loss

Class: CompactRegressionTree

Regression error

Syntax

L = loss(tree,X,Y)
[L,se] = loss(tree,X,Y)
[L,se,NLeaf] = loss(tree,X,Y)
[L,se,NLeaf,bestlevel] = loss(tree,X,Y)
L = loss(tree,X,Y,Name,Value)

Description

L = loss(tree,X,Y) returns the mean squared error between the predictions of tree to the data in X, compared to the true responses Y.

[L,se] = loss(tree,X,Y) returns the standard error of the loss.

[L,se,NLeaf] = loss(tree,X,Y) returns the number of leaves (terminal nodes) in the tree.

[L,se,NLeaf,bestlevel] = loss(tree,X,Y) returns the optimal pruning level for tree.

L = loss(tree,X,Y,Name,Value) computes the error in prediction with additional options specified by one or more Name,Value pair arguments.

Input Arguments

 tree Regression tree created with fitrtree, or the compact method. X A matrix of predictor values. Each column of X represents one variable, and each row represents one observation. Y A numeric column vector with the same number of rows as X. Each entry in Y is the response to the data in the corresponding row of X.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

 'LossFun' Function handle for loss, or the string 'mse' representing mean-squared error. If you pass a function handle fun, loss calls fun as: fun(Y,Yfit,W) Y is the vector of true responses.Yfit is the vector of predicted responses.W is the observation weights. If you pass W, the elements are normalized to sum to 1. All the vectors have the same number of rows as Y. Default: 'mse' 'Subtrees' A vector of nonnegative integers in ascending order or 'all'. If you specify a vector, then all elements must be at least 0 and at most max(tree.PruneList). 0 indicates the full, unpruned tree and max(tree.PruneList) indicates the a completely pruned tree (i.e., just the root node). If you specify 'all', then CompactRegressionTree.loss operates on all subtrees (i.e., the entire pruning sequence). This specification is equivalent to using 0:max(tree.PruneList). CompactRegressionTree.loss prunes tree to each level indicated in Subtrees, and then estimates the corresponding output arguments. The size of Subtrees determines the size of some output arguments. To invoke Subtrees, the properties PruneList and PruneAlpha of tree must be nonempty. In other words, grow tree by setting 'Prune','on', or by pruning tree using prune. Default: 0 'TreeSize' A string, either: 'se' — loss returns bestlevel that corresponds to the smallest tree whose mean squared error (MSE) is within one standard error of the minimum MSE.'min' — loss returns bestlevel that corresponds to the minimal MSE tree. 'Weights' Numeric vector of observation weights with the same number of elements as Y. Default: ones(size(Y))

Output Arguments

 L Classification error, a vector the length of Subtrees. The error for each tree is the mean squared error, weighted with Weights. If you include LossFun, L reflects the loss calculated with LossFun. se Standard error of loss, a vector the length of Subtrees. NLeaf Number of leaves (terminal nodes) in the pruned subtrees, a vector the length of Subtrees. bestlevel A scalar whose value depends on TreeSize: TreeSize = 'se' — loss returns the highest pruning level with loss within one standard deviation of the minimum (L+se, where L and se relate to the smallest value in Subtrees).TreeSize = 'min' — loss returns the element of Subtrees with smallest loss, usually the smallest element of Subtrees.

Definitions

Mean Squared Error

The mean squared error m of the predictions f(Xn) with weight vector w is

$m=\frac{\sum {w}_{n}{\left(f\left({X}_{n}\right)-{Y}_{n}\right)}^{2}}{\sum {w}_{n}}.$

Examples

collapse all

Compute the In-Sample MSE

Load the carsmall data set. Consider Displacement, Horsepower, and Weight as predictors of the response MPG.

X = [Displacement Horsepower Weight];

Grow a regression tree using all observations.

tree = fitrtree(X,MPG);

Estimate the in-sample MSE.

L = loss(tree,X,MPG)
L =

4.8952

Find the Pruning Level Yielding the Optimal In-sample Loss

Load the carsmall data set. Consider Displacement, Horsepower, and Weight as predictors of the response MPG.

X = [Displacement Horsepower Weight];

Grow a regression tree using all observations.

Mdl = fitrtree(X,MPG);

View the regression tree.

view(Mdl,'Mode','graph');

Find the best pruning level that yields the optimal in-sample loss.

[L,se,NLeaf,bestLevel] = loss(Mdl,X,MPG,'Subtrees','all');
bestLevel
bestLevel =

1

The best pruning level is level 1.

Prune the tree to level 1.

pruneMdl = prune(Mdl,'Level',bestLevel);
view(pruneMdl,'Mode','graph');

Examine the MSE for Each Subtree

Unpruned decision trees tend to overfit. One way to balance model complexity and out-of-sample performance is to prune a tree (or restrict its growth) so that in-sample and out-of-sample performance are satisfactory.

Load the carsmall data set. Consider Displacement, Horsepower, and Weight as predictors of the response MPG.

X = [Displacement Horsepower Weight];
Y = MPG;

Partition the data into training (50%) and validation (50%) sets.

n = size(X,1);
rng(1) % For reproducibility
idxTrn = false(n,1);
idxTrn(randsample(n,round(0.5*n))) = true; % Training set logical indices
idxVal = idxTrn == false;                  % Validation set logical indices

Grow a regression tree using the training set.

Mdl = fitrtree(X(idxTrn,:),Y(idxTrn));

View the regression tree.

view(Mdl,'Mode','graph');

The regression tree has seven pruning levels. Level 0 is the full, unpruned tree (as displayed). Level 7 is just the root node (i.e., no splits).

Examine the training sample MSE for each subtree (or pruning level) excluding the highest level.

m = max(Mdl.PruneList) - 1;
trnLoss = resubLoss(Mdl,'SubTrees',0:m)
trnLoss =

5.9789
6.2768
6.8316
7.5209
8.3951
10.7452
14.8445

• The MSE for the full, unpruned tree is about 6 units.

• The MSE for the tree pruned to level 1 is about 6.3 units.

• The MSE for the tree pruned to level 6 (i.e., a stump) is about 14.8 units.

Examine the validation sample MSE at each level excluding the highest level.

valLoss = loss(Mdl,X(idxVal,:),Y(idxVal),'SubTrees',0:m)
valLoss =

32.1205
31.5035
32.0541
30.8183
26.3535
30.0137
38.4695

• The MSE for the full, unpruned tree (level 0) is about 32.1 units.

• The MSE for the tree pruned to level 4 is about 26.4 units.

• The MSE for the tree pruned to level 5 is about 30.0 units.

• The MSE for the tree pruned to level 6 (i.e., a stump) is about 38.5 units.

To balance model complexity and out-of-sample performance, consider pruning Mdl to level 4.

pruneMdl = prune(Mdl,'Level',4);
view(pruneMdl,'Mode','graph')