Content deleted Content added
adding == Log-log linear regression models == section |
→Log-log linear regression models: adding the error term |
||
Line 86:
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>
Taking the logarithm of both sides, we get:
:<math>\log(y) = \log(a) + b \cdot \log(x) + \epsilon</math>
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 <math>\epsilon \sim Normal(\mu, \sigma^2)</math>, and <math>e^\epsilon \sim Log-Normal(\mu, \sigma^2)</math>.
[[File:Visualizing Loglog Normal Data.png|thumb|Figure 1: Visualizing Loglog Normal Data]]
Line 107:
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.
== Applications ==
| |||