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# 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 with integer values from 0 (full unpruned tree) to the maximal pruning level max(tree.PruneList). You can set Subtrees to 'all', meaning the entire pruning sequence. 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

Find the loss of a regression tree predictor of the carsmall data to find MPG as a function of engine displacement, horsepower, and vehicle weight:

```load carsmall
X = [Displacement Horsepower Weight];
tree = fitrtree(X,MPG);
L = loss(tree,X,MPG)

L =
4.8952```

Find the pruning level that gives the optimal level of loss for the carsmall data:

```load carsmall
X = [Displacement Horsepower Weight];
tree = fitrtree(X,MPG);
[L,se,NLeaf,bestlevel] = loss(tree,X,MPG,'Subtrees','all');
bestlevel

bestlevel =
4```