Current Periodicity. Indicates the periodicity (if any) currently defined for the active dataset. The current periodicity is given as an integerfor example, 12 for annual periodicity, with each case representing a month. The value None is displayed if no periodicity has been set. Seasonal models require a periodicity. You can set the periodicity from the Define Dates dialog box. Dependent Variable Transformation. You can specify a transformation performed on
None. No transformation is performed. Square root. Square root transformation. Natural log. Natural log transformation. Include constant in model. Inclusion of a constant is standard unless you are sure that the overall mean series value is 0. Excluding the constant is recommended when differencing is applied.

18 Chapter 2

Transfer Functions in Custom ARIMA Models
Figure 2-6 ARIMA Criteria dialog box, Transfer Function tab
The Transfer Function tab (only present if independent variables are specified) allows you to define transfer functions for any or all of the independent variables specified on the Variables tab. Transfer functions allow you to specify the manner in which past values of independent (predictor) variables are used to forecast future values of the dependent series.
Transfer Function Orders. Enter values for the various components of the transfer
function into the corresponding cells of the Structure grid. All values must be non-negative integers. For numerator and denominator components, the value represents the maximum order. All positive lower orders will be included in the model. In addition, order 0 is always included for numerator components. For example, if you specify 2 for numerator, the model includes orders 2, 1, and 0. If you specify 3 for

19 Time Series Modeler

denominator, the model includes orders 3, 2, and 1. Cells in the Seasonal column are only enabled if a periodicity has been defined for the active dataset (see Current Periodicity below).
Numerator. The numerator order of the transfer function. Specifies which previous
values from the selected independent (predictor) series are used to predict current values of the dependent series. For example, a numerator order of 1 specifies that the value of an independent series one time period in the pastas well as the current value of the independent seriesis used to predict the current value of each dependent series.
Denominator. The denominator order of the transfer function. Specifies how
deviations from the series mean, for previous values of the selected independent (predictor) series, are used to predict current values of the dependent series. For example, a denominator order of 1 specifies that deviations from the mean value of an independent series one time period in the past be considered when predicting the current value of each dependent series.

estimated model. You can select one or more of the following for inclusion in the plot:
Observed values. The observed values of the dependent series. Forecasts. The model predicted values for the forecast period. Fit values. The model predicted values for the estimation period. Confidence intervals for forecasts. The confidence intervals for the forecast period. Confidence intervals for fit values. The confidence intervals for the estimation

period.

Residual autocorrelation function (ACF). Displays a plot of residual autocorrelations for

each estimated model.

Residual partial autocorrelation function (PACF). Displays a plot of residual partial
autocorrelations for each estimated model.

26 Chapter 2

Limiting Output to the Best- or Poorest-Fitting Models
Figure 2-10 Time Series Modeler, Output Filter tab
The Output Filter tab provides options for restricting both tabular and chart output to a subset of the estimated models. You can choose to limit output to the best-fitting and/or the poorest-fitting models according to fit criteria you provide. By default, all estimated models are included in the output.
Best-fitting models. Select (check) this option to include the best-fitting models in the
output. Select a goodness-of-fit measure and specify the number of models to include. Selecting this option does not preclude also selecting the poorest-fitting models. In that case, the output will consist of the poorest-fitting models as well as the best-fitting ones.

27 Time Series Modeler

Fixed number of models. Specifies that results are displayed for the n best-fitting
models. If the number exceeds the number of estimated models, all models are displayed.
Percentage of total number of models. Specifies that results are displayed for models
with goodness-of-fit values in the top n percent across all estimated models.
Poorest-fitting models. Select (check) this option to include the poorest-fitting models in the output. Select a goodness-of-fit measure and specify the number of models to include. Selecting this option does not preclude also selecting the best-fitting models. In that case, the output will consist of the best-fitting models as well as the poorest-fitting ones. Fixed number of models. Specifies that results are displayed for the n poorest-fitting
with goodness-of-fit values in the bottom n percent across all estimated models.
Goodness of Fit Measure. Select the goodness-of-fit measure to use for filtering models.
The default is stationary R square.

28 Chapter 2

Saving Model Predictions and Model Specifications
Figure 2-11 Time Series Modeler, Save tab
The Save tab allows you to save model predictions as new variables in the active dataset and save model specifications to an external file in XML format.

