resubLoss
Resubstitution classification loss for discriminant analysis classifier
Description
Examples
Resubstituted Classification Error
Compute the resubstituted classification error for the Fisher iris data.
Create a classification model for the Fisher iris data.
load fisheriris
mdl = fitcdiscr(meas,species);
Compute the resubstituted classification error.
L = resubLoss(mdl)
L = 0.0200
Input Arguments
mdl
— Trained discriminant analysis classifier
ClassificationDiscriminant
model object
Trained discriminant analysis classifier, specified as a ClassificationDiscriminant
model object. To create a discriminant analysis
classifier, use fitcdiscr
.
lossf
— Loss function
"mincost"
(default) | "binodeviance"
| "classifcost"
| "classiferror"
| "exponential"
| "hinge"
| "logit"
| "quadratic"
| function handle
Loss function, specified as a built-in loss function name or function handle.
The following table lists the available loss functions. Specify one using the corresponding character vector or string scalar.
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. Discriminant analysis models return posterior probabilities as classification scores by default (seepredict
).Specify your own function using function handle notation.
Suppose that
n
be the number of observations inX
andK
be the number of distinct classes (numel(mdl.ClassNames)
). Your function must have this signaturewhere:lossvalue =
lossfun
(C,S,W,Cost)The output argument
lossvalue
is a scalar.You choose the function name (
lossfun
).C
is ann
-by-K
logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order inmdl.ClassNames
.Construct
C
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. If you passW
, the software normalizes them to sum to1
.Cost
is a K-by-K
numeric matrix of misclassification costs. For example,Cost = ones(K) - eye(K)
specifies a cost of0
for correct classification, and1
for misclassification.
Specify your function using
LossFun=@
.lossfun
For more details on loss functions, see Classification Loss.
Data Types: char
| string
| function_handle
Output Arguments
L
— Classification error
number
Classification error, returned as a number. The meaning of the error depends on the
values in weights
and lossfun
. See Classification Loss.
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.
For binary classification:
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.
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 theClassNames
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 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).
Posterior Probability
The posterior probability that a point x belongs to class k is the product of the prior probability and the multivariate normal density. The density function of the multivariate normal with 1-by-d mean μ_{k} and d-by-d covariance Σ_{k} at a 1-by-d point x is
$$P\left(x|k\right)=\frac{1}{{\left({\left(2\pi \right)}^{d}\left|{\Sigma}_{k}\right|\right)}^{1/2}}\mathrm{exp}\left(-\frac{1}{2}\left(x-{\mu}_{k}\right){\Sigma}_{k}^{-1}{\left(x-{\mu}_{k}\right)}^{T}\right),$$
where $$\left|{\Sigma}_{k}\right|$$ is the determinant of Σ_{k}, and $${\Sigma}_{k}^{-1}$$ is the inverse matrix.
Let P(k) represent the prior probability of class k. Then the posterior probability that an observation x is of class k is
$$\widehat{P}\left(k|x\right)=\frac{P\left(x|k\right)P\left(k\right)}{P\left(x\right)},$$
where P(x) is a normalization constant, the sum over k of P(x|k)P(k).
Prior Probability
The prior probability is one of three choices:
'uniform'
— The prior probability of classk
is one over the total number of classes.'empirical'
— The prior probability of classk
is the number of training samples of classk
divided by the total number of training samples.Custom — The prior probability of class
k
is thek
th element of theprior
vector. Seefitcdiscr
.
After creating a classification model (Mdl
)
you can set the prior using dot notation:
Mdl.Prior = v;
where v
is a vector of positive elements
representing the frequency with which each element occurs. You do
not need to retrain the classifier when you set a new prior.
Cost
The matrix of expected costs per observation is defined in Cost.
Version History
Introduced in R2011bR2023b: Observations with missing predictor values are used in resubstitution and cross-validation computations
Starting in R2023b, the following classification model object functions use observations with missing predictor values as part of resubstitution ("resub") and cross-validation ("kfold") computations for classification edges, losses, margins, and predictions.
In previous releases, the software omitted observations with missing predictor values from the resubstitution and cross-validation computations.
See Also
Classes
Functions
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