Smoothing Methods
As a first step in time series analysis it is often helpful to neutralize the effects of the random irregular component and thereby more clearly visualize the behavior of the other three components. Perhaps the simplest method of summarizing a time series is known as a moving average. As the name suggests, the method relies on computing means. We begin by computing the average of the first few observations from, say, y1 through ym. That average then corresponds either to the chronological center of the first m observations or serves as the "prediction" for ym+1. We then move along in the series, computing the average of observations y2through ym+1. This continues until we've passed through the entire sample, continually taking averages of m observations at a time, but always a different group of observations.
Moving averages are commonly used, perhaps chiefly for their simplicity and computability if not for their usefulness as a forecasting method. In JMP, simple moving averages are featured among the control chart methods discussed in Chapter 19 of this book. To introduce the concept of a smoothing method, we'll turn first to three techniques known as exponential smoothing methods. JMP offers six variations of exponential smoothing, and we'll consider three of them.
Unlike moving averages, in which each estimate relies on a relatively small subsample of the data, exponential smoothing uses a weighted combination of all prior values to compute each successive estimate. The computed values incorporate a weight, or smoothing constant, traditionally selected through an iterative trial-and-error process. In exponential smoothing, each forecast is a weighted average of the prior period's observed and forecast values. Your principal text probably presents the formulas used in various exponential smoothing methods, so we'll move directly into JMP.
Simple Exponential Smoothing
Simple (or single) exponential smoothing is most appropriate when the time series is essentially stationary, showing no particularly strong trend or seasonal components. As such it is not well-suited to this particular time series, and we'll see why. The software spares us the exploration for an optimal smoothing constant, and can locate that weight for us. Let's see how we can use the method with the same set of Indian production data.
Click on the red triangle next to Time Series BasicGoods, and select Smoothing Model → Simple Exponential Smoothing.
At the bottom of the Time Series report window, you will find the new panel shown in Figure 17.3. By default JMP reports 95% confidence intervals for the forecast values, and constrains the smoothing constant to be a value between 0 and 1. Just click Estimate.
Figure 17.3. Specifying a Simple Exponential Smoothing Model
Your results should look like Figure 17.4. The statistics in the model summary enable us to evaluate the fit and performance of the model, and are directly comparable to subsequent models that we'll build. In this introductory discussion of time series analysis, we'll focus on just four of the summary measures:
| Variance Estimate | Sometimes called MSE, a measure of the variation in the irregular component. Models with small variance are preferred. |
| RSquare | Goodness of fit measure, just as in regression. Values close to 1 are preferred. |
| MAPE | Mean Absolute Percentage Error. Expresses the average forecast error as a percentage of the observed values. Models with small MAPE are preferred. |
| MAE | Mean Absolute Error (sometimes called MAD—mean absolute deviation). Average of the absolute value of forecast errors. Models with small MAE are preferred. |
Under Parameter Estimates we also see the optimal smoothing constant value of .3845. This value minimizes the model variance and other measures of forecast error.
Figure 17.4. A Simple Exponential Smoothing Model
In the Forecast graph, we see that the model does smooth the series, and to the far right we see the forecasts for the next three periods represented by the short red line bracketed by the blue confidence interval. We can save the forecast values in a new data table as follows:
Click the red triangle next to Model: Simple Exponential Smoothing, and select Save Columns.
This creates a new worksheet containing the observed and forecast values, as well as residuals, standard errors, and confidence limits. We had requested forecasts for three periods into the future. Because the original data table contained three observations that we hid and excluded from calculations, we actually get forecasts for those three months and then three additional months beyond.
Notice that each of the forecasts is the same value, represented in the graph by the horizontal red line—reflecting the problem with applying this simple approach to forecasting a series with a strong trend. For that, we need a method that captures the trend component. Holt's method does just that.
Linear Exponential Smoothing (Holt's Method)
Linear exponential smoothing, developed by Charles Holt in 1957, expands the model of exponential smoothing by incorporating a linear trend element. Operationally, we proceed very much as we did in generating a simple exponential smoothing model.
Again, click the red triangle and select Smoothing Models → Linear Exponential Smoothing.
In your graph of the linear exponential smoothing model, notice that the forecast for the next three periods does rise slightly, in contrast to the simple model in which the forecasts are flat. This method estimates two optimal smoothing weights, one for the level of the index and another for the trend. We also find the same standard set of summary statistics reported, and near the top of the results window we see a model comparison panel, as shown in Figure 17.5.
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Each time we fit a different model, JMP adds a row to this table, enabling easy comparison of the different methods. |
Figure 17.5. Comparing the Two Models
In this case we find that the linear exponential smoothing method has a smaller variance than simple exponential smoothing. On the other hand, the two methods have the same RSquare, and the simple model has slightly better MAPE and MAE.
In both of the smoothed graphs, we do see oscillations that might indicate a seasonal component. Neither simple nor linear exponential smoothing incorporates a seasonal component, but Winters' method does offer an explicit consideration of seasons.
Winters' Method
In time series estimation, there are two fundamental approaches to seasonality. One set of models is additive, treating each season as routinely deviating by a fixed amount above or below the general trend. The other family of models is multiplicative, positing that seasonal variation is a fixed percentage of the overall trend. In JMP, the implementation of Winters' method, named for Peter Winters who proposed it in 1960, is an additive model. Let's see how it works, and as a starting point we'll assume there are four seasons per year, each consisting of three months.
Click the red triangle and select Smoothing Models → Linear Exponential Smoothing.
The resulting parameter estimates and graph are shown in Figure 17.6. The graph is enlarged to more clearly show the forecast for the next three months. Notice in the parameter estimates that there are now three weights, corresponding to the three time series components. In the graph, we see a more refined set of future estimates that continue the upward trend, but show a dip in production in the second month down the road.
Figure 17.6. Results of Winters' Method
Now look at the comparison of the three models on your screen. Of the three models, the Winters' model has the smallest variance and highest RSquare. On the other hand, the simple exponential smoothing model still has the lowest MAPE and MAE.