Example. You are an inventory manager with a major retailer, and responsible for each of 5,000 products. Youve used the Expert Modeler to create models that forecast sales for each product three months into the future. Your data warehouse is refreshed each month with actual sales data which youd like to use to produce monthly updated forecasts. The Apply Time Series Models procedure allows you to accomplish this using the original models, and simply reestimating model parameters to account for the new data. Statistics. Goodness-of-fit measures: stationary R-square, R-square (R2), root mean
square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), maximum absolute error (MaxAE), maximum absolute percentage error (MaxAPE), normalized Bayesian information criterion (BIC). Residuals: autocorrelation function, partial autocorrelation function, Ljung-Box Q.
Plots. Summary plots across all models: histograms of stationary R-square, R-square (R2), root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), maximum absolute error (MaxAE), maximum absolute percentage error (MaxAPE), normalized Bayesian information criterion (BIC); box plots of residual autocorrelations and partial autocorrelations. Results for individual models: forecast values, fit values, observed values, upper and lower confidence limits, residual autocorrelations and partial autocorrelations. Apply Time Series Models Data Considerations Data. Variables (dependent and independent) to which models will be applied should be numeric.

34 Chapter 3

Assumptions. Models are applied to variables in the active dataset with the same names as the variables specified in the model. All such variables are treated as time series, meaning that each case represents a time point, with successive cases separated by a constant time interval. Forecasts. For producing forecasts using models with independent (predictor)
variables, the active dataset should contain values of these variables for all cases in the forecast period. If model parameters are reestimated, then independent variables should not contain any missing values in the estimation period.
The Apply Time Series Models procedure requires that the periodicity, if any, of the active dataset matches the periodicity of the models to be applied. If youre simply forecasting using the same dataset (perhaps with new or revised data) as that used to the build the model, then this condition will be satisfied. If no periodicity exists for the active dataset, you will be given the opportunity to navigate to the Define Dates dialog box to create one. If, however, the models were created without specifying a periodicity, then the active dataset should also be without one.

Seasonal Decomposition Save
Figure 4-2 Season Save dialog box
Create Variables. Allows you to choose how to treat new variables. Add to file. The new series created by Seasonal Decomposition are saved as regular
variables in your active dataset. Variable names are formed from a three-letter prefix, an underscore, and a number.
Replace existing. The new series created by Seasonal Decomposition are saved
as temporary variables in your active dataset. At the same time, any existing temporary variables created by the Trends procedures are dropped. Variable names are formed from a three-letter prefix, a pound sign (#), and a number.
Do not create. The new series are not added to the active dataset. New Variable Names
The Seasonal Decomposition procedure creates four new variables (series), with the following three-letter prefixes, for each series specified:
SAF. Seasonal adjustment factors. These values indicate the effect of each period
on the level of the series.

52 Chapter 4

SAS. Seasonally adjusted series. These are the values obtained after removing the seasonal variation of a series. STC. Smoothed trend-cycle components. These values show the trend and cyclical
behavior present in the series.
ERR. Residual or error values. The values that remain after the seasonal, trend,
and cycle components have been removed from the series.
SEASON Command Additional Features
The SPSS command language also allows you to: Specify any periodicity within the SEASON command rather than select one of the alternatives offered by the Define Dates procedure. See the SPSS Command Syntax Reference for complete syntax information.
The Spectral Plots procedure is used to identify periodic behavior in time series. Instead of analyzing the variation from one time point to the next, it analyzes the variation of the series as a whole into periodic components of different frequencies. Smooth series have stronger periodic components at low frequencies; random variation (white noise) spreads the component strength over all frequencies. Series that include missing data cannot be analyzed with this procedure.

one series leads or lags the other.
By frequency. All plots are produced by frequency, ranging from frequency 0 (the
constant or mean term) to frequency 0.5 (the term for a cycle of two observations).
By period. All plots are produced by period, ranging from 2 (the term for a cycle of two observations) to a period equal to the number of observations (the constant or mean term). Period is displayed on a logarithmic scale.
SPECTRA Command Additional Features
The SPSS command language also allows you to: Save computed spectral analysis variables to the active dataset for later use. Specify custom weights for the spectral window. Produce plots by both frequency and period. Print a complete listing of each value shown in the plot. See the SPSS Command Syntax Reference for complete syntax information.

