Gavin Howe
Power laws describe a wide range of physical, biological, and human-made phenomena, including the sizes of craters on the moon, the distribution of income, and the growth of compound interest. 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. Power laws are typically non-linear relationships, but they can be transformed into linear relationships by plotting the variables on logarithmic axes. This transformation allows for a visual representation of the data that highlights the power law relationship between the variables.
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What You'll Learn
Log-normal distributions
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 those quantities. For example, the area of a square has a power-law relationship with the length of its side.
A log-normal distribution is a statistical distribution of logarithmic values from a related normal distribution. It is a continuous probability distribution of a random variable whose logarithm is normally distributed. Log-normal distributions are often mistaken for power-law distributions. A log-normal distribution can be translated to a normal distribution and vice versa using associated logarithmic calculations. For instance, if the random variable X is log-normally distributed, then Y = ln X has a normal distribution.
A log-normal distribution is positively skewed with long right tails due to low mean values and high variances in the random variables. The log-normal distribution is a right-skewed continuous probability distribution, meaning it has a long tail towards the right. It is used for modelling various natural phenomena such as income distributions, the length of chess games, or the time to repair a maintainable system.
The probability content of a log-normal distribution in any arbitrary domain can be computed to the desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method. A log-normal distribution can be scaled using a different base, which affects the shape of the lognormal distribution. Overall, the log-normal distribution plots the log of random variables from a normal distribution curve.
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Compound interest
The Rule of 72 is another way to estimate compound interest. If you divide 72 by your rate of return, you find out how long it will take your money to double in value. For example, if you have $100 that was earning a 4% return, it would grow to $200 in 18 years (72 / 4 = 18).
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Inverse power law
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 inverse power law (IPL) is a commonly used model for non-thermal accelerated stresses. It is also one of the most important and widely used models for mechanical systems. The inverse power law appears as a straight line when plotted on a log-log graph.
The inverse power law relationship is used in accelerated life testing (ALT) to aid in the design and validation process. ALT exploits the fatigue properties of materials and tests components at higher stress levels than would be experienced during normal operation. The inverse power law usually considers high voltage as the accelerated variable. The data collected from an ALT can be used to determine the reliability of a component at a given stress level.
The inverse power law relationship can be used to develop basic life-stressor models with little effort. These models can then be used to assess a system's suitability for "off-design" conditions and its sensitivity to its "inputs". This can help in developing more robust components. The inverse power law can also be used to simplify the analysis of time-varying stress tests by using a constant stress model.
The inverse power law can be applied to other areas beyond mechanical systems. For example, in battery technology, the logarithms of battery capacity fade and the increase in resistance evolve linearly over ageing. The slopes identified from the straight lines correspond to battery ageing rates.
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Power-law probability distribution
Power laws describe a wide range of physical, biological, and human-made phenomena. These include the sizes of craters on the moon, the foraging patterns of various species, the population sizes of cities in a region, and many other examples. 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. One quantity varies as a power of another, and this 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.
Power laws can be used to describe a phenomenon where a small number of items is clustered at the top of a distribution (or the bottom), taking up most of the resources. This implies that small occurrences are common, while larger occurrences are rare. For instance, in income distribution, there are very few billionaires, and most people have modest savings. Power laws can also be used to understand compound interest, where the relationship between the amount of time a sum of money is left in an account and the amount accumulated at the end is non-linear.
To identify a power-law probability distribution, a standard method is to use a logarithmic scale on both axes. If there is a power relationship between two variables, a linear relationship will be observed between their logarithms. It is important to note that graphical methods like linear regression may not be suitable for checking power laws. Instead, using logarithmic scales can provide a good fit for the data. However, discrepancies may still exist, and understanding these deviations can offer insights into the underlying system, such as the health of a website in the case of user activity data.
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Logarithmic plot of variables
A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences in the magnitudes of the numbers involved. Unlike a linear scale, where each unit of distance corresponds to the same increment, each unit of length on a log scale is a multiple of some base value raised to a power. Logarithmic scales are typically in base 10 unless otherwise specified. This means that instead of counting 1, 2, 3, 4, etc., or 10, 20, 30, 40, etc., a logarithmic scale counts by powers of 10 (10, 100, 1000, etc.).
Logarithmic scales are useful when data grows or decreases exponentially. For example, exponential growth curves are often depicted on a log scale graph. Log scales are also commonly used for the frequency level in music, with units such as cent, minor second, major second, and octave. Logarithmic scales are also used for several industrial measures, such as standard values for resistors, the American wire gauge, and the Birmingham gauge.
In science and engineering, a log-log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions appear as straight lines in a log-log graph, with the exponent corresponding to the slope and the coefficient corresponding to the intercept. Log-log plots are useful for visualizing log-log linear regression models with log-normal or log-logistic errors. After log-transforming the dependent and independent variables, a simple linear regression model can be fitted, with the errors becoming homoscedastic.
Log-log plots are widely used in various fields, including economics, biology, and physics, where many phenomena exhibit power-law behaviour. They are also useful in regression analysis when dealing with heteroscedastic data, as the log transformation can help stabilize the variance. However, it's important to note that simple log-log plots may not be sufficient for validating power laws, and more sophisticated statistics may be required.
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Frequently asked questions
A power-law relationship 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.
A power-law relationship can be made linear by plotting the variables on logarithmic axes.
Compound interest is a power-law relationship because the relationship between the amount of time a sum of money is left in an account and the amount accumulated at the end is non-linear.
