This report appears only when a form of cross validation is selected as a Validation Method in the Model Launch control panel. It shows summary statistics for models fit, using from 0 to the maximum number of extracted factors, as specified in the Model Launch control panel. The report also provides a plot of Root Mean PRESS values. See Root Mean PRESS Plot. An optimum number of factors is identified using the minimum Root Mean PRESS statistic.
Figure 5.9 Cross Validation Report
When the Standardize X option is selected, the standardization is applied once to the entire data table. It is not reapplied to the individual training sets. However, when any combination of the Centering or Scaling options are selected, this combination of selections is applied to each cross validation training set. Cross validation proceeds by using the training sets, which are individually centered and scaled if these options are selected. The square root of the average of the PRESS values across all responses. For details, see Root Mean PRESS.
Test statistic for the van der Voet test, which tests whether models with different numbers of extracted factors differ significantly from the optimum model. The null hypothesis for each van der Voet T2 test states that the model based on the corresponding number of factors does not differ from the optimum model. The alternative hypothesis is that the model does differ from the optimum model. For more details, see van der Voet T2.
Q2
Indicator of the predictive ability of models with the given number of factors or fewer. For a given number of factors, f, Cumulative Q2 is defined as follows:
R2X
Percent of X variation explained by the specified factor. A component with a large R2X explains a large amount of the variation in the X variables. See Calculation of R2X and R2Y When Validation Is Used.
Percent of X variation explained by the model with the given number of factors. This is the sum of the R2X values for i = 1 to the given number of factors.
R2Y
Percent of Y variation explained by the specified factor. A component with a large R2Y explains a large amount of the variation in the Y variables. See Calculation of R2X and R2Y When Validation Is Used.
Percent of Y variation explained by the model with the given number of factors. This is the Sum of the R2Y values for i = 1 to the given number of factors.
The statistics Q2 and Cumulative R2Y both measure the predictive ability of the model, but in different ways.
|
•
|
Cumulative R2Y increases as the number of factors increases. This is because, as factors are added to the model, more variation is explained.
|
|
•
|
Q2 tends to increase and then decrease, or at least discontinue increasing, as the number of factors increases. This is because, as more factors are added, the model becomes tuned to the training set and does not generalize well to new data, causing the PRESS statistic to decrease.
|
Analysis of Q2 and Cumulative R2Y provides an alternative to using the van der Voet test for determining how many factors to include in your model. Select a number of factors for which Q2 is large and has not started decreasing. You also want Cumulative R2Y to be large.
Figure 5.10 shows plots of Cumulative R2Y and Q2 against the number of factors for the Penta.jmp data table, using Leave-One-Out as the validation method. Cumulative R2Y increases and levels off for about four factors. The statistic Q2 is largest for two factors and then begins to level off. The plot suggests that a model with two factors will explain a large portion of the variation in Y without overfitting the data.
This bar chart shows the number of factors along the horizontal axis and the Root Mean PRESS values on the vertical axis. It is equivalent to the horizontal bar chart that appears to the right of the Root Mean PRESS column in the Cross Validation report. See Figure 5.9.
For a specified number of factors, a, Root Mean PRESS is calculated as follows:
|
1.
|
Fit a model with a factors to each training set.
|
|
3.
|
|
4.
|
Root Mean PRESS for a factors is the square root of the average of the PRESS values across all responses.
|
|
5.
|
The statistic Q2 is defined as 1 - PRESS / SSY. The PRESS statistic is the predicted error sum of squares averaged across all responses for the model developed based on the training data, but evaluated on the validation set. The value of SSY is the sum of squares for Y averaged across all responses and based on the observations in the validation set.
The statistic Q2 in the Cross Validation report is computed in the following ways, depending on the selected Validation Method:
Q2 is the average of the values 1 - PRESS / SSY computed for the validation sets based on the models constructed by leaving out one observation at a time.
Q2 is the average of the values 1 - PRESS / SSY computed for the validation sets based on the K models constructed by leaving out each of the K folds.
Q2 is the value of 1 - PRESS / SSY computed for the validation set based on the model constructed using the single set of training data.
The statistics R2X and R2Y in the Cross Validation report are computed in the following ways, depending on the selected Validation Method:
R2X is the average of the Percent Variation Explained for X Effects for the models constructed by leaving out one observation at a time.
R2X is the average of the Percent Variation Explained for X Effects for the K models constructed by leaving out each fold.
R2X is the Percent Variation Explained for X Effects for the model constructed using the training data.