Part II: Examples

Bulk Forecasting with the Expert Modeler
An analyst for a national broadband provider is required to produce forecasts of user subscriptions in order to predict utilization of bandwidth. Forecasts are needed for each of the 85 local markets that make up the national subscriber base. Monthly historical data is collected in broadband_1.sav, found in the \tutorial\sample_files\ folder where SPSS was installed. In this example, you will use the Expert Modeler to produce forecasts for the next three months for each of the 85 local markets, saving the generated models to an external XML file. Once you are finished, you might want to work through the next example, Bulk Reforecasting by Applying Saved Models in Chapter 7 on p. 73, which applies the saved models to an updated dataset in order to extend the forecasts by another three months without having to rebuild the models.

Examining Your Data

It is always a good idea to have a feel for the nature of your data before building a model. Does the data exhibit seasonal variations? Although the Expert Modeler will automatically find the best seasonal or non-seasonal model for each series, you can often obtain faster results by limiting the search to non-seasonal models when seasonality is not present in your data. Without examining the data for each of the 85 local markets, we can get a rough picture by plotting the total number of subscribers over all markets.
E From the menus choose: Graphs Sequence.
60 Chapter 6 Figure 6-1 Sequence Charts dialog box
E Select Total Number of Subscribers and move it into the Variables list. E Select Date and move it into the Time Axis Labels box. E Click OK.

61 Bulk Forecasting with the Expert Modeler Figure 6-2 Total number of broadband subscribers across all markets
The series exhibits a very smooth upward trend with no hint of seasonal variations. There might be individual series with seasonality, but it appears that seasonality is not a prominent feature of the data in general. Of course you should inspect each of the series before ruling out seasonal models. You can then separate out series exhibiting seasonality and model them separately. In the present case, inspection of the 85 series would show that none exhibit seasonality.

Running the Analysis

To use the Expert Modeler:
E From the menus choose: Analyze Time Series Create Models.
62 Chapter 6 Figure 6-3 Time Series Modeler dialog box
E Select Subscribers for Market 1 through Subscribers for Market 85 for dependent

variables.

E Verify that Expert Modeler is selected in the Method drop-down list. The Expert
Modeler will automatically find the best-fitting model for each of the dependent variable series. The set of cases used to estimate the model is referred to as the estimation period. By default, it includes all of the cases in the active dataset. You can set the estimation period by selecting Based on time or case range in the Select Cases dialog box. For this example, we will stick with the default.
63 Bulk Forecasting with the Expert Modeler
Notice also that the default forecast period starts after the end of the estimation period and goes through to the last case in the active dataset. If you are forecasting beyond the last case, you will need to extend the forecast period. This is done from the Options tab as you will see later on in this example.
E Click Criteria. Figure 6-4 Expert Modeler Criteria dialog box, Model tab
E Deselect Expert Modeler considers seasonal models in the Model Type group.
Although the data is monthly and the current periodicity is 12, we have seen that the data does not exhibit any seasonality, so there is no need to consider seasonal models. This reduces the space of models searched by the Expert Modeler and can significantly reduce computing time.
64 Chapter 6 E Click Continue. E Click the Options tab on the Time Series Modeler dialog box. Figure 6-5 Time Series Modeler, Options tab
E Select First case after end of estimation period through a specified date in the Forecast

Period group.

E Select Mean absolute percentage error and Maximum absolute percentage error in the
Plots for Comparing Models group.

68 Chapter 6

Absolute percentage error is a measure of how much a dependent series varies from its model-predicted level. By examining the mean and maximum across all models, you can get an indication of the uncertainty in your predictions. And looking at summary plots of percentage errors, rather than absolute errors, is advisable since the dependent series represent subscriber numbers for markets of varying sizes.
E Click OK in the Time Series Modeler dialog box.

