Decisions :: SAS(R) Enterprise Miner(TM) 12.1 Extension Nodes: Developer’s Guide

Each modeling node can make a decision for each case in a scoring data set, based on numerical consequences specified via a decision matrix and cost variables or cost constants. The decision matrix can specify profit, loss, or revenue. In the GUI, the decision matrix is provided via the Target Profile. With a previously scored data set containing posterior probabilities, decisions can also be made using PROC DECIDE, which reads the decision matrix from a decision data set.

When you use decision processing, the modeling nodes compute summary statistics giving the total and average profit or loss for each model. These profit and loss summary statistics are useful for selecting models. To use these summary statistics for model selection, you must specify numeric consequences for making each decision for each value of the target variable. It is your responsibility to provide reasonable numbers for the decision consequences based on your particular application.

In some applications, the numeric consequences of each decision might not all be known at the time you are training the model. Hence, you might want to perform what-if analyses to explore the effects of different decision consequences using the Model Comparison node. In particular, when one of the decisions is to "do nothing," the profit charts in the Model Comparison node provide a convenient way to see the effect of applying different thresholds for the do-nothing decision.

To use profit charts, the do-nothing decision should not be included in the decision matrix; the Model Comparison node will implicitly supply a do-nothing decision when computing the profit charts. When you omit the do-nothing decision from the profit matrix so you can obtain profit charts, you should not use the profit and loss summary statistics for comparing models, since these summary statistics will not incorporate the implicit do-nothing decision. This topic is discussed further in Decision Thresholds and Profit Charts.

The decision matrix contains columns (decision variables) corresponding to each decision, and rows (observations) corresponding to target values. The values of the decision variables represent target-specific consequences, which might be profit, loss, or revenue. These consequences are the same for all cases being scored. A decision data set might contain prior probabilities in addition to the decision matrix.

For a categorical target variable, there should be one row for each class. The value in the decision matrix located at a given row and column specifies the consequence of making the decision corresponding to the column when the target value corresponds to the row. The decision matrix is allowed to contain rows for classes that do not appear in the data being analyzed. For a profit or revenue matrix, the decision is chosen to maximize the expected profit. For a loss matrix, the decision is chosen to minimize the expected loss.

For an interval target variable, each row defines a knot in a piecewise linear spline function. The consequence of making a decision is computed by linear interpolation in the corresponding column of the decision matrix. If the predicted target value is outside the range of knots in the decision matrix, the consequence of a decision is computed by linear extrapolation. Decisions are made to maximize the predicted profit or minimize the predicted loss.

For each decision, there might also be either a cost variable or a numeric cost constant. The values of cost variables represent case-specific consequences, which are always treated as costs. These consequences do not depend on the target values of the cases being scored. Costs are used for computing return on investment as (revenue-cost)/cost.

Cost variables can be specified only if the decision matrix contains revenue, not profit or loss. Hence, if revenues and costs are specified, profits are computed as revenue minus cost. If revenues are specified without costs, the costs are assumed to be zero. The interpretation of consequences as profits, losses, revenues, and costs is needed only to compute return on investment. You can specify values in the decision matrix that are target-specific consequences but that might have some practical interpretation other than profit, loss, or revenue. Likewise, you can specify values for the cost variables that are case-specific consequences but that might have some practical interpretation other than costs. If the revenue/ cost interpretation is not applicable, the values computed for return on investment might not be meaningful. There are some restrictions on the use of cost variables in the Decision Tree node; see the documentation on the Decision Tree node for more information.

In principle, consequences need not be the sum of target-specific and case-specific terms, but SAS Enterprise Miner does not support such nonadditive consequences.

For a categorical target variable, you can use a decision matrix to classify cases by specifying the same number of decisions as classes and having each decision correspond to one class. However, there is no requirement for the number of decisions to equal the number of classes except for ordinal target variables in the Decision Tree node.

