loss
Classification loss for Gaussian kernel classification model
Description
returns the classification loss for the model L
= loss(Mdl
,Tbl
,ResponseVarName
)Mdl
using the
predictor data in Tbl
and the true class labels in
Tbl.ResponseVarName
.
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 classification loss function and observation weights. Then,
L
= loss(___,Name,Value
)loss
returns the weighted classification loss using the
specified loss function.
Note
If the predictor data in X
or Tbl
contains
any missing values and LossFun
is not set to
"classifcost"
, "classiferror"
, or
"mincost"
, the loss
function can
return NaN. For more details, see loss can return NaN for predictor data with missing values.
Examples
Estimate Test-Set Classification Loss
Load the ionosphere
data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b'
) or good ('g'
).
load ionosphere
Partition the data set into training and test sets. Specify a 15% holdout sample for the test set.
rng('default') % For reproducibility Partition = cvpartition(Y,'Holdout',0.15); trainingInds = training(Partition); % Indices for the training set testInds = test(Partition); % Indices for the test set
Train a binary kernel classification model using the training set.
Mdl = fitckernel(X(trainingInds,:),Y(trainingInds));
Estimate the training-set classification error and the test-set classification error.
ceTrain = loss(Mdl,X(trainingInds,:),Y(trainingInds))
ceTrain = 0.0067
ceTest = loss(Mdl,X(testInds,:),Y(testInds))
ceTest = 0.1140
Specify Custom Classification Loss
Load the ionosphere
data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b'
) or good ('g'
).
load ionosphere
Partition the data set into training and test sets. Specify a 15% holdout sample for the test set.
rng('default') % For reproducibility Partition = cvpartition(Y,'Holdout',0.15); trainingInds = training(Partition); % Indices for the training set testInds = test(Partition); % Indices for the test set
Train a binary kernel classification model using the training set.
Mdl = fitckernel(X(trainingInds,:),Y(trainingInds));
Create an anonymous function that measures linear loss, that is,
$$L=\frac{\sum _{j}-{w}_{j}{y}_{j}{f}_{j}}{\sum _{j}{w}_{j}}.$$
$${w}_{j}$$ is the weight for observation j, $${y}_{j}$$ is response j (-1 for the negative class, and 1 otherwise), and $${f}_{j}$$ is the raw classification score of observation j.
linearloss = @(C,S,W,Cost)sum(-W.*sum(S.*C,2))/sum(W);
Custom loss functions must be written in a particular form. For rules on writing a custom loss function, see the 'LossFun'
name-value pair argument.
Estimate the training-set classification loss and the test-set classification loss using the linear loss function.
ceTrain = loss(Mdl,X(trainingInds,:),Y(trainingInds),'LossFun',linearloss)
ceTrain = -1.0851
ceTest = loss(Mdl,X(testInds,:),Y(testInds),'LossFun',linearloss)
ceTest = -0.7821
Input Arguments
Mdl
— Binary kernel classification model
ClassificationKernel
model object
Binary kernel classification model, specified as a ClassificationKernel
model object. You can create a
ClassificationKernel
model object using fitckernel
.
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; logical or numeric vector; or cell array of character vectors.
The data type of
Y
must be the same as the data type ofMdl.ClassNames
. (The software treats string arrays as cell arrays of character vectors.)The distinct classes in
Y
must be a subset ofMdl.ClassNames
.If
Y
is a character array, then each element must correspond to one row of the array.The length of
Y
must be equal to the number of observations inX
orTbl
.
Data Types: categorical
| char
| string
| logical
| single
| double
| cell
Tbl
— Sample data
table
Sample data used to train the model, 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 Tbl
contains the response variable used to train Mdl
, then you do not need to specify ResponseVarName
or Y
.
If you train Mdl
using sample data contained in a table, then the input
data for loss
must also be in a 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 specify it 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
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: L =
loss(Mdl,X,Y,'LossFun','quadratic','Weights',weights)
returns the
weighted classification loss using the quadratic loss function.
