kfoldMargin

Classification margins for observations not used in training

Syntax

m = kfoldMargin(CVMdl)

m = kfoldMargin(CVMdl,Name,Value)

Description

m = kfoldMargin(CVMdl) returns the cross-validated classification margins obtained by CVMdl, which is a cross-validated, error-correcting output codes (ECOC) model composed of linear classification models. That is, for every fold, kfoldMargin estimates the classification margins for observations that it holds out when it trains using all other observations.

m contains classification margins for each regularization strength in the linear classification models that comprise CVMdl.

m = kfoldMargin(CVMdl,Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, specify a decoding scheme 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:

1. Specifying any one of the cross-validation, name-value pair arguments, for example, CrossVal

2. Setting the name-value pair argument Learners to 'linear' or a linear classification model template returned by templateLinear

To obtain estimates, kfoldMargin applies the same data used to cross-validate the ECOC model (X and Y).

Name-Value 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.

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 y_j is a class label for a particular binary learner (in the set {-1,1,0}), s_j is the score for observation j, and g(y_j,s_j) 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 y_j = 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 form

  bLoss = customFunction(M,s)

  where:

  • M is the K-by-L coding matrix stored in Mdl.CodingMatrix.

  • s is the 1-by-L 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.

  • L 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'

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

Output Arguments

m — Cross-validated classification margins

numeric vector | numeric matrix

Cross-validated classification margins, returned as a numeric vector or matrix.

m is n-by-L, where n is the number of observations in X and L is the number of regularization strengths in Mdl (that is, numel(Mdl.Lambda)).

m(i,j) is the cross-validated classification margin of observation i using the ECOC model, composed of linear classification models, that has regularization strength Mdl.Lambda(j).

Examples

Estimate k-Fold Cross-Validation Margins

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';

Cross-validate a multiclass, linear classification model.

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. You can alter the number of folds using the 'KFold' name-value pair argument.

Estimate the k-fold margins.

m = kfoldMargin(CVMdl);
size(m)

ans = 1×2

       31572           1

m is a 31572-by-1 vector. m(j) is the average of the out-of-fold margins for observation j.

Plot the k-fold margins using box plots.

figure;
boxplot(m);
h = gca;
h.YLim = [-5 5];
title('Distribution of Cross-Validated Margins')

Feature Selection Using k-fold Margins

One way to perform feature selection is to compare k-fold margins from multiple models. Based solely on this criterion, the classifier with the larger margins is the better classifier.

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Margins, and orient the predictor data so that observations correspond to columns.

load nlpdata
Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';
X = X';

Create these two data sets:

• fullX contains all predictors.

• partX contains 1/2 of the predictors chosen at random.

rng(1); % For reproducibility
p = size(X,1); % Number of predictors
halfPredIdx = randsample(p,ceil(0.5*p));
fullX = X;
partX = X(halfPredIdx,:);

Create a linear classification model template that specifies optimizing the objective function using SpaRSA.

t = templateLinear('Solver','sparsa');

Cross-validate two ECOC models composed of binary, linear classification models: one that uses the all of the predictors and one that uses half of the predictors. Indicate that observations correspond to columns.

CVMdl = fitcecoc(fullX,Y,'Learners',t,'CrossVal','on',...
    'ObservationsIn','columns');
PCVMdl = fitcecoc(partX,Y,'Learners',t,'CrossVal','on',...
    'ObservationsIn','columns');

CVMdl and PCVMdl are ClassificationPartitionedLinearECOC models.

Estimate the k-fold margins for each classifier. Plot the distribution of the k-fold margins sets using box plots.

fullMargins = kfoldMargin(CVMdl);
partMargins = kfoldMargin(PCVMdl);

figure;
boxplot([fullMargins partMargins],'Labels',...
    {'All Predictors','Half of the Predictors'});
h = gca;
h.YLim = [-1 1];
title('Distribution of Cross-Validated Margins')

The distributions of the k-fold margins of the two classifiers are similar.

