oobLoss

Out-of-bag classification error

Syntax

L = oobLoss(ens)
L = oobLoss(ens,Name,Value)


Description

L = oobLoss(ens) returns the classification error for ens computed for out-of-bag data.

L = oobLoss(ens,Name,Value) computes error 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

ens	A classification bagged ensemble, constructed with fitcensemble.

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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'learners'	Indices of weak learners in the ensemble ranging from 1 to NumTrained. oobLoss uses only these learners for calculating loss. Default: 1:NumTrained
'lossfun'	Loss function, specified as the comma-separated pair consisting of 'LossFun' and a built-in, loss-function name or function handle. The following table lists the available loss functions. Specify one using its corresponding character vector or string scalar. Value Description 'binodeviance' Binomial deviance 'classiferror' Classification error 'exponential' Exponential 'hinge' Hinge 'logit' Logistic 'mincost' Minimal expected misclassification cost (for classification scores that are posterior probabilities) 'quadratic' Quadratic 'mincost' is appropriate for classification scores that are posterior probabilities. Bagged ensembles return posterior probabilities as classification scores by default. Specify your own function using function handle notation. Suppose that n be the number of observations in X and K be the number of distinct classes (numel(ens.ClassNames), ens is the input model). Your function must have this signature lossvalue = lossfun(C,S,W,Cost) where: The output argument lossvalue is a scalar. You choose the function name (lossfun). C is an n-by-K logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order in ens.ClassNames. Construct 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 ens.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 them 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. Specify your function using 'LossFun',@lossfun. For more details on loss functions, see Classification Loss. Default: 'classiferror'
'mode'	Character vector or string scalar representing the meaning of the output L: 'ensemble' — L is a scalar value, the loss for the entire ensemble. 'individual' — L is a vector with one element per trained learner. 'cumulative' — L is a vector in which element J is obtained by using learners 1:J from the input list of learners. Default: 'ensemble'

Output Arguments

L	Classification loss of the out-of-bag observations, a scalar. L can be a vector, or can represent a different quantity, depending on the name-value settings.

Examples

Estimate Out-Of-Bag Error

Load Fisher's iris data set.

load fisheriris

Grow a bag of 100 classification trees.

ens = fitcensemble(meas,species,'Method','Bag');

Estimate the out-of-bag classification error.

L = oobLoss(ens)

L = 0.0400

More About

Out of Bag

Bagging, which stands for “bootstrap aggregation”, is a type of ensemble learning. To bag a weak learner such as a decision tree on a dataset, fitcensemble generates many bootstrap replicas of the dataset and grows decision trees on these replicas. fitcensemble obtains each bootstrap replica by randomly selecting N observations out of N with replacement, where N is the dataset size. To find the predicted response of a trained ensemble, predict take an average over predictions from individual trees.

Drawing N out of N observations with replacement omits on average 37% (1/e) of observations for each decision tree. These are "out-of-bag" observations. For each observation, oobLoss estimates the out-of-bag prediction by averaging over predictions from all trees in the ensemble for which this observation is out of bag. It then compares the computed prediction against the true response for this observation. It calculates the out-of-bag error by comparing the out-of-bag predicted responses against the true responses for all observations used for training. This out-of-bag average is an unbiased estimator of the true ensemble error.

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, respectively.

  • f(X_j) is the raw 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. The software also normalizes the prior probabilities so they sum to 1. 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 pair argument.

Loss Function	Value of LossFun	Equation
Binomial deviance	'binodeviance'	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left\{1+\exp \left[-2m_j\right]\right\}}.$$
Exponential loss	'exponential'	$$L={\displaystyle \sum _{j=1}^n{w}_j\exp \left(-m_j\right)}.$$
Classification error	'classiferror'	$$L={\displaystyle \sum _{j=1}^n{w}_j}I\left\{{\widehat{y}}_j\ne y_j\right\}.$$ The classification error is the weighted fraction of misclassified observations where $${\widehat{y}}_j$$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.
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 cost	'mincost'	The software computes the weighted minimal cost using this procedure for observations j = 1,...,n. Estimate the 1-by-K vector of expected classification costs for observation j: $${\gamma }_j=f{\left(X_j\right)}^{\prime }C.$$ f(Xj) is the column vector of class posterior probabilities for binary and multiclass classification. C is the cost matrix stored by the input model in the Cost property. For observation j, predict the class label corresponding to the minimum expected classification cost: $${\widehat{y}}_j=\underset{j=1,\mathrm{...},K}{\min }\left({\gamma }_j\right).$$ Using C, identify the cost incurred (cj) for making the prediction. The weighted, average, minimum cost loss is $$L={\displaystyle \sum _{j=1}^n{w}_j{c}_j}.$$
Quadratic loss	'quadratic'	$$L={\displaystyle \sum _{j=1}^n{w}_j{\left(1-m_j\right)}^2}.$$

This figure compares the loss functions (except 'mincost') for one observation over m. Some functions are normalized to pass through [0,1].