7237:, syllables, or whole-word units. A lexicon decoding system is applied to constrain the paths investigated, so only words in the system's lexicon (word dictionary) are investigated. Similar to the lexicon decoding, the system path is further constrained by the rules of grammar and syntax. Finally, semantic analysis is applied and the system outputs the recognized utterance. A limitation of many HMM applications to speech recognition is that the current state only depends on the state at the previous time-step, which is unrealistic for speech as dependencies are often several time-steps in duration. The Baum–Welch algorithm also has extensive applications in solving HMMs used in the field of speech synthesis.
2906:
7331:(CNVs) are an abundant form of genome structure variation in humans. A discrete-valued bivariate HMM (dbHMM) was used assigning chromosomal regions to seven distinct states: unaffected regions, deletions, duplications and four transition states. Solving this model using Baum-Welch demonstrated the ability to predict the location of CNV breakpoint to approximately 300 bp from
7320:. While most integrated gene-finding software (at the time of GENSCANs release) assumed input sequences contained exactly one gene, GENSCAN solves a general case where partial, complete, or multiple genes (or even no gene at all) is present. GENSCAN was shown to exactly predict exon location with 90% accuracy with 80% specificity compared to an annotated database.
4925:
chicken lays eggs, and that this state only depends on the state on the previous day. Now we don't know the state at the initial starting point, we don't know the transition probabilities between the two states and we don't know the probability that the chicken lays an egg given a particular state. To start we first guess the transition and emission matrices.
2482:
2392:
7253:. HMMs and as a consequence the Baum–Welch algorithm have also been used to identify spoken phrases in encrypted VoIP calls. In addition HMM cryptanalysis is an important tool for automated investigations of cache-timing data. It allows for the automatic discovery of critical algorithm state, for example key values.
7232:
in 1975. Continuous speech recognition occurs by the following steps, modeled by a HMM. Feature analysis is first undertaken on temporal and/or spectral features of the speech signal. This produces an observation vector. The feature is then compared to all sequences of the speech recognition units.
4924:
Suppose we have a chicken from which we collect eggs at noon every day. Now whether or not the chicken has laid eggs for collection depends on some unknown factors that are hidden. We can however (for simplicity) assume that the chicken is always in one of two states that influence whether the
4781:
4557:
2901:{\displaystyle \xi _{ij}(t)=P(X_{t}=i,X_{t+1}=j\mid Y,\theta )={\frac {P(X_{t}=i,X_{t+1}=j,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)a_{ij}\beta _{j}(t+1)b_{j}(y_{t+1})}{\sum _{k=1}^{N}\sum _{w=1}^{N}\alpha _{k}(t)a_{kw}\beta _{w}(t+1)b_{w}(y_{t+1})}},}
3640:
2144:
4911:
3761:
7316:(1 Mbp) long. GENSCAN utilizes a general inhomogeneous, three periodic, fifth order Markov model of DNA coding regions. Additionally, this model accounts for differences in gene density and structure (such as intron lengths) that occur in different
3426:
5241:
1818:
1313:
1627:
3958:
2128:
4563:
1104:
4364:
7295:
and accuracy compared with its predecessors with regard to predicting translation initiation sites, demonstrating an average 99% accuracy in locating 3' locations compared to confirmed genes in prokaryotes.
978:
8167:
8162:
8157:
4370:
1178:
with random initial conditions. They can also be set using prior information about the parameters if it is available; this can speed up the algorithm and also steer it toward the desired local maximum.
673:
350:
1493:
2387:{\displaystyle \gamma _{i}(t)=P(X_{t}=i\mid Y,\theta )={\frac {P(X_{t}=i,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)\beta _{i}(t)}{\sum _{j=1}^{N}\alpha _{j}(t)\beta _{j}(t)}},}
6951:
6956:
This allows us to calculate the emission matrix as described above in the algorithm, by adding up the probabilities for the respective observed sequences. We then repeat for if N came from
3473:
6093:
7677:
4792:
3236:
1632:
Since this series converges exponentially to zero, the algorithm will numerically underflow for longer sequences. However, this can be avoided in a slightly modified algorithm by scaling
4274:
3651:
1372:
4022:
1870:
7249:. In data security an observer would like to extract information from a data stream without knowing all the parameters of the transmission. This can involve reverse engineering a
440:
229:
8087:; Urban, Alexander; Grubert, Fabien; Du, Jiang; Royce, Thomas; Starr, Peter; Zhong, Guoneng; Emanuel, Beverly; Weissman, Sherman; Snyder, Michael; Gerstein, Marg (12 June 2007).
4068:
1176:
1025:
811:
3812:
1994:
4167:
7777:
Tokuda, Keiichi; Yoshimura, Takayoshi; Masuko, Takashi; Kobayashi, Takao; Kitamura, Tadashi (2000). "Speech
Parameter Generation Algorithms for HMM-Based Speech Synthesis".
3074:
4209:
1946:
3113:
3287:
7811:
Wright, Charles; Ballard, Lucas; Coull, Scott; Monrose, Fabian; Masson, Gerald (2008). "Spot me if you can: Uncovering spoken phrases in encrypted VoIP conversations".
756:
3176:
3153:
3035:
2475:
1650:
1124:
570:
1670:
7208:
7177:
7014:
6983:
6916:
6865:
6834:
6799:
6768:
6728:
6697:
6662:
6631:
6591:
6560:
6525:
6494:
6451:
6277:
6246:
6215:
6184:
6153:
6122:
6058:
6027:
5978:
5947:
5904:
5873:
5830:
5799:
5756:
5725:
5682:
5651:
5608:
5577:
5534:
5503:
5460:
5429:
5386:
5355:
5309:
5278:
5105:
5074:
3839:
865:
838:
730:
703:
517:
470:
134:
4108:
2995:
379:
4128:
4088:
3879:
3859:
3279:
3259:
3133:
3015:
2969:
2949:
2929:
2455:
2435:
2415:
1910:
1890:
1412:
1392:
537:
490:
249:
174:
154:
5112:
1683:
1190:
7514:
Jelinek, Frederick; Bahl, Lalit R.; Mercer, Robert L. (May 1975). "Design of a linguistic statistical decoder for the recognition of continuous speech".
1499:
4776:{\displaystyle b_{i}^{*}(v_{k})={\frac {\sum _{r=1}^{R}\sum _{t=1}^{T}1_{y_{tr}=v_{k}}\gamma _{ir}(t)}{\sum _{r=1}^{R}\sum _{t=1}^{T}\gamma _{ir}(t)}},}
7245:
The Baum–Welch algorithm is often used to estimate the parameters of HMMs in deciphering hidden or noisy information and consequently is often used in
3893:
2000:
64:
1030:
7644:
99:− 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state.
8149:
4282:
7313:
7619:
492:
possible values. We also assume the observation given the "hidden" state is time independent. The probability of a certain observation
8206:
873:
8163:
An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology
4552:{\displaystyle a_{ij}^{*}={\frac {\sum _{r=1}^{R}\sum _{t=1}^{T-1}\xi _{ijr}(t)}{\sum _{r=1}^{R}\sum _{t=1}^{T-1}\gamma _{ir}(t)}},}
7352:
7605:
A Gentle
Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models
36:
578:
7843:
7796:
Dingel, Janis; Hagenauer, Joachim (24 June 2007). "Parameter
Estimation of a Convolutional Encoder from Noisy Observations".
7587:
71:. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular
257:
5047:
The next step is to estimate a new transition matrix. For example, the probability of the sequence NN and the state being
1420:
7732:
Rabiner, Lawrence (February 1989). "A Tutorial on Hidden Markov Models and
Selected Applications in Speech Recognition".
8235:
3635:{\displaystyle b_{i}^{*}(v_{k})={\frac {\sum _{t=1}^{T}1_{y_{t}=v_{k}}\gamma _{i}(t)}{\sum _{t=1}^{T}\gamma _{i}(t)}},}
63:. The algorithm and the Hidden Markov models were first described in a series of articles by Baum and his peers at the
6923:
4906:{\displaystyle 1_{y_{tr}=v_{k}}={\begin{cases}1&{\text{if }}y_{t,r}=v_{k},\\0&{\text{otherwise}}\end{cases}}}
6065:
7417:
3756:{\displaystyle 1_{y_{t}=v_{k}}={\begin{cases}1&{\text{if }}y_{t}=v_{k},\\0&{\text{otherwise}}\end{cases}}}
3184:
4214:
8089:"Systematic prediction and validation of breakpoints associated with copy-number variations in the human genome"
4070:. In this case, the information from all of the observed sequences must be used in the update of the parameters
8230:
7972:
7356:
1318:
8190:"Comparing and evaluating HMM ensemble training algorithms using train and test and condition number criteria"
7374:
7365:
7541:
Bishop, Martin J.; Thompson, Elizabeth A. (20 July 1986). "Maximum likelihood alignment of DNA sequences".
7400:
7390:
3975:
1823:
8158:
A Maximization
Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains
387:
179:
7651:
7430:
4027:
1137:
986:
761:
44:
7492:
6281:
transition probabilities and normalize so they add to 1. This gives us the updated transition matrix:
3769:
1954:
4133:
8012:
7998:
Burge, Chris; Karlin, Samuel (1997). "Prediction of
Complete Gene Structures in Human Genomic DNA".
7746:
4837:
3693:
3421:{\displaystyle a_{ij}^{*}={\frac {\sum _{t=1}^{T-1}\xi _{ij}(t)}{\sum _{t=1}^{T-1}\gamma _{i}(t)}},}
3043:
7411:
7339:
than previously possible, allowing the study of CNV population frequencies. It also demonstrated a
7292:
4172:
1915:
67:
in the late 1960s and early 1970s. One of the first major applications of HMMs was to the field of
3079:
7386:
8174:
The
Shannon Lecture by Welch, which speaks to how the algorithm can be implemented efficiently:
8240:
8177:
8007:
7741:
735:
75:. They have since become an important tool in the probabilistic modeling of genomic sequences.
20:
7340:
7336:
7328:
5044:
This gives us a set of observed transitions between days: NN, NN, NN, NN, NE, EE, EN, NN, NN
3161:
3138:
3020:
2460:
1635:
1109:
542:
106:
estimate of the parameters of a hidden Markov model given a set of observed feature vectors.
24:
7335:. This magnitude of resolution enables more precise correlations between different CNVs and
1655:
8100:
7186:
7155:
6992:
6961:
6894:
6843:
6812:
6777:
6746:
6706:
6675:
6640:
6609:
6569:
6538:
6503:
6472:
6429:
6255:
6224:
6193:
6162:
6131:
6100:
6036:
6005:
5956:
5925:
5882:
5851:
5808:
5777:
5734:
5703:
5660:
5629:
5586:
5555:
5512:
5481:
5438:
5407:
5364:
5333:
5287:
5256:
5083:
5052:
3817:
843:
816:
708:
681:
495:
448:
112:
8203:
7150:
To estimate the initial probabilities we assume all sequences start with the hidden state
4093:
8:
7447:
7378:
7317:
7215:
Finally we repeat these steps until the resulting probabilities converge satisfactorily.
5041:
We then take a set of observations (E = eggs, N = no eggs): N, N, N, N, N, E, E, N, N, N
2974:
358:
84:
40:
8197:
8104:
5236:{\displaystyle P(S_{1})\cdot P(N|S_{1})\cdot P(S_{1}\rightarrow S_{2})\cdot P(N|S_{2}).}
1813:{\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )}
8215:
8123:
8088:
7948:
7923:
7759:
7462:
7457:
7407:
6095:(referred to as "Pseudo probabilities" in the following tables). We then calculate the
4113:
4073:
3864:
3844:
3264:
3244:
3118:
3000:
2954:
2934:
2914:
2440:
2420:
2400:
1895:
1875:
1397:
1377:
522:
475:
234:
159:
139:
103:
8061:
8044:
7885:
7860:
7426:
8128:
8066:
8025:
7953:
7890:
7839:
7583:
7558:
7554:
7442:
3451:, but to any state including itself. This is equivalent to the number of times state
1308:{\displaystyle \alpha _{i}(t)=P(Y_{1}=y_{1},\ldots ,Y_{t}=y_{t},X_{t}=i\mid \theta )}
251:, which leads to the definition of the time-independent stochastic transition matrix
68:
8189:
7939:
7763:
7620:"A Tutorial on Hidden Markov Models and Selected Applications in Speech recognition"
8118:
8108:
8056:
8017:
7943:
7935:
7922:
Delcher, Arthur; Bratke, Kirsten A.; Powers, Edwin C.; Salzberg, Steven L. (2007).
