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Minimizing square of deviation / curve fitting

  1. May 22, 2009 #1
    1. The problem statement, all variables and given/known data

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

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

    [tex]d_i^2=(y_i-y)^2=(y-bx_i^2-a)^2[/tex]

    3. The attempt at a solution

    If I understand correctly, I want to minimize [tex]\sum{d_i^2}[/tex] 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:

    [tex]\sum_{i}^n y_i = an + \sum_{i}^n bx_i^2 [/tex]

    [tex]\sum_{i}^n{x_i^2y_i}=a\sum_{i}^n x_i^2 + b \sum_{i}^n x_n^4[/tex]


    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. jcsd
  3. May 22, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    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.
     
  4. May 22, 2009 #3
    Solving the system using Cramer's rule is very simple.

    [tex] 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} [/tex]


    [tex] 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} [/tex]

    and then calculate the sums using a spreadsheet
     
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