Tag Archives: Filters

Trend extraction and Detrending

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.

Assume that m_{t} is the filtered version of the y_{t} series. Then

m_{t} = M_{n}(L)y_{t} =\sum_{j=-r}^{j=r} w_{j}y_{t-j}


The weights of a moving average filters add up to one i.e.  M_{n}(L)=1. The simplest such filter is the uniform moving average for which:

w_{j} = \frac{1}{n} \; \; \; \; j=-r,...,r


The gain of such filter is

M_{n}(e^{-i \lambda}) = \left| \sum_{j=-r}^{r} \frac{1}{n} e^{-ij\lambda} \right| = \left| \frac{1}{n} \left( 1+2 \sum_{j=1}^{r} cos \lambda_{j} \right) \right| = \left| \frac{sin (n \lambda /2)}{n sin (\lambda /2)} \right|




Gain of uniform moving average filter

The uniform moving average filter (applied on artificial data)

The Moving Average filter removes a cycle of period n together with its harmonics.

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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}


\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.


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