Basically this exercise is about an easy and common curve fitting, which is used on a set of raw data points. The book 'Mathematische Formelsammlung' by Lothar Papula was referred for this exercise and its particular chapter is 'Ausgleichs- oder Regressionsparabel'.
Your mission, should you choose to accept it, is to fit a parabolic curve of the form y(x) = a*x^2 + b*x + c, to the given set of raw data points.
At the end of this post, I've included the link to download the M-script file.
At the end of this post, I've included the link to download the M-script file.
Raw data
x = [-3:1:5]';
y = [-3.94 -1.47 -4.66 -3.21 0.84 7.24 15.58 27.94 42.59]';
[n,m] = size(x);
Calculation
The curve parameter a, b and c can be determined from the method of least squares (German- Methode der kleinsten Quadrate), which uses linear equation system with 3 equations and 3 unknowns.
x4 = x.^4;
x3 = x.^3;
x2 = x.^2;
sx4 = sum(x4);
sx3 = sum(x3);
sx2 = sum(x2);
sx = sum(x);
sy = sum(y);
x2y = x2.*y;
xy = x.*y;
sx2y = sum(x2y);
sxy = sum(xy);
Formation of the known matrices
The targetted matrix equation is as follows: A*K = vekb. Out of those 3 matrices, A and vekb are known.
The targetted matrix equation is as follows: A*K = vekb. Out of those 3 matrices, A and vekb are known.
A = [sx4 sx3 sx2;...
sx3 sx2 sx;...
sx2 sx n];
vekb = [sx2y; sxy; sy];
Determination of the curve parameter
The matrix K consists of the parameters - a, b and c, that we sought after. Simply multiply the inverse of A and vekb, both gained from the previous step.
K = inv(A)*vekb;
Generation of new data points
By using the gained parameters in K, new points can now be plotted.
By using the gained parameters in K, new points can now be plotted.
yneu = K(1)*x.*x + K(2)*x + K(3);
Plotting
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| The displaying of the result. |
M-Script code:
MV_Blatt_1_Aufgb5.m
p/s: just in case that you're not using MATLAB or Octave, that file can also be viewed in normal text editor =P