7880:
7872:
7831:
7751:
7714:
7550:
7523:
7276:
1622:{\displaystyle \alpha _{i}(t+1)=b_{i}(y_{t+1})\sum _{j=1}^{N}\alpha _{j}(t)a_{ji}.}
7908:
7396:
8210:
8185:
An alternative to the Baum–Welch algorithm, the
Viterbi Path Counting algorithm:
7835:
7577:
3953:{\displaystyle P(Y\mid \theta _{\text{final}})>P(Y\mid \theta _{\text{true}})}
2123:{\displaystyle \beta _{i}(t)=\sum _{j=1}^{N}\beta _{j}(t+1)a_{ij}b_{j}(y_{t+1}).}
88:
56:
7579:
Biological
Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids
91:" and observed discrete random variables. It relies on the assumption that the
8168:
Statistical
Inference for Probabilistic Functions of Finite State Markov Chains
8093:
Proceedings of the National Academy of Sciences of the United States of America
7718:
7702:
7467:
7332:
7229:
3884:
These steps are now repeated iteratively until a desired level of convergence.
1099:{\displaystyle \theta ^{*}=\operatorname {arg\,max} _{\theta }P(Y\mid \theta )}
60:
28:
8224:
8084:
7527:
7472:
7250:
7246:
3115:
are the same ; they represent the probability of making the observation
8113:
8192:, Pattern Analysis and Applications, vol. 6, no. 4, pp. 327–336, 2003.
8132:
8021:
7957:
7876:
7452:
7382:
3814:
is the expected number of times the output observations have been equal to
2138:
We can now calculate the temporary variables, according to Bayes' theorem:
8070:
8029:
7894:
7562:
7779:
IEEE International Conference on Acoustics, Speech, and Signal Processing
7280:
8148:
A comprehensive review of HMM methods and software in bioinformatics –
7859:
Salzberg, Steven; Delcher, Arthur L.; Kasif, Simon; White, Owen (1998).
7607:. Berkeley, CA: International Computer Science Institute. pp. 7–13.
4359:{\displaystyle \pi _{i}^{*}={\frac {\sum _{r=1}^{R}\gamma _{ir}(1)}{R}}}
7830:. Lecture Notes in Computer Science. Vol. 5912. pp. 667–684.
7309:
8198:
An Interactive Spreadsheet for Teaching the Forward-Backward Algorithm
102:
The Baum–Welch algorithm uses the well known EM algorithm to find the
7826:
Brumley, Bob; Hakala, Risto (2009). "Cache-Timing Template Attacks".
7683:. Aeronautics and Astronautics, Massachusetts Institute of Technology
7377:
library that implements Baum-Welch on various kind of Markov Models (
7275:(Gene Locator and Interpolated Markov ModelER) software was an early
4024:. However, in many situations, there are several sequences observed:
3439:
compared to the expected total number of transitions away from state
7976:
7755:
7283:
DNA. GLIMMER uses Interpolated Markov Models (IMMs) to identify the
3972:
The algorithm described thus far assumes a single observed sequence
7371:
7234:
72:
7650:. Johns Hopkins Bloomberg School of Public Health. Archived from
7305:
7291:. The latest release (GLIMMER3) has been shown to have increased
7288:
7272:
7228:
Hidden Markov Models were first applied to speech recognition by
7861:"Microbial gene identification using interpolated Markov Models"
7924:"Identifying bacterial genes and endosymbiont DNA with Glimmer"
7421:
973:{\displaystyle Y=(Y_{1}=y_{1},Y_{2}=y_{2},\ldots ,Y_{T}=y_{T})}
7911:. Johns Hopkins University - Center for Computational Biology.
7776:
7368:
bindings that supports both discrete and continuous emissions.
7707:
IEEE Transactions on Acoustics, Speech, and Signal Processing
7212:. Again we then normalize to give an updated initial vector.
7921:
7284:
5251:
Highest probability of observing that sequence if state is
4899:
3890:
It is possible to over-fit a particular data set. That is,
3749:
7181:
and calculate the highest probability and then repeat for
7858:
7810:
7678:"Intro to Hidden Markov Models: the Baum-Welch Algorithm"
7279:
program used for the identification of coding regions in
8180:, IEEE Information Theory Society Newsletter, Dec. 2003.
3443:. To clarify, the number of transitions away from state
668:{\displaystyle b_{j}(y_{i})=P(Y_{t}=y_{i}\mid X_{t}=j).}
8200:(spreadsheet and article with step-by-step walkthrough)
3431:
which is the expected number of transitions from state
1820:
that is the probability of the ending partial sequence
55:
The Baum–Welch algorithm was named after its inventors
345:{\displaystyle A=\{a_{ij}\}=P(X_{t}=j\mid X_{t-1}=i).}
7189:
7158:
6995:
6964:
6926:
6897:
6846:
6815:
6780:
6749:
6709:
6678:
6643:
6612:
6572:
6541:
6506:
6475:
6432:
6258:
6227:
6196:
6165:
6134:
6103:
6068:
6039:
6008:
5959:
5928:
5885:
5854:
5811:
5780:
5737:
5706:
5663:
5632:
5589:
5558:
5515:
5484:
5441:
5410:
5367:
5336:
5290:
5259:
5115:
5086:
5055:
4795:
4566:
4373:
4285:
4217:
4175:
4136:
4116:
4096:
4076:
4030:
3978:
3896:
3867:
3847:
3820:
3772:
3654:
3476:
3290:
3267:
3247:
3187:
3164:
3141:
3121:
3082:
3046:
3023:
3003:
2977:
2957:
2937:
2917:
2485:
2463:
2443:
2423:
2403:
2147:
2003:
1957:
1918:
1898:
1878:
1826:
1686:
1658:
1638:
1502:
1423:
1400:
1380:
1321:
1193:
1140:
1112:
1033:
1027:. The Baum–Welch algorithm finds a local maximum for
989:
876:
846:
819:
764:
738:
711:
684:
581:
545:
525:
498:
478:
451:
390:
361:
260:
237:
182:
162:
142:
115:
8083:
7813:
IEEE International Symposium on Security and Privacy
1488:{\displaystyle \alpha _{i}(1)=\pi _{i}b_{i}(y_{1}),}
87:
describes the joint probability of a collection of "
47:
to compute the statistics for the expectation step.
7361:
1126:that maximize the probability of the observation).
7798:IEEE International Symposium on Information Theory
7323:
7202:
7171:
7008:
6977:
6945:
6910:
6859:
6828:
6793:
6762:
6722:
6691:
6656:
6625:
6585:
6554:
6519:
6488:
6445:
6271:
6240:
6209:
6178:
6147:
6116:
6087:
6052:
6021:
5972:
5941:
5898:
5867:
5824:
5793:
5750:
5719:
5676:
5645:
5602:
5571:
5528:
5497:
5454:
5423:
5380:
5349:
5303:
5272:
5235:
5099:
5068:
4905:
4775:
4551:
4358:
4268:
4203:
4161:
4122:
4102:
4082:
4062:
4016:
3952:
3873:
3853:
3833:
3806:
3755:
3634:
3420:
3273:
3253:
3230:
3170:
3147:
3127:
3107:
3068:
3029:
3009:
2989:
2963:
2943:
2923:
2900:
2469:
2449:
2429:
2409:
2386:
2122:
1988:
1940:
1904:
1884:
1864:
1812:
1664:
1644:
1621:
1487:
1406:
1386:
1366:
1307:
1170:
1118:
1098:
1019:
972:
859:
832:
805:
750:
724:
697:
667:
564:
531:
511:
484:
464:
434:
373:
344:
243:
223:
168:
148:
128:
8178:Hidden Markov Models and the Baum–Welch Algorithm
7513:
7308:webserver is a gene locator capable of analyzing
7256:
6413:Next, we want to estimate a new emission matrix,
3861:over the expected total number of times in state
65:IDA Center for Communications Research, Princeton
8222:
7795:
7341:direct inheritance pattern for a particular CNV
6946:{\displaystyle {\frac {0.2394}{0.2730}}=0.8769}
6457:Highest Probability of observing that sequence
5315:Highest Probability of observing that sequence
3447:does not mean transitions to a different state
3241:which is the expected frequency spent in state
678:Taking into account all the possible values of
7540:
6422:Highest probability of observing that sequence
983:Thus we can describe a hidden Markov chain by
8204:Formal derivation of the Baum–Welch algorithm
7852:
7825:
6330:New Transition Matrix (Pseudo Probabilities)
6088:{\displaystyle {\frac {0.22}{2.4234}}=0.0908}
1315:, the probability of seeing the observations
6371:New Transition Matrix (After Normalization)
5915:0.024 = 0.2 × 0.3 × 0.5 × 0.8
5841:0.024 = 0.2 × 0.3 × 0.5 × 0.8
5767:0.056 = 0.2 × 0.7 × 0.5 × 0.8
5693:0.014 = 0.2 × 0.7 × 0.5 × 0.2
5619:0.006 = 0.2 × 0.3 × 0.5 × 0.2
5545:0.024 = 0.2 × 0.3 × 0.5 × 0.8
5471:0.024 = 0.2 × 0.3 × 0.5 × 0.8
5397:0.024 = 0.2 × 0.3 × 0.5 × 0.8
5323:0.024 = 0.2 × 0.3 × 0.5 × 0.8
3231:{\displaystyle \pi _{i}^{*}=\gamma _{i}(1),}
800:
771:
283:
267:
8043:Burge, Christopher; Karlin, Samuel (1998).
8042:
7997:
7901:
4269:{\displaystyle y_{1,r},\ldots ,y_{N_{r},r}}
3158:The parameters of the hidden Markov model
2911:which is the probability of being in state
2397:which is the probability of being in state
8216:Implementation of the Baum–Welch algorithm
7108:New Emission Matrix (After Normalization)
355:The initial state distribution (i.e. when
156:possible values (i.e. We assume there are
136:be a discrete hidden random variable with
8122:
8112:
8060:
8011:
7947:
7884:
7745:
7705:(1975). "The DRAGON system—An overview".
7569:
2997:respectively given the observed sequence
1367:{\displaystyle y_{1},y_{2},\ldots ,y_{t}}
1058:
39:used to find the unknown parameters of a
8188:Davis, Richard I. A.; Lovell, Brian C.;
7909:"Glimmer: Microbial Gene-Finding System"
7828:Advances in Cryptology – ASIACRYPT 2009
7731:
7617:
7516:IEEE Transactions on Information Theory
6889:The new estimate for the E coming from
840:belongs to all the possible states and
8223:
7602:
7575:
8049:Current Opinion in Structural Biology
7701:
7493:"First Hand: The Hidden Markov Model"
7223:
4276:, the parameters can now be updated:
4017:{\displaystyle Y=y_{1},\ldots ,y_{N}}
3967:
1865:{\displaystyle y_{t+1},\ldots ,y_{T}}
1675:
7675:
1182:
870:An observation sequence is given by
435:{\displaystyle \pi _{i}=P(X_{1}=i).}
224:{\displaystyle P(X_{t}\mid X_{t-1})}
7618:Rabiner, Lawrence (February 1989).
7490:
4063:{\displaystyle Y_{1},\ldots ,Y_{R}}
13:
8045:"Finding the Genes in Genomic DNA"
7346:
4130:. Assuming that you have computed
1171:{\displaystyle \theta =(A,B,\pi )}
1065:
1062:
1059:
1055:
1052:
1049:
1020:{\displaystyle \theta =(A,B,\pi )}
806:{\displaystyle B=\{b_{j}(y_{i})\}}
37:expectation–maximization algorithm
14:
8252:
8142:
7970:
7645:"Baum-Welch and HMM applications"
7576:Durbin, Richard (23 April 1998).
3455:is observed in the sequence from
1672:in the backward procedure below.
867:belongs to all the observations.
8154:Early HMM publications by Baum:
7261:
7240:
7067:New Emission Matrix (Estimates)
3807:{\displaystyle b_{i}^{*}(v_{k})}
1989:{\displaystyle \beta _{i}(T)=1,}
176:states in total). We assume the
8077:
8036:
7991:
7973:"The GENSCAN Web Server at MIT"
7964:
7915:
7819:
7804:
7789:
7770:
7725:
7324:Copy-number variation detection
7218:
4162:{\displaystyle \gamma _{ir}(t)}
95:-th hidden variable given the (
7695:
7669:
7637:
7611:
7596:
7582:. Cambridge University Press.
