kfoldLoss
Classification loss for observations not used in training
Description
returns
the cross-validated classification
error rates estimated by the cross-validated, error-correcting
output codes (ECOC) model composed of linear classification models L
= kfoldLoss(CVMdl
)CVMdl
.
That is, for every fold, kfoldLoss
estimates the
classification error rate for observations that it holds out when
it trains using all other observations. kfoldLoss
applies
the same data used create CVMdl
(see fitcecoc
).
L
contains a classification loss for each
regularization strength in the linear classification models that compose CVMdl
.
uses
additional options specified by one or more L
= kfoldLoss(CVMdl
,Name,Value
)Name,Value
pair
arguments. For example, specify a decoding scheme, which folds to
use for the loss calculation, or verbosity level.
Input Arguments
CVMdl
— Cross-validated, ECOC model composed of linear classification models
ClassificationPartitionedLinearECOC
model
object
Cross-validated, ECOC model composed of linear classification
models, specified as a ClassificationPartitionedLinearECOC
model
object. You can create a ClassificationPartitionedLinearECOC
model
using fitcecoc
and by:
Specifying any one of the cross-validation, name-value pair arguments, for example,
CrossVal
Setting the name-value pair argument
Learners
to'linear'
or a linear classification model template returned bytemplateLinear
To obtain estimates, kfoldLoss applies the same data used
to cross-validate the ECOC model (X
and Y
).
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.
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 contains names and descriptions of the built-in functions, where yj is the class label for a particular binary learner (in the set {-1,1,0}), sj is the score for observation j, and g(yj,sj) is the binary loss formula.
Value Description Score Domain g(yj,sj) "binodeviance"
Binomial deviance (–∞,∞) log[1 + exp(–2yjsj)]/[2log(2)] "exponential"
Exponential (–∞,∞) exp(–yjsj)/2 "hamming"
Hamming [0,1] or (–∞,∞) [1 – sign(yjsj)]/2 "hinge"
Hinge (–∞,∞) max(0,1 – yjsj)/2 "linear"
Linear (–∞,∞) (1 – yjsj)/2 "logit"
Logistic (–∞,∞) log[1 + exp(–yjsj)]/[2log(2)] "quadratic"
Quadratic [0,1] [1 – yj(2sj – 1)]2/2 The software normalizes the binary losses such that the loss is 0.5 when yj = 0. Also, the software calculates the mean binary loss for each class.
For a custom binary loss function, e.g.,
customFunction
, specify its function handle'BinaryLoss',@customFunction
.customFunction
should have this formwhere:bLoss = customFunction(M,s)
M
is the K-by-B coding matrix stored inMdl.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.
By default, if all binary learners are linear classification models using:
SVM, then
BinaryLoss
is'hinge'
Logistic regression, then
BinaryLoss
is'quadratic'
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'
Folds
— Fold indices to use for classification-score prediction
1:CVMdl.KFold
(default) | numeric vector of positive integers
Fold indices to use for classification-score prediction, specified as a numeric vector of
positive integers. The elements of Folds
must range from
1
through CVMdl.KFold
.
Example: Folds=[1 4 10]
Data Types: single
| double
LossFun
— Loss function
'classiferror'
(default) | 'classifcost'
| function handle
Loss function, specified as 'classiferror'
,
'classifcost'
, or a function handle.
You can:
Specify the built-in function
'classiferror'
, then the loss function is the classification error.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.Specify your own function using function handle notation.
For what follows,
n
is the number of observations in the training data (CVMdl.NumObservations
) andK
is the number of classes (numel(CVMdl.ClassNames)
). Your function needs the signaturelossvalue =
, where:lossfun
(C,S,W,Cost)The output argument
lossvalue
is a scalar.You choose the function name (
lossfun
).C
is ann
-by-K
logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order inCVMdl.ClassNames
.Construct
C
by settingC(p,q) = 1
if observationp
is in classq
, for each row. Set every element of rowp
to0
.S
is ann
-by-K
numeric matrix of negated loss values for classes. Each row corresponds to an observation. The column order corresponds to the class order inCVMdl.ClassNames
.S
resembles the output argumentNegLoss
ofkfoldPredict
.W
is ann
-by-1 numeric vector of observation weights. If you passW
, the software normalizes its elements to sum to1
.Cost
is aK
-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: function_handle
| char
| string
Mode
— Loss aggregation level
"average"
(default) | "individual"
Loss aggregation level, specified as "average"
or
"individual"
.
Value | Description |
---|---|
"average" | Returns losses averaged over all folds |
"individual" | Returns losses for each fold |
Example: Mode="individual"
Options
— Estimation options
[]
(default) | structure array
Estimation options, specified as a structure array as returned by statset
.
