# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

# loss

Classification error

## Syntax

`L = loss(ens,tbl,ResponseVarName)L = loss(ens,tbl,Y)L = loss(ens,X,Y)L = loss(___,Name,Value)`

## Description

`L = loss(ens,tbl,ResponseVarName)` returns the classification error for ensemble `ens` computed using table of predictors `tbl` and true class labels `tbl.ResponseVarName`.

`L = loss(ens,tbl,Y)` returns the classification error for ensemble `ens` computed using table of predictors `tbl` and true class labels `Y`.

`L = loss(ens,X,Y)` returns the classification error for ensemble `ens` computed using matrix of predictors `X` and true class labels `Y`.

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

When computing the loss, `loss` normalizes the class probabilities in `ResponseVarName` or `Y` to the class probabilities used for training, stored in the `Prior` property of `ens`.

## Input Arguments

 `ens` Classification ensemble created with `fitensemble`, or a compact classification ensemble created with `compact`. `tbl` 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 the model. Multi-column variables and cell arrays other than cell arrays of character vectors are not allowed. If you trained `ens` using sample data contained in a `table`, then the input data for this method must also be in a table. `ResponseVarName` Response variable name, specified as the name of a variable in `tbl`. The response variable must be a numeric vector. You must specify `ResponseVarName` as a character vector. For example, if the response variable `Y` is stored as `tbl.Y`, then specify it as `'Y'`. Otherwise, the software treats all columns of `tbl`, including `Y`, as predictors when training the model. `X` Matrix of data to classify. Each row of `X` represents one observation, and each column represents one predictor. `X` must have the same number of columns as the data used to train `ens`. `X` should have the same number of rows as the number of elements in `Y`. If you trained `ens` using sample data contained in a matrix, then the input data for this method must also be in a matrix. `Y` Classification of `tbl` or `X`. `Y` should be of the same type as the classification used to train `ens`, and its number of elements should equal the number of rows of `tbl` or `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`.

`'learners'`

Indices of weak learners in the ensemble ranging from `1` to `ens``.NumTrained`. `loss` 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 lists available loss functions. Specify one using its corresponding character vector.

ValueDescription
`'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 and subspace ensembles return posterior probabilities by default (`ens.Method` is `'Bag'` or `'Subspace'`).

• If the ensemble method is `'AdaBoostM1'`, `'AdaBoostM2'`, `GentleBoost`, or `'LogitBoost'`, then, to use posterior probabilities as classification scores, you must specify the double-logit score transform by entering

`ens.ScoreTransform = 'doublelogit';`

• For all other ensemble methods, the software does not support posterior probabilities as classification scores.

• 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'`

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'`

`'UseObsForLearner'`

A logical matrix of size `N`-by-`T`, where:

When `UseObsForLearner(i,j)` is `true`, learner `j` is used in predicting the class of row `i` of `X`.

Default: `true(N,T)`

`'weights'`

Vector of observation weights, with nonnegative entries. The length of `weights` must equal the number of rows in `X`. When you specify weights, `loss` normalizes the weights so that observation weights in each class sum to the prior probability of that class.

Default: `ones(size(X,1),1)`

## Output Arguments

 `L` Classification loss, by default the fraction of misclassified data. `L` can be a vector, and can mean different things, depending on the name-value pair settings.

## Definitions

### Classification Loss

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

Suppose that:

• L is the weighted average classification loss.

• n is the sample size.

• For binary classification:

• yj is the observed class label. The software codes it as –1 or 1 indicating the negative or positive class, respectively.

• f(Xj) is the raw classification score for observation (row) j of the predictor data X.

• mj = yjf(Xj) is the classification score for classifying observation j into the class corresponding to yj. Positive values of mj indicate correct classification and do not contribute much to the average loss. Negative values of mj indicate incorrect classification and contribute to the average loss.

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

• yj* is a vector of K – 1 zeros, and a 1 in the position corresponding to the true, observed class yj. For example, if the true class of the second observation is the third class and K = 4, then y*2 = [0 0 1 0]′. The order of the classes corresponds to the order in the `ClassNames` property of the input model.

• f(Xj) 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.

• mj = yj*f(Xj). Therefore, mj is the scalar classification score that the model predicts for the true, observed class.

• The weight for observation j is wj. 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,

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

The supported loss functions are:

• Binomial deviance, specified using `'LossFun','binodeviance'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$`
• Exponential loss, specified using `'LossFun','exponential'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$`
• Classification error, specified using `'LossFun','classiferror'`. It is the weighted fraction of misclassified observations, with equation

`$L=\sum _{j=1}^{n}{w}_{j}I\left\{{\stackrel{^}{y}}_{j}\ne {y}_{j}\right\}.$`

${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.

• Hinge loss, specified using `'LossFun','hinge'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$`
• Logit loss, specified using `'LossFun','logit'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).$`
• Minimal cost, specified using `'LossFun','mincost'`. The software computes the weighted minimal cost using this procedure for observations j = 1,...,n:

1. 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 the input model stores in the property `Cost`.

2. For observation j, predict the class label corresponding to the minimum, expected classification cost:

`${\stackrel{^}{y}}_{j}=\underset{j=1,...,K}{\mathrm{min}}\left({\gamma }_{j}\right).$`
3. Using C, identify the cost incurred (cj) for making the prediction.

The weighted, average, minimum cost loss is

`$L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.$`
• Quadratic loss, specified using `'LossFun','quadratic'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}.$`

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

## Examples

expand all

Load Fisher's iris data set.

```load fisheriris ```

Boost 100 classification trees using AdaBoostM2.

```ens = fitensemble(meas,species,'AdaBoostM2',100,'Tree'); ```

Estimate the classification error of the model using the training observations.

```L = loss(ens,meas,species) ```
```L = 0.0333 ```

Alternatively, if `ens` is not compact, then you can estimate the training-sample classification error by passing `ens` to `resubLoss`.