loss

Regression error

Syntax

L = loss(tree,Tbl,ResponseVarName)

L = loss(tree,Tbl,Y)

L = loss(tree,X,Y)

L = loss(___,Name,Value)

[L,se,NLeaf,bestlevel]
= loss(___)

Description

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

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

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 = loss(___,Name,Value) computes the error in prediction with additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes.

[L,se,NLeaf,bestlevel] = loss(___) also returns the standard error of the loss (se), the number of leaves (terminal nodes) in the tree (NLeaf), and the optimal pruning level for tree (bestlevel).

Input Arguments

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`tree` — Trained regression tree
`RegressionTree` object | `CompactRegressionTree` object

Trained regression tree, specified as a RegressionTree object constructed by fitrtree or a CompactRegressionTree object constructed by compact.

`Tbl` — sample data
table

Sample data, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Tbl must contain all of the predictors used to train tree. Optionally, Tbl can contain additional columns for the response variable and observation weights. 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 this method must also be in a table.

Data Types: table

`X` — Predictor values
numeric matrix

Predictor values, 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 elements in Y.

Data Types: single | double

`ResponseVarName` — Response variable name
name of a variable in `Tbl`

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.

If you specify ResponseVarName, then you must do so 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.ResponseVarName, as predictors.

Data Types: char | string

`Y` — Response data
numeric column vector

Response data, specified as 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.

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.

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

Loss function, specified as the comma-separated pair consisting of 'LossFun' and a function handle for loss, or '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 observed 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.

Example: 'LossFun','mse'

Data Types: function_handle | char | string

`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 loss operates on all subtrees (i.e., the entire pruning sequence). This specification is equivalent to using 0:max(tree.PruneList).

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.

Example: 'Subtrees','all'

Data Types: single | double | char | string

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

Tree size, specified as the comma-separated pair consisting of 'TreeSize' and one of the following:

'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.

Example: 'TreeSize','min'

`Weights` — Observation weights
`ones(size(X,1),1)` (default) | vector of scalar values | name of a variable in `Tbl`

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a vector of scalar values. The software weights the observations in each row of X or Tbl with the corresponding value in Weights. The size of Weights must equal the number of rows in X or Tbl.

If you specify the input data as a table Tbl, then Weights can be the name of a variable in Tbl that contains a numeric vector. In this case, you must specify Weights as a variable name. For example, if weights vector W is stored as Tbl.W, then specify Weights as 'W'. Otherwise, the software treats all columns of Tbl, including W, as predictors when training the model.

Data Types: single | double | char | string

Output Arguments

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`L` — Classification error
vector of scalar values

Classification error, returned as 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
vector of scalar values

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

`NLeaf` — Number of leaf nodes
vector of integer values

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

`bestlevel` — Best pruning level
scalar value

Best pruning level as defined in the TreeSize name-value pair, returned as 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.

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

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);

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 60 objects of type line, text. One or more of the lines displays its values using only markers

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');

Figure Regression tree viewer contains an axes object and other objects of type uimenu, uicontrol. The axes object contains 57 objects of type line, text. One or more of the lines displays its values using only markers

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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Mean Squared Error

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

$m = \frac{\sum w_{n} {(f (X_{n}) - Y_{n})}^{2}}{\sum w_{n}} .$

Extended Capabilities

Tall Arrays
Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

Only one output is supported.
You can use models trained on either in-memory or tall data with this function.

For more information, see Tall Arrays.

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

Usage notes and limitations:

The loss function does not support decision tree models trained with surrogate splits.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

loss

Syntax

Description

Input Arguments

tree — Trained regression tree RegressionTree object | CompactRegressionTree object

Tbl — sample data table

X — Predictor values numeric matrix

ResponseVarName — Response variable name name of a variable in Tbl

Y — Response data numeric column vector

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'

Weights — Observation weights ones(size(X,1),1) (default) | vector of scalar values | name of a variable in Tbl

Output Arguments

L — Classification error vector of scalar values

se — Standard error of loss vector of scalar values

NLeaf — Number of leaf nodes vector of integer values

bestlevel — Best pruning level scalar value

Examples

Compute the In-Sample MSE

Find the Pruning Level Yielding the Optimal In-sample Loss

Examine the MSE for Each Subtree

More About

Mean Squared Error

Extended Capabilities

Tall Arrays Calculate with arrays that have more rows than fit in memory.

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

See Also

WeChat

`tree` — Trained regression tree
`RegressionTree` object | `CompactRegressionTree` object

`Tbl` — sample data
table

`X` — Predictor values
numeric matrix

`ResponseVarName` — Response variable name
name of a variable in `Tbl`

`Y` — Response data
numeric column vector

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

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

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

`Weights` — Observation weights
`ones(size(X,1),1)` (default) | vector of scalar values | name of a variable in `Tbl`

`L` — Classification error
vector of scalar values

`se` — Standard error of loss
vector of scalar values

`NLeaf` — Number of leaf nodes
vector of integer values

`bestlevel` — Best pruning level
scalar value

Tall Arrays
Calculate with arrays that have more rows than fit in memory.

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