To invoke parallel computing you need a Parallel Computing Toolbox™ license.
Example: Options=statset(UseParallel=true)
Data Types: struct
Verbose
— Verbosity level
0
(default) | 1
Verbosity level, specified as 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
Output Arguments
L
— Cross-validated classification losses
numeric scalar | numeric vector | numeric matrix
Cross-validated classification losses, returned
as a numeric scalar, vector, or matrix. The interpretation of L
depends
on LossFun
.
Let R
be the number of regularizations
strengths is the cross-validated models (CVMdl.Trained{1}.BinaryLearners{1}.Lambda
)
and F
be the number of folds (stored in CVMdl.KFold
).
If
Mode
is'average'
, thenL
is a 1-by-R
vector.L(
is the average classification loss over all folds of the cross-validated model that uses regularization strengthj
)j
.Otherwise,
L
is aF
-by-R
matrix.L(
is the classification loss for foldi
,j
)i
of the cross-validated model that uses regularization strengthj
.
Examples
Estimate k-Fold Cross-Validation Classification Error
Load the NLP data set.
load nlpdata
X
is a sparse matrix of predictor data, and Y
is a categorical vector of class labels.
Cross-validate an ECOC model of linear classification models.
rng(1); % For reproducibility CVMdl = fitcecoc(X,Y,'Learner','linear','CrossVal','on');
CVMdl
is a ClassificationPartitionedLinearECOC
model. By default, the software implements 10-fold cross validation.
Estimate the average of the out-of-fold classification error rates.
ce = kfoldLoss(CVMdl)
ce = 0.0958
Alternatively, you can obtain the per-fold classification error rates by specifying the name-value pair 'Mode','individual'
in kfoldLoss
.
Specify Custom Classification Loss
Load the NLP data set. Transpose the predictor data.
load nlpdata
X = X';
For simplicity, use the label 'others' for all observations in Y
that are not 'simulink'
, 'dsp'
, or 'comm'
.
Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';
Create a linear classification model template that specifies optimizing the objective function using SpaRSA.
t = templateLinear('Solver','sparsa');
Cross-validate an ECOC model of linear classification models using 5-fold cross-validation. Optimize the objective function using SpaRSA. Specify that the predictor observations correspond to columns.
rng(1); % For reproducibility CVMdl = fitcecoc(X,Y,'Learners',t,'KFold',5,'ObservationsIn','columns'); CMdl1 = CVMdl.Trained{1}
CMdl1 = CompactClassificationECOC ResponseName: 'Y' ClassNames: [comm dsp simulink others] ScoreTransform: 'none' BinaryLearners: {6x1 cell} CodingMatrix: [4x6 double]
CVMdl
is a ClassificationPartitionedLinearECOC
model. It contains the property Trained
, which is a 5-by-1 cell array holding a CompactClassificationECOC
model that the software trained using the training set of each fold.
Create a function that takes the minimal loss for each observation, and then averages the minimal losses across all observations. Because the function does not use the class-identifier matrix (C
), observation weights (W
), and classification cost (Cost
), use ~
to have kfoldLoss
ignore their positions.
lossfun = @(~,S,~,~)mean(min(-S,[],2));
Estimate the average cross-validated classification loss using the minimal loss per observation function. Also, obtain the loss for each fold.
ce = kfoldLoss(CVMdl,'LossFun',lossfun)
ce = 0.0485
ceFold = kfoldLoss(CVMdl,'LossFun',lossfun,'Mode','individual')
ceFold = 5×1
0.0488
0.0511
0.0496
0.0479
0.0452
Find Good Lasso Penalty Using Cross-Validation
To determine a good lasso-penalty strength for an ECOC model composed of linear classification models that use logistic regression learners, implement 5-fold cross-validation.
Load the NLP data set.
load nlpdata
X
is a sparse matrix of predictor data, and Y
is a categorical vector of class labels.
For simplicity, use the label 'others' for all observations in Y
that are not 'simulink'
, 'dsp'
, or 'comm'
.
Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';
Create a set of 11 logarithmically-spaced regularization strengths from through .
Lambda = logspace(-7,-2,11);
Create a linear classification model template that specifies to use logistic regression learners, use lasso penalties with strengths in Lambda
, train using SpaRSA, and lower the tolerance on the gradient of the objective function to 1e-8
.
t = templateLinear('Learner','logistic','Solver','sparsa',... 'Regularization','lasso','Lambda',Lambda,'GradientTolerance',1e-8);
Cross-validate the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns.
X = X'; rng(10); % For reproducibility CVMdl = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns','KFold',5);
CVMdl
is a ClassificationPartitionedLinearECOC
model.
