Log–log plot: Difference between revisions

Revision as of 22:31, 25 November 2024

Revision as of 22:36, 25 November 2024

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Log–log plots are often use for visualizing log-log linear regression models with (roughly) [[log-normal]], or [[Log-logistic distribution|Log-logistic]], errors. In such models, after log-transforming the dependent and independent variables, a [[Simple linear regression]] model can be fitted, with the errors becoming [[Homoscedasticity|homoscedastic]]. 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., [[heteroscedasticity|heteroscedastic]] error).

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:

:<math>y = a \cdot x^b \cdot e^\epsilon</math>

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:<math>\log(y) = \log(a) + b \cdot \log(x) + \epsilon</math>

This is a [[linear equation]] in the logarithms of `<math>x`</math> and `<math>y`</math>, with `<math>\log(a)`</math> as the intercept and `<math>b`</math> as the slope. In which <math>\epsilon \sim \textrm{Normal}(\mu, \sigma^2)</math>, and <math>e^\epsilon \sim \textrm{Log-Normal}(\mu, \sigma^2)</math>.

[[File:Visualizing Loglog Normal Data.png|thumb|Figure 1: Visualizing Loglog Normal Data]]