
22038:
21937:
21889:
21862:
21835:
21793:
21766:
21739:
21693:
21650:
21618:
21598:
21569:
21528:
21500:
21480:
21417:
21338:
21315:
21145:
20956:
20851:
20773:
20406:
20383:
20199:
19971:
19932:
19912:
19892:
19872:
19822:
19802:
19744:
19724:
19675:
19651:
19645:
is going to fall in. We can model our uncertainty of
19631:
19582:
19558:
19501:
19466:
19446:
19440:
observations of a fixed but unknown scalar parameter
19426:
19406:
19374:
19347:
19222:
19216:. Computing the minimum mean square error then gives
19186:
19153:
19114:
19079:
18859:
18829:
18776:
18581:
18548:
18122:
18092:
18059:
17994:
17922:
17883:
17856:
17687:
17599:
17519:
17492:
17465:
17425:
17327:
17250:
17217:
17191:
17171:
17127:
17087:
17061:
17041:
16990:
16977:{\displaystyle {\hat {x}}_{k+1}^{(0)}={\hat {x}}_{k}}
16915:
16844:
16800:
16576:
16427:
16185:
16161:
16135:
16115:
16088:
15991:
15964:
15841:
15736:
15677:
15657:
15615:
15588:
15559:
15526:
15411:
15360:
15199:
15159:
15126:
14975:
14930:
14894:
14867:
14836:
14805:
14737:
14591:
14512:
14398:
14368:
14277:
14149:
14017:
13984:
13948:
13755:
13373:
13343:
13160:
13127:
13084:
13042:
13015:
12986:
12946:
12904:
12877:
12848:
12767:
12714:
12701:{\displaystyle {\tilde {y}}_{k}=y_{k}-{\bar {y}}_{k}}
12643:
12616:
12549:
12522:
12316:
12221:
12170:
11995:
11790:
11711:
11657:
11581:
11561:
11541:
11436:
11275:
11252:
11182:
11142:
11007:
10984:
10932:
10905:
10825:
10774:
10704:
10386:
10356:
10310:
10290:
10207:
10187:
10143:
10081:
10061:
10015:
9995:
9989:
as the weight matrix. Moreover, if the components of
9960:
9862:
9842:
9801:
9671:
9639:
9578:
9551:
9531:
9427:
9299:
9218:
9159:
8965:
8914:
8881:
8852:
8825:
8752:
8724:
8700:
8521:
8366:
8334:
8318:{\displaystyle W=C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}.}
8233:
8210:
8151:
8081:
8027:
7988:
7950:
7930:
7910:
7875:
7839:
7808:
7776:
7749:
7725:
7693:
7673:
7635:
7537:
7508:
7488:
7438:
7394:
7368:
7236:
7114:
7088:
7068:
7002:
6866:
6690:
6667:
6647:
6591:
6525:
6432:
6402:
6348:
5642:
5552:
5522:
5502:
5239:
5180:
5151:
5043:
4973:
4940:
4889:
4862:
4832:
4772:
4749:
4352:
4299:
4209:
4111:
4092:{\displaystyle {\hat {x}}=W(y-{\bar {y}})+{\bar {x}}}
4031:
3981:
3934:
3875:
3843:
3778:
3755:
3735:
3691:
3633:
3509:
3483:
3463:
3433:
3346:
3286:
3180:
3154:
3127:
3107:
3087:
3057:
3007:
2960:
2900:
2843:
2820:
2800:
2776:
2756:
2736:
2716:
2696:
2620:
2579:
2559:
2539:
2519:
2466:
2446:
2426:
2406:
2356:
2326:
2306:
2277:
2253:
2233:
2142:
2111:
2069:
1957:
1837:
1808:
1726:
1659:
1615:
1595:
1562:
1529:
1412:
1282:
1221:
1184:
1087:
1040:
1020:
944:
855:
665:
598:
578:
558:
538:
518:
333:
278:
258:
238:
200:
171:
151:
125:
105:
24530:
Detection, Estimation, and
Modulation Theory, Part I
14722:
12832:{\displaystyle C_{{\tilde {Y}}_{k}}=C_{Y_{k}|X_{k}}}
11781:, its mean is given by the previous MMSE estimate,
10275:
8067:{\displaystyle \operatorname {E} \{y\}=A{\bar {x}},}
7865:
Linear MMSE estimator for linear observation process
5623:{\displaystyle C_{\hat {X}}=C_{XY}C_{Y}^{-1}C_{YX}.}
4934:, the expression can also be re-written in terms of
3417:{\displaystyle C_{\hat {X}}=C_{XY}C_{Y}^{-1}C_{YX},}
2098:{\displaystyle {\hat {x}}_{\operatorname {MMSE} }-x}
24425:
Prediction and
Improved Estimation in Linear Models
17877:such that it will yield an optimal linear estimate
10272:, which is identical to ridge regression solution.
10009:are uncorrelated and have equal variance such that
6509:{\displaystyle C_{e}=C_{X}-C_{XY}C_{Y}^{-1}C_{YX}.}
3729:, where we are required to find the expression for
3044:{\displaystyle {\bar {y}}=\operatorname {E} \{y\},}
1081:is the posterior mean, the error covariance matrix
24443:
24422:
24366:Minimum Mean Squared Error Estimators the original
24245:
24079:
24010:
23937:
23813:
23778:
23758:
23719:
23680:
23653:
23626:
23599:
23564:
23438:
23289:
23177:
23154:
22970:
22935:
22893:
22670:
22638:
22309:
22171:
22144:
22117:
22021:
21917:
21875:
21848:
21821:
21779:
21752:
21725:
21679:
21636:
21604:
21584:
21555:
21506:
21486:
21463:
21400:
21321:
21298:
21125:
20937:
20837:
20756:
20389:
20366:
20179:
19954:
19918:
19898:
19878:
19858:
19808:
19788:
19730:
19710:
19657:
19637:
19617:
19564:
19544:
19487:
19452:
19432:
19412:
19380:
19360:
19333:
19208:
19172:
19139:
19101:
19062:
18842:
18815:
18759:
18564:
18531:
18105:
18078:
18045:
17980:
17905:
17869:
17839:
17670:
17585:
17505:
17478:
17451:
17394:
17313:
17236:
17203:
17177:
17157:
17113:
17073:
17047:
17027:
16976:
16901:
16830:
16783:
16561:
16412:
16167:
16147:
16121:
16101:
16066:
15977:
15947:
15824:
15722:
15663:
15621:
15601:
15574:
15545:
15509:
15386:
15343:
15178:
15153:is the new scalar observation and the gain factor
15145:
15109:
14958:
14912:
14880:
14849:
14818:
14791:
14704:
14576:
14497:
14381:
14351:
14262:
14134:
13997:
13970:
13931:
13735:
13356:
13337:Thus, we have the new estimate as new observation
13326:
13143:
13113:
13070:
13028:
13001:
12972:
12932:
12890:
12863:
12831:
12753:
12700:
12629:
12602:
12535:
12504:
12299:
12207:
12150:
11974:
11773:
11694:
11643:
11567:
11547:
11527:
11408:
11258:
11238:
11164:
11122:
10990:
10970:
10918:
10887:
10811:
10760:
10687:
10369:
10342:
10296:
10264:
10201:are uncorrelated and have equal variance, we have
10193:
10173:
10129:
10067:
10047:
10001:
9981:
9946:
9848:
9828:
9772:
9654:
9625:
9564:
9537:
9514:
9407:
9282:
9204:
9142:
8940:
8900:
8858:
8838:
8807:
8730:
8706:
8671:
8501:
8349:
8317:
8216:
8193:
8136:
8066:
8010:
7974:
7936:
7916:
7896:
7845:
7821:
7782:
7762:
7739:. A more numerically stable method is provided by
7731:
7699:
7679:
7659:
7621:
7523:
7494:
7474:
7424:
7380:
7350:
7216:
7094:
7074:
7050:
6985:
6851:
6673:
6653:
6622:
6574:
6508:
6415:
6388:
6331:
5622:
5535:
5508:
5485:
5222:
5166:
5134:
5023:
4956:
4926:
4875:
4848:
4815:
4755:
4732:
4335:
4285:
4192:
4091:
4015:{\displaystyle {\bar {y}}=\operatorname {E} \{y\}}
4014:
3968:{\displaystyle {\bar {x}}=\operatorname {E} \{x\}}
3967:
3917:
3858:
3826:
3761:
3741:
3721:
3667:
3618:
3489:
3469:
3449:
3416:
3331:
3271:
3160:
3140:
3113:
3093:
3073:
3043:
2994:{\displaystyle {\bar {x}}=\operatorname {E} \{x\}}
2993:
2943:
2885:
2826:
2806:
2782:
2762:
2750:. So although it may be convenient to assume that
2742:
2722:
2702:
2679:
2603:
2565:
2545:
2525:
2506:{\displaystyle \operatorname {E} \{x\mid y\}=Wy+b}
2505:
2452:
2432:
2412:
2380:
2332:
2312:
2292:
2259:
2239:
2213:
2126:
2097:
2043:
1940:
1823:
1790:
1706:
1645:
1601:
1568:
1544:
1511:
1382:
1255:
1205:
1170:
1073:
1026:
1002:
915:
835:
648:
584:
564:
544:
524:
498:
305:
264:
244:
224:
183:
157:
137:
111:
24444:Lehmann, E. L.; Casella, G. (1998). "Chapter 4".
21471:and the LMMSE of the estimate will tend to zero.
21266:
21197:
21100:
21031:
18086:as a scalar quantity. The autocorrelation matrix
7227:or as square root of the ratio of two variances
24568:
19236:
14861:-by-1 random column vector to be estimated, and
14352:{\displaystyle C_{e_{k}}=(I-W_{k}A)C_{e_{k-1}}.}
12603:{\displaystyle {\bar {y}}_{k}=A{\hat {x}}_{k-1}}
10350:, Bayes' rule gives us the posterior density of
9153:which can be established by post-multiplying by
4856:is cross-covariance matrix between X and Y, and
4102:and the expression for estimation error becomes
2622:
24420:
14577:{\displaystyle W_{k}=C_{e_{k}}A^{T}C_{Z}^{-1},}
7475:{\displaystyle \sigma _{e}^{2}=\sigma _{X}^{2}}
21411:while the variance will be unaffected by data
12516:The difference between the predicted value of
11774:{\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})}
11644:{\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})}
10888:{\displaystyle p(x_{k}|y_{1},\ldots ,y_{k-1})}
10265:{\displaystyle W=(A^{T}A+\lambda I)^{-1}A^{T}}
9283:{\displaystyle (A^{T}C_{Z}^{-1}A+C_{X}^{-1}),}
4203:From the orthogonality principle, we can have
17314:{\displaystyle C_{e_{k+1}}^{(m)}=C_{e_{k+1}}}
11535:However, the mean and covariance matrices of
7719:can be used to solve the matrix equation for
7707:, as given by the equation of straight line.
6681:are scalars, the above relations simplify to
2604:{\displaystyle \operatorname {E} \{x\mid y\}}
2381:{\displaystyle \operatorname {E} \{x\mid y\}}
24508:
23999:
23986:
23974:
23961:
22083:
22070:
22058:
22045:
20947:Similarly, the variance of the estimator is
20190:Thus, the linear MMSE estimator is given by
20139:
20123:
20041:
20025:
19989:
19983:
19789:{\displaystyle \sigma _{X}^{2}=x_{0}^{2}/3.}
18053:as a row vector, and the estimated variable
16029:
15997:
15939:
15907:
15816:
15798:
15781:
15759:
15717:
15683:
11239:{\displaystyle p(x_{k}|y_{1},\ldots ,y_{k})}
10761:{\displaystyle p(x_{k}|y_{1},\ldots ,y_{k})}
8690:estimate is that the number of observations
8040:
8034:
7963:
7957:
6617:
6604:
6280:
6222:
6207:
6149:
6130:
6036:
6014:
5956:
5922:
5864:
5841:
5747:
5728:
5670:
5430:
5372:
5350:
5274:
5230:, we can also obtain its auto-covariance as
5202:
5187:
4681:
4623:
4611:
4553:
4531:
4437:
4421:
4363:
4274:
4216:
4009:
4003:
3962:
3956:
3818:
3812:
3800:
3785:
3659:
3646:
3308:
3293:
3035:
3029:
2988:
2982:
2598:
2586:
2485:
2473:
2375:
2363:
2202:
2149:
2029:
1964:
1935:
1923:
1901:
1848:
1779:
1733:
1374:
1368:
1356:
1353:
1341:
1332:
1320:
1289:
1165:
1107:
1074:{\displaystyle {\hat {x}}_{\mathrm {MMSE} }}
994:
982:
827:
809:
776:
760:
743:
727:
699:
683:
490:
432:
415:
357:
19552:. Depending on context it will be clear if
16902:{\displaystyle C_{e_{k+1}}^{(0)}=C_{e_{k}}}
11423:-th time step is the posterior density of (
8328:Putting everything into the expression for
24489:
24018:. Thus, we can combine the two sounds as
21136:Thus the MMSE of this linear estimator is
19625:to be the range within which the value of
17850:then our task is to find the coefficients
14792:{\displaystyle y_{k}=a_{k}^{T}x_{k}+z_{k}}
272:. The estimation error vector is given by
24527:
7975:{\displaystyle \operatorname {E} \{z\}=0}
7770:is a symmetric positive definite matrix,
6575:{\displaystyle C_{e}=C_{X}-C_{\hat {X}}.}
3918:{\displaystyle b={\bar {x}}-W{\bar {y}},}
2886:{\displaystyle b={\bar {x}}-W{\bar {y}},}
1888:
1874:
24404:Probability Theory: The Logic of Science
12980:, respectively. Also, we should replace
8808:{\displaystyle (AC_{X}A^{T}+C_{Z})^{-1}}
8137:{\displaystyle C_{Y}=AC_{X}A^{T}+C_{Z},}
6519:This we can recognize to be the same as
4022:. Thus we can re-write the estimator as
3680:Derivation using orthogonality principle
2344:
24362:"Minimum Mean Squared Error Estimators"
24080:{\displaystyle y=w_{1}y_{1}+w_{2}y_{2}}
13978:based on measurements generating space
12307:and the covariance matrix is as before
10819:is called the likelihood function, and
5024:{\displaystyle W^{T}=C_{Y}^{-1}C_{YX}.}
145:hidden random vector variable, and let
14:
24569:
24546:
24401:
22936:{\displaystyle \sigma _{\hat {X}}^{2}}
15651:. However, since the estimation error
9626:{\displaystyle W=C_{e}A^{T}C_{Z}^{-1}}
4883:is auto-covariance matrix of Y. Since
21726:{\displaystyle \sigma _{X}^{2}=1/12.}
19859:{\displaystyle N(0,\sigma _{Z}^{2}I)}
17028:{\displaystyle C_{Z_{k+1}}^{(\ell )}}
10971:{\displaystyle y_{1},\ldots ,y_{k-1}}
9633:, we get a simplified expression for
7944:is random noise vector with the mean
7790:can be solved twice as fast with the
7425:{\displaystyle {\hat {x}}={\bar {x}}}
1178:is equal to the posterior covariance
24577:Statistical deviation and dispersion
24492:Optimization by Vector Space Methods
23688:, each with zero mean and variances
21918:{\displaystyle \sigma _{Z_{2}}^{2}.}
21822:{\displaystyle \sigma _{Z_{1}}^{2}.}
13114:{\displaystyle C_{{\tilde {Y}}_{k}}}
10174:{\displaystyle C_{X}^{-1}=\lambda I}
10130:{\displaystyle W=(A^{T}A)^{-1}A^{T}}
24509:Moon, T.K.; Stirling, W.C. (2000).
24462:
24359:
24278:Minimum-variance unbiased estimator
23759:{\displaystyle \sigma _{Z_{2}}^{2}}
23720:{\displaystyle \sigma _{Z_{1}}^{2}}
16831:{\displaystyle \ell =1,2,\ldots ,m}
14888:is scalar noise term with variance
10343:{\displaystyle y_{1},\ldots ,y_{k}}
10048:{\displaystyle C_{Z}=\sigma ^{2}I,}
9205:{\displaystyle (AC_{X}A^{T}+C_{Z})}
8951:
6423:in terms of covariance matrices as
4816:{\displaystyle W=C_{XY}C_{Y}^{-1}.}
3457:is cross-covariance matrix between
3081:is cross-covariance matrix between
2944:{\displaystyle W=C_{XY}C_{Y}^{-1}.}
2394:stochastic gradient descent methods
252:is any function of the measurement
85:minimum-variance unbiased estimator
24:
24421:Bibby, J.; Toutenburg, H. (1977).
24352:
23980:
23955:
23466:
23463:
23460:
23457:
23454:
23001:
22998:
22995:
22992:
22989:
22064:
22039:
21928:As with previous example, we have
20117:
20019:
19977:
19576:or a vector. Suppose that we know
19249:
17706:
15993:
15903:
15794:
15755:
15738:
15679:
12716:
11878:
11814:
8194:{\displaystyle C_{XY}=C_{X}A^{T}.}
8028:
7951:
6636:
6389:{\displaystyle W=C_{XY}C_{Y}^{-1}}
6216:
6143:
6030:
5950:
5858:
5741:
5664:
5366:
5268:
5181:
4617:
4547:
4431:
4357:
4343:. Here the left-hand-side term is
4210:
3997:
3950:
3806:
3779:
3287:
3023:
2976:
2580:
2467:
2357:
2143:
1958:
1932:
1895:
1840:
1727:
1463:
1362:
1335:
1326:
1283:
1101:
1065:
1062:
1059:
1056:
976:
803:
754:
710:
677:
599:
525:{\displaystyle \operatorname {E} }
519:
426:
351:
25:
24598:
18816:{\displaystyle C_{Y}W^{T}=C_{YX}}
17158:{\displaystyle A_{k+1}^{(\ell )}}
15402:error covariance matrix given by
14723:Special case: scalar observations
13746:and the new error covariance as
10768:is called the posterior density,
10276:Sequential linear MMSE estimation
8815:exists; this is the case for any
7794:, while for large sparse systems
7660:{\displaystyle \sigma _{e}^{2}=0}
7060:Pearson's correlation coefficient
5174:is itself a random variable with
4927:{\displaystyle C_{XY}=C_{YX}^{T}}
4336:{\displaystyle g(y)=y-{\bar {y}}}
23814:{\displaystyle \sigma _{X}^{2}.}
22971:{\displaystyle \sigma _{X}^{2}.}
19180:as the optimal coefficients for
14959:{\displaystyle {\hat {x}}_{k+1}}
13071:{\displaystyle {\bar {y}}_{k-1}}
12933:{\displaystyle {\hat {x}}_{k-1}}
11133:This is the Markov assumption.
8512:Lastly, the error covariance is
24551:(5th ed.). Prentice Hall.
24513:(1st ed.). Prentice Hall.
22320:where the weights are given by
14913:{\displaystyle \sigma _{k}^{2}}
9545:can now be written in terms of
6641:For the special case when both
3722:{\displaystyle {\hat {x}}=Wy+b}
2646:
2640:
2440:is a simple linear function of
1646:{\displaystyle {\hat {x}}=g(y)}
24406:. Cambridge University Press.
24340:
24315:
23278:
23266:
23260:
23238:
23198:
23036:
22921:
22698:
22662:
22298:
22286:
22280:
22258:
22242:
22236:
22214:
22195:
22095:
21680:{\displaystyle {\bar {x}}=1/2}
21657:
21631:
21619:
21547:
21535:
21428:
21345:
21180:
20967:
20858:
20826:
20780:
20741:
20417:
20342:
20292:
20210:
19853:
19826:
19705:
19676:
19612:
19583:
19533:
19508:
19278:
19255:
19231:
19225:
19209:{\displaystyle {\hat {z}}_{4}}
19194:
18707:
18681:
18671:
18645:
18635:
18609:
18449:
18423:
18415:
18389:
18381:
18355:
18345:
18319:
18311:
18285:
18277:
18251:
18241:
18215:
18207:
18181:
18173:
18147:
18040:
18001:
17969:
17929:
17906:{\displaystyle {\hat {z}}_{4}}
17891:
17728:
17712:
17700:
17694:
17659:
17606:
17527:
17374:
17359:
17353:
17335:
17280:
17274:
17150:
17144:
17020:
17014:
16962:
16947:
16941:
16923:
16874:
16868:
16778:
16773:
16761:
16749:
16737:
16731:
16707:
16701:
16682:
16677:
16671:
16647:
16635:
16623:
16608:
16602:
16584:
16554:
16548:
16528:
16523:
16517:
16496:
16490:
16465:
16457:
16451:
16404:
16396:
16390:
16371:
16366:
16360:
16338:
16332:
16308:
16302:
16269:
16263:
16242:
16236:
16208:
16202:
16052:
16017:
15917:
15877:
15849:
15807:
15769:
15747:
15711:
15693:
15645:recursive least squares filter
15566:
15484:
15438:
15190:-by-1 column vector given by
15104:
15092:
15042:
15011:
14983:
14938:
14696:
14678:
14652:
14621:
14599:
14320:
14298:
14254:
14236:
14210:
14179:
14157:
14117:
14064:
13971:{\displaystyle {\hat {x}}_{1}}
13956:
13888:
13835:
13726:
13708:
13682:
13670:
13617:
13563:
13543:
13531:
13508:
13486:
13460:
13411:
13385:
13267:
13191:
13097:
13050:
12993:
12912:
12855:
12813:
12780:
12754:{\displaystyle \mathrm {E} =0}
12742:
12730:
12720:
12686:
12651:
12582:
12557:
12379:
12337:
12279:
12254:
12229:
12208:{\displaystyle p(y_{k}|x_{k})}
12202:
12188:
12174:
12078:
12012:
11954:
11941:
11902:
11882:
11871:
11832:
11818:
11798:
11768:
11729:
11715:
11695:{\displaystyle p(y_{k}|x_{k})}
11689:
11675:
11661:
11638:
11599:
11585:
11516:
11504:
11498:
11483:
11443:
11400:
11361:
11341:
11332:
11293:
11279:
11233:
11200:
11186:
11165:{\displaystyle {\hat {x}}_{k}}
11150:
11114:
11100:
11086:
11077:
11025:
11011:
10882:
10843:
10829:
10812:{\displaystyle p(y_{k}|x_{k})}
10806:
10792:
10778:
10755:
10722:
10708:
10675:
10636:
10622:
10616:
10602:
10588:
10572:
10533:
10519:
10513:
10468:
10454:
10441:
10408:
10394:
10240:
10214:
10105:
10088:
9904:
9869:
9761:
9749:
9743:
9725:
9678:
9646:
9497:
9441:
9362:
9306:
9274:
9219:
9199:
9160:
9097:
9041:
9026:
8986:
8793:
8753:
8641:
8601:
8558:
8490:
8478:
8472:
8454:
8442:
8402:
8373:
8341:
8300:
8260:
8055:
7710:
7613:
7601:
7595:
7580:
7544:
7416:
7401:
7319:
7191:
7143:
7128:
6962:
6943:
6840:
6828:
6822:
6807:
6771:
6759:
6753:
6738:
6697:
6562:
6271:
6264:
6249:
6246:
6240:
6225:
6198:
6191:
6176:
6173:
6167:
6152:
6121:
6114:
6099:
6096:
6093:
6087:
6072:
6066:
6060:
6045:
6039:
6005:
5998:
5983:
5980:
5968:
5959:
5913:
5906:
5891:
5888:
5876:
5867:
5832:
5828:
5822:
5807:
5801:
5795:
5780:
5774:
5771:
5759:
5750:
5719:
5706:
5697:
5694:
5682:
5673:
5563:
5421:
5414:
5399:
5396:
5390:
5375:
5341:
5334:
5319:
5310:
5307:
5301:
5286:
5277:
5254:
5214:
5196:
5158:
5123:
5111:
5105:
5090:
5050:
4672:
4665:
4650:
4647:
4641:
4626:
4602:
4595:
4580:
4577:
4571:
4556:
4522:
4515:
4500:
4497:
4494:
4488:
4473:
4467:
4461:
4446:
4440:
4412:
4405:
4390:
4387:
4375:
4366:
4327:
4309:
4303:
4265:
4258:
4243:
4240:
4228:
4219:
4184:
4178:
4163:
4157:
4151:
4136:
4118:
4083:
4071:
4065:
4050:
4038:
3988:
3941:
3906:
3888:
3850:
3794:
3698:
3546:
3357:
3320:
3302:
3260:
3248:
3242:
3227:
3187:
3014:
2967:
2874:
2856:
2653:
2190:
2180:
2162:
2152:
2118:
2077:
2026:
2020:
2007:
1991:
1985:
1967:
1914:
1907:
1884:
1860:
1854:
1818:
1812:
1776:
1770:
1764:
1746:
1736:
1698:
1692:
1667:
1653:is an optimal estimator, i.e.
1640:
1634:
1622:
1576:. Thus, the MMSE estimator is
1539:
1533:
1498:
1492:
1448:
1430:
1420:
1317:
1311:
1299:
1244:
1194:
1156:
1143:
1134:
1131:
1119:
1110:
1048:
970:
964:
952:
897:
881:
875:
863:
632:
619:
610:
487:
475:
466:
457:
444:
435:
406:
393:
384:
381:
369:
360:
306:{\displaystyle e={\hat {x}}-x}
291:
219:
213:
207:
13:
1:
22978:Thus, the LMMSE is given by
19140:{\displaystyle w_{2}=-0.142,}
18542:The cross correlation matrix
16984:. The intermediate variables
3148:is auto-covariance matrix of
1256:{\displaystyle C_{e}=C_{X|Y}}
926:
225:{\displaystyle {\hat {x}}(y)}
94:
73:
24582:Point estimation performance
24364:. Connexions. Archived from
23574:
21883:with zero mean and variance
21787:with zero mean and variance
21517:
19886:is an identity matrix. Also
19395:
17410:
17055:-th diagonal element of the
15649:stochastic gradient descents
11419:Thus, the prior density for
10075:is an identity matrix, then
9829:{\displaystyle C_{X}^{-1}=0}
7687:is completely determined by
3837:Plugging the expression for
7:
24256:
19816:be normally distributed as
19173:{\displaystyle w_{3}=.5714}
19102:{\displaystyle w_{1}=2.57,}
18850:and pre-multiplying to get
17452:{\displaystyle z_{1},z_{2}}
17405:
17114:{\displaystyle C_{Z_{k+1}}}
16838:, using the initial values
15387:{\displaystyle C_{e_{k+1}}}
12973:{\displaystyle C_{e_{k-1}}}
12637:gives the prediction error
10926:from previous observations
8941:{\displaystyle AC_{X}A^{T}}
7524:{\displaystyle \rho =\pm 1}
5496:Putting the expression for
2794:The expression for optimal
1831:in closed, linear subspace
10:
24603:
24471:. Prentice Hall. pp.
24448:(2nd ed.). Springer.
24446:Theory of Point Estimation
24323:"Mean Squared Error (MSE)"
23446:and the LMMSE is given by
22671:{\displaystyle {\hat {x}}}
19962:. It is easy to see that
18770:We now solve the equation
17593:. If the random variables
15575:{\displaystyle {\hat {x}}}
13002:{\displaystyle {\bar {y}}}
12864:{\displaystyle {\bar {x}}}
9982:{\displaystyle C_{Z}^{-1}}
9655:{\displaystyle {\hat {x}}}
8350:{\displaystyle {\hat {x}}}
7502:. On the other hand, when
5167:{\displaystyle {\hat {x}}}
3859:{\displaystyle {\hat {x}}}
2127:{\displaystyle {\hat {x}}}
18:Minimum mean squared error
24528:Van Trees, H. L. (1968).
24327:www.probabilitycourse.com
23301:-th pollster is given by
17204:{\displaystyle m\times n}
17074:{\displaystyle m\times m}
16148:{\displaystyle m\times 1}
15978:{\displaystyle \eta _{k}}
15641:least mean squares filter
12610:, and its observed value
8718:, (i.e. the dimension of
8694:, (i.e. the dimension of
7859:Wiener–Kolmogorov filters
7796:conjugate gradient method
7743:method. Since the matrix
184:{\displaystyle m\times 1}
138:{\displaystyle n\times 1}
40:minimum mean square error
24309:
19955:{\displaystyle C_{XZ}=0}
19368:to compute the value of
10895:is the prior density of
8011:{\displaystyle C_{XZ}=0}
2513:, where the measurement
1578:asymptotically efficient
64:Wiener–Kolmogorov filter
24494:(1st ed.). Wiley.