Dissect CVMdl
, and each model within it.
numECOCModels = numel(CVMdl.Trained)
numECOCModels = 5
ECOCMdl1 = CVMdl.Trained{1}
ECOCMdl1 = CompactClassificationECOC ResponseName: 'Y' ClassNames: [comm dsp simulink others] ScoreTransform: 'none' BinaryLearners: {6×1 cell} CodingMatrix: [4×6 double] Properties, Methods
numCLModels = numel(ECOCMdl1.BinaryLearners)
numCLModels = 6
CLMdl1 = ECOCMdl1.BinaryLearners{1}
CLMdl1 = ClassificationLinear ResponseName: 'Y' ClassNames: [-1 1] ScoreTransform: 'logit' Beta: [34023×11 double] Bias: [-0.3169 -0.3169 -0.3168 -0.3168 -0.3168 -0.3167 -0.1725 -0.0805 -0.1762 -0.3450 -0.5174] Lambda: [1.0000e-07 3.1623e-07 1.0000e-06 3.1623e-06 1.0000e-05 3.1623e-05 1.0000e-04 3.1623e-04 1.0000e-03 0.0032 0.0100] Learner: 'logistic' Properties, Methods
Because fitcecoc
implements 5-fold cross-validation, CVMdl
contains a 5-by-1 cell array of CompactClassificationECOC
models that the software trains on each fold. The BinaryLearners
property of each CompactClassificationECOC
model contains the ClassificationLinear
models. The number of ClassificationLinear
models within each compact ECOC model depends on the number of distinct labels and coding design. Because Lambda
is a sequence of regularization strengths, you can think of CLMdl1
as 11 models, one for each regularization strength in Lambda
.
Determine how well the models generalize by plotting the averages of the 5-fold classification error for each regularization strength. Identify the regularization strength that minimizes the generalization error over the grid.
ce = kfoldLoss(CVMdl); figure; plot(log10(Lambda),log10(ce)) [~,minCEIdx] = min(ce); minLambda = Lambda(minCEIdx); hold on plot(log10(minLambda),log10(ce(minCEIdx)),'ro'); ylabel('log_{10} 5-fold classification error') xlabel('log_{10} Lambda') legend('MSE','Min classification error') hold off
Train an ECOC model composed of linear classification model using the entire data set, and specify the minimal regularization strength.
t = templateLinear('Learner','logistic','Solver','sparsa',... 'Regularization','lasso','Lambda',minLambda,'GradientTolerance',1e-8); MdlFinal = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns');
To estimate labels for new observations, pass MdlFinal
and the new data to predict
.
More About
Classification Error
The classification error has the form
where:
wj is the weight for observation j. The software renormalizes the weights to sum to 1.
ej = 1 if the predicted class of observation 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
where:
wj is the weight for observation j. The software renormalizes the weights to sum to 1.
is the user-specified cost of classifying an observation into class when its true class is yj.
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. The decoding scheme of an ECOC model specifies how the software aggregates the binary losses and determines the predicted class for each observation.
Assume the following:
mkj is element (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.
sj is the score of binary learner j for an observation.
g is the binary loss function.
is the predicted class for the 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.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.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 yj is a class label for a particular binary learner (in the set {–1,1,0}), sj is the score for observation j, and g(yj,sj) is the binary loss function.
Value | Description | Score Domain | g(yj,sj) |
---|---|---|---|
"binodeviance" | Binomial deviance | (–∞,∞) | log[1 + exp(–2yjsj)]/[2log(2)] |
"exponential" | Exponential | (–∞,∞) | exp(–yjsj)/2 |
"hamming" | Hamming | [0,1] or (–∞,∞) | [1 – sign(yjsj)]/2 |
"hinge" | Hinge | (–∞,∞) | max(0,1 – yjsj)/2 |
"linear" | Linear | (–∞,∞) | (1 – yjsj)/2 |
"logit" | Logistic | (–∞,∞) | log[1 + exp(–yjsj)]/[2log(2)] |
"quadratic" | Quadratic | [0,1] | [1 – yj(2sj – 1)]2/2 |
The software normalizes binary losses so that the loss is 0.5 when yj = 0, and aggregates using the average of the binary learners [1].
Do not confuse the binary loss with the overall classification loss (specified by the
LossFun
name-value argument of the kfoldLoss
and
kfoldPredict
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
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
:
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™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2016aR2023b: Observations with missing predictor values are used in resubstitution and cross-validation computations
Starting in R2023b, the following classification model object functions use observations with missing predictor values as part of resubstitution ("resub") and cross-validation ("kfold") computations for classification edges, losses, margins, and predictions.
In previous releases, the software omitted observations with missing predictor values from the resubstitution and cross-validation computations.
See Also
ClassificationPartitionedLinearECOC
| ClassificationECOC
| ClassificationLinear
| loss
| kfoldPredict
| fitcecoc
| statset
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