183:
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133:
3075:{\displaystyle {\begin{aligned}A&={\frac {F_{0}/x_{0}^{m}}{m+1}}\cdot (x_{1}^{m+1}-x_{0}^{m+1})\\\log A&=\log \left\\&=\log {\frac {F_{0}}{m+1}}-\log {\frac {1}{x_{0}^{m}}}+\log(x_{1}^{m+1}-x_{0}^{m+1})\\&=\log {\frac {F_{0}}{m+1}}+\log \left({\frac {x_{1}^{m+1}-x_{0}^{m+1}}{x_{0}^{m}}}\right)\\&=\log {\frac {F_{0}}{m+1}}+\log \left({\frac {x_{1}^{m}}{x_{0}^{m}}}\cdot x_{1}-{\frac {x_{0}^{m+1}}{x_{0}^{m}}}\right)\end{aligned}}}
36:
4012:- MALE) calculated over a sliding window of size 28 on the x-axis. The y-axis gives the error, plotted against the independent variable (x). Each error metric is represented by a different color, with the corresponding smoothed line overlaying the original line (since this is just simulated data, the error estimation is a bit jumpy). These error metrics provide a measure of the noise as it varies across different x values.
3219:
3630:{\displaystyle {\begin{aligned}A_{(m=-1)}&=\int _{x_{0}}^{x_{1}}F(x)\,dx=\int _{x_{0}}^{x_{1}}{\frac {\mathrm {constant} }{x}}\,dx={\frac {F_{0}}{x_{0}^{-1}}}\int _{x_{0}}^{x_{1}}{\frac {dx}{x}}=F_{0}\cdot x_{0}\cdot {\ln x}{\Big |}_{x_{0}}^{x_{1}}\\A_{(m=-1)}&=F_{0}\cdot x_{0}\cdot \ln {\frac {x_{1}}{x_{0}}}\end{aligned}}}
3968:
Figure 1 illustrates how this looks. It presents two plots generated using 10,000 simulated points. The left plot, titled 'Concave Line with Log-Normal Noise', displays a scatter plot of the observed data (y) against the independent variable (x). The red line represents the 'Median line', while the
3972:
When both variables are log-transformed, as shown in the right plot of Figure 1, titled 'Log-Log Linear Line with Normal Noise', the relationship becomes linear. This plot also displays a scatter plot of the observed data against the independent variable, but after both axes are on a logarithmic
4015:
Log-log linear models are widely used in various fields, including economics, biology, and physics, where many phenomena exhibit power-law behavior. They are also useful in regression analysis when dealing with heteroscedastic data, as the log transformation can help to stabilize the variance.
2303:
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1968:
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As above, in a log-log linear model the relationship between the variables is expressed as a power law. Every unit change in the independent variable will result in a constant percentage change in the dependent variable. The model is expressed as:
4614:, as these tests may have low likelihood of rejecting power laws in the presence of other true functional forms. While simple log–log plots may be instructive in detecting possible power laws, and have been used dating back to
2411:
1629:
250:– appear as straight lines in a log–log graph, with the exponent corresponding to the slope, and the coefficient corresponding to the intercept. Thus these graphs are very useful for recognizing these relationships and
3988:
The transformation from the left plot to the right plot in Figure 1 also demonstrates the effect of the log transformation on the distribution of noise in the data. In the left plot, the noise appears to follow a
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However, going in the other direction – observing that data appears as an approximate line on a log–log scale and concluding that the data follows a power law – is not always valid.
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4610:) may be invalid, as the assumptions of the linear regression model, such as Gaussian error, may not be satisfied; in addition, tests of fit of the log–log form may exhibit low
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To calculate the area under a continuous, straight-line segment of a log–log plot (or estimating an area of an almost-straight line), take the function defined previously
424:
299:
389:
248:
4437:
4457:
459:
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4005:
3660:. This model is useful when dealing with data that exhibits exponential growth or decay, while the errors continue to grow as the independent value grows (i.e.,
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These graphs are also extremely useful when data are gathered by varying the control variable along an exponential function, in which case the control variable
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is more naturally represented on a log scale, so that the data points are evenly spaced, rather than compressed at the low end. The output variable
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blue line is the 'Mean line'. This plot illustrates a dataset with a power-law relationship between the variables, represented by a concave line.
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3993:, which is right-skewed and can be difficult to work with. In the right plot, after the log transformation, the noise appears to follow a
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to the power of the slope of the straight line of its log–log graph. Specifically, a straight line on a log–log plot containing points (
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and integrate it. Since it is only operating on a definite integral (two defined endpoints), the area A under the plot takes the form
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514:
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72:
2298:{\displaystyle A(x)=\int _{x_{0}}^{x_{1}}F(x)\,dx=\left.{\frac {\mathrm {constant} }{m+1}}\cdot x^{m+1}\right|_{x_{0}}^{x_{1}}}
3732:
586:
3815:
1063:{\displaystyle m={\frac {\log(F_{2})-\log(F_{1})}{\log(x_{2})-\log(x_{1})}}={\frac {\log(F_{2}/F_{1})}{\log(x_{2}/x_{1})}},}
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79:
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This is a linear equation in the logarithms of `x` and `y`, with `log(a)` as the intercept and `b` as the slope. In which
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In fact, many other functional forms appear approximately linear on the log–log scale, and simply evaluating the
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This normalization of noise is further analyzed in Figure 2, which presents a line plot of three error metrics (
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scale. Here, both the mean and median lines are the same (red) line. This transformation allows us to fit a
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1963:{\displaystyle F(x)={F_{0}}\left({\frac {x}{x_{0}}}\right)^{\frac {\log(F_{1}/F_{0})}{\log(x_{1}/x_{0})}},}
1110:. The formula also provides a negative slope, as can be seen from the following property of the logarithm:
1106:). The figure at right illustrates the formula. Notice that the slope in the example of the figure is
4713:
3649:
4761:
Clauset, A.; Shalizi, C. R.; Newman, M. E. J. (2009). "Power-Law
Distributions in Empirical Data".
