# Minimizing square of deviation / curve fitting

1. May 22, 2009

### exmachina

1. The problem statement, all variables and given/known data

Given some data set, (x,y), fit to the the curve $$y=bx^2+a$$ by minimizing the square of the deviation. Preferred to use matrices.

2. Relevant equations
The deviation for the ith data point is simply:

$$d_i^2=(y_i-y)^2=(y-bx_i^2-a)^2$$

3. The attempt at a solution

If I understand correctly, I want to minimize $$\sum{d_i^2}$$ by differentiating wrt to a and b and setting to zero to find the local minima. So far I have differentiated to yield the following system of equations:

$$\sum_{i}^n y_i = an + \sum_{i}^n bx_i^2$$

$$\sum_{i}^n{x_i^2y_i}=a\sum_{i}^n x_i^2 + b \sum_{i}^n x_n^4$$

So I'm stuck using either Cramer's Rule, LU, or GE. Cramer's Rule is the easiest to implement, but I don't know how much slower it will be compared to LU/GE. I have around 300 data points

Last edited: May 22, 2009
2. May 22, 2009

### Dick

It doesn't really matter what method you use, there are only two unknowns, a and b. It's a 2x2 system. Once you have found the summations of the various powers of your data points, it's easy.

3. May 22, 2009

### Random Variable

Solving the system using Cramer's rule is very simple.

$$a = \frac { \sum x^{4}_{i} \sum y_{i} - \sum x^{2}_{i} \sum x^{2}_{i} y_{i}} {n \sum x^{4}_{i }- ( \sum x^{2}_{i})^2}$$

$$b = \frac { n \sum x^{2}_{i} y_{i} - \sum x^{2}_{i} \sum y_{i}} {n \sum x^{4}_{i }- ( \sum x^{2}_{i})^2}$$

and then calculate the sums using a spreadsheet