7534:
7507:
7484:
7287:and distinguish them from the
7266:
7257:Applications in bioinformatics
6000:Thus the new estimate for the
5227:
5213:
5206:
5197:
5184:
5171:
5162:
5148:
5141:
5132:
5119:
4764:
4758:
4698:
4692:
4595:
4582:
4540:
4534:
4468:
4462:
4347:
4341:
4198:
4192:
4156:
4150:
3947:
3928:
3919:
3900:
3801:
3788:
3766:is an indicator function, and
3623:
3617:
3581:
3575:
3505:
3492:
3409:
3403:
3361:
3355:
3222:
3216:
3102:
3096:
3069:{\displaystyle \gamma _{i}(t)}
3063:
3057:
2889:
2870:
2857:
2845:
2819:
2813:
2756:
2737:
2724:
2712:
2686:
2680:
2658:
2646:
2638:
2582:
2570:
2514:
2505:
2499:
2375:
2369:
2356:
2350:
2314:
2308:
2295:
2289:
2267:
2255:
2247:
2216:
2204:
2173:
2164:
2158:
2114:
2095:
2069:
2057:
2020:
2014:
1974:
1968:
1935:
1929:
1807:
1712:
1703:
1697:
1600:
1594:
1560:
1541:
1525:
1513:
1479:
1466:
1440:
1434:
1302:
1219:
1210:
1204:
1165:
1147:
1093:
1081:
1014:
996:
967:
883:
797:
784:
659:
614:
605:
592:
426:
407:
336:
292:
218:
186:
78:
1:
8062:10.1016/s0959-440x(98)80069-9
7940:10.1093/bioinformatics/btm009
7495:. IEEE Global History Network
7478:
7299:
7022:
6987:and for if N and E came from
6424:if E is assumed to come from
6285:
4929:
4204:{\displaystyle \xi _{ijr}(t)}
1941:{\displaystyle \beta _{i}(t)}
1414:. This is found recursively:
8150:Profile Hidden Markov Models
8000:Journal of Molecular Biology
7836:10.1007/978-3-642-10366-7_39
7555:10.1016/0022-2836(86)90289-5
7543:Journal of Molecular Biology
7312:sequences up to one million
3964:guarantee a global maximum.
3108:{\displaystyle \xi _{ij}(t)}
2437:given the observed sequence
1129:
7:
7436:
7020:
6283:
5109:is given by the following,
4927:
43:(HMM). It makes use of the
10:
8257:
7719:10.1109/TASSP.1975.1162650
4919:
3960:. The algorithm also does
445:The observation variables
50:
45:forward-backward algorithm
8236:Bioinformatics algorithms
7625:. Proceedings of the IEEE
6456:
6421:
5314:
4916:is an indicator function
2133:
1106:(i.e. the HMM parameters
751:{\displaystyle N\times K}
35:is a special case of the
7603:Bilmes, Jeff A. (1998).
7528:10.1109/tit.1975.1055384
16:Algorithm in mathematics
8114:10.1073/pnas.0703834104
7734:Proceedings of the IEEE
7406:HMMFit function in the
7333:micro-array experiments
3171:{\displaystyle \theta }
3148:{\displaystyle \theta }
3030:{\displaystyle \theta }
2470:{\displaystyle \theta }
1645:{\displaystyle \alpha }
1119:{\displaystyle \theta }
565:{\displaystyle X_{t}=j}
231:is independent of time
8022:10.1006/jmbi.1997.0951
7865:Nucleic Acids Research
7329:Copy-number variations
7204:
7173:
7010:
6979:
6947:
6912:
6861:
6830:
6795:
6764:
6724:
6693:
6658:
6627:
6587:
6556:
6521:
6490:
6447:
6289:Old Transition Matrix
6273:
6242:
6211:
6180:
6149:
6118:
6089:
6054:
6023:
5974:
5943:
5900:
5869:
5826:
5795:
5752:
5721:
5678:
5647:
5604:
5573:
5530:
5499:
5456:
5425:
5382:
5351:
5305:
5274:
5237:
5101:
5070:
4907:
4777:
4744:
4723:
4645:
4624:
4553:
4520:
4493:
4445:
4418:
4360:
4327:
4270:
4205:
4163:
4124:
4104:
4084:
4064:
4018:
3954:
3875:
3855:
3835:
3808:
3757:
3636:
3606:
3534:
3422:
3392:
3341:
3275:
3255:
3232:
3172:
3149:
3129:
3109:
3070:
3031:
3011:
2991:
2965:
2945:
2925:
2902:
2802:
2781:
2471:
2451:
2431:
2411:
2388:
2339:
2124:
2046:
1990:
1942:
1906:
1886:
1866:
1814:
1666:
1665:{\displaystyle \beta }
1646:
1623:
1583:
1489:
1408:
1388:
1368:
1309:
1172:
1120:
1100:
1021:
974:
861:
834:
807:
752:
726:
699:
669:
566:
533:
513:
486:
466:
436:
375:
346:
245:
225:
170:
150:
130:
21:electrical engineering
8231:Randomized algorithms
7397:HiddenMarkovModels.jl
7233:These units could be
7205:
7203:{\displaystyle S_{2}}
7174:
7172:{\displaystyle S_{1}}
7011:
7009:{\displaystyle S_{2}}
6980:
6978:{\displaystyle S_{1}}
6948:
6913:
6911:{\displaystyle S_{1}}
6862:
6860:{\displaystyle S_{2}}
6831:
6829:{\displaystyle S_{2}}
6796:
6794:{\displaystyle S_{2}}
6765:
6763:{\displaystyle S_{1}}
6725:
6723:{\displaystyle S_{1}}
6694:
6692:{\displaystyle S_{1}}
6659:
6657:{\displaystyle S_{1}}
6628:
6626:{\displaystyle S_{1}}
6588:
6586:{\displaystyle S_{1}}
6557:
6555:{\displaystyle S_{2}}
6522:
6520:{\displaystyle S_{1}}
6491:
6489:{\displaystyle S_{2}}
6448:
6446:{\displaystyle S_{1}}
6274:
6272:{\displaystyle S_{1}}
6243:
6241:{\displaystyle S_{1}}
6212:
6210:{\displaystyle S_{2}}
6181:
6179:{\displaystyle S_{2}}
6150:
6148:{\displaystyle S_{1}}
6119:
6117:{\displaystyle S_{2}}
6090:
6055:
6053:{\displaystyle S_{2}}
6024:
6022:{\displaystyle S_{1}}
5975:
5973:{\displaystyle S_{2}}
5944:
5942:{\displaystyle S_{2}}
5901:
5899:{\displaystyle S_{2}}
5870:
5868:{\displaystyle S_{2}}
5827:
5825:{\displaystyle S_{2}}
5796:
5794:{\displaystyle S_{2}}
5753:
5751:{\displaystyle S_{1}}
5722:
5720:{\displaystyle S_{1}}
5679:
5677:{\displaystyle S_{1}}
5648:
5646:{\displaystyle S_{2}}
5605:
5603:{\displaystyle S_{2}}
5574:
5572:{\displaystyle S_{2}}
5531:
5529:{\displaystyle S_{2}}
5500:
5498:{\displaystyle S_{2}}
5457:
5455:{\displaystyle S_{2}}
5426:
5424:{\displaystyle S_{2}}
5383:
5381:{\displaystyle S_{2}}
5352:
5350:{\displaystyle S_{2}}
5306:
5304:{\displaystyle S_{2}}
5275:
5273:{\displaystyle S_{1}}
5238:
5102:
5100:{\displaystyle S_{2}}
5071:
5069:{\displaystyle S_{1}}
4908:
4778:
4724:
4703:
4625:
4604:
4554:
4494:
4473:
4419:
4398:
4361:
4307:
4271:
4206:
4164:
4125:
4105:
4085:
4065:
4019:
3955:
3876:
3856:
3836:
3834:{\displaystyle v_{k}}
3809:
3758:
3637:
3586:
3514:
3423:
3366:
3315:
3276:
3256:
3233:
3173:
3150:
3135:given the parameters
3130:
3110:
3071:
3032:
3012:
2992:
2966:
2946:
2926:
2903:
2782:
2761:
2472:
2452:
2432:
2412:
2389:
2319:
2125:
2026:
1991:
1943:
1907:
1887:
1872:given starting state
1867:
1815:
1667:
1647:
1624:
1563:
1490:
1409:
1389:
1369:
1310:
1173:
1121:
1101:
1022:
975:
862:
860:{\displaystyle y_{i}}
835:
833:{\displaystyle b_{j}}
808:
753:
727:
725:{\displaystyle X_{t}}
700:
698:{\displaystyle Y_{t}}
670:
567:
534:
514:
512:{\displaystyle y_{i}}
487:
467:
465:{\displaystyle Y_{t}}
437:
376:
347:
246:
226:
171:
151:
131:
129:{\displaystyle X_{t}}
25:statistical computing
7971:Burge, Christopher.
7877:10.1093/nar/26.2.544
7187:
7156:
7026:Old Emission Matrix
6993:
6962:
6924:
6895:
6844:
6813:
6778:
6747:
6707:
6676:
6641:
6610:
6570:
6539:
6504:
6473:
6430:
6256:
6225:
6194:
6163:
6132:
6101:
6066:
6037:
6006:
5957:
5926:
5883:
5852:
5809:
5778:
5735:
5704:
5661:
5630:
5587:
5556:
5513:
5482:
5439:
5408:
5365:
5334:
5288:
5257:
5113:
5084:
5053:
4793:
4564:
4371:
4283:
4215:
4173:
4134:
4114:
4103:{\displaystyle \pi }
4094:
4074:
4028:
3976:
3894:
3865:
3845:
3818:
3770:
3652:
3474:
3288:
3265:
3245:
3185:
3178:can now be updated:
3162:
3139:
3119:
3080:
3044:
3040:The denominators of
3021:
3001:
2975:
2955:
2935:
2915:
2483:
2461:
2441:
2421:
2401:
2145:
2001:
1955:
1916:
1896:
1876:
1824:
1684:
1656:
1636:
1500:
1421:
1398:
1378:
1319:
1191:
1138:
1110:
1031:
987:
874:
844:
817:
762:
736:
709:
682:
579:
543:
523:
496:
476:
449:
388:
359:
258:
235:
180:
160:
140:
113:
33:Baum–Welch algorithm
8105:2007PNAS..10410110K
7979:on 6 September 2013
7491:Rabiner, Lawrence.
7448:Hidden Markov model
7109:
7068:
7027:
6372:
6331:
6290:
5016:
4975:
4934:
4581:
4391:
4300:
3787:
3491:
3308:
3202:
2990:{\displaystyle t+1}
2457:and the parameters
1652:in the forward and
1374:and being in state
374:{\displaystyle t=1}
85:hidden Markov model
73:genetic information
41:hidden Markov model
8209:2012-02-28 at the
7676:Frazzoli, Emilio.