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will have a straight line as its log–log graph representation, where the slope of the line is
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93:
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A log-log plot condensing information that spans more than one order of magnitude along both axes
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46:
17:
186:
Comparison of Linear, Concave, and Convex
Functions\nIn original (left) and log10 (right) scales
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4180:
394:
265:
254:. Any base can be used for the logarithm, though most commonly base 10 (common logs) are used.
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need to be estimated from numerical data. Specifications such as this are used frequently in
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3652:, errors. In such models, after log-transforming the dependent and independent variables, a
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Log–log plots are often use for visualizing log-log linear regression models with (roughly)
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Rearranging the original equation and plugging in the fixed point values, it is found that
8:
4309:
3994:
4784:
4796:
4770:
4678:), so a log-log plot is useful for estimating the reaction parameters from experiment.
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model (which can then be transformed back to the original scale - as the median line).
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in the 1890s, validation as a power laws requires more sophisticated statistics.
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are parameters to be estimated. Taking logs gives the linear regression equation
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2048:
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Note the logarithmic scale markings on each of the axes, and that the log
1624:{\displaystyle {\frac {F_{1}}{F_{0}}}=\left({\frac {x_{1}}{x_{0}}}\right)^{m}}
4826:
4687:
4671:
1265:), somewhere on the straight line in the above graph, and further some other
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4641:), or its logarithm can also be taken, yielding the log–log graph (log
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on an alternative, higher yielding asset in excess of that on money,
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The above procedure now is reversed to find the form of the function
35:
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4407:
is the number of hours of labor employed in production per month,
4585:
191:
1381:{\displaystyle m={\frac {\log(F_{1}/F_{0})}{\log(x_{1}/x_{0})}}}
426:
which corresponds to using a log–log graph, yields the equation
4411:
is the number of hours of physical capital utilized per month,
1225:) using its (assumed) known log–log plot. To find the function
3950:{\displaystyle e^{\epsilon }\sim Log-Normal(\mu ,\sigma ^{2})}
2375:{\displaystyle \mathrm {constant} ={\frac {F_{0}}{x_{0}^{m}}}}
2049:
Finding the area under a straight-line segment of log–log plot
665:
To find the slope of the plot, two points are selected on the
301:
taking the logarithm of the equation (with any base) yields:
4140:
1970:
Of course, the inverse is true too: any function of the form
1212:
4415:
is an error term assumed to be lognormally distributed, and
2202:
4403:
is the quantity of output that can be produced per month,
4392:{\displaystyle Q_{t}=AN_{t}^{\alpha }K_{t}^{\beta }U_{t},}
1534:{\displaystyle \log(F_{1}/F_{0})=m\log(x_{1}/x_{0})=\log.}
3984:
Figure 2: Sliding Window Error
Metrics Loglog Normal Data
1283:) on the same graph. Then from the slope formula above:
3802:{\displaystyle \log(y)=\log(a)+b\cdot \log(x)+\epsilon }
1697:{\displaystyle F_{1}={\frac {F_{0}}{x_{0}^{m}}}\,x^{m},}
4319:
Another economic example is the estimation of a firm's
4055:, in which it can be assumed that money demand at time
2384:
Substituting back into the integral, you find that for
206:
is a two-dimensional graph of numerical data that uses
4530:{\displaystyle q_{t}=a+\alpha n_{t}+\beta k_{t}+u_{t}}
3868:{\displaystyle \epsilon \sim Normal(\mu ,\sigma ^{2})}
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1203:{\displaystyle \log(x_{1}/x_{2})=-\log(x_{2}/x_{1}).}
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The equation for a line on a log–log scale would be:
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Log–log regression can also be used to estimate the
2120:{\displaystyle F(x)=\mathrm {constant} \cdot x^{m}.}
1771:{\displaystyle F(x)=\mathrm {constant} \cdot x^{m}.}
3639:
2034:{\displaystyle F(x)=\mathrm {constant} \cdot x^{m}}
60:. Unsourced material may be challenged and removed.
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3716:{\displaystyle y=a\cdot x^{b}\cdot e^{\epsilon }}
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3203:{\displaystyle A={\frac {F_{0}}{m+1}}\cdot \left}
4824:
4745:Graphs on Logarithmic and Semi-Logarithmic Paper
4674:on concentration takes the form of a power law (
661:Finding the slope of a log–log plot using ratios
4629:can either be represented linearly, yielding a
4183:parameters to be estimated. Taking logs yields
4128:{\displaystyle M_{t}=AR_{t}^{b}Y_{t}^{c}U_{t},}
1548:. Therefore, the logs can be inverted to find:
27:2D graphic with logarithmic scales on both axes
3656:model can be fitted, with the errors becoming
577:{\displaystyle \log _{10}F(x)=m\log _{10}x+b,}
3997:, which is easier to reason about and model.
4670:, the general form of the dependence of the
4257:{\displaystyle m_{t}=a+br_{t}+cy_{t}+u_{t},}
4032:These graphs are useful when the parameters
3726:Taking the logarithm of both sides, we get:
170:axes (where the logarithms are 0) are where
4323:, which is the right side of the equation
4171:is a scale parameter to be estimated, and
1213:Finding the function from the log–log plot
652:
257:
210:on both the horizontal and vertical axes.
4774:
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4754:
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1680:
120:Learn how and when to remove this message
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3964:Figure 1: Visualizing Loglog Normal Data
3959:
656:
649:is the intercept point on the log plot.