For example, suppose there are three classes designated red, blue, and green. For an identity decision matrix, the average profit is equal to the correct-classification rate:

Profit Matrix to Compute the Correct-Classification Rate
Target Value:	Decision:
	Red	Blue	Green
Red	1	0	0
Blue	0	1	0
Green	0	0	1

To obtain the misclassification rate, you can specify a loss matrix with zeros on the diagonal and ones everywhere else:

Loss Matrix to Compute the Misclassification Rate
Target Value:	Decision:
	Red	Blue	Green
Red	0	1	1
Blue	1	0	1
Green	1	1	0

If it is 20 times more important to classify red cases correctly than blue or green cases, you can specify a diagonal profit matrix with a profit of 20 for classifying red cases correctly and a profit of one for classifying blue or green cases correctly:

Profit Matrix for Detecting a Rare (Red) Class
Target Value:	Decision:
	Red	Blue	Green
Red	20	0	0
Blue	0	1	0
Green	0	0	1

When you use a diagonal profit matrix, the decisions depend only on the products of the prior probabilities and the corresponding profits, not on their separate values. Hence, for any given combination of priors and diagonal profit matrix, you can make any change to the priors (other than replacing a zero with a nonzero value) and find a corresponding change to the diagonal profit matrix that leaves the decisions unchanged, even though the expected profit for each case might change.

Similarly, for any given combination of priors and diagonal profit matrix, you can find a set of priors that will yield the same decisions when used with an identity profit matrix. Therefore, using a diagonal profit matrix does not provide you with any power in decision making that could not be achieved with no profit matrix by choosing appropriate priors (although the profit matrix might provide an advantage in interpretability). Furthermore, any two by two decision matrix can be transformed into a diagonal profit matrix as discussed in the following section on Decision Thresholds and Profit Charts.

When the decision matrix is three by three or larger, it might not be possible to diagonalize the profit matrix, and some nondiagonal profit matrices will produce effects that could not be achieved by manipulating the priors. To show the effect of a nondiagonalizable decision matrix, the data in the upper left plot of the following figure were generated to have three classes, shown as red circles, blue crosses, and green triangles.

Each class has 100 training cases with a bivariate normal distribution. The training data were used to fit a linear logistic regression model using the Neural Network engine. The posterior probabilities are shown in the upper right plot. Classification according to the posterior probabilities yields linear classification boundaries as shown in the lower left plot. Use of a nondiagonalizable decision matrix causes the decision boundaries in the lower right plot to be rotated in comparison with the classification boundaries, and the decision boundaries are curved rather than linear.

The decision matrix that produced the curved decision boundaries is shown in the following table:

Nondiagonal Profit Matrix
Target Value:	Decision:
	Red	Blue	Green
Red	4	0	3
Blue	3	4	0
Green	0	3	4

In each row, the two profit values for misclassification are different. Hence, it is impossible to diagonalize the matrix by adding a constant to each row. Consider the blue row. The greatest profit is for a correct assignment into blue, but there is also a smaller but still substantial profit for assignment into red. There is no profit for assigning red into blue, so the red-blue decision boundary is moved toward the blue mean in comparison with the classification boundary based on posterior probabilities. The following figure shows the effect of the same nondiagonal profit matrix on a quadratic logistic regression model.

For the Neural Network and Regression nodes, a separate decision is made for each case. For the Decision Tree node, a common decision is made for all cases in the same leaf of the tree, so when different cases have different costs, the average cost in the leaf is used in place of the individual costs for each case. That is, the profit equals the revenue minus the average cost among all training cases in the same leaf. Hence, a single decision is assigned to all cases in the same leaf of a tree.

The decision alternative assigned to a validation, test, or scoring case ignores any cost associated with the case. The new data are assumed similar to the training data in cost as well as predictive relations. However, the actual cost values for each case are used for the investment cost, ROI, and quantities that depend on the actual target value.

Decision and cost matrices do not affect:

Estimating parameters in the Regression node
Learning weights in the Neural Network node
Growing (as opposed to pruning) trees in the Decision Tree node unless the target is ordinal
Residuals, which are based on posteriors before adjustment for priors
Error functions such as deviance or likelihood
Fit statistics such as MSE based on residuals or error functions
Posterior probabilities
Classification
Misclassification rate.

Growing trees in the Decision Tree node when the target is ordinal
Decisions
Expected profit or loss
Profit and loss summary statistics, including the relative contribution of each class.

Decision and cost matrices will by default affect the following processes if and only if there are two or more decisions:

Choice of models in the Regression node
Early stopping in the Neural Network node
Pruning trees in the Decision Tree node.