7463:Speech recognition
7458:Maximum likelihood
7337:across populations
7224:Speech recognition
7200:
7169:
7107:
7066:
7025:
7006:
6975:
6943:
6908:
6857:
6826:
6791:
6760:
6720:
6689:
6654:
6623:
6583:
6552:
6517:
6486:
6443:
6370:
6329:
6288:
6269:
6238:
6207:
6176:
6145:
6114:
6085:
6062:transition is now
6050:
6019:
5970:
5939:
5896:
5865:
5822:
5791:
5748:
5717:
5674:
5643:
5600:
5569:
5526:
5495:
5452:
5421:
5378:
5347:
5301:
5270:
5233:
5097:
5066:
5014:
4973:
4932:
4903:
4898:
4773:
4567:
4549:
4374:
4356:
4286:
4266:
4211:for each sequence
4201:
4159:
4120:
4100:
4080:
4060:
4014:
3968:Multiple sequences
3950:
3871:
3851:
3831:
3804:
3773:
3753:
3748:
3632:
3477:
3459: = 1 to
3418:
3291:
3271:
3251:
3228:
3188:
3168:
3145:
3125:
3105:
3066:
3027:
3007:
2987:
2961:
2941:
2921:
2898:
2467:
2447:
2427:
2407:
2384:
2120:
1986:
1938:
1902:
1882:
1862:
1810:
1676:Backward procedure
1662:
1642:
1619:
1485:
1404:
1384:
1364:
1305:
1168:
1116:
1096:
1017:
970:
857:
830:
803:
748:
722:
695:
665:
562:
529:
509:
482:
462:
432:
371:
342:
241:
221:
166:
146:
126:
104:maximum likelihood
7845:978-3-642-10365-0
7589:978-0-521-62041-3
7443:Viterbi algorithm
7148:
7147:
7144:
7143:
7103:
7102:
7062:
7061:
6935:
6887:
6886:
6419:Observed Sequence
6411:
6410:
6407:
6406:
6366:
6365:
6325:
6324:
6077:
5998:
5997:
5248:Observed sequence
5039:
5038:
5035:
5034:
5010:
5009:
4969:
4968:
4894:
4848:
4768:
4544:
4354:
4123:{\displaystyle b}
4083:{\displaystyle A}
3944:
3916:
3874:{\displaystyle i}
3854:{\displaystyle i}
3744:
3704:
3627:
3413:
3274:{\displaystyle 1}
3254:{\displaystyle i}
3128:{\displaystyle Y}
3010:{\displaystyle Y}
2964:{\displaystyle t}
2944:{\displaystyle j}
2924:{\displaystyle i}
2893:
2662:
2450:{\displaystyle Y}
2430:{\displaystyle t}
2410:{\displaystyle i}
2379:
2271:
1905:{\displaystyle t}
1885:{\displaystyle i}
1407:{\displaystyle t}
1387:{\displaystyle i}
1183:Forward procedure
532:{\displaystyle t}
485:{\displaystyle K}
244:{\displaystyle t}
169:{\displaystyle N}
149:{\displaystyle N}
69:speech processing
8248:
8137:
8136:
8126:
8116:
8081:
8075:
8074:
8064:
8040:
8034:
8033:
8015:
7995:
7989:
7988:
7986:
7984:
7975:. Archived from
7968:
7962:
7961:
7951:
7919:
7913:
7912:
7905:
7899:
7898:
7888:
7856:
7850:
7849:
7823:
7817:
7816:
7808:
7802:
7801:
7793:
7787:
7786:
7774:
7768:
7767:
7749:
7729:
7723:
7722:
7699:
7693:
7692:
7690:
7688:
7682:
7673:
7667:
7666:
7664:
7662:
7656:
7649:
7641:
7635:
7634:
7632:
7630:
7624:
7615:
7609:
7608:
7600:
7594:
7593:
7573:
7567:
7566:
7538:
7532:
7531:
7511:
7505:
7504:
7502:
7500:
7488:
7211:
7209:
7207:
7206:
7201:
7199:
7198:
7180:
7178:
7176:
7175:
7170:
7168:
7167:
7110:
7106:
7069:
7065:
7028:
7024:
7021:
7017:
7015:
7013:
7012:
7007:
7005:
7004:
6986:
6984:
6982:
6981:
6976:
6974:
6973:
6952:
6950:
6949:
6944:
6936:
6928:
6920:emission is now
6919:
6917:
6915:
6914:
6909:
6907:
6906:
6868:
6866:
6864:
6863:
6858:
6856:
6855:
6837:
6835:
6833:
6832:
6827:
6825:
6824:
6802:
6800:
6798:
6797:
6792:
6790:
6789:
6771:
6769:
6767:
6766:
6761:
6759:
6758:
6731:
6729:
6727:
6726:
6721:
6719:
6718:
6700:
6698:
6696:
6695:
6690:
6688:
6687:
6665:
6663:
6661:
6660:
6655:
6653:
6652:
6634:
6632:
6630:
6629:
6624:
6622:
6621:
6594:
6592:
6590:
6589:
6584:
6582:
6581:
6563:
6561:
6559:
6558:
6553:
6551:
6550:
6528:
6526:
6524:
6523:
6518:
6516:
6515:
6497:
6495:
6493:
6492:
6487:
6485:
6484:
6454:
6452:
6450:
6449:
6444:
6442:
6441:
6416:
6415:
6373:
6369:
6332:
6328:
6291:
6287:
6284:
6280:
6278:
6276:
6275:
6270:
6268:
6267:
6249:
6247:
6245:
6244:
6239:
6237:
6236:
6218:
6216:
6214:
6213:
6208:
6206:
6205:
6187:
6185:
6183:
6182:
6177:
6175:
6174:
6156:
6154:
6152:
6151:
6146:
6144:
6143:
6125:
6123:
6121:
6120:
6115:
6113:
6112:
6094:
6092:
6091:
6086:
6078:
6070:
6061:
6059:
6057:
6056:
6051:
6049:
6048:
6030:
6028:
6026:
6025:
6020:
6018:
6017:
5981:
5979:
5977:
5976:
5971:
5969:
5968:
5950:
5948:
5946:
5945:
5940:
5938:
5937:
5907:
5905:
5903:
5902:
5897:
5895:
5894:
5876:
5874:
5872:
5871:
5866:
5864:
5863:
5833:
5831:
5829:
5828:
5823:
5821:
5820:
5802:
5800:
5798:
5797:
5792:
5790:
5789:
5759:
5757:
5755:
5754:
5749:
5747:
5746:
5728:
5726:
5724:
5723:
5718:
5716:
5715:
5685:
5683:
5681:
5680:
5675:
5673:
5672:
5654:
5652:
5650:
5649:
5644:
5642:
5641:
5611:
5609:
5607:
5606:
5601:
5599:
5598:
5580:
5578:
5576:
5575:
5570:
5568:
5567:
5537:
5535:
5533:
5532:
5527:
5525:
5524:
5506:
5504:
5502:
5501:
5496:
5494:
5493:
5463:
5461:
5459:
5458:
5453:
5451:
5450:
5432:
5430:
5428:
5427:
5422:
5420:
5419:
5389:
5387:
5385:
5384:
5379:
5377:
5376:
5358:
5356:
5354:
5353:
5348:
5346:
5345:
5312:
5310:
5308:
5307:
5302:
5300:
5299:
5281:
5279:
5277:
5276:
5271:
5269:
5268:
5245:
5244:
5242:
5240:
5239:
5234:
5226:
5225:
5216:
5196:
5195:
5183:
5182:
5161:
5160:
5151:
5131:
5130:
5108:
5106:
5104:
5103:
5098:
5096:
5095:
5077:
5075:
5073:
5072:
5067:
5065:
5064:
5017:
5013:
4976:
4972:
4935:
4931:
4928:
4912:
4910:
4909:
4904:
4902:
4901:
4895:
4892:
4878:
4877:
4865:
4864:
4849:
4846:
4828:
4827:
4826:
4825:
4813:
4812:
4782:
4780:
4779:
4774:
4769:
4767:
4757:
4756:
4743:
4738:
4722:
4717:
4701:
4691:
4690:
4678:
4677:
4676:
4675:
4663:
4662:
4644:
4639:
4623:
4618:
4602:
4594:
4593:
4580:
4575:
4558:
4556:
4555:
4550:
4545:
4543:
4533:
4532:
4519:
4508:
4492:
4487:
4471:
4461:
4460:
4444:
4433:
4417:
4412:
4396:
4390:
4385:
4365:
4363:
4362:
4357:
4355:
4350:
4340:
4339:
4326:
4321:
4305:
4299:
4294:
4275:
4273:
4272:
4267:
4265:
4264:
4257:
4256:
4233:
4232:
4210:
4208:
4207:
4202:
4191:
4190:
4168:
4166:
4165:
4160:
4149:
4148:
4129:
4127:
4126:
4121:
4109:
4107:
4106:
4101:
4089:
4087:
4086:
4081:
4069:
4067:
4066:
4061:
4059:
4058:
4040:
4039:
4023:
4021:
4020:
4015:
4013:
4012:
3994:
3993:
3959:
3957:
3956:
3951:
3946:
3945:
3942:
3918:
3917:
3914:
3880:
3878:
3877:
3872:
3860:
3858:
3857:
3852:
3840:
3838:
3837:
3832:
3830:
3829:
3813:
3811:
3810:
3805:
3800:
3799:
3786:
3781:
3762:
3760:
3759:
3754:
3752:
3751:
3745:
3742:
3728:
3727:
3715:
3714:
3705:
3702:
3684:
3683:
3682:
3681:
3669:
3668:
3641:
3639:
3638:
3633:
3628:
3626:
3616:
3615:
3605:
3600:
3584:
3574:
3573:
3564:
3563:
3562:
3561:
3549:
3548:
3533:
3528:
3512:
3504:
3503:
3490:
3485:
3467: − 1.
3427:
3425:
3424:
3419:
3414:
3412:
3402:
3401:
3391:
3380:
3364:
3354:
3353:
3340:
3329:
3313:
3307:
3302:
3280:
3278:
3277:
3272:
3260:
3258:
3257:
3252:
3237:
3235:
3234:
3229:
3215:
3214:
3201:
3196:
3177:
3175:
3174:
3169:
3154:
3152:
3151:
3146:
3134:
3132:
3131:
3126:
3114:
3112:
3111:
3106:
3095:
3094:
3075:
3073:
3072:
3067:
3056:
3055:
3036:
3034:
3033:
3028:
3016:
3014:
3013:
3008:
2996:
2994:
2993:
2988:
2970:
2968:
2967:
2962:
2950:
2948:
2947:
2942:
2930:
2928:
2927:
2922:
2907:
2905:
2904:
2899:
2894:
2892:
2888:
2887:
2869:
2868:
2844:
2843:
2834:
2833:
2812:
2811:
2801:
2796:
2780:
2775:
2759:
2755:
2754:
2736:
2735:
2711:
2710:
2701:
2700:
2679:
2678:
2668:
2663:
2661:
2641:
2619:
2618:
2594:
2593:
2577:
2551:
2550:
2526:
2525:
2498:
2497:
2476:
2474:
2473:
2468:
2456:
2454:
2453:
2448:
2436:
2434:
2433:
2428:
2416:
2414:
2413:
2408:
2393:
2391:
2390:
2385:
2380:
2378:
2368:
2367:
2349:
2348:
2338:
2333:
2317:
2307:
2306:
2288:
2287:
2277:
2272:
2270:
2250:
2228:
2227:
2211:
2185:
2184:
2157:
2156:
2129:
2127:
2126:
2121:
2113:
2112:
2094:
2093:
2084:
2083:
2056:
2055:
2045:
2040:
2013:
2012:
1995:
1993:
1992:
1987:
1967:
1966:
1947:
1945:
1944:
1939:
1928:
1927:
1911:
1909:
1908:
1903:
1891:
1889:
1888:
1883:
1871:
1869:
1868:
1863:
1861:
1860:
1842:
1841:
1819:
1817:
1816:
1811:
1794:
1793:
1781:
1780:
1768:
1767:
1749:
1748:
1730:
1729:
1696:
1695:
1671:
1669:
1668:
1663:
1651:
1649:
1648:
1643:
1628:
1626:
1625:
1620:
1615:
1614:
1593:
1592:
1582:
1577:
1559:
1558:
1540:
1539:
1512:
1511:
1494:
1492:
1491:
1486:
1478:
1477:
1465:
1464:
1455:
1454:
1433:
1432:
1413:
1411:
1410:
1405:
1393:
1391:
1390:
1385:
1373:
1371:
1370:
1365:
1363:
1362:
1344:
1343:
1331:
1330:
1314:
1312:
1311:
1306:
1289:
1288:
1276:
1275:
1263:
1262:
1244:
1243:
1231:
1230:
1203:
1202:
1177:
1175:
1174:
1169:
1125:
1123:
1122:
1117:
1105:
1103:
1102:
1097:
1074:
1073:
1068:
1043:
1042:
1026:
1024:
1023:
1018:
979:
977:
976:
971:
966:
965:
953:
952:
934:
933:
921:
920:
908:
907:
895:
894:
866:
864:
863:
858:
856:
855:
839:
837:
836:
831:
829:
828:
812:
810:
809:
804:
796:
795:
783:
782:
757:
755:
754:
749:
732:, we obtain the
731:
729:
728:
723:
721:
720:
704:
702:
701:
696:
694:
693:
674:
672:
671:
666:
652:
651:
639:
638:
626:
625:
604:
603:
591:
590:
571:
569:
568:
563:
555:
554:
538:
536:
535:
530:
518:
516:
515:
510:
508:
507:
491:
489:
488:
483:
472:can take one of
471:
469:
468:
463:
461:
460:
441:
439:
438:
433:
419:
418:
400:
399:
380:
378:
377:
372:
351:
349:
348:
343:
329:
328:
304:
303:
282:
281:
250:
248:
247:
242:
230:
228:
227:
222:
217:
216:
198:
197:
175:
173:
172:
167:
155:
153:
152:
147:
135:
133:
132:
127:
125:
124:
8256:
8255:
8251:
8250:
8249:
8247:
8246:
8245:
8221:
8220:
8211:Wayback Machine
8145:
8140:
8099:(24): 10110–5.
8082:
8078:
8041:
8037:
8013:10.1.1.115.3107
7996:
7992:
7982:
7980:
7969:
7965:
7920:
7916:
7907:
7906:
7902:
7857:
7853:
7846:
7824:
7820:
7809:
7805:
7794:
7790:
7775:
7771:
7756:10.1109/5.18626
7747:10.1.1.381.3454
7730:
7726:
7703:Baker, James K.
7700:
7696:
7686:
7684:
7680:
7674:
7670:
7660:
7658:
7654:
7647:
7643:
7642:
7638:
7628:
7626:
7622:
7616:
7612:
7601:
7597:
7590:
7574:
7570:
7539:
7535:
7512:
7508:
7498:
7496:
7489:
7485:
7481:
7439:
7364:C library with
7349:
7347:Implementations
7326:
7302:
7269:
7264:
7259:
7251:channel encoder
7243:
7226:
7221:
7194:
7190:
7188:
7185:
7184:
7182:
7163:
7159:
7157:
7154:
7153:
7151:
7018:and normalize.
