loss

Classification loss for classification tree model

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 classification loss L for the trained classification tree model tree using the predictor data in table Tbl and the true class labels in Tbl.ResponseVarName. The interpretation of L depends on the loss function (LossFun) and weighting scheme (Weights). In general, better classifiers yield smaller classification loss values.

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

L = loss(tree,X,Y) uses the predictor data in matrix X and the true class labels 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 or observation weights.

[L,SE,Nleaf,BestLevel] = loss(___) also returns the standard errors of the classification 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

Compute the In-Sample Classification Error

Compute the resubstituted classification error for the ionosphere data set.

load ionosphere
tree = fitctree(X,Y);
L = loss(tree,X,Y)

L = 
0.0114

Examine the Classification Error for Each Subtree

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 Fisher's iris data set. Partition the data into training (50%) and validation (50%) sets.

load fisheriris
n = size(meas,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 classification tree using the training set.

Mdl = fitctree(meas(idxTrn,:),species(idxTrn));

View the classification tree.

view(Mdl,'Mode','graph');

The classification tree has four pruning levels. Level 0 is the full, unpruned tree (as displayed). Level 3 is just the root node (i.e., no splits).

Examine the training sample classification error for each subtree (or pruning level) excluding the highest level.

m = max(Mdl.PruneList) - 1;
trnLoss = resubLoss(Mdl,'Subtrees',0:m)

trnLoss = 3×1

    0.0267
    0.0533
    0.3067

• The full, unpruned tree misclassifies about 2.7% of the training observations.

• The tree pruned to level 1 misclassifies about 5.3% of the training observations.

• The tree pruned to level 2 (i.e., a stump) misclassifies about 30.6% of the training observations.

Examine the validation sample classification error at each level excluding the highest level.

valLoss = loss(Mdl,meas(idxVal,:),species(idxVal),'Subtrees',0:m)

valLoss = 3×1

    0.0369
    0.0237
    0.3067

• The full, unpruned tree misclassifies about 3.7% of the validation observations.

• The tree pruned to level 1 misclassifies about 2.4% of the validation observations.

• The tree pruned to level 2 (i.e., a stump) misclassifies about 30.7% of the validation observations.

To balance model complexity and out-of-sample performance, consider pruning Mdl to level 1.

pruneMdl = prune(Mdl,'Level',1);
view(pruneMdl,'Mode','graph')

Input Arguments

tree — Trained classification tree

ClassificationTree model object | CompactClassificationTree model object

Trained classification tree, specified as a ClassificationTree model object trained with fitctree, or a CompactClassificationTree model object created with 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. 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 train tree using sample data contained in a table, then the input data for loss must also be in a table.

Data Types: table

ResponseVarName — Response variable name

name of 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.

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.

The response variable must be a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

Data Types: char | string

Y — Class labels

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class labels, specified as a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. Y must be of the same type as the class labels used to train tree, and its number of elements must equal the number of rows of X.

Data Types: categorical | char | string | logical | single | double | cell

X — Predictor data

numeric matrix

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,LossFun="exponential") specifies to use an exponential loss function.

LossFun — Loss function

"mincost" (default) | "binodeviance" | "classifcost" | "classiferror" | "exponential" | "hinge" | "logit" | "quadratic" | function handle

Loss function, specified as a built-in loss function name or a function handle.

The following table describes the values for the built-in loss functions.

Value	Description
"binodeviance"	Binomial deviance
"classifcost"	Observed misclassification cost
"classiferror"	Misclassified rate in decimal
"exponential"	Exponential loss
"hinge"	Hinge loss
"logit"	Logistic loss
"mincost"	Minimal expected misclassification cost (for classification scores that are posterior probabilities)
"quadratic"	Quadratic loss

"mincost" is appropriate for classification scores that are posterior probabilities. Classification trees return posterior probabilities as classification scores by default (see predict).

