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resubLoss

Resubstitution loss for regression tree model

Description

L = resubLoss(tree) returns the resubstitution loss computed for the data used by fitrtree to create tree. By default, resubLoss uses the mean squared error to compute L.

example

L = resubLoss(tree,Name=Value) specifies additional options using one or more name-value arguments. For example, you can specify the loss function, the pruning level, and the tree size that resubLoss uses to calculate the loss.

example

[L,SE,Nleaf,BestLevel] = resubLoss(___) also returns the standard error of the loss, the number of leaf nodes in the trees of the pruning sequence, and the best pruning level as defined in the TreeSize name-value argument. By default, BestLevel is the pruning level that gives the loss within one standard deviation of the minimal loss.

Examples

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

load carsmall
X = [Displacement Horsepower Weight];

Grow a regression tree using all observations.

Mdl = fitrtree(X,MPG);

Compute the resubstitution MSE.

resubLoss(Mdl)
ans = 
4.8952

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.

load carsmall
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");

Figure Regression tree viewer contains an axes object and other objects of type uimenu, uicontrol. The axes object contains 27 objects of type line, text. One or more of the lines displays its values using only markers

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 = 7×1

    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 = 7×1

   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")

Figure Regression tree viewer contains an axes object and other objects of type uimenu, uicontrol. The axes object contains 15 objects of type line, text. One or more of the lines displays its values using only markers

Input Arguments

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Regression tree model, specified as a RegressionTree model object trained with fitrtree.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: L = resubloss(tree,Subtrees="all") prunes all subtrees.

Loss function, specified as "mse" (mean squared error) or as a function handle. If you pass a function handle fun, resubLoss calls it as

fun(Y,Yfit,W)

where Y, Yfit, and W are numeric vectors of the same length.

  • Y is the observed response.

  • Yfit is the predicted response.

  • W is the observation weights.

The returned value of fun(Y,Yfit,W) must be a scalar.

Example: LossFun="mse"

Example: LossFun=@Lossfun

Data Types: char | string | function_handle

Pruning level, specified as 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 completely pruned tree (that is, just the root node).

If you specify "all", then resubLoss operates on all subtrees, meaning the entire pruning sequence. This specification is equivalent to using 0:max(tree.PruneList).

resubLoss prunes tree to each level specified by Subtrees, and then estimates the corresponding output arguments. The size of Subtrees determines the size of some output arguments.

For the function to invoke Subtrees, the properties PruneList and PruneAlpha of tree must be nonempty. In other words, grow tree by setting Prune="on" when you use fitrtree, or by pruning tree using prune.

Example: Subtrees="all"

Data Types: single | double | char | string

Tree size, specified as one of these values:

  • "se"resubLoss returns the best pruning level (BestLevel), which corresponds to the highest pruning level with the loss within one standard deviation of the minimum (L+se, where L and se relate to the smallest value in Subtrees).

  • "min"resubLoss returns the best pruning level, which corresponds to the element of Subtrees with the smallest loss. This element is usually the smallest element of Subtrees.

Example: TreeSize="min"

Data Types: char | string

Output Arguments

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Resubstitution loss, returned as a numeric vector of positive values that has the same length as Subtrees. The loss for each tree is the mean squared error. If you specify LossFun, then L reflects the loss calculated with LossFun.

Standard error of loss, returned as a numeric vector of positive values that has the same length as Subtrees.

Number of leaf nodes in the pruned subtrees, returned as a numeric vector of nonnegative integers that has the same length as Subtrees. Leaf nodes are terminal nodes, which give responses, not splits.

Best pruning level, returned as a numeric scalar whose value depends on TreeSize:

  • When TreeSize is "se", the loss function returns the highest pruning level whose loss is within one standard deviation of the minimum (L+se, where L and se relate to the smallest value in Subtrees).

  • When TreeSize is "min", the loss function returns the element of Subtrees with the smallest loss, usually the smallest element of Subtrees.

More About

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Loss Functions

The built-in loss function is "mse", meaning mean squared error.

To write your own loss function, create a function file of the form

function loss = lossfun(Y,Yfit,W)
  • N is the number of rows of tree.X.

  • Y is an N-element vector representing the observed response.

  • Yfit is an N-element vector representing the predicted responses.

  • W is an N-element vector representing the observation weights.

  • The output loss should be a scalar.

Pass the function handle @lossfun as the value of the LossFun name-value argument.

Extended Capabilities

Version History

Introduced in R2011a