LossFun
— Loss function
'classiferror'
(default) | 'binodeviance'
| 'classifcost'
| 'exponential'
| 'hinge'
| 'logit'
| 'mincost'
| 'quadratic'
| function handle
Loss function, specified as the comma-separated pair consisting of
'LossFun'
and a built-in loss function name or a
function handle.
This table lists the available loss functions. Specify one using its corresponding value.
Value Description "binodeviance"
Binomial deviance "classifcost"
Observed misclassification cost "classiferror"
Misclassified rate in decimal "exponential"
Exponential loss "hinge"
Hinge loss "logit"
Logistic loss "mincost"
Minimal expected misclassification cost (for classification scores that are posterior probabilities) "quadratic"
Quadratic loss 'mincost'
is appropriate for classification scores that are posterior probabilities. For kernel classification models, logistic regression learners return posterior probabilities as classification scores by default, but SVM learners do not (seepredict
).To specify a custom loss function, use function handle notation. The function must have this form:
lossvalue =
lossfun
(C,S,W,Cost)The output argument
lossvalue
is a scalar.You specify the function name (
lossfun
).C
is ann
-by-K
logical matrix with rows indicating the class to which the corresponding observation belongs.n
is the number of observations inTbl
orX
, andK
is the number of distinct classes (numel(Mdl.ClassNames)
). The column order corresponds to the class order inMdl.ClassNames
. CreateC
by settingC(p,q) = 1
, if observationp
is in classq
, for each row. Set all other elements of rowp
to0
.S
is ann
-by-K
numeric matrix of classification scores. The column order corresponds to the class order inMdl.ClassNames
.S
is a matrix of classification scores, similar to the output ofpredict
.W
is ann
-by-1 numeric vector of observation weights.Cost
is aK
-by-K
numeric matrix of misclassification costs. For example,Cost = ones(K) – eye(K)
specifies a cost of0
for correct classification and1
for misclassification.
Example: 'LossFun',@
lossfun
Data Types: char
| string
| function_handle
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
Weights
is a numeric vector, then the size ofWeights
must be equal to the number of rows inX
orTbl
.If
Weights
is the name of a variable inTbl
, you must specifyWeights
as a character vector or string scalar. For example, if the weights are stored asTbl.W
, then specifyWeights
as'W'
. Otherwise, the software treats all columns ofTbl
, includingTbl.W
, as predictors.
If you supply weights, loss
computes the weighted
classification loss and normalizes the weights to sum up to
the value of the prior probability in the respective class.
Data Types: double
| single
| char
| string
Output Arguments
L
— Classification loss
numeric scalar
Classification loss, returned as a numeric scalar. The
interpretation of L
depends on
Weights
and LossFun
.
More About
Classification Loss
Classification loss functions measure the predictive inaccuracy of classification models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.
Consider the following scenario.
L is the weighted average classification loss.
n is the sample size.
y_{j} is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the
ClassNames
property), respectively.f(X_{j}) is the positive-class classification score for observation (row) j of the predictor data X.
m_{j} = y_{j}f(X_{j}) is the classification score for classifying observation j into the class corresponding to y_{j}. Positive values of m_{j} indicate correct classification and do not contribute much to the average loss. Negative values of m_{j} indicate incorrect classification and contribute significantly to the average loss.
The weight for observation j is w_{j}. The software normalizes the observation weights so that they sum to the corresponding prior class probability stored in the
Prior
property. Therefore,$$\sum _{j=1}^{n}{w}_{j}}=1.$$
Given this scenario, the following table describes the supported loss functions that you can specify by using the LossFun
name-value argument.
