surrogateAssociation

Mean predictive measure of association for surrogate splits in regression tree

Syntax

ma = surrogateAssociation(tree) ma = surrogateAssociation(tree,N)

Description

ma = surrogateAssociation(tree) returns a matrix of predictive measures of association for the predictors in tree.

ma = surrogateAssociation(tree,N) returns a matrix of predictive measures of association averaged over the nodes in vector N.

Input Arguments

`tree`	A regression tree constructed with `fitrtree`, or a compact regression tree constructed with `compact`.
`N`	Vector of node numbers in `tree`.

Output Arguments

ma

ma = surrogateAssociation(tree) returns a P-by-P matrix, where P is the number of predictors in tree. 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.
ma = surrogateAssociation(tree,N) returns a P-by-P representing the predictive measure of association between variables averaged over nodes in the vector N. N contains node numbers from 1 to max(tree.NumNodes).

Examples

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Estimate Predictive Measures of Association for Surrogate Splits

Open Live Script

Load the carsmall data set. Specify Displacement, Horsepower, and Weight as predictor variables.

load carsmall
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 tree.

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

More About

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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 x_j and x_k are predictor variables j and k, respectively, and j ≠ k. At node t, the predictive measure of association between the optimal split x_j < u and a surrogate split x_k < v is

$λ_{j k} = \frac{min (P_{L}, P_{R}) - (1 - P_{L_{j} L_{k}} - P_{R_{j} R_{k}})}{min (P_{L}, P_{R})} .$

P_L is the proportion of observations in node t, such that x_j < u. The subscript L stands for the left child of node t.
P_R is the proportion of observations in node t, such that x_j ≥ u. The subscript R stands for the right child of node t.
$P_{L_{j} L_{k}}$ is the proportion of observations at node t, such that x_j < u and x_k < v.
$P_{R_{j} R_{k}}$ is the proportion of observations at node t, such that x_j ≥ u and x_k ≥ v.
Observations with missing values for x_j or x_k do not contribute to the proportion calculations.

λ_jk is a value in (–∞,1]. If λ_jk > 0, then x_k < v is a worthwhile surrogate split for x_j < 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.

Algorithms

Element ma(i,j) is the predictive measure of association averaged over surrogate splits on predictor j for which 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 i and surrogate splits on predictor j and dividing by the total number of optimal splits on predictor i, including splits for which the predictive measure of association between predictors i and j is negative.

Extended Capabilities

GPU Arrays
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).