7000:
6996:
6994:
6991:
6990:
6988:
6969:
6965:
6963:
6960:
6959:
6957:
6927:
6925:
6922:
6921:
6902:
6898:
6896:
6893:
6892:
6890:
6851:
6847:
6845:
6842:
6841:
6839:
6820:
6816:
6814:
6811:
6810:
6808:
6785:
6781:
6779:
6776:
6775:
6773:
6754:
6750:
6748:
6745:
6744:
6742:
6714:
6710:
6708:
6705:
6704:
6702:
6683:
6679:
6677:
6674:
6673:
6671:
6648:
6644:
6642:
6639:
6638:
6636:
6617:
6613:
6611:
6608:
6607:
6605:
6577:
6573:
6571:
6568:
6567:
6565:
6546:
6542:
6540:
6537:
6536:
6534:
6511:
6507:
6505:
6502:
6501:
6499:
6480:
6476:
6474:
6471:
6470:
6468:
6437:
6433:
6431:
6428:
6427:
6425:
6423:
6263:
6259:
6257:
6254:
6253:
6251:
6232:
6228:
6226:
6223:
6222:
6220:
6201:
6197:
6195:
6192:
6191:
6189:
6170:
6166:
6164:
6161:
6160:
6158:
6139:
6135:
6133:
6130:
6129:
6127:
6108:
6104:
6102:
6099:
6098:
6096:
6069:
6067:
6064:
6063:
6044:
6040:
6038:
6035:
6034:
6032:
6013:
6009:
6007:
6004:
6003:
6001:
5964:
5960:
5958:
5955:
5954:
5952:
5933:
5929:
5927:
5924:
5923:
5921:
5890:
5886:
5884:
5881:
5880:
5878:
5859:
5855:
5853:
5850:
5849:
5847:
5816:
5812:
5810:
5807:
5806:
5804:
5785:
5781:
5779:
5776:
5775:
5773:
5742:
5738:
5736:
5733:
5732:
5730:
5711:
5707:
5705:
5702:
5701:
5699:
5668:
5664:
5662:
5659:
5658:
5656:
5637:
5633:
5631:
5628:
5627:
5625:
5594:
5590:
5588:
5585:
5584:
5582:
5563:
5559:
5557:
5554:
5553:
5551:
5520:
5516:
5514:
5511:
5510:
5508:
5489:
5485:
5483:
5480:
5479:
5477:
5446:
5442:
5440:
5437:
5436:
5434:
5415:
5411:
5409:
5406:
5405:
5403:
5372:
5368:
5366:
5363:
5362:
5360:
5341:
5337:
5335:
5332:
5331:
5329:
5295:
5291:
5289:
5286:
5285:
5283:
5264:
5260:
5258:
5255:
5254:
5252:
5221:
5217:
5212:
5191:
5187:
5178:
5174:
5156:
5152:
5147:
5126:
5122:
5114:
5111:
5110:
5091:
5087:
5085:
5082:
5081:
5079:
5060:
5056:
5054:
5051:
5050:
5048:
4922:
4897:
4896:
4891:
4889:
4883:
4882:
4873:
4869:
4854:
4850:
4845:
4843:
4833:
4832:
4821:
4817:
4805:
4801:
4800:
4796:
4794:
4791:
4790:
4749:
4745:
4739:
4728:
4718:
4707:
4702:
4683:
4679:
4671:
4667:
4655:
4651:
4650:
4646:
4640:
4629:
4619:
4608:
4603:
4601:
4589:
4585:
4576:
4571:
4565:
4562:
4561:
4525:
4521:
4509:
4498:
4488:
4477:
4472:
4450:
4446:
4434:
4423:
4413:
4402:
4397:
4395:
4386:
4378:
4372:
4369:
4368:
4332:
4328:
4322:
4311:
4306:
4304:
4295:
4290:
4284:
4281:
4280:
4252:
4248:
4247:
4243:
4222:
4218:
4216:
4213:
4212:
4180:
4176:
4174:
4171:
4170:
4141:
4137:
4135:
4132:
4131:
4115:
4112:
4111:
4095:
4092:
4091:
4075:
4072:
4071:
4054:
4050:
4035:
4031:
4029:
4026:
4025:
4008:
4004:
3989:
3985:
3977:
3974:
3973:
3970:
3941:
3937:
3913:
3909:
3895:
3892:
3891:
3866:
3863:
3862:
3846:
3843:
3842:
3841:while in state
3825:
3821:
3819:
3816:
3815:
3795:
3791:
3782:
3777:
3771:
3768:
3767:
3747:
3746:
3741:
3739:
3733:
3732:
3723:
3719:
3710:
3706:
3701:
3699:
3689:
3688:
3677:
3673:
3664:
3660:
3659:
3655:
3653:
3650:
3649:
3611:
3607:
3601:
3590:
3585:
3569:
3565:
3557:
3553:
3544:
3540:
3539:
3535:
3529:
3518:
3513:
3511:
3499:
3495:
3486:
3481:
3475:
3472:
3471:
3397:
3393:
3381:
3370:
3365:
3346:
3342:
3330:
3319:
3314:
3312:
3303:
3295:
3289:
3286:
3285:
3266:
3263:
3262:
3246:
3243:
3242:
3210:
3206:
3197:
3192:
3186:
3183:
3182:
3163:
3160:
3159:
3140:
3137:
3136:
3120:
3117:
3116:
3087:
3083:
3081:
3078:
3077:
3051:
3047:
3045:
3042:
3041:
3022:
3019:
3018:
3017:and parameters
3002:
2999:
2998:
2976:
2973:
2972:
2956:
2953:
2952:
2936:
2933:
2932:
2916:
2913:
2912:
2877:
2873:
2864:
2860:
2839:
2835:
2826:
2822:
2807:
2803:
2797:
2786:
2776:
2765:
2760:
2744:
2740:
2731:
2727:
2706:
2702:
2693:
2689:
2674:
2670:
2669:
2667:
2642:
2608:
2604:
2589:
2585:
2578:
2576:
2540:
2536:
2521:
2517:
2490:
2486:
2484:
2481:
2480:
2462:
2459:
2458:
2442:
2439:
2438:
2422:
2419:
2418:
2402:
2399:
2398:
2363:
2359:
2344:
2340:
2334:
2323:
2318:
2302:
2298:
2283:
2279:
2278:
2276:
2251:
2223:
2219:
2212:
2210:
2180:
2176:
2152:
2148:
2146:
2143:
2142:
2136:
2102:
2098:
2089:
2085:
2076:
2072:
2051:
2047:
2041:
2030:
2008:
2004:
2002:
1999:
1998:
1962:
1958:
1956:
1953:
1952:
1923:
1919:
1917:
1914:
1913:
1912:. We calculate
1897:
1894:
1893:
1877:
1874:
1873:
1856:
1852:
1831:
1827:
1825:
1822:
1821:
1789:
1785:
1776:
1772:
1763:
1759:
1738:
1734:
1719:
1715:
1691:
1687:
1685:
1682:
1681:
1678:
1657:
1654:
1653:
1637:
1634:
1633:
1607:
1603:
1588:
1584:
1578:
1567:
1548:
1544:
1535:
1531:
1507:
1503:
1501:
1498:
1497:
1473:
1469:
1460:
1456:
1450:
1446:
1428:
1424:
1422:
1419:
1418:
1399:
1396:
1395:
1379:
1376:
1375:
1358:
1354:
1339:
1335:
1326:
1322:
1320:
1317:
1316:
1284:
1280:
1271:
1267:
1258:
1254:
1239:
1235:
1226:
1222:
1198:
1194:
1192:
1189:
1188:
1185:
1139:
1136:
1135:
1132:
1111:
1108:
1107:
1069:
1048:
1047:
1038:
1034:
1032:
1029:
1028:
988:
985:
984:
961:
957:
948:
944:
929:
925:
916:
912:
903:
899:
890:
886:
875:
872:
871:
851:
847:
845:
842:
841:
824:
820:
818:
815:
814:
791:
787:
778:
774:
763:
760:
759:
737:
734:
733:
716:
712:
710:
707:
706:
689:
685:
683:
680:
679:
647:
643:
634:
630:
621:
617:
599:
595:
586:
582:
580:
577:
576:
550:
546:
544:
541:
540:
524:
521:
520:
503:
499:
497:
494:
493:
477:
474:
473:
456:
452:
450:
447:
446:
414:
410:
395:
391:
389:
386:
385:
360:
357:
356:
318:
314:
299:
295:
274:
270:
259:
256:
255:
236:
233:
232:
206:
202:
193:
189:
181:
178:
177:
161:
158:
157:
141:
138:
137:
120:
116:
114:
111:
110:
81:
57:Leonard E. Baum
53:
17:
12:
11:
5:
8254:
8244:
8243:
8238:
8233:
8219:
8218:
8213:
8201:
8195:
8194:
8193:
8183:
8182:
8181:
8172:
8171:
8170:
8165:
8160:
8152:
8144:
8143:External links
8141:
8139:
8138:
8076:
8055:(3): 346–354.
8035:
7990:
7963:
7934:(6): 673–679.
7928:Bioinformatics
7914:
7900:
7871:(2): 544–548.
7851:
7844:
7818:
7803:
7788:
7769:
7740:(2): 257–286.
7724:
7694:
7668:
7636:
7610:
7595:
7588:
7568:
7533:
7506:
7482:
7480:
7477:
7476:
7475:
7470:
7468:Bioinformatics
7465:
7460:
7455:
7450:
7445:
7438:
7435:
7434:
7433:
7424:
7415:
7404:
7394:
7369:
7359:
7348:
7345:
7325:
7322:
7301:
7298:
7285:coding regions
7268:
7265:
7263:
7260:
7258:
7255:
7242:
7239:
7230:James K. Baker
7225:
7222:
7220:
7217:
7197:
7193:
7166:
7162:
7146:
7145:
7142:
7141:
7138:
7135:
7131:
7130:
7127:
7124:
7120:
7119:
7116:
7113:
7104:
7101:
7100:
7097:
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7086:
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7049:
7048:
7045:
7042:
7038:
7037:
7034:
7031:
7003:
6999:
6972:
6968:
6942:
6939:
6934:
6931:
6905:
6901:
6885:
6884:
6882:
6879:
6877:
6874:
6870:
6869:
6854:
6850:
6823:
6819:
6806:
6803:
6788:
6784:
6757:
6753:
6740:
6737:
6733:
6732:
6717:
6713:
6686:
6682:
6669:
6666:
6651:
6647:
6620:
6616:
6603:
6600:
6596:
6595:
6580:
6576:
6549:
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6532:
6529:
6514:
6510:
6483:
6479:
6466:
6463:
6459:
6458:
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6440:
6436:
6420:
6409:
6408:
6405:
6404:
6401:
6398:
6394:
6393:
6390:
6387:
6383:
6382:
6379:
6376:
6367:
6364:
6363:
6360:
6357:
6353:
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6346:
6342:
6341:
6338:
6335:
6326:
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6322:
6319:
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6311:
6308:
6305:
6301:
6300:
6297:
6294:
6266:
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6235:
6231:
6204:
6200:
6173:
6169:
6142:
6138:
6111:
6107:
6084:
6081:
6076:
6073:
6047:
6043:
6016:
6012:
5996:
5995:
5993:
5990:
5987:
5983:
5982:
5967:
5963:
5936:
5932:
5919:
5916:
5913:
5909:
5908:
5893:
5889:
5862:
5858:
5845:
5842:
5839:
5835:
5834:
5819:
5815:
5788:
5784:
5771:
5768:
5765:
5761:
5760:
5745:
5741:
5714:
5710:
5697:
5694:
5691:
5687:
5686:
5671:
5667:
5640:
5636:
5623:
5620:
5617:
5613:
5612:
5597:
5593:
5566:
5562:
5549:
5546:
5543:
5539:
5538:
5523:
5519:
5492:
5488:
5475:
5472:
5469:
5465:
5464:
5449:
5445:
5418:
5414:
5401:
5398:
5395:
5391:
5390:
5375:
5371:
5344:
5340:
5327:
5324:
5321:
5317:
5316:
5313:
5298:
5294:
5267:
5263:
5249:
5232:
5229:
5224:
5220:
5215:
5211:
5208:
5205:
5202:
5199:
5194:
5190:
5186:
5181:
5177:
5173:
5170:
5167:
5164:
5159:
5155:
5150:
5146:
5143:
5140:
5137:
5134:
5129:
5125:
5121:
5118:
5094:
5090:
5063:
5059:
5037:
5036:
5033:
5032:
5029:
5025:
5024:
5021:
5011:
5008:
5007:
5004:
5001:
4997:
4996:
4993:
4990:
4986:
4985:
4982:
4979:
4970:
4967:
4966:
4963:
4960:
4956:
4955:
4952:
4949:
4945:
4944:
4941:
4938:
4921:
4918:
4914:
4913:
4900:
4890:
4888:
4885:
4884:
4881:
4876:
4872:
4868:
4863:
4860:
4857:
4853:
4844:
4842:
4839:
4838:
4836:
4831:
4824:
4820:
4816:
4811:
4808:
4804:
4799:
4784:
4783:
4772:
4766:
4763:
4760:
4755:
4752:
4748:
4742:
4737:
4734:
4731:
4727:
4721:
4716:
4713:
4710:
4706:
4700:
4697:
4694:
4689:
4686:
4682:
4674:
4670:
4666:
4661:
4658:
4654:
4649:
4643:
4638:
4635:
4632:
4628:
4622:
4617:
4614:
4611:
4607:
4600:
4597:
4592:
4588:
4584:
4579:
4574:
4570:
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4531:
4528:
4524:
4518:
4515:
4512:
4507:
4504:
4501:
4497:
4491:
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4483:
4480:
4476:
4470:
4467:
4464:
4459:
4456:
4453:
4449:
4443:
4440:
4437:
4432:
4429:
4426:
4422:
4416:
4411:
4408:
4405:
4401:
4394:
4389:
4384:
4381:
4377:
4366:
4353:
4349:
4346:
4343:
4338:
4335:
4331:
4325:
4320:
4317:
4314:
4310:
4303:
4298:
4293:
4289:
4263:
4260:
4255:
4251:
4246:
4242:
4239:
4236:
4231:
4228:
4225:
4221:
4200:
4197:
4194:
4189:
4186:
4183:
4179:
4158:
4155:
4152:
4147:
4144:
4140:
4119:
4099:
4079:
4057:
4053:
4049:
4046:
4043:
4038:
4034:
4011:
4007:
4003:
4000:
3997:
3992:
3988:
3984:
3981:
3969:
3966:
3949:
3940:
3936:
3933:
3930:
3927:
3924:
3921:
3912:
3908:
3905:
3902:
3899:
3870:
3850:
3828:
3824:
3803:
3798:
3794:
3790:
3785:
3780:
3776:
3764:
3763:
3750:
3740:
3738:
3735:
3734:
3731:
3726:
3722:
3718:
3713:
3709:
3700:
3698:
3695:
3694:
3692:
3687:
3680:
3676:
3672:
3667:
3663:
3658:
3643:
3642:
3631:
3625:
3622:
3619:
3614:
3610:
3604:
3599:
3596:
3593:
3589:
3583:
3580:
3577:
3572:
3568:
3560:
3556:
3552:
3547:
3543:
3538:
3532:
3527:
3524:
3521:
3517:
3510:
3507:
3502:
3498:
3494:
3489:
3484:
3480:
3429:
3428:
3417:
3411:
3408:
3405:
3400:
3396:
3390:
3387:
3384:
3379:
3376:
3373:
3369:
3363:
3360:
3357:
3352:
3349:
3345:
3339:
3336:
3333:
3328:
3325:
3322:
3318:
3311:
3306:
3301:
3298:
3294:
3270:
3250:
3239:
3238:
3227:
3224:
3221:
3218:
3213:
3209:
3205:
3200:
3195:
3191:
3167:
3144:
3124:
3104:
3101:
3098:
3093:
3090:
3086:
3065:
3062:
3059:
3054:
3050:
3026:
3006:
2986:
2983:
2980:
2960:
2940:
2920:
2909:
2908:
2897:
2891:
2886:
2883:
2880:
2876:
2872:
2867:
2863:
2859:
2856:
2853:
2850:
2847:
2842:
2838:
2832:
2829:
2825:
2821:
2818:
2815:
2810:
2806:
2800:
2795:
2792:
2789:
2785:
2779:
2774:
2771:
2768:
2764:
2758:
2753:
2750:
2747:
2743:
2739:
2734:
2730:
2726:
2723:
2720:
2717:
2714:
2709:
2705:
2699:
2696:
2692:
2688:
2685:
2682:
2677:
2673:
2666:
2660:
2657:
2654:
2651:
2648:
2645:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2617:
2614:
2611:
2607:
2603:
2600:
2597:
2592:
2588:
2584:
2581:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2549:
2546:
2543:
2539:
2535:
2532:
2529:
2524:
2520:
2516:
2513:
2510:
2507:
2504:
2501:
2496:
2493:
2489:
2466:
2446:
2426:
2406:
2395:
2394:
2383:
2377:
2374:
2371:
2366:
2362:
2358:
2355:
2352:
2347:
2343:
2337:
2332:
2329:
2326:
2322:
2316:
2313:
2310:
2305:
2301:
2297:
2294:
2291:
2286:
2282:
2275:
2269:
2266:
2263:
2260:
2257:
2254:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2226:
2222:
2218:
2215:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2183:
2179:
2175:
2172:
2169:
2166:
2163:
2160:
2155:
2151:
2135:
2132:
2131:
2130:
2119:
2116:
2111:
2108:
2105:
2101:
2097:
2092:
2088:
2082:
2079:
2075:
2071:
2068:
2065:
2062:
2059:
2054:
2050:
2044:
2039:
2036:
2033:
2029:
2025:
2022:
2019:
2016:
2011:
2007:
1996:
1985:
1982:
1979:
1976:
1973:
1970:
1965:
1961:
1937:
1934:
1931:
1926:
1922:
1901:
1881:
1859:
1855:
1851:
1848:
1845:
1840:
1837:
1834:
1830:
1809:
1806:
1803:
1800:
1797:
1792:
1788:
1784:
1779:
1775:
1771:
1766:
1762:
1758:
1755:
1752:
1747:
1744:
1741:
1737:
1733:
1728:
1725:
1722:
1718:
1714:
1711:
1708:
1705:
1702:
1699:
1694:
1690:
1677:
1674:
1661:
1641:
1630:
1629:
1618:
1613:
1610:
1606:
1602:
1599:
1596:
1591:
1587:
1581:
1576:
1573:
1570:
1566:
1562:
1557:
1554:
1551:
1547:
1543:
1538:
1534:
1530:
1527:
1524:
1521:
1518:
1515:
1510:
1506:
1495:
1484:
1481:
1476:
1472:
1468:
1463:
1459:
1453:
1449:
1445:
1442:
1439:
1436:
1431:
1427:
1403:
1383:
1361:
1357:
1353:
1350:
1347:
1342:
1338:
1334:
1329:
1325:
1304:
1301:
1298:
1295:
1292:
1287:
1283:
1279:
1274:
1270:
1266:
1261:
1257:
1253:
1250:
1247:
1242:
1238:
1234:
1229:
1225:
1221:
1218:
1215:
1212:
1209:
1206:
1201:
1197:
1184:
1181:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1131:
1128:
1115:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1072:
1067:
1064:
1061:
1057:
1054:
1051:
1046:
1041:
1037:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
969:
964:
960:
956:
951:
947:
943:
940:
937:
932:
928:
924:
919:
915:
911:
906:
902:
898:
893:
889:
885:
882:
879:
854:
850:
827:
823:
802:
799:
794:
790:
786:
781:
777:
773:
770:
767:
747:
744:
741:
719:
715:
692:
688:
676:
675:
664:
661:
658:
655:
650:
646:
642:
637:
633:
629:
624:
620:
616:
613:
610:
607:
602:
598:
594:
589:
585:
561:
558:
553:
549:
528:
506:
502:
481:
459:
455:
443:
442:
431:
428:
425:
422:
417:
413:
409:
406:
403:
398:
394:
381:) is given by
370:
367:
364:
353:
352:
341:
338:
335:
332:
327:
324:
321:
317:
313:
310:
307:
302:
298:
294:
291:
288:
285:
280:
277:
273:
269:
266:
263:
240:
220:
215:
212:
209:
205:
201:
196:
192:
188:
185:
165:
145:
123:
119:
80:
77:
61:Lloyd R. Welch
52:
49:
29:bioinformatics
15:
9:
6:
4:
3:
2:
8253:
8242:
8241:Markov models
8239:
8237:
8234:
8232:
8229:
8228:
8226:
8217:
8214:
8212:
8208:
8205:
8202:
8199:
8196:
8191:
8187:
8186:
8184:
8179:
8176:
8175:
8173:
8169:
8166:
8164:
8161:
8159:
8156:
8155:
8153:
8151:
8147:
8146:
8134:
8130:
8125:
8120:
8115:
8110:
8106:
8102:
8098:
8094:
8090:
8086:
8080:
8072:
8068:
8063:
8058:
8054:
8050:
8046:
8039:
8031:
8027:
8023:
8019:
8014:
8009:
8005:
8001:
7994:
7978:
7974:
7967:
7959:
7955:
7950:
7945:
7941:
7937:
7933:
7929:
7925:
7918:
7910:
7904:
7896:
7892:
7887:
7882:
7878:
7874:
7870:
7866:
7862:
7855:
7847:
7841:
7837:
7833:
7829:
7822:
7814:
7807:
7799:
7792:
7784:
7780:
7773:
7765:
7761:
7757:
7753:
7748:
7743:
7739:
7735:
7728:
7720:
7716:
7712:
7708:
7704:
7698:
7679:
7672:
7657:on 2021-04-14
7653:
7646:
7640:
7621:
7614:
7606:
7599:
7591:
7585:
7581:
7580:
7572:
7564:
7560:
7556:
7552:
7549:(2): 159–65.