491: = 0, so, reversing the logs,
181:
131:
4312:. This equation can be estimated using
634:{\displaystyle F(x)=x^{m}\cdot 10^{b},}
14:
4825:
4751:
3216: = −1, the integral becomes
350:{\displaystyle \log y=k\log x+\log a.}
846:{\displaystyle \log=m\log(x_{2})+b.}
761:{\displaystyle \log=m\log(x_{1})+b,}
58:adding citations to reliable sources
29:
4724:Variance-stabilizing transformation
4663:of a system) is also log–log plot.
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483:is the intercept on the (log
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4833:Logarithmic scales of measurement
4811:
4047:One example is the estimation of
4719:Data transformation (statistics)
4321:Cobb–Douglas production function
3640:Log-log linear regression models
857:is found taking the difference:
34:
4838:Statistical charts and diagrams
4163:is an error term assumed to be
4019:
4010:Mean Absolute Logarithmic Error
487:)-axis, meaning where log
45:needs additional citations for
4818:Non-Newtonian calculus website
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683:. Using the below equation:
4604:coefficient of determination
506:
214:– relationships of the form
7:
4681:
10:
4859:
1814:) will have the function:
471:is the slope of the line (
262:Given a monomial equation
4714:Log-logistic distribution
4602:on logged data using the
4584:of a naturally occurring
419:{\displaystyle Y=\log y,}
294:{\displaystyle y=ax^{k},}
4139:is the real quantity of
3975:Simple linear regression
3654:Simple linear regression
384:{\displaystyle X=\log x}
243:{\displaystyle y=ax^{k}}
4709:Log-normal distribution
4432:{\displaystyle \alpha }
4165:lognormally distributed
3991:log-normal distribution
653:Slope of a log–log plot
499:value corresponding to
258:Relation with monomials
4843:Non-Newtonian calculus
4531:
4453:
4452:{\displaystyle \beta }
4433:
4393:
4314:ordinary least squares
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4029:
4006:Root Mean Square Error
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455:
454:{\displaystyle Y=mX+b}
420:
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252:estimating parameters
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185:
135:
4690:(lin–log or log–lin)
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4310:normally distributed
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4143:held by the public,
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54:improve this article
4785:2009SIAMR..51..661C
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4051:functions based on
4002:Mean Absolute Error
3995:normal distribution
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208:logarithmic scales
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136:A log–log plot of
4793:10.1137/070710111
4747:(www.intmath.com)
4668:chemical kinetics
4612:statistical power
4600:linear regression
4582:fractal dimension
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1704:which means that
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1541:Notice that 10 =
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1254:is shorthand for
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1077:is shorthand for
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645:is the slope and
178:themselves are 1.
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16:(Redirected from
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4704:Log-linear model
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2713:
2702:
2700:
2689:
2688:
2679:
2665:
2661:
2657:
2652:
2641:
2628:
2617:
2602:
2600:
2589:
2587:
2582:
2573:
2568:
2567:
2557:
2520:
2509:
2496:
2485:
2470:
2468:
2457:
2455:
2450:
2441:
2436:
2435:
2425:
2381:
2379:
2378:
2373:
2371:
2368:
2363:
2354:
2353:
2344:
2339:
2304:
2302:
2301:
2296:
2293:
2292:
2291:
2281:
2280:
2279:
2269:
2265:
2264:
2263:
2245:
2243:
2232:
2206:
2176:
2175:
2174:
2164:
2163:
2162:
2126:
2124:
2123:
2118:
2113:
2112:
2100:
2040:
2038:
2037:
2032:
2030:
2029:
2017:
1969:
1967:
1966:
1961:
1956:
1955:
1953:
1949:
1948:
1939:
1934:
1933:
1914:
1910:
1909:
1900:
1895:
1894:
1875:
1873:
1869:
1867:
1866:
1854:
1847:
1846:
1845:
1778:In other words,
1777:
1775:
1774:
1769:
1764:
1763:
1751:
1703:
1701:
1700:
1695:
1690:
1689:
1679:
1676:
1671:
1662:
1661:
1652:
1647:
1646:
1630:
1628:
1627:
1622:
1620:
1619:
1614:
1610:
1608:
1607:
1598:
1597:
1588:
1578:
1576:
1575:
1566:
1565:
1556:
1540:
1538:
1537:
1532:
1524:
1523:
1514:
1513:
1504:
1499:
1498:
1471:
1470:
1461:
1456:
1455:
1428:
1427:
1418:
1413:
1412:
1387:
1385:
1384:
1379:
1377:
1375:
1371:
1370:
1361:
1356:
1355:
1336:
1332:
1331:
1322:
1317:
1316:
1297:
1209:
1207:
1206:
1201:
1193:
1192:
1183:
1178:
1177:
1150:
1149:
1140:
1135:
1134:
1069:
1067:
1066:
1061:
1056:
1054:
1050:
1049:
1040:
1035:
1034:
1015:
1011:
1010:
1001:
996:
995:
976:
971:
969:
965:
964:
940:
939:
920:
916:
915:
891:
890:
871:
852:
850:
849:
844:
830:
829:
799:
798:
767:
765:
764:
759:
745:
744:
714:
713:
640:
638:
637:
632:
627:
626:
614:
613:
583:
581:
580:
575:
558:
557:
527:
526:
503: = 1.
