Gavin Howe
Fitting data to a power law involves modelling the relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other. This is done without considering the initial size of the quantities. For example, the area of a square has a power-law relationship with the length of its side, since if the length is doubled, the area is multiplied by four. Power laws are used to model a wide range of physical, biological, and human-made phenomena, such as the sizes of craters on the moon and cloud sizes. When fitting data to a power law, it is important to validate the power-law relation by testing many orthogonal predictions of a particular generative mechanism against the data. Simply fitting a power-law relation to the data is not sufficient. Various methods can be used to fit a power law to data, such as using non-linear least squares on the original data in linear space, or taking the natural logarithm of both sides of the equation to produce a log-log plot.
| Characteristics | Values |
|---|---|
| Definition | A power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent. |
| Data Fitting | Simply fitting a power-law relation to a particular kind of data is not considered a rational approach. |
| Validation | One method to validate a power-law relation tests many orthogonal predictions of a particular generative mechanism against data. |
| Cumulative Frequency | Using cumulative frequency allows one to put data gathered from different scales on the same diagram. |
| Survival Function | The survival function preserves the linear signature on doubly logarithmic axes. |
| Maximum Likelihood Estimator | When estimating exponents of a power-law distribution, a maximum likelihood estimator is recommended. |
| Log-Log Plot | Plotting log y against log x (natural logarithm) should result in a straight line. |
| Non-Linear Least Squares | Fit a power law using non-linear least squares to the original data in linear space without any transformation. |
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What You'll Learn
- The power law is a functional relationship between two quantities
- Simply fitting a power-law relation to a particular kind of data is not considered a rational approach
- The cumulative frequency allows one to put data gathered from different scales on the same diagram
- The maximum likelihood estimator is recommended when estimating exponents of a power-law distribution
- The log-log plot is produced by taking the natural logarithm of both sides of the equation
The power law is a functional relationship between two quantities
In statistics, a power law is a functional relationship between two quantities. A power law is a relationship in which a relative change in one quantity gives rise to a proportional relative change in the other quantity, independent of the initial size of those quantities. In other words, one quantity varies as a power of another.
For example, the area of a square has a power-law relationship with the length of its side. If the length is doubled, the area is multiplied by 2^2 (2 squared); if the length is tripled, the area is multiplied by 3^2, and so on. This relationship holds regardless of the initial length of the square's side. Similarly, if we double the length of a side of a cube, we multiply the volume of the cube by a factor of 8 (2^3).
Power laws are found in a wide range of physical, biological, and human-made phenomena. These include the sizes of craters on the moon and solar flares, cloud sizes, foraging patterns of species, neuronal population activity patterns, and many other quantities.
When fitting data to a power law, it is important to note that simply fitting a power-law relation to a particular type of data is not sufficient. Demonstrating that data follows a power law requires more than just fitting a model to the data. This is because superficially similar distributions may arise for different reasons, and different models will yield different predictions. One method to validate a power-law relation is to test many orthogonal predictions of a particular generative mechanism against the data.
Additionally, the choice of the fitting method depends on which method fits the data better. For example, the survival function is more robust to biases in the data and preserves the linear signature on doubly logarithmic axes. While fitting a power law to data, the maximum likelihood estimator is recommended to estimate the exponents of the power-law distribution.
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Simply fitting a power-law relation to a particular kind of data is not considered a rational approach
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent. The change is independent of the initial size of the quantities. For example, the area of a square has a power-law relationship with the length of its side. If the length is doubled, the area is multiplied by 4, and if the length is tripled, the area is multiplied by 9, and so on. The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes. These include the sizes of craters on the moon, cloud sizes, and the foraging patterns of various species.
Power-law relations are attractive for many theoretical reasons. However, demonstrating that data follows a power-law relation requires more than simply fitting a particular model to the data. This is because superficially similar distributions may arise for significantly different reasons, and different models yield different predictions. One method to validate a power-law relation is to test many orthogonal predictions of a particular generative mechanism against data.
Empirical distributions can only fit a power law for a limited range of values. This is because a pure power law would allow for arbitrarily large or small values. The power-law model does not obey the statistical completeness paradigm, especially in terms of probability bounds, which are suspected to cause typical bending and/or flattening phenomena in the high- and low-frequency graphical segments. One attribute of power laws is their scale invariance. Given a relation of the form c^{-k}, all power laws with a particular scaling exponent are equivalent up to constant factors, as each is simply a scaled version of the others.
When fitting a power law to data, the survival function representation is favoured over the probability density function (PDF) while using the linear least square method. However, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power-law distribution, the maximum likelihood estimator is recommended. The choice of whether to use a linear, power, or exponential law to fit a given data set is dictated solely by which function fits the data better.
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The cumulative frequency allows one to put data gathered from different scales on the same diagram
The use of cumulative frequency allows for the combination of data gathered from different scales onto a single diagram. This is achieved by plotting the upper-class/lower-class boundary with the cumulative frequency. Cumulative frequency is calculated by adding each frequency from a frequency distribution table to the sum of its predecessors. This table can be constructed from stem-and-leaf plots or directly from the data. The final value in the table will always be equal to the total for all observations.
