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.