460:
458:
457:
452:
425:
423:
422:
417:
390:
388:
387:
382:
356:
354:
353:
348:
300:
298:
297:
292:
287:
286:
249:
247:
246:
241:
239:
238:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
4858:
4857:
4853:
4852:
4851:
4849:
4848:
4847:
4823:
4822:
4814:
4809:
4808:
4759:
4752:
4741:
4737:
4732:
4684:
4596:goodness of fit
4521:
4517:
4508:
4504:
4492:
4488:
4470:
4466:
4464:
4461:
4460:
4444:
4441:
4440:
4424:
4421:
4420:
4380:
4376:
4370:
4365:
4355:
4350:
4334:
4330:
4328:
4325:
4324:
4245:
4241:
4232:
4228:
4216:
4212:
4194:
4190:
4188:
4185:
4184:
4116:
4112:
4106:
4101:
4091:
4086:
4070:
4066:
4064:
4061:
4060:
4022:
3938:
3934:
3886:
3882:
3880:
3877:
3876:
3856:
3852:
3817:
3814:
3813:
3734:
3731:
3730:
3707:
3703:
3694:
3690:
3676:
3673:
3672:
3662:heteroscedastic
3642:
3624:
3623:
3615:
3611:
3605:
3601:
3599:
3584:
3580:
3571:
3567:
3560:
3539:
3535:
3532:
3531:
3523:
3519:
3518:
3511:
3507:
3506:
3500:
3499:
3487:
3478:
3474:
3465:
3461:
3445:
3443:
3435:
3431:
3430:
3423:
3419:
3418:
3403:
3398:
3388:
3384:
3382:
3341:
3339:
3331:
3327:
3326:
3319:
3315:
3314:
3280:
3276:
3275:
3268:
3264:
3263:
3252:
3231:
3227:
3223:
3221:
3218:
3217:
3189:
3185:
3176:
3164:
3160:
3154:
3150:
3148:
3144:
3143:
3134:
3130:
3129:
3125:
3109:
3103:
3099:
3097:
3089:
3086:
3085:
3069:
3068:
3055:
3050:
3034:
3029:
3023:
3014:
3010:
2999:
2994:
2984:
2979:
2973:
2972:
2968:
2946:
2940:
2936:
2934:
2919:
2918:
2906:
2901:
2884:
2879:
2860:
2855:
2850:
2848:
2844:
2822:
2816:
2812:
2810:
2795:
2794:
2779:
2774:
2755:
2750:
2726:
2721:
2712:
2690:
2684:
2680:
2678:
2663:
2662:
2642:
2637:
2618:
2613:
2590:
2583:
2578:
2569:
2563:
2559:
2558:
2556:
2555:
2551:
2538:
2526:
2525:
2510:
2505:
2486:
2481:
2458:
2451:
2446:
2437:
2431:
2427:
2426:
2424:
2417:
2410:
2408:
2405:
2404:
2401:
2394:
2364:
2359:
2349:
2345:
2343:
2314:
2312:
2309:
2308:
2287:
2283:
2282:
2275:
2271:
2270:
2253:
2249:
2233:
2207:
2205:
2204:
2201:
2170:
2166:
2165:
2158:
2154:
2153:
2132:
2129:
2128:
2108:
2104:
2075:
2058:
2055:
2054:
2051:
2025:
2021:
1992:
1975:
1972:
1971:
1944:
1940:
1935:
1929:
1925:
1915:
1905:
1901:
1896:
1890:
1886:
1876:
1874:
1862:
1858:
1853:
1849:
1848:
1841:
1837:
1836:
1819:
1816:
1815:
1813:
1806:
1799:
1792:
1759:
1755:
1726:
1709:
1706:
1705:
1685:
1681:
1672:
1667:
1657:
1653:
1651:
1642:
1638:
1636:
1633:
1632:
1615:
1603:
1599:
1593:
1589:
1587:
1583:
1582:
1571:
1567:
1561:
1557:
1555:
1553:
1550:
1549:
1547:
1519:
1515:
1509:
1505:
1500:
1494:
1490:
1466:
1462:
1457:
1451:
1447:
1423:
1419:
1414:
1408:
1404:
1393:
1390:
1389:
1388:which leads to
1366:
1362:
1357:
1351:
1347:
1337:
1327:
1323:
1318:
1312:
1308:
1298:
1296:
1288:
1285:
1284:
1282:
1275:
1267:arbitrary point
1264:
1253:
1246:
1239:
1215:
1188:
1184:
1179:
1173:
1169:
1145:
1141:
1136:
1130:
1126:
1115:
1112:
1111:
1105:
1094:
1087:
1076:
1045:
1041:
1036:
1030:
1026:
1016:
1006:
1002:
997:
991:
987:
977:
975:
960:
956:
935:
931:
921:
911:
907:
886:
882:
872:
870:
862:
859:
858:
825:
821:
794:
790:
773:
770:
769:
740:
736:
709:
705:
688:
685:
684:
682:
675:
655:
622:
618:
609:
605:
588:
585:
584:
553:
549:
522:
518:
516:
513:
512:
509:
431:
428:
427:
396:
393:
392:
364:
361:
360:
306:
303:
302:
282:
278:
267:
264:
263:
260:
234:
230:
219:
216:
215:
212:Power functions
161:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
4856:
4846:
4845:
4840:
4835:
4821:
4820:
4813:
4812:External links
4810:
4807:
4806:
4769:(4): 661–703.
