Mean predictive measure of association for surrogate splits in regression tree
Estimate Predictive Measures of Association for Surrogate Splits
carsmall data set. Specify
Weight as predictor variables.
X = [Displacement Horsepower Weight];
Grow a regression tree using
MPG as the response. Specify to use surrogate splits for missing values.
tree = fitrtree(X,MPG,surrogate="on");
Find the mean predictive measure of association between the predictor variables.
ma = surrogateAssociation(tree)
ma = 3×3
1.0000 0.2167 0.5083
0.4521 1.0000 0.3769
0.2540 0.2659 1.0000
Find the mean predictive measure of association averaged over the odd-numbered nodes in
N = 1:2:tree.NumNodes; ma = surrogateAssociation(tree,N)
ma = 3×3
1.0000 0.1250 0.6875
0.5632 1.0000 0.5861
0.3333 0.3148 1.0000
ma — Predictive measures of association
Predictive measures of association, returned as a numeric matrix of size
P matrix, where
P is the number of predictors in
tree. The element
ma(i,j) is the
predictive measure of
association between the optimal split on variable
i and a surrogate split on variable
j. For more details, see Algorithms.
When you specify node numbers, the predictive measure of association
between variables is averaged over nodes in the vector
Predictive Measure of Association
The predictive measure of association is a value that indicates the similarity between decision rules that split observations. Among all possible decision splits that are compared to the optimal split (found by growing the tree), the best surrogate decision split yields the maximum predictive measure of association. The second-best surrogate split has the second-largest predictive measure of association.
Suppose xj and xk are predictor variables j and k, respectively, and j ≠ k. At node t, the predictive measure of association between the optimal split xj < u and a surrogate split xk < v is
PL is the proportion of observations in node t, such that xj < u. The subscript L stands for the left child of node t.
PR is the proportion of observations in node t, such that xj ≥ u. The subscript R stands for the right child of node t.
is the proportion of observations at node t, such that xj < u and xk < v.
is the proportion of observations at node t, such that xj ≥ u and xk ≥ v.
Observations with missing values for xj or xk do not contribute to the proportion calculations.
λjk is a value in (–∞,1]. If λjk > 0, then xk < v is a worthwhile surrogate split for xj < u.
Surrogate Decision Splits
A surrogate decision split is an alternative to the optimal decision split at a given node in a decision tree. The optimal split is found by growing the tree; the surrogate split uses a similar or correlated predictor variable and split criterion.
When the value of the optimal split predictor for an observation is missing, the observation is sent to the left or right child node using the best surrogate predictor. When the value of the best surrogate split predictor for the observation is also missing, the observation is sent to the left or right child node using the second-best surrogate predictor, and so on. Candidate splits are sorted in descending order by their predictive measure of association.
ma(i,j) is the predictive measure
of association averaged over surrogate splits on predictor
i is the optimal split predictor.
This average is computed by summing positive values of the predictive
measure of association over optimal splits on predictor
surrogate splits on predictor
j and dividing by
the total number of optimal splits on predictor
including splits for which the predictive measure of association between
j is negative.
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Introduced in R2011a