Class: ClassificationDiscriminant
Classification error by resubstitution
L = resubLoss(obj)
L = resubLoss(obj,Name,Value)
returns
the resubstitution loss, meaning the loss computed for the data that L
= resubLoss(obj
)fitcdiscr
used to create obj
.
returns
loss statistics with additional options specified by one or more L
= resubLoss(obj
,Name,Value
)Name,Value
pair
arguments.

Discriminant analysis classifier, produced using 
Specify optional
commaseparated 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
.
'LossFun'
— Loss function'classiferror'
(default)  'binodeviance'
 'exponential'
 'hinge'
 'logit'
 'mincost'
 'quadratic'
 function handleLoss function, specified as the commaseparated pair consisting
of 'LossFun'
and a builtin, lossfunction 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 
'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.
Discriminant analysis models return posterior probabilities
as classification scores by default (see predict
).
Specify your own function using function handle notation.
Suppose that n
be the number of observations
in X
and K
be the number of
distinct classes (numel(obj.ClassNames)
). 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
byK
logical
matrix with rows indicating which class the corresponding observation
belongs. The column order corresponds to the class order in obj.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
byK
numeric
matrix of classification scores. The column order corresponds to the
class order in obj.ClassNames
. S
is
a matrix of classification scores, similar to the output of predict
.
W
is an n
by1
numeric vector of observation weights. If you pass W
,
the software normalizes them to sum to 1
.
Cost
is a KbyK
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.
Data Types: char
 string
 function_handle

Classification error, a scalar. The meaning of the error depends
on the values in 
Compute the resubstituted classification error for the Fisher iris data:
load fisheriris obj = fitcdiscr(meas,species); L = resubLoss(obj) L = 0.0200
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'
namevalue 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 posterior probability that a point z belongs to class j is the product of the prior probability and the multivariate normal density. The density function of the multivariate normal with mean μ_{j} and covariance Σ_{j} at a point z is
$$P\left(xk\right)=\frac{1}{{\left(2\pi \left{\Sigma}_{k}\right\right)}^{1/2}}\mathrm{exp}\left(\frac{1}{2}{\left(x{\mu}_{k}\right)}^{T}{\Sigma}_{k}^{1}\left(x{\mu}_{k}\right)\right),$$
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
$$\widehat{P}\left(kx\right)=\frac{P\left(xk\right)P\left(k\right)}{P\left(x\right)},$$
where P(x) is a normalization constant, the sum over k of P(xk)P(k).
The prior probability is one of three choices:
'uniform'
— The prior probability
of class k
is one over the total number of classes.
'empirical'
— The prior
probability of class k
is the number of training
samples of class k
divided by the total number
of training samples.
Custom — The prior probability of class k
is
the k
th element of the prior
vector.
See fitcdiscr
.
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.
The matrix of expected costs per observation is defined in Cost.
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