Savitzky–Golay filter

Algorithm applied on a gaussian peak with random noise, with a degree 3 and a width of 9 points. Smoothing (top), first derivation (middle), second derivation (bottom).

The Savitzky–Golay smoothing filter is a filter that essentially performs a local polynomial regression (of degree k) on a series of values (of at least k+1 points which are treated as being equally spaced in the series) to determine the smoothed value for each point. The main advantage of this approach is that it tends to preserve features of the distribution such as relative maxima, minima and width, which are usually 'flattened' by other adjacent averaging techniques (like moving averages, for example).

This filtering algorithm was first described in 1964 by Abraham Savitzky and Marcel J. E. Golay.

Savitzky and Golay's paper is one of the most widely cited papers in the journal Analytical Chemistry and is classed by that journal as one of its "10 seminal papers" saying "it can be argued that the dawn of the computer-controlled analytical instrument can be traced to this article".

Case of a constant step

Let us consider a collection of data (x_i, y_i)_{1 ≤ i ≤ n}. When the step x_i - x_{i - 1} is constant, the Savitzky-Golay smoothing is a finite impulse response filter (FIR). For example, if we use a third degree polynomial and a window size of 5 points, then the value of the smoothed curve, y_sm, and the value of the first and second derivative, y'_sm and y''_sm, can be calculated by the following linear combinations:

$\textstyle y_{\mathrm {sm} \ i}={\frac {1}{35}}(-3\cdot y_{i-2}+12\cdot y_{i-1}+17\cdot y_{i}+12\cdot y_{i+l}-3\cdot y_{i+2})$ ;

$\textstyle y'_{\mathrm {sm} \ i}={\frac {1}{10}}(-2\cdot y_{i-2}-1\cdot y_{i-1}+0\cdot y_{i}+1\cdot y_{i+l}+2\cdot y_{i+2})$ ;

$\textstyle y_{\mathrm {sm} \ i}^{\prime \prime }={\frac {1}{7}}(2\cdot y_{i-2}-1\cdot y_{i-1}-2\cdot y_{i}-1\cdot y_{i+l}+2\cdot y_{i+2})$.

Some coefficients are tabulated.

Convolution coefficients for smoothing

Degree	2/3 (quadratic/cubic)			4/5 (quartic/quintic)
Window size	5	7	9	7	9
-4			-21		15
-3		-2	14	5	-55
-2	-3	3	39	-30	30
-1	12	6	54	75	135
0	17	7	59	131	179
1	12	6	54	75	135
2	-3	3	39	-30	30
3		-2	14	5	-55
4			-21		15
Normalisation	35	21	231	231	429

Convolution coefficients for 1^st derivative

Degree	2 (quadratic)			3/4 (cubic/quartic)
Window size	5	7	9	5	7	9
-4			-4			86
-3		-3	-3		22	-142
-2	-2	-2	-2	1	-67	-193
-1	-1	-1	-1	-8	-58	-126
0	0	0	0	0	0	0
1	1	1	1	8	58	126
2	2	2	2	-1	67	193
3		3	3		-22	142
4			4			-86
Normalisation	10	28	60	12	252	1,188

Convolution coefficients for 2^nd derivative

Degree	2/3 (quadratic/cubic)			4/5 (quartic/quintic)
Window size	5	7	9	5	7	9
-4			28			−4,158
-3		5	7		-117	12,243
-2	2	0	-8	-3	603	4,983
-1	-1	-3	-17	48	-171	−6,963
0	-2	-4	-20	-90	-630	−12,210
1	-1	-3	-17	48	-171	−6,963
2	2	0	-8	-3	603	4,983
3		5	7		-117	12,243
4			28			−4,158
Normalisation	7	42	462	36	1,188	56,628

See also

• Numerical smoothing and differentiation

External links

• Savitzky-Golay filter in Fundamentals of Statistics