Data-reduction using the Least square polynomial

Sometimes you have to operate on a more or less large set of data. Manipulating such sets is generally difficult.

Suppose your result should be a straight line in a plane area, but of course your data only moves around. You guess that there is a linear tendency, but the computer doesn't. Here the example of a 25 data-set:

x y x2 xy
1 0,66496424 1 0,66496424
2 3,94075718 4 7,88151436
3 3,35247829 9 10,0574349
4 3,86552513 16 15,4621005
5 4,49854689 25 22,4927345
6 5,80525828 36 34,8315497
7 5,81779177 49 40,7245424
8 6,10924665 64 48,8739732
9 6,48756951 81 58,3881256
10 5,59638624 100 55,9638624
11 7,28547502 121 80,1402252
12 9,0264179 144 108,317015
13 8,02644055 169 104,343727
14 10,0489198 196 140,684877
15 9,21802309 225 138,270346
16 10,9809859 256 175,695774
17 8,63910735 289 146,864825
18 9,44954277 324 170,09177
19 12,3358053 361 234,380301
20 11,5203612 400 230,407224
21 11,4433213 441 240,309747
22 12,0350019 484 264,770042
23 13,729675 529 315,782525
24 12,7141765 576 305,140236
25 12,7032674 625 317,581685
325 205,295045 5525 3268,12112

 

Suppose you have N data points (xi, yi). The equation for the straight line is y = c1 + c2x, where c1 and c2 are unknown.

The error of the real yi-value compared to the straight line is declared as ei =  yi - (c1 + c2x)

To make sure big errors have a more important impact on the computation, we consider the sum of the squared errors:

As we want to find the values c1 and c2 that will make sure S will be minimized, we set the partial derivates of S respective to c1 and c2 equal to zero:

Rearranging these equations we get:

Again rearranging we have:

To extract c1 and c2 we must only solve the system of two simultaneous equations.

Our example:

25c1 + 325c2 = 205.295045

325c1 + 5525c2 = 3268.12112

Using the determinant-procedure we get:

This technic may be generalyzed as:

y = c1 + c2x + c2x2 + ... + cm+1xm+1

if m=1 we fit the data in a straight line, if m=2 it is a parabola...

The coefficients are determined by solving the m+1 simultaneous equations:

Of course solving more simutaneous equations requires an adequate algorithm such as the Gauss-Jordan Elimination.

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