loss
Classification loss for neural network classifier
Description
returns the classification loss
for the trained neural network classifier L
= loss(Mdl
,Tbl
,ResponseVarName
)Mdl
using the predictor
data in table Tbl
and the class labels in the
ResponseVarName
table variable.
L
is returned as a scalar value that represents the
classification error by default.
specifies options using one or more name-value arguments in addition to any of the input
argument combinations in previous syntaxes. For example, you can specify that columns in
the predictor data correspond to observations, specify the loss function, or supply
observation weights.L
= loss(___,Name,Value
)
Note
If the predictor data in X
or Tbl
contains
any missing values and LossFun
is not set to
"classifcost"
, "classiferror"
, or
"mincost"
, the loss
function can
return NaN. For more details, see loss can return NaN for predictor data with missing values.
Examples
Test Set Classification Error of Neural Network
Calculate the test set classification error of a neural network classifier.
Load the patients
data set. Create a table from the data set. Each row corresponds to one patient, and each column corresponds to a diagnostic variable. Use the Smoker
variable as the response variable, and the rest of the variables as predictors.
load patients
tbl = table(Diastolic,Systolic,Gender,Height,Weight,Age,Smoker);
Separate the data into a training set tblTrain
and a test set tblTest
by using a stratified holdout partition. The software reserves approximately 30% of the observations for the test data set and uses the rest of the observations for the training data set.
rng("default") % For reproducibility of the partition c = cvpartition(tbl.Smoker,"Holdout",0.30); trainingIndices = training(c); testIndices = test(c); tblTrain = tbl(trainingIndices,:); tblTest = tbl(testIndices,:);
Train a neural network classifier using the training set. Specify the Smoker
column of tblTrain
as the response variable. Specify to standardize the numeric predictors.
Mdl = fitcnet(tblTrain,"Smoker", ... "Standardize",true);
Calculate the test set classification error. Classification error is the default loss type for neural network classifiers.
testError = loss(Mdl,tblTest,"Smoker")
testError = 0.0671
testAccuracy = 1 - testError
testAccuracy = 0.9329
The neural network model correctly classifies approximately 93% of the test set observations.
Select Features to Include in Neural Network Classifier
Perform feature selection by comparing test set classification margins, edges, errors, and predictions. Compare the test set metrics for a model trained using all the predictors to the test set metrics for a model trained using only a subset of the predictors.
Load the sample file fisheriris.csv
, which contains iris data including sepal length, sepal width, petal length, petal width, and species type. Read the file into a table.
fishertable = readtable('fisheriris.csv');
Separate the data into a training set trainTbl
and a test set testTbl
by using a stratified holdout partition. The software reserves approximately 30% of the observations for the test data set and uses the rest of the observations for the training data set.
rng("default") c = cvpartition(fishertable.Species,"Holdout",0.3); trainTbl = fishertable(training(c),:); testTbl = fishertable(test(c),:);
Train one neural network classifier using all the predictors in the training set, and train another classifier using all the predictors except PetalWidth
. For both models, specify Species
as the response variable, and standardize the predictors.
allMdl = fitcnet(trainTbl,"Species","Standardize",true); subsetMdl = fitcnet(trainTbl,"Species ~ SepalLength + SepalWidth + PetalLength", ... "Standardize",true);
Calculate the test set classification margins for the two models. Because the test set includes only 45 observations, display the margins using bar graphs.
For each observation, the classification margin is the difference between the classification score for the true class and the maximal score for the false classes. Because neural network classifiers return classification scores that are posterior probabilities, margin values close to 1 indicate confident classifications and negative margin values indicate misclassifications.
tiledlayout(2,1) % Top axes ax1 = nexttile; allMargins = margin(allMdl,testTbl); bar(ax1,allMargins) xlabel(ax1,"Observation") ylabel(ax1,"Margin") title(ax1,"All Predictors") % Bottom axes ax2 = nexttile; subsetMargins = margin(subsetMdl,testTbl); bar(ax2,subsetMargins) xlabel(ax2,"Observation") ylabel(ax2,"Margin") title(ax2,"Subset of Predictors")
Compare the test set classification edge, or mean of the classification margins, of the two models.
allEdge = edge(allMdl,testTbl)
allEdge = 0.8198
subsetEdge = edge(subsetMdl,testTbl)
subsetEdge = 0.9556
Based on the test set classification margins and edges, the model trained on a subset of the predictors seems to outperform the model trained on all the predictors.
Compare the test set classification error of the two models.
allError = loss(allMdl,testTbl); allAccuracy = 1-allError
allAccuracy = 0.9111
subsetError = loss(subsetMdl,testTbl); subsetAccuracy = 1-subsetError
subsetAccuracy = 0.9778
Again, the model trained using only a subset of the predictors seems to perform better than the model trained using all the predictors.
Visualize the test set classification results using confusion matrices.
allLabels = predict(allMdl,testTbl);
figure
confusionchart(testTbl.Species,allLabels)
title("All Predictors")
subsetLabels = predict(subsetMdl,testTbl);
figure
confusionchart(testTbl.Species,subsetLabels)
title("Subset of Predictors")
The model trained using all the predictors misclassifies four of the test set observations. The model trained using a subset of the predictors misclassifies only one of the test set observations.
Given the test set performance of the two models, consider using the model trained using all the predictors except PetalWidth
.
Input Arguments
Mdl
— Trained neural network classifier
ClassificationNeuralNetwork
model object | CompactClassificationNeuralNetwork
model object
Trained neural network classifier, specified as a ClassificationNeuralNetwork
model object or CompactClassificationNeuralNetwork
model object returned by fitcnet
or
compact
,
respectively.
Tbl
— Sample data
table
Sample 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 an additional column for the response variable. Tbl
must contain all of the predictors used to train Mdl
. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.
If
Tbl
contains the response variable used to trainMdl
, then you do not need to specifyResponseVarName
orY
.If you trained
Mdl
using sample data contained in a table, then the input data forloss
must also be in a table.If you set
'Standardize',true
infitcnet
when trainingMdl
, then the software standardizes the numeric columns of the predictor data using the corresponding means and standard deviations.
Data Types: table
ResponseVarName
— Response variable name
name of variable in Tbl
Response variable name, specified as the name of a variable in Tbl
. If Tbl
contains the response variable used to train Mdl
, then you do not need to specify ResponseVarName
.
If you specify ResponseVarName
, then you must specify it as a character
vector or string scalar. For example, if the response variable is stored as
Tbl.Y
, then specify ResponseVarName
as
'Y'
. Otherwise, the software treats all columns of
Tbl
, including Tbl.Y
, as predictors.
The response variable must be a categorical, character, or string array; a logical or numeric vector; or a 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
Y
— Class labels
categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors
Class labels, specified as a categorical, character, or string array; logical or numeric vector; or cell array of character vectors.
The data type of
Y
must be the same as the data type ofMdl.ClassNames
. (The software treats string arrays as cell arrays of character vectors.)The distinct classes in
Y
must be a subset ofMdl.ClassNames
.If
