# Methods and formulas for partial dependence plots in Fit Model and Discover Key Predictors with TreeNet® Classification

###### Note

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## One predictor partial dependence plots

Assume there are m predictors in a training data set, denoted as x1, x2, ..., xm. First, sort the distinct values of predictor x1 in the training data set in increasing order. Denote x11 as the first distinct value of x1. Then, x11 is the x-coordinate for the leftmost point on the plot.

The y-coordinate at x1 = x11 equals
TermDescription
Nthe total number of rows in the training data set the observed values for in the training data set
jeach individual row of the J rows the fitted value from the model when x1 = x11, x2 = x2j,...., xm = xmj

Replacing x11 by each of the distinct values of x1, we get the y-coordinates for the rest of the points on the plot. The calculations for the rest of the predictors are done similarly.

Calculations of all the y-coordinates for all distinct values of x can be time consuming with large data sets. For TreeNet®, there is a faster way to do the calculations. Refer to Friedman, J. H. (2001). Greedy function approximation: A gradient boosting machine. The Annals of Statistics, 29(5), page 1221.

The calculations for multinomial response case are similar. Here the fitted value is from the model for each individual class.

## Two predictor partial dependence plots

Assume there are m predictors in a training data set, denoted as x1, x2, ..., xm. First, sort the distinct values of predictors x1, x2 in the training data set in increasing order. Denote x11, x21 as one of the distinct pairs. Then, each pair makes the x and y-coordinates for a point on the surface plot.

The z-coordinate at x1 = x11, x2 = x21 equals
TermDescription
Nthe total number of rows in the training data set that all share the commonality of x1 = x11, x2 = x21 the observed values for in the training data set
jeach individual row of the J rows the fitted value from the model when x1 = x11, x2 = x21, x3 = x3j...., xm = xmj

The completion of the calculations for all distinct value combinations of x1 and x2 produces all the z-coordinates for the contour or surface plot. For large data sets, the calculations for all distinct pairs of x and y are time consuming. For TreeNet® models, there is a faster way to do the calculations. Refer to Friedman, J. H. (2001). Greedy function approximation: A gradient boosting machine. The Annals of Statistics, 29(5), page 1221.

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