# loss

Classification loss for multiclass error-correcting output codes (ECOC) model

## Description

returns the classification loss (`L`

= loss(`Mdl`

,`tbl`

,`ResponseVarName`

)`L`

), a scalar representing how well
the trained multiclass error-correcting output codes (ECOC) model `Mdl`

classifies the predictor data in `tbl`

compared to the true class
labels in `tbl.ResponseVarName`

. By default, `loss`

uses the classification error to compute
`L`

.

specifies options using one or more name-value pair arguments in addition to any of the
input argument combinations in previous syntaxes. For example, you can specify a decoding
scheme, classification loss function, and verbosity level.`L`

= loss(___,`Name,Value`

)

## Examples

### Determine Test-Sample Loss of ECOC Model

Load Fisher's iris data set. Specify the predictor data `X`

, the response data `Y`

, and the order of the classes in `Y`

.

load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y); % Class order rng(1); % For reproducibility

Train an ECOC model using SVM binary classifiers. Specify a 15% holdout sample, standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',true); PMdl = fitcecoc(X,Y,'Holdout',0.15,'Learners',t,'ClassNames',classOrder); Mdl = PMdl.Trained{1}; % Extract trained, compact classifier

`PMdl`

is a `ClassificationPartitionedECOC`

model. It has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

model that the software trained using the training set.

Estimate the test-sample classification error, which is the default classification loss.

```
testInds = test(PMdl.Partition); % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds,:);
L = loss(Mdl,XTest,YTest)
```

L = 0

The ECOC model correctly classifies all irises in the test sample.

### Determine ECOC Model Quality Using Custom Loss

Determine the quality of an ECOC model by using a custom loss function that considers the minimal binary loss for each observation.

Load Fisher's iris data set. Specify the predictor data `X`

, the response data `Y`

, and the order of the classes in `Y`

.

load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y); % Class order rng(1) % For reproducibility

Train an ECOC model using SVM binary classifiers. Specify a 15% holdout sample, standardize the predictors using an SVM template, and define the class order.

t = templateSVM('Standardize',true); PMdl = fitcecoc(X,Y,'Holdout',0.15,'Learners',t,'ClassNames',classOrder); Mdl = PMdl.Trained{1}; % Extract trained, compact classifier

`PMdl`

is a `ClassificationPartitionedECOC`

model. It has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

model that the software trained using the training set.

Create a function that takes the minimal loss for each observation, then averages the minimal losses for all observations. `S`

corresponds to the `NegLoss`

output of `predict`

.

lossfun = @(~,S,~,~)mean(min(-S,[],2));

Compute the test-sample custom loss.

testInds = test(PMdl.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); loss(Mdl,XTest,YTest,'LossFun',lossfun)

ans = 0.0049

The average minimal binary loss for the test-sample observations is `0.0033`

.

## Input Arguments

`Mdl`

— Full or compact multiclass ECOC model

`ClassificationECOC`

model object | `CompactClassificationECOC`

model
object

Full or compact multiclass ECOC model, specified as a
`ClassificationECOC`

or
`CompactClassificationECOC`

model
object.

To create a full or compact ECOC model, see `ClassificationECOC`

or `CompactClassificationECOC`

.

`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 `Mdl`

. Multicolumn variables and cell arrays other than cell
arrays of character vectors are not allowed.

If you train `Mdl`

using sample data contained in a
`table`

, then the input data for `loss`

must also be in a table.

When training `Mdl`

, assume that you set
`'Standardize',true`

for a template object specified in the
`'Learners'`

name-value pair argument of `fitcecoc`

. In
this case, for the corresponding binary learner `j`

, the software standardizes
the columns of the new predictor data using the corresponding means in
`Mdl.BinaryLearner{j}.Mu`

and standard deviations in
`Mdl.BinaryLearner{j}.Sigma`

.

**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
`Mdl`

, 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.y`

, then specify `ResponseVarName`

as
`'y'`

. Otherwise, the software treats all columns of
`tbl`

, including `tbl.y`

, 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`

`X`

— Predictor data

numeric matrix

Predictor data, specified as a numeric matrix.

Each row of `X`

corresponds to one observation, and each column corresponds
to one variable. The variables in the columns of
`X`

must be the same as the
variables that trained the classifier
`Mdl`

.

The number of rows in `X`

must equal the number of rows in
`Y`

.

When training `Mdl`

, assume that you set
`'Standardize',true`

for a template object specified in the
`'Learners'`

name-value pair argument of `fitcecoc`

. In
this case, for the corresponding binary learner `j`

, the software standardizes
the columns of the new predictor data using the corresponding means in
`Mdl.BinaryLearner{j}.Mu`

and standard deviations in
`Mdl.BinaryLearner{j}.Sigma`

.

