# An Embarrassingly Simple Insight into Polynomial Regression

Joe Thorley · 2018-11-16 · 1 minute read

Linear regression is of course defined by the certain relationship

$\mu = \alpha + \beta \cdot x$ and the uncertain relationship

$y \sim~ N(\mu, \sigma)$ where $$\alpha$$ is the intercept and $$\beta$$ is the slope.

For example with $$\alpha = 0$$ and $$\beta = 10$$ the deterministic relationship can be represented as follows

beta <- 10
x <- 0:10
mu <- beta * x
plot(x, mu, type = "l")

I was just thinking about how I would like the slope, ie $$\beta$$, to vary with $$x$$ and came up with the following certain relationship $\mu = \alpha + (\beta + \beta_2 \cdot x) \cdot x$ which with $$\beta_2 = -0.5$$ gives

beta2 <- -0.5
mu <- (beta + beta2 * x) * x
plot(x, mu, type = "l")

As I was admiring it I suddenly realized that it can be very simply rearranged to give $\mu = \alpha + \beta \cdot x + \beta_2 \cdot x^2$ which is of course the standard polynomial relationship!

While this will be blindingly obvious to many it has somehow managed to elude me for many years. I’m not sure if I should feel exhilarated for having seen it or humiliated for having taken so long…