Simple Moving Average

Moving average is the simplest technique for smoothing. The operator determines a time window within which the values are averaged. For the next value the window is lagged one step ahead and the mean is again calculated. The time window will then be 

In our example, two centered windows are applied. With the symmetrical window with 5 steps (5 months were averaged) a smoothing effect can be seen (Fig. 6-4). The most extreme points are cut off and yearly periods are indicated. 

As a second possibility, a symmetrical window with 13 steps was also chosen (Fig. 6-5). This size is larger than one period in the series, because there are twelve months in the year. A good starting point for estimating a trend can therefore be achieved. In this application, the negative trend of the last three years is detectable without seasonal fluctuations. 

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An example is the relatively simple moving average filter. In case of a digitized signal, the values of a fixed (odd) number of data points (a window) are added and divided by the number of points. The result is a new value of the center point. Then the window shifts one point and the procedure, which can be considered as a convolution of the sipal with a rectangular pulse function, repeats. Of course, other functions like a triangle, an exponential and a Gaussian, can be used. 

FIGURE 10.13 FIR signal processing with a simple moving-average filter. 

FIGURE 10.15 Transfer functions for FIR filters shown as a function of window size. Transfer functions for simple, moving-average filters with windows of 5 (...), 10 (-), and... 

The simple moving average technique can be represented mathematically by... 

Simple moving-average formulae which can be applied to oscillatory shear data to recover estimates of the relaxation spectrum have been reported [Davies and Anderssen, 1998]. These formulae represent an improvement over previous commercial software in that they take into accoimt the limits imposed by sampling localization and yield accurate spectra very rapidly on a PC. 

The output of the simple moving average filter is the average of the M -F 1 most recent values of x(n). Intuitively, this corresponds to a smoothed version of the input, but its operation is more appropriately described by calculating the frequency response of the filter. First, however, the z-domain representation of the filter is introduced in analogy to the s- (or Laplace-) domain representation of analog filters. The z transform of a causal discrete-time signal x(n) is defined by... 

The z transform of a delayed version of x( ), namely, x n — k) with k a positive integer, is found to be given by z X z). This result can be used to relate the z transform of the output y n) of the simple moving average filter to its input... 

FIGURE 8.95 The magnitude and phase response of the simple moving average filter with M = 7. 

The magnitude and phase response of the simple moving average filter, with M = 7, are calculated from j (g 27r/) shown in Fig. 8.95. The filter is seen clearly to act as a crude low-pass, smoothing filter with a linear-phase response. The sampHng frequency periodicity in the magnitude and phase response is a property of discrete-time systems. The hnear-phase response is due to the term in... 

The z-domain transfer function is shown to be the ratio of two Mth-order polynomials in z, namely, JV(z) and D z). The values of z for which N z) = 0 are termed the zeros of the filter, whereas those for which D z) = 0 are the poles. The poles of such an FIR filter are at the origin, that is, z = 0, in the z plane. The positions of the zeros are determined by the weighting coefficients, that is, foji, fc = 0,1,..., M. The poles and zeros in the z plane for the simple moving average filter are shown in Fig. 8.96. The zeros, marked with a circle, are coincident with the unit circle, that is, the contour in the z plane for which... 

Anderssen, R. S., Davies, A. R. Simple moving-average formulae for the direct recovery of the relaxation spectrum. /. Rheol. (2001) 45, pp. 1-27... 

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