24284:Orthogonality principle
24094:-th weight is given as
23165:In general, if we have
19796:. Let the noise vector
19390:orthogonality principle
18079:{\displaystyle x=z_{4}}
17988:, the estimator matrix
17244:. The final values are
17237:{\displaystyle A_{k+1}}
15633:Alternative approaches:
15546:{\displaystyle w_{k+1}}
15179:{\displaystyle w_{k+1}}
15146:{\displaystyle y_{k+1}}
9212:and pre-multiplying by
8901:{\displaystyle C_{Z}=0}
7833:. This can happen when
7381:{\displaystyle \rho =0}
1587:orthogonality principle
1398:asymptotically unbiased
1206:{\displaystyle C_{X|Y}}
24549:Adaptive Filter Theory
24304:Zero-forcing equalizer
24247:
24081:
24012:
23939:
23815:
23780:
23760:
23721:
23682:
23655:
23628:
23601:
23566:
23499:
23440:
23376:
23291:
23227:
23179:
23156:
22972:
22937:
22895:
22672:
22640:
22311:
22173:
22146:
22119:
22023:
21919:
21877:
21850:
21823:
21781:
21754:
21727:
21681:
21638:
21606:
21586:
21557:
21556:{\displaystyle x\in .}
21508:
21488:
21465:
21402:
21384:
21323:
21300:
21127:
20939:
20915:
20839:
20838:{\displaystyle y=^{T}}
20758:
20391:
20368:
20181:
19956:
19920:
19900:
19880:
19860:
19810:
19790:
19738:will have variance of
19732:
19712:
19659:
19639:
19619:
19566:
19546:
19545:{\displaystyle 1=^{T}}
19489:
19488:{\displaystyle y=1x+z}
19454:
19434:
19414:
19382:
19362:
19335:
19210:
19174:
19141:
19103:
19064:
18844:
18817:
18761:
18566:
18565:{\displaystyle C_{YX}}
18533:
18107:
18080:
18047:
17982:
17981:{\displaystyle y=^{T}}
17907:
17871:
17841:
17672:
17671:{\displaystyle z=^{T}}
17587:
17562:
17507:
17480:
17453:
17396:
17315:
17238:
17205:
17179:
17159:
17115:
17075:
17049:
17029:
16978:
16903:
16832:
16785:
16563:
16414:
16169:
16155:measurement vector as
16149:
16123:
16103:
16068:
15979:
15949:
15826:
15724:
15665:
15623:
15603:
15576:
15547:
15511:
15388:
15345:
15180:
15147:
15111:
14960:
14914:
14882:
14851:
14820:
14793:
14706:
14578:
14499:
14383:
14353:
14264:
14136:
13999:
13972:
13933:
13737:
13358:
13328:
13145:
13144:{\displaystyle C_{XY}}
13115:
13072:
13030:
13003:
12974:
12934:
12892:
12865:
12833:
12755:
12702:
12631:
12604:
12537:
12506:
12301:
12209:
12152:
11976:
11775:
11705:For the prior density
11696:
11645:
11569:
11549:
11529:
11410:
11260:
11240:
11166:
11124:
10992:
10972:
10920:
10889:
10813:
10762:
10689:
10371:
10344:
10298:
10266:
10195:
10175:
10131:
10069:
10049:
10003:
9983:
9948:
9850:
9830:
9795:. In particular, when
9774:
9656:
9627:
9566:
9539:
9516:
9409:
9284:
9206:
9144:
8942:
8902:
8860:
8840:
8809:
8732:
8708:
8673:
8503:
8351:
8319:
8218:
8195:
8138:
8068:
8012:
7976:
7938:
7924:is a known matrix and
7918:
7898:
7897:{\displaystyle y=Ax+z}
7847:
7823:
7802:is a fast method when
7792:Cholesky decomposition
7784:
7764:
7733:
7701:
7681:
7661:
7623:
7525:
7496:
7476:
7426:
7382:
7352:
7218:
7096:
7076:
7052:
6987:
6853:
6675:
6655:
6624:
6576:
6510:
6417:
6390:
6333:
5624:
5537:
5510:
5487:
5224:
5168:
5136:
5025:
4958:
4957:{\displaystyle C_{YX}}
4928:
4877:
4850:
4849:{\displaystyle C_{XY}}
4817:
4757:
4734:
4337:
4287:
4194:
4093:
4016:
3969:
3919:
3860:
3828:
3763:
3743:
3723:
3669:
3620:
3491:
3471:
3451:
3450:{\displaystyle C_{YX}}
3418:
3333:
3273:
3162:
3142:
3115:
3095:
3075:
3074:{\displaystyle C_{XY}}
3045:
2995:
2945:
2887:
2828:
2808:
2784:
2764:
2744:
2724:
2704:
2681:
2605:
2567:
2547:
2527:
2507:
2454:
2434:
2414:
2382:
2334:
2314:
2294:
2261:
2241:
2215:
2128:
2099:
2045:
1942:
1825:
1792:
1708:
1647:
1603:
1570:
1546:
1513:
1396:The MMSE estimator is
1384:
1257:
1207:
1172:
1075:
1028:
1004:
917:
837:
802:
650:
586:
566:
546:
526:
500:
317:(MSE) is given by the
307:
266:
246:
226:
185:
159:
139:
113:
24547:Haykin, S.O. (2013).
24402:Jaynes, E.T. (2003).
24248:
24082:
24013:
23940:
23816:
23781:
23761:
23722:
23683:
23681:{\displaystyle z_{2}}
23656:
23654:{\displaystyle z_{1}}
23629:
23627:{\displaystyle a_{2}}
23602:
23600:{\displaystyle a_{1}}
23567:
23479:
23441:
23356:
23297:where the weight for
23292:
23207:
23180:
23157:
22973:
22938:
22896:
22673:
22641:
22312:
22174:
22172:{\displaystyle y_{2}}
22147:
22145:{\displaystyle y_{1}}
22120:
22024:
21920:
21878:
21876:{\displaystyle z_{2}}
21851:
21849:{\displaystyle y_{2}}
21824:
21782:
21780:{\displaystyle z_{1}}
21755:
21753:{\displaystyle y_{1}}
21728:
21682:
21639:
21607:
21587:
21558:
21509:
21489:
21466:
21403:
21364:
21324:
21301:
21128:
20940:
20895:
20840:
20759:
20392:
20369:
20182:
19957:
19921:
19901:
19881:
19861:
19811:
19791:
19733:
19713:
19660:
19640:
19620:
19567:
19547:
19490:
19455:
19435:
19415:
19383:
19363:
19361:{\displaystyle C_{Y}}
19336:
19211:
19175:
19142:
19104:
19065:
18845:
18843:{\displaystyle C_{Y}}
18818:
18762:
18567:
18534:
18108:
18106:{\displaystyle C_{Y}}
18081:
18048:
17983:
17908:
17872:
17870:{\displaystyle w_{i}}
17842:
17673:
17588:
17542:
17508:
17506:{\displaystyle z_{4}}
17481:
17479:{\displaystyle z_{3}}
17454:
17397:
17316:
17239:
17206:
17180:
17178:{\displaystyle \ell }
17160:
17116:
17076:
17050:
17048:{\displaystyle \ell }
17030:
16979:
16904:
16833:
16786:
16564:
16415:
16170:
16150:
16124:
16104:
16102:{\displaystyle C_{Z}}
16069:
15980:
15950:
15827:
15725:
15666:
15624:
15604:
15602:{\displaystyle C_{e}}
15577:
15548:
15512:
15389:
15346:
15181:
15148:
15112:
14961:
14915:
14883:
14881:{\displaystyle z_{k}}
14852:
14850:{\displaystyle x_{k}}
14821:
14819:{\displaystyle a_{k}}
14794:
14707:
14579:
14500:
14384:
14382:{\displaystyle W_{k}}
14354:
14265:
14137:
14000:
13998:{\displaystyle Y_{1}}
13973:
13934:
13738:
13359:
13357:{\displaystyle y_{k}}
13329:
13146:
13121:. Lastly, we replace
13116:
13073:
13031:
13029:{\displaystyle C_{Y}}
13004:
12975:
12935:
12893:
12891:{\displaystyle C_{X}}
12866:
12834:
12756:
12703:
12632:
12630:{\displaystyle y_{k}}
12605:
12538:
12536:{\displaystyle Y_{k}}
12507:
12302:
12210:
12153:
11977:
11776:
11697:
11646:
11570:
11550:
11530:
11411:
11261:
11241:
11167:
11125:
10993:
10973:
10921:
10919:{\displaystyle y_{k}}
10890:
10814:
10763:
10690:
10372:
10370:{\displaystyle x_{k}}
10345:
10299:
10267:
10196:
10176:
10132:
10070:
10050:
10004:
9984:
9949:
9851:
9831:
9793:Gauss–Markov estimate
9775:
9657:
9628:
9567:
9565:{\displaystyle C_{e}}
9540:
9517:
9410:
9285:
9207:
9145:
8943:
8908:, because as long as
8903:
8861:
8841:
8839:{\displaystyle C_{Z}}
8810:
8733:
8709:
8674:
8504:
8352:
8320:
8219:
8196:
8139:
8069:
8013:
7982:and cross-covariance
7977:
7939:
7919:
7899:
7855:wide sense stationary
7848:
7824:
7822:{\displaystyle C_{Y}}
7785:
7765:
7763:{\displaystyle C_{Y}}
7734:
7715:Standard method like
7702:
7682:
7662:
7624:
7526:
7497:
7477:
7427:
7383:
7353:
7219:
7097:
7077:
7053:
6988:
6854:
6676:
6656:
6625:
6577:
6511:
6418:
6416:{\displaystyle C_{e}}
6391:
6334:
5625:
5538:
5536:{\displaystyle W^{T}}
5511:
5488:
5225:
5169:
5137:
5026:
4959:
4929:
4878:
4876:{\displaystyle C_{Y}}
4851:
4818:
4758:
4735:
4338:
4288:
4195:
4094:
4017:
3970:
3920:
3861:
3829:
3764:
3744:
3724:
3670:
3621:
3492:
3472:
3452:
3419:
3334:
3274:
3163:
3143:
3141:{\displaystyle C_{Y}}
3116:
3096:
3076:
3046:
2996:
2946:
2888:
2829:
2809:
2785:
2765:
2745:
2725:
2705:
2682:
2606:
2568:
2548:
2528:
2508:
2455:
2435:
2415:
2383:
2345:Linear MMSE estimator
2335:
2315:
2295:
2262:
2242:
2216:
2129:
2100:
2046:
1943:
1826:
1793:
1709:
1648:
1604:
1571:
1547:
1514:
1385:
1258:
1208:
1173:
1076:
1029:
1005:
918:
838:
782:
651:
587:
567:
547:
527:
501:
308:
267:
247:
227:
186:
160:
140:
114:
24101:
24025:
23952:
23831:
23790:
23770:
23731:
23692:
23665:
23638:
23611:
23584:
23450:
23305:
23189:
23169:
22985:
22947:
22908:
22685:
22653:
22327:
22186:
22156:
22129:
22036:
21935:
21887:
21860:
21833:
21791:
21764:
21737:
21691:
21648:
21644:so that its mean is
21616:
21596:
21585:{\displaystyle 1-x.}
21567:
21526:
21498:
21478:
21415:
21336:
21313:
21143:
20954:
20849:
20771:
20404:
20381:
20197:
19969:
19930:
19926:are independent and
19910:
19890:
19870:
19820:
19800:
19742:
19722:
19673:
19667:uniform distribution
19649:
19629:
19580:
19556:
19499:
19464:
19444:
19424:
19404:
19372:
19345:
19220:
19184:
19151:
19112:
19077:
18857:
18827:
18774:
18579:
18546:
18120:
18090:
18057:
17992:
17920:
17881:
17854:
17685:
17597:
17517:
17490:
17463:
17423:
17325:
17248:
17215:
17189:
17169:
17125:
17085:
17059:
17039:
16988:
16913:
16842:
16798:
16574:
16425:
16183:
16159:
16133:
16113:
16086:
15989:
15962:
15839:
15734:
15675:
15655:
15613:
15586:
15557:
15524:
15409:
15358:
15197:
15157:
15124:
14973:
14928:
14892:
14865:
14834:
14803:
14735:
14589:
14510:
14396:
14366:
14275:
14147:
14015:
13982:
13946:
13753:
13371:
13341:
13158:
13125:
13082:
13040:
13013:
12984:
12944:
12902:
12875:
12846:
12765:
12712:
12641:
12614:
12547:
12520:
12314:
12219:
12168:
11993:
11788:
11709:
11655:
11579:
11559:
11539:
11434:
11273:
11250:
11180:
11140:
11005:
10982:
10930:
10903:
10823:
10772:
10702:
10384:
10354:
10308:
10288:
10205:
10185:
10141:
10079:
10059:
10013:
9993:
9958:
9860:
9840:
9799:
9789:weighed least square
9669:
9637:
9576:
9549:
9529:
9425:
9297:
9216:
9157:
8963:
8912:
8879:
8850:
8823:
8750:
8722:
8698:
8519:
8364:
8332:
8231:
8224:further modifies to
8208:
8149:
8079:
8025:
7986:
7948:
7928:
7908:
7873:
7837:
7806:
7774:
7747:
7723:
7691:
7671:
7633:
7535:
7506:
7486:
7436:
7392:
7366:
7234:
7112:
7086:
7066:
7000:
6864:
6688:
6665:
6645:
6589:
6523:
6430:
6400:
6346:
5640:
5550:
5520:
5500:
5237:
5178:
5149:
5041:
4971:
4938:
4887:
4860:
4830:
4770:
4747:
4350:
4297:
4207:
4109:
4029:
3979:
3932:
3873:
3841:
3776:
3753:
3733:
3689:
3631:
3507:
3481:
3461:
3431:
3344:
3284:
3178:
3152:
3125:
3105:
3085:
3055:
3005:
2958:
2898:
2841:
2818:
2798:
2774:
2754:
2734:
2714:
2694:
2618:
2577:
2557:
2537:
2533:is a random vector,
2517:
2464:
2444:
2424:
2404:
2354:
2324:
2304:
2293:{\displaystyle Wy+b}
2275:
2251:
2231:
2140:
2109:
2067:
1955:
1835:
1824:{\displaystyle g(y)}
1806:
1724:
1657:
1613:
1593:
1560:
1545:{\displaystyle I(x)}
1527:
1410:
1280:
1219:
1182:
1085:
1038:
1018:
942:
853:
663:
596:
576:
556:
536:
516:
331:
276:
256:
236:
198:
169:
149:
123:
103:
24532:. New York: Wiley.
24463:Kay, S. M. (1993).
24236:
24210:
24183:
24156:
23807:
23755:
23716:
23555:
23529:
23432:
23406:
23353:
23145:
23119:
23086:
23047:
23022:
22964:
22932:
22887:
22869:
22843:
22810:
22778:
22745:
22709:
22625:
22599:
22566:
22534:
22474:
22448:
22415:
22383:
21911:
21815:
21708:
21457:
21439:
21287:
21252:
21234:
21218:
21191:
21166:
21119:
21086:
21068:
21052:
21012:
20978:
20723:
20705:
20689:
20633:
20608:
20580:
20542:
20496:
20465:
20337:
20309:
20281:
20253:
20159:
20089:
20061:
19849:
19777:
19759:
19245:
18877:
17363:
17284:
17154:
17024:
16951:
16878:
16777:
16741:
16711:
16681:
16651:
16612:
16558:
16527:
16500:
16461:
16403:
16370:
16342:
16312:
16276:
16246:
16212:
15483:
15301:
15277:
15084:
14909:
14765:
14570:
14488:
14457:
14423:
13507:
11482:
10161:
9978:
9943:
9899:
9819:
9724:
9622:
9495:
9471:
9401:
9360:
9336:
9273:
9249:
9136:
9095:
9071:
7798:is more effective.
7650:
7471:
7453:
7345:
7330:
7301:
7285:
7267:
6979:
6937:
6922:
6899:
6881:
6735:
6489:
6385:
5603:
5145:Since the estimate
5089:
5004:
4923:
4809:
3599:
3397:
3226:
2937:
2390:Monte Carlo methods
1984:
1459:
826:
24346:Moon and Stirling.
24268:Mean squared error
24263:Bayesian estimator
24243:
24222:
24189:
24169:
24168:
24135:
24077:
24008:
23935:
23933:
23811:
23793:
23776:
23766:respectively. Let
23756:
23734:
23717:
23695:
23678:
23651:
23624:
23597:
23562:
23541:
23508:
23436:
23418:
23385:
23332:
23287:
23175:
23152:
23131:
23098:
23065:
23026:
23008:
22968:
22950:
22933:
22911:
22891:
22873:
22855:
22822:
22789:
22757:
22724:
22688:
22668:
22636:
22634:
22611:
22578:
22545:
22513:
22460:
22427:
22394:
22362:
22307:
22169:
22142:
22115:
22019:
22017:
21915:
21890:
21873:
21846:
21819:
21794:
21777:
21750:
21723:
21694:
21677:
21634:
21602:
21582:
21553:
21504:
21484:
21461:
21443:
21418:
21398:
21319:
21296:
21273:
21238:
21220:
21204:
21170:
21152:
21123:
21105:
21072:
21054:
21038:
20995:
20957:
20935:
20835:
20754:
20752:
20709:
20691:
20675:
20619:
20594:
20566:
20528:
20482:
20451:
20387:
20364:
20362:
20323:
20295:
20267:
20236:
20177:
20175:
20145:
20075:
20047:
19952:
19916:
19896:
19876:
19856:
19835:
19806:
19786:
19763:
19745:
19728:
19708:
19655:
19635:
19615:
19562:
19542:
19485:
19450:
19430:
19410:
19400:Consider a vector
19378:
19358:
19331:
19223:
19206:
19170:
19137:
19099:
19060:
19038:
18999:
18966:
18860:
18840:
18813:
18757:
18748:
18712:
18562:
18529:
18520:
18454:
18103:
18076:
18046:{\displaystyle W=}
18043:
17978:
17903:
17867:
17837:
17828:
17668:
17583:
17503:
17476:
17449:
17392:
17328:
17311:
17251:
17234:
17201:
17175:
17155:
17128:
17111:
17071:
17045:
17025:
16991:
16974:
16916:
16899:
16845:
16828:
16781:
16742:
16715:
16685:
16655:
16616:
16577:
16559:
16531:
16501:
16474:
16428:
16410:
16374:
16343:
16316:
16279:
16247:
16219:
16186:
16165:
16145:
16119:
16099:
16064:
15975:
15945:
15822:
15720:
15661:
15619:
15599:
15572:
15543:
15507:
15463:
15384:
15341:
15281:
15257:
15176:
15143:
15107:
15064:
14956:
14910:
14895:
14878:
14847:
14816:
14789:
14751:
14702:
14574:
14553:
14495:
14471:
14427:
14399:
14379:
14349:
14260:
14132:
13995:
13968:
13929:
13733:
13731:
13474:
13354:
13324:
13322:
13141:
13111:
13068:
13026:
12999:
12970:
12930:
12888:
12861:
12829:
12751:
12698:
12627:
12600:
12533:
12502:
12500:
12297:
12205:
12148:
11972:
11771:
11692:
11641:
11565:
11545:
11525:
11465:
11406:
11256:
11236:
11162:
11136:The MMSE estimate
11120:
10988:
10968:
10916:
10885:
10809:
10758:
10685:
10683:
10367:
10340:
10294:
10262:
10191:
10171:
10144:
10127:
10065:
10045:
9999:
9979:
9961:
9944:
9926:
9882:
9846:
9826:
9802:
9770:
9707:
9652:
9623:
9605:
9562:
9535:
9512:
9478:
9454:
9405:
9384:
9343:
9319:
9280:
9256:
9232:
9202:
9140:
9119:
9078:
9054:
8938:
8898:
8856:
8836:
8819:if, for instance,
8805:
8728:
8704:
8669:
8499:
8347:
8315:
8214:
8191:
8134:
8064:
8008:
7972:
7934:
7914:
7894:
7843:
7819:
7800:Levinson recursion
7780:
7760:
7729:
7697:
7677:
7657:
7636:
7619:
7521:
7492:
7472:
7457:
7439:
7422:
7378:
7348:
7331:
7309:
7287:
7271:
7253:
7214:
7092:
7072:
7048:
6983:
6965:
6923:
6905:
6885:
6867:
6849:
6721:
6671:
6651:
6620:
6572:
6506:
6472:
6413:
6396:, we can re-write
6386:
6368:
6329:
6327:
5936:
5929:
5620:
5586:
5533:
5506:
5483:
5481:
5220:
5164:
5132:
5072:
5021:
4987:
4954:
4924:
4906:
4873:
4846:
4813:
4792:
4753:
4730:
4728:
4333:
4283:
4190:
4089:
4012:
3965:
3915:
3856:
3824:
3759:
3739:
3719:
3665:
3616:
3582:
3487:
3467:
3447:
3414:
3380:
3329:
3269:
3209:
3158:
3138:
3111:
3091:
3071:
3041:
2991:
2941:
2920:
2883:
2824:
2804:
2780:
2760:
2740:
2720:
2700:
2677:
2636:
2601:
2563:
2543:
2523:
2503:
2450:
2430:
2410:
2378:
2330:
2310:
2290:
2257:
2237:
2211:
2124:
2105:and the estimator
2095:
2041:
1970:
1938:
1821:
1788:
1704:
1643:
1599:
1566:
1554:Fisher information
1542:
1509:
1380:
1253:
1203:
1168:
1071:
1024:
1000:
913:
833:
812:
646:
582:
562:
542:
522:
496:
315:mean squared error
303:
262:
242:
222:
181:
155:
135:
109:
52:dependent variable
24587:Signal estimation
24299:Linear prediction
24238:
24159:
23779:{\displaystyle x}
23557:
23434:
23281:
23263:
23201:
23178:{\displaystyle N}
23147:
23039:
22924:
22871:
22701:
22665:
22627:
22476:
22301:
22283:
22239:
22198:
22098:
21660:
21605:{\displaystyle x}
21507:{\displaystyle x}
21487:{\displaystyle x}
21431:
21362:
21348:
21322:{\displaystyle N}
21291:
21262:
21183:
21096:
20970:
20930:
20887:
20861:
20744:
20733:
20634:
20609:
20581:
20543:
20497:
20466:
20420:
20390:{\displaystyle W}
20213:
19919:{\displaystyle z}
19899:{\displaystyle x}
19879:{\displaystyle I}
19809:{\displaystyle z}
19731:{\displaystyle x}
19669:over an interval
19658:{\displaystyle x}
19638:{\displaystyle x}
19565:{\displaystyle 1}
19453:{\displaystyle x}
19433:{\displaystyle N}
19420:formed by taking
19413:{\displaystyle y}
19381:{\displaystyle W}
19197:
17894:
17530:
17417:linear prediction
17377:
17338:
16965:
16926:
16752:
16626:
16587:
16408:
16168:{\displaystyle m}
16122:{\displaystyle y}
16055:
16020:
15920:
15880:
15852:
15810:
15772:
15750:
15714:
15696:
15664:{\displaystyle e}
15622:{\displaystyle x}
15569:
15336:
15095:
15014:
14986:
14941:
14681:
14624:
14602:
14239:
14182:
14160:
13959:
13711:
13566:
13534:
13489:
13463:
13414:
13388:
13270:
13100:
13053:
12996:
12915:
12858:
12783:
12733:
12689:
12654:
12585:
12560:
12282:
12257:
12232:
11957:
11801:
11568:{\displaystyle Y}
11548:{\displaystyle X}
11519:
11501:
11446:
11259:{\displaystyle x}
11153:
10991:{\displaystyle x}
10297:{\displaystyle k}
10194:{\displaystyle z}
10068:{\displaystyle I}
10002:{\displaystyle z}
9849:{\displaystyle x}
9764:
9746:
9681:
9649:
9538:{\displaystyle W}
8859:{\displaystyle x}
8731:{\displaystyle x}
8707:{\displaystyle y}
8561:
8493:
8475:
8376:
8344:
8217:{\displaystyle W}
8058:
7937:{\displaystyle z}
7917:{\displaystyle A}
7846:{\displaystyle y}
7783:{\displaystyle W}
7732:{\displaystyle W}
7717:Gauss elimination
7700:{\displaystyle y}
7680:{\displaystyle x}
7616:
7598:
7578:
7547:
7495:{\displaystyle x}
7419:
7404:
7346:
7322:
7302:
7208:
7194:
7160:
7146:
7131:
7095:{\displaystyle y}
7075:{\displaystyle x}
7046:
6938:
6843:
6825:
6805:
6774:
6756:
6736:
6700:
6674:{\displaystyle y}
6654:{\displaystyle x}
6565:
6267:
6243:
6194:
6170:
6117:
6090:
6063:
6001:
5971:
5909:
5879:
5856:
5854:
5825:
5798:
5762:
5709:
5685:
5566:
5509:{\displaystyle W}
5417:
5393:
5337:
5322:
5304:
5289:
5257:
5217:
5199:
5161:
5126:
5108:
5053:
4756:{\displaystyle W}
4668:
4644:
4598:
4574:
4518:
4491:
4464:
4408:
4378:
4330:
4261:
4231:
4181:
4154:
4121:
4086:
4068:
4041:
3991:
3944:
3909:
3891:
3866:in above, we get
3853:
3797:
3762:{\displaystyle b}
3742:{\displaystyle W}
3701:
3549:
3490:{\displaystyle x}
3470:{\displaystyle y}
3360:
3323:
3305:
3263:
3245:
3190:
3161:{\displaystyle y}
3114:{\displaystyle y}
3094:{\displaystyle x}
3017:
2970:
2877:
2859:
2827:{\displaystyle W}
2807:{\displaystyle b}
2783:{\displaystyle y}
2763:{\displaystyle x}
2743:{\displaystyle y}
2723:{\displaystyle x}
2703:{\displaystyle x}
2656:
2644:
2621:
2566:{\displaystyle b}
2546:{\displaystyle W}
2526:{\displaystyle y}
2453:{\displaystyle y}
2433:{\displaystyle y}
2413:{\displaystyle x}
2333:{\displaystyle b}
2313:{\displaystyle W}
2260:{\displaystyle y}
2240:{\displaystyle x}
2193:
2165:
2121:
2080:
1749:
1670:
1625:
1602:{\displaystyle x}
1569:{\displaystyle x}
1460:
1433:
1418:
1302:
1146:
1122:
1051:
1027:{\displaystyle x}
955:
900:
866:
622:
585:{\displaystyle x}
565:{\displaystyle y}
545:{\displaystyle x}
478:
447:
396:
372:
323:covariance matrix
294:
265:{\displaystyle y}
245:{\displaystyle x}
210:
158:{\displaystyle y}
112:{\displaystyle x}
48:mean square error
36:signal processing
16:(Redirected from
24594:
24562:
24543:
24524:
24505:
24486:
24470:
24459:
24440:
24428:
24417:
24398:
24392:
24387:
24385:
24377:
24375:
24373:
24347:
24344:
24338:
24337:
24335:
24333:
24319:
24252:
24250:
24249:
24244:
24239:
24237:
24235:
24230:
24221:
24209:
24204:
24203:
24202:
24188:
24182:
24177:
24167:
24157:
24155:
24150:
24149:
24148:
24134:
24129:
24128:
24118:
24113:
24112:
24086:
24084:
24083:
24078:
24076:
24075:
24066:
24065:
24053:
24052:
24043:
24042:
24017:
24015:
24014:
24009:
23998:
23997:
23973:
23972:
23944:
23942:
23941:
23936:
23934:
23927:
23926:
23911:
23910:
