Savitzky-Golay filter

Animation of the smoothing of noisy data by a Savitzky-Golay filter (3rd degree polynomial, 9 point wide sliding window).
• blue curve: raw data;
• blue points: data points determined by the smoothing;
• yellow curve: regression polynomial used to determine the current point;
• red curve: section of the regression polynomial in the area of the sliding window.

The Savitzky-Golay filter is a mathematical smoothing filter used in signal processing . It was first described in 1964 by Abraham Savitzky and Marcel JE Golay .

It essentially performs a polynomial regression ( k th degree) over a series of values (on at least k + 1 support points, which are treated as equidistant) in order to determine a smoothed value for each point. One advantage of the Savitzky-Golay filter is that, unlike other smoothing filters, components of high frequencies are not simply cut off, but are included in the calculation. As a result, the filter shows excellent properties of the distribution, such as relative maxima, minima and scatter, which are usually falsified by flattening or shifting by conventional methods such as the formation of the moving average .

As already mentioned, the filter uses a variable window width and variable smoothing factors; these values have a decisive influence on the effectiveness of the filter. By adapting the coefficients, the filter can act not only like a polynomial smoothing, but also like a sliding mean value formation or even a smooth derivation.

The Savitzky-Golay filter is used in spectroscopy . The first publication by Savitzky and Golay is considered by some authors to be one of the most important and most cited basic publications in the field of computer-aided numerics .

literature

Per-Olof Persson, Gilbert Strang: Smoothing by Savitzky-Golay and Legendre Filters . In: David S. Gilliam (Ed.): Mathematical systems theory in biology, communications, computation, and finance . Springer, 2003, ISBN 978-0-387-40319-9 , pp. 301 ( limited preview in Google Book search).
William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling: Numerical Recipes in C: The Art of Scientific Computing, Second Edition . 2nd Edition. Cambridge University Press, 1992, ISBN 0-521-43108-5 , Section 14.8 Savitzky-Golay Smoothing Filters , pp. 650–655 ( section as PDF - further explanations and source text for the calculation in C).

Web links

Hans Lohninger: Savitzky-Golay filter . Basics of Statistics, March 19, 2011 ( coefficients ).
Savitzky-Golay filter in Fundamentals of Statistics (English)

Individual evidence

↑ Abraham. Savitzky, MJE Golay: Smoothing and Differentiation of Data by Simplified Least Squares Procedures . In: Analytical Chemistry . tape 36 , no. 8 , June 1, 1964, pp. 1627-1639 , doi : 10.1021 / ac60214a047 .
^ Hans Lohninger: Savitzky-Golay filter - coefficients . Basics of Statistics, March 19, 2011.
↑ James Riordon, Elizabeth Zubritsky, Alan Newman: Top 10 Articles . In: Analytical Chemistry . tape 72 , no. 9 , 2000, pp. 324A-329 A , doi : 10.1021 / ac002801q .

[1] Abraham. Savitzky, MJE Golay: Smoothing and Differentiation of Data by Simplified Least Squares Procedures . In: Analytical Chemistry . tape 36 , no. 8 , June 1, 1964, pp. 1627-1639 , doi : 10.1021 / ac60214a047 .

[2] Hans Lohninger: Savitzky-Golay filter - coefficients . Basics of Statistics, March 19, 2011.

[analytchemsem-3] James Riordon, Elizabeth Zubritsky, Alan Newman: Top 10 Articles . In: Analytical Chemistry . tape 72 , no. 9 , 2000, pp. 324A-329 A , doi : 10.1021 / ac002801q .