7548:
7544:
7537:
7529:
7525:
7521:
7517:
7510:
7494:
7487:
7483:
7474:
7473:Cryptanalysis
7471:
7469:
7466:
7464:
7461:
7459:
7456:
7454:
7451:
7449:
7446:
7444:
7441:
7440:
7432:
7428:
7425:
7423:
7419:
7416:
7413:
7409:
7405:
7402:
7398:
7395:
7392:
7388:
7384:
7380:
7376:
7373:
7370:
7367:
7363:
7360:
7358:
7354:
7351:
7350:
7344:
7342:
7338:
7334:
7330:
7321:
7319:
7315:
7311:
7307:
7297:
7294:
7290:
7289:noncoding DNA
7286:
7282:
7278:
7274:
7262:Finding genes
7254:
7252:
7248:
7247:cryptanalysis
7241:Cryptanalysis
7238:
7236:
7231:
7216:
7213:
7195:
7191:
7164:
7160:
7139:
7136:
7133:
7132:
7128:
7125:
7122:
7121:
7117:
7114:
7112:
7111:
7105:
7098:
7095:
7092:
7091:
7087:
7084:
7081:
7080:
7076:
7073:
7071:
7070:
7064:
7057:
7054:
7051:
7050:
7046:
7043:
7040:
7039:
7035:
7032:
7030:
7029:
7023:
7019:
7001:
6997:
6970:
6966:
6954:
6940:
6937:
6932:
6929:
6903:
6899:
6883:
6880:
6878:
6875:
6872:
6871:
6852:
6848:
6821:
6817:
6807:
6804:
6786:
6782:
6755:
6751:
6741:
6738:
6735:
6734:
6715:
6711:
6684:
6680:
6670:
6667:
6649:
6645:
6618:
6614:
6604:
6601:
6598:
6597:
6578:
6574:
6547:
6543:
6533:
6530:
6512:
6508:
6481:
6477:
6467:
6464:
6461:
6460:
6438:
6434:
6418:
6417:
6414:
6402:
6399:
6396:
6395:
6391:
6388:
6385:
6384:
6380:
6377:
6375:
6374:
6368:
6361:
6358:
6355:
6354:
6350:
6347:
6344:
6343:
6339:
6336:
6334:
6333:
6327:
6320:
6317:
6314:
6313:
6309:
6306:
6303:
6302:
6298:
6295:
6293:
6292:
6286:
6282:
6264:
6260:
6233:
6229:
6202:
6198:
6171:
6167:
6140:
6136:
6109:
6105:
6082:
6079:
6074:
6071:
6045:
6041:
6014:
6010:
5994:
5991:
5988:
5985:
5984:
5965:
5961:
5934:
5930:
5920:
5917:
5914:
5911:
5910:
5891:
5887:
5860:
5856:
5846:
5843:
5840:
5837:
5836:
5817:
5813:
5786:
5782:
5772:
5769:
5766:
5763:
5762:
5743:
5739:
5712:
5708:
5698:
5695:
5692:
5689:
5688:
5669:
5665:
5638:
5634:
5624:
5621:
5618:
5615:
5614:
5595:
5591:
5564:
5560:
5550:
5547:
5544:
5541:
5540:
5521:
5517:
5490:
5486:
5476:
5473:
5470:
5467:
5466:
5447:
5443:
5416:
5412:
5402:
5399:
5396:
5393:
5392:
5373:
5369:
5342:
5338:
5328:
5325:
5322:
5319:
5318:
5296:
5292:
5265:
5261:
5250:
5247:
5246:
5243:
5230:
5222:
5218:
5209:
5203:
5200:
5192:
5188:
5179:
5175:
5168:
5165:
5157:
5153:
5144:
5138:
5135:
5127:
5123:
5116:
5092:
5088:
5061:
5057:
5045:
5042:
5030:
5027:
5026:
5022:
5019:
5018:
5012:
5005:
5002:
4999:
4998:
4994:
4991:
4988:
4987:
4983:
4980:
4978:
4977:
4971:
4964:
4961:
4958:
4957:
4953:
4950:
4947:
4946:
4942:
4939:
4937:
4936:
4930:
4926:
4917:
4886:
4879:
4874:
4870:
4866:
4861:
4858:
4855:
4851:
4840:
4834:
4829:
4822:
4818:
4814:
4809:
4806:
4802:
4797:
4789:
4788:
4787:
4770:
4761:
4753:
4750:
4746:
4740:
4735:
4732:
4729:
4725:
4719:
4714:
4711:
4708:
4704:
4695:
4687:
4684:
4680:
4672:
4668:
4664:
4659:
4656:
4652:
4647:
4641:
4636:
4633:
4630:
4626:
4620:
4615:
4612:
4609:
4605:
4598:
4590:
4586:
4577:
4572:
4568:
4560:
4546:
4537:
4529:
4526:
4522:
4516:
4513:
4510:
4505:
4502:
4499:
4495:
4489:
4484:
4481:
4478:
4474:
4465:
4457:
4454:
4451:
4447:
4441:
4438:
4435:
4430:
4427:
4424:
4420:
4414:
4409:
4406:
4403:
4399:
4392:
4387:
4382:
4379:
4375:
4367:
4351:
4344:
4336:
4333:
4329:
4323:
4318:
4315:
4312:
4308:
4301:
4296:
4291:
4287:
4279:
4278:
4277:
4261:
4258:
4253:
4249:
4244:
4240:
4237:
4234:
4229:
4226:
4223:
4219:
4195:
4187:
4184:
4181:
4177:
4153:
4145:
4142:
4138:
4117:
4097:
4077:
4055:
4051:
4047:
4044:
4041:
4036:
4032:
4009:
4005:
4001:
3998:
3995:
3990:
3986:
3982:
3979:
3965:
3963:
3938:
3934:
3931:
3925:
3922:
3910:
3906:
3903:
3897:
3889:
3885:
3882:
3868:
3848:
3826:
3822:
3796:
3792:
3783:
3778:
3774:
3736:
3729:
3724:
3720:
3716:
3711:
3707:
3696:
3690:
3685:
3678:
3674:
3670:
3665:
3661:
3656:
3648:
3647:
3646:
3629:
3620:
3612:
3608:
3602:
3597:
3594:
3591:
3587:
3578:
3570:
3566:
3558:
3554:
3550:
3545:
3541:
3536:
3530:
3525:
3522:
3519:
3515:
3508:
3500:
3496:
3487:
3482:
3478:
3470:
3469:
3468:
3466:
3463: =
3462:
3458:
3454:
3450:
3446:
3442:
3438:
3434:
3415:
3406:
3398:
3394:
3388:
3385:
3382:
3377:
3374:
3371:
3367:
3358:
3350:
3347:
3343:
3337:
3334:
3331:
3326:
3323:
3320:
3316:
3309:
3304:
3299:
3296:
3292:
3284:
3283:
3282:
3268:
3248:
3225:
3219:
3211:
3207:
3203:
3198:
3193:
3189:
3181:
3180:
3179:
3165:
3156:
3142:
3122:
3099:
3091:
3088:
3084:
3060:
3052:
3048:
3038:
3024:
3004:
2984:
2981:
2978:
2958:
2938:
2918:
2895:
2884:
2881:
2878:
2874:
2865:
2861:
2854:
2851:
2848:
2840:
2836:
2830:
2827:
2823:
2816:
2808:
2804:
2798:
2793:
2790:
2787:
2783:
2777:
2772:
2769:
2766:
2762:
2751:
2748:
2745:
2741:
2732:
2728:
2721:
2718:
2715:
2707:
2703:
2697:
2694:
2690:
2683:
2675:
2671:
2664:
2655:
2652:
2649:
2643:
2635:
2632:
2629:
2626:
2623:
2620:
2615:
2612:
2609:
2605:
2601:
2598:
2595:
2590:
2586:
2579:
2573:
2567:
2564:
2561:
2558:
2555:
2552:
2547:
2544:
2541:
2537:
2533:
2530:
2527:
2522:
2518:
2511:
2508:
2502:
2494:
2491:
2487:
2479:
2478:
2477:
2464:
2444:
2424:
2404:
2381:
2372:
2364:
2360:
2353:
2345:
2341:
2335:
2330:
2327:
2324:
2320:
2311:
2303:
2299:
2292:
2284:
2280:
2273:
2264:
2261:
2258:
2252:
2244:
2241:
2238:
2235:
2232:
2229:
2224:
2220:
2213:
2207:
2201:
2198:
2195:
2192:
2189:
2186:
2181:
2177:
2170:
2167:
2161:
2153:
2149:
2141:
2140:
2139:
2117:
2109:
2106:
2103:
2099:
2090:
2086:
2080:
2077:
2073:
2066:
2063:
2060:
2052:
2048:
2042:
2037:
2034:
2031:
2027:
2023:
2017:
2009:
2005:
1997:
1983:
1980:
1977:
1971:
1963:
1959:
1951:
1950:
1949:
1932:
1924:
1920:
1899:
1879:
1857:
1853:
1849:
1846:
1843:
1838:
1835:
1832:
1828:
1804:
1801:
1798:
1795:
1790:
1786:
1782:
1777:
1773:
1769:
1764:
1760:
1756:
1753:
1750:
1745:
1742:
1739:
1735:
1731:
1726:
1723:
1720:
1716:
1709:
1706:
1700:
1692:
1688:
1673:
1659:
1639:
1616:
1611:
1608:
1604:
1597:
1589:
1585:
1579:
1574:
1571:
1568:
1564:
1555:
1552:
1549:
1545:
1536:
1532:
1528:
1522:
1519:
1516:
1508:
1504:
1496:
1482:
1474:
1470:
1461:
1457:
1451:
1447:
1443:
1437:
1429:
1425:
1417:
1416:
1415:
1401:
1381:
1359:
1355:
1351:
1348:
1345:
1340:
1336:
1332:
1327:
1323:
1299:
1296:
1293:
1290:
1285:
1281:
1277:
1272:
1268:
1264:
1259:
1255:
1251:
1248:
1245:
1240:
1236:
1232:
1227:
1223:
1216:
1213:
1207:
1199:
1195:
1180:
1162:
1159:
1156:
1153:
1150:
1144:
1141:
1127:
1113:
1090:
1087:
1084:
1078:
1075:
1070:
1044:
1039:
1035:
1011:
1008:
1005:
1002:
999:
993:
990:
981:
962:
958:
954:
949:
945:
941:
938:
935:
930:
926:
922:
917:
913:
909:
904:
900:
896:
891:
887:
880:
877:
868:
852:
848:
825:
821:
792:
788:
779:
775:
768:
765:
745:
742:
739:
717:
713:
690:
686:
662:
656:
653:
648:
644:
640:
635:
631:
627:
622:
618:
611:
608:
600:
596:
587:
583:
575:
574:
573:
559:
556:
551:
547:
526:
504:
500:
479:
457:
453:
429:
423:
420:
415:
411:
404:
401:
396:
392:
384:
383:
382:
368:
365:
362:
339:
333:
330:
325:
322:
319:
315:
311:
308:
305:
300:
296:
289:
286:
278:
275:
271:
264:
261:
254:
253:
252:
238:
213:
210:
207:
203:
199:
194:
190:
183:
163:
143:
121:
117:
107:
105:
100:
98:
94:
90:
86:
76:
74:
70:
66:
62:
58:
48:
46:
42:
38:
34:
30:
26:
22:
8096:
8092:
8079:
8052:
8048:
8038:
8006:(1): 78–94.
8003:
7999:
7993:
7981:. Retrieved
7977:the original
7966:
7931:
7927:
7917:
7903:
7868:
7864:
7854:
7827:
7821:
7812:
7806:
7797:
7791:
7782:
7778:
7772:
7737:
7733:
7727:
7710:
7706:
7697:
7685:. Retrieved
7671:
7659:. Retrieved
7652:the original
7639:
7627:. Retrieved
7613:
7604:
7598:
7578:
7571:
7546:
7542:
7536:
7522:(3): 250–6.
7519:
7515:
7509:
7497:. Retrieved
7486:
7453:EM algorithm
7410:package for
7399:package for
7327:
7303:
7277:gene-finding
7270:
7244:
7227:
7219:Applications
7214:
7149:
6955:
6888:
6412:
5999:
5046:
5043:
5040:
4923:
4915:
4785:
3971:
3961:
3887:
3886:
3883:
3765:
3644:
3464:
3460:
3456:
3452:
3448:
3444:
3440:
3436:
3432:
3430:
3240:
3157:
3039:
2910:
2396:
2137:
1679:
1631:
1186:
1133:
982:
869:
677:
572:is given by
444:
354:
108:
101:
96:
92:
82:
54:
32:
18:
8085:Korbel, Jan
7629:29 November
7293:specificity
7281:prokaryotic
7267:Prokaryotic
4933:Transition
79:Description
8225:Categories
7661:11 October
7479:References
7353:Accord.NET
7314:base-pairs
7310:eukaryotic
7300:Eukaryotic
539:for state
8008:CiteSeerX
7983:2 October
7742:CiteSeerX
7713:: 24–29.
7687:2 October
7499:2 October
7318:isochores
5201:⋅
5185:→
5166:⋅
5136:⋅
4974:Emission
4893:otherwise
4747:γ
4726:∑
4705:∑
4681:γ
4627:∑
4606:∑
4578:∗
4523:γ
4514:−
4496:∑
4475:∑
4448:ξ
4439:−
4421:∑
4400:∑
4388:∗
4330:γ
4309:∑
4297:∗
4288:π
4238:…
4178:ξ
4139:γ
4098:π
4045:…
3999:…
3939:θ
3935:∣
3911:θ
3907:∣
3784:∗
3743:otherwise
3609:γ
3588:∑
3567:γ
3516:∑
3488:∗
3435:to state
3395:γ
3386:−
3368:∑
3344:ξ
3335:−
3317:∑
3305:∗
3208:γ
3199:∗
3190:π
3166:θ
3143:θ
3085:ξ
3049:γ
3025:θ
2951:at times
2837:β
2805:α
2784:∑
2763:∑
2704:β
2672:α
2656:θ
2653:∣
2636:θ
2633:∣
2568:θ
2559:∣
2488:ξ
2465:θ
2361:β
2342:α
2321:∑
2300:β
2281:α
2265:θ
2262:∣
2245:θ
2242:∣
2202:θ
2193:∣
2150:γ
2049:β
2028:∑
2006:β
1960:β
1921:β
1847:…
1805:θ
1783:∣
1754:…
1689:β
1660:β
1640:α
1586:α
1565:∑
1505:α
1448:π
1426:α
1349:…
1300:θ
1297:∣
1249:…
1196:α
1163:π
1142:θ
1130:Algorithm
1114:θ
1091:θ
1088:∣
1076:
1071:θ
1040:∗
1036:θ
1012:π
991:θ
939:…
743:×
641:∣
393:π
323:−
312:∣
211:−
200:∣
8207:Archived
8133:17551006
7958:17237039
7764:13618539
7437:See also
7418:hmmtrain
7235:phonemes
7134:State 2
7123:State 1
7093:State 2
7082:State 1
7052:State 2
7041:State 1
6397:State 2
6386:State 1
6381:State 2
6356:State 2
6345:State 1
6340:State 2
6315:State 2
6304:State 1
6299:State 2
5028:State 2
5020:State 1
5015:Initial
5000:State 2
4989:State 1
4959:State 2
4948:State 1
4943:State 2
4847:if
3703:if
3261:at time
2417:at time
1892:at time
1394:at time
519:at time
8124:1891248
8101:Bibcode
8071:9666331
8030:9149143
7949:2387122
7895:9421513
7563:3641921
7427:rustbio
7306:GENSCAN
7273:GLIMMER
7210:
7183:
7179:
7152:
7140:0.4248
7129:0.9559
7115:No Eggs
7099:0.7385
7088:0.8769
7074:No Eggs
7033:No Eggs
7016:
6989:
6985:
6958:
6918:
6891:
6867:
6840:
6836:
6809:
6801:
6774:
6770:
6743:
6730:
6703:
6699:
6672:
6664:
6637:
6633:
6606:
6593:
6566:
6562:
6535:
6527:
6500:
6496:
6469:
6453:
6426:
6403:0.8167
6392:0.6027
6378:State 1
6362:0.9705
6351:0.0908
6337:State 1
6296:State 1
6279:
6252:
6248:
6221:
6217:
6190:
6186:
6159:
6155:
6128:
6124:
6097:
6060:
6033:
6029:
6002:
5980:
5953:
5949:
5922:
5906:
5879:
5875:
5848:
5832:
5805:
5801:
5774:
5758:
5731:
5727:
5700:
5684:
5657:
5653:
5626:
5610:
5583:
5579:
5552:
5536:
5509:
5505:
5478:
5462:
5435:
5431:
5404:
5388:
5361:
5357:
5330:
5311:
5284:
5280:
5253:
5107:
5080:
5076:
5049:
4981:No Eggs
4940:State 1
4920:Example
758:matrix
51:History
8131:
8121:
8069:
8028:
8010:
7956:
7946:
7893:
7886:147303
7883:
7842:
7762:
7744:
7586:
7561:
7422:MATLAB
7375:Python
7372:Jajapy
7366:Python
7137:0.5752
7126:0.0441
7096:1.0000
7085:0.0404
6941:0.8769
6933:0.2730
6930:0.2394
6881:0.2730
6876:0.2394
6873:Total
6805:0.0896
6739:0.0560
6668:0.0490
6602:0.0490
6531:0.1344
6465:0.1344
6400:0.1833
6389:0.3973
6359:0.2179
6348:0.0598
6083:0.0908
6075:2.4234
5992:2.4234
5986:Total
5918:0.3584
5844:0.3584
5770:0.0896
5696:0.0490
5622:0.1344
5548:0.3584
5474:0.3584
5400:0.3584
5326:0.3584
4786:where
4110:, and
3645:where
2134:Update
813:where
89:hidden
31:, the
7760:S2CID
7681:(PDF)
7655:(PDF)
7648:(PDF)
7623:(PDF)
7401:Julia
7118:Eggs
7077:Eggs
7036:Eggs
5282:then
5078:then
4984:Eggs
3915:final
3888:Note:
8129:PMID
8067:PMID
8026:PMID
7985:2013
7954:PMID
7891:PMID
7840:ISBN
7689:2013
7663:2019
7631:2019
7584:ISBN
7559:PMID
7501:2013
7431:Rust
7408:RHmm
7391:CTMC
7362:ghmm
7304:The
7271:The
7058:0.2
7047:0.7
6321:0.7
6310:0.5
6219:and
6072:0.22
5989:0.22
5031:0.8
5023:0.2
5006:0.2
4995:0.7
4965:0.7
4954:0.5
4169:and
3943:true
3923:>
3076:and
2971:and
2931:and
1948:as,
1680:Let
1187:Let
1134:Set
705:and
109:Let
59:and
27:and
8119:PMC
8109:doi
8097:104
8057:doi
8018:doi
8004:268
7944:PMC
7936:doi
7881:PMC
7873:doi
7832:doi
7752:doi
7715:doi
7551:doi
7547:190
7524:doi
7429:in
7420:in
7387:MDP
7379:HMM
7355:in
7055:0.8
7044:0.3
6318:0.3
6307:0.5
6250:to
6188:to
6126:to
6031:to
5003:0.8
4992:0.3
4962:0.3
4951:0.5
3962:not
19:In
8227::
8127:.