23894:
23893:
23880:
23879:
23864:
23863:
23847:
23846:
23820:
23818:
23817:
23812:
23806:
23801:
23785:
23783:
23782:
23777:
23765:
23763:
23762:
23757:
23754:
23749:
23748:
23747:
23726:
23724:
23723:
23718:
23715:
23710:
23709:
23708:
23687:
23685:
23684:
23679:
23677:
23676:
23660:
23658:
23657:
23652:
23650:
23649:
23633:
23631:
23630:
23625:
23623:
23622:
23606:
23604:
23603:
23598:
23596:
23595:
23571:
23569:
23568:
23563:
23558:
23556:
23554:
23549:
23540:
23528:
23523:
23522:
23521:
23507:
23498:
23493:
23474:
23469:
23445:
23443:
23442:
23437:
23435:
23433:
23431:
23426:
23417:
23405:
23400:
23399:
23398:
23384:
23375:
23370:
23354:
23352:
23347:
23346:
23345:
23331:
23322:
23317:
23316:
23296:
23294:
23293:
23288:
23283:
23282:
23274:
23265:
23264:
23256:
23250:
23249:
23237:
23236:
23226:
23221:
23203:
23202:
23194:
23185:pollsters, then
23184:
23182:
23181:
23176:
23161:
23159:
23158:
23153:
23148:
23146:
23144:
23139:
23130:
23118:
23113:
23112:
23111:
23097:
23085:
23080:
23079:
23078:
23064:
23052:
23046:
23041:
23040:
23032:
23021:
23016:
23004:
22977:
22975:
22974:
22969:
22963:
22958:
22942:
22940:
22939:
22934:
22931:
22926:
22925:
22917:
22900:
22898:
22897:
22892:
22886:
22881:
22872:
22870:
22868:
22863:
22854:
22842:
22837:
22836:
22835:
22821:
22809:
22804:
22803:
22802:
22788:
22779:
22777:
22772:
22771:
22770:
22756:
22744:
22739:
22738:
22737:
22723:
22714:
22708:
22703:
22702:
22694:
22677:
22675:
22674:
22669:
22667:
22666:
22658:
22645:
22643:
22642:
22637:
22635:
22628:
22626:
22624:
22619:
22610:
22598:
22593:
22592:
22591:
22577:
22565:
22560:
22559:
22558:
22544:
22535:
22533:
22528:
22527:
22526:
22512:
22503:
22494:
22493:
22477:
22475:
22473:
22468:
22459:
22447:
22442:
22441:
22440:
22426:
22414:
22409:
22408:
22407:
22393:
22384:
22382:
22377:
22376:
22375:
22361:
22352:
22343:
22342:
22316:
22314:
22313:
22308:
22303:
22302:
22294:
22285:
22284:
22276:
22270:
22269:
22257:
22256:
22241:
22240:
22232:
22226:
22225:
22213:
22212:
22200:
22199:
22191:
22178:
22176:
22175:
22170:
22168:
22167:
22151:
22149:
22148:
22143:
22141:
22140:
22124:
22122:
22121:
22116:
22111:
22100:
22099:
22091:
22082:
22081:
22057:
22056:
22028:
22026:
22025:
22020:
22018:
22011:
22010:
21988:
21987:
21974:
21973:
21951:
21950:
21924:
21922:
21921:
21916:
21910:
21905:
21904:
21903:
21882:
21880:
21879:
21874:
21872:
21871:
21855:
21853:
21852:
21847:
21845:
21844:
21828:
21826:
21825:
21820:
21814:
21809:
21808:
21807:
21786:
21784:
21783:
21778:
21776:
21775:
21759:
21757:
21756:
21751:
21749:
21748:
21732:
21730:
21729:
21724:
21719:
21707:
21702:
21687:and variance is
21686:
21684:
21683:
21678:
21673:
21662:
21661:
21653:
21643:
21641:
21640:
21637:{\displaystyle }
21635:
21611:
21609:
21608:
21603:
21591:
21589:
21588:
21583:
21562:
21560:
21559:
21554:
21513:
21511:
21510:
21505:
21493:
21491:
21490:
21485:
21470:
21468:
21467:
21462:
21456:
21451:
21438:
21433:
21432:
21424:
21407:
21405:
21404:
21399:
21394:
21393:
21383:
21378:
21363:
21355:
21350:
21349:
21341:
21328:
21326:
21325:
21320:
21305:
21303:
21302:
21297:
21292:
21286:
21281:
21272:
21270:
21269:
21263:
21261:
21257:
21251:
21246:
21233:
21228:
21217:
21212:
21203:
21201:
21200:
21190:
21185:
21184:
21176:
21165:
21160:
21132:
21130:
21129:
21124:
21118:
21113:
21104:
21103:
21097:
21095:
21091:
21085:
21080:
21067:
21062:
21051:
21046:
21037:
21035:
21034:
21025:
21024:
21011:
21003:
20994:
20993:
20977:
20972:
20971:
20963:
20944:
20942:
20941:
20936:
20931:
20926:
20925:
20924:
20914:
20909:
20893:
20888:
20883:
20879:
20878:
20868:
20863:
20862:
20854:
20844:
20842:
20841:
20836:
20834:
20833:
20824:
20823:
20805:
20804:
20792:
20791:
20763:
20761:
20760:
20755:
20753:
20746:
20745:
20737:
20734:
20732:
20728:
20722:
20717:
20704:
20699:
20688:
20683:
20674:
20666:
20659:
20658:
20649:
20648:
20640:
20636:
20635:
20632:
20627:
20615:
20610:
20607:
20602:
20590:
20582:
20579:
20574:
20562:
20554:
20544:
20541:
20536:
20524:
20522:
20521:
20512:
20511:
20503:
20499:
20498:
20495:
20490:
20478:
20467:
20464:
20459:
20447:
20445:
20444:
20422:
20421:
20413:
20396:
20394:
20393:
20388:
20373:
20371:
20370:
20365:
20363:
20353:
20352:
20336:
20331:
20319:
20318:
20308:
20303:
20291:
20290:
20280:
20275:
20260:
20252:
20244:
20235:
20234:
20215:
20214:
20206:
20186:
20184:
20183:
20178:
20176:
20169:
20168:
20158:
20153:
20138:
20137:
20113:
20112:
20099:
20088:
20083:
20071:
20070:
20060:
20055:
20040:
20039:
20015:
20014:
20004:
19975:
19961:
19959:
19958:
19953:
19945:
19944:
19925:
19923:
19922:
19917:
19905:
19903:
19902:
19897:
19885:
19883:
19882:
19877:
19865:
19863:
19862:
19857:
19848:
19843:
19815:
19813:
19812:
19807:
19795:
19793:
19792:
19787:
19782:
19776:
19771:
19758:
19753:
19737:
19735:
19734:
19729:
19717:
19715:
19714:
19711:{\displaystyle }
19709:
19704:
19703:
19691:
19690:
19664:
19662:
19661:
19656:
19644:
19642:
19641:
19636:
19624:
19622:
19621:
19618:{\displaystyle }
19616:
19611:
19610:
19598:
19597:
19571:
19569:
19568:
19563:
19551:
19549:
19548:
19543:
19541:
19540:
19494:
19492:
19491:
19486:
19459:
19457:
19456:
19451:
19439:
19437:
19436:
19431:
19419:
19417:
19416:
19411:
19387:
19385:
19384:
19379:
19367:
19365:
19364:
19359:
19357:
19356:
19340:
19338:
19337:
19332:
19324:
19323:
19299:
19298:
19277:
19276:
19267:
19266:
19244:
19239:
19234:
19215:
19213:
19212:
19207:
19205:
19204:
19199:
19198:
19190:
19179:
19177:
19176:
19171:
19163:
19162:
19146:
19144:
19143:
19138:
19124:
19123:
19108:
19106:
19105:
19100:
19089:
19088:
19069:
19067:
19066:
19061:
19056:
19055:
19043:
19039:
19004:
19000:
18971:
18967:
18890:
18889:
18876:
18868:
18849:
18847:
18846:
18841:
18839:
18838:
18822:
18820:
18819:
18814:
18812:
18811:
18796:
18795:
18786:
18785:
18766:
18764:
18763:
18758:
18753:
18749:
18717:
18713:
18706:
18705:
18693:
18692:
18670:
18669:
18657:
18656:
18634:
18633:
18621:
18620:
18594:
18593:
18571:
18569:
18568:
18563:
18561:
18560:
18538:
18536:
18535:
18530:
18525:
18521:
18459:
18455:
18448:
18447:
18435:
18434:
18414:
18413:
18401:
18400:
18380:
18379:
18367:
18366:
18344:
18343:
18331:
18330:
18310:
18309:
18297:
18296:
18276:
18275:
18263:
18262:
18240:
18239:
18227:
18226:
18206:
18205:
18193:
18192:
18172:
18171:
18159:
18158:
18132:
18131:
18112:
18110:
18109:
18104:
18102:
18101:
18085:
18083:
18082:
18077:
18075:
18074:
18052:
18050:
18049:
18044:
18039:
18038:
18026:
18025:
18013:
18012:
17987:
17985:
17984:
17979:
17977:
17976:
17967:
17966:
17954:
17953:
17941:
17940:
17912:
17910:
17909:
17904:
17902:
17901:
17896:
17895:
17887:
17876:
17874:
17873:
17868:
17866:
17865:
17846:
17844:
17843:
17838:
17833:
17829:
17727:
17726:
17677:
17675:
17674:
17669:
17667:
17666:
17657:
17656:
17644:
17643:
17631:
17630:
17618:
17617:
17592:
17590:
17589:
17584:
17582:
17581:
17572:
17571:
17561:
17556:
17538:
17537:
17532:
17531:
17523:
17512:
17510:
17509:
17504:
17502:
17501:
17485:
17483:
17482:
17477:
17475:
17474:
17458:
17456:
17455:
17450:
17448:
17447:
17435:
17434:
17415:We shall take a
17401:
17399:
17398:
17393:
17391:
17390:
17379:
17378:
17370:
17362:
17351:
17340:
17339:
17331:
17320:
17318:
17317:
17312:
17310:
17309:
17308:
17307:
17283:
17272:
17271:
17270:
17243:
17241:
17240:
17235:
17233:
17232:
17210:
17208:
17207:
17202:
17184:
17182:
17181:
17176:
17164:
17162:
17161:
17156:
17153:
17142:
17120:
17118:
17117:
17112:
17110:
17109:
17108:
17107:
17081:diagonal matrix
17080:
17078:
17077:
17072:
17054:
17052:
17051:
17046:
17034:
17032:
17031:
17026:
17023:
17012:
17011:
17010:
16983:
16981:
16980:
16975:
16973:
16972:
16967:
16966:
16958:
16950:
16939:
16928:
16927:
16919:
16908:
16906:
16905:
16900:
16898:
16897:
16896:
16895:
16877:
16866:
16865:
16864:
16837:
16835:
16834:
16829:
16790:
16788:
16787:
16782:
16776:
16759:
16754:
16753:
16745:
16740:
16729:
16710:
16699:
16680:
16669:
16650:
16633:
16628:
16627:
16619:
16611:
16600:
16589:
16588:
16580:
16568:
16566:
16565:
16560:
16557:
16546:
16545:
16544:
16526:
16515:
16499:
16488:
16460:
16449:
16448:
16447:
16419:
16417:
16416:
16411:
16409:
16407:
16402:
16388:
16369:
16358:
16357:
16356:
16341:
16330:
16311:
16300:
16299:
16298:
16277:
16275:
16261:
16245:
16234:
16233:
16232:
16217:
16211:
16200:
16174:
16172:
16171:
16166:
16154:
16152:
16151:
16146:
16128:
16126:
16125:
16120:
16108:
16106:
16105:
16100:
16098:
16097:
16073:
16071:
16070:
16065:
16063:
16062:
16057:
16056:
16048:
16044:
16043:
16028:
16027:
16022:
16021:
16013:
16009:
16008:
15996:
15984:
15982:
15981:
15976:
15974:
15973:
15954:
15952:
15951:
15946:
15938:
15937:
15928:
15927:
15922:
15921:
15913:
15906:
15901:
15900:
15888:
15887:
15882:
15881:
15873:
15866:
15865:
15854:
15853:
15845:
15831:
15829:
15828:
15823:
15812:
15811:
15803:
15797:
15780:
15779:
15774:
15773:
15765:
15758:
15753:
15752:
15751:
15743:
15729:
15727:
15726:
15721:
15716:
15715:
15707:
15704:
15703:
15698:
15697:
15689:
15682:
15670:
15668:
15667:
15662:
15637:adaptive filters
15628:
15626:
15625:
15620:
15608:
15606:
15605:
15600:
15598:
15597:
15581:
15579:
15578:
15573:
15571:
15570:
15562:
15552:
15550:
15549:
15544:
15542:
15541:
15516:
15514:
15513:
15508:
15503:
15502:
15501:
15500:
15482:
15477:
15462:
15461:
15434:
15433:
15432:
15431:
15393:
15391:
15390:
15385:
15383:
15382:
15381:
15380:
15350:
15348:
15347:
15342:
15337:
15335:
15334:
15333:
15318:
15317:
15316:
15315:
15300:
15295:
15276:
15271:
15255:
15254:
15253:
15238:
15237:
15236:
15235:
15220:
15215:
15214:
15185:
15183:
15182:
15177:
15175:
15174:
15152:
15150:
15149:
15144:
15142:
15141:
15116:
15114:
15113:
15108:
15103:
15102:
15097:
15096:
15088:
15083:
15078:
15060:
15059:
15041:
15040:
15022:
15021:
15016:
15015:
15007:
15000:
14999:
14988:
14987:
14979:
14965:
14963:
14962:
14957:
14955:
14954:
14943:
14942:
14934:
14919:
14917:
14916:
14911:
14908:
14903:
14887:
14885:
14884:
14879:
14877:
14876:
14856:
14854:
14853:
14848:
14846:
14845:
14825:
14823:
14822:
14817:
14815:
14814:
14798:
14796:
14795:
14790:
14788:
14787:
14775:
14774:
14764:
14759:
14747:
14746:
14711:
14709:
14708:
14703:
14695:
14694:
14683:
14682:
14674:
14664:
14663:
14651:
14650:
14638:
14637:
14626:
14625:
14617:
14610:
14609:
14604:
14603:
14595:
14583:
14581:
14580:
14575:
14569:
14561:
14552:
14551:
14542:
14541:
14540:
14539:
14522:
14521:
14504:
14502:
14501:
14496:
14487:
14479:
14470:
14469:
14456:
14448:
14447:
14446:
14422:
14414:
14413:
14412:
14388:
14386:
14385:
14380:
14378:
14377:
14358:
14356:
14355:
14350:
14345:
14344:
14343:
14342:
14316:
14315:
14294:
14293:
14292:
14291:
14269:
14267:
14266:
14261:
14253:
14252:
14241:
14240:
14232:
14222:
14221:
14209:
14208:
14196:
14195:
14184:
14183:
14175:
14168:
14167:
14162:
14161:
14153:
14141:
14139:
14138:
14133:
14128:
14127:
14115:
14114:
14102:
14101:
14092:
14091:
14090:
14089:
14063:
14062:
14053:
14052:
14051:
14050:
14027:
14026:
14004:
14002:
14001:
13996:
13994:
13993:
13977:
13975:
13974:
13969:
13967:
13966:
13961:
13960:
13952:
13938:
13936:
13935:
13930:
13925:
13924:
13923:
13922:
13899:
13898:
13886:
13885:
13873:
13872:
13863:
13862:
13861:
13860:
13834:
13833:
13824:
13823:
13822:
13821:
13798:
13797:
13796:
13795:
13772:
13771:
13770:
13769:
13742:
13740:
13739:
13734:
13732:
13725:
13724:
13713:
13712:
13704:
13694:
13693:
13681:
13680:
13668:
13667:
13655:
13654:
13645:
13644:
13643:
13642:
13616:
13615:
13606:
13605:
13604:
13603:
13580:
13579:
13568:
13567:
13559:
13549:
13542:
13541:
13536:
13535:
13527:
13520:
13519:
13506:
13498:
13497:
13496:
13491:
13490:
13482:
13473:
13472:
13471:
13470:
13465:
13464:
13456:
13452:
13451:
13428:
13427:
13416:
13415:
13407:
13396:
13395:
13390:
13389:
13381:
13363:
13361:
13360:
13355:
13353:
13352:
13333:
13331:
13330:
13325:
13323:
13316:
13315:
13306:
13305:
13304:
13303:
13280:
13279:
13278:
13277:
13272:
13271:
13263:
13259:
13258:
13231:
13230:
13229:
13228:
13204:
13203:
13194:
13189:
13188:
13179:
13178:
13150:
13148:
13147:
13142:
13140:
13139:
13120:
13118:
13117:
13112:
13110:
13109:
13108:
13107:
13102:
13101:
13093:
13077:
13075:
13074:
13069:
13067:
13066:
13055:
13054:
13046:
13035:
13033:
13032:
13027:
13025:
13024:
13008:
13006:
13005:
13000:
12998:
12997:
12989:
12979:
12977:
12976:
12971:
12969:
12968:
12967:
12966:
12939:
12937:
12936:
12931:
12929:
12928:
12917:
12916:
12908:
12897:
12895:
12894:
12889:
12887:
12886:
12870:
12868:
12867:
12862:
12860:
12859:
12851:
12838:
12836:
12835:
12830:
12828:
12827:
12826:
12825:
12816:
12811:
12810:
12793:
12792:
12791:
12790:
12785:
12784:
12776:
12760:
12758:
12757:
12752:
12741:
12740:
12735:
12734:
12726:
12719:
12707:
12705:
12704:
12699:
12697:
12696:
12691:
12690:
12682:
12675:
12674:
12662:
12661:
12656:
12655:
12647:
12636:
12634:
12633:
12628:
12626:
12625:
12609:
12607:
12606:
12601:
12599:
12598:
12587:
12586:
12578:
12568:
12567:
12562:
12561:
12553:
12542:
12540:
12539:
12534:
12532:
12531:
12511:
12509:
12508:
12503:
12501:
12494:
12493:
12481:
12480:
12471:
12470:
12469:
12468:
12442:
12441:
12429:
12428:
12419:
12418:
12417:
12416:
12392:
12391:
12382:
12377:
12376:
12352:
12351:
12350:
12349:
12340:
12335:
12334:
12306:
12304:
12303:
12298:
12296:
12295:
12284:
12283:
12275:
12265:
12264:
12259:
12258:
12250:
12240:
12239:
12234:
12233:
12225:
12214:
12212:
12211:
12206:
12201:
12200:
12191:
12186:
12185:
12157:
12155:
12154:
12149:
12144:
12143:
12142:
12141:
12118:
12117:
12116:
12115:
12091:
12090:
12081:
12076:
12075:
12052:
12051:
12050:
12049:
12025:
12024:
12015:
12010:
12009:
11981:
11979:
11978:
11973:
11971:
11970:
11959:
11958:
11950:
11940:
11939:
11915:
11914:
11905:
11900:
11899:
11881:
11870:
11869:
11845:
11844:
11835:
11830:
11829:
11817:
11809:
11808:
11803:
11802:
11794:
11780:
11778:
11777:
11772:
11767:
11766:
11742:
11741:
11732:
11727:
11726:
11702:, respectively.
11701:
11699:
11698:
11693:
11688:
11687:
11678:
11673:
11672:
11650:
11648:
11647:
11642:
11637:
11636:
11612:
11611:
11602:
11597:
11596:
11574:
11572:
11571:
11566:
11554:
11552:
11551:
11546:
11534:
11532:
11531:
11526:
11521:
11520:
11512:
11503:
11502:
11494:
11481:
11473:
11464:
11463:
11448:
11447:
11439:
11415:
11413:
11412:
11407:
11399:
11398:
11374:
11373:
11364:
11359:
11358:
11331:
11330:
11306:
11305:
11296:
11291:
11290:
11265:
11263:
11262:
11257:
11245:
11243:
11242:
11237:
11232:
11231:
11213:
11212:
11203:
11198:
11197:
11171:
11169:
11168:
11163:
11161:
11160:
11155:
11154:
11146:
11129:
11127:
11126:
11121:
11113:
11112:
11103:
11098:
11097:
11076:
11075:
11051:
11050:
11038:
11037:
11028:
11023:
11022:
10997:
10995:
10994:
10989:
10977:
10975:
10974:
10969:
10967:
10966:
10942:
10941:
10925:
10923:
10922:
10917:
10915:
10914:
10894:
10892:
10891:
10886:
10881:
10880:
10856:
10855:
10846:
10841:
10840:
10818:
10816:
10815:
10810:
10805:
10804:
10795:
10790:
10789:
10767:
10765:
10764:
10759:
10754:
10753:
10735:
10734:
10725:
10720:
10719:
10694:
10692:
10691:
10686:
10684:
10674:
10673:
10649:
10648:
10639:
10634:
10633:
10615:
10614:
10605:
10600:
10599:
10578:
10571:
10570:
10546:
10545:
10536:
10531:
10530:
10512:
10511:
10487:
10486:
10471:
10466:
10465:
10440:
10439:
10421:
10420:
10411:
10406:
10405:
10376:
10374:
10373:
10368:
10366:
10365:
10349:
10347:
10346:
10341:
10339:
10338:
10320:
10319:
10303:
10301:
10300:
10295:
10271:
10269:
10268:
10263:
10261:
10260:
10251:
10250:
10226:
10225:
10200:
10198:
10197:
10192:
10180:
10178:
10177:
10172:
10160:
10152:
10136:
10134:
10133:
10128:
10126:
10125:
10116:
10115:
10100:
10099:
10074:
10072:
10071:
10066:
10054:
10052:
10051:
10046:
10038:
10037:
10025:
10024:
10008:
10006:
10005:
10000:
9988:
9986:
9985:
9980:
9977:
9969:
9953:
9951:
9950:
9945:
9942:
9934:
9925:
9924:
9915:
9914:
9898:
9890:
9881:
9880:
9855:
9853:
9852:
9847:
9835:
9833:
9832:
9827:
9818:
9810:
9785:ridge regression
9779:
9777:
9776:
9771:
9766:
9765:
9757:
9748:
9747:
9739:
9723:
9715:
9706:
9705:
9696:
9695:
9683:
9682:
9674:
9661:
9659:
9658:
9653:
9651:
9650:
9642:
9632:
9630:
9629:
9624:
9621:
9613:
9604:
9603:
9594:
9593:
9571:
9569:
9568:
9563:
9561:
9560:
9544:
9542:
9541:
9536:
9521:
9519:
9518:
9513:
9508:
9507:
9494:
9486:
9470:
9462:
9453:
9452:
9437:
9436:
9414:
9412:
9411:
9406:
9400:
9392:
9383:
9382:
9373:
9372:
9359:
9351:
9335:
9327:
9318:
9317:
9289:
9287:
9286:
9281:
9272:
9264:
9248:
9240:
9231:
9230:
9211:
9209:
9208:
9203:
9198:
9197:
9185:
9184:
9175:
9174:
9149:
9147:
9146:
9141:
9135:
9127:
9118:
9117:
9108:
9107:
9094:
9086:
9070:
9062:
9053:
9052:
9037:
9036:
9024:
9023:
9011:
9010:
9001:
9000:
8985:
8984:
8975:
8974:
8952:Alternative form
8947:
8945:
8944:
8939:
8937:
8936:
8927:
8926:
8907:
8905:
8904:
8899:
8891:
8890:
8865:
8863:
8862:
8857:
8845:
8843:
8842:
8837:
8835:
8834:
8814:
8812:
8811:
8806:
8804:
8803:
8791:
8790:
8778:
8777:
8768:
8767:
8737:
8735:
8734:
8729:
8713:
8711:
8710:
8705:
8678:
8676:
8675:
8670:
8665:
8664:
8652:
8651:
8639:
8638:
8626:
8625:
8616:
8615:
8600:
8599:
8590:
8589:
8577:
8576:
8564:
8563:
8562:
8554:
8544:
8543:
8531:
8530:
8508:
8506:
8505:
8500:
8495:
8494:
8486:
8477:
8476:
8468:
8453:
8452:
8440:
8439:
8427:
8426:
8417:
8416:
8401:
8400:
8391:
8390:
8378:
8377:
8369:
8356:
8354:
8353:
8348:
8346:
8345:
8337:
8324:
8322:
8321:
8316:
8311:
8310:
8298:
8297:
8285:
8284:
8275:
8274:
8259:
8258:
8249:
8248:
8223:
8221:
8220:
8215:
8200:
8198:
8197:
8192:
8187:
8186:
8177:
8176:
8164:
8163:
8143:
8141:
8140:
8135:
8130:
8129:
8117:
8116:
8107:
8106:
8091:
8090:
8073:
8071:
8070:
8065:
8060:
8059:
8051:
8017:
8015:
8014:
8009:
8001:
8000:
7981:
7979:
7978:
7973:
7943:
7941:
7940:
7935:
7923:
7921:
7920:
7915:
7903:
7901:
7900:
7895:
7852:
7850:
7849:
7844:
7828:
7826:
7825:
7820:
7818:
7817:
7789:
7787:
7786:
7781:
7769:
7767:
7766:
7761:
7759:
7758:
7741:QR decomposition
7738:
7736:
7735:
7730:
7706:
7704:
7703:
7698:
7686:
7684:
7683:
7678:
7666:
7664:
7663:
7658:
7649:
7644:
7628:
7626:
7625:
7620:
7618:
7617:
7609:
7600:
7599:
7591:
7579:
7577:
7576:
7567:
7566:
7554:
7549:
7548:
7540:
7530:
7528:
7527:
7522:
7501:
7499:
7498:
7493:
7481:
7479:
7478:
7473:
7470:
7465:
7452:
7447:
7431:
7429:
7428:
7423:
7421:
7420:
7412:
7406:
7405:
7397:
7387:
7385:
7384:
7379:
7357:
7355:
7354:
7349:
7347:
7344:
7339:
7329:
7324:
7323:
7315:
7308:
7303:
7300:
7295:
7286:
7284:
7279:
7266:
7261:
7251:
7246:
7245:
7223:
7221:
7220:
7215:
7213:
7209:
7207:
7206:
7197:
7196:
7195:
7187:
7177:
7165:
7161:
7159:
7158:
7149:
7148:
7147:
7139:
7133:
7132:
7124:
7120:
7101:
7099:
7098:
7093:
7081:
7079:
7078:
7073:
7057:
7055:
7054:
7049:
7047:
7045:
7044:
7043:
7034:
7033:
7023:
7022:
7010:
6992:
6990:
6989:
6984:
6978:
6973:
6961:
6960:
6939:
6936:
6931:
6921:
6916:
6904:
6898:
6893:
6880:
6875:
6858:
6856:
6855:
6850:
6845:
6844:
6836:
6827:
6826:
6818:
6806:
6804:
6803:
6794:
6793:
6784:
6776:
6775:
6767:
6758:
6757:
6749:
6737:
6734:
6729:
6720:
6719:
6707:
6702:
6701:
6693:
6680:
6678:
6677:
6672:
6660:
6658:
6657:
6652:
6629:
6627:
6626:
6621:
6616:
6615:
6581:
6579:
6578:
6573:
6568:
6567:
6566:
6558:
6548:
6547:
6535:
6534:
6515:
6513:
6512:
6507:
6502:
6501:
6488:
6480:
6471:
6470:
6455:
6454:
6442:
6441:
6422:
6420:
6419:
6414:
6412:
6411:
6395:
6393:
6392:
6387:
6384:
6376:
6367:
6366:
6338:
6336:
6335:
6330:
6328:
6321:
6320:
6302:
6301:
6286:
6279:
6278:
6269:
6268:
6260:
6245:
6244:
6236:
6206:
6205:
6196:
6195:
6187:
6172:
6171:
6163:
6136:
6129:
6128:
6119:
6118:
6110:
6092:
6091:
6083:
6065:
6064:
6056:
6020:
6013:
6012:
6003:
6002:
5994:
5973:
5972:
5964:
5946:
5945:
5935:
5930:
5925:
5921:
5920:
5911:
5910:
5902:
5881:
5880:
5872:
5847:
5840:
5839:
5827:
5826:
5818:
5800:
5799:
5791:
5764:
5763:
5755:
5734:
5727:
5726:
5711:
5710:
5702:
5687:
5686:
5678:
5656:
5655:
5629:
5627:
5626:
5621:
5616:
5615:
5602:
5594:
5585:
5584:
5569:
5568:
5567:
5559:
5542:
5540:
5539:
5534:
5532:
5531:
5515:
5513:
5512:
5507:
5492:
5490:
5489:
5484:
5482:
5475:
5474:
5465:
5464:
5446:
5442:
5441:
5429:
5428:
5419:
5418:
5410:
5395:
5394:
5386:
5356:
5349:
5348:
5339:
5338:
5330:
5324:
5323:
5315:
5306:
5305:
5297:
5291:
5290:
5282:
5260:
5259:
5258:
5250:
5229:
5227:
5226:
5221:
5219:
5218:
5210:
5201:
5200:
5192:
5173:
5171:
5170:
5165:
5163:
5162:
5154:
5141:
5139:
5138:
5133:
5128:
5127:
5119:
5110:
5109:
5101:
5088:
5080:
5071:
5070:
5055:
5054:
5046:
5030:
5028:
5027:
5022:
5017:
5016:
5003:
4995:
4983:
4982:
4963:
4961:
4960:
4955:
4953:
4952:
4933:
4931:
4930:
4925:
4922:
4917:
4902:
4901:
4882:
4880:
4879:
4874:
4872:
4871:
4855:
4853:
4852:
4847:
4845:
4844:
4822:
4820:
4819:
4814:
4808:
4800:
4791:
4790:
4762:
4760:
4759:
4754:
4739:
4737:
4736:
4731:
4729:
4722:
4721:
4706:
4705:
4687:
4680:
4679:
4670:
4669:
4661:
4646:
4645:
4637:
4610:
4609:
4600:
4599:
4591:
4576:
4575:
4567:
4537:
4530:
4529:
4520:
4519:
4511:
4493:
4492:
4484:
4466:
4465:
4457:
4420:
4419:
4410:
4409:
4401:
4380:
4379:
4371:
4342:
4340:
4339:
4334:
4332:
4331:
4323:
4293:, where we take
4292:
4290:
4289:
4284:
4273:
4272:
4263:
4262:
4254:
4233:
4232:
4224:
4199:
4197:
4196:
4191:
4183:
4182:
4174:
4156:
4155:
4147:
4123:
4122:
4114:
4098:
4096:
4095:
4090:
4088:
4087:
4079:
4070:
4069:
4061:
4043:
4042:
4034:
4021:
4019:
4018:
4013:
3993:
3992:
3984:
3974:
3972:
3971:
3966:
3946:
3945:
3937:
3924:
3922:
3921:
3916:
3911:
3910:
3902:
3893:
3892:
3884:
3865:
3863:
3862:
3857:
3855:
3854:
3846:
3833:
3831:
3830:
3825:
3799:
3798:
3790:
3768:
3766:
3765:
3760:
3748:
3746:
3745:
3740:
3728:
3726:
3725:
3720:
3703:
3702:
3694:
3674:
3672:
3671:
3666:
3658:
3657:
3625:
3623:
3622:
3617:
3612:
3611:
3598:
3590:
3581:
3580:
3565:
3564:
3552:
3551:
3550:
3542:
3532:
3531:
3519:
3518:
3496:
3494:
3493:
3488:
3476:
3474:
3473:
3468:
3456:
3454:
3453:
3448:
3446:
3445:
3423:
3421:
3420:
3415:
3410:
3409:
3396:
3388:
3379:
3378:
3363:
3362:
3361:
3353:
3338:
3336:
3335:
3330:
3325:
3324:
3316:
3307:
3306:
3298:
3278:
3276:
3275:
3270:
3265:
3264:
3256:
3247:
3246:
3238:
3225:
3217:
3208:
3207:
3192:
3191:
3183:
3167:
3165:
3164:
3159:
3147:
3145:
3144:
3139:
3137:
3136:
3120:
3118:
3117:
3112:
3100:
3098:
3097:
3092:
3080:
3078:
3077:
3072:
3070:
3069:
3050:
3048:
3047:
3042:
3019:
3018:
3010:
3000:
2998:
2997:
2992:
2972:
2971:
2963:
2950:
2948:
2947:
2942:
2936:
2928:
2919:
2918:
2892:
2890:
2889:
2884:
2879:
2878:
2870:
2861:
2860:
2852:
2833:
2831:
2830:
2825:
2813:
2811:
2810:
2805:
2789:
2787:
2786:
2781:
2769:
2767:
2766:
2761:
2749:
2747:
2746:
2741:
2729:
2727:
2726:
2721:
2709:
2707:
2706:
2701:
2686:
2684:
2683:
2678:
2658:
2657:
2649:
2645:
2642:
2635:
2610:
2608:
2607:
2602:
2572:
2570:
2569:
2564:
2553:is a matrix and
2552:
2550:
2549:
2544:
2532:
2530:
2529:
2524:
2512:
2510:
2509:
2504:
2459:
2457:
2456:
2451:
2439:
2437:
2436:
2431:
2419:
2417:
2416:
2411:
2387:
2385:
2384:
2379:
2339:
2337:
2336:
2331:
2319:
2317:
2316:
2311:
2299:
2297:
2296:
2291:
2269:jointly Gaussian
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2220:
2218:
2217:
2212:
2201:
2200:
2195:
2194:
2186:
2173:
2172:
2167:
2166:
2158:
2134:should be zero,
2133:
2131:
2130:
2125:
2123:
2122:
2114:
2104:
2102:
2101:
2096:
2088:
2087:
2082:
2081:
2073:
2050:
2048:
2047:
2042:
2019:
2018:
2006:
2005:
1983:
1978:
1947:
1945:
1944:
1939:
1922:
1921:
1891:
1883:
1882:
1877:
1844:
1843:
1830:
1828:
1827:
1822:
1797:
1795:
1794:
1789:
1757:
1756:
1751:
1750:
1742:
1713:
1711:
1710:
1705:
1691:
1690:
1678:
1677:
1672:
1671:
1663:
1652:
1650:
1649:
1644:
1627:
1626:
1618:
1608:
1606:
1605:
1600:
1575:
1573:
1572:
1567:
1551:
1549:
1548:
1543:
1518:
1516:
1515:
1510:
1505:
1501:
1491:
1490:
1467:
1466:
1451:
1441:
1440:
1435:
1434:
1426:
1419:
1414:
1389:
1387:
1386:
1381:
1310:
1309:
1304:
1303:
1295:
1262:
1260:
1259:
1254:
1252:
1251:
1247:
1231:
1230:
1212:
1210:
1209:
1204:
1202:
1201:
1197:
1177:
1175:
1174:
1169:
1164:
1163:
1148:
1147:
1139:
1124:
1123:
1115:
1097:
1096:
1080:
1078:
1077:
1072:
1070:
1069:
1068:
1053:
1052:
1044:
1033:
1031:
1030:
1025:
1009:
1007:
1006:
1001:
963:
962:
957:
956:
948:
922:
920:
919:
914:
903:
902:
901:
893:
874:
873:
868:
867:
859:
842:
840:
839:
834:
825:
820:
801:
796:
772:
771:
750:
746:
742:
741:
706:
702:
698:
697:
655:
653:
652:
647:
645:
641:
640:
639:
624:
623:
615:
591:
589:
588:
583:
571:
569:
568:
563:
551:
549:
548:
543:
531:
529:
528:
523:
505:
503:
502:
497:
480:
479:
471:
465:
464:
449:
448:
440:
422:
418:
414:
413:
398:
397:
389:
374:
373:
365:
312:
310:
309:
304:
296:
295:
287:
271:
269:
268:
263:
251:
249:
248:
243:
231:
229:
228:
223:
212:
211:
203:
190:
188:
187:
182:
164:
162:
161:
156:
144:
142:
141:
136:
118:
116:
115:
110:
21:
24602:
24601:
24597:
24596:
24595:
24593:
24592:
24591:
24567:
24566:
24565:
24559:
24540:
24521:
24502:
24483:
24456:
24437:
24414:
24390:
24388:
24379:
24378:
24371:
24369:
24368:on 25 July 2008
24355:
24353:Further reading
24350:
24345:
24341:
24331:
24329:
24321:
24320:
24316:
24312:
24259:
24231:
24226:
24217:
24205:
24198:
24194:
24193:
24184:
24178:
24173:
24163:
24158:
24151:
24144:
24140:
24139:
24130:
24124:
24120:
24119:
24117:
24108:
24104:
24102:
24099:
24098:
24071:
24067:
24061:
24057:
24048:
24044:
24038:
24034:
24026:
24023:
24022:
23993:
23989:
23968:
23964:
23953:
23950:
23949:
23932:
23931:
23922:
23918:
23906:
23902:
23895:
23889:
23885:
23882:
23881:
23875:
23871:
23859:
23855:
23848:
23842:
23838:
23834:
23832:
23829:
23828:
23802:
23797:
23791:
23788:
23787:
23771:
23768:
23767:
23750:
23743:
23739:
23738:
23732:
23729:
23728:
23711:
23704:
23700:
23699:
23693:
23690:
23689:
23672:
23668:
23666:
23663:
23662:
23645:
23641:
23639:
23636:
23635:
23618:
23614:
23612:
23609:
23608:
23591:
23587:
23585:
23582:
23581:
23577:
23550:
23545:
23536:
23524:
23517:
23513:
23512:
23503:
23494:
23483:
23478:
23473:
23453:
23451:
23448:
23447:
23427:
23422:
23413:
23401:
23394:
23390:
23389:
23380:
23371:
23360:
23355:
23348:
23341:
23337:
23336:
23327:
23323:
23321:
23312:
23308:
23306:
23303:
23302:
23273:
23272:
23255:
23254:
23245:
23241:
23232:
23228:
23222:
23211:
23193:
23192:
23190:
23187:
23186:
23170:
23167:
23166:
23140:
23135:
23126:
23114:
23107:
23103:
23102:
23093:
23081:
23074:
23070:
23069:
23060:
23056:
23051:
23042:
23031:
23030:
23017:
23012:
22988:
22986:
22983:
22982:
22959:
22954:
22948:
22945:
22944:
22927:
22916:
22915:
22909:
22906:
22905:
22882:
22877:
22864:
22859:
22850:
22838:
22831:
22827:
22826:
22817:
22805:
22798:
22794:
22793:
22784:
22780:
22773:
22766:
22762:
22761:
22752:
22740:
22733:
22729:
22728:
22719:
22715:
22713:
22704:
22693:
22692:
22686:
22683:
22682:
22657:
22656:
22654:
22651:
22650:
22633:
22632:
22620:
22615:
22606:
22594:
22587:
22583:
22582:
22573:
22561:
22554:
22550:
22549:
22540:
22536:
22529:
22522:
22518:
22517:
22508:
22504:
22502:
22495:
22489:
22485:
22482:
22481:
22469:
22464:
22455:
22443:
22436:
22432:
22431:
22422:
22410:
22403:
22399:
22398:
22389:
22385:
22378:
22371:
22367:
22366:
22357:
22353:
22351:
22344:
22338:
22334:
22330:
22328:
22325:
22324:
22293:
22292:
22275:
22274:
22265:
22261:
22252:
22248:
22231:
22230:
22221:
22217:
22208:
22204:
22190:
22189:
22187:
22184:
22183:
22163:
22159:
22157:
22154:
22153:
22136:
22132:
22130:
22127:
22126:
22107:
22090:
22089:
22077:
22073:
22052:
22048:
22037:
22034:
22033:
22032:Here, both the
22016:
22015:
22006:
22002:
21989:
21983:
21979:
21976:
21975:
21969:
21965:
21952:
21946:
21942:
21938:
21936:
21933:
21932:
21906:
21899:
21895:
21894:
21888:
21885:
21884:
21867:
21863:
21861:
21858:
21857:
21840:
21836:
21834:
21831:
21830:
21810:
21803:
21799:
21798:
21792:
21789:
21788:
21771:
21767:
21765:
21762:
21761:
21744:
21740:
21738:
21735:
21734:
21715:
21703:
21698:
21692:
21689:
21688:
21669:
21652:
21651:
21649:
21646:
21645:
21617:
21614:
21613:
21597:
21594:
21593:
21568:
21565:
21564:
21527:
21524:
21523:
21520:
21499:
21496:
21495:
21479:
21476:
21475:
21452:
21447:
21434:
21423:
21422:
21416:
21413:
21412:
21389:
21385:
21379:
21368:
21354:
21340:
21339:
21337:
21334:
21333:
21314:
21311:
21310:
21309:For very large
21282:
21277:
21271:
21265:
21264:
21253:
21247:
21242:
21229:
21224:
21219:
21213:
21208:
21202:
21196:
21195:
21186:
21175:
21174:
21161:
21156:
21144:
21141:
21140:
21114:
21109:
21099:
21098:
21087:
21081:
21076:
21063:
21058:
21053:
21047:
21042:
21036:
21030:
21029:
21017:
21013:
21004:
20999:
20986:
20982:
20973:
20962:
20961:
20955:
20952:
20951:
20920:
20916:
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20332:
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20314:
20310:
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20299:
20286:
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19868:
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9755:
9738:
9737:
9716:
9711:
9701:
9697:
9691:
9687:
9673:
9672:
9670:
9667:
9666:
9641:
9640:
9638:
9635:
9634:
9614:
9609:
9599:
9595:
9589:
9585:
9577:
9574:
9573:
9556:
9552:
9550:
9547:
9546:
9530:
9527:
9526:
9500:
9496:
9487:
9482:
9463:
9458:
9448:
9444:
9432:
9428:
9426:
9423:
9422:
9393:
9388:
9378:
9374:
9365:
9361:
9352:
9347:
9328:
9323:
9313:
9309:
9298:
9295:
9294:
9265:
9260:
9241:
9236:
9226:
9222:
9217:
9214:
9213:
9193:
9189:
9180:
9176:
9170:
9166:
9158:
9155:
9154:
9128:
9123:
9113:
9109:
9100:
9096:
9087:
9082:
9063:
9058:
9048:
9044:
9029:
9025:
9019:
9015:
9006:
9002:
8996:
8992:
8980:
8976:
8970:
8966:
8964:
8961:
8960:
8954:
8932:
8928:
8922:
8918:
8913:
8910:
8909:
8886:
8882:
8880:
8877:
8876:
8851:
8848:
8847:
8830:
8826:
8824:
8821:
8820:
8796:
8792:
8786:
8782:
8773:
8769:
8763:
8759:
8751:
8748:
8747:
8723:
8720:
8719:
8699:
8696:
8695:
8660:
8656:
8644:
8640:
8634:
8630:
8621:
8617:
8611:
8607:
8595:
8591:
8585:
8581:
8572:
8568:
8553:
8552:
8548:
8539:
8535:
8526:
8522:
8520:
8517:
8516:
8485:
8484:
8467:
8466:
8445:
8441:
8435:
8431:
8422:
8418:
8412:
8408:
8396:
8392:
8386:
8382:
8368:
8367:
8365:
8362:
8361:
8336:
8335:
8333:
8330:
8329:
8303:
8299:
8293:
8289:
8280:
8276:
8270:
8266:
8254:
8250:
8244:
8240:
8232:
8229:
8228:
8209:
8206:
8205:
8182:
8178:
8172:
8168:
8156:
8152:
8150:
8147:
8146:
8125:
8121:
8112:
8108:
8102:
8098:
8086:
8082:
8080:
8077:
8076:
8050:
8049:
8026:
8023:
8022:
7993:
7989:
7987:
7984:
7983:
7949:
7946:
7945:
7929:
7926:
7925:
7909:
7906:
7905:
7874:
7871:
7870:
7867:
7838:
7835:
7834:
7831:Toeplitz matrix
7813:
7809:
7807:
7804:
7803:
7775:
7772:
7771:
7754:
7750:
7748:
7745:
7744:
7724:
7721:
7720:
7713:
7692:
7689:
7688:
7672:
7669:
7668:
7645:
7640:
7634:
7631:
7630:
7608:
7607:
7590:
7589:
7572:
7568:
7559:
7555:
7553:
7539:
7538:
7536:
7533:
7532:
7507:
7504:
7503:
7487:
7484:
7483:
7466:
7461:
7448:
7443:
7437:
7434:
7433:
7411:
7410:
7396:
7395:
7393:
7390:
7389:
7367:
7364:
7363:
7340:
7335:
7325:
7314:
7313:
7307:
7296:
7291:
7280:
7275:
7262:
7257:
7252:
7250:
7241:
7237:
7235:
7232:
7231:
7202:
7198:
7186:
7185:
7178:
7176:
7172:
7154:
7150:
7138:
7137:
7123:
7122:
7121:
7119:
7115:
7113:
7110:
7109:
7087:
7084:
7083:
7067:
7064:
7063:
7039:
7035:
7029:
7025:
7024:
7015:
7011:
7009:
7001:
6998:
6997:
6974:
6969:
6956:
6952:
6932:
6927:
6917:
6909:
6903:
6894:
6889:
6876:
6871:
6865:
6862:
6861:
6835:
6834:
6817:
6816:
6799:
6795:
6789:
6785:
6783:
6766:
6765:
6748:
6747:
6730:
6725:
6712:
6708:
6706:
6692:
6691:
6689:
6686:
6685:
6666:
6663:
6662:
6646:
6643:
6642:
6639:
6637:Univariate case
6634:
6633:
6611:
6607:
6590:
6587:
6586:
6557:
6556:
6552:
6543:
6539:
6530:
6526:
6524:
6521:
6520:
6494:
6490:
6481:
6476:
6463:
6459:
6450:
6446:
6437:
6433:
6431:
6428:
6427:
6407:
6403:
6401:
6398:
6397:
6377:
6372:
6359:
6355:
6347:
6344:
6343:
6326:
6325:
6313:
6309:
6297:
6293:
6284:
6283:
6274:
6270:
6259:
6258:
6235:
6234:
6201:
6197:
6186:
6185:
6162:
6161:
6134:
6133:
6124:
6120:
6109:
6108:
6082:
6081:
6055:
6054:
6018:
6017:
6008:
6004:
5993:
5992:
5963:
5962:
5941:
5937:
5931:
5916:
5912:
5901:
5900:
5871:
5870:
5857:
5855:
5845:
5844:
5835:
5831:
5817:
5816:
5790:
5789:
5754:
5753:
5732:
5731:
5722:
5718:
5701:
5700:
5677:
5676:
5657:
5651:
5647:
5643:
5641:
5638:
5637:
5608:
5604:
5595:
5590:
5577:
5573:
5558:
5557:
5553:
5551:
5548:
5547:
5527:
5523:
5521:
5518:
5517:
5501:
5498:
5497:
5480:
5479:
5470:
5466:
5460:
5456:
5444:
5443:
5437:
5433:
5424:
5420:
5409:
5408:
5385:
5384:
5354:
5353:
5344:
5340:
5329:
5328:
5314:
5313:
5296:
5295:
5281:
5280:
5261:
5249:
5248:
5244:
5240:
5238:
5235:
5234:
5209:
5208:
5191:
5190:
5179:
5176:
5175:
5153:
5152:
5150:
5147:
5146:
5118:
5117:
5100:
5099:
5081:
5076:
5063:
5059:
5045:
5044:
5042:
5039:
5038:
5009:
5005:
4996:
4991:
4978:
4974:
4972:
4969:
4968:
4945:
4941:
4939:
4936:
4935:
4918:
4910:
4894:
4890:
4888:
4885:
4884:
4867:
4863:
4861:
4858:
4857:
4837:
4833:
4831:
4828:
4827:
4801:
4796:
4783:
4779:
4771:
4768:
4767:
4748:
4745:
4744:
4727:
4726:
4714:
4710:
4701:
4697:
4685:
4684:
4675:
4671:
4660:
4659:
4636:
4635:
4605:
4601:
4590:
4589:
4566:
4565:
4535:
4534:
4525:
4521:
4510:
4509:
4483:
4482:
4456:
4455:
4424:
4415:
4411:
4400:
4399:
4370:
4369:
4353:
4351:
4348:
4347:
4322:
4321:
4298:
4295:
4294:
4268:
4264:
4253:
4252:
4223:
4222:
4208:
4205:
4204:
4173:
4172:
4146:
4145:
4113:
4112:
4110:
4107:
4106:
4078:
4077:
4060:
4059:
4033:
4032:
4030:
4027:
4026:
3983:
3982:
3980:
3977:
3976:
3936:
3935:
3933:
3930:
3929:
3901:
3900:
3883:
3882:
3874:
3871:
3870:
3845:
3844:
3842:
3839:
3838:
3789:
3788:
3777:
3774:
3773:
3754:
3751:
3750:
3734:
3731:
3730:
3693:
3692:
3690:
3687:
3686:
3682:
3681:
3653:
3649:
3632:
3629:
3628:
3604:
3600:
3591:
3586:
3573:
3569:
3560:
3556:
3541:
3540:
3536:
3527:
3523:
3514:
3510:
3508:
3505:
3504:
3482:
3479:
3478:
3462:
3459:
3458:
3438:
3434:
3432:
3429:
3428:
3402:
3398:
3389:
3384:
3371:
3367:
3352:
3351:
3347:
3345:
3342:
3341:
3315:
3314:
3297:
3296:
3285:
3282:
3281:
3255:
3254:
3237:
3236:
3218:
3213:
3200:
3196:
3182:
3181:
3179:
3176:
3175:
3153:
3150:
3149:
3132:
3128:
3126:
3123:
3122:
3106:
3103:
3102:
3086:
3083:
3082:
3062:
3058:
3056:
3053:
3052:
3009:
3008:
3006:
3003:
3002:
2962:
2961:
2959:
2956:
2955:
2929:
2924:
2911:
2907:
2899:
2896:
2895:
2869:
2868:
2851:
2850:
2842:
2839:
2838:
2819:
2816:
2815:
2799:
2796:
2795:
2775:
2772:
2771:
2755:
2752:
2751:
2735:
2732:
2731:
2715:
2712:
2711:
2695:
2692:
2691:
2648:
2647:
2641:
2625:
2619:
2616:
2615:
2578:
2575:
2574:
2558:
2555:
2554:
2538:
2535:
2534:
2518:
2515:
2514:
2465:
2462:
2461:
2445:
2442:
2441:
2425:
2422:
2421:
2405:
2402:
2401:
2355:
2352:
2351:
2347:
2325:
2322:
2321:
2305:
2302:
2301:
2276:
2273:
2272:
2252:
2249:
2248:
2232:
2229:
2228:
2196:
2185:
2184:
2183:
2168:
2157:
2156:
2155:
2141:
2138:
2137:
2113:
2112:
2110:
2107:
2106:
2083:
2072:
2071:
2070:
2068:
2065:
2064:
2014:
2010:
2001:
1997:
1979:
1974:
1956:
1953:
1952:
1917:
1913:
1887:
1878:
1873:
1872:
1839:
1838:
1836:
1833:
1832:
1807:
1804:
1803:
1752:
1741:
1740:
1739:
1725:
1722:
1721:
1686:
1682:
1673:
1662:
1661:
1660:
1658:
1655:
1654:
1617:
1616:
1614:
1611:
1610:
1594:
1591:
1590:
1561:
1558:
1557:
1528:
1525:
1524:
1483:
1479:
1472:
1468:
1462:
1461:
1436:
1425:
1424:
1423:
1413:
1411:
1408:
1407:
1305:
1294:
1293:
1292:
1281:
1278:
1277:
1243:
1239:
1235:
1226:
1222:
1220:
1217:
1216:
1193:
1189:
1185:
1183:
1180:
1179:
1159:
1155:
1138:
1137:
1114:
1113:
1092:
1088:
1086:
1083:
1082:
1055:
1054:
1043:
1042:
1041:
1039:
1036:
1035:
1019:
1016:
1015:
958:
947:
946:
945:
943:
940:
939:
929:
892:
891:
887:
869:
858:
857:
856:
854:
851:
850:
821:
816:
797:
786:
767:
763:
737:
733:
720:
716:
693:
689:
676:
672:
664:
661:
660:
635:
631:
614:
613:
609:
605:
597:
594:
593:
577:
574:
573:
557:
554:
553:
552:conditioned on
537:
534:
533:
517:
514:
513:
470:
469:
460:
456:
439:
438:
409:
405:
388:
387:
364:
363:
350:
346:
332:
329:
328:
286:
285:
277:
274:
273:
257:
254:
253:
237:
234:
233:
202:
201:
199:
196:
195:
170:
167:
166:
150:
147:
146:
124:
121:
120:
104:
101:
100:
97:
76:
28:
23:
22:
15:
12:
11:
5:
24600:
24590:
24589:
24584:
24579:
24564:
24563:
24558:978-0132671453
24557:
24544:
24538:
24525:
24520:978-0201361865
24519:
24506:
24501:978-0471181170
24500:
24487:
24481:
24460:
24454:
24441:
24435:
24418:
24413:978-0521592710
24412:
24399:
24356:
24354:
24351:
24349:
24348:
24339:
24313:
24311:
24308:
24307:
24306:
24301:
24296:
24291:
24286:
24281:
24275:
24270:
24265:
24258:
24255:
24254:
24253:
24242:
24234:
24229:
24225:
24220:
24216:
24213:
24208:
24201:
24197:
24192:
24187:
24181:
24176:
24172:
24166:
24162:
24154:
24147:
24143:
24138:
24133:
24127:
24123:
24116:
24111:
24107:
24088:
24087:
24074:
24070:
24064:
24060:
24056:
24051:
24047:
24041:
24037:
24033:
24030:
24007:
24004:
24001:
23996:
23992:
23988:
23985:
23982:
23979:
23976:
23971:
23967:
23963:
23960:
23957:
23948:Here both the
23946:
23945:
23930:
23925:
23921:
23917:
23914:
23909:
23905:
23901:
23898:
23896:
23892:
23888:
23884:
23883:
23878:
23874:
23870:
23867:
23862:
23858:
23854:
23851:
23849:
23845:
23841:
23837:
23836:
23810:
23805:
23800:
23796:
23775:
23753:
23746:
23742:
23737:
23714:
23707:
23703:
23698:
23675:
23671:
23648:
23644:
23621:
23617:
23594:
23590:
23576:
23573:
23561:
23553:
23548:
23544:
23539:
23535:
23532:
23527:
23520:
23516:
23511:
23506:
23502:
23497:
23492:
23489:
23486:
23482:
23477:
23472:
23468:
23465:
23462:
23459:
23456:
23430:
23425:
23421:
23416:
23412:
23409:
23404:
23397:
23393:
23388:
23383:
23379:
23374:
23369:
23366:
23363:
23359:
23351:
23344:
23340:
23335:
23330:
23326:
23320:
23315:
23311:
23286:
23280:
23277:
23271:
23268:
23262:
23259:
23253:
23248:
23244:
23240:
23235:
23231:
23225:
23220:
23217:
23214:
23210:
23206:
23200:
23197:
23174:
23163:
23162:
23151:
23143:
23138:
23134:
23129:
23125:
23122:
23117:
23110:
23106:
23101:
23096:
23092:
23089:
23084:
23077:
23073:
23068:
23063:
23059:
23055:
23050:
23045:
23038:
23035:
23029:
23025:
23020:
23015:
23011:
23007:
23003:
23000:
22997:
22994:
22991:
22967:
22962:
22957:
22953:
22930:
22923:
22920:
22914:
22902:
22901:
22890:
22885:
22880:
22876:
22867:
22862:
22858:
22853:
22849:
22846:
22841:
22834:
22830:
22825:
22820:
22816:
22813:
22808:
22801:
22797:
22792:
22787:
22783:
22776:
22769:
22765:
22760:
22755:
22751:
22748:
22743:
22736:
22732:
22727:
22722:
22718:
22712:
22707:
22700:
22697:
22691:
22664:
22661:
22647:
22646:
22631:
22623:
22618:
22614:
22609:
22605:
22602:
22597:
22590:
22586:
22581:
22576:
22572:
22569:
22564:
22557:
22553:
22548:
22543:
22539:
22532:
22525:
22521:
22516:
22511:
22507:
22501:
22498:
22496:
22492:
22488:
22484:
22483:
22480:
22472:
22467:
22463:
22458:
22454:
22451:
22446:
22439:
22435:
22430:
22425:
22421:
22418:
22413:
22406:
22402:
22397:
22392:
22388:
22381:
22374:
22370:
22365:
22360:
22356:
22350:
22347:
22345:
22341:
22337:
22333:
22332:
22318:
22317:
22306:
22300:
22297:
22291:
22288:
22282:
22279:
22273:
22268:
22264:
22260:
22255:
22251:
22247:
22244:
22238:
22235:
22229:
22224:
22220:
22216:
22211:
22207:
22203:
22197:
22194:
22166:
22162:
22139:
22135:
22114:
22110:
22106:
22103:
22097:
22094:
22088:
22085:
22080:
22076:
22072:
22069:
22066:
22063:
22060:
22055:
22051:
22047:
22044:
22041:
22030:
22029:
22014:
22009:
22005:
22001:
21998:
21995:
21992:
21990:
21986:
21982:
21978:
21977:
21972:
21968:
21964:
21961:
21958:
21955:
21953:
21949:
21945:
21941:
21940:
21914:
21909:
21902:
21898:
21893:
21870:
21866:
21856:with an error
21843:
21839:
21818:
21813:
21806:
21802:
21797:
21774:
21770:
21747:
21743:
21722:
21718:
21714:
21711:
21706:
21701:
21697:
21676:
21672:
21668:
21665:
21659:
21656:
21633:
21630:
21627:
21624:
21621:
21601:
21592:We shall take
21581:
21578:
21575:
21572:
21552:
21549:
21546:
21543:
21540:
21537:
21534:
21531:
21519:
21516:
21503:
21483:
21460:
21455:
21450:
21446:
21442:
21437:
21430:
21427:
21421:
21409:
21408:
21397:
21392:
21388:
21382:
21377:
21374:
21371:
21367:
21361:
21358:
21353:
21347:
21344:
21318:
21307:
21306:
21295:
21290:
21285:
21280:
21276:
21268:
21260:
21256:
21250:
21245:
21241:
21237:
21232:
21227:
21223:
21216:
21211:
21207:
21199:
21194:
21189:
21182:
21179:
21173:
21169:
21164:
21159:
21155:
21151:
21148:
21134:
21133:
21122:
21117:
21112:
21108:
21102:
21094:
21090:
21084:
21079:
21075:
21071:
21066:
21061:
21057:
21050:
21045:
21041:
21033:
21028:
21023:
21020:
21016:
21010:
21007:
21002:
20998:
20992:
20989:
20985:
20981:
20976:
20969:
20966:
20960:
20934:
20929:
20923:
20919:
20913:
20908:
20905:
20902:
20898:
20891:
20886:
20882:
20877:
20873:
20866:
20860:
20857:
20832:
20828:
20822:
20818:
20814:
20811:
20808:
20803:
20799:
20795:
20790:
20786:
20782:
20779:
20776:
20765:
20764:
20749:
20743:
20740:
20731:
20727:
20721:
20716:
20712:
20708:
20703:
20698:
20694:
20687:
20682:
20678:
20672:
20669:
20667:
20665:
20662:
20657:
20653:
20647:
20644:
20639:
20631:
20626:
20622:
20618:
20613:
20606:
20601:
20597:
20593:
20587:
20578:
20573:
20569:
20565:
20560:
20557:
20555:
20553:
20550:
20547:
20540:
20535:
20531:
20527:
20520:
20516:
20510:
20507:
20502:
20494:
20489:
20485:
20481:
20476:
20473:
20470:
20463:
20458:
20454:
20450:
20443:
20439:
20434:
20429:
20426:
20424:
20419:
20416:
20410:
20409:
20386:
20375:
20374:
20359:
20356:
20351:
20348:
20344:
20340:
20335:
20330:
20326:
20322:
20317:
20313:
20307:
20302:
20298:
20294:
20289:
20285:
20279:
20274:
20270:
20266:
20263:
20261:
20259:
20256:
20251:
20248:
20243:
20239:
20233:
20230:
20226:
20222:
20219:
20217:
20212:
20209:
20203:
20202:
20188:
20187:
20172:
20167:
20163:
20157:
20152:
20148:
20144:
20141:
20136:
20132:
20128:
20125:
20122:
20119:
20116:
20111:
20108:
20104:
20100:
20098:
20095:
20092:
20087:
20082:
20078:
20074:
20069:
20065:
20059:
20054:
20050:
20046:
20043:
20038:
20034:
20030:
20027:
20024:
20021:
20018:
20013:
20009:
20005:
20003:
20000:
19997:
19994:
19991:
19988:
19985:
19982:
19979:
19976:
19974:
19951:
19948:
19943:
19940:
19936:
19915:
19895:
19875:
19855:
19852:
19847:
19842:
19838:
19834:
19831:
19828:
19825:
19805:
19785:
19781:
19775:
19770:
19766:
19762:
19757:
19752:
19748:
19727:
19707:
19702:
19698:
19694:
19689:
19685:
19681:
19678:
19654:
19634:
19614:
19609:
19605:
19601:
19596:
19592:
19588:
19585:
19561:
19539:
19535:
19531:
19528:
19525:
19522:
19519:
19516:
19513:
19510:
19507:
19504:
19484:
19481:
19478:
19475:
19472:
19469:
19449:
19429:
19409:
19397:
19394:
19377:
19355:
19351:
19330:
19327:
19322:
19319:
19315:
19311:
19308:
19305:
19302:
19297:
19294:
19290:
19286:
19283:
19280:
19275:
19271:
19265:
19261:
19257:
19254:
19251:
19248:
19243:
19238:
19233:
19230:
19227:
19203:
19196:
19193:
19169:
19166:
19161:
19157:
19136:
19133:
19130:
19127:
19122:
19118:
19098:
19095:
19092:
19087:
19083:
19071:
19070:
19059:
19054:
19050:
19046:
19042:
19035:
19032:
19031:
19028:
19025:
19022:
19021:
19018:
19015:
19014:
19011:
19007:
19003:
18996:
18993:
18992:
18989:
18986:
18985:
18982:
18979:
18978:
18975:
18970:
18963:
18960:
18957:
18955:
18952:
18950:
18947:
18944:
18943:
18940:
18937:
18935:
18932:
18930:
18927:
18924:
18923:
18920:
18917:
18914:
18912:
18909:
18906:
18904:
18901:
18900:
18897:
18893:
18888:
18885:
18881:
18875:
18872:
18867:
18863:
18837:
18833:
18810:
18807:
18803:
18799:
18794:
18790:
18784:
18780:
18768:
18767:
18756:
18752:
18745:
18742:
18741:
18738:
18735:
18734:
18731:
18728:
18727:
18724:
18720:
18716:
18709:
18704:
18700:
18696:
18691:
18687:
18683:
18680:
18677:
18676:
18673:
18668:
18664:
18660:
18655:
18651:
18647:
18644:
18641:
18640:
18637:
18632:
18628:
18624:
18619:
18615:
18611:
18608:
18605:
18604:
18601:
18597:
18592:
18589:
18585:
18572:is defined as
18559:
18556:
18552:
18540:
18539:
18528:
18524:
18517:
18514:
18512:
18509:
18507:
18504:
18503:
18500:
18497:
18495:
18492:
18490:
18487:
18486:
18483:
18480:
18478:
18475:
18473:
18470:
18469:
18466:
18462:
18458:
18451:
18446:
18442:
18438:
18433:
18429:
18425:
18422:
18419:
18417:
18412:
18408:
18404:
18399:
18395:
18391:
18388:
18385:
18383:
18378:
18374:
18370:
18365:
18361:
18357:
18354:
18351:
18350:
18347:
18342:
18338:
18334:
18329:
18325:
18321:
18318:
18315:
18313:
18308:
18304:
18300:
18295:
18291:
18287:
18284:
18281:
18279:
18274:
18270:
18266:
18261:
18257:
18253:
18250:
18247:
18246:
18243:
18238:
18234:
18230:
18225:
18221:
18217:
18214:
18211:
18209:
18204:
18200:
18196:
18191:
18187:
18183:
18180:
18177:
18175:
18170:
18166:
18162:
18157:
18153:
18149:
18146:
18143:
18142:
18139:
18135:
18130:
18126:
18113:is defined as
18100:
18096:
18073:
18069:
18065:
18062:
18042:
18037:
18033:
18029:
18024:
18020:
18016:
18011:
18007:
18003:
18000:
17997:
17975:
17971:
17965:
17961:
17957:
17952:
17948:
17944:
17939:
17935:
17931:
17928:
17925:
17900:
17893:
17890:
17864:
17860:
17848:
17847:
17836:
17832:
17825:
17822:
17820:
17817:
17815:
17812:
17810:
17807:
17806:
17803:
17800:
17798:
17795:
17793:
17790:
17788:
17785:
17784:
17781:
17778:
17776:
17773:
17771:
17768:
17766:
17763:
17762:
17759:
17756:
17754:
17751:
17749:
17746:
17744:
17741:
17740:
17737:
17733:
17730:
17725:
17721:
17717:
17714:
17711:
17708:
17705:
17702:
17699:
17696:
17693:
17690:
17665:
17661:
17655:
17651:
17647:
17642:
17638:
17634:
17629:
17625:
17621:
17616:
17612:
17608:
17605:
17602:
17580:
17576:
17570:
17566:
17560:
17555:
17552:
17549:
17545:
17541:
17536:
17529:
17526:
17500:
17496:
17473:
17469:
17446:
17442:
17438:
17433:
17429:
17412:
17409:
17407:
17404:
17389:
17386:
17383:
17376:
17373:
17366:
17361:
17358:
17355:
17350:
17347:
17344:
17337:
17334:
17306:
17303:
17300:
17296:
17291:
17287:
17282:
17279:
17276:
17269:
17266:
17263:
17259:
17254:
17231:
17228:
17225:
17221:
17200:
17197:
17194:
17174:
17152:
17149:
17146:
17141:
17138:
17135:
17131:
17106:
17103:
17100:
17096:
17091:
17070:
17067:
17064:
17044:
17022:
17019:
17016:
17009:
17006:
17003:
16999:
16994:
16971:
16964:
16961:
16954:
16949:
16946:
16943:
16938:
16935:
16932:
16925:
16922:
16894:
16890:
16885:
16881:
16876:
16873:
16870:
16863:
16860:
16857:
16853:
16848:
16827:
16824:
16821:
16818:
16815:
16812:
16809:
16806:
16803:
16792:
16791:
16780:
16775:
16772:
16769:
16766:
16763:
16758:
16751:
16748:
16739:
16736:
16733:
16728:
16725:
16722:
16718:
16714:
16709:
16706:
16703:
16698:
16695:
16692:
16688:
16684:
16679:
16676:
16673:
16668:
16665:
16662:
16658:
16654:
16649:
16646:
16643:
16640:
16637:
16632:
16625:
16622:
16615:
16610:
16607:
16604:
16599:
16596:
16593:
16586:
16583:
16569:
16556:
16553:
16550:
16543:
16539:
16534:
16530:
16525:
16522:
16519:
16514:
16511:
16508:
16504:
16498:
16495:
16492:
16487:
16484:
16481:
16477:
16473:
16470:
16467:
16464:
16459:
16456:
16453:
16446:
16443:
16440:
16436:
16431:
16420:
16406:
16401:
16398:
16395:
16392:
16387:
16384:
16381:
16377:
16373:
16368:
16365:
16362:
16355:
16351:
16346:
16340:
16337:
16334:
16329:
16326:
16323:
16319:
16315:
16310:
16307:
16304:
16297:
16294:
16291:
16287:
16282:
16274:
16271:
16268:
16265:
16260:
16257:
16254:
16250:
16244:
16241:
16238:
16231:
16227:
16222:
16215:
16210:
16207:
16204:
16199:
16196:
16193:
16189:
16164:
16144:
16141:
16138:
16118:
16096:
16092:
16079:
16076:
16061:
16054:
16051:
16042:
16038:
16034:
16031:
16026:
16019:
16016:
16007:
16003:
15999:
15995:
15972:
15968:
15956:
15955:
15944:
15941:
15936:
15932:
15926:
15919:
15916:
15909:
15905:
15899:
15895:
15891:
15886:
15879:
15876:
15869:
15864:
15861:
15858:
15851:
15848:
15821:
15818:
15815:
15809:
15806:
15800:
15796:
15792:
15789:
15786:
15783:
15778:
15771:
15768:
15761:
15757:
15749:
15746:
15740:
15719:
15713:
15710:
15702:
15695:
15692:
15685:
15681:
15660:
15618:
15596:
15592:
15568:
15565:
15540:
15537:
15534:
15530:
15518:
15517:
15506:
15499:
15495:
15490:
15486:
15481:
15476:
15473:
15470:
15466:
15460:
15457:
15454:
15450:
15446:
15443:
15440:
15437:
15430:
15427:
15424:
15420:
15415:
15379:
15376:
15373:
15369:
15364:
15352:
15351:
15340:
15332:
15329:
15326:
15322:
15314:
15310:
15305:
15299:
15294:
15291:
15288:
15284:
15280:
15275:
15270:
15267:
15264:
15260:
15252:
15249:
15246:
15242:
15234:
15230:
15225:
15218:
15213:
15210:
15207:
15203:
15173:
15170:
15167:
15163:
15140:
15137:
15134:
15130:
15118:
15117:
15106:
15101:
15094:
15091:
15082:
15077:
15074:
15071:
15067:
15063:
15058:
15055:
15052:
15048:
15044:
15039:
15036:
15033:
15029:
15025:
15020:
15013:
15010:
15003:
14998:
14995:
14992:
14985:
14982:
14953:
14950:
14947:
14940:
14937:
14907:
14902:
14898:
14875:
14871:
14844:
14840:
14813:
14809:
14786:
14782:
14778:
14773:
14769:
14763:
14758:
14754:
14750:
14745:
14741:
14724:
14721:
14713:
14712:
14701:
14698:
14693:
14690:
14687:
14680:
14677:
14670:
14667:
14662:
14658:
14654:
14649:
14645:
14641:
14636:
14633:
14630:
14623:
14620:
14613:
14608:
14601:
14598:
14584:
14573:
14568:
14565:
14560:
14556:
14550:
14546:
14538:
14534:
14529:
14525:
14520:
14516:
14505:
14494:
14491:
14486:
14483:
14478:
14474:
14468:
14464:
14460:
14455:
14452:
14445:
14442:
14439:
14435:
14430:
14426:
14421:
14418:
14411:
14407:
14402:
14376:
14372:
14360:
14359:
14348:
14341:
14338:
14335:
14331:
14326:
14322:
14319:
14314:
14310:
14306:
14303:
14300:
14297:
14290:
14286:
14281:
14270:
14259:
14256:
14251:
14248:
14245:
14238:
14235:
14228:
14225:
14220:
14216:
14212:
14207:
14203:
14199:
14194:
14191:
14188:
14181:
14178:
14171:
14166:
14159:
14156:
14142:
14131:
14126:
14123:
14119:
14113:
14109:
14105:
14100:
14096:
14088:
14085:
14082:
14078:
14073:
14069:
14066:
14061:
14057:
14049:
14046:
14043:
14039:
14034:
14030:
14025:
14021:
13992:
13988:
13965:
13958:
13955:
13940:
13939:
13928:
13921:
13918:
13915:
13911:
13906:
13902:
13897:
13894:
13890:
13884:
13880:
13876:
13871:
13867:
13859:
13856:
13853:
13849:
13844:
13840:
13837:
13832:
13828:
13820:
13817:
13814:
13810:
13805:
13801:
13794:
13791:
13788:
13784:
13779:
13775:
13768:
13764:
13759:
13744:
13743:
13728:
13723:
13720:
13717:
13710:
13707:
13700:
13697:
13692:
13688:
13684:
13679:
13676:
13672:
13666:
13662:
13658:
13653:
13649:
13641:
13638:
13635:
13631:
13626:
13622:
13619:
13614:
13610:
13602:
13599:
13596:
13592:
13587:
13583:
13578:
13575:
13572:
13565:
13562:
13555:
13552:
13550:
13548:
13545:
13540:
13533:
13530:
13523:
13518:
13514:
13510:
13505:
13502:
13495:
13488:
13485:
13477:
13469:
13462:
13459:
13450:
13447:
13444:
13440:
13435:
13431:
13426:
13423:
13420:
13413:
13410:
13403:
13400:
13398:
13394:
13387:
13384:
13377:
13376:
13351:
13347:
13335:
13334:
13319:
13314:
13310:
13302:
13299:
13296:
13292:
13287:
13283:
13276:
13269:
13266:
13257:
13254:
13251:
13247:
13242:
13238:
13235:
13233:
13227:
13224:
13221:
13217:
13213:
13210:
13207:
13202:
13198:
13193:
13187:
13183:
13177:
13173:
13168:
13164:
13163:
13138:
13135:
13131:
13106:
13099:
13096:
13088:
13065:
13062:
13059:
13052:
13049:
13023:
13019:
12995:
12992:
12965:
12962:
12959:
12955:
12950:
12927:
12924:
12921:
12914:
12911:
12885:
12881:
12857:
12854:
12824:
12820:
12815:
12809:
12805:
12800:
12796:
12789:
12782:
12779:
12771:
12750:
12747:
12744:
12739:
12732:
12729:
12722:
12718:
12695:
12688:
12685:
12678:
12673:
12669:
12665:
12660:
12653:
12650:
12624:
12620:
12597:
12594:
12591:
12584:
12581:
12574:
12571:
12566:
12559:
12556:
12543:, as given by
12530:
12526:
12514:
12513:
12497:
12492:
12488:
12484:
12479:
12475:
12467:
12464:
12461:
12457:
12452:
12448:
12445:
12440:
12436:
12432:
12427:
12423:
12415:
12412:
12409:
12405:
12401:
12398:
12395:
12390:
12386:
12381:
12375:
12371:
12366:
12362:
12359:
12356:
12354:
12348:
12344:
12339:
12333:
12329:
12324:
12320:
12319:
12294:
12291:
12288:
12281:
12278:
12271:
12268:
12263:
12256:
12253:
12246:
12243:
12238:
12231:
12228:
12204:
12199:
12195:
12190:
12184:
12180:
12176:
12173:
12159:
12158:
12147:
12140:
12137:
12134:
12130:
12125:
12121:
12114:
12111:
12108:
12104:
12100:
12097:
12094:
12089:
12085:
12080:
12074:
12071:
12068:
12064:
12059:
12055:
12048:
12045:
12042:
12038:
12034:
12031:
12028:
12023:
12019:
12014:
12008:
12004:
11999:
11984:
11983:
11969:
11966:
11963:
11956:
11953:
11946:
11943:
11938:
11935:
11932:
11928:
11924:
11921:
11918:
11913:
11909:
11904:
11898:
11895:
11892:
11888:
11884:
11880:
11876:
11873:
11868:
11865:
11862:
11858:
11854:
11851:
11848:
11843:
11839:
11834:
11828:
11824:
11820:
11816:
11812:
11807:
11800:
11797:
11770:
11765:
11762:
11759:
11755:
11751:
11748:
11745:
11740:
11736:
11731:
11725:
11721:
11717:
11714:
11691:
11686:
11682:
11677:
11671:
11667:
11663:
11660:
11640:
11635:
11632:
11629:
11625:
11621:
11618:
11615:
11610:
11606:
11601:
11595:
11591:
11587:
11584:
11564:
11544:
11524:
11518:
11515:
11509:
11506:
11500:
11497:
11491:
11488:
11485:
11480:
11477:
11472:
11468:
11462:
11459:
11455:
11451:
11445:
11442:
11417:
11416:
11405:
11402:
11397:
11394:
11391:
11387:
11383:
11380:
11377:
11372:
11368:
11363:
11357:
11354:
11351:
11347:
11343:
11340:
11337:
11334:
11329:
11326:
11323:
11319:
11315:
11312:
11309:
11304:
11300:
11295:
11289:
11285:
11281:
11278:
11255:
11235:
11230:
11226:
11222:
11219:
11216:
11211:
11207:
11202:
11196:
11192:
11188:
11185:
11159:
11152:
11149:
11131:
11130:
11119:
11116:
11111:
11107:
11102:
11096:
11092:
11088:
11085:
11082:
11079:
11074:
11071:
11068:
11064:
11060:
11057:
11054:
11049:
11045:
11041:
11036:
11032:
11027:
11021:
11017:
11013:
11010:
10987:
10965:
10962:
10959:
10955:
10951:
10948:
10945:
10940:
10936:
10913:
10909:
10884:
10879:
10876:
10873:
10869:
10865:
10862:
10859:
10854:
10850:
10845:
10839:
10835:
10831:
10828:
10808:
10803:
10799:
10794:
10788:
10784:
10780:
10777:
10757:
10752:
10748:
10744:
10741:
10738:
10733:
10729:
10724:
10718:
10714:
10710:
10707:
10696:
10695:
10680:
10677:
10672:
10669:
10666:
10662:
10658:
10655:
10652:
10647:
10643:
10638:
10632:
10628:
10624:
10621:
10618:
10613:
10609:
10604:
10598:
10594:
10590:
10587:
10584:
10581:
10579:
10577:
10574:
10569:
10566:
10563:
10559:
10555:
10552:
10549:
10544:
10540:
10535:
10529:
10525:
10521:
10518:
10515:
10510:
10507:
10504:
10500:
10496:
10493:
10490:
10485:
10481:
10477:
10474:
10470:
10464:
10460:
10456:
10453:
10450:
10447:
10445:
10443:
10438:
10434:
10430:
10427:
10424:
10419:
10415:
10410:
10404:
10400:
10396:
10393:
10390:
10389:
10364:
10360:
10337:
10333:
10329:
10326:
10323:
10318:
10314:
10304:observations,
10293:
10277:
10274:
10259:
10255:
10249:
10246:
10242:
10238:
10235:
10232:
10229:
10224:
10220:
10216:
10213:
10210:
10190:
10170:
10167:
10164:
10159:
10156:
10151:
10147:
10124:
10120:
10114:
10111:
10107:
10103:
10098:
10094:
10090:
10087:
10084:
10064:
10044:
10041:
10036:
10032:
10028:
10023:
10019:
9998:
9976:
9973:
9968:
9964:
9941:
9938:
9933:
9929:
9923:
9919:
9913:
9910:
9906:
9902:
9897:
9894:
9889:
9885:
9879:
9875:
9871:
9868:
9865:
9845:
9825:
9822:
9817:
9814:
9809:
9805:
9781:
9780:
9769:
9763:
9760:
9754:
9751:
9745:
9742:
9736:
9733:
9730:
9727:
9722:
9719:
9714:
9710:
9704:
9700:
9694:
9690:
9686:
9680:
9677:
9648:
9645:
9620:
9617:
9612:
9608:
9602:
9598:
9592:
9588:
9584:
9581:
9559:
9555:
9534:
9523:
9522:
9511:
9506:
9503:
9499:
9493:
9490:
9485:
9481:
9477:
9474:
9469:
9466:
9461:
9457:
9451:
9447:
9443:
9440:
9435:
9431:
9416:
9415:
9404:
9399:
9396:
9391:
9387:
9381:
9377:
9371:
9368:
9364:
9358:
9355:
9350:
9346:
9342:
9339:
9334:
9331:
9326:
9322:
9316:
9312:
9308:
9305:
9302:
9279:
9276:
9271:
9268:
9263:
9259:
9255:
9252:
9247:
9244:
9239:
9235:
9229:
9225:
9221:
9201:
9196:
9192:
9188:
9183:
9179:
9173:
9169:
9165:
9162:
9151:
9150:
9139:
9134:
9131:
9126:
9122:
9116:
9112:
9106:
9103:
9099:
9093:
9090:
9085:
9081:
9077:
9074:
9069:
9066:
9061:
9057:
9051:
9047:
9043:
9040:
9035:
9032:
9028:
9022:
9018:
9014:
9009:
9005:
8999:
8995:
8991:
8988:
8983:
8979:
8973:
8969:
8953:
8950:
8935:
8931:
8925:
8921:
8917:
8897:
8894:
8889:
8885:
8855:
8833:
8829:
8802:
8799:
8795:
8789:
8785:
8781:
8776:
8772:
8766:
8762:
8758:
8755:
8727:
8703:
8680:
8679:
8668:
8663:
8659:
8655:
8650:
8647:
8643:
8637:
8633:
8629:
8624:
8620:
8614:
8610:
8606:
8603:
8598:
8594:
8588:
8584:
8580:
8575:
8571:
8567:
8560:
8557:
8551:
8547:
8542:
8538:
8534:
8529:
8525:
8510:
8509:
8498:
8492:
8489:
8483:
8480:
8474:
8471:
8465:
8462:
8459:
8456:
8451:
8448:
8444:
8438:
8434:
8430:
8425:
8421:
8415:
8411:
8407:
8404:
8399:
8395:
8389:
8385:
8381:
8375:
8372:
8343:
8340:
8326:
8325:
8314:
8309:
8306:
8302:
8296:
8292:
8288:
8283:
8279:
8273:
8269:
8265:
8262:
8257:
8253:
8247:
8243:
8239:
8236:
8213:
8202:
8201:
8190:
8185:
8181:
8175:
8171:
8167:
8162:
8159:
8155:
8144:
8133:
8128:
8124:
8120:
8115:
8111:
8105:
8101:
8097:
8094:
8089:
8085:
8074:
8063:
8057:
8054:
8048:
8045:
8042:
8039:
8036:
8033:
8030:
8007:
8004:
7999:
7996:
7992:
7971:
7968:
7965:
7962:
7959:
7956:
7953:
7933:
7913:
7893:
7890:
7887:
7884:
7881:
7878:
7866:
7863:
7842:
7816:
7812:
7779:
7757:
7753:
7728:
7712:
7709:
7696:
7676:
7656:
7653:
7648:
7643:
7639:
7615:
7612:
7606:
7603:
7597:
7594:
7588:
7585:
7582:
7575:
7571:
7565:
7562:
7558:
7552:
7546:
7543:
7520:
7517:
7514:
7511:
7491:
7469:
7464:
7460:
7456:
7451:
7446:
7442:
7418:
7415:
7409:
7403:
7400:
7377:
7374:
7371:
7360:
7359:
7343:
7338:
7334:
7328:
7321:
7318:
7312:
7306:
7299:
7294:
7290:
7283:
7278:
7274:
7270:
7265:
7260:
7256:
7249:
7244:
7240:
7225:
7224:
7212:
7205:
7201:
7193:
7190:
7184:
7181:
7175:
7171:
7168:
7164:
7157:
7153:
7145:
7142:
7136:
7130:
7127:
7118:
7091:
7071:
7042:
7038:
7032:
7028:
7021:
7018:
7014:
7008:
7005:
6994:
6993:
6982:
6977:
6972:
6968:
6964:
6959:
6955:
6951:
6948:
6945:
6942:
6935:
6930:
6926:
6920:
6915:
6912:
6908:
6902:
6897:
6892:
6888:
6884:
6879:
6874:
6870:
6859:
6848:
6842:
6839:
6833:
6830:
6824:
6821:
6815:
6812:
6809:
6802:
6798:
6792:
6788:
6782:
6779:
6773:
6770:
6764:
6761:
6755:
6752:
6746:
6743:
6740:
6733:
6728:
6724:
6718:
6715:
6711:
6705:
6699:
6696:
6670:
6650:
6638:
6635:
6632:
6631:
6619:
6614:
6610:
6606:
6603:
6600:
6597:
6594:
6571:
6564:
6561:
6555:
6551:
6546:
6542:
6538:
6533:
6529:
6517:
6516:
6505:
6500:
6497:
6493:
6487:
6484:
6479:
6475:
6469:
6466:
6462:
6458:
6453:
6449:
6445:
6440:
6436:
6410:
6406:
6383:
6380:
6375:
6371:
6365:
6362:
6358:
6354:
6351:
6340:
6339:
6324:
6319:
6316:
6312:
6308:
6305:
6300:
6296:
6292:
6289:
6287:
6285:
6282:
6277:
6273:
6266:
6263:
6257:
6254:
6251:
6248:
6242:
6239:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6209:
6204:
6200:
6193:
6190:
6184:
6181:
6178:
6175:
6169:
6166:
6160:
6157:
6154:
6151:
6148:
6145:
6142:
6139:
6137:
6135:
6132:
6127:
6123:
6116:
6113:
6107:
6104:
6101:
6098:
6095:
6089:
6086:
6080:
6077:
6074:
6071:
6068:
6062:
6059:
6053:
6050:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6026:
6023:
6021:
6019:
6016:
6011:
6007:
6000:
5997:
5991:
5988:
5985:
5982:
5979:
5976:
5970:
5967:
5961:
5958:
5955:
5952:
5949:
5944:
5940:
5934:
5928:
5924:
5919:
5915:
5908:
5905:
5899:
5896:
5893:
5890:
5887:
5884:
5878:
5875:
5869:
5866:
5863:
5860:
5853:
5850:
5848:
5846:
5843:
5838:
5834:
5830:
5824:
5821:
5815:
5812:
5809:
5806:
5803:
5797:
5794:
5788:
5785:
5782:
5779:
5776:
5773:
5770:
5767:
5761:
5758:
5752:
5749:
5746:
5743:
5740:
5737:
5735:
5733:
5730:
5725:
5721:
5717:
5714:
5708:
5705:
5699:
5696:
5693:
5690:
5684:
5681:
5675:
5672:
5669:
5666:
5663:
5660:
5658:
5654:
5650:
5646:
5645:
5631:
5630:
5619:
5614:
5611:
5607:
5601:
5598:
5593:
5589:
5583:
5580:
5576:
5572:
5565:
5562:
5556:
5530:
5526:
5505:
5494:
5493:
5478:
5473:
5469:
5463:
5459:
5455:
5452:
5449:
5447:
5445:
5440:
5436:
5432:
5427:
5423:
5416:
5413:
5407:
5404:
5401:
5398:
5392:
5389:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5359:
5357:
5355:
5352:
5347:
5343:
5336:
5333:
5327:
5321:
5318:
5312:
5309:
5303:
5300:
5294:
5288:
5285:
5279:
5276:
5273:
5270:
5267:
5264:
5262:
5256:
5253:
5247:
5243:
5242:
5216:
5213:
5207:
5204:
5198:
5195:
5189:
5186:
5183:
5160:
5157:
5143:
5142:
5131:
5125:
5122:
5116:
5113:
5107:
5104:
5098:
5095:
5092:
5087:
5084:
5079:
5075:
5069:
5066:
5062:
5058:
5052:
5049:
5032:
5031:
5020:
5015:
5012:
5008:
5002:
4999:
4994:
4990:
4986:
4981:
4977:
4951:
4948:
4944:
4921:
4916:
4913:
4909:
4905:
4900:
4897:
4893:
4870:
4866:
4843:
4840:
4836:
4824:
4823:
4812:
4807:
4804:
4799:
4795:
4789:
4786:
4782:
4778:
4775:
4752:
4741:
4740:
4725:
4720:
4717:
4713:
4709:
4704:
4700:
4696:
4693:
4690:
4688:
4686:
4683:
4678:
4674:
4667:
4664:
4658:
4655:
4652:
4649:
4643:
4640:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4608:
4604:
4597:
4594:
4588:
4585:
4582:
4579:
4573:
4570:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4538:
4536:
4533:
4528:
4524:
4517:
4514:
4508:
4505:
4502:
4499:
4496:
4490:
4487:
4481:
4478:
4475:
4472:
4469:
4463:
4460:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4425:
4423:
4418:
4414:
4407:
4404:
4398:
4395:
4392:
4389:
4386:
4383:
4377:
4374:
4368:
4365:
4362:
4359:
4356:
4355:
4329:
4326:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4282:
4279:
4276:
4271:
4267:
4260:
4257:
4251:
4248:
4245:
4242:
4239:
4236:
4230:
4227:
4221:
4218:
4215:
4212:
4201:
4200:
4189:
4186:
4180:
4177:
4171:
4168:
4165:
4162:
4159:
4153:
4150:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4120:
4117:
4100:
4099:
4085:
4082:
4076:
4073:
4067:
4064:
4058:
4055:
4052:
4049:
4046:
4040:
4037:
4011:
4008:
4005:
4002:
3999:
3996:
3990:
3987:
3964:
3961:
3958:
3955:
3952:
3949:
3943:
3940:
3926:
3925:
3914:
3908:
3905:
3899:
3896:
3890:
3887:
3881:
3878:
3852:
3849:
3835:
3834:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3796:
3793:
3787:
3784:
3781:
3758:
3738:
3718:
3715:
3712:
3709:
3706:
3700:
3697:
3683:
3679:
3678:
3677:
3676:
3675:
3664:
3661:
3656:
3652:
3648:
3645:
3642:
3639:
3636:
3626:
3615:
3610:
3607:
3603:
3597:
3594:
3589:
3585:
3579:
3576:
3572:
3568:
3563:
3559:
3555:
3548:
3545:
3539:
3535:
3530:
3526:
3522:
3517:
3513:
3486:
3466:
3444:
3441:
3437:
3425:
3424:
3413:
3408:
3405:
3401:
3395:
3392:
3387:
3383:
3377:
3374:
3370:
3366:
3359:
3356:
3350:
3339:
3328:
3322:
3319:
3313:
3310:
3304:
3301:
3295:
3292:
3289:
3279:
3268:
3262:
3259:
3253:
3250:
3244:
3241:
3235:
3232:
3229:
3224:
3221:
3216:
3212:
3206:
3203:
3199:
3195:
3189:
3186:
3157:
3135:
3131:
3110:
3090:
3068:
3065:
3061:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3016:
3013:
2990:
2987:
2984:
2981:
2978:
2975:
2969:
2966:
2952:
2951:
2940:
2935:
2932:
2927:
2923:
2917:
2914:
2910:
2906:
2903:
2893:
2882:
2876:
2873:
2867:
2864:
2858:
2855:
2849:
2846:
2834:is given by:
2823:
2803:
2779:
2759:
2739:
2719:
2699:
2688:
2687:
2676:
2673:
2670:
2667:
2664:
2661:
2655:
2652:
2639:
2634:
2631:
2628:
2624:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2562:
2542:
2522:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2449:
2429:
2409:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2346:
2343:
2342:
2341:
2329:
2309:
2289:
2286:
2283:
2280:
2256:
2236:
2224:
2223:
2222:
2221:
2210:
2207:
2204:
2199:
2192:
2189:
2182:
2179:
2176:
2171:
2164:
2161:
2154:
2151:
2148:
2145:
2120:
2117:
2094:
2091:
2086:
2079:
2076:
2053:
2052:
2051:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2017:
2013:
2009:
2004:
2000:
1996:
1993:
1990:
1987:
1982:
1977:
1973:
1969:
1966:
1963:
1960:
1937:
1934:
1931:
1928:
1925:
1920:
1916:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1890:
1886:
1881:
1876:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1842:
1820:
1817:
1814:
1811:
1800:
1799:
1798:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1755:
1748:
1745:
1738:
1735:
1732:
1729:
1716:
1715:
1714:if and only if
1703:
1700:
1697:
1694:
1689:
1685:
1681:
1676:
1669:
1666:
1642:
1639:
1636:
1633:
1630:
1624:
1621:
1598:
1582:
1581:
1565:
1541:
1538:
1535:
1532:
1521:
1520:
1519:
1508:
1504:
1500:
1497:
1494:
1489:
1486:
1482:
1478:
1475:
1471:
1465:
1458:
1454:
1450:
1447:
1444:
1439:
1432:
1429:
1422:
1417:
1402:
1401:
1393:
1392:
1391:
1390:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1308:
1301:
1298:
1291:
1288:
1285:
1272:
1271:
1267:
1266:
1265:
1264:
1250:
1246:
1242:
1238:
1234:
1229:
1225:
1200:
1196:
1192:
1188:
1167:
1162:
1158:
1154:
1151:
1145:
1142:
1136:
1133:
1130:
1127:
1121:
1118:
1112:
1109:
1106:
1103:
1100:
1095:
1091:
1067:
1064:
1061:
1058:
1050:
1047:
1023:
1012:
1011:
1010:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
961:
954:
951:
934:
933:
928:
925:
924:
923:
912:
909:
906:
899:
896:
890:
886:
883:
880:
877:
872:
865:
862:
844:
843:
832:
829:
824:
819:
815:
811:
808:
805:
800:
795:
792:
789:
785:
781:
778:
775:
770:
766:
762:
759:
756:
753:
749:
745:
740:
736:
732:
729:
726:
723:
719:
715:
712:
709:
705:
701:
696:
692:
688:
685:
682:
679:
675:
671:
668:
644:
638:
634:
630:
627:
621:
618:
612:
608:
604:
601:
581:
561:
541:
532:is taken over
521:
507:
506:
495:
492:
489:
486:
483:
477:
474:
468:
463:
459:
455:
452:
446:
443:
437:
434:
431:
428:
425:
421:
417:
412:
408:
404:
401:
395:
392:
386:
383:
380:
377:
371:
368:
362:
359:
356:
353:
349:
345:
342:
339:
336:
302:
299:
293:
290:
284:
281:
261:
241:
221:
218:
215:
209:
206:
180:
177:
174:
154:
134:
131:
128:
108:
96:
93:
75:
72:
26:
9:
6:
4:
3:
2:
24599:
24588:
24585:
24583:
24580:
24578:
24575:
24574:
24572:
24560:
24554:
24550:
24545:
24541:
24539:0-471-09517-6
24535:
24531:
24526:
24522:
24516:
24512:
24507:
24503:
24497:
24493:
24488:
24484:
24482:0-13-042268-1
24478:
24474:
24469:
24468:
24461:
24457:
24455:0-387-98502-6
24451:
24447:
24442:
24438:
24436:9780471016564
24432:
24427:
24426:
24419:
24415:
24409:
24405:
24400:
24396:
24383:
24367:
24363:
24358:
24357:
24343:
24328:
24324:
24318:
24314:
24305:
24302:
24300:
24297:
24295:
24294:Kalman filter
24292:
24290:
24289:Wiener filter
24287:
24285:
24282:
24279:
24276:
24274:
24273:Least squares
24271:
24269:
24266:
24264:
24261:
24260:
24240:
24232:
24227:
24223:
24218:
24214:
24211:
24206:
24199:
24195:
24190:
24185:
24179:
24174:
24170:
24164:
24160:
24152:
24145:
24141:
24136:
24131:
24125:
24121:
24114:
24109:
24105:
24097:
24096:
24095:
24093:
24072:
24068:
24062:
24058:
24054:
24049:
24045:
24039:
24035:
24031:
24028:
24021:
24020:
24019:
24005:
24002:
23994:
23990:
23983:
23977:
23969:
23965:
23958:
23928:
23923:
23919:
23915:
23912:
23907:
23903:
23899:
23897:
23890:
23886:
23876:
23872:
23868:
23865:
23860:
23856:
23852:
23850:
23843:
23839:
23827:
23826:
23825:
23822:
23808:
23803:
23798:
23794:
23773:
23751:
23744:
23740:
23735:
23712:
23705:
23701:
23696:
23673:
23669:
23646:
23642:
23619:
23615:
23592:
23588:
23572:
23559:
23551:
23546:
23542:
23537:
23533:
23530:
23525:
23518:
23514:
23509:
23504:
23500:
23495:
23490:
23487:
23484:
23480:
23475:
23470:
23428:
23423:
23419:
23414:
23410:
23407:
23402:
23395:
23391:
23386:
23381:
23377:
23372:
23367:
23364:
23361:
23357:
23349:
23342:
23338:
23333:
23328:
23324:
23318:
23313:
23309:
23300:
23284:
23275:
23269:
23257:
23251:
23246:
23242:
23233:
23229:
23223:
23218:
23215:
23212:
23208:
23204:
23195:
23172:
23149:
23141:
23136:
23132:
23127:
23123:
23120:
23115:
23108:
23104:
23099:
23094:
23090:
23087:
23082:
23075:
23071:
23066:
23061:
23057:
23053:
23048:
23043:
23033:
23027:
23023:
23018:
23013:
23009:
23005:
22981:
22980:
22979:
22965:
22960:
22955:
22951:
22943:smaller than
22928:
22918:
22912:
22888:
22883:
22878:
22874:
22865:
22860:
22856:
22851:
22847:
22844:
22839:
22832:
22828:
22823:
22818:
22814:
22811:
22806:
22799:
22795:
22790:
22785:
22781:
22774:
22767:
22763:
22758:
22753:
22749:
22746:
22741:
22734:
22730:
22725:
22720:
22716:
22710:
22705:
22695:
22689:
22681:
22680:
22679:
22659:
22629:
22621:
22616:
22612:
22607:
22603:
22600:
22595:
22588:
22584:
22579:
22574:
22570:
22567:
22562:
22555:
22551:
22546:
22541:
22537:
22530:
22523:
22519:
22514:
22509:
22505:
22499:
22497:
22490:
22486:
22478:
22470:
22465:
22461:
22456:
22452:
22449:
22444:
22437:
22433:
22428:
22423:
22419:
22416:
22411:
22404:
22400:
22395:
22390:
22386:
22379:
22372:
22368:
22363:
22358:
22354:
22348:
22346:
22339:
22335:
22323:
22322:
22321:
22304:
22295:
22289:
22277:
22271:
22266:
22262:
22253:
22249:
22245:
22233:
22227:
22222:
22218:
22209:
22205:
22201:
22192:
22182:
22181:
22180:
22164:
22160:
22137:
22133:
22112:
22108:
22104:
22101:
22092:
22086:
22078:
22074:
22067:
22061:
22053:
22049:
22042:
22012:
22007:
22003:
21999:
21996:
21993:
21991:
21984:
21980:
21970:
21966:
21962:
21959:
21956:
21954:
21947:
21943:
21931:
21930:
21929:
21926:
21912:
21907:
21900:
21896:
21891:
21868:
21864:
21841:
21837:
21816:
21811:
21804:
21800:
21795:
21772:
21768:
21745:
21741:
21720:
21716:
21712:
21709:
21704:
21699:
21695:
21674:
21670:
21666:
21663:
21654:
21628:
21625:
21622:
21599:
21579:
21576:
21573:
21570:
21550:
21544:
21541:
21538:
21532:
21529:
21515:
21501:
21481:
21472:
21458:
21453:
21448:
21444:
21440:
21435:
21425:
21419:
21395:
21390:
21386:
21380:
21375:
21372:
21369:
21365:
21359:
21356:
21351:
21342:
21332:
21331:
21330:
21316:
21293:
21288:
21283:
21278:
21274:
21258:
21254:
21248:
21243:
21239:
21235:
21230:
21225:
21221:
21214:
21209:
21205:
21192:
21187:
21177:
21171:
21167:
21162:
21157:
21153:
21149:
21146:
21139:
21138:
21137:
21120:
21115:
21110:
21106:
21092:
21088:
21082:
21077:
21073:
21069:
21064:
21059:
21055:
21048:
21043:
21039:
21026:
21021:
21018:
21014:
21008:
21005:
21000:
20996:
20990:
20987:
20983:
20979:
20974:
20964:
20958:
20950:
20949:
20948:
20945:
20932:
20927:
20921:
20917:
20911:
20906:
20903:
20900:
20896:
20889:
20884:
20880:
20875:
20871:
20864:
20855:
20830:
20820:
20816:
20812:
20809:
20806:
20801:
20797:
20793:
20788:
20784:
20777:
20774:
20747:
20738:
20729:
20725:
20719:
20714:
20710:
20706:
20701:
20696:
20692:
20685:
20680:
20676:
20670:
20668:
20660:
20655:
20651:
20645:
20642:
20637:
20629:
20624:
20620:
20616:
20611:
20604:
20599:
20595:
20591:
20585:
20576:
20571:
20567:
20563:
20558:
20556:
20548:
20545:
20538:
20533:
20529:
20525:
20518:
20514:
20508:
20505:
20500:
20492:
20487:
20483:
20479:
20474:
20471:
20468:
20461:
20456:
20452:
20448:
20441:
20437:
20432:
20427:
20425:
20414:
20400:
20399:
20398:
20384:
20357:
20354:
20349:
20346:
20338:
20333:
20328:
20324:
20320:
20315:
20311:
20305:
20300:
20296:
20287:
20283:
20277:
20272:
20268:
20264:
20262:
20254:
20249:
20246:
20241:
20237:
20231:
20228:
20224:
20220:
20218:
20207:
20193:
20192:
20191:
20170:
20165:
20161:
20155:
20150:
20146:
20142:
20134:
20130:
20126:
20120:
20114:
20109:
20106:
20102:
20093:
20090:
20085:
20080:
20076:
20072:
20067:
20063:
20057:
20052:
20048:
20044:
20036:
20032:
20028:
20022:
20016:
20011:
20007:
19998:
19995:
19992:
19986:
19980:
19965:
19964:
19963:
19949:
19946:
19941:
19938:
19934:
19913:
19893:
19873:
19850:
19845:
19840:
19836:
19832:
19829:
19823:
19803:
19783:
19779:
19773:
19768:
19764:
19760:
19755:
19750:
19746:
19725:
19700:
19696:
19692:
19687:
19683:
19679:
19668:
19665:by an aprior
19652:
19632:
19607:
19603:
19599:
19594:
19590:
19586:
19575:
19572:represents a
19559:
19537:
19529:
19526:
19523:
19520:
19517:
19514:
19511:
19505:
19502:
19482:
19479:
19476:
19473:
19470:
19467:
19447:
19427:
19407:
19393:
19391:
19375:
19353:
19349:
19328:
19325:
19320:
19317:
19313:
19309:
19306:
19303:
19300:
19295:
19292:
19288:
19284:
19281:
19273:
19269:
19263:
19259:
19252:
19246:
19241:
19228:
19201:
19191:
19167:
19164:
19159:
19155:
19134:
19131:
19128:
19125:
19120:
19116:
19096:
19093:
19090:
19085:
19081:
19057:
19052:
19048:
19044:
19040:
19033:
19026:
19023:
19016:
19009:
19005:
19001:
18994:
18987:
18980:
18973:
18968:
18961:
18958:
18953:
18948:
18945:
18938:
18933:
18928:
18925:
18918:
18915:
18910:
18907:
18902:
18895:
18891:
18886:
18883:
18879:
18873:
18870:
18865:
18861:
18853:
18852:
18851:
18835:
18831:
18823:by inverting
18808:
18805:
18801:
18797:
18792:
18788:
18782:
18778:
18754:
18750:
18743:
18736:
18729:
18722:
18718:
18714:
18702:
18698:
18694:
18689:
18685:
18678:
18666:
18662:
18658:
18653:
18649:
18642:
18630:
18626:
18622:
18617:
18613:
18606:
18599:
18595:
18590:
18587:
18583:
18575:
18574:
18573:
18557:
18554:
18550:
18526:
18522:
18515:
18510:
18505:
18498:
18493:
18488:
18481:
18476:
18471:
18464:
18460:
18456:
18444:
18440:
18436:
18431:
18427:
18420:
18410:
18406:
18402:
18397:
18393:
18386:
18376:
18372:
18368:
18363:
18359:
18352:
18340:
18336:
18332:
18327:
18323:
18316:
18306:
18302:
18298:
18293:
18289:
18282:
18272:
18268:
18264:
18259:
18255:
18248:
18236:
18232:
18228:
18223:
18219:
18212:
18202:
18198:
18194:
18189:
18185:
18178:
18168:
18164:
18160:
18155:
18151:
18144:
18137:
18133:
18128:
18124:
18116:
18115:
18114:
18098:
18094:
18071:
18067:
18063:
18060:
18035:
18031:
18027:
18022:
18018:
18014:
18009:
18005:
17998:
17995:
17973:
17963:
17959:
17955:
17950:
17946:
17942:
17937:
17933:
17926:
17923:
17914:
17898:
17888:
17862:
17858:
17834:
17830:
17823:
17818:
17813:
17808:
17801:
17796:
17791:
17786:
17779:
17774:
17769:
17764:
17757:
17752:
17747:
17742:
17735:
17731:
17723:
17719:
17715:
17709:
17703:
17697:
17691:
17688:
17681:
17680:
17679:
17663:
17653:
17649:
17645:
17640:
17636:
17632:
17627:
17623:
17619:
17614:
17610:
17603:
17600:
17578:
17574:
17568:
17564:
17558:
17553:
17550:
17547:
17543:
17539:
17534:
17524:
17498:
17494:
17471:
17467:
17444:
17440:
17436:
17431:
17427:
17418:
17403:
17387:
17384:
17381:
17371:
17364:
17356:
17348:
17345:
17342:
17332:
17304:
17301:
17298:
17294:
17289:
17285:
17277:
17267:
17264:
17261:
17257:
17252:
17229:
17226:
17223:
17219:
17198:
17195:
17192:
17172:
17147:
17139:
17136:
17133:
17129:
17104:
17101:
17098:
17094:
17089:
17068:
17065:
17062:
17042:
17017:
17007:
17004:
17001:
16997:
16992:
16969:
16959:
16952:
16944:
16936:
16933:
16930:
16920:
16892:
16888:
16883:
16879:
16871:
16861:
16858:
16855:
16851:
16846:
16825:
16822:
16819:
16816:
16813:
16810:
16807:
16804:
16801:
16770:
16767:
16764:
16756:
16746:
16734:
16726:
16723:
16720:
16716:
16712:
16704:
16696:
16693:
16690:
16686:
16674:
16666:
16663:
16660:
16656:
16652:
16644:
16641:
16638:
16630:
16620:
16613:
16605:
16597:
16594:
16591:
16581:
16570:
16551:
16541:
16537:
16532:
16520:
16512:
16509:
16506:
16502:
16493:
16485:
16482:
16479:
16475:
16471:
16468:
16462:
16454:
16444:
16441:
16438:
16434:
16429:
16421:
16399:
16393:
16385:
16382:
16379:
16375:
16363:
16353:
16349:
16344:
16335:
16327:
16324:
16321:
16317:
16313:
16305:
16295:
16292:
16289:
16285:
16280:
16272:
16266:
16258:
16255:
16252:
16248:
16239:
16229:
16225:
16220:
16213:
16205:
16197:
16194:
16191:
16187:
16179:
16178:
16177:
16162:
16142:
16139:
16136:
16116:
16094:
16090:
16075:
16059:
16049:
16040:
16036:
16032:
16024:
16014:
16005:
16001:
15970:
15966:
15942:
15934:
15930:
15924:
15914:
15897:
15893:
15889:
15884:
15874:
15867:
15862:
15859:
15856:
15846:
15835:
15834:
15833:
15819:
15813:
15804:
15790:
15787:
15784:
15776:
15766:
15744:
15708:
15700:
15690:
15658:
15650:
15646:
15642:
15638:
15634:
15630:
15616:
15594:
15590:
15563:
15538:
15535:
15532:
15528:
15504:
15497:
15493:
15488:
15479:
15474:
15471:
15468:
15464:
15458:
15455:
15452:
15448:
15444:
15441:
15435:
15428:
15425:
15422:
15418:
15413:
15405:
15404:
15403:
15401:
15397:
15377:
15374:
15371:
15367:
15362:
15338:
15330:
15327:
15324:
15320:
15312:
15308:
15303:
15297:
15292:
15289:
15286:
15282:
15278:
15273:
15268:
15265:
15262:
15258:
15250:
15247:
15244:
15240:
15232:
15228:
15223:
15216:
15211:
15208:
15205:
15201:
15193:
15192:
15191:
15189:
15171:
15168:
15165:
15161:
15138:
15135:
15132:
15128:
15099:
15089:
15080:
15075:
15072:
15069:
15065:
15061:
15056:
15053:
15050:
15046:
15037:
15034:
15031:
15027:
15023:
15018:
15008:
15001:
14996:
14993:
14990:
14980:
14969:
14968:
14967:
14951:
14948:
14945:
14935:
14923:
14905:
14900:
14896:
14873:
14869:
14860:
14842:
14838:
14829:
14811:
14807:
14784:
14780:
14776:
14771:
14767:
14761:
14756:
14752:
14748:
14743:
14739:
14730:
14720:
14718:
14717:Kalman filter
14699:
14691:
14688:
14685:
14675:
14668:
14665:
14660:
14656:
14647:
14643:
14639:
14634:
14631:
14628:
14618:
14611:
14606:
14596:
14585:
14571:
14566:
14563:
14558:
14554:
14548:
14544:
14536:
14532:
14527:
14523:
14518:
14514:
14506:
14492:
14489:
14484:
14481:
14476:
14472:
14466:
14462:
14458:
14453:
14450:
14443:
14440:
14437:
14433:
14428:
14424:
14419:
14416:
14409:
14405:
14400:
14392:
14391:
14390:
14374:
14370:
14346:
14339:
14336:
14333:
14329:
14324:
14317:
14312:
14308:
14304:
14301:
14295:
14288:
14284:
14279:
14271:
14257:
14249:
14246:
14243:
14233:
14226:
14223:
14218:
14214:
14205:
14201:
14197:
14192:
14189:
14186:
14176:
14169:
14164:
14154:
14143:
14129:
14124:
14121:
14111:
14107:
14103:
14098:
14094:
14086:
14083:
14080:
14076:
14071:
14067:
14059:
14055:
14047:
14044:
14041:
14037:
14032:
14028:
14023:
14019:
14011:
14010:
14009:
14006:
13990:
13986:
13963:
13953:
13926:
13919:
13916:
13913:
13909:
13904:
13900:
13895:
13892:
13882:
13878:
13874:
13869:
13865:
13857:
13854:
13851:
13847:
13842:
13838:
13830:
13826:
13818:
13815:
13812:
13808:
13803:
13799:
13792:
13789:
13786:
13782:
13777:
13773:
13766:
13762:
13757:
13749:
13748:
13747:
13721:
13718:
13715:
13705:
13698:
13695:
13690:
13686:
13677:
13674:
13664:
13660:
13656:
13651:
13647:
13639:
13636:
13633:
13629:
13624:
13620:
13612:
13608:
13600:
13597:
13594:
13590:
13585:
13581:
13576:
13573:
13570:
13560:
13553:
13551:
13538:
13528:
13521:
13516:
13512:
13503:
13500:
13493:
13483:
13475:
13467:
13457:
13448:
13445:
13442:
13438:
13433:
13429:
13424:
13421:
13418:
13408:
13401:
13399:
13392:
13382:
13367:
13366:
13365:
13349:
13345:
13317:
13312:
13308:
13300:
13297:
13294:
13290:
13285:
13281:
13274:
13264:
13255:
13252:
13249:
13245:
13240:
13236:
13234:
13225:
13222:
13219:
13215:
13211:
13208:
13205:
13200:
13196:
13185:
13181:
13175:
13171:
13166:
13154:
13153:
13152:
13136:
13133:
13129:
13104:
13094:
13086:
13063:
13060:
13057:
13047:
13021:
13017:
12990:
