# loss

Regression error for regression tree model

## Description

example

L = loss(tree,Tbl,ResponseVarName) returns the mean squared error (MSE) L for the trained regression tree model tree using the predictor data in table Tbl and the true responses in Tbl.ResponseVarName. The interpretation of L depends on the loss function (LossFun) and weighting scheme (Weights).

L = loss(tree,Tbl,Y) uses the predictor data in table Tbl and the true responses in Y.

L = loss(tree,X,Y) uses the predictor data in matrix X and the true responses in Y.

L = loss(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, you can specify the loss function and observation weights.

example

[L,SE,Nleaf,BestLevel] = loss(___) also returns the standard error of the loss, number of leaf nodes in the trees of the pruning sequence, and best pruning level as defined in the TreeSize name-value argument.

## Examples

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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

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");

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 = 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")

## Input Arguments

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Regression tree model, specified as a RegressionTree model object trained with fitrtree, or a CompactRegressionTree model object created with compact.

Sample data, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Optionally, Tbl can contain additional columns for the response variable and observation weights. Tbl must contain all the predictors used to train tree. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If Tbl contains the response variable used to train tree, then you do not need to specify ResponseVarName or Y.

If you trained tree using sample data contained in a table, then the input data for loss must also be in a table.

Data Types: table

Response variable name, specified as the name of a variable in Tbl. If Tbl contains the response variable used to train tree, then you do not need to specify ResponseVarName.

You must specify ResponseVarName as a character vector or string scalar. For example, if the response variable is stored as Tbl.Response, then specify it as "Response". Otherwise, the software treats all columns of Tbl, including Tbl.Response, as predictors.

Data Types: char | string

Response data, specified as a numeric column vector with the same number of rows as Tbl or X. Each entry in Y is the response to the data in the corresponding row of Tbl or X.

Data Types: single | double

Predictor data, specified as a numeric matrix. Each column of X represents one variable, and each row represents one observation.

X must have the same number of columns as the data used to train tree. X must have the same number of rows as the number of rows in Y.

Data Types: single | double

### 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 = loss(tree,X,Y,Subtrees="all") specifies to prune all subtrees.

Loss function, specified as "mse" (mean squared error) or as a function handle. If you pass a function handle fun, loss 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 loss operates on all subtrees, meaning the entire pruning sequence. This specification is equivalent to using 0:max(tree.PruneList).

loss 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 the following:

• "se" — The loss function returns BestLevel corresponding to the smallest tree whose MSE is within one standard error of the minimum MSE.

• "min" — The loss function returns BestLevel corresponding to the minimal MSE tree.

Example: TreeSize="min"

Data Types: char | string

Observation weights, specified as a numeric vector of positive values or the name of a variable in Tbl.

If you specify Weights as a numeric vector, then the size of Weights must be equal to the number of rows in X or Tbl.

If you specify Weights as the name of a variable in Tbl, you must do so as a character vector or string scalar. For example, if the weights are stored as Tbl.W, then specify Weights as "W". Otherwise, the software treats all columns of Tbl, including Tbl.W, as predictors.

Example: Weights="W"

Data Types: single | double | char | string

## Output Arguments

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

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

Number of leaf nodes in the pruned subtrees, returned as a numeric vector 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.

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### Mean Squared Error

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

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

## Version History

Introduced in R2011a