How do I calculate the regression equation of $y = ax^2$ (find the coefficient $a$ that gives the smallest sum of squares of errors)?

Question

The input is a group of points $(x,y)$. I am trying to find the regression equation of $y = ax^2$ (I am trying to find the coefficient $a$ that gives the smallest sum of squares of errors from the input points. How do I find this?

What I have tried

I tried to calculate coefficient a like above, but the result was not what I expected.

Thank you.

The formula looks OK. I've tried to write a short script in python, and it yields the correct result — Andrei, Feb 08 '23 at 21:46
Thank you all for confirming that my solution was correct. I just realized that I was doing the math wrong. I was doing (ΣyΣx^2)/(Σx^4), instead of (Σ(yx^2))/(Σx^4). A stupid mistake haha. I got the correct result now! Thank you! — Sung Woo Han, Feb 09 '23 at 23:28

score 1 · Accepted Answer · answered Feb 08 '23 at 21:57

1

Your regression equation is modeling the response $y$ as a linear function of $x^2$, with no intercept. So formally it is the same problem as this one, except you replace every occurrence of $x_i$ with $x_i^2$. The result will agree with your calculation.

answered Feb 08 '23 at 21:57

grand_chat

38,951

Thank you for your answer! – Sung Woo Han Feb 09 '23 at 23:29

How do I calculate the regression equation of $y = ax^2$ (find the coefficient $a$ that gives the smallest sum of squares of errors)?

1 Answers1