Data Linearization

What is it?

Data linearization is the process of taking non-linear data and transforming it to linear. This is most commonly used in Statistics, to fit non-linear data to linear models.

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Linearizing $y=a\ast x^{\alpha }$

For $y=a\ast x^{\alpha }$ models, we can apply Logarithms to linearize the data.

$$\begin{array}{rl}& \ln y=\ln a\ast x^a\\ & \ln y=\ln a+a\ast \ln x\end{array}$$

The idea is that we can now collect $(\ln x_i,\ln y_i)$, and the model will behave like a linear equation. In fact, this is the idea behind every linearization method demonstrated here.

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Linearizing $y=a\sqrt{x}$

For $y=a\sqrt{x}$ models, we can just transform $\sqrt{x}$ to a different variable, like $t$. That way, $y=at$, and we collect $(\sqrt{x_i},y_i)$.

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Linearizing $y=a\ast e^{bx}$

For $y=a\ast e^{bx}$ models, one can apply Logarithms once again:

$$\begin{array}{rl}& \ln y=\ln a\ast e^{bx}\\ & \ln y=\ln a+\ln e^{bx}\\ & \ln y=\ln a+bx\end{array}$$

That way, we can collect $(x_i,\ln y_i)$ data.