**Data Types: **`double`

| `single`

`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 have the same
data type as `Mdl.ClassNames`

. (The software treats string arrays as cell arrays of character
vectors.)

The number of rows in `Y`

must equal the number of rows in
`tbl`

or `X`

.

**Data Types: **`categorical`

| `char`

| `string`

| `logical`

| `single`

| `double`

| `cell`

### 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: **`loss(Mdl,X,Y,'BinaryLoss','hinge','LossFun',@lossfun)`

specifies `'hinge'`

as the binary learner loss function and the custom
function handle `@lossfun`

as the overall loss function.

`BinaryLoss`

— Binary learner loss function

`'hamming'`

| `'linear'`

| `'logit'`

| `'exponential'`

| `'binodeviance'`

| `'hinge'`

| `'quadratic'`

| function handle

Binary learner loss function, specified as the comma-separated pair consisting of
`'BinaryLoss'`

and a built-in loss function name or function handle.

This table describes the built-in functions, where

*y*is the class label for a particular binary learner (in the set {–1,1,0}),_{j}*s*is the score for observation_{j}*j*, and*g*(*y*,_{j}*s*) is the binary loss formula._{j}Value Description Score Domain *g*(*y*,_{j}*s*)_{j}`'binodeviance'`

Binomial deviance (–∞,∞) log[1 + exp(–2 *y*)]/[2log(2)]_{j}s_{j}`'exponential'`

Exponential (–∞,∞) exp(– *y*)/2_{j}s_{j}`'hamming'`

Hamming [0,1] or (–∞,∞) [1 – sign( *y*)]/2_{j}s_{j}`'hinge'`

Hinge (–∞,∞) max(0,1 – *y*)/2_{j}s_{j}`'linear'`

Linear (–∞,∞) (1 – *y*)/2_{j}s_{j}`'logit'`

Logistic (–∞,∞) log[1 + exp(– *y*)]/[2log(2)]_{j}s_{j}`'quadratic'`

Quadratic [0,1] [1 – *y*(2_{j}*s*– 1)]_{j}^{2}/2The software normalizes binary losses so that the loss is 0.5 when

*y*= 0. Also, the software calculates the mean binary loss for each class._{j}For a custom binary loss function, for example

`customFunction`

, specify its function handle`'BinaryLoss',@customFunction`

.`customFunction`

has this form:bLoss = customFunction(M,s)

`M`

is the*K*-by-*B*coding matrix stored in`Mdl.CodingMatrix`

.`s`

is the 1-by-*B*row vector of classification scores.`bLoss`

is the classification loss. This scalar aggregates the binary losses for every learner in a particular class. For example, you can use the mean binary loss to aggregate the loss over the learners for each class.*K*is the number of classes.*B*is the number of binary learners.

For an example of passing a custom binary loss function, see Predict Test-Sample Labels of ECOC Model Using Custom Binary Loss Function.

The default `BinaryLoss`

value depends on the score ranges returned by the
binary learners. This table identifies what some default `BinaryLoss`

values are when you use the default score transform (`ScoreTransform`

property of the model is `'none'`

).

Assumption | Default Value |
---|---|

All binary learners are any of the following: Classification decision trees Discriminant analysis models *k*-nearest neighbor modelsLinear or kernel classification models of logistic regression learners Naive Bayes models
| `'quadratic'` |

All binary learners are SVMs or linear or kernel classification models of SVM learners. | `'hinge'` |

All binary learners are ensembles trained by
`AdaboostM1` or
`GentleBoost` . | `'exponential'` |

All binary learners are ensembles trained by
`LogitBoost` . | `'binodeviance'` |

You specify to predict class posterior probabilities by setting
`'FitPosterior',true` in `fitcecoc` . | `'quadratic'` |

Binary learners are heterogeneous and use different loss functions. | `'hamming'` |

To check the default value, use dot notation to display the `BinaryLoss`

property of the trained model at the command line.

**Example: **`'BinaryLoss','binodeviance'`

**Data Types: **`char`

| `string`

| `function_handle`

`Decoding`

— Decoding scheme

`'lossweighted'`

(default) | `'lossbased'`

Decoding scheme that aggregates the binary losses, specified as the comma-separated pair
consisting of `'Decoding'`

and `'lossweighted'`

or
`'lossbased'`

. For more information, see Binary Loss.

