Classification loss for multiclass error-correcting output codes (ECOC) model
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
)
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 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 = 0.0033
The average minimal binary loss for the test-sample observations is 0.0033
.
Mdl
— Full or compact multiclass ECOC modelClassificationECOC
model object | CompactClassificationECOC
model
objectFull 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 dataSample 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 nametbl
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 dataPredictor 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 labelsClass 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
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
.
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 handleBinary 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 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 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 binary losses so 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, for example
customFunction
, specify its function handle
'BinaryLoss',@customFunction
.
customFunction
has this form:
bLoss = customFunction(M,s)
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.
The default BinaryLoss
value depends on the score ranges returned
by the binary learners. This table describes some default
BinaryLoss
values based on the given assumptions.
Assumption | Default Value |
---|---|
All binary learners are SVMs or either 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' |
All binary learners are linear or kernel classification models of
logistic regression learners. Or, you specify to predict class
posterior probabilities by setting
'FitPosterior',true in fitcecoc . | 'quadratic' |
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) | function handleLoss function, specified as the comma-separated pair consisting of
'LossFun'
and 'classiferror'
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.
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 level0
(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 weightsones(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
L
— Classification lossClassification 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.
The classification error is a binary classification error measure that 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.
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:
mkj is element (k,j) of the coding design matrix M (that is, the code corresponding to class k of binary learner j).
sj is the score of binary learner j for an observation.
g is the binary loss function.
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,
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,
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 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).
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 yj = 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.
[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.
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.
To run in parallel, set the 'UseParallel'
option to true
.
Set the 'UseParallel'
field of the options structure to true
using statset
and specify the 'Options'
name-value pair argument in the call to this function.
For example: 'Options',statset('UseParallel',true)
For more information, see the 'Options'
name-value pair argument.
For more general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
ClassificationECOC
| CompactClassificationECOC
| fitcecoc
| predict
| resubLoss
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