In a cumulative frequency diagram, the cumulative frequency is plotted on the vertical axis, and the class boundaries are plotted on the horizontal axis. The horizontal axis typically represents the variable being measured, with values increasing from left to right. The vertical axis expresses the cumulative frequency as absolute counts or percentages (cumulative relative frequency). When using percentages, the maximum value will be 100%, representing the entire dataset.
The ability to combine data from different scales in a single diagram is particularly useful for comparing multiple datasets. For example, data from samples of different lengths or scales (e.g., from outcrop and microscope observations) can be plotted on the same graph, facilitating intuitive comparisons between the datasets. This feature of cumulative frequency diagrams provides a powerful tool for statistical analysis and data interpretation.
Additionally, cumulative frequency graphs offer a visual representation of data distribution, making it easier to understand how values accumulate across the entire range. This advantage over other statistical representations enhances the interpretation of data and can provide insights into whether the data follows a normal distribution. By examining the shape of the cumulative frequency curve, analysts can assess the normality of the data.
In summary, the cumulative frequency allows for the combination of data from different scales on a single diagram by plotting the class boundaries on the horizontal axis and the cumulative frequency on the vertical axis. This capability enables effective comparisons between multiple datasets and provides a visual representation of data distribution, making cumulative frequency a valuable tool in statistical analysis and data interpretation.
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The maximum likelihood estimator is recommended when estimating exponents of a power-law distribution
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent. Diverse systems with the same critical exponents display identical scaling behaviour as they approach criticality and are said to share the same fundamental dynamics. Power-law relations are attractive for many theoretical reasons, but demonstrating that data follows a power-law relation requires more than simply fitting a particular model to the data. Simply fitting a power-law relation to a particular type of data is not considered a rational approach.
Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods. The maximum likelihood estimator (MLE) for alpha is simple if given the value for x_min. However, there is no simple expression for estimating x_min as the likelihood is increasing in x_min, corresponding to throwing out more and more of the data. Therefore, another method is needed.
In the case of a power law with an exponential cutoff, finding exact expressions is much harder. The estimators of lambda and alpha are coupled (due to the normalization constant), so finding them sequentially does not work. We have to use numerical methods. The maximum-likelihood estimators and Kolmogorov-Smirnov (K-S) statistics in widespread use are unexpectedly sensitive to ubiquitous errors in data such as measurement noise, quantization noise, heaping, and censorship of small values. This sensitivity causes spurious rejection of power laws and biases parameter estimates even in arbitrarily large samples, which explains inconsistencies between theory and data.
Logarithmic binning by powers of λ > 1 attenuates these errors in a manner analogous to noise averaging in normal statistics, and λ tunes a trade-off between accuracy and precision in estimation. Binning also removes potentially misleading within-scale information while preserving information about the shape of a distribution over powers of λ. Some amount of binning can improve sensitivity and specificity of K-S tests without any cost, while more extreme binning tunes a trade-off between sensitivity and specificity. When estimating exponents of a power-law distribution, the maximum likelihood estimator is recommended.
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The log-log plot is produced by taking the natural logarithm of both sides of the equation
In statistics, a power law describes the relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent. The change is independent of the initial size of those quantities. For example, the area of a square has a power-law relationship with the length of its side.
When fitting data to a power law, one method involves creating a log-log plot by taking the natural logarithm of both sides of the equation. This technique is particularly useful when dealing with complex functions in calculus. By applying the natural logarithm (ln) to both sides of the equation, we can transform the relationship into a linear representation. In other words, we plot ln(y) versus ln(x), resulting in a straight-line dependence. This approach simplifies the analysis and allows for easier interpretation of the data.
The validity of taking the log of both sides of an equation stems from the fact that a logarithm is a function. If we have two equal expressions, let's say "thing1" and "thing2," and we apply a function that includes both of these in its domain, we can assert that f(thing1) equals f(thing2). This principle extends to logarithmic functions, where log(thing1) equals log(thing2) if thing1 equals thing2. However, it's important to exercise caution and ensure that the values being logarithmed fall within the domain of the logarithm function.
Taking the natural logarithm of both sides of an equation can help simplify complex expressions and bring exponential terms down, making them more manageable. For example, consider the equation $(60-X)ln(1.006)+(60-2X)(ln(2))(ln(1.006))=ln(3.823)$. By applying natural logarithms to both sides, we can transform the equation and isolate the unknown variable. However, it's important to be mindful of specific rules, such as the distinction between $\ln(a+b)$ and $\ln(ab)$.
In summary, creating a log-log plot by taking the natural logarithm of both sides of the equation is a valuable technique when fitting data to a power law. It allows for the linearization of the relationship and simplifies the analysis of complex functions. By following the principles of logarithmic functions and their properties, we can effectively utilize this method to gain insights from data that exhibits power-law behaviour.
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Frequently asked questions
A power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent.
Fitting a power law to data is done using non-linear least squares on the original data in linear space. Simply fitting a power-law relation to a particular kind of data is not considered a rational approach.
Plotting log y against log x (it doesn't matter which base of logs) is a good way to visualise power laws. This will result in a straight line.
