This example analyzes the gas furnace data (seriesJ.jmp) from Box and Jenkins. To begin the analysis, select Input Gas Rate as the Input List and Output CO2 as the Y, Time Series. The launch dialog should appear as in Series J Launch Dialog.
Series J Launch Dialog
This preliminary report shows diagnostic information and groups the analysis in two main parts. The first part, under Time Series Output CO2, contains analyses of the output series, while the Input Time Series Panel, contains analyses on the input series. The latter may include more than one series.
Series J Report
Each report section has its own set of commands. For the output (top) series, the commands are accessible from the red triangle on the outermost outline bar (Transfer Function Analysis). For the input (bottom) series, the red triangle is located on the inner outline bar (Input Series: Input Gas Rate).
Output and Input Series Menus shows these two command sets. Note their organization. Both start with a Graph command. The next set of commands are for exploration. The third set is for model building. The fourth set includes functions that control the platform.
Both parts give basic diagnostics, including the sample mean (Mean), sample standard deviation (Std), and series length (N).
In addition, the platform tests for stationarity using Augmented Dickey-Fuller (ADF) tests.
Basic diagnostics also include the autocorrelation and partial autocorrelation functions, as well as the Ljung-Box Q-statistic and p-values, found under the Time Series Basic Diagnostics outline node.
The Cross Correlation command adds a cross-correlation plot to the report. The length of the plot is twice that of an autocorrelation plot, or .
The plot includes plots of the output series versus all input series, in both numerical and graphical forms. The blue lines indicate two standard errors.
Building a transfer function model is quite similar to building an ARIMA model, in that it is an iterative process of exploring, fitting, and comparing.
Before building a model and during the data exploration process, it is sometimes useful to prewhiten the data. This means find an adequate model for the input series, apply the model to the output, and get residuals from both series. Compute cross-correlations from residual series and identify the proper orders for the transfer function polynomials.
To prewhiten the input series, select the Prewhitening command. This brings up a dialog similar to the ARIMA dialog where you specify a stochastic model for the input series. For our SeriesJ example, we use an ARMA(2,2) prewhitening model, as shown in Prewhitening Dialog.
Click Estimate to reveal the Prewhitening plot.
Yt denotes the output series
X1 to Xm denote m input series
et represents the noise series
X1, t–d1 indicates the series X1 is indexed by t with a d1-step lag
μ represents the mean level of the model
ϕ(B) and θ(B) represent autoregressive and moving average polynomials from an ARIMA model
ωk(B) and δk(B) represent numerator and denominator factors (or polynomials) for individual transfer functions, with k representing an index for the 1 to m individual inputs.
Each polynomial in the above model can contain two parts, either nonseasonal, seasonal, or a product of the two as in seasonal ARIMA. When specifying a model, leave the default 0 for any part that you do not want in the model.
Select Transfer Function to bring up the model specification dialog.
contains specifications for the noise series. Lowercase letters are coefficients for non-seasonal polynomials, and uppercase letters for seasonal ones.
specifies polynomials related to the input series.The first three orders deal with non-seasonal polynomials. The next four are for seasonal polynomials. The final is for an input lag.
specifies whether μ is zero or not.
Using the information from prewhitening, we specify the model as shown in Transfer Function Specification Dialog.
The analysis report is titled Transfer Function Model and is indexed sequentially. Results for the Series J example are shown in Series J Transfer Function Reports.
shows the parameter estimates and is similar to the ARIMA version. In addition, the Variable column shows the correspondence between series names and parameters. The table is followed by the formula of the model. Note the notation B is for the backshift operator.
provides a forecasting graph based on a specified confidence interval. The functionality changes based on the number entered in the Forecast Periods box.
If the number of Forecast Periods is less than or equal to the Input Lag, the forecasting box shows the forecast for the number of periods. A confidence interval around the prediction is shown in blue, and this confidence interval can be changed by entering a number in the Confidence Interval box above the graph.
If the number of forecast periods is larger than the number of lags (say, eight in our example), the presentation is a little different.
Here, you manipulate lagged values of the series by entering values into the edit boxes next to the series, or by manipulating the sliders. As before, the confidence interval can also be changed. The results of your changes are reflected in real time in the Interactive Forecasting graph.
creates a new data table containing the input and output series, a time column, predicted output with standard errors, residuals, and 95% confidence limits.
The model comparison table works like its ARIMA counterpart by accumulating statistics on the models you specify.
A regression model with serially correlated errors can be specified by including regressors in the model and not specifying any polynomial orders.