Savitzky-Golay Algorithm

Local Algorithm - One-Dimensional Algorithm

Savitzky-Golay algorithm performs a local polynomial regression on a given sequence of values. The basic formula (for filter width $L = 2M+1$ and polynomial order $k$) is stated as follows:

$$\hat{Y}[n] = \frac{1}{2M+1} \sum_{k=-M}^M A[k] Y[n-k] \, \text{,}$$

where $Y[n]$ and $\hat{Y}[n]$ denote the raw and processed data sequences, respectively. The values of $A[k]$, the Savitzky-Golay coefficient vector, depends on the choice of polynomial order $k$. Note that the Savitzky-Golay coefficient vector can be pre-computed based on the idea to make for each point a local least-square polynomial fit.

Input Parameters

Parameter	Type	Constraint	Description	Remarks
$Y[n]$	$Y[n] \in \mathbb R^N$	$N \in \mathbb{N}$	Input data sequence of length $N$	The algorithm assumes that input values contain no outliers and improper values such as ‘nan’, ‘inf’, ‘null’.
$L$	$L \in \mathbb N$	$L = 2M + 1, \quad M \in \mathbb{N}$
$k$	$k \in \mathbb N$

Output Parameters

Parameter	Type	Constraint	Description	Remarks
$\hat{Y}[n]$	$\hat{Y}[n] \in \mathbb R^N$	$N \in \mathbb{N}$	Output data sequence of length $N$

Tool Support

• Matlab

  For details refer to the online documentation of the function ‘sgolayfilt’.

Single Steps using the Algorithm

• Data Denoising with Savitzky-Golay Smoothing

References

• A. Savitzky, M.J.E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedures, Analytical Chemistry, vol. 36, Issue 8, pp 1627-1639, 1964.
• S.J. Orfanidis, Introduction to Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1996.