Model Summary Charts

Figure 6-9 Histogram of mean absolute percentage error
This histogram displays the mean absolute percentage error (MAPE) across all models. It shows that all models display a mean uncertainty of roughly 1%.
69 Bulk Forecasting with the Expert Modeler Figure 6-10 Histogram of maximum absolute percentage error
This histogram displays the maximum absolute percentage error (MaxAPE) across all models and is useful for imagining a worst-case scenario for your forecasts. It shows that the largest percentage error for each model falls in the range of 1 to 5%. Do these values represent an acceptable amount of uncertainty? This is a situation in which your business sense comes into play because acceptable risk will change from problem to problem.

70 Chapter 6

Model Predictions
Figure 6-11 New variables containing model predictions
The Data Editor shows the new variables containing the model predictions. Although only two are shown here, there are 85 new variables, one for each of the 85 dependent series. The variable names consist of the default prefix Predicted, followed by the name of the associated dependent variable (for example, Market_1), followed by a model identifier (for example, Model_1). Three new cases, containing the forecasts for January 2004 through March 2004, have been added to the dataset, along with automatically generated date labels. Each of the new variables contains the model predictions for the estimation period (January 1999 through December 2003), allowing you to see how well the model fits the known values.
Figure 6-12 Forecast table
71 Bulk Forecasting with the Expert Modeler

Modeler will automatically find the best-fitting seasonal or non-seasonal model for the dependent variable series.
E Click Criteria and then click the Outliers tab. Figure 8-4 Expert Modeler Criteria dialog box, Outliers tab
E Select Detect outliers automatically and leave the default selections for the types of
outliers to detect. Our visual inspection of the data suggested that there may be outliers. With the current choices, the Expert Modeler will search for the most common outlier types and incorporate any outliers into the final model. Outlier detection can add significantly to the computing time needed by the Expert Modeler, so it is a feature that should

86 Chapter 8

be used with some discretion, particularly when modeling many series at once. By default, outliers are not detected.
E Click Continue. E Click the Save tab on the Time Series Modeler dialog box. Figure 8-5 Time Series Modeler, Save tab
You will want to save the estimated model to an external XML file so that you can experiment with different values of the predictorsusing the Apply Time Series Models procedurewithout having to rebuild the model.
87 Using the Expert Modeler to Determine Significant Predictors E Navigate to the folder where you would like to save the XML model file, enter a filename, and click Save.
E Click the Statistics tab. Figure 8-6 Time Series Modeler, Statistics tab
E Select Parameter estimates.
This option produces a table displaying all of the parameters, including the significant predictors, for the model chosen by the Expert Modeler.
88 Chapter 8 Figure 8-7 Time Series Modeler, Plots tab

E Deselect Forecasts.

In the current example, we are only interested in determining the significant predictors and building a model. We will not be doing any forecasting.

E Select Fit values.

This option displays the predicted values in the period used to estimate the model. This period is referred to as the estimation period, and it includes all cases in the active dataset for this example. These values provide an indication of how well the model fits the observed values, so they are referred to as fit values. The resulting plot will consist of both the observed values and the fit values.
89 Using the Expert Modeler to Determine Significant Predictors

Series Plot

Figure 8-8 Predicted and observed values
The predicted values show good agreement with the observed values, indicating that the model has satisfactory predictive ability. Notice how well the model predicts the seasonal peaks. And it does a good job of capturing the upward trend of the data.

90 Chapter 8

Model Description Table
Figure 8-9 Model Description table
The model description table contains an entry for each estimated model and includes both a model identifier and the model type. The model identifier consists of the name (or label) of the associated dependent variable and a system-assigned name. In the current example, the dependent variable is Sales of Mens Clothing and the system-assigned name is Model_1. The Time Series Modeler supports both exponential smoothing and ARIMA models. Exponential smoothing model types are listed by their commonly used names such as Holt and Winters Additive. ARIMA model types are listed using the standard notation of ARIMA(p,d,q)(P,D,Q), where p is the order of autoregression, d is the order of differencing (or integration), and q is the order of moving-average, and (P,D,Q) are their seasonal counterparts. The Expert Modeler has determined that sales of mens clothing is best described by a seasonal ARIMA model with one order of differencing. The seasonal nature of the model accounts for the seasonal peaks that we saw in the series plot, and the single order of differencing reflects the upward trend that was evident in the data.

