The HPFOREST Procedure

The formulas for adjusting the probabilities and statistics assume that the distribution of the inputs for a target class is the same in the training data as in the population of interest. The class proportions can differ, but the distribution of input values within a class must be the same.

The formula for converting probabilities from the training sample to the population of interest is

where

The circumflex above a population statistic such as indicates that the statistic is estimated from training data. A population statistic without a circumflex indicates that the statistic is computed from population data.

Let S denote a statistic that can be expressed as the average of a loss metric, Loss(y, p(y)), between the actual target value and the predicted probabilities. Average square error and misclassification rate are examples.

The probability p(y) might be adjusted or unadjusted, and the average might be taken over the training data or the population of interest. An optional argument to S indicates whether the probabilities are adjusted, and an optional suffix indicates the sample for evaluating the statistic. Thus,

To compute the estimate of a population statistic , multiply the terms for target class j by the proportion of class j observations in the population as

where

The class j formulas for the average square error and misclassification rate are

where