Centered Moving Average

Local Algorithm - One-Dimensional Algorithm

Centered Moving Average algorithm replaces each original data value by the average over its neighbor values. The choice of filter width has a great impact on the final results. The basic formula (for filter width \(L = 2M+1\)) is stated as follows:

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

where \(x[n]\) and \(\hat{x}[n]\) denote raw data and processed data, respectively.

Input Parameters

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

Output Parameters

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

Tool Support

• Excel

• Matlab

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

Single Steps using the Algorithm

• Data Denoising with Centered Moving Average

References

• B. Jaehne, Digitale Bildverarbeitung, 5th ed. Berlin: Springer Verlag, 2002.
• K.D. Kammeyer and K. Kroschel, Digitale Signalverarbeitung, 5th ed. Stuttgart: Teubner, 2002.