Loss Function | Value of LossFun | Equation |
---|---|---|
Binomial deviance | "binodeviance" | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |
Observed misclassification cost | "classifcost" | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}{c}_{{y}_{j}{\widehat{y}}_{j}},$$ where $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the maximal score, and $${c}_{{y}_{j}{\widehat{y}}_{j}}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_{j}$$ when its true class is y_{j}. |
Misclassified rate in decimal | "classiferror" | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\},$$ where I{·} is the indicator function. |
Cross-entropy loss | "crossentropy" |
The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^{n}\frac{{\tilde{w}}_{j}\mathrm{log}({m}_{j})}{Kn}},$$ where the weights $${\tilde{w}}_{j}$$ are normalized to sum to n instead of 1. |
Exponential loss | "exponential" | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |
Hinge loss | "hinge" | $$L={\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$$ |
Logit loss | "logit" | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right)}.$$ |
Minimal expected misclassification cost | "mincost" |
The software computes the weighted minimal expected classification cost using this procedure for observations j = 1,...,n.
The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ |
Quadratic loss | "quadratic" | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}}.$$ |
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"
, "classiferror"
, and
"mincost"
are identical. For a model with a nondefault cost matrix,
the "classifcost"
loss is equivalent to the "mincost"
loss most of the time. These losses can be different if prediction into the class with
maximal posterior probability is different from prediction into the class with minimal
expected cost. Note that "mincost"
is appropriate only if classification
scores are posterior probabilities.
This figure compares the loss functions (except "classifcost"
,
"crossentropy"
, and "mincost"
) over the score
m for one observation. Some functions are normalized to pass through
the point (0,1).
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
The
loss
function supports tall arrays with the following usage
notes and limitations:
loss
does not support talltable
data.
For more information, see Tall Arrays.
Version History
Introduced in R2017bR2022a: loss
returns a different value for a model with a nondefault cost matrix
If you specify a nondefault cost matrix when you train the input model object, the loss
function returns a different value compared to previous releases.
The loss
function uses the prior
probabilities stored in the Prior
property to normalize the observation
weights of the input data. Also, the function uses the cost matrix stored in the
Cost
property if you specify the LossFun
name-value
argument as "classifcost"
or "mincost"
. The way the
function uses the Prior
and Cost
property values has not
changed. However, the property values stored in the input model object have changed for a model
with a nondefault cost matrix, so the function might return a different value.
For details about the property value change, see Cost property stores the user-specified cost matrix.
If you want the software to handle the cost matrix, prior
probabilities, and observation weights in the same way as in previous releases, adjust the prior
probabilities and observation weights for the nondefault cost matrix, as described in Adjust Prior Probabilities and Observation Weights for Misclassification Cost Matrix. Then, when you train a
classification model, specify the adjusted prior probabilities and observation weights by using
the Prior
and Weights
name-value arguments, respectively,
and use the default cost matrix.
R2022a: loss
can return NaN for predictor data with missing values
The loss
function no longer omits an observation with a
NaN score when computing the weighted average classification loss. Therefore,
loss
can now return NaN when the predictor data
X
or the predictor variables in Tbl
contain any missing values, and the name-value argument LossFun
is
not specified as "classifcost"
, "classiferror"
, or
"mincost"
. In most cases, if the test set observations do not
contain missing predictors, the loss
function does not
return NaN.
This change improves the automatic selection of a classification model when you use
fitcauto
.
Before this change, the software might select a model (expected to best classify new
data) with few non-NaN predictors.
If loss
in your code returns NaN, you can update your code
to avoid this result by doing one of the following:
Remove or replace the missing values by using
rmmissing
orfillmissing
, respectively.Specify the name-value argument
LossFun
as"classifcost"
,"classiferror"
, or"mincost"
.
The following table shows the classification models for which the
loss
object function might return NaN. For more details,
see the Compatibility Considerations for each loss
function.
Model Type | Full or Compact Model Object | loss Object
Function |
---|---|---|
Discriminant analysis classification model | ClassificationDiscriminant , CompactClassificationDiscriminant | loss |
Ensemble of learners for classification | ClassificationEnsemble , CompactClassificationEnsemble | loss |
Gaussian kernel classification model | ClassificationKernel | loss |
k-nearest neighbor classification model | ClassificationKNN | loss |
Linear classification model | ClassificationLinear | loss |
Neural network classification model | ClassificationNeuralNetwork , CompactClassificationNeuralNetwork | loss |
Support vector machine (SVM) classification model | loss |
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