Find classification error for support vector machine (SVM) classifier
returns the classification error (see Classification Loss), a
scalar representing how well the trained support vector machine (SVM) classifier
(L
= loss(SVMModel
,TBL
,ResponseVarName
)SVMModel
) classifies the predictor data in table
TBL
compared to the true class labels in
TBL.ResponseVarName
.
loss
normalizes the class probabilities in
TBL.ResponseVarName
to the prior class probabilities that
fitcsvm
used for training, stored
in the Prior
property of SVMModel
.
The classification loss (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.
specifies options using one or more name-value pair arguments in addition to the
input arguments in previous syntaxes. For example, you can specify the loss
function and the classification weights.L
= loss(___,Name,Value
)
Load the ionosphere
data set.
load ionosphere rng(1); % For reproducibility
Train an SVM classifier. Specify a 15% holdout sample for testing, standardize the data, and specify that 'g'
is the positive class.
CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},... 'Standardize',true); CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier testInds = test(CVSVMModel.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:);
CVSVMModel
is a ClassificationPartitionedModel
classifier. It contains the property Trained
, which is a 1-by-1 cell array holding a CompactClassificationSVM
classifier that the software trained using the training set.
Determine how well the algorithm generalizes by estimating the test sample classification error.
L = loss(CompactSVMModel,XTest,YTest)
L = 0.0787
The SVM classifier misclassifies approximately 8% of the test sample.
Load the ionosphere
data set.
load ionosphere rng(1); % For reproducibility
Train an SVM classifier. Specify a 15% holdout sample for testing, standardize the data, and specify that 'g'
is the positive class.
CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},... 'Standardize',true); CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier testInds = test(CVSVMModel.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:);
CVSVMModel
is a ClassificationPartitionedModel
classifier. It contains the property Trained
, which is a 1-by-1 cell array holding a CompactClassificationSVM
classifier that the software trained using the training set.
Determine how well the algorithm generalizes by estimating the test sample hinge loss.
L = loss(CompactSVMModel,XTest,YTest,'LossFun','hinge')
L = 0.2998
The hinge loss is approximately 0.3. Classifiers with hinge losses close to 0 are preferred.
SVMModel
— SVM classification modelClassificationSVM
model object | CompactClassificationSVM
model objectSVM classification model, specified as a ClassificationSVM
model
object or CompactClassificationSVM
model
object returned by fitcsvm
or compact
,
respectively.
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 of the
predictors used to train SVMModel
.
Multicolumn variables and cell arrays other than cell arrays of
character vectors are not allowed.
If TBL
contains the response variable
used to train SVMModel
, then you do not need
to specify ResponseVarName
or Y
.
If you trained SVMModel
using sample data contained in a table, then the
input data for loss
must also be in
a table.
If you set 'Standardize',true
in fitcsvm
when training SVMModel
, then the software
standardizes the columns of the predictor data using the
corresponding means in SVMModel.Mu
and the
standard deviations in SVMModel.Sigma
.
Data Types: table
ResponseVarName
— Response variable nameTBL
Response variable name, specified as the name of a variable in
TBL
.
You must specify ResponseVarName
as a character vector
or string scalar. For example, if the response variable Y
is stored as TBL.Y
, then specify
ResponseVarName
as 'Y'
.
Otherwise, the software treats all columns of TBL
,
including Y
, as predictors when training the
model.
The response variable must be a categorical, character, or string array, logical or numeric vector, or 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 (also known as an instance
or example), and each column corresponds to one variable (also known as a feature). The
variables in the columns of X
must be the same as the variables
that trained the SVMModel
classifier.
The length of Y
and the number of rows in X
must be
equal.
If you set 'Standardize',true
in fitcsvm
to train SVMModel
, then the software
standardizes the columns of X
using the corresponding means in
SVMModel.Mu
and the standard deviations in
SVMModel.Sigma
.
Data Types: double
| single
Y
— Class labelsClass labels, specified as a categorical, character, or string array, logical or numeric
vector, or cell array of character vectors. Y
must be the same as the data type of
SVMModel.ClassNames
. (The software treats string arrays as cell arrays of character
vectors.)
The length of Y
must equal the number of rows in TBL
or the number of rows in X
.
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(SVMModel,TBL,Y,'Weights',W)
weighs the
observations in each row of TBL
using the corresponding weight in
each row of the variable W
in
TBL
.'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 character vector or string scalar.
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. You
can specify to use posterior probabilities as classification
scores for SVM models by setting
'FitPosterior',true
when you
cross-validate the model using fitcsvm
.
Specify your own function by using function handle notation.
Suppose that n
is the number of
observations in X
, and
K
is the number of distinct classes
(numel(SVMModel.ClassNames)
) used to
create the input model (SVMModel
). 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
SVMModel.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, similar
to the output of predict
. The column order corresponds
to the class order in
SVMModel.ClassNames
.
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.
Specify your function using
'LossFun',@
.lossfun
For more details on loss functions, see Classification Loss.
Example: 'LossFun','binodeviance'
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
. The software weighs the
observations in each row of X
or
TBL
with the corresponding weight in
Weights
.
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, then the software
normalizes Weights
to sum up to the value of the
prior probability in the respective class.
Example: 'Weights','W'
Data Types: single
| double
| char
| string
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.
For binary classification:
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 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.
For algorithms that support multiclass classification (that is, K ≥ 3):
y_{j}^{*}
is a vector of K – 1 zeros, with 1 in the
position corresponding to the true, observed class
y_{j}. For example,
if the true class of the second observation is the third class and
K = 4, then
y^{*}_{2}
= [0 0 1 0]′. The order of the classes corresponds to the order in
the ClassNames
property of the input
model.
f(X_{j})
is the length K vector of class scores for
observation j of the predictor data
X. The order of the scores corresponds to the
order of the classes in the ClassNames
property
of the input model.
m_{j} = y_{j}^{*}′f(X_{j}). Therefore, m_{j} is the scalar classification score that the model predicts for the true, observed class.
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 they sum to 1. 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 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' | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$ It 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' | Minimal cost. 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 'mincost'
) for one
observation over m. Some functions are normalized to pass through [0,1].
The SVM classification score for classifying observation x is the signed distance from x to the decision boundary ranging from -∞ to +∞. A positive score for a class indicates that x is predicted to be in that class. A negative score indicates otherwise.
The positive class classification score $$f(x)$$ is the trained SVM classification function. $$f(x)$$ is also the numerical, predicted response for x, or the score for predicting x into the positive class.
$$f(x)={\displaystyle \sum _{j=1}^{n}{\alpha}_{j}}{y}_{j}G({x}_{j},x)+b,$$
where $$({\alpha}_{1},\mathrm{...},{\alpha}_{n},b)$$ are the estimated SVM parameters, $$G({x}_{j},x)$$ is the dot product in the predictor space between x and the support vectors, and the sum includes the training set observations. The negative class classification score for x, or the score for predicting x into the negative class, is –f(x).
If G(x_{j},x) = x_{j}′x (the linear kernel), then the score function reduces to
$$f\left(x\right)=\left(x/s\right)\prime \beta +b.$$
s is the kernel scale and β is the vector of fitted linear coefficients.
For more details, see Understanding Support Vector Machines.
[1] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, second edition. Springer, New York, 2008.
This function fully supports tall arrays. For more information, see Tall Arrays (MATLAB).
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
ClassificationSVM
| CompactClassificationSVM
| fitcsvm
| predict
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