Moving averages

Autoregressive Integrated Moving Average Model (ARIMA) ... 

Another approach to smoothing is to use the arithmetic moving average, which is represented by the following equation ... 

Figure 3.11. Smoothing a noisy signal. The synthetic, noise-free signal is given at the top. After the addition of noise by means of the Monte Carlo technique, the panels in the second row are obtained (little noise, left, five times as much noise, right). A seven-point Savitzky-Golay filter of order 2 (third row) and a seven-point moving average (bottom row) filter are...

The moving average can be explained as before. After having obtained the first set of averages, the train is moved by one point and the operation is repeated. This simply provides more points to the picture. Wider bins result in more averaging. There is a trade-off between an increase of the signal-to-... 

A general approach was developed by G.E.P. Box and G.M. Jenkins (S) which combines these various methods into an analysis which permits choice of the most appropriate model, checks the forecast precision, and allows for interpretation. The Box-Jenkins analysis is an autoregressive integrated moving average model (ARIMA). This approach, as implemented in the MINITAB computer program is one used for the analyses reported here. 

There are three adjustable parameters in a Box-Jenkins analysis, one each for autoregression, differencing, and moving average terms. Corrections for cyclical behavior may be added as three optional terms. The approach is flexible, and provides much information. 

The ARIMA analysis evaluates the autocorrelation functions to determine the order of the appropriate moving average and the need for differencing. An appropriate model is chosen and the fit to the data is constructed followed by a careful analysis of the residuals. The parameters are adjusted and the fit is checked again. The process is applied iteratively until the errors are minimized or the model fails to converge. 

As a consequence, a moving average in the time domain is a multiplication in the Fourier domain, namely ... 

Fig. 40.22. Distortion (hJhn) of a Gaussian peak for various window sizes (indicated within parentheses). (a) Moving average, (b) Polynomial smoothing.

The convolution or smoothing function, h f), used in moving averaging is a simple block function. However, one could try and derive somewhat more complex convolution functions giving a better signal-to-noise ratio with less deformation of the underlying deterministic signal. 

Multiplication of this 4x8 transformation matrix with the 8x1 column vector of the signal results in 4 wavelet transform coefficients or N/2 coefficients for a data vector of length N. For c, = C2 = Cj = C4 = 1, these wavelet transform coefficients are equivalent to the moving average of the signal over 4 data points. Consequently,... 

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

For an IIR filter, the parameter T in Eq. (9) tends to infinity. IIR filters can be represented as a function of previous filter outputs and often can be computed with fewer multiplications and reduced data storage requirements compared to a FIR filter. A popular example of an IIR filter is the exponentially weighted moving average (EWMA) or exponential smoothing, which is represented as... 

To illustrate behaviors of different filters, consider a moving average filter that averages over 11 terms. Such a filter has the frequency response shown in Fig. 8. Note that this filter has a relatively low gain of 0.55 at the break-point frequency of 0.05 cycles per minute. So in the range of... 

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