Find Good Lasso Penalty Using k-fold Margins

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare distributions of k-fold margins.

Load the NLP data set. Preprocess the data as in Feature Selection Using k-fold Margins.

load nlpdata
Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';
X = X';

Create a set of 11 logarithmically-spaced regularization strengths from $$10^{-8}$$ through $$10^1$$.

Lambda = logspace(-8,1,11);

Create a linear classification model template that specifies using logistic regression with a lasso penalty, using each of the regularization strengths, optimizing the objective function using SpaRSA, and reducing 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 an ECOC model composed of binary, linear classification models using 5-fold cross-validation and that

rng(10); % For reproducibility
CVMdl = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns','KFold',5)

CVMdl = 
  ClassificationPartitionedLinearECOC
    CrossValidatedModel: 'LinearECOC'
           ResponseName: 'Y'
        NumObservations: 31572
                  KFold: 5
              Partition: [1x1 cvpartition]
             ClassNames: [comm    dsp    simulink    others]
         ScoreTransform: 'none'


  Properties, Methods

CVMdl is a ClassificationPartitionedLinearECOC model.

Estimate the k-fold margins for each regularization strength. The scores for logistic regression are in [0,1]. Apply the quadratic binary loss.

m = kfoldMargin(CVMdl,'BinaryLoss','quadratic');
size(m)

ans = 1×2

       31572          11

m is a 31572-by-11 matrix of cross-validated margins for each observation. The columns correspond to the regularization strengths.

Plot the k-fold margins for each regularization strength.

figure;
boxplot(m)
ylabel('Cross-validated margins')
xlabel('Lambda indices')

Several values of Lambda yield similarly high margin distribution centers with low spreads. Higher values of Lambda lead to predictor variable sparsity, which is a good quality of a classifier.

Choose the regularization strength that occurs just before the margin distribution center starts decreasing and spread starts increasing.

LambdaFinal = Lambda(5);

Train an ECOC model composed of linear classification model using the entire data set and specify the regularization strength LambdaFinal.

t = templateLinear('Learner','logistic','Solver','sparsa',...
    'Regularization','lasso','Lambda',Lambda(5),'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

Binary Loss

A 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_kj is element (k,j) of the coding design matrix M (that is, the code corresponding to class k of binary learner j).

• s_j is the score of binary learner j for an observation.

• g is the binary loss function.

• $$\widehat{k}$$ is the predicted class for the observation.

In loss-based decoding [Escalera et al.], the class producing the minimum sum of the binary losses over binary learners determines the predicted class of an observation, that is,

$$\widehat{k}=\underset{k}{\text{argmin}}{\displaystyle \sum _{j=1}^L\left|m_{kj}\right|g}(m_{kj},s_j).$$

In loss-weighted decoding [Escalera et al.], the class producing the minimum average of the binary losses over binary learners determines the predicted class of an observation, that is,

$$\widehat{k}=\underset{k}{\text{argmin}}\frac{{\displaystyle \sum _{j=1}^L\left|m_{kj}\right|g}(m_{kj},s_j)}{{\displaystyle \sum }_{j=1}^L\left|m_{kj}\right|}.$$

Allwein et al. suggest that loss-weighted decoding improves classification accuracy by keeping loss values for all classes in the same dynamic range.

This table summarizes the supported loss functions, where y_j is a class label for a particular binary learner (in the set {–1,1,0}), s_j is the score for observation j, and g(y_j,s_j).

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 such that the loss is 0.5 when y_j = 0, and aggregates using the average of the binary learners [Allwein et al.].

Do not confuse the binary loss with the overall classification loss (specified by the 'LossFun' name-value pair argument of the loss and predict object functions), which measures how well an ECOC classifier performs as a whole.

Classification Margin

The classification margin is, for each observation, the difference between the negative loss for the true class and the maximal negative loss among the false classes. If the margins are on the same scale, then they serve as a classification confidence measure. Among multiple classifiers, those that yield greater margins are better.

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. “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.

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

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.

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

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