Classification loss for Gaussian kernel classification model
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.
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
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
Mdl
— Binary kernel classification modelClassificationKernel
model objectBinary kernel classification model, specified as a ClassificationKernel
model object. You can create a
ClassificationKernel
model object using fitckernel
.
Y
— Class labelsClass 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 of Mdl.ClassNames
. (The software treats string arrays as cell arrays of character
vectors.)
The distinct classes in Y
must
be a subset of Mdl.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 in X
or
Tbl
.
Data Types: categorical
| char
| string
| logical
| single
| double
| cell
Tbl
— Sample dataSample 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 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 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
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
.
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'
| 'exponential'
| 'hinge'
| 'logit'
| 'mincost'
| 'quadratic'
| function handleLoss 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 |
'classiferror' | Classification error |
'exponential' | Exponential |
'hinge' | Hinge |
'logit' | Logistic |
'mincost' | Minimal expected misclassification cost (for classification scores that are posterior probabilities) |
'quadratic' | Quadratic |
'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 (see predict
).
Specify your own function by using function handle notation.
Let n
be the number of observations in X
and
K
be the number of distinct
classes (numel(Mdl.ClassNames)
, where
Mdl
is the input model). Your
function must have this signature:
lossvalue = lossfun
(C,S,W,Cost)
The output argument
lossvalue
is a scalar.
You choose 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 classification scores. The
column order corresponds to the class order in
Mdl.ClassNames
.
S
is a matrix of classification
scores, similar to the output of predict
.
W
is an
n
-by-1 numeric vector of
observation weights. If you pass
W
, the software normalizes the
weights 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.
Example: 'LossFun',@
lossfun
Data Types: char
| string
| function_handle
'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 Weights
is a numeric vector, then the
size of Weights
must be equal to the number
of rows in X
or
Tbl
.
If Weights
is the name of a variable in
Tbl
, you must specify
Weights
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 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
L
— Classification lossClassification loss, returned as a numeric scalar. The
interpretation of L
depends on
Weights
and LossFun
.
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.
Suppose the following:
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, respectively.
f(X_{j}) is the raw classification score for the transformed observation (row) j of the predictor data X using feature expansion.
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 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. The software also normalizes the prior probabilities so that they sum to 1. Therefore,
$$\sum _{j=1}^{n}{w}_{j}}=1.$$
This table describes the supported loss functions that you can specify by using the
'LossFun'
name-value pair argument.
Loss Function | Value of LossFun | Equation |
---|---|---|
Binomial deviance | 'binodeviance' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |
Exponential loss | 'exponential' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |
Classification error | 'classiferror' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$ The classification error is the weighted fraction of misclassified observations where $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function. |
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 cost | 'mincost' | The software computes the weighted minimal cost using this procedure for observations j = 1,...,n.
The weighted, average, minimum cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ |
Quadratic loss | 'quadratic' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}}.$$ |
This figure compares the loss functions (except minimal cost) for one observation over m. Some functions are normalized to pass through [0,1].
Usage notes and limitations:
loss
does not support tall table
data.
For more information, see Tall Arrays.
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