Backward Moving Average

Local Algorithm - One-Dimensional Algorithm

Backward 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 $L = M+1$) is stated as follows:

$$\hat{x}[n] = \frac{1}{M+1} \sum_{k=-M}^0 x[n+k] \, \text{,}$$

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’.
$L$	$L \in \mathbb N$	$L = M + 1, \quad M \in \mathbb{N}$

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 Backward 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.