Find classification loss for support vector machine (SVM) classifier by resubstitution
returns the classification loss by
resubstitution (L), the in-sample classification loss, for the support vector
machine (SVM) classifier L
= resubLoss(SVMModel
)SVMModel
using the training data stored
in SVMModel.X
and the corresponding class labels stored in
SVMModel.Y
.
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.
Load the ionosphere
data set.
load ionosphere
Train an SVM classifier. Standardize the data and specify that 'g'
is the positive class.
SVMModel = fitcsvm(X,Y,'ClassNames',{'b','g'},'Standardize',true);
SVMModel
is a trained ClassificationSVM
classifier.
Estimate the resubstitution loss (that is, the in-sample classification error).
L = resubLoss(SVMModel)
L = 0.0570
The SVM classifier misclassifies 5.7% of the training sample radar returns.
Load the ionosphere
data set.
load ionosphere
Train an SVM classifier. Standardize the data and specify that 'g'
is the positive class.
SVMModel = fitcsvm(X,Y,'ClassNames',{'b','g'},'Standardize',true);
SVMModel
is a trained ClassificationSVM
classifier.
Estimate the in-sample hinge loss.
L = resubLoss(SVMModel,'LossFun','hinge')
L = 0.1603
The hinge loss is 0.1603
. Classifiers with hinge losses close to 0 are preferred.
SVMModel
— Full, trained SVM classifierClassificationSVM
classifierFull, trained SVM classifier, specified as a ClassificationSVM
model trained with fitcsvm
.
lossFun
— Loss function'classiferror'
(default) | 'binodeviance'
| 'exponential'
| 'hinge'
| 'logit'
| 'mincost'
| 'quadratic'
| function handleLoss function, specified as 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
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.
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