Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more.  ## Savitzky-Golay Filter - Mathematical Details

A polynomial filter can be considered as piece-by-piece fitting of a polynomial function to the signal. The fitting is done by a least squares (LS) estimate between the X matrix and the y vector:
y = Xb.
The standard LS solution is given by:
b = (XTX)-1XTy.
The estimated values that are used for the smoothing are: = Xb = X(XTX)-1XTy = Hy.

The product H = X(XTX)-1XTy is also called the hat matrix and is the same for any y for a given polynomial. So it can be calculated once and stored for latter application. This is what Savitzky and Golay have done for polynomials of various orders, and pieces of different length n (original publication and corrections  ).

Note: The (n+1)th row of the H matrix gives the tabulated coefficients for the Savitzky-Golay filters. We only use the estimate for the middle point of the moving window for smoothing. The other rows are used only for the smoothing of the endpoint of the signal, when there are fewer data points left than the window size 2n+1.

The matrix X is the so-called Vandermond matrix. When we want to fit a polynomial function of order p, we get a series of equations in the following form:

yi = bp . xip + bp-1 . xip-1 + ... b1 . xi1 + b0 . xi0 , for i = 1 .. 2n+1.

In the case of n=1, x = [-1 0 +1] and we have:  ,

resulting in the filtering coefficients b = [0 1 0]. In this case, actually no calculations would be performed, but rather the original values would be taken. For n=5 and p=3, the filtering coefficients are b= [-0.0857 0.3429 0.4857 0.3429 -0.0857].

As a special case, we can also derive the moving average, i.e. when we want to fit a constant line (a zeroth order polynomial).  