4750:
4734:
4733:
4731:
4728:
4727:
4726:
4721:
4716:
4711:
4706:
4701:
4696:
4691:
4683:
4680:
4524:
4520:
4516:
4511:
4507:
4503:
4500:
4495:
4491:
4487:
4484:
4481:
4478:
4473:
4469:
4448:
4428:
4388:
4383:
4379:
4373:
4368:
4364:
4358:
4353:
4349:
4345:
4342:
4337:
4333:
4253:
4248:
4244:
4240:
4235:
4231:
4227:
4224:
4219:
4215:
4211:
4208:
4205:
4202:
4197:
4193:
4149:rate of return
4124:
4119:
4115:
4109:
4104:
4100:
4094:
4089:
4085:
4081:
4078:
4073:
4069:
4021:
4018:
3946:
3941:
3937:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3889:
3885:
3864:
3859:
3855:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3810:
3809:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3724:
3723:
3710:
3706:
3702:
3697:
3693:
3689:
3686:
3683:
3680:
3641:
3638:
3618:
3614:
3608:
3604:
3598:
3595:
3592:
3587:
3583:
3579:
3574:
3570:
3566:
3563:
3561:
3557:
3554:
3551:
3548:
3545:
3542:
3538:
3534:
3533:
3526:
3522:
3514:
3510:
3503:
3496:
3493:
3490:
3486:
3481:
3477:
3473:
3468:
3464:
3460:
3455:
3451:
3448:
3438:
3434:
3426:
3422:
3417:
3409:
3406:
3401:
3397:
3391:
3387:
3381:
3378:
3375:
3369:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3334:
3330:
3322:
3318:
3313:
3309:
3306:
3303:
3299:
3296:
3293:
3290:
3283:
3279:
3271:
3267:
3262:
3258:
3255:
3253:
3249:
3246:
3243:
3240:
3237:
3234:
3230:
3226:
3225:
3198:
3192:
3188:
3184:
3179:
3174:
3167:
3163:
3157:
3153:
3147:
3142:
3137:
3133:
3128:
3124:
3118:
3115:
3112:
3106:
3102:
3096:
3093:
3066:
3058:
3053:
3049:
3043:
3040:
3037:
3032:
3028:
3022:
3017:
3013:
3009:
3002:
2997:
2993:
2987:
2982:
2978:
2971:
2967:
2964:
2961:
2955:
2952:
2949:
2943:
2939:
2933:
2930:
2927:
2924:
2922:
2920:
2916:
2909:
2904:
2900:
2893:
2890:
2887:
2882:
2878:
2874:
2869:
2866:
2863:
2858:
2854:
2847:
2843:
2840:
2837:
2831:
2828:
2825:
2819:
2815:
2809:
2806:
2803:
2800:
2798:
2796:
2793:
2788:
2785:
2782:
2777:
2773:
2769:
2764:
2761:
2758:
2753:
2749:
2745:
2742:
2739:
2736:
2729:
2724:
2720:
2716:
2711:
2708:
2705:
2699:
2696:
2693:
2687:
2683:
2677:
2674:
2671:
2668:
2666:
2664:
2660:
2656:
2651:
2648:
2645:
2640:
2636:
2632:
2627:
2624:
2621:
2616:
2612:
2608:
2605:
2599:
2596:
2593:
2586:
2581:
2577:
2572:
2566:
2562:
2554:
2550:
2547:
2544:
2541:
2539:
2537:
2534:
2531:
2528:
2527:
2524:
2519:
2516:
2513:
2508:
2504:
2500:
2495:
2492:
2489:
2484:
2480:
2476:
2473:
2467:
2464:
2461:
2454:
2449:
2445:
2440:
2434:
2430:
2423:
2420:
2418:
2416:
2413:
2412:
2399:
2392:
2367:
2362:
2358:
2352:
2348:
2342:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2290:
2286:
2278:
2274:
2268:
2262:
2259:
2256:
2252:
2248:
2242:
2239:
2236:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2203:
2199:
2196:
2193:
2189:
2186:
2183:
2180:
2173:
2169:
2161:
2157:
2152:
2148:
2145:
2142:
2139:
2136:
2116:
2111:
2107:
2103:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2074:
2071:
2068:
2065:
2062:
2050:
2047:
2028:
2024:
2020:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1991:
1988:
1985:
1982:
1979:
1959:
1952:
1947:
1943:
1938:
1932:
1928:
1924:
1921:
1918:
1913:
1908:
1904:
1899:
1893:
1889:
1885:
1882:
1879:
1872:
1865:
1861:
1857:
1852:
1844:
1840:
1835:
1832:
1829:
1826:
1823:
1811:
1804:
1797:
1790:
1767:
1762:
1758:
1754:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1725:
1722:
1719:
1716:
1713:
1693:
1688:
1684:
1675:
1670:
1666:
1660:
1656:
1650:
1645:
1641:
1618:
1613:
1606:
1602:
1596:
1592:
1586:
1581:
1574:
1570:
1564:
1560:
1545:
1530:
1527:
1522:
1518:
1512:
1508:
1503:
1497:
1493:
1489:
1486:
1483:
1480:
1477:
1474:
1469:
1465:
1460:
1454:
1450:
1446:
1443:
1440:
1437:
1434:
1431:
1426:
1422:
1417:
1411:
1407:
1403:
1400:
1397:
1374:
1369:
1365:
1360:
1354:
1350:
1346:
1343:
1340:
1335:
1330:
1326:
1321:
1315:
1311:
1307:
1304:
1301:
1295:
1292:
1280:
1273:
1262:
1251:
1244:
1237:
1214:
1211:
1199:
1196:
1191:
1187:
1182:
1176:
1172:
1168:
1165:
1162:
1159:
1156:
1153:
1148:
1144:
1139:
1133:
1129:
1125:
1122:
1119:
1103:
1092:
1085:
1074:
1059:
1053:
1048:
1044:
1039:
1033:
1029:
1025:
1022:
1019:
1014:
1009:
1005:
1000:
994:
990:
986:
983:
980:
974:
968:
963:
959:
955:
952:
949:
946:
943:
938:
934:
930:
927:
924:
919:
914:
910:
906:
903:
900:
897:
894:
889:
885:
881:
878:
875:
869:
866:
842:
839:
836:
833:
828:
824:
820:
817:
814:
811:
808:
805:
802:
797:
793:
789:
786:
783:
780:
777:
757:
754:
751:
748:
743:
739:
735:
732:
729:
726:
723:
720:
717:
712:
708:
704:
701:
698:
695:
692:
680:
673:
654:
651:
630:
625:
621:
617:
612:
608:
604:
601:
598:
595:
592:
573:
570:
567:
564:
561:
556:
552:
548:
545:
542:
539:
536:
533:
530:
525:
521:
508:
505:
450:
447:
444:
441:
438:
435:
415:
412:
409:
406:
403:
400:
380:
377:
374:
371:
368:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
290:
285:
281:
277:
274:
271:
259:
256:
237:
233:
229:
226:
223:
144: (blue),
128:
127:
69:"Log–log plot"
42:
40:
33:
26:
9:
6:
4:
3:
2:
4855:
4844:
4841:
4839:
4836:
4834:
4831:
4830:
4828:
4819:
4816:
4815:
4802:
4798:
4794:
4790:
4786:
4782:
4777:
4772:
4768:
4764:
4757:
4755:
4748:
4746:
4739:
4735:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4692:
4689:
4688:Semi-log plot
4686:
4685:
4679:
4677:
4673:
4672:reaction rate
4669:
4664:
4662:
4658:
4654:
4650:
4648:
4644:
4640:
4636:
4632:
4631:lin–log graph
4628:
4624:
4619:
4617:
4613:
4609:
4605:
4601:
4597:
4592:
4589:
4587:
4583:
4578:
4576:
4572:
4568:
4564:
4560:
4556:
4552:
4548:
4544:
4540:
4522:
4518:
4514:
4509:
4505:
4501:
4498:
4493:
4489:
4485:
4482:
4479:
4476:
4471:
4467:
4446:
4426:
4418:
4414:
4410:
4406:
4402:
4386:
4381:
4377:
4371:
4366:
4362:
4356:
4351:
4347:
4343:
4340:
4335:
4331:
4322:
4317:
4315:
4311:
4307:
4303:
4299:
4295:
4291:
4287:
4283:
4279:
4275:
4271:
4267:
4251:
4246:
4242:
4238:
4233:
4229:
4225:
4222:
4217:
4213:
4209:
4206:
4203:
4200:
4195:
4191:
4182:
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4150:
4146:
4142:
4138:
4122:
4117:
4113:
4107:
4102:
4098:
4092:
4087:
4083:
4079:
4076:
4071:
4067:
4058:
4054:
4050:
4045:
4043:
4039:
4035:
4026:
4017:
4013:
4011:
4007:
4003:
3998:
3996:
3992:
3982:
3978:
3976:
3970:
3962:
3958:
3939:
3935:
3931:
3928:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3887:
3883:
3857:
3853:
3849:
3846:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3796:
3793:
3787:
3781:
3778:
3775:
3772:
3769:
3763:
3757:
3754:
3751:
3745:
3739:
3736:
3729:
3728:
3727:
3708:
3704:
3700:
3695:
3691:
3687:
3684:
3681:
3678:
3671:
3670:
3669:
3665:
3663:
3659:
3658:homoscedastic
3655:
3651:
3647:
3637:
3616:
3612:
3606:
3602:
3596:
3593:
3590:
3585:
3581:
3577:
3572:
3568:
3564:
3562:
3552:
3549:
3546:
3543:
3536:
3524:
3520:
3512:
3508:
3494:
3491:
3488:
3484:
3479:
3475:
3471:
3466:
3462:
3458:
3453:
3449:
3446:
3436:
3432:
3424:
3420:
3415:
3407:
3404:
3399:
3395:
3389:
3385:
3379:
3376:
3373:
3367:
3332:
3328:
3320:
3316:
3311:
3307:
3304:
3301:
3294:
3288:
3281:
3277:
3269:
3265:
3260:
3256:
3254:
3244:
3241:
3238:
3235:
3228:
3215:
3210:
3196:
3190:
3186:
3182:
3177:
3172:
3165:
3161:
3155:
3151:
3145:
3140:
3135:
3131:
3126:
3122:
3116:
3113:
3110:
3104:
3100:
3094:
3091:
3082:
3064:
3056:
3051:
3047:
3041:
3038:
3035:
3030:
3026:
3020:
3015:
3011:
3007:
3000:
2995:
2991:
2985:
2980:
2976:
2969:
2965:
2962:
2959:
2953:
2950:
2947:
2941:
2937:
2931:
2928:
2925:
2923:
2914:
2907:
2902:
2898:
2891:
2888:
2885:
2880:
2876:
2872:
2867:
2864:
2861:
2856:
2852:
2845:
2841:
2838:
2835:
2829:
2826:
2823:
2817:
2813:
2807:
2804:
2801:
2799:
2786:
2783:
2780:
2775:
2771:
2767:
2762:
2759:
2756:
2751:
2747:
2740:
2737:
2734:
2727:
2722:
2718:
2714:
2709:
2706:
2703:
2697:
2694:
2691:
2685:
2681:
2675:
2672:
2669:
2667:
2658:
2649:
2646:
2643:
2638:
2634:
2630:
2625:
2622:
2619:
2614:
2610:
2603:
2597:
2594:
2591:
2584:
2579:
2575:
2570:
2564:
2560:
2552:
2548:
2545:
2542:
2540:
2535:
2532:
2529:
2517:
2514:
2511:
2506:
2502:
2498:
2493:
2490:
2487:
2482:
2478:
2471:
2465:
2462:
2459:
2452:
2447:
2443:
2438:
2432:
2428:
2421:
2419:
2414:
2402:
2398:
2391:
2387:
2382:
2365:
2360:
2356:
2350:
2346:
2340:
2305:
2288:
2284:
2276:
2272:
2266:
2260:
2257:
2254:
2250:
2246:
2240:
2237:
2234:
2197:
2194:
2191:
2184:
2178:
2171:
2167:
2159:
2155:
2150:
2146:
2140:
2134:
2114:
2109:
2105:
2101:
2072:
2066:
2060:
2046:
2044:
2026:
2022:
2018:
1989:
1983:
1977:
1957:
1945:
1941:
1936:
1930:
1926:
1919:
1916:
1906:
1902:
1897:
1891:
1887:
1880:
1877:
1870:
1863:
1859:
1855:
1850:
1842:
1838:
1833:
1827:
1821:
1810:
1803:
1796:
1789:
1785:
1781:
1765:
1760:
1756:
1752:
1723:
1717:
1711:
1691:
1686:
1682:
1673:
1668:
1664:
1658:
1654:
1648:
1643:
1639:
1616:
1611:
1604:
1600:
1594:
1590:
1584:
1579:
1572:
1568:
1562:
1558:
1544:
1528:
1520:
1510:
1506:
1501:
1495:
1491:
1481:
1478:
1475:
1467:
1463:
1458:
1452:
1448:
1441:
1438:
1435:
1432:
1424:
1420:
1415:
1409:
1405:
1398:
1395:
1367:
1363:
1358:
1352:
1348:
1341:
1338:
1328:
1324:
1319:
1313:
1309:
1302:
1299:
1293:
1290:
1279:
1272:
1268:
1261:
1257:
1250:
1243:
1236:
1232:
1229:, pick some
1228:
1224:
1220:
1210:
1197:
1189:
1185:
1180:
1174:
1170:
1163:
1160:
1157:
1154:
1146:
1142:
1137:
1131:
1127:
1120:
1117:
1109:
1102:
1098:
1091:
1084:
1080:
1073:
1057:
1046:
1042:
1037:
1031:
1027:
1020:
1017:
1007:
1003:
998:
992:
988:
981:
978:
972:
961:
957:
950:
947:
944:
936:
932:
925:
922:
912:
908:
901:
898:
895:
887:
883:
876:
873:
867:
864:
856:
840:
837:
834:
826:
822:
815:
812:
809:
806:
795:
791:
784:
778:
775:
755:
752:
749:
741:
737:
730:
727:
724:
721:
710:
706:
699:
693:
690:
679:
672:
668:
659:
650:
648:
644:
628:
623:
619:
615:
610:
606:
602:
596:
590:
571:
568:
565:
562:
559:
554:
550:
546:
543:
537:
531:
528:
523:
519:
504:
502:
498:
494:
490:
486:
482:
478:
474:
470:
467: =
466:
461:
448:
445:
442:
439:
436:
433:
413:
410:
407:
404:
401:
398:
378:
375:
372:
369:
366:
357:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
288:
283:
279:
275:
272:
269:
255:
253:
235:
231:
227:
224:
221:
213:
209:
205:
201:
200:log–log graph
197:
193:
184:
177:
173:
169:
166:and log
165:
159:
156: =
155:
151:
148: =
147:
143:
140: =
139:
134:
124:
121:
113:
110:December 2009
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
4766:
4762:
4744:
4738:
4665:
4651:
4646:
4642:
4638:
4634:
4626:
4622:
4620:
4607:
4593:
4590:
4579:
4574:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4538:
4416:
4412:
4408:
4404:
4400:
4318:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4277:
4273:
4269:
4265:
4176:
4172:
4168:
4160:
4152:
4144:
4136:
4059:is given by
4056:
4049:money demand
4046:
4037:
4033:
4031:
4020:Applications
4014:
4008:- RMSE, and
3999:
3987:
3971:
3967:
3811:
3725:
3666:
3650:Log-logistic
3643:
3213:
3211:
3083:
2403:
2396:
2389:
2385:
2383:
2306:
2052:
2042:
1808:
1801:
1794:
1787:
1783:
1779:
1542:
1277:
1270:
1266:
1259:
1255:
1248:
1241:
1234:
1230:
1226:
1222:
1218:
1216:
1107:
1100:
1096:
1089:
1082:
1078:
1071:
854:
677:
670:
666:
664:
646:
642:
510:
500:
496:
492:
488:
484:
480:
476:
468:
464:
462:
358:
261:
204:log–log plot
203:
199:
189:
175:
171:
167:
163:
160: (red).
157:
153:
149:
145:
141:
137:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
4763:SIAM Review
4645:, log
4157:real income
3084:Therefore,
1231:fixed point
669:-axis, say
196:engineering
4827:Categories
4743:M. Bourne
4730:References
4633:(log
4181:elasticity
3646:log-normal
853:The slope
80:newspapers
4776:0706.1062
4694:Power law
4653:Bode plot
4502:β
4486:α
4447:β
4427:α
4399:in which
4372:β
4357:α
4042:economics
3936:σ
3929:μ
3905:−
3893:∼
3888:ϵ
3854:σ
3847:μ
3823:∼
3820:ϵ
3797:ϵ
3782:
3776:⋅
3758:
3740:
3709:ϵ
3701:⋅
3688:⋅
3597:
3591:⋅
3578:⋅
3550:−
3492:
3485:⋅
3472:⋅
3416:∫
3405:−
3312:∫
3261:∫
3242:−
3183:−
3141:⋅
3123:⋅
3021:−
3008:⋅
2966:
2932:
2873:−
2842:
2808:
2768:−
2741:
2710:
2704:−
2676:
2631:−
2604:⋅
2549:
2533:
2499:−
2472:⋅
2247:⋅
2151:∫
2102:⋅
2019:⋅
1920:
1881:
1753:⋅
1482:
1442:
1399:
1342:
1303:
1247:), where
1164:
1158:−
1121:
1021:
982:
951:
945:−
926:
902:
896:−
877:
816:
779:
731:
694:
616:⋅
560:
529:
507:Equations
408:
376:
339:
327:
312:
4699:Zipf law
4682:See also
3664:error).
1108:negative
473:gradient
359:Setting
4801:9155618
4781:Bibcode
4659:of the
4586:fractal
4147:is the
4004:- MAE,
1807:,
1800:) and (
1793:,
495:is the
192:science
94:scholar
18:Log-log
4799:
4616:Pareto
4573:= log
4569:, and
4565:= log
4557:= log
4549:= log
4541:= log
4537:where
4439:, and
4308:being
4300:= log
4296:, and
4292:= log
4284:= log
4276:= log
4268:= log
4264:where
4135:where
3875:, and
1088:) and
1070:where
641:where
475:) and
463:where
96:
89:
82:
75:
67:
4797:S2CID
4771:arXiv
4657:graph
4598:of a
4304:with
4141:money
3648:, or
2388:over
101:JSTOR
87:books
4179:are
4175:and
4036:and
3212:For
768:and
676:and
391:and
198:, a
194:and
174:and
73:news
4789:doi
4666:In
4655:(a
4649:).