Model Statistics Table

Figure 8-10 Model Statistics table
The model statistics table provides summary information and goodness-of-fit statistics for each estimated model. Results for each model are labeled with the model identifier provided in the model description table. First, notice that the model contains two predictors out of the five candidate predictors that you originally specified. So it appears that the Expert Modeler has identified two independent variables that may prove useful for forecasting.
91 Using the Expert Modeler to Determine Significant Predictors
Although the Time Series Modeler offers a number of different goodness-of-fit statistics, we opted only for the stationary R-squared value. This statistic provides an estimate of the proportion of the total variation in the series that is explained by the model and is preferable to ordinary R-squared when there is a trend or seasonal pattern, as is the case here. Larger values of stationary R-squared (up to a maximum value of 1) indicate better fit. A value of 0.948 means that the model does an excellent job of explaining the observed variation in the series. The Ljung-Box statistic, also known as the modified Box-Pierce statistic, provides an indication of whether the model is correctly specified. A significance value less than 0.05 implies that there is structure in the observed series which is not accounted for by the model. The value of 0.984 shown here is not significant, so we can be confident that the model is correctly specified. The Expert Modeler detected nine points that were considered to be outliers. Each of these points has been modeled appropriately, so there is no need for you to remove them from the series.

E From the menus choose: Graphs Time Series Autocorrelations.
111 Seasonal Decomposition Figure 10-4 Autocorrelations dialog box
E Select men and move it into the Variables list. E Click OK. Figure 10-5 Autocorrelation plot for men

112 Chapter 10

The autocorrelation function shows a significant peak at a lag of 1 with a long exponential taila typical pattern for time series. The significant peak at a lag of 12 suggests the presence of an annual seasonal component in the data. Examination of the partial autocorrelation function will allow a more definitive conclusion.
Figure 10-6 Partial autocorrelation plot for men
The significant peak at a lag of 12 in the partial autocorrelation function confirms the presence of an annual seasonal component in the data. To set an annual periodicity:
E From the menus choose: Data Define Dates.
113 Seasonal Decomposition Figure 10-7 Define Dates dialog box
E Select Years, months in the Cases Are list. E Enter 1989 for the year and 1 for the month. E Click OK.
This sets the periodicity to 12 and creates a set of date variables that are designed to work with Trends procedures.
To run the Seasonal Decomposition procedure:
114 Chapter 10 Figure 10-8 Seasonal Decomposition dialog box
E Select men and move it into the Variables list. E Select Multiplicative in the Model group. E Click OK.

Understanding the Output

The Seasonal Decomposition procedure creates four new variables for each of the original variables analyzed by the procedure. By default, the new variables are added to the active data set. The new series have names beginning with the following prefixes:
SAF. Seasonal adjustment factors, representing seasonal variation. For the multiplicative model, the value 1 represents the absence of seasonal variation; for the additive model, the value 0 represents the absence of seasonal variation. SAS. Seasonally adjusted series, representing the original series with seasonal
variations removed. Working with a seasonally adjusted series, for example, allows a trend component to be isolated and analyzed independent of any seasonal component.
STC. Smoothed trend-cycle component, which is a smoothed version of the seasonally
adjusted series that shows both trend and cyclic components.
ERR. The residual component of the series for a particular observation.
115 Seasonal Decomposition
For the present case, the seasonally adjusted series is the most appropriate, because it represents the original series with the seasonal variations removed.

E Select Sales of Mens Clothing and move it into the Variables list. E Select Spectral density in the Plot group. E Click OK.