**Example: **`'Decoding','lossbased'`

`LossFun`

— Loss function

`'classiferror'`

(default) | `'classifcost'`

| function handle

Loss function, specified as `'classiferror'`

,
`'classifcost'`

, or a function handle.

Specify the built-in function

`'classiferror'`

. In this case, the loss function is the classification error, which is the proportion of misclassified observations.Specify the built-in function

`'classifcost'`

. In this case, the loss function is the observed misclassification cost. 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'`

and`'classiferror'`

are identical.Or, specify your own function using function handle notation.

Assume that

`n = size(X,1)`

is the sample size and`K`

is the number of classes. 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`Mdl.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 negated loss values for the classes. Each row corresponds to an observation. The column order corresponds to the class order in`Mdl.ClassNames`

. The input`S`

resembles the output argument`NegLoss`

of`predict`

.`W`

is an`n`

-by-1 numeric vector of observation weights. If you pass`W`

, the software normalizes its elements 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`

.

**Data Types: **`char`

| `string`

| `function_handle`

`ObservationsIn`

— Predictor data observation dimension

`'rows'`

(default) | `'columns'`

Predictor data observation dimension, specified as the comma-separated pair consisting of
`'ObservationsIn'`

and `'columns'`

or
`'rows'`

. `Mdl.BinaryLearners`

must contain
`ClassificationLinear`

models.

**Note**

If you orient your predictor matrix so that
observations correspond to columns and specify
`'ObservationsIn','columns'`

, you
can experience a significant reduction in
execution time. You cannot specify
`'ObservationsIn','columns'`

for
predictor data in a table.

`Options`

— Estimation options

`[]`

(default) | structure array returned by `statset`

Estimation options, specified as the comma-separated pair consisting
of `'Options'`

and a structure array returned by `statset`

.

To invoke parallel computing:

You need a Parallel Computing Toolbox™ license.

Specify

`'Options',statset('UseParallel',true)`

.

`Verbose`

— Verbosity level

`0`

(default) | `1`

Verbosity level, specified as the comma-separated pair consisting of
`'Verbose'`

and `0`

or `1`

.
`Verbose`

controls the number of diagnostic messages that the
software displays in the Command Window.

If `Verbose`

is `0`

, then the software does not display
diagnostic messages. Otherwise, the software displays diagnostic messages.

**Example: **`'Verbose',1`

**Data Types: **`single`

| `double`

`Weights`

— Observation weights

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

(default) | numeric vector | name of variable in `tbl`

Observation weights, specified as the comma-separated pair consisting of
`'Weights'`

and a numeric vector or the name of a variable in
`tbl`

. If you supply weights, then `loss`

computes the weighted loss.

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`

, you must do so as a character vector or string scalar. For
example, if the weights are stored as `tbl.w`

, then specify
`Weights`

as `'w'`

. Otherwise, the software
treats all columns of `tbl`

, including `tbl.w`

,
as predictors.

If you do not specify your own loss function (using `LossFun`

),
then the software normalizes `Weights`

to sum up to the value of
the prior probability in the respective class.

**Data Types: **`single`

| `double`

| `char`

| `string`

## Output Arguments

`L`

— Classification loss

numeric scalar | numeric row vector

Classification loss, returned as a numeric scalar or row vector.
`L`

is a generalization or resubstitution quality measure. Its
interpretation depends on the loss function and weighting scheme, but in general, better
classifiers yield smaller classification loss values.

If `Mdl.BinaryLearners`

contains `ClassificationLinear`

models, then `L`

is a
1-by-*ℓ* vector, where *ℓ* is the number of
regularization strengths in the linear classification models
(`numel(Mdl.BinaryLearners{1}.Lambda)`

). The value
`L(j)`

is the loss for the model trained using regularization
strength `Mdl.BinaryLearners{1}.Lambda(j)`

.

Otherwise, `L`

is a scalar value.

## More About

### Classification Error

The *classification error* has the form

$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{e}_{j}},$$

where:

*w*is the weight for observation_{j}*j*. The software renormalizes the weights to sum to 1.*e*= 1 if the predicted class of observation_{j}*j*differs from its true class, and 0 otherwise.

In other words, the classification error is the proportion of observations misclassified by the classifier.

### Observed Misclassification Cost

The *observed misclassification cost* has the form

$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}{c}_{{y}_{j}{\widehat{y}}_{j}},$$

where:

*w*is the weight for observation_{j}*j*. The software renormalizes the weights to sum to 1.$${c}_{{y}_{j}{\widehat{y}}_{j}}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_{j}$$ when its true class is

*y*._{j}

### Binary Loss

The *binary loss* is a function of the class and
classification score that determines how well a binary learner classifies an observation
into the class.

