Thursday, June 21, 2012

Linear regression and logarithms don't mix?

You have a power-law model \(y=a_0 x^{a_1}\), and a bunch of experimental data points \[\begin{bmatrix}x_1 & y_1 \\ x_2 & y_2 \\ \vdots & \vdots \\ x_n & y_n\end{bmatrix}.\]
You want to estimate \(a_0\) and \(a_1\), it is tempting to take the logarithm of both sides \(\log y = \log a_0 + a_1 \log x\), and perform linear regression on suitably transformed experimental data \[\begin{bmatrix}\log x_1 & \log y_1 \\ \log x_2 & \log y_2 \\ \vdots & \vdots \\ \log x_n & \log y_n\end{bmatrix}.\]
Beware! You may get something different from what you expect! And your answers might not mean much.

Perhaps, you should be doing maximum likelihood instead. Here is a nice tutorial (pdf) on it.

PS: Part of the motivation for this post was this.

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