MARSplines Example

At this stage, you can change the model factors, e.g. Maximum number of basis functions, Penalty, etc., located on the Startup Panel - Options tab. For example, you may want to adjust the maximum number of basis functions that can be added during model building, or you can decrease the penalty for adding basis functions. It is recommended that you set the maximum number of basis functions to as high a number as possible so that STATISTICA MARSplines can search for as many combinations as possible. To build even more complex models, you may want to increase the Degree of interactions among the input variables.

It is also recommended that you always prune your model by selecting the Apply pruning check box. This will reduce the chance of overfitting the data. Refer to the Introductory Overview for further details concerning the MARSplines method.

In the Summary box at the top of the Results dialog, you can see the specifications of the MARSplines model you just built, including the number of terms and basis functions present in the model, and the generalized cross validation error after training is complete (see also the Technical Notes for computational details). You will also see the choices you made in the Startup Panel displayed here for your reference including the dependent and independent variable list. When independent variables are highlighted in red, it means they have been selected in the formation of the basis functions for the final model. This is useful information since it identifies the important (relevant) predictor variables in the model. In this example, note that the predictor AGE is not identified as significant or important by the algorithm and, thus, does not participate in the formation of the basis functions.

The coefficients spreadsheet provides the full details of the MARSplines terms together with the corresponding model coefficients. It also indicates the type of each basis function and the order interactions in each term. In the spreadsheet above, the bias term (a constant) consists of the intercept coefficient. The first term consists of the basis function ( POP_CNG - 7.1). Note that this basis function is highlighted in red indicating it is of type (x-t)+. The third term contains the product of (TAX_RATE - 0.4) and (7.5 - PT_PHONE). The last term consists of products of three basis functions and therefore its degree of interaction is three. In summary, the MARSplines model is:

PT_POOR = 23.211 - 0.197*max(0,POP_CHNG - 7.1) - 0.435*max(0,PT_PHONE - 75) + 1.14*max(0,TAX_RATE -0.4)*( 0,75 - PT_PHONE) + 0.0003*max(0,N_EMPLD  - 1070)*max(0,75 - PT_PHONE)*max(0,PR_RURAL - 5.9)

You can place the equation above into a standard STATISTICA Report by clicking the Equation button on the MARSplines Results dialog - Quick tab.

For example, shown above is a scatterplot of the predicted and observed values plotted against the values in variable N_EMPLD. In general, this type of plot will give you an effective way of comparing model predictions with the observed data. To produce the graph shown above, select N_EMPLD from the X-axis list and Obsd. (Observed) and Pred. (Predicted) from the Y-axis list. Then click the Graphs of X and Y button.

The plot shown here was further customized using the graphics brushing tools; specifically, the point for the apparent outlier case (County) Shelby was labeled, to illustrate a major advantage of the MARSplines method over simple multiple linear regression or other parameterized regression models. Even though Shelby is a clear outlier, the overall fit of the model does not appear to be seriously degraded by this case. Unlike in multiple regression (see also the Multiple Regression Example 1: Standard Regression Analysis), where outliers can greatly affect the overall fit, outliers can easily be accommodated in MARSplines because of the "piecewise-regression-like" nature of the method (see also the Introductory Overview for details).