How to find point in polynomial regresion

Question

I have the following data set:

Hz        duration of ease-in
324       1.139
390       1.134
403       1.167
410       1.1
423.4     1.1
693.5     0.766
1040      0.567
1134      0.567
1480      0.434

I know how to find linear regression: Intercept (a): 1.3739956457219 Slope (b): -0.0006973690931099.

So, Y = a + bX

But as I've gathered more data points it appears to be on a curve. Is there a simple formula to solve for a correlation on a polynomial curve?

Answer 1

Put $\text{Hz}$ in one column, $\text{Hz}^2$ in the next, then regress $y$ on $\text{Hz}$ and $\text{Hz}^2$. Just linear regression on two predictors rather than one. If you know how to do multiple linear regression, that's it.

If you put a column of $1$s as the first column of the design matrix $D$, and $\text{Hz}$ as the next column, and $\text{Hz}^2$ as the next, getting an $n\times 3$ matrix with three linearly independent columns, then the least-squares estimators of the three coefficients in the model are the elements of the $3\times 1$ column vector $(D^T D)^{-1}D^T Y$, where $Y$ is the $n\times 1$ column that you labeled "duration of ease-in". So if you know matrix algebra, then that's it.

In this case, it appears that the second- and higher-degree terms do not differ from $0$ in a statistically significant way.