3779:log
3755:log
3737:log
2963:log
2929:log
2839:log
2805:log
2738:log
2707:log
2673:log
2546:log
2530:log
2395:to
1917:log
1878:log
1631:or
1479:log
1439:log
1396:log
1339:log
1300:log
1161:log
1118:log
1018:log
979:log
948:log
923:log
899:log
874:log
813:log
776:log
728:log
691:log
551:log
520:log
405:log
373:log
336:log
324:log
309:log
202:or
190:In
56:by
4829::
4795:.
4787:.
4779:.
4767:51
4765:.
4753:^
4637:,
4588:.
4577:.
4561:,
4553:,
4545:,
4419:,
4316:.
4288:,
4280:,
4272:,
4167:,
4159:,
4044:.
3957:.
3594:ln
3489:ln
2045:.
1276:,
1240:,
620:10
555:10
524:10
4803:.
4791::
4783::
4773::
4647:y
4643:x
4639:y
4635:x
4627:y
4623:x
4608:R
4606:(
4575:U
4571:u
4567:K
4563:k
4559:N
4555:n
4551:A
4547:a
4543:Q
4539:q
4523:t
4519:u
4515:+
4510:t
4506:k
4499:+
4494:t
4490:n
4483:+
4480:a
4477:=
4472:t
4468:q
4417:A
4413:U
4409:K
4405:N
4401:Q
4387:,
4382:t
4378:U
4367:t
4363:K
4352:t
4348:N
4344:A
4341:=
4336:t
4332:Q
4306:u
4302:U
4298:u
4294:Y
4290:y
4286:R
4282:r
4278:A
4274:a
4270:M
4266:m
4252:,
4247:t
4243:u
4239:+
4234:t
4230:y
4226:c
4223:+
4218:t
4214:r
4210:b
4207:+
4204:a
4201:=
4196:t
4192:m
4177:c
4173:b
4169:A
4161:U
4153:Y
4145:R
4137:M
4123:,
4118:t
4114:U
4108:c
4103:t
4099:Y
4093:b
4088:t
4084:R
4080:A
4077:=
4072:t
4068:M
4057:t
4038:b
4034:a
3945:)
3940:2
3932:,
3926:(
3923:l
3920:a
3917:m
3914:r
3911:o
3908:N
3902:g
3899:o
3896:L
3884:e
3863:)
3858:2
3850:,
3844:(
3841:l
3838:a
3835:m
3832:r
3829:o
3826:N
3794:+
3791:)
3788:x
3785:(
3773:b
3770:+
3767:)
3764:a
3761:(
3752:=
3749:)
3746:y
3743:(
3705:e
3696:b
3692:x
3685:a
3682:=
3679:y
3617:0
3613:x
3607:1
3603:x
3586:0
3582:x
3573:0
3569:F
3565:=
3556:)
3553:1
3547:=
3544:m
3541:(
3537:A
3525:1
3521:x
3513:0
3509:x
3502:|
3495:x
3480:0
3476:x
3467:0
3463:F
3459:=
3454:x
3450:x
3447:d
3437:1
3433:x
3425:0
3421:x
3408:1
3400:0
3396:x
3390:0
3386:F
3380:=
3377:x
3374:d
3368:x
3364:t
3361:n
3358:a
3355:t
3352:s
3349:n
3346:o
3343:c
3333:1
3329:x
3321:0
3317:x
3308:=
3305:x
3302:d
3298:)
3295:x
3292:(
3289:F
3282:1
3278:x
3270:0
3266:x
3257:=
3248:)
3245:1
3239:=
3236:m
3233:(
3229:A
3214:m
3197:]
3191:0
3187:x
3178:m
3173:)
3166:0
3162:x
3156:1
3152:x
3146:(
3136:1
3132:x
3127:[
3117:1
3114:+
3111:m
3105:0
3101:F
3095:=
3092:A
3065:)
3057:m
3052:0
3048:x
3042:1
3039:+
3036:m
3031:0
3027:x
3016:1
3012:x
3001:m
2996:0
2992:x
2986:m
2981:1
2977:x
2970:(
2960:+
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2926:=
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2172:1
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2115:.
2110:m
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2073:=
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2043:m
2027:m
2023:x
2015:t
2012:n
2009:a
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1984:x
1981:(
1978:F
1958:,
1951:)
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1937:/
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1923:(
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1834:=
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1402:(
1373:)
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1320:/
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1128:x
1124:(
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1038:/
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1024:(
1013:)
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999:/
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985:(
973:=
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962:1
958:x
954:(
942:)
937:2
933:x
929:(
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913:1
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905:(
893:)
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880:(
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855:m
841:.
838:b
835:+
832:)
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823:x
819:(
810:m
807:=
804:]
801:)
796:2
792:x
788:(
785:F
782:[
756:,
753:b
750:+
747:)
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738:x
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719:]
716:)
711:1
707:x
703:(
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697:[
681:2
678:x
674:1
671:x
667:x
647:b
643:m
629:,
624:b
611:m
607:x
603:=
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597:x
594:(
591:F
572:,
569:b
566:+
563:x
547:m
544:=
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538:x
535:(
532:F
501:x
497:y
493:a
489:x
485:y
481:a
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469:k
465:m
449:b
446:+
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440:m
437:=
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402:=
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370:=
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345:.
342:a
333:+
330:x
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318:=
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225:=
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176:y
172:x
168:y
164:x
158:x
154:y
150:x
146:y
142:x
138:y
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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