Specialized Models • Gaussian Process • The Gaussian Process Report
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The Gaussian Process Report
After clicking OK from the launch dialog, the following report appears.
Gaussian Process Default Report
Actual by Predicted Plot
The Actual by Predicted plot shows the actual Y values on the y-axis and the jackknife predicted values on the x-axis. One measure of goodness-of-fit is how well the points lie along the 45 degree diagonal line.
The jackknife values are really pseudo-jackknife values because they are not refit unless the row is excluded. Therefore, the correlation parameters still have the contribution of that row in them, but the prediction formula does not. If the row is excluded, neither the correlation parameters nor the prediction formula have the contribution.
Model Report
The Model Report shows a functional ANOVA table for the model parameters that the platform estimates. Specifically, it is an analysis of variance table, but the variation is computed using a function-driven method.
The Total Variation is the integrated variability over the entire experimental space.
For each covariate, we can create a marginal prediction formula by averaging the overall prediction formula over the values of all the other factors. The functional main effect of X1 is the integrated total variation due to X1 alone. In this case, we see that 37.6% of the variation in Y is due to X1.
The ratio of (Functional X1 effect)/(Total Variation) is the value listed as the Main Effect in the Model report. A similar ratio exists for each factor in the model.
Functional interaction effects, computed in a similar way, are also listed in the Model Report table.
Summing the value for main effect and all interaction terms gives the Total Sensitivity, the amount of influence a factor and all its two-way interactions have on the response variable.
Mu, Theta, and Sigma
The Gaussian correlation structure uses the product exponential correlation function with a power of 2 as the estimated model. This comes with the assumptions that Y is Normally distributed with mean μ and covariance matrix σ2R. The R matrix is composed of elements
In the Model report, μ is the Normal distribution mean, σ2 is the Normal Distribution parameter, and the Theta column corresponds to the values of θk in the definition of R.
These parameters are all fitted via maximum likelihood.
Note: If you see Nugget parameters set to avoid singular variance matrix, JMP has added a ridge parameter to the variance matrix so that it is invertible.
The Cubic correlation structure also assumes that Y is Normally distributed with mean μ and covariance matrix σ2R. The R matrix is composed of elements
     d = xik–xjk
where
For more information see Santer (2003). The theta parameter used in the cubic correlation is the reciprocal of the parameter used in the literature. The reason is so that when a parameter (theta) has no effect on the model, then it has a value of zero, instead of infinity.
Marginal Model Plots
The Marginal Model plots are shown in Marginal Model Plots.
Marginal Model Plots
These plots show the average value of each factor across all other factors. In this two-dimensional example, we examine slices of X1 from –1 to 1, and plot the average value at each point.