Least Squares Regression for Nonlinear Equations

If we can re-write our non-linear equation in a linear form, we can apply linear regression:

$$\begin{array}{rl}y& ={\alpha }_1{e}^{{\beta }_{1}x}\\ \ln y& =\ln {\alpha }_1+{\beta }_{1}x\\ y& =A+B_{1}x\end{array}$$

Linearization Examples

Exponential function

Take $\ln$ on both sides of equation.

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

Solve for $A$ and $b$ and transform back:

$$y=(e^A)e^{bx}$$

Power function

Use base 10 $\log$.

$$\begin{array}{rl}\log y& =\log a+b\log x\\ Y& =A+bx\end{array}$$

Solve and transform back:

$$y=(10^A)x^b$$

Fraction thing

Do everything in terms of $\frac{1}{y}$.

$$\begin{array}{rl}y& =a\left(\frac{x}{b+x}\right)\\ \frac{1}{y}& =\frac{1}{a}\left(\frac{b}{x}+1\right)\\ \frac{1}{y}& =\frac{1}{a}+\frac{b}{a}\cdot \frac{1}{x}\\ Y& =A+Bx\end{array}$$

Solve and transform back:

$$y=\left(\frac{1}{A}\right)\left(\frac{X}{B/A+x}\right)$$