Reply to grant about a regression problem

posted by .

This is a reply to the question posted here

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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