The LinearFilter function applies linear filter to a set of observations. By default, convolution method is used:

`X'`[i] = Sum(X[i+1-j]*Y[j], j = 1..m);

${\mathrm{X\text{'}}}_i=\sum _{j=1}^m{X}_{i+1-j}{Y}_j$

where m is the size of the filter. For $i<m$ the set of initial values will be used. By default, X is padded on the left with $m-1$ zeros. Option initial can be used to specify the initial values.

Recursive filter is defined as follows:

`X'`[i] = X[i]*Y[1]+Sum(`X'`[i+1-j]*Y[j], j = 2..m);

${\mathrm{X\text{'}}}_i=X_i{Y}_1+\left(\sum _{j=2}^m{\mathrm{X\text{'}}}_{i+1-j}{Y}_j\right)$

The first parameter X is a single data sample - given as e.g. a Vector. Each value represents an individual observation.

The second parameter Y is the filter - given as e.g. a Vector. Each value represents a filter coefficient.

The options argument can contain one or more of the options shown below.

ignore=truefalse -- This option is used to specify how to handle non-numeric data. If ignore is set to true all non-numeric items in X will be ignored. Missing values are allowed in the data set but not in the filter.

initial=deduce, or Vector -- This option specifies the initial values in reverse order. The default is a set of $m-1$ zeros.

recursive=truefalse -- If this option is set to true then recursive filter will be used.

$\mathrm{with}\left(\mathrm{Statistics}\right)\&colon;$

$A≔⟨\mathrm{seq}\left(\sin \left(i\right)\&comma;i=1..9\right)\&comma;\mathrm{undefined}⟩\&colon;$

$\mathrm{LinearFilter}\left(A\&comma;⟨1.\&comma;0.3\&comma;0.2⟩\right)$

$\left[\begin{array}{c}0.841470984807897\\ 1.16173872226805\\ 0.582203433069151\\ \mathrm{-0.532607007524832}\\ \mathrm{-1.15774102164354}\\ \mathrm{-0.718453279659453}\\ 0.381377094326484\\ 1.13057112659923\\ 0.840323278972529\\ Float\left(\mathrm{undefined}\right)\end{array}\right]$

$\mathrm{LinearFilter}\left(A\&comma;⟨1.\&comma;0.3\&comma;0.2⟩\&comma;\mathrm{ignore}\right)$

$\left[\begin{array}{c}0.841470984807897\\ 1.16173872226805\\ 0.582203433069151\\ \mathrm{-0.532607007524832}\\ \mathrm{-1.15774102164354}\\ \mathrm{-0.718453279659453}\\ 0.381377094326484\\ 1.13057112659923\\ 0.840323278972529\end{array}\right]$

$\mathrm{LinearFilter}\left(A\&comma;⟨1.\&comma;0.3\&comma;0.2⟩\&comma;\mathrm{ignore}\&comma;\mathrm{initial}=⟨3\&comma;2\&comma;1⟩\right)$

$\left[\begin{array}{c}2.14147098480790\\ 1.76173872226805\\ 0.582203433069151\\ \mathrm{-0.532607007524832}\\ \mathrm{-1.15774102164354}\\ \mathrm{-0.718453279659453}\\ 0.381377094326484\\ 1.13057112659923\\ 0.840323278972529\end{array}\right]$

$\mathrm{LinearFilter}\left(A\&comma;⟨1.\&comma;0.3\&comma;0.2⟩\&comma;\mathrm{ignore}\&comma;\mathrm{recursive}\right)$

$\left[\begin{array}{c}0.841470984807897\\ 1.16173872226805\\ 0.657935821701862\\ \mathrm{-0.327074004343760}\\ \mathrm{-0.925459311625894}\\ \mathrm{-0.622468092555446}\\ 0.285154308626976\\ 0.950410920700385\\ 0.754272623177268\end{array}\right]$