121 Spectral Plots

Understanding the Periodogram and Spectral Density

Figure 11-2 Periodogram

The plot of the periodogram shows a sequence of peaks that stand out from the background noise, with the lowest frequency peak at a frequency of just less than 0.1. You suspect that the data contain an annual periodic component, so consider the contribution that an annual component would make to the periodogram. Each of the data points in the time series represents a month, so an annual periodicity corresponds to a period of 12 in the current data set. Because period and frequency are reciprocals of each other, a period of 12 corresponds to a frequency of 1/12 (or 0.083). So an annual component implies a peak in the periodogram at 0.083, which seems consistent with the presence of the peak just below a frequency of 0.1.
122 Chapter 11 Figure 11-3 Univariate statistics table
The univariate statistics table contains the data points that are used to plot the periodogram. Notice that, for frequencies of less than 0.1, the largest value in the Periodogram column occurs at a frequency of 0.08333precisely what you expect to find if there is an annual periodic component. This information confirms the identification of the lowest frequency peak with an annual periodic component. But what about the other peaks at higher frequencies?
123 Spectral Plots Figure 11-4 Spectral density
The remaining peaks are best analyzed with the spectral density function, which is simply a smoothed version of the periodogram. Smoothing provides a means of eliminating the background noise from a periodogram, allowing the underlying structure to be more clearly isolated. The spectral density consists of five distinct peaks that appear to be equally spaced. The lowest frequency peak simply represents the smoothed version of the peak at 0.08333. To understand the significance of the four higher frequency peaks, remember that the periodogram is calculated by modeling the time series as the sum of cosine and sine functions. Periodic components that have the shape of a sine or cosine function (sinusoidal) show up in the periodogram as single peaks. Periodic components that are not sinusoidal show up as a series of equally spaced peaks of different heights, with the lowest frequency peak in the series occurring at the frequency of the periodic component. So the four higher frequency peaks in the spectral density simply indicate that the annual periodic component is not sinusoidal. You have now accounted for all of the discernible structure in the spectral density plot and conclude that the data contain a single periodic component with a period of 12 months.

same units as the dependent series. Like MaxAPE, it is useful for imagining the worst-case scenario for your forecasts. Maximum absolute error and maximum

126 Appendix A

absolute percentage error may occur at different series points--for example, when the absolute error for a large series value is slightly larger than the absolute error for a small series value. In that case, the maximum absolute error will occur at the larger series value and the maximum absolute percentage error will occur at the smaller series value.
Normalized BIC. Normalized Bayesian Information Criterion. A general measure
of the overall fit of a model that attempts to account for model complexity. It is a score based upon the mean square error and includes a penalty for the number of parameters in the model and the length of the series. The penalty removes the advantage of models with more parameters, making the statistic easy to compare across different models for the same series.

Outlier Types

This section provides definitions of the outlier types used in time series modeling.
Additive. An outlier that affects a single observation. For example, a data coding
error might be identified as an additive outlier.
Level shift. An outlier that shifts all observations by a constant, starting at a
particular series point. A level shift could result from a change in policy.
Innovational. An outlier that acts as an addition to the noise term at a particular
series point. For stationary series, an innovational outlier affects several observations. For nonstationary series, it may affect every observation starting at a particular series point.
Transient. An outlier whose impact decays exponentially to 0. Seasonal additive. An outlier that affects a particular observation and all
subsequent observations separated from it by one or more seasonal periods. All such observations are affected equally. A seasonal additive outlier might occur if, beginning in a certain year, sales are higher every January.
Local trend. An outlier that starts a local trend at a particular series point. Additive patch. A group of two or more consecutive additive outliers. Selecting this
outlier type results in the detection of individual additive outliers in addition to patches of them.

Guide to ACF/PACF Plots

The plots shown here are those of pure or theoretical ARIMA processes. Here are some general guidelines for identifying the process: Nonstationary series have an ACF that remains significant for half a dozen or more lags, rather than quickly declining to 0. You must difference such a series until it is stationary before you can identify the process. Autoregressive processes have an exponentially declining ACF and spikes in the first one or more lags of the PACF. The number of spikes indicates the order of the autoregression. Moving average processes have spikes in the first one or more lags of the ACF and an exponentially declining PACF. The number of spikes indicates the order of the moving average. Mixed (ARMA) processes typically show exponential declines in both the ACF and the PACF. At the identification stage, you do not need to worry about the sign of the ACF or PACF, or about the speed with which an exponentially declining ACF or PACF approaches 0. These depend upon the sign and actual value of the AR and MA coefficients. In some instances, an exponentially declining ACF alternates between positive and negative values. ACF and PACF plots from real data are never as clean as the plots shown here. You must learn to pick out what is essential in any given plot. Always check the ACF and PACF of the residuals, in case your identification is wrong. Bear in mind that: Seasonal processes show these patterns at the seasonal lags (the multiples of the seasonal period).