Weighted Backward Moving Average

Local Algorithm - One-Dimensional Algorithm

Weighted Backward Moving Average algorithm is based on the Backward Moving Average algorithm. Differently, varying weights are assigned to the values within the filter width. The basic formula (for filter width \(L = M+1\)) is stated as follows:

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

where \(x[n]\), \(\hat{x}[n]\) and \(w[k]\) denote raw data, processed data and weights, 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’.
\(L\)	\(L \in \mathbb N\)	\(L = M + 1, \quad M \in \mathbb{N}\)
\(w[k]\)	\(w[k] \in \mathbb{R}^L\)	\(w[k] \geq 0, \quad \sum w[k] = 1\)	Weighting vector of of length \(L\)

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 Weighted Backward Moving Average