resubLoss

Regression error by resubstitution

Syntax

L = resubLoss(tree) L = resubLoss(tree,Name,Value) L = resubLoss(tree,'Subtrees',subtreevector) [L,se] = resubLoss(tree,'Subtrees',subtreevector) [L,se,NLeaf] = resubLoss(tree,'Subtrees',subtreevector) [L,se,NLeaf,bestlevel] = resubLoss(tree,'Subtrees',subtreevector) [L,...] = resubLoss(tree,'Subtrees',subtreevector,Name,Value)

Description

L = resubLoss(tree) returns the resubstitution loss, meaning the loss computed for the data that fitrtree used to create tree.

L = resubLoss(tree,Name,Value) returns the loss with additional options specified by one or more Name,Value pair arguments. You can specify several name-value pair arguments in any order as Name1,Value1,…,NameN,ValueN.

L = resubLoss(tree,'Subtrees',subtreevector) returns a vector of mean squared errors for the trees in the pruning sequence subtreevector.

[L,se] = resubLoss(tree,'Subtrees',subtreevector) returns the vector of standard errors of the classification errors.

[L,se,NLeaf] = resubLoss(tree,'Subtrees',subtreevector) returns the vector of numbers of leaf nodes in the trees of the pruning sequence.

[L,se,NLeaf,bestlevel] = resubLoss(tree,'Subtrees',subtreevector) returns the best pruning level as defined in the TreeSize name-value pair. By default, bestlevel is the pruning level that gives loss within one standard deviation of minimal loss.

[L,...] = resubLoss(tree,'Subtrees',subtreevector,Name,Value) returns loss statistics with additional options specified by one or more Name,Value pair arguments. You can specify several name-value pair arguments in any order as Name1,Value1,…,NameN,ValueN.

Input Arguments

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`tree` — Regression tree
`RegressionTree` model object

A regression tree (RegressionTree model object) constructed using 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.

`LossFun` — Loss function
`'mse'` (default) | function handle

Loss function, specified as a function handle or 'mse' meaning mean squared error.

You can write your own loss function in the syntax described in Loss Functions.

Data Types: char | string | function_handle

Name,Value arguments associated with pruning subtrees:

`Subtrees` — Pruning level
0 (default) | vector of nonnegative integers | `'all'`

Pruning level, specified as the comma-separated pair consisting of 'Subtrees' and 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 (i.e., just the root node).

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

resubLoss 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.

Example: 'Subtrees','all'

Data Types: single | double | char | string

`TreeSize` — Tree size
`'se'` (default) | `'min'`

Tree size, specified as one of the following:

'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).
'min' — loss returns the element of Subtrees with smallest loss, usually the smallest element of Subtrees.

Output Arguments

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`L` — Regression loss
numeric vector of positive values

Regression loss (mean squared error), a vector the length of Subtrees. The meaning of the error depends on the values in Weights and LossFun.

`se` — Standard error of loss
numeric vector of positive values

Standard error of loss, a vector the length of Subtrees.

`NLeaf` — Number of leaves
numeric vector of nonnegative integers

Number of leaves (terminal nodes) in the pruned subtrees, a vector the length of Subtrees.

`bestlevel` — Optimal pruning level
nonnegative numeric scalar

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.

Examples

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Compute the In-Sample MSE

Open Live Script

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

Examine the MSE for Each Subtree

Open Live Script

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

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 pair.

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).

resubLoss

Syntax

Description

Input Arguments

tree — Regression tree RegressionTree model object

Name-Value Arguments

LossFun — Loss function 'mse' (default) | function handle

Subtrees — Pruning level 0 (default) | vector of nonnegative integers | 'all'

TreeSize — Tree size 'se' (default) | 'min'

Output Arguments

L — Regression loss numeric vector of positive values

se — Standard error of loss numeric vector of positive values

NLeaf — Number of leaves numeric vector of nonnegative integers

bestlevel — Optimal pruning level nonnegative numeric scalar

Examples

Compute the In-Sample MSE

Examine the MSE for Each Subtree

More About

Loss Functions

Extended Capabilities

GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

See Also

WeChat

`tree` — Regression tree
`RegressionTree` model object

`LossFun` — Loss function
`'mse'` (default) | function handle

`Subtrees` — Pruning level
0 (default) | vector of nonnegative integers | `'all'`

`TreeSize` — Tree size
`'se'` (default) | `'min'`

`L` — Regression loss
numeric vector of positive values

`se` — Standard error of loss
numeric vector of positive values

`NLeaf` — Number of leaves
numeric vector of nonnegative integers

`bestlevel` — Optimal pruning level
nonnegative numeric scalar

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.