SAVITZKY-GOLAY FILTERS

The computation of the susceptibilities of a system from experimental data involves specific problems that have to be addressed. Neville's algorithm is suitable for the computation of derivatives from nearly exact data. However, when the level of noise is significant, alternative procedures must be used. One of them is the Savitzky-Golay algorithm for smoothing and differentiation.

The idea is to fit equally-spaced data points within a moving window, , with , to a polynomial of order . For this, a least-squares procedure can be used. Since the polynomial is of the form , we must solve the matrix equation

(1)

where is the matrix of powers

(2)

is the vector of coefficients of the polynomial and is the vector of data. Note that is the distance of the point to the central point in the window of data.

The matrix has more rows than columns and, hence, the system (1) is overdetermined, as long as the rank is , i.e., full-rank. In order to obtain the least-squares solution, one must solve the normal equations:

(3)

That is,

(4)

The key point of the Savitzky-Golay method is the realization that, because eq. (3) is linear, eq. (4) can be expressed as

(5)

where , a Savitzky-Golay matrix, does not depend on the vector of data. Therefore, there exist universal sets of Savitzky-Golay coefficients that transform any vector of data into the vector of polynomial coefficients for the best fit, . This means that one need not repeat the least-squares fit for every vector of data, but it is possible to precalculate , once and have been specified.

Moreover, the polynomial coefficients are related to the first derivatives, plus the zeroth-order one, at the central point in the window, :

(6)

is the interval between points (in the context of biochemical susceptibilities, a perturbation magnitude, i.e., a variation of a concentration). As a consequence of eq. (6), if the goal is smoothing the data (zeroth-order term), only the first row of is needed, for the first derivative, the second row suffices, etc. This can be written as a convolution (linear combination),

(7)

from which the filtered order derivative can be calculated.

Now, we note that, if , then , with and , where is Kronecker's delta. Substituting in eq. (4), we get

(8)

These are the Savitzky-Golay coefficients for smoothing and differentiation. They satisfy the normalization condition

(9)

REFERENCE

A. Savitzky and M.J.E. Golay "Smoothing and differentiation of data by simplified least squares procedures" Anal. Chem., 36, 1627-1639 (1964)

Copyright © 2003-2013 Antonio Sánchez Torralba
Last modified:
Monday, 04-Mar-2013 11:58:25 CET