Formulas will be presented first for the Neural Network and Regression nodes. Let:

t

be an index for target values (classes)

d

be an index for decisions

i

be an index for cases

n_d

be the number of decisions

Class(t)

be the set of indices of cases belonging to target t

Profit(t,d)

be the profit for making decision d when the target is t

Loss(t,d)

be the loss for making decision d when the target is t

Revenue(t,d)

be the revenue for making decision d when the target is t

Cost(i,d)

be the cost for making decision d for case i

Q(i,t,d)

be the combined consequences for making decision d when the target is t for case i

Prior(t)

be the prior probability for target t

Paw(t)

be the prior-adjustment weight for target t

Post(i,t)

be the posterior probability of target t for case i

F(i)

be the frequency for case i

T(i)

be the index of the actual target value for case i

A(i,d)

be the expected profit of decision d for case i

B(i)

be the best possible profit for case i based on the actual target value

C(i)

be the computed profit for case i based on the actual target value

D(i)

be the index of the decision chosen by the model for case i

E(i)

be the expected profit for case i of the decision chosen by the model

IC(i)

be the investment cost for case i for the decision chosen by the model

ROI(i)

be the return on investment for case i for the decision chosen by the model.

These quantities are related by the following formulas:

When the target variable is categorical, the expected profit for decision d in case i is:

For each case i, the decision is made by choosing D(i) to be the value of d that maximizes the expected profit:

If two or more decisions are tied for maximum expected profit, the first decision in the user-specified list of decisions is chosen.

The expected profit E(i) is the expected combined consequence for the chosen decision D(i), computed as a weighted average over the target values of the combined consequences, using the posterior probabilities as weights:

The expected loss is the negative of expected profit.

Note that E(i) and D(i) can be computed without knowing the target index T(i). When T(i) is known, two more quantities useful for evaluating the model can also be computed. C(i) is the profit computed from the target value using the decision chosen by the model:

The loss computed from the target value is the negative of C(i). C(i) is the most important variable for assessing and comparing models. The best possible profit for any of the decisions, which is an upper bound for C(i), is:

The best possible loss is the negative of B(i).

When revenue and cost are specified, investment cost is:

And return on investment is:

For an interval target variable, let:

Y(i)

be the actual target value for case i

P(i)

be the predicted target value for case i

K(t)

be the knot value for the row of the decision matrix.

For interval targets, the predicted value is assumed to be accurate enough that no integration over the predictive distribution is required. Define the functions:

Then the decision is made by maximizing the expected profit:

The expected profit for the chosen decision is:

When Y(i) is known, the profit computed from the target value using the decision chosen by the model is:

And the best possible profit for any of the decisions is:

For both categorical and interval targets, the summary statistics for decision processing with profit and revenue matrices are computed by summation over cases with nonmissing cost values. If no adjustment for prior probabilities is used, the sums are weighted only by the case frequencies. Hence, total profit and average profit are given by the following formulas:

For loss matrices, total loss and average loss are the negatives of total profit and average profit, respectively.

If total and average profit are adjusted for prior probabilities, an additional weight Paw(t)is used:

Total and average profit are then given by:

If any class with a positive prior probability has a total frequency of zero, total and average profit and loss cannot be computed and are assigned missing values. Note that the adjustment of total and average profit and loss is done only if you explicitly specify prior probabilities; the adjustment is not done when the implicit priors based on the training set proportions are used.

The adjustment for prior probabilities is not done for fit statistics such as SSE, deviance, likelihood, or misclassification rate. For example, consider the situation shown in the following table:

	Proportion In		Unconditional Misclassification Rate
Class	Operational Data	Training Data	Prior Probability	Conditional Misclassification Rate	Unadjusted	Adjusted
Rare	0.1	0.5	0.1	0.8	0.5 * 0.8 + 0.5 * 0.2 = 0.50	0.1 * 0.8 + 0.9 * 0.2 = 0.26
Common	0.9	0.5	0.9	0.2

There is a rare class comprising 10% of the operational data, and a common class comprising 90%. For reasons discussed in the section below on Detecting Rare Classes, you might want to train using a balanced sample with 50% from each class. To obtain correct posterior probabilities and decisions, you specify prior probabilities of .1 and .9 that are equal to the operational proportions of the two classes.

Suppose the conditional misclassification rate for the common class is low, just 20%, but the conditional misclassification rate for the rare class is high, 80%. If it is important to detect the rare class accurately, these misclassification rates are poor.

The unconditional misclassification rate computed using the training proportions without adjustment for priors is a mediocre 50%. But adjusting for priors, the unconditional misclassification rate is apparently much better at only 26%. Hence, the adjusted misclassification rate is misleading.

For the Decision Tree node, the following modifications to the formulas are required. Let Leaf(i) be the set of indices of cases in the same leaf as case i. Then:

The combined consequences are:

For a categorical target, the decision is:

And the expected profit is:

For an interval target:

The decision is:

And the expected profit is:

The other formulas are unchanged.