kfoldEdge
Classification edge for observations not used for training
Description
returns
the cross-validated classification
edges obtained by the cross-validated, error-correcting output
codes (ECOC) model composed of linear classification models e
= kfoldEdge(CVMdl
)CVMdl
.
That is, for every fold, kfoldEdge
estimates the
classification edge for observations that it holds out when it trains
using all other observations.
e
contains a classification edge for each
regularization strength in the linear classification models that comprise CVMdl
.
uses
additional options specified by one or more e
= kfoldEdge(CVMdl
,Name,Value
)Name,Value
pair
arguments. For example, specify a decoding scheme, which folds to
use for the edge 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, kfoldEdge 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
Mode
— Edge aggregation level
"average"
(default) | "individual"
Edge aggregation level, specified as "average"
or
"individual"
.
Value | Description |
---|---|
"average" | Returns classification edges averaged over all folds |
"individual" | Returns classification edges 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
e
— Cross-validated classification edges
numeric scalar | numeric vector | numeric matrix
Cross-validated classification edges, returned as a numeric scalar, vector, or matrix.
Let L
be the number of regularization
strengths in the cross-validated models (that is, L is numel(CVMdl.Trained{1}.BinaryLearners{1}.Lambda)
)
and F
be the number of folds (stored in CVMdl.KFold
).
If
Mode
is'average'
, thene
is a 1-by-L
vector.e(
is the average classification edge over all folds of the cross-validated model that uses regularization strengthj
)j
.Otherwise,
e
is aF
-by-L
matrix.e(
is the classification edge for foldi
,j
)i
of the cross-validated model that uses regularization strengthj
.
Examples
Estimate k-Fold Cross-Validation Edge
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 average of the out-of-fold edges.
e = kfoldEdge(CVMdl)
e = 1.4464
Alternatively, you can obtain the per-fold edges by specifying the name-value pair 'Mode','individual'
in kfoldEdge
.
Feature Selection Using k-fold Edges
One way to perform feature selection is to compare k-fold edges from multiple models. Based solely on this criterion, the classifier with the highest edge is the best classifier.
Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Edge, 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 a 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 to optimize 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 edge for each classifier.
fullEdge = kfoldEdge(CVMdl)
fullEdge = 0.6181
partEdge = kfoldEdge(PCVMdl)
partEdge = 0.5235
Based on the k-fold edges, the classifier that uses all of the predictors is the better model.
Find Good Lasso Penalty Using k-fold Edge
To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare k-fold edges.
Load the NLP data set. Preprocess the data as in Feature Selection Using k-fold Edges.
load nlpdata Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others'; X = X';
Create a set of 8 logarithmically-spaced regularization strengths from through .
Lambda = logspace(-8,1,8);
Create a linear classification model template that specifies to use logistic regression with a lasso penalty, use each of the regularization strengths, optimize the objective function using SpaRSA, and reduce 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'
CVMdl
is a ClassificationPartitionedLinearECOC
model.
Estimate the edges for each fold and regularization strength.
eFolds = kfoldEdge(CVMdl,'Mode','individual')
eFolds = 5×8
0.5520 0.5524 0.5528 0.5508 0.4936 0.2933 0.1029 0.0853
0.5241 0.5255 0.5259 0.5260 0.4773 0.2945 0.1052 0.0868
0.5281 0.5298 0.5301 0.5302 0.4786 0.2880 0.1034 0.0868
0.5389 0.5404 0.5408 0.5373 0.4833 0.2910 0.1022 0.0854
0.5497 0.5552 0.5586 0.5571 0.4936 0.2949 0.1027 0.0849
eFolds
is a 5-by-8 matrix of edges. Rows correspond to folds and columns correspond to regularization strengths in Lambda
. You can use eFolds
to identify ill-performing folds, that is, unusually low edges.
Estimate the average edge over all folds for each regularization strength.
e = kfoldEdge(CVMdl)
e = 1×8
0.5386 0.5407 0.5417 0.5403 0.4853 0.2923 0.1033 0.0858
Determine how well the models generalize by plotting the averages of the 5-fold edge for each regularization strength. Identify the regularization strength that maximizes the 5-fold edge over the grid.
figure plot(log10(Lambda),log10(e),'-o') [~, maxEIdx] = max(e); maxLambda = Lambda(maxEIdx); hold on plot(log10(maxLambda),log10(e(maxEIdx)),'ro') ylabel('log_{10} 5-fold edge') xlabel('log_{10} Lambda') legend('Edge','Max edge') hold off
Several values of Lambda
yield similarly high edges. Greater regularization strength values lead to predictor variable sparsity, which is a good quality of a classifier.
Choose the regularization strength that occurs just before the edge starts decreasing.
LambdaFinal = Lambda(4);
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',LambdaFinal,'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
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.
Classification Edge
The classification edge is the weighted mean of the classification margins.
One way to choose among multiple classifiers, for example to perform feature selection, is to choose the classifier that yields the greatest edge.
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. “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
| kfoldMargin
| edge
| kfoldPredict
| fitcecoc
| statset
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