8117:.
8107:.
8095:.
8091:.
8065:.
8051:.
8047:.
8024:.
8016:.
8002:.
7952:.
7942:.
7932:23
7930:.
7926:.
7889:.
7879:.
7869:26
7867:.
7863:.
7838:.
7781:.
7758:.
7750:.
7738:77
7736:.
7711:23
7709:.
7557:.
7545:.
7520:21
7518:.
7393:).
7389:,
7385:,
7383:MC
7381:,
7357:C#
7343:.
6953:.
6736:EN
6599:EE
6462:NE
6157:,
5912:NN
5838:NN
5764:EN
5690:EE
5616:NE
5542:NN
5468:NN
5394:NN
5320:NN
4090:,
3881:.
3281:.
3155:.
3037:.
980:.
83:A
23:,
8135:.
8111::
8103::
8073:.
8059::
8053:8
8032:.
8020::
7987:.
7960:.
7938::
7897:.
7875::
7848:.
7834::
7815:.
7800:.
7785:.
7783:3
7766:.
7754::
7721:.
7717::
7691:.
7665:.
7633:.
7592:.
7565:.
7553::
7530:.
7526::
7503:.
7414:.
7412:R
7403:.
7196:2
7192:S
7165:1
7161:S
7002:2
6998:S
6971:1
6967:S
6938:=
6904:1
6900:S
6853:2
6849:S
6838:,
6822:2
6818:S
6787:2
6783:S
6772:,
6756:1
6752:S
6716:1
6712:S
6701:,
6685:1
6681:S
6650:1
6646:S
6635:,
6619:1
6615:S
6579:1
6575:S
6564:,
6548:2
6544:S
6513:1
6509:S
6498:,
6482:2
6478:S
6439:1
6435:S
6265:1
6261:S
6234:1
6230:S
6203:2
6199:S
6172:2
6168:S
6141:1
6137:S
6110:2
6106:S
6080:=
6046:2
6042:S
6015:1
6011:S
5966:2
5962:S
5951:,
5935:2
5931:S
5892:2
5888:S
5877:,
5861:2
5857:S
5818:2
5814:S
5803:,
5787:2
5783:S
5744:1
5740:S
5729:,
5713:1
5709:S
5670:1
5666:S
5655:,
5639:2
5635:S
5596:2
5592:S
5581:,
5565:2
5561:S
5522:2
5518:S
5507:,
5491:2
5487:S
5448:2
5444:S
5433:,
5417:2
5413:S
5374:2
5370:S
5359:,
5343:2
5339:S
5297:2
5293:S
5266:1
5262:S
5231:.
5228:)
5223:2
5219:S
5214:|
5210:N
5207:(
5204:P
5198:)
5193:2
5189:S
5180:1
5176:S
5172:(
5169:P
5163:)
5158:1
5154:S
5149:|
5145:N
5142:(
5139:P
5133:)
5128:1
5124:S
5120:(
5117:P
5093:2
5089:S
5062:1
5058:S
4887:0
4880:,
4875:k
4871:v
4867:=
4862:r
4859:,
4856:t
4852:y
4841:1
4835:{
4830:=
4823:k
4819:v
4815:=
4810:r
4807:t
4803:y
4798:1
4771:,
4765:)
4762:t
4759:(
4754:r
4751:i
4741:T
4736:1
4733:=
4730:t
4720:R
4715:1
4712:=
4709:r
4699:)
4696:t
4693:(
4688:r
4685:i
4673:k
4669:v
4665:=
4660:r
4657:t
4653:y
4648:1
4642:T
4637:1
4634:=
4631:t
4621:R
4616:1
4613:=
4610:r
4599:=
4596:)
4591:k
4587:v
4583:(
4573:i
4569:b
4547:,
4541:)
4538:t
4535:(
4530:r
4527:i
4517:1
4511:T
4506:1
4503:=
4500:t
4490:R
4485:1
4482:=
4479:r
4469:)
4466:t
4463:(
4458:r
4455:j
4452:i
4442:1
4436:T
4431:1
4428:=
4425:t
4415:R
4410:1
4407:=
4404:r
4393:=
4383:j
4380:i
4376:a
4352:R
4348:)
4345:1
4342:(
4337:r
4334:i
4324:R
4319:1
4316:=
4313:r
4302:=
4292:i
4262:r
4259:,
4254:r
4250:N
4245:y
4241:,
4235:,
4230:r
4227:,
4224:1
4220:y
4199:)
4196:t
4193:(
4188:r
4185:j
4182:i
4157:)
4154:t
4151:(
4146:r
4143:i
4118:b
4078:A
4056:R
4052:Y
4048:,
4042:,
4037:1
4033:Y
4010:N
4006:y
4002:,
3996:,
3991:1
3987:y
3983:=
3980:Y
3948:)
3932:Y
3929:(
3926:P
3920:)
3904:Y
3901:(
3898:P
3869:i
3849:i
3827:k
3823:v
3802:)
3797:k
3793:v
3789:(
3779:i
3775:b
3737:0
3730:,
3725:k
3721:v
3717:=
3712:t
3708:y
3697:1
3691:{
3686:=
3679:k
3675:v
3671:=
3666:t
3662:y
3657:1
3630:,
3624:)
3621:t
3618:(
3613:i
3603:T
3598:1
3595:=
3592:t
3582:)
3579:t
3576:(
3571:i
3559:k
3555:v
3551:=
3546:t
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3526:1
3523:=
3520:t
3509:=
3506:)
3501:k
3497:v
3493:(
3483:i
3479:b
3465:T
3461:t
3457:t
3453:i
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3445:i
3441:i
3437:j
3433:i
3416:,
3410:)
3407:t
3404:(
3399:i
3389:1
3383:T
3378:1
3375:=
3372:t
3362:)
3359:t
3356:(
3351:j
3348:i
3338:1
3332:T
3327:1
3324:=
3321:t
3310:=
3300:j
3297:i
3293:a
3269:1
3249:i
3226:,
3223:)
3220:1
3217:(
3212:i
3204:=
3194:i
3123:Y
3103:)
3100:t
3097:(
3092:j
3089:i
3064:)
3061:t
3058:(
3053:i
3005:Y
2985:1
2982:+
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2959:t
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2919:i
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2890:)
2885:1
2882:+
2879:t
2875:y
2871:(
2866:w
2862:b
2858:)
2855:1
2852:+
2849:t
2846:(
2841:w
2831:w
2828:k
2824:a
2820:)
2817:t
2814:(
2809:k
2799:N
2794:1
2791:=
2788:w
2778:N
2773:1
2770:=
2767:k
2757:)
2752:1
2749:+
2746:t
2742:y
2738:(
2733:j
2729:b
2725:)
2722:1
2719:+
2716:t
2713:(
2708:j
2698:j
2695:i
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2687:)
2684:t
2681:(
2676:i
2665:=
2659:)
2650:Y
2647:(
2644:P
2639:)
2630:Y
2627:,
2624:j
2621:=
2616:1
2613:+
2610:t
2606:X
2602:,
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2596:=
2591:t
2587:X
2583:(
2580:P
2574:=
2571:)
2565:,
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2556:j
2553:=
2548:1
2545:+
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2531:i
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2523:t
2519:X
2515:(
2512:P
2509:=
2506:)
2503:t
2500:(
2495:j
2492:i
2445:Y
2425:t
2405:i
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2376:)
2373:t
2370:(
2365:j
2357:)
2354:t
2351:(
2346:j
2336:N
2331:1
2328:=
2325:j
2315:)
2312:t
2309:(
2304:i
2296:)
2293:t
2290:(
2285:i
2274:=
2268:)
2259:Y
2256:(
2253:P
2248:)
2239:Y
2236:,
2233:i
2230:=
2225:t
2221:X
2217:(
2214:P
2208:=
2205:)
2199:,
2196:Y
2190:i
2187:=
2182:t
2178:X
2174:(
2171:P
2168:=
2165:)
2162:t
2159:(
2154:i
2118:.
2115:)
2110:1
2107:+
2104:t
2100:y
2096:(
2091:j
2087:b
2081:j
2078:i
2074:a
2070:)
2067:1
2064:+
2061:t
2058:(
2053:j
2043:N
2038:1
2035:=
2032:j
2024:=
2021:)
2018:t
2015:(
2010:i
1984:,
1981:1
1978:=
1975:)
1972:T
1969:(
1964:i
1936:)
1933:t
1930:(
1925:i
1900:t
1880:i
1858:T
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1850:,
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1839:1
1836:+
1833:t
1829:y
1808:)
1802:,
1799:i
1796:=
1791:t
1787:X
1778:T
1774:y
1770:=
1765:T
1761:Y
1757:,
1751:,
1746:1
1743:+
1740:t
1736:y
1732:=
1727:1
1724:+
1721:t
1717:Y
1713:(
1710:P
1707:=
1704:)
1701:t
1698:(
1693:i
1617:.
1612:i
1609:j
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1601:)
1598:t
1595:(
1590:j
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1575:1
1572:=
1569:j
1561:)
1556:1
1553:+
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1542:(
1537:i
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1529:=
1526:)
1523:1
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1514:(
1509:i
1483:,
1480:)
1475:1
1471:y
1467:(
1462:i
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1438:1
1435:(
1430:i
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1346:,
1341:2
1337:y
1333:,
1328:1
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1303:)
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1291:=
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1278:,
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1265:=
1260:t
1256:Y
1252:,
1246:,
1241:1
1237:y
1233:=
1228:1
1224:Y
1220:(
1217:P
1214:=
1211:)
1208:t
1205:(
1200:i
1166:)
1160:,
1157:B
1154:,
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1145:=
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1079:P
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1063:a
1060:m
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1015:)
1009:,
1006:B
1003:,
1000:A
997:(
994:=
968:)
963:T
959:y
955:=
950:T
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942:,
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931:2
927:y
923:=
918:2
914:Y
910:,
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881:=
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801:}
798:)
793:i
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780:j
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654:=
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284:}
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164:N
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122:t
118:X
97:i
93:i
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