Reply to grant about a regression problem
posted by Count Iblis .
This is a reply to the question posted here
http://www.jiskha.com/display.cgi?id=1178989522
As I explained there, you can find the parameters by defining:
x1 = Sin(0.49 t)
x2 = Cos(0.49 t)
and treat this as an ordinary linear regression problem. If we denote the height by Y, then we want to find parameters A, B, and C such that Y is best approximated by:
Y = A + B X1 + C X2
In regression analysis one chooses the parameters such that the sum of the squared deviations is minimized. This yields the follwing 3 equations:
Sum_{data points} [y  A  B X1  C X2 ] = 0
Sum_{data points} X1 [y  A  B X1  C X2 ] = 0
Sum_{data points} X2 [y  A  B X1  C X2 ] = 0
So, all you have to do is to calculate the summations:
Sum Y = 81.3
Sum X1 = 0.0508842
Sum X2 = 1.17151
Sum X1^2 = 6.42279
Sum X2^2 = 7.57721
Sum X1 X2 = 0.0502369
Sum X1 Y = 0.837748
Sum X2 Y = 15.4769
And solve the equations (N = 14, the number of data points):
Sum Y  N A  B Sum X1  C Sum X2 = 0
Sum Y X1  A SumX1 B Sum X1^2  C Sum X2 X1 = 0
Sum Y X2  A SumX2 B Sum X1 X2  C Sum X2^2 = 0
You then find:
A = 6.0569
B = 0.155126
C = 2.97792
You can rewrite the fit function as:
6.05 + 2.98 Sin(0.49 t  1.62)
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