Specify your own function using function handle notation. Suppose that n is the number of observations in X, and K is the number of distinct classes (numel(tree.ClassNames)). Your function must have the signature

lossvalue = lossfun(C,S,W,Cost)

where:

• The output argument lossvalue is a scalar.

• You specify the function name (lossfun).

• C is an n-by-K logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order in tree.ClassNames.

  Create C by setting C(p,q) = 1, if observation p is in class q, for each row. Set all other elements of row p to 0.

• S is an n-by-K numeric matrix of classification scores. The column order corresponds to the class order in tree.ClassNames. S is a matrix of classification scores, similar to the output of predict.

• W is an n-by-1 numeric vector of observation weights. If you pass W, the software normalizes the weights to sum to 1.

• Cost is a K-by-K numeric matrix of misclassification costs. For example, Cost = ones(K) - eye(K) specifies a cost of 0 for correct classification and 1 for misclassification.

For more details on the loss functions, see Classification Loss.

Example: LossFun="binodeviance"

Example: LossFun=@lossfun

Data Types: char | string | function_handle

weights — Observation weights

ones(size(X,1),1) (default) | name of a variable in Tbl | numeric vector

Observation weights, specified as a numeric vector 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, then the name must be a character vector or string scalar. For example, if the weights are stored as Tbl.W, then specify it as "W". Otherwise, the software treats all columns of Tbl, including Tbl.W, as predictors.

loss normalizes the weights so that the observation weights in each class sum to the prior probability of that class. When you specify Weights, loss computes the weighted classification loss.

Example: Weights="W"

Data Types: single | double | char | string

Subtrees — Pruning level

0 (default) | vector of nonnegative integers | "all"

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 (in other words, 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 fitctree, 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 these values:

• "se" — loss 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" — loss 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

L — Classification loss

numeric vector

Classification loss, returned as a numeric vector that has the same length as Subtrees. The meaning of the error depends on the values in Weights and LossFun.

SE — Standard error of loss

numeric vector

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

Nleaf — Number of leaf nodes

vector of integer values

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

BestLevel — Best pruning level

numeric scalar

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

Classification Loss

Classification loss functions measure the predictive inaccuracy of classification models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

Consider the following scenario.

• L is the weighted average classification loss.

• n is the sample size.

• For binary classification:

  • y_j is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the ClassNames property), respectively.

  • f(X_j) is the positive-class classification score for observation (row) j of the predictor data X.

  • m_j = y_jf(X_j) is the classification score for classifying observation j into the class corresponding to y_j. Positive values of m_j indicate correct classification and do not contribute much to the average loss. Negative values of m_j indicate incorrect classification and contribute significantly to the average loss.

• For algorithms that support multiclass classification (that is, K ≥ 3):

  • y_j^* is a vector of K – 1 zeros, with 1 in the position corresponding to the true, observed class y_j. For example, if the true class of the second observation is the third class and K = 4, then y₂^* = [0 0 1 0]′. The order of the classes corresponds to the order in the ClassNames property of the input model.

  • f(X_j) is the length K vector of class scores for observation j of the predictor data X. The order of the scores corresponds to the order of the classes in the ClassNames property of the input model.

  • m_j = y_j^*′f(X_j). Therefore, m_j is the scalar classification score that the model predicts for the true, observed class.

• The weight for observation j is w_j. The software normalizes the observation weights so that they sum to the corresponding prior class probability stored in the Prior property. Therefore,

  $${\displaystyle \sum _{j=1}^n{w}_j}=1.$$

Given this scenario, the following table describes the supported loss functions that you can specify by using the LossFun name-value argument.

