Loss of k-nearest neighbor classifier by resubstitution
returns the classification loss by resubstitution,
which is the loss computed for the data used by L
= resubLoss(mdl
)fitcknn
to create mdl
.
The classification loss (L
) is a numeric scalar, whose
interpretation depends on the loss function and the observation weights in
mdl
.
Create a k-nearest neighbor classifier for the Fisher iris data, where $$k$$ = 5.
Load the Fisher iris data set.
load fisheriris
Create a classifier for five nearest neighbors.
mdl = fitcknn(meas,species,'NumNeighbors',5);
Examine the resubstitution loss of the classifier.
L = resubLoss(mdl)
L = 0.0333
The classifier predicts incorrect classifications for 1/30 of its training data.
mdl
— k-nearest neighbor classifier modelClassificationKNN
objectk-nearest neighbor classifier model, specified as a
ClassificationKNN
object.
lossfun
— Loss function'mincost'
(default) | 'binodeviance'
| 'classiferror'
| 'exponential'
| 'hinge'
| 'logit'
| 'quadratic'
| function handleLoss function, specified as a built-in loss function name or a function handle.
The following table lists the available loss functions.
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. By
default, k-nearest neighbor models return
posterior probabilities as classification scores (see predict
).
You can specify a function handle for a custom loss function
using @
(for example,
@lossfun
). Let n be
the number of observations in X
and
K be the number of distinct classes
(numel(mdl.ClassNames)
). Your custom loss
function must have this form:
function lossvalue = lossfun(C,S,W,Cost)
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
. The argument
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.
The output argument lossvalue
is a scalar.
For more details on loss functions, see Classification Loss.
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].
Two costs are associated with KNN classification: the true misclassification cost per class and the expected misclassification cost per observation.
You can set the true misclassification cost per class by using the 'Cost'
name-value pair argument when you run fitcknn
. The value Cost(i,j)
is the cost of classifying
an observation into class j
if its true class is i
. By
default, Cost(i,j) = 1
if i ~= j
, and
Cost(i,j) = 0
if i = j
. In other words, the cost
is 0
for correct classification and 1
for incorrect
classification.
Two costs are associated with KNN classification: the true misclassification cost per class
and the expected misclassification cost per observation. The third output of resubPredict
is the expected misclassification cost per
observation.
Suppose you have Nobs
observations that you classified with a trained
classifier mdl
, and you have K
classes. The
command
[label,score,cost] = resubPredict(mdl)
returns a matrix cost
of size
Nobs
-by-K
, among other outputs. Each row of the
cost
matrix contains the expected (average) cost of classifying the
observation into each of the K
classes. cost(n,j)
is
$$\sum _{i=1}^{K}\widehat{P}\left(i|X(n)\right)C\left(j|i\right)},$$
where
K is the number of classes.
$$\widehat{P}\left(i|X(n)\right)$$ is the posterior probability of class i for observation X(n).
$$C\left(j|i\right)$$ is the true misclassification cost of classifying an observation as j when its true class is i.
ClassificationKNN
| fitcknn
| resubEdge
| resubMargin
| resubPredict
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