Some notes on Linear Filters

Let \{ X_t \} and  \{ Y_t \} be two stationary time series related by:

 

X_{t} = M_{n}(L)Y_{t} =\sum_{j=- \infty}^{j=\infty} g_{j}Y_{t-j}

where

\sum_{j=- \infty}^{j=\infty}|g_{j}| < \infty and  \sum_{j=- \infty}^{j=\infty}|g_{j}|^2 < \infty

 

\{ X_t \} is the filtered version of  \{ Y_t \} and M_{n}(L) is the filter. The effect of a linear filter is to change the importance of various cyclical components of the series and/or induce a shift with regard to the position in time.

 

 

Note that

{M_{n}(1) = 1}: Trend Extraction
{M_{n}(1) = 0}: Detrending

A Linear time-invariant filter demonstrates the following property:

\displaystyle X_{t}=M(L)Y_{t} = M(L) \Psi (L) \epsilon_{t}

 

 

We can show that the spectral density function f of the filtered series is equal to

\displaystyle f_{y}(\lambda)=\underbrace{|M(e^{-i\lambda})|^{2}}_{| \Gamma ( \omega) |^{2}}f_{x}(\lambda)

 

where | \Gamma ( \omega) |^{2}| is the Transfer Function:

\displaystyle | \Gamma ( \omega) |^{2} =\left| \sum_{j=-r}^{r} g_{j}e^{-i j \omega} \right| ^2

 

\displaystyle \Gamma ( \omega) is the Frequency Response Function:

\displaystyle \Gamma ( \omega) = M_{n}(e^{-i \omega})=\sum_{j=-r}^{r} g_{j}e^{-i j \omega}

 

and \displaystyle M_{n}(\omega) is the Gain:

\displaystyle M_{n}(\omega) = |M_{n}(e^{-i \omega}) |

 

Τhe Gain indicates the factor by which the amplitude of a cyclical component changes as a result of applying the linear filter. On the other side the phase Ph(λ) measures the shift in time

\displaystyle  \boxed{Ph(\lambda)=tan^{-1}[-M^{\dotplus}(\lambda)/M^{*}(\lambda)], \;  \; \; \; 0\leqslant\lambda\leqslant\pi }

where

\displaystyle M(e^{-i\lambda})=iM^{\dotplus}(\lambda)+M^{*}(\lambda)

 

Note finally that if k multiple filters are applied consecutively

\displaystyle X_{t}=M_{k}(L)...M_{2}(L)M_{1}(L)Y_{t}

their combined effect is simply given by the product of the respective frequency response functions. Hence:

\displaystyle M(e^{-i\lambda})=M_{k}(e^{-i\lambda})...M_{2}(e^{-i\lambda})M_{1}(e^{-i\lambda})

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One thought on “Some notes on Linear Filters

  1. […] Filters may be applied to a time series for a variety of reasons. Suppose that a time series consists of a long-term movement, a trend, upon which is superimposed an irregular component. A moving average filter will smooth the series revealing the trend more clearly. […]

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