Exponential smoothing

The principle of exponential averaging has been introduced in Chapters 7 and 20, and is given by the following equation  

The exponential shape of the filter follows directly by elaborating eq. (40.14) for a few consecutive data points (see Table 40.4). From this table we can see that a smoothed data point at time i is the average of all data points measured before, weighted with an exponentially decaying weight X ) with d the distance of that data point from the measurement to be smoothed. Such shapes are also found for electronic filters with a given time constant. The effect of exponential smoothing is visualized in the plot of the a , and values (Fig. 40.25) listed in 

The exponential shape of the moving average filter (x = (1 - X.)x,- + Xx, i) i Measurement Smoothed value 

Simple exponential smoothing of a time series means the representation of a value at a specific time by an exponential weighted sum of recent values  

Umax — number of cases which join the series a - smoothing parameter 

Low values of a (minimum value zero) introduce a long memory effect, higher a (maximum value 1) create a short memory. Fig. 6-6 explains the effect of the parameter a in a small simulated model time series. 

To estimate a for the best fitted time series it is necessary to calculate the sum of squared residuals. The best fit is that with a minimum sum of squared residuals. 

Time series without systematic changes (trend or seasonal fluctuations), i.e. with a fixed level, are best approximated by the mean of the series, i.e. a = 0. The mean over the full time range gives a minimum sum of squared differences between the mean and the original series (squared residues). All cases have the same weight, because a is equal to zero. 

---

Fig. 40.25. Effect of exponential smoothing on the data points listed in Table 40.3 (solid line original data dotted line smoothed data).

Table 40.3. As one can see, the filter introduces a slower response to stepwise changes of the signal, as if it were measured with an instrument with a large response time. Because fluctuations are smoothed, the standard deviation of the signal is decreased, in this example from 2.58 to 1.95. A Gaussian peak is broadened and becomes asymmetric by exponential smoothing (Fig. 40.26).

Input mapping methods can be divided into univariate, multivariate, and probabalistic methods. Univariate methods analyze the inputs by extracting the relationship between the measurements. These methods include various types of single-scale and multiscale filtering such as exponential smoothing, wavelet thresholding, and median filtering. Multivariate methods analyze... 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

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... 

The starting point for the computation of the exponential smoothing model with trend and seasonal effects is the additive component model ... 

The trend component is, therefore, a weighted mean from the previous slopes of the series. The seasonal component, sea(t), is the weighted mean from the seasonal deviations from the mean seasonal value of the series. This means that all monthly values are averaged in year x. This is the starting point for calculating the seasonal component, sea(t), e.g. for January. It is the deviation of the January value from the total mean for one year. The seasonal components for each month were smoothed exponentially, smoothing over all of the years. 

Exponential smoothing is intended for calculation of one step ahead forecasts. All further forecasts x t+2), a(/+3),. .. relate to the recent forecasted value x(t+ ), x t+2),. .. and also, in dependence on the value of the smoothing parameter, to more recent, real values ... 

The technique of seasonal decomposition uses the same additive and multiplicative models as in exponential smoothing, but without the smoothing procedure. 

The moving average is a process similar to exponential smoothing. The exponential smoothing method (see also Section 6.4.2) has exponentially decreasing coefficients of the recent values. In a MA model the single coefficients b1 b2,. .., b were calculated by minimization of the sum of squared errors. 

Because these are forecasted limits, the same exponential smoothing can be applied ... 

Fig. 3 Filter used in (a) mean filtering and (b) exponential smoothing.

Fig. 14 (a) Bumps signal with mean shift and white noise oj varianee 0.5. (h) OLMS filtering using Haar (MSE = 0.1635J, (e) mean filtering (MSE = 0.2530J, (d) exponential smoothing (MSE - 0.2237). 

Averages, medians and differences are but three methods which can be used for data smoothing. The advantage of these methods over other methods such as exponential smoothing is that these methods are easily applied to most sets of data. 

Many of the big chain stores have a database and computer software that can predict the requirements for the days to be scheduled, based on annual history and recent history that wUl account for trends. Exponential smoothing, running averages, or other methods can be used to predict the daily demand. Adjustments must be made based on weather conditions in the past, special promotions, or other predictable influences. 

---