Loss Function	Value of LossFun	Equation
Binomial deviance	"binodeviance"	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left\{1+\exp \left[-2m_j\right]\right\}}.$$
Observed misclassification cost	"classifcost"	$$L={\displaystyle \sum _{j=1}^n{w}_j}{c}_{y_j{\widehat{y}}_j},$$ where $${\widehat{y}}_j$$ is the class label corresponding to the class with the maximal score, and $$c_{y_j{\widehat{y}}_j}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_j$$ when its true class is yj.
Misclassified rate in decimal	"classiferror"	$$L={\displaystyle \sum _{j=1}^n{w}_j}I\left\{{\widehat{y}}_j\ne y_j\right\},$$ where I{·} is the indicator function.
Cross-entropy loss	"crossentropy"	"crossentropy" is appropriate only for neural network models. The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^n\frac{{\tilde{w}}_j\log (m_j)}{Kn}},$$ where the weights $${\tilde{w}}_j$$ are normalized to sum to n instead of 1.
Exponential loss	"exponential"	$$L={\displaystyle \sum _{j=1}^n{w}_j\exp \left(-m_j\right)}.$$
Hinge loss	"hinge"	$$L={\displaystyle \sum }_{j=1}^n{w}_j\max \left\{0,1-m_j\right\}.$$
Logit loss	"logit"	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left(1+\exp \left(-m_j\right)\right)}.$$
Minimal expected misclassification cost	"mincost"	"mincost" is appropriate only if classification scores are posterior probabilities. The software computes the weighted minimal expected classification cost using this procedure for observations j = 1,...,n. Estimate the expected misclassification cost of classifying the observation Xj into the class k: $${\gamma }_{jk}={\left(f{\left(X_j\right)}^{\prime }C\right)}_k.$$ f(Xj) is the column vector of class posterior probabilities for the observation Xj. C is the cost matrix stored in the Cost property of the model. For observation j, predict the class label corresponding to the minimal expected misclassification cost: $${\widehat{y}}_j=\underset{k=1,\mathrm{...},K}{\text{argmin}}{\gamma }_{jk}.$$ Using C, identify the cost incurred (cj) for making the prediction. The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^n{w}_j{c}_j}.$$
Quadratic loss	"quadratic"	$$L={\displaystyle \sum _{j=1}^n{w}_j{\left(1-m_j\right)}^2}.$$

If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the loss values for "classifcost", "classiferror", and "mincost" are identical. For a model with a nondefault cost matrix, the "classifcost" loss is equivalent to the "mincost" loss most of the time. These losses can be different if prediction into the class with maximal posterior probability is different from prediction into the class with minimal expected cost. Note that "mincost" is appropriate only if classification scores are posterior probabilities.

This figure compares the loss functions (except "classifcost", "crossentropy", and "mincost") over the score m for one observation. Some functions are normalized to pass through the point (0,1).

True Misclassification Cost

The true misclassification cost is the cost of classifying an observation into an incorrect class.

You can set the true misclassification cost per class by using the Cost name-value argument when you create the classifier. Cost(i,j) is the cost of classifying an observation into class j when its true class is i. By default, Cost(i,j)=1 if i~=j, and Cost(i,j)=0 if i=j. In other words, the cost is 0 for correct classification and 1 for incorrect classification.

Expected Misclassification Cost

The expected misclassification cost per observation is an averaged cost of classifying the observation into each class.

Suppose you have Nobs observations that you want to classify with a trained classifier, and you have K classes. You place the observations into a matrix X with one observation per row.

The expected cost matrix CE has size Nobs-by-K. Each row of CE contains the expected (average) cost of classifying the observation into each of the K classes. CE(n,k) is

$${\displaystyle \sum _{i=1}^K\widehat{P}\left(i|X(n)\right)C\left(k|i\right)},$$

where:

• K is the number of classes.

• $$\widehat{P}\left(i|X(n)\right)$$ is the posterior probability of class i for observation X(n).

• $$C\left(k|i\right)$$ is the true misclassification cost of classifying an observation as k when its true class is i.

Score (tree)

For trees, the score of a classification of a leaf node is the posterior probability of the classification at that node. The posterior probability of the classification at a node is the number of training sequences that lead to that node with the classification, divided by the number of training sequences that lead to that node.

For an example, see Posterior Probability Definition for Classification Tree.

Extended Capabilities

Tall Arrays

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

The loss function supports tall arrays with the following 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).

Version History

Introduced in R2011a