Suppose the following:

*m*is element (_{kj}*k*,*j*) of the coding design matrix*M*—that is, the code corresponding to class*k*of binary learner*j*.*M*is a*K*-by-*B*matrix, where*K*is the number of classes, and*B*is the number of binary learners.*s*is the score of binary learner_{j}*j*for an observation.*g*is the binary loss function.$$\widehat{k}$$ is the predicted class for the observation.

The decoding scheme of an ECOC model specifies how the software aggregates the binary losses and determines the predicted class for each observation. The software supports two decoding schemes:

*Loss-based decoding*[2] (`Decoding`

is`'lossbased'`

) — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over all binary learners.$$\widehat{k}=\underset{k}{\text{argmin}}\frac{1}{B}{\displaystyle \sum _{j=1}^{B}\left|{m}_{kj}\right|g}({m}_{kj},{s}_{j}).$$

*Loss-weighted decoding*[3] (`Decoding`

is`'lossweighted'`

) — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over the binary learners for the corresponding class.$$\widehat{k}=\underset{k}{\text{argmin}}\frac{{\displaystyle \sum _{j=1}^{B}\left|{m}_{kj}\right|g}({m}_{kj},{s}_{j})}{{\displaystyle \sum}_{j=1}^{B}\left|{m}_{kj}\right|}.$$

The denominator corresponds to the number of binary learners for class

*k*. [1] suggests that loss-weighted decoding improves classification accuracy by keeping loss values for all classes in the same dynamic range.

The `predict`

, `resubPredict`

, and
`kfoldPredict`

functions return the negated value of the objective
function of `argmin`

as the second output argument
(`NegLoss`

) for each observation and class.

This table summarizes the supported binary loss functions, where
*y _{j}* is a class label for a particular
binary learner (in the set {–1,1,0}),

*s*is the score for observation

_{j}*j*, and

*g*(

*y*,

_{j}*s*) is the binary loss function.

_{j}Value | Description | Score Domain | g(y,_{j}s)_{j} |
---|---|---|---|

`"binodeviance"` | Binomial deviance | (–∞,∞) | log[1 +
exp(–2y)]/[2log(2)]_{j}s_{j} |

`"exponential"` | Exponential | (–∞,∞) | exp(–y)/2_{j}s_{j} |

`"hamming"` | Hamming | [0,1] or (–∞,∞) | [1 – sign(y)]/2_{j}s_{j} |

`"hinge"` | Hinge | (–∞,∞) | max(0,1 – y)/2_{j}s_{j} |

`"linear"` | Linear | (–∞,∞) | (1 – y)/2_{j}s_{j} |

`"logit"` | Logistic | (–∞,∞) | log[1 +
exp(–y)]/[2log(2)]_{j}s_{j} |

`"quadratic"` | Quadratic | [0,1] | [1 – y(2_{j}s –
1)]_{j}^{2}/2 |

The software normalizes binary losses so that the loss is 0.5 when
*y _{j}* = 0, and aggregates using the average
of the binary learners.

Do not confuse the binary loss with the overall classification loss (specified by the
`LossFun`

name-value argument of the `loss`

and
`predict`

object functions), which measures how well an ECOC classifier
performs as a whole.

## References

[1] Allwein, E., R. Schapire, and Y. Singer. “Reducing multiclass to binary: A unifying approach for margin classiﬁers.” *Journal of Machine Learning Research*. Vol. 1, 2000, pp. 113–141.

[2] Escalera, S., O. Pujol, and P. Radeva. “Separability of ternary codes for sparse designs of error-correcting output codes.” *Pattern Recog. Lett.*, Vol. 30, Issue 3, 2009, pp. 285–297.

[3] Escalera, S., O. Pujol, and P. Radeva. “On the decoding process in ternary error-correcting output codes.” *IEEE Transactions on Pattern Analysis and Machine Intelligence*. Vol. 32, Issue 7, 2010, pp. 120–134.

## Extended Capabilities

### Tall Arrays

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

Usage notes and limitations:

`loss`

does not support tall`table`

data when`Mdl`

contains kernel or linear binary learners.

For more information, see Tall Arrays.

### Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, specify the `'Options'`

name-value argument in the call
to this function and set the `'UseParallel'`

field of the options
structure to `true`

using `statset`

.

For example: `'Options',statset('UseParallel',true)`

For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

### 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 models trained using decision tree learners with surrogate splits.The

`loss`

function does not support models trained using SVM learners.

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

## Version History

**Introduced in R2014b**

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