如何将非线性回归方程转换为可最小二乘求解的线性回归方程

Let's walk through exactly how to turn this nonlinear regression equation into a linear form that's compatible with least squares estimation. Here's the full step-by-step derivation:

First, expand the original nonlinear equation by expanding the squared term and distributing the coefficients:
\begin{align}
y &= a(x-b) + c(x-b)^2 \
&= ax - ab + c(x^2 - 2bx + b^2) \
&= ax - ab + cx^2 - 2cbx + cb^2
\end{align}

Next, we group terms by powers of $x$ and define new linear coefficients to rewrite the equation in a standard linear regression structure:
\begin{align}
y &= \underbrace{-ab + cb^2}{\beta_0} + x\underbrace{(a - 2cb)}{\beta_1} + x^2\underbrace{c}_{\beta_2}
\end{align}

By substituting these new $\beta$ coefficients, we get a standard multiple linear regression model that we can directly solve using least squares:
$$ y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \epsilon_i, \quad i=1,...,n $$

One important note: you'll notice that $\beta_0$ and $\beta_1$ are not independent—they both depend on the original parameters $a$, $b$, and $c$. This means while we can estimate $\beta_0$, $\beta_1$, and $\beta_2$ easily with least squares, converting back to the original $a$, $b$, $c$ will require solving a system of equations since the mapping isn't directly one-to-one.

内容的提问来源于stack exchange，提问作者Tzur Roy