12963:
12960:
12957:
12953:
12948:
12925:
12922:
12919:
12909:
12883:
12879:
12852:
12840:
12822:
12818:
12807:
12803:
12798:
12794:
12787:
12777:
12769:
12748:
12745:
12737:
12727:
12693:
12683:
12676:
12671:
12667:
12663:
12658:
12648:
12622:
12618:
12595:
12592:
12589:
12579:
12572:
12569:
12564:
12554:
12528:
12524:
12495:
12490:
12486:
12482:
12477:
12473:
12465:
12462:
12459:
12455:
12450:
12446:
12443:
12438:
12434:
12430:
12425:
12421:
12413:
12410:
12407:
12403:
12399:
12396:
12393:
12388:
12384:
12373:
12369:
12364:
12360:
12357:
12355:
12346:
12342:
12331:
12327:
12322:
12310:
12309:
12308:
12292:
12289:
12286:
12276:
12269:
12266:
12261:
12251:
12244:
12241:
12236:
12226:
12197:
12193:
12182:
12178:
12171:
12162:
12145:
12138:
12135:
12132:
12128:
12123:
12119:
12112:
12109:
12106:
12102:
12098:
12095:
12092:
12087:
12083:
12072:
12069:
12066:
12062:
12057:
12053:
12046:
12043:
12040:
12036:
12032:
12029:
12026:
12021:
12017:
12006:
12002:
11997:
11989:
11988:
11987:
11967:
11964:
11961:
11951:
11944:
11936:
11933:
11930:
11926:
11922:
11919:
11916:
11911:
11907:
11896:
11893:
11890:
11886:
11874:
11866:
11863:
11860:
11856:
11852:
11849:
11846:
11841:
11837:
11826:
11822:
11810:
11805:
11795:
11784:
11783:
11782:
11763:
11760:
11757:
11753:
11749:
11746:
11743:
11738:
11734:
11723:
11719:
11712:
11703:
11684:
11680:
11669:
11665:
11658:
11633:
11630:
11627:
11623:
11619:
11616:
11613:
11608:
11604:
11593:
11589:
11582:
11562:
11542:
11522:
11513:
11507:
11495:
11489:
11486:
11478:
11475:
11470:
11466:
11460:
11457:
11453:
11449:
11440:
11428:
11426:
11422:
11403:
11395:
11392:
11389:
11385:
11381:
11378:
11375:
11370:
11366:
11355:
11352:
11349:
11345:
11338:
11335:
11327:
11324:
11321:
11317:
11313:
11310:
11307:
11302:
11298:
11287:
11283:
11276:
11269:
11268:
11267:
11253:
11228:
11224:
11220:
11217:
11214:
11209:
11205:
11194:
11190:
11183:
11175:
11157:
11147:
11134:
11117:
11109:
11105:
11094:
11090:
11083:
11080:
11072:
11069:
11066:
11062:
11058:
11055:
11052:
11047:
11043:
11039:
11034:
11030:
11019:
11015:
11008:
11001:
11000:
10999:
10985:
10963:
10960:
10957:
10953:
10949:
10946:
10943:
10938:
10934:
10911:
10907:
10898:
10877:
10874:
10871:
10867:
10863:
10860:
10857:
10852:
10848:
10837:
10833:
10826:
10801:
10797:
10786:
10782:
10775:
10750:
10746:
10742:
10739:
10736:
10731:
10727:
10716:
10712:
10705:
10678:
10670:
10667:
10664:
10660:
10656:
10653:
10650:
10645:
10641:
10630:
10626:
10619:
10611:
10607:
10596:
10592:
10585:
10582:
10580:
10567:
10564:
10561:
10557:
10553:
10550:
10547:
10542:
10538:
10527:
10523:
10516:
10508:
10505:
10502:
10498:
10494:
10491:
10488:
10483:
10479:
10475:
10472:
10462:
10458:
10451:
10448:
10446:
10436:
10432:
10428:
10425:
10422:
10417:
10413:
10402:
10398:
10391:
10380:
10379:
10378:
10362:
10358:
10335:
10331:
10327:
10324:
10321:
10316:
10312:
10291:
10282:
10273:
10257:
10253:
10247:
10244:
10236:
10233:
10230:
10227:
10222:
10218:
10211:
10208:
10188:
10168:
10165:
10162:
10157:
10154:
10149:
10145:
10122:
10118:
10112:
10109:
10101:
10096:
10092:
10085:
10082:
10062:
10042:
10039:
10034:
10030:
10026:
10021:
10017:
9996:
9974:
9971:
9966:
9962:
9939:
9936:
9931:
9927:
9921:
9917:
9911:
9908:
9900:
9895:
9892:
9887:
9883:
9877:
9873:
9866:
9863:
9856:, the result
9843:
9823:
9820:
9815:
9812:
9807:
9803:
9794:
9790:
9786:
9767:
9758:
9752:
9740:
9734:
9731:
9728:
9720:
9717:
9712:
9708:
9702:
9698:
9692:
9688:
9684:
9675:
9665:
9664:
9663:
9643:
9618:
9615:
9610:
9606:
9600:
9596:
9590:
9586:
9582:
9579:
9557:
9553:
9532:
9509:
9504:
9501:
9491:
9488:
9483:
9479:
9475:
9472:
9467:
9464:
9459:
9455:
9449:
9445:
9438:
9433:
9429:
9421:
9420:
9419:
9402:
9397:
9394:
9389:
9385:
9379:
9375:
9369:
9366:
9356:
9353:
9348:
9344:
9340:
9337:
9332:
9329:
9324:
9320:
9314:
9310:
9303:
9300:
9293:
9292:
9291:
9277:
9269:
9266:
9261:
9257:
9253:
9250:
9245:
9242:
9237:
9233:
9227:
9223:
9194:
9190:
9186:
9181:
9177:
9171:
9167:
9163:
9137:
9132:
9129:
9124:
9120:
9114:
9110:
9104:
9101:
9091:
9088:
9083:
9079:
9075:
9072:
9067:
9064:
9059:
9055:
9049:
9045:
9038:
9033:
9030:
9020:
9016:
9012:
9007:
9003:
8997:
8993:
8989:
8981:
8977:
8971:
8967:
8959:
8958:
8957:
8949:
8933:
8929:
8923:
8919:
8915:
8895:
8892:
8887:
8883:
8874:
8870:
8853:
8831:
8827:
8818:
8800:
8797:
8787:
8783:
8779:
8774:
8770:
8764:
8760:
8756:
8745:
8741:
8725:
8717:
8701:
8693:
8689:
8685:
8684:least squares
8666:
8661:
8657:
8653:
8648:
8645:
8635:
8631:
8627:
8622:
8618:
8612:
8608:
8604:
8596:
8592:
8586:
8582:
8578:
8573:
8569:
8565:
8555:
8549:
8545:
8540:
8536:
8532:
8527:
8523:
8515:
8514:
8513:
8496:
8487:
8481:
8469:
8463:
8460:
8457:
8449:
8446:
8436:
8432:
8428:
8423:
8419:
8413:
8409:
8405:
8397:
8393:
8387:
8383:
8379:
8370:
8360:
8359:
8358:
8338:
8312:
8307:
8304:
8294:
8290:
8286:
8281:
8277:
8271:
8267:
8263:
8255:
8251:
8245:
8241:
8237:
8234:
8227:
8226:
8225:
8211:
8188:
8183:
8179:
8173:
8169:
8165:
8160:
8157:
8153:
8145:
8131:
8126:
8122:
8118:
8113:
8109:
8103:
8099:
8095:
8092:
8087:
8083:
8075:
8061:
8052:
8046:
8043:
8037:
8031:
8021:
8020:
8019:
8005:
8002:
7997:
7994:
7990:
7969:
7966:
7960:
7954:
7931:
7911:
7891:
7888:
7885:
7882:
7879:
7876:
7862:
7860:
7856:
7840:
7832:
7814:
7810:
7801:
7797:
7793:
7777:
7755:
7751:
7742:
7726:
7718:
7708:
7694:
7674:
7654:
7651:
7646:
7641:
7637:
7610:
7604:
7592:
7586:
7583:
7573:
7569:
7563:
7560:
7556:
7550:
7541:
7518:
7515:
7512:
7509:
7489:
7467:
7462:
7458:
7454:
7449:
7444:
7440:
7413:
7407:
7398:
7375:
7372:
7369:
7341:
7336:
7332:
7326:
7316:
7310:
7304:
7297:
7292:
7288:
7281:
7276:
7272:
7268:
7263:
7258:
7254:
7247:
7242:
7238:
7230:
7229:
7228:
7210:
7203:
7199:
7188:
7182:
7179:
7173:
7169:
7166:
7162:
7155:
7151:
7140:
7134:
7125:
7116:
7108:
7107:
7106:
7103:
7089:
7069:
7061:
7040:
7036:
7030:
7026:
7019:
7016:
7012:
7006:
7003:
6980:
6975:
6970:
6966:
6957:
6953:
6949:
6946:
6940:
6933:
6928:
6924:
6918:
6913:
6910:
6906:
6900:
6895:
6890:
6886:
6882:
6877:
6872:
6868:
6860:
6846:
6837:
6831:
6819:
6813:
6810:
6800:
6796:
6790:
6786:
6780:
6777:
6768:
6762:
6750:
6744:
6741:
6731:
6726:
6722:
6716:
6713:
6709:
6703:
6694:
6684:
6683:
6682:
6668:
6648:
6612:
6608:
6601:
6598:
6595:
6592:
6585:
6584:
6583:
6569:
6559:
6553:
6549:
6544:
6540:
6536:
6531:
6527:
6503:
6498:
6495:
6491:
6485:
6482:
6477:
6473:
6467:
6464:
6460:
6456:
6451:
6447:
6443:
6438:
6434:
6426:
6425:
6424:
6408:
6404:
6381:
6378:
6373:
6369:
6363:
6360:
6356:
6352:
6349:
6322:
6317:
6314:
6310:
6306:
6303:
6298:
6294:
6290:
6288:
6275:
6261:
6255:
6252:
6237:
6231:
6228:
6219:
6213:
6210:
6202:
6188:
6182:
6179:
6164:
6158:
6155:
6146:
6140:
6138:
6125:
6111:
6105:
6102:
6084:
6078:
6075:
6069:
6057:
6051:
6048:
6042:
6033:
6027:
6024:
6022:
6009:
5995:
5989:
5986:
5977:
5974:
5965:
5953:
5947:
5942:
5938:
5932:
5926:
5917:
5903:
5897:
5894:
5885:
5882:
5873:
5861:
5851:
5849:
5836:
5819:
5813:
5810:
5804:
5792:
5786:
5783:
5777:
5768:
5765:
5756:
5744:
5738:
5736:
5723:
5715:
5712:
5703:
5691:
5688:
5679:
5667:
5661:
5659:
5652:
5648:
5636:
5635:
5634:
5617:
5612:
5609:
5605:
5599:
5596:
5591:
5587:
5581:
5578:
5574:
5570:
5560:
5554:
5546:
5545:
5544:
5528:
5524:
5503:
5476:
5471:
5467:
5461:
5457:
5453:
5450:
5448:
5438:
5434:
5425:
5411:
5405:
5402:
5387:
5381:
5378:
5369:
5363:
5360:
5358:
5345:
5331:
5325:
5316:
5298:
5292:
5283:
5271:
5265:
5263:
5251:
5245:
5233:
5232:
5231:
5211:
5205:
5193:
5184:
5155:
5129:
5120:
5114:
5102:
5096:
5093:
5085:
5082:
5077:
5073:
5067:
5064:
5060:
5056:
5047:
5037:
5036:
5035:
5018:
5013:
5010:
5006:
5000:
4997:
4992:
4988:
4984:
4979:
4975:
4967:
4966:
4965:
4949:
4946:
4942:
4919:
4914:
4911:
4907:
4903:
4898:
4895:
4891:
4868:
4864:
4841:
4838:
4834:
4810:
4805:
4802:
4797:
4793:
4787:
4784:
4780:
4776:
4773:
4766:
4765:
4764:
4750:
4723:
4718:
4715:
4711:
4707:
4702:
4698:
4694:
4691:
4689:
4676:
4662:
4656:
4653:
4638:
4632:
4629:
4620:
4614:
4606:
4592:
4586:
4583:
4568:
4562:
4559:
4550:
4544:
4541:
4539:
4526:
4512:
4506:
4503:
4485:
4479:
4476:
4470:
4458:
4452:
4449:
4443:
4434:
4428:
4426:
4416:
4402:
4396:
4393:
4384:
4381:
4372:
4360:
4346:
4345:
4344:
4324:
4318:
4315:
4312:
4306:
4300:
4280:
4277:
4269:
4255:
4249:
4246:
4237:
4234:
4225:
4213:
4187:
4175:
4169:
4166:
4160:
4148:
4142:
4139:
4133:
4130:
4127:
4124:
4115:
4105:
4104:
4103:
4080:
4074:
4062:
4056:
4053:
4047:
4044:
4035:
4025:
4024:
4023:
4006:
4000:
3994:
3985:
3959:
3953:
3947:
3938:
3912:
3903:
3897:
3894:
3885:
3879:
3876:
3869:
3868:
3867:
3847:
3821:
3815:
3809:
3803:
3791:
3782:
3772:
3771:
3770:
3756:
3736:
3716:
3713:
3710:
3707:
3704:
3695:
3662:
3654:
3650:
3643:
3640:
3637:
3634:
3627:
3613:
3608:
3605:
3601:
3595:
3592:
3587:
3583:
3577:
3574:
3570:
3566:
3561:
3557:
3553:
3543:
3537:
3533:
3528:
3524:
3520:
3515:
3511:
3503:
3502:
3501:
3498:
3484:
3464:
3442:
3439:
3435:
3411:
3406:
3403:
3399:
3393:
3390:
3385:
3381:
3375:
3372:
3368:
3364:
3354:
3348:
3340:
3326:
3317:
3311:
3299:
3290:
3280:
3266:
3257:
3251:
3239:
3233:
3230:
3222:
3219:
3214:
3210:
3204:
3201:
3197:
3193:
3184:
3174:
3173:
3172:
3169:
3155:
3133:
3129:
3108:
3088:
3066:
3063:
3059:
3038:
3032:
3026:
3020:
3011:
2985:
2979:
2973:
2964:
2938:
2933:
2930:
2925:
2921:
2915:
2912:
2908:
2904:
2901:
2894:
2880:
2871:
2865:
2862:
2853:
2847:
2844:
2837:
2836:
2835:
2821:
2801:
2792:
2777:
2757:
2737:
2717:
2697:
2674:
2671:
2668:
2665:
2662:
2659:
2650:
2637:
2632:
2629:
2626:
2614:
2613:
2612:
2595:
2592:
2589:
2583:
2560:
2540:
2520:
2500:
2497:
2494:
2491:
2488:
2482:
2479:
2476:
2470:
2447:
2427:
2407:
2397:
2395:
2391:
2372:
2369:
2366:
2360:
2327:
2320:and constant
2307:
2287:
2284:
2281:
2278:
2270:
2254:
2234:
2226:
2225:
2208:
2205:
2197:
2187:
2177:
2174:
2169:
2159:
2146:
2136:
2135:
2115:
2092:
2089:
2084:
2074:
2062:
2058:
2054:
2038:
2035:
2032:
2023:
2015:
2011:
2002:
1998:
1994:
1988:
1980:
1975:
1971:
1961:
1951:
1950:
1929:
1926:
1918:
1910:
1904:
1898:
1892:
1879:
1869:
1866:
1863:
1857:
1851:
1845:
1815:
1809:
1801:
1785:
1782:
1773:
1767:
1761:
1758:
1753:
1743:
1730:
1720:
1719:
1718:
1717:
1701:
1695:
1687:
1683:
1679:
1674:
1664:
1637:
1631:
1628:
1619:
1596:
1588:
1584:
1583:
1579:
1563:
1555:
1536:
1530:
1522:
1506:
1502:
1495:
1487:
1484:
1480:
1476:
1473:
1469:
1456:
1452:
1445:
1442:
1437:
1427:
1415:
1406:
1405:
1404:
1403:
1399:
1395:
1394:
1377:
1371:
1365:
1359:
1350:
1347:
1344:
1338:
1329:
1323:
1314:
1306:
1296:
1286:
1276:
1275:
1274:
1273:
1269:
1268:
1248:
1240:
1236:
1232:
1227:
1223:
1215:
1214:
1198:
1190:
1186:
1160:
1152:
1149:
1140:
1128:
1125:
1116:
1104:
1098:
1093:
1089:
1045:
1021:
1013:
997:
991:
988:
985:
979:
973:
967:
959:
949:
938:
937:
936:
935:
931:
930:
910:
907:
904:
894:
888:
884:
878:
870:
860:
849:
848:
847:
830:
822:
817:
813:
806:
798:
793:
790:
787:
783:
779:
773:
768:
764:
757:
751:
747:
738:
734:
730:
724:
721:
717:
713:
707:
703:
694:
690:
686:
680:
673:
669:
666:
659:
658:
657:
642:
636:
628:
625:
616:
606:
602:
579:
559:
539:
512:
493:
484:
481:
472:
461:
453:
450:
441:
429:
423:
419:
410:
402:
399:
390:
378:
375:
366:
354:
347:
343:
340:
337:
334:
327:
326:
325:
324:
320:
316:
300:
297:
288:
282:
279:
259:
239:
216:
204:
194:
178:
175:
172:
152:
132:
129:
126:
106:
92:
90:
89:Bayes theorem
86:
81:
71:
69:
68:Kalman filter
65:
61:
60:loss function
57:
53:
49:
45:
41:
37:
33:
19:
24548:
24529:
24510:
24491:
24466:
24445:
24424:
24403:
24370:. Retrieved
24360:Johnson, D.
24342:
24330:. Retrieved
24326:
24317:
24091:
24089:
23947:
23823:
23578:
23298:
23164:
22904:which makes
22903:
22678:is given by
22648:
22319:
22031:
21927:
21521:
21473:
21410:
21308:
21135:
20946:
20766:
20376:
20189:
19399:
19072:
18769:
18541:
17915:
17849:
17414:
16793:
16081:
15957:
15632:
15631:
15519:
15399:
15395:
15353:
15187:
15119:
14921:
14858:
14827:
14728:
14726:
14714:
14361:
14007:
13941:
13745:
13336:
12841:
12515:
12215:is given by
12163:
12160:
11985:
11704:
11429:
11424:
11420:
11418:
11173:
11135:
11132:
10896:
10697:
10283:
10279:
9782:
9524:
9417:
9152:
8955:
8872:
8868:
8816:
8743:
8739:
8715:
8691:
8688:Gauss–Markov
8681:
8511:
8327:
8203:
7868:
7714:
7361:
7226:
7104:
6995:
6640:
6518:
6341:
5632:
5495:
5144:
5033:
4825:
4742:
4202:
4101:
3927:
3836:
3684:
3499:
3426:
3170:
2953:
2793:
2689:
2398:
2348:
2060:
2056:
845:
508:
98:
77:
43:
39:
29:
19718:, and thus
19073:So we have
17185:-th row of
14362:The matrix
13364:arrives as
7711:Computation
2300:for matrix
511:expectation
24571:Categories
24391:|url=
24090:where the
20767:where for
17513:such that
11172:given the
9290:to obtain
7829:is also a
7531:, we have
7388:, we have
3427:where the
927:Properties
509:where the
95:Definition
74:Motivation
32:statistics
24429:. Wiley.
24372:8 January
24224:σ
24191:σ
24161:∑
24137:σ
23984:
23959:
23795:σ
23736:σ
23697:σ
23575:Example 4
23543:σ
23510:σ
23481:∑
23420:σ
23387:σ
23358:∑
23334:σ
23279:¯
23261:¯
23252:−
23209:∑
23199:^
23133:σ
23100:σ
23067:σ
23037:^
23028:σ
23024:−
23010:σ
22952:σ
22922:^
22913:σ
22875:σ
22857:σ
22824:σ
22791:σ
22759:σ
22726:σ
22699:^
22690:σ
22663:^
22613:σ
22580:σ
22547:σ
22515:σ
22462:σ
22429:σ
22396:σ
22364:σ
22299:¯
22281:¯
22272:−
22237:¯
22228:−
22196:^
22096:¯
22068:
22043:
21892:σ
21796:σ
21696:σ
21658:¯
21574:−
21533:∈
21518:Example 3
21445:σ
21429:^
21420:σ
21366:∑
21346:^
21275:σ
21240:σ
21222:σ
21206:σ
21181:^
21172:σ
21168:−
21154:σ
21107:σ
21074:σ
21056:σ
21040:σ
21006:−
20968:^
20959:σ
20897:∑
20859:¯
20810:…
20742:¯
20711:σ
20693:σ
20677:σ
20643:−
20621:σ
20596:σ
20568:σ
20530:σ
20506:−
20484:σ
20453:σ
20418:^
20347:−
20325:σ
20297:σ
20269:σ
20247:−
20211:^
20147:σ
20121:
20077:σ
20049:σ
20023:
19981:
19837:σ
19747:σ
19680:−
19587:−
19524:…
19396:Example 2
19307:−
19282:−
19253:
19195:^
19129:−
19024:−
18959:−
18946:−
18926:−
18916:−
18908:−
18871:−
17892:^
17710:
17692:
17544:∑
17528:^
17411:Example 1
17375:^
17336:^
17196:×
17173:ℓ
17148:ℓ
17066:×
17043:ℓ
17018:ℓ
16963:^
16924:^
16820:…
16802:ℓ
16768:−
16765:ℓ
16750:^
16735:ℓ
16713:−
16705:ℓ
16675:ℓ
16642:−
16639:ℓ
16624:^
16606:ℓ
16585:^
16552:ℓ
16521:ℓ
16494:ℓ
16472:−
16455:ℓ
16394:ℓ
16364:ℓ
16336:ℓ
16306:ℓ
16267:ℓ
16240:ℓ
16206:ℓ
16140:×
16053:~
16033:≈
16018:~
15967:η
15918:~
15894:η
15878:^
15850:^
15808:~
15788:−
15770:~
15748:^
15739:∇
15712:~
15694:~
15567:^
15445:−
15259:σ
15093:^
15062:−
15012:^
14984:^
14939:^
14920:. After (
14897:σ
14689:−
14679:^
14666:−
14632:−
14622:^
14600:^
14564:−
14482:−
14451:−
14441:−
14417:−
14337:−
14305:−
14247:−
14237:^
14224:−
14190:−
14180:^
14158:^
14122:−
14084:−
14045:−
13957:^
13917:−
13893:−
13855:−
13816:−
13800:−
13790:−
13719:−
13709:^
13696:−
13675:−
13637:−
13598:−
13574:−
13564:^
13532:¯
13522:−
13501:−
13487:~
13461:~
13446:−
13422:−
13412:^
13386:^
13298:−
13268:~
13253:−
13223:−
13209:…
13098:~
13061:−
13051:¯
12994:¯
12961:−
12923:−
12913:^
12856:¯
12781:~
12731:~
12687:¯
12677:−
12652:~
12593:−
12583:^
12558:¯
12463:−
12411:−
12397:…
12290:−
12280:^
12255:¯
12230:¯
12136:−
12110:−
12096:…
12070:−
12044:−
12030:…
11965:−
11955:^
11934:−
11920:…
11894:−
11864:−
11850:…
11799:¯
11761:−
11747:…
11631:−
11617:…
11517:¯
11499:¯
11490:−
11476:−
11444:^
11393:−
11379:…
11353:−
11325:−
11311:…
11218:…
11151:^
11070:−
11056:…
10961:−
10947:…
10875:−
10861:…
10740:…
10668:−
10654:…
10565:−
10551:…
10506:−
10492:…
10449:∝
10426:…
10325:…
10245:−
10234:λ
10166:λ
10155:−
10110:−
10031:σ
9972:−
9937:−
9909:−
9893:−
9813:−
9762:¯
9744:¯
9732:−
9718:−
9679:^
9647:^
9616:−
9502:−
9489:−
9465:−
9395:−
9367:−
9354:−
9330:−
9267:−
9243:−
9130:−
9102:−
9089:−
9065:−
9031:−
8798:−
8646:−
8579:−
8559:^
8546:−
8491:¯
8473:¯
8461:−
8447:−
8374:^
8357:, we get
8342:^
8305:−
8056:¯
8032:
7955:
7638:σ
7614:¯
7596:¯
7587:−
7570:σ
7557:σ
7545:^
7516:±
7510:ρ
7459:σ
7441:σ
7417:¯
7402:^
7370:ρ
7333:σ
7320:^
7311:σ
7289:σ
7273:σ
7269:−
7255:σ
7239:ρ
7200:σ
7192:¯
7183:−
7170:ρ
7152:σ
7144:¯
7135:−
7129:^
7037:σ
7027:σ
7013:σ
7004:ρ
6967:σ
6954:ρ
6950:−
6925:σ
6907:σ
6901:−
6887:σ
6869:σ
6841:¯
6823:¯
6814:−
6797:σ
6787:σ
6781:ρ
6772:¯
6754:¯
6745:−
6723:σ
6710:σ
6698:^
6602:
6563:^
6550:−
6483:−
6457:−
6379:−
6304:−
6265:¯
6256:−
6241:¯
6232:−
6220:
6211:−
6192:¯
6183:−
6168:¯
6159:−
6147:
6115:¯
6106:−
6088:¯
6079:−
6070:−
6061:¯
6052:−
6034:
6028:−
5999:¯
5990:−
5975:−
5969:^
5954:
5948:−
5927:⏟
5907:¯
5898:−
5883:−
5877:^
5862:
5823:¯
5814:−
5805:−
5796:¯
5787:−
5766:−
5760:^
5745:
5713:−
5707:^
5689:−
5683:^
5668:
5597:−
5564:^
5543:, we get
5415:¯
5406:−
5391:¯
5382:−
5370:
5335:¯
5326:−
5320:^
5302:¯
5293:−
5287:^
5272:
5255:^
5215:¯
5197:^
5185:
5159:^
5124:¯
5106:¯
5097:−
5083:−
5051:^
4998:−
4803:−
4708:−
4666:¯
4657:−
4642:¯
4633:−
4621:
4615:−
4596:¯
4587:−
4572:¯
4563:−
4551:
4516:¯
4507:−
4489:¯
4480:−
4471:−
4462:¯
4453:−
4435:
4406:¯
4397:−
4382:−
4376:^
4361:
4328:¯
4319:−
4259:¯
4250:−
4235:−
4229:^
4214:
4179:¯
4170:−
4161:−
4152:¯
4143:−
4125:−
4119:^
4084:¯
4066:¯
4057:−
4039:^
4001:
3989:¯
3954:
3942:¯
3907:¯
3895:−
3889:¯
3851:^
3810:
3795:^
3783:
3699:^
3644:
3593:−
3567:−
3547:^
3534:−
3391:−
3358:^
3321:¯
3303:^
3291:
3261:¯
3243:¯
3234:−
3220:−
3188:^
3027:
3015:¯
2980:
2968:¯
2931:−
2875:¯
2863:−
2857:¯
2654:^
2593:∣
2584:
2480:∣
2471:
2370:∣
2361:
2191:^
2175:−
2163:^
2147:
2119:^
2090:−
2078:^
1995:−
1981:∗
1962:
1933:∞
1899:
1885:→
1864:∣
1759:−
1747:^
1731:
1688:∗
1668:^
1623:^
1485:−
1443:−
1431:^
1366:
1348:∣
1339:
1330:
1300:^
1287:
1150:−
1144:^
1126:−
1120:^
1105:
1049:^
989:∣
980:
953:^
905:
898:^
864:^
807:
784:∑
758:
725:
714:
681:
670:
626:−
620:^
603:
482:−
476:^
451:−
445:^
430:
400:−
394:^
376:−
370:^
355:
344:
321:of error
298:−
292:^
208:^
193:estimator
176:×
130:×
54:. In the
24382:cite web
24257:See also
20845:we have
19495:, where
19232:‖
19226:‖
17406:Examples
17121:; while
14799:, where
10181:and the
7904:, where
7062:between
2055:for all
1802:for all
1453:→
1213:matrix,
313:and its
80:Bayesian
56:Bayesian
24393:value (
17211:matrix
17165:is the
17035:is the
8746:matrix
7667:. Here
7058:is the
1589:: When
1552:is the
572:. When
24555:
24536:
24517:
24498:
24479:
24475:–350.
24452:
24433:
24410:
24389:Check
24280:(MVUE)
19866:where
19574:scalar
19034:0.5714
18962:0.1429
18954:0.2857
18939:0.2857
16794:where
15958:where
15120:where
10978:given
10055:where
9525:Since
6996:where
3928:where
3121:, the
2954:where
2420:given
1523:where
889:argmin
24332:9 May
24310:Notes
21147:LMMSE
19329:.2857
19168:.5714
19132:0.142
19027:0.142
18949:0.142
18934:0.428
18919:0.142
8871:<
7853:is a
7362:When
6593:LMMSE
3635:LMMSE
319:trace
165:be a
119:be a
24553:ISBN
24534:ISBN
24515:ISBN
24496:ISBN
24477:ISBN
24450:ISBN
24431:ISBN
24408:ISBN
24395:help
24374:2013
24334:2017
23727:and
23661:and
23607:and
22152:and
19906:and
19147:and
19094:2.57
19017:2.57
18929:1.71
18911:1.71
18903:4.85
17459:and
17321:and
16909:and
15643:and
15582:and
15398:-by-
15354:The
14966:as:
13151:by
13078:and
13009:and
12940:and
12871:and
12761:and
11555:and
10998:as
10698:The
9791:and
9418:and
8742:-by-
8686:and
7629:and
7432:and
7082:and
6661:and
5516:and
4826:The
3975:and
3749:and
3477:and
3101:and
3051:the
2814:and
2770:and
2730:and
2643:s.t.
2267:are
2247:and
2170:MMSE
2085:MMSE
2059:and
1927:<
1754:MMSE
1675:MMSE
1585:The
1438:MMSE
1307:MMSE
960:MMSE
871:MMSE
99:Let
66:and
44:MMSE
38:, a
34:and
24473:344
22179:as
21721:12.
20397:as
19237:min
17689:cov
15394:is
15186:is
14857:is
14826:is
13036:by
12898:by
10377:as
9662:as
9572:as
7102:.
4964:as
4763:as
2638:MSE
2623:min
2227:If
1556:of
908:MSE
335:MSE
232:of
30:In
24573::
24386::
24384:}}
24380:{{
24325:.
20312:11
20064:11
19784:3.
19392:.
19304:15
18995:10
18744:10
17913:.
17824:15
17819:10
17802:10
17402:.
15629:.
12839:.
9787:,
7861:.
6599:tr
3641:tr
3497:.
3168:.
3001:,
2460:,
2209:0.
722:tr
667:tr
341:tr
70:.
24561:.
24542:.
24523:.
24504:.
24485:.
24458:.
24439:.
24416:.
24397:)
24376:.
24336:.
24241:.
24233:2
24228:X
24219:/
24215:1
24212:+
24207:2
24200:j
24196:Z
24186:/
24180:2
24175:j
24171:a
24165:j
24153:2
24146:i
24142:Z
24132:/
24126:i
24122:a
24115:=
24110:i
24106:w
24092:i
24073:2
24069:y
24063:2
24059:w
24055:+
24050:1
24046:y
24040:1
24036:w
24032:=
24029:y
24006:0
24003:=
24000:}
23995:2
23991:y
23987:{
23981:E
23978:=
23975:}
23970:1
23966:y
23962:{
23956:E
23929:.
23924:2
23920:z
23916:+
23913:x
23908:2
23904:a
23900:=
23891:2
23887:y
23877:1
23873:z
23869:+
23866:x
23861:1
23857:a
23853:=
23844:1
23840:y
23809:.
23804:2
23799:X
23774:x
23752:2
23745:2
23741:Z
23713:2
23706:1
23702:Z
23674:2
23670:z
23647:1
23643:z
23620:2
23616:a
23593:1
23589:a
23560:.
23552:2
23547:X
23538:/
23534:1
23531:+
23526:2
23519:j
23515:Z
23505:/
23501:1
23496:N
23491:1
23488:=
23485:j
23476:1
23471:=
23467:E
23464:S
23461:M
23458:M
23455:L
23429:2
23424:X
23415:/
23411:1
23408:+
23403:2
23396:j
23392:Z
23382:/
23378:1
23373:N
23368:1
23365:=
23362:j
23350:2
23343:i
23339:Z
23329:/
23325:1
23319:=
23314:i
23310:w
23299:i
23285:,
23276:x
23270:+
23267:)
23258:x
23247:i
23243:y
23239:(
23234:i
23230:w
23224:N
23219:1
23216:=
23213:i
23205:=
23196:x
23173:N
23150:.
23142:2
23137:X
23128:/
23124:1
23121:+
23116:2
23109:2
23105:Z
23095:/
23091:1
23088:+
23083:2
23076:1
23072:Z
23062:/
23058:1
23054:1
23049:=
23044:2
23034:X
23019:2
23014:X
23006:=
23002:E
22999:S
22996:M
22993:M
22990:L
22966:.
22961:2
22956:X
22929:2
22919:X
22889:,
22884:2
22879:X
22866:2
22861:X
22852:/
22848:1
22845:+
22840:2
22833:2
22829:Z
22819:/
22815:1
22812:+
22807:2
22800:1
22796:Z
22786:/
22782:1
22775:2
22768:2
22764:Z
22754:/
22750:1
22747:+
22742:2
22735:1
22731:Z
22721:/
22717:1
22711:=
22706:2
22696:X
22660:x
22630:.
22622:2
22617:X
22608:/
22604:1
22601:+
22596:2
22589:2
22585:Z
22575:/
22571:1
22568:+
22563:2
22556:1
22552:Z
22542:/
22538:1
22531:2
22524:2
22520:Z
22510:/
22506:1
22500:=
22491:2
22487:w
22479:,
22471:2
22466:X
22457:/
22453:1
22450:+
22445:2
22438:2
22434:Z
22424:/
22420:1
22417:+
22412:2
22405:1
22401:Z
22391:/
22387:1
22380:2
22373:1
22369:Z
22359:/
22355:1
22349:=
22340:1
22336:w
22305:,
22296:x
22290:+
22287:)
22278:x
22267:2
22263:y
22259:(
22254:2
22250:w
22246:+
22243:)
22234:x
22223:1
22219:y
22215:(
22210:1
22206:w
22202:=
22193:x
22165:2
22161:y
22138:1
22134:y
22113:2
22109:/
22105:1
22102:=
22093:x
22087:=
22084:}
22079:2
22075:y
22071:{
22065:E
22062:=
22059:}
22054:1
22050:y
22046:{
22040:E
22013:.
22008:2
22004:z
22000:+
21997:x
21994:=
21985:2
21981:y
21971:1
21967:z
21963:+
21960:x
21957:=
21948:1
21944:y
21913:.
21908:2
21901:2
21897:Z
21869:2
21865:z
21842:2
21838:y
21817:.
21812:2
21805:1
21801:Z
21773:1
21769:z
21746:1
21742:y
21717:/
21713:1
21710:=
21705:2
21700:X
21675:2
21671:/
21667:1
21664:=
21655:x
21632:]
21629:1
21626:,
21623:0
21620:[
21600:x
21580:.
21577:x
21571:1
21551:.
21548:]
21545:1
21542:,
21539:0
21536:[
21530:x
21502:x
21482:x
21459:,
21454:2
21449:X
21441:=
21436:2
21426:X
21396:,
21391:i
21387:y
21381:N
21376:1
21373:=
21370:i
21360:N
21357:1
21352:=
21343:x
21317:N
21294:.
21289:N
21284:2
21279:X
21267:)
21259:N
21255:/
21249:2
21244:Z
21236:+
21231:2
21226:X
21215:2
21210:Z
21198:(
21193:=
21188:2
21178:X
21163:2
21158:X
21150:=
21121:.
21116:2
21111:X
21101:)
21093:N
21089:/
21083:2
21078:Z
21070:+
21065:2
21060:X
21049:2
21044:X
21032:(
21027:=
21022:X
21019:Y
21015:C
21009:1
21001:Y
20997:C
20991:Y
20988:X
20984:C
20980:=
20975:2
20965:X
20933:.
20928:N
20922:i
20918:y
20912:N
20907:1
20904:=
20901:i
20890:=
20885:N
20881:y
20876:T
20872:1
20865:=
20856:y
20831:T
20827:]
20821:N
20817:y
20813:,
20807:,
20802:2
20798:y
20794:,
20789:1
20785:y
20781:[
20778:=
20775:y
20748:,
20739:y
20730:N
20726:/
20720:2
20715:Z
20707:+
20702:2
20697:X
20686:2
20681:X
20671:=
20661:y
20656:T
20652:1
20646:1
20638:)
20630:2
20625:X
20617:1
20612:+
20605:2
20600:Z
20592:N
20586:(
20577:2
20572:Z
20564:1
20559:=
20549:y
20546:I
20539:2
20534:Z
20526:1
20519:T
20515:1
20509:1
20501:)
20493:2
20488:X
20480:1
20475:+
20472:1
20469:I
20462:2
20457:Z
20449:1
20442:T
20438:1
20433:(
20428:=
20415:x
20385:W
20358:.
20355:y
20350:1
20343:)
20339:I
20334:2
20329:Z
20321:+
20316:T
20306:2
20301:X
20293:(
20288:T
20284:1
20278:2
20273:X
20265:=
20255:y
20250:1
20242:Y
20238:C
20232:Y
20229:X
20225:C
20221:=
20208:x
20171:.
20166:T
20162:1
20156:2
20151:X
20143:=
20140:}
20135:T
20131:y
20127:x
20124:{
20118:E
20115:=
20110:Y
20107:X
20103:C
20094:,
20091:I
20086:2
20081:Z
20073:+
20068:T
20058:2
20053:X
20045:=
20042:}
20037:T
20033:y
20029:y
20026:{
20020:E
20017:=
20012:Y
20008:C
19999:,
19996:0
19993:=
19990:}
19987:y
19984:{
19978:E
19950:0
19947:=
19942:Z
19939:X
19935:C
19914:z
19894:x
19874:I
19854:)
19851:I
19846:2
19841:Z
19833:,
19830:0
19827:(
19824:N
19804:z
19780:/
19774:2
19769:0
19765:x
19761:=
19756:2
19751:X
19726:x
19706:]
19701:0
19697:x
19693:,
19688:0
19684:x
19677:[
19653:x
19633:x
19613:]
19608:0
19604:x
19600:,
19595:0
19591:x
19584:[
19560:1
19538:T
19534:]
19530:1
19527:,
19521:,
19518:1
19515:,
19512:1
19509:[
19506:=
19503:1
19483:z
19480:+
19477:x
19474:1
19471:=
19468:y
19448:x
19428:N
19408:y
19376:W
19354:Y
19350:C
19326:=
19321:X
19318:Y
19314:C
19310:W
19301:=
19296:X
19293:Y
19289:C
19285:W
19279:]
19274:4
19270:z
19264:4
19260:z
19256:[
19250:E
19247:=
19242:2
19229:e
19202:4
19192:z
19165:=
19160:3
19156:w
19135:,
19126:=
19121:2
19117:w
19097:,
19091:=
19086:1
19082:w
19058:.
19053:T
19049:W
19045:=
19041:]
19010:[
19006:=
19002:]
18988:9
18981:4
18974:[
18969:]
18896:[
18892:=
18887:X
18884:Y
18880:C
18874:1
18866:Y
18862:C
18836:Y
18832:C
18809:X
18806:Y
18802:C
18798:=
18793:T
18789:W
18783:Y
18779:C
18755:.
18751:]
18737:9
18730:4
18723:[
18719:=
18715:]
18708:]
18703:3
18699:z
18695:,
18690:4
18686:z
18682:[
18679:E
18672:]
18667:2
18663:z
18659:,
18654:4
18650:z
18646:[
18643:E
18636:]
18631:1
18627:z
18623:,
18618:4
18614:z
18610:[
18607:E
18600:[
18596:=
18591:X
18588:Y
18584:C
18558:X
18555:Y
18551:C
18527:.
18523:]
18516:6
18511:8
18506:3
18499:8
18494:5
18489:2
18482:3
18477:2
18472:1
18465:[
18461:=
18457:]
18450:]
18445:3
18441:z
18437:,
18432:3
18428:z
18424:[
18421:E
18416:]
18411:3
18407:z
18403:,
18398:2
18394:z
18390:[
18387:E
18382:]
18377:3
18373:z
18369:,
18364:1
18360:z
18356:[
18353:E
18346:]
18341:2
18337:z
18333:,
18328:3
18324:z
18320:[
18317:E
18312:]
18307:2
18303:z
18299:,
18294:2
18290:z
18286:[
18283:E
18278:]
18273:2
18269:z
18265:,
18260:1
18256:z
18252:[
18249:E
18242:]
18237:1
18233:z
18229:,
18224:3
18220:z
18216:[
18213:E
18208:]
18203:1
18199:z
18195:,
18190:2
18186:z
18182:[
18179:E
18174:]
18169:1
18165:z
18161:,
18156:1
18152:z
18148:[
18145:E
18138:[
18134:=
18129:Y
18125:C
18099:Y
18095:C
18072:4
18068:z
18064:=
18061:x
18041:]
18036:3
18032:w
18028:,
18023:2
18019:w
18015:,
18010:1
18006:w
18002:[
17999:=
17996:W
17974:T
17970:]
17964:3
17960:z
17956:,
17951:2
17947:z
17943:,
17938:1
17934:z
17930:[
17927:=
17924:y
17899:4
17889:z
17863:i
17859:w
17835:,
17831:]
17814:9
17809:4
17797:6
17792:8
17787:3
17780:9
17775:8
17770:5
17765:2
17758:4
17753:3
17748:2
17743:1
17736:[
17732:=
17729:]
17724:T
17720:z
17716:z
17713:[
17707:E
17704:=
17701:)
17698:Z
17695:(
17664:T
17660:]
17654:4
17650:z
17646:,
17641:3
17637:z
17633:,
17628:2
17624:z
17620:,
17615:1
17611:z
17607:[
17604:=
17601:z
17579:i
17575:z
17569:i
17565:w
17559:3
17554:1
17551:=
17548:i
17540:=
17535:4
17525:z
17499:4
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17472:3
17468:z
17445:2
17441:z
17437:,
17432:1
17428:z
17388:1
17385:+
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17372:x
17365:=
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17357:m
17354:(
17349:1
17346:+
17343:k
17333:x
17305:1
17302:+
17299:k
17295:e
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17286:=
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17275:(
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17063:m
17021:)
17015:(
17008:1
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17002:k
16998:Z
16993:C
16970:k
16960:x
16953:=
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16945:0
16942:(
16937:1
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16931:k
16921:x
16893:k
16889:e
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16880:=
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16872:0
16869:(
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16859:+
16856:k
16852:e
16847:C
16826:m
16823:,
16817:,
16814:2
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16808:1
16805:=
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16774:)
16771:1
16762:(
16757:k
16747:x
16738:)
16732:(
16727:1
16724:+
16721:k
16717:A
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16667:1
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16645:1
16636:(
16631:k
16621:x
16614:=
16609:)
16603:(
16598:1
16595:+
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16582:x
16555:)
16549:(
16542:k
16538:e
16533:C
16529:)
16524:)
16518:(
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16507:k
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16497:)
16491:(
16486:1
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16480:k
16476:w
16469:I
16466:(
16463:=
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16452:(
16445:1
16442:+
16439:k
16435:e
16430:C
16405:)
16400:T
16397:)
16391:(
16386:1
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16376:A
16372:(
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16286:Z
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16273:T
16270:)
16264:(
16259:1
16256:+
16253:k
16249:A
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16237:(
16230:k
16226:e
16221:C
16214:=
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16203:(
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16188:w
16163:m
16143:1
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16117:y
16095:Z
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16060:k
16050:y
16041:k
16037:a
16030:}
16025:k
16015:y
16006:k
16002:a
15998:{
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15943:,
15940:}
15935:k
15931:a
15925:k
15915:y
15908:{
15904:E
15898:k
15890:+
15885:k
15875:x
15868:=
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15860:+
15857:k
15847:x
15820:.
15817:}
15814:a
15805:y
15799:{
15795:E
15791:2
15785:=
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15777:2
15767:y
15760:{
15756:E
15745:x
15718:}
15709:y
15701:T
15691:y
15684:{
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15617:x
15595:e
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15564:x
15539:1
15536:+
15533:k
15529:w
15505:.
15498:k
15494:e
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15485:)
15480:T
15475:1
15472:+
15469:k
15465:a
15459:1
15456:+
15453:k
15449:w
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15439:(
15436:=
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15426:+
15423:k
15419:e
15414:C
15400:n
15396:n
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15372:k
15368:e
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15339:.
15331:1
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15325:k
15321:a
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15293:1
15290:+
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15269:1
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15263:k
15251:1
15248:+
15245:k
15241:a
15233:k
15229:e
15224:C
15217:=
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15206:k
15202:w
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15169:+
15166:k
15162:w
15139:1
15136:+
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15129:y
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15100:k
15090:x
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15076:1
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15070:k
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15057:1
15054:+
15051:k
15047:y
15043:(
15038:1
15035:+
15032:k
15028:w
15024:+
15019:k
15009:x
15002:=
14997:1
14994:+
14991:k
14981:x
14952:1
14949:+
14946:k
14936:x
14922:k
14906:2
14901:k
14874:k
14870:z
14859:n
14843:k
14839:x
14828:n
14812:k
14808:a
14785:k
14781:z
14777:+
14772:k
14768:x
14762:T
14757:k
14753:a
14749:=
14744:k
14740:y
14729:k
14700:,
14697:)
14692:1
14686:k
14676:x
14669:A
14661:k
14657:y
14653:(
14648:k
14644:W
14640:+
14635:1
14629:k
14619:x
14612:=
14607:k
14597:x
14572:,
14567:1
14559:Z
14555:C
14549:T
14545:A
14537:k
14533:e
14528:C
14524:=
14519:k
14515:W
14493:,
14490:A
14485:1
14477:Z
14473:C
14467:T
14463:A
14459:+
14454:1
14444:1
14438:k
14434:e
14429:C
14425:=
14420:1
14410:k
14406:e
14401:C
14375:k
14371:W
14347:.
14340:1
14334:k
14330:e
14325:C
14321:)
14318:A
14313:k
14309:W
14302:I
14299:(
14296:=
14289:k
14285:e
14280:C
14258:,
14255:)
14250:1
14244:k
14234:x
14227:A
14219:k
14215:y
14211:(
14206:k
14202:W
14198:+
14193:1
14187:k
14177:x
14170:=
14165:k
14155:x
14130:,
14125:1
14118:)
14112:Z
14108:C
14104:+
14099:T
14095:A
14087:1
14081:k
14077:e
14072:C
14068:A
14065:(
14060:T
14056:A
14048:1
14042:k
14038:e
14033:C
14029:=
14024:k
14020:W
13991:1
13987:Y
13964:1
13954:x
13927:.
13920:1
13914:k
13910:e
13905:C
13901:A
13896:1
13889:)
13883:Z
13879:C
13875:+
13870:T
13866:A
13858:1
13852:k
13848:e
13843:C
13839:A
13836:(
13831:T
13827:A
13819:1
13813:k
13809:e
13804:C
13793:1
13787:k
13783:e
13778:C
13774:=
13767:k
13763:e
13758:C
13727:)
13722:1
13716:k
13706:x
13699:A
13691:k
13687:y
13683:(
13678:1
13671:)
13665:Z
13661:C
13657:+
13652:T
13648:A
13640:1
13634:k
13630:e
13625:C
13621:A
13618:(
13613:T
13609:A
13601:1
13595:k
13591:e
13586:C
13582:+
13577:1
13571:k
13561:x
13554:=
13544:)
13539:k
13529:y
13517:k
13513:y
13509:(
13504:1
13494:k
13484:Y
13476:C
13468:k
13458:Y
13449:1
13443:k
13439:e
13434:C
13430:+
13425:1
13419:k
13409:x
13402:=
13393:k
13383:x
13350:k
13346:y
13318:.
13313:T
13309:A
13301:1
13295:k
13291:e
13286:C
13282:=
13275:k
13265:Y
13256:1
13250:k
13246:e
13241:C
13237:=
13226:1
13220:k
13216:Y
13212:,
13206:,
13201:1
13197:Y
13192:|
13186:k
13182:Y
13176:k
13172:X
13167:C
13137:Y
13134:X
13130:C
13105:k
13095:Y
13087:C
13064:1
13058:k
13048:y
13022:Y
13018:C
12991:y
12964:1
12958:k
12954:e
12949:C
12926:1
12920:k
12910:x
12884:X
12880:C
12853:x
12823:k
12819:X
12814:|
12808:k
12804:Y
12799:C
12795:=
12788:k
12778:Y
12770:C
12749:0
12746:=
12743:]
12738:k
12728:y
12721:[
12717:E
12694:k
12684:y
12672:k
12668:y
12664:=
12659:k
12649:y
12623:k
12619:y
12596:1
12590:k
12580:x
12573:A
12570:=
12565:k
12555:y
12529:k
12525:Y
12512:.
12496:.
12491:Z
12487:C
12483:+
12478:T
12474:A
12466:1
12460:k
12456:e
12451:C
12447:A
12444:=
12439:Z
12435:C
12431:+
12426:T
12422:A
12414:1
12408:k
12404:Y
12400:,
12394:,
12389:1
12385:Y
12380:|
12374:k
12370:X
12365:C
12361:A
12358:=
12347:k
12343:X
12338:|
12332:k
12328:Y
12323:C
12293:1
12287:k
12277:x
12270:A
12267:=
12262:k
12252:x
12245:A
12242:=
12237:k
12227:y
12203:)
12198:k
12194:x
12189:|
12183:k
12179:y
12175:(
12172:p
12146:,
12139:1
12133:k
12129:e
12124:C
12120:=
12113:1
12107:k
12103:Y
12099:,
12093:,
12088:1
12084:Y
12079:|
12073:1
12067:k
12063:X
12058:C
12054:=
12047:1
12041:k
12037:Y
12033:,
12027:,
12022:1
12018:Y
12013:|
12007:k
12003:X
11998:C
11982:,
11968:1
11962:k
11952:x
11945:=
11942:]
11937:1
11931:k
11927:y
11923:,
11917:,
11912:1
11908:y
11903:|
11897:1
11891:k
11887:x
11883:[
11879:E
11875:=
11872:]
11867:1
11861:k
11857:y
11853:,
11847:,
11842:1
11838:y
11833:|
11827:k
11823:x
11819:[
11815:E
11811:=
11806:k
11796:x
11769:)
11764:1
11758:k
11754:y
11750:,
11744:,
11739:1
11735:y
11730:|
11724:k
11720:x
11716:(
11713:p
11690:)
11685:k
11681:x
11676:|
11670:k
11666:y
11662:(
11659:p
11639:)
11634:1
11628:k
11624:y
11620:,
11614:,
11609:1
11605:y
11600:|
11594:k
11590:x
11586:(
11583:p
11563:Y
11543:X
11523:.
11514:x
11508:+
11505:)
11496:y
11487:y
11484:(
11479:1
11471:Y
11467:C
11461:Y
11458:X
11454:C
11450:=
11441:x
11425:k
11421:k
11404:.
11401:)
11396:1
11390:k
11386:y
11382:,
11376:,
11371:1
11367:y
11362:|
11356:1
11350:k
11346:x
11342:(
11339:p
11336:=
11333:)
11328:1
11322:k
11318:y
11314:,
11308:,
11303:1
11299:y
11294:|
11288:k
11284:x
11280:(
11277:p
11254:x
11234:)
11229:k
11225:y
11221:,
11215:,
11210:1
11206:y
11201:|
11195:k
11191:x
11187:(
11184:p
11174:k
11158:k
11148:x
11118:.
11115:)
11110:k
11106:x
11101:|
11095:k
11091:y
11087:(
11084:p
11081:=
11078:)
11073:1
11067:k
11063:y
11059:,
11053:,
11048:1
11044:y
11040:,
11035:k
11031:x
11026:|
11020:k
11016:y
11012:(
11009:p
10986:x
10964:1
10958:k
10954:y
10950:,
10944:,
10939:1
10935:y
10912:k
10908:y
10897:k
10883:)
10878:1
10872:k
10868:y
10864:,
10858:,
10853:1
10849:y
10844:|
10838:k
10834:x
10830:(
10827:p
10807:)
10802:k
10798:x
10793:|
10787:k
10783:y
10779:(
10776:p
10756:)
10751:k
10747:y
10743:,
10737:,
10732:1
10728:y
10723:|
10717:k
10713:x
10709:(
10706:p
10679:.
10676:)
10671:1
10665:k
10661:y
10657:,
10651:,
10646:1
10642:y
10637:|
10631:k
10627:x
10623:(
10620:p
10617:)
10612:k
10608:x
10603:|
10597:k
10593:y
10589:(
10586:p
10583:=
10573:)
10568:1
10562:k
10558:y
10554:,
10548:,
10543:1
10539:y
10534:|
10528:k
10524:x
10520:(
10517:p
10514:)
10509:1
10503:k
10499:y
10495:,
10489:,
10484:1
10480:y
10476:,
10473:x
10469:|
10463:k
10459:y
10455:(
10452:p
10442:)
10437:k
10433:y
10429:,
10423:,
10418:1
10414:y
10409:|
10403:k
10399:x
10395:(
10392:p
10363:k
10359:x
10336:k
10332:y
10328:,
10322:,
10317:1
10313:y
10292:k
10258:T
10254:A
10248:1
10241:)
10237:I
10231:+
10228:A
10223:T
10219:A
10215:(
10212:=
10209:W
10189:z
10169:I
10163:=
10158:1
10150:X
10146:C
10123:T
10119:A
10113:1
10106:)
10102:A
10097:T
10093:A
10089:(
10086:=
10083:W
10063:I
10043:,
10040:I
10035:2
10027:=
10022:Z
10018:C
9997:z
9975:1
9967:Z
9963:C
9940:1
9932:Z
9928:C
9922:T
9918:A
9912:1
9905:)
9901:A
9896:1
9888:Z
9884:C
9878:T
9874:A
9870:(
9867:=
9864:W
9844:x
9824:0
9821:=
9816:1
9808:X
9804:C
9768:.
9759:x
9753:+
9750:)
9741:x
9735:A
9729:y
9726:(
9721:1
9713:Z
9709:C
9703:T
9699:A
9693:e
9689:C
9685:=
9676:x
9644:x
9619:1
9611:Z
9607:C
9601:T
9597:A
9591:e
9587:C
9583:=
9580:W
9558:e
9554:C
9533:W
9510:.
9505:1
9498:)
9492:1
9484:X
9480:C
9476:+
9473:A
9468:1
9460:Z
9456:C
9450:T
9446:A
9442:(
9439:=
9434:e
9430:C
9403:,
9398:1
9390:Z
9386:C
9380:T
9376:A
9370:1
9363:)
9357:1
9349:X
9345:C
9341:+
9338:A
9333:1
9325:Z
9321:C
9315:T
9311:A
9307:(
9304:=
9301:W
9278:,
9275:)
9270:1
9262:X
9258:C
9254:+
9251:A
9246:1
9238:Z
9234:C
9228:T
9224:A
9220:(
9200:)
9195:Z
9191:C
9187:+
9182:T
9178:A
9172:X
9168:C
9164:A
9161:(
9138:,
9133:1
9125:Z
9121:C
9115:T
9111:A
9105:1
9098:)
9092:1
9084:X
9080:C
9076:+
9073:A
9068:1
9060:Z
9056:C
9050:T
9046:A
9042:(
9039:=
9034:1
9027:)
9021:Z
9017:C
9013:+
9008:T
9004:A
8998:X
8994:C
8990:A
8987:(
8982:T
8978:A
8972:X
8968:C
8934:T
8930:A
8924:X
8920:C
8916:A
8896:0
8893:=
8888:Z
8884:C
8873:n
8869:m
8854:x
8832:Z
8828:C
8817:m
8801:1
8794:)
8788:Z
8784:C
8780:+
8775:T
8771:A
8765:X
8761:C
8757:A
8754:(
8744:m
8740:m
8726:x
8716:n
8702:y
8692:m
8667:.
8662:X
8658:C
8654:A
8649:1
8642:)
8636:Z
8632:C
8628:+
8623:T
8619:A
8613:X
8609:C
8605:A
8602:(
8597:T
8593:A
8587:X
8583:C
8574:X
8570:C
8566:=
8556:X
8550:C
8541:X
8537:C
8533:=
8528:e
8524:C
8497:.
8488:x
8482:+
8479:)
8470:x
8464:A
8458:y
8455:(
8450:1
8443:)
8437:Z
8433:C
8429:+
8424:T
8420:A
8414:X
8410:C
8406:A
8403:(
8398:T
8394:A
8388:X
8384:C
8380:=
8371:x
8339:x
8313:.
8308:1
8301:)
8295:Z
8291:C
8287:+
8282:T
8278:A
8272:X
8268:C
8264:A
8261:(
8256:T
8252:A
8246:X
8242:C
8238:=
8235:W
8212:W
8189:.
8184:T
8180:A
8174:X
8170:C
8166:=
8161:Y
8158:X
8154:C
8132:,
8127:Z
8123:C
8119:+
8114:T
8110:A
8104:X
8100:C
8096:A
8093:=
8088:Y
8084:C
8062:,
8053:x
8047:A
8044:=
8041:}
8038:y
8035:{
8029:E
8006:0
8003:=
7998:Z
7995:X
7991:C
7970:0
7967:=
7964:}
7961:z
7958:{
7952:E
7932:z
7912:A
7892:z
7889:+
7886:x
7883:A
7880:=
7877:y
7841:y
7815:Y
7811:C
7778:W
7756:Y
7752:C
7727:W
7695:y
7675:x
7655:0
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1324:=
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1290:{
1284:E
1263:.
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1199:Y
1195:|
1191:X
1187:C
1166:}
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1157:)
1153:x
1141:x
1135:(
1132:)
1129:x
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1111:(
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744:}
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637:2
633:)
629:x
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407:)
403:x
391:x
385:(
382:)
379:x
367:x
361:(
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348:{
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289:x
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240:x
220:)
217:y
214:(
205:x
179:1
173:m
153:y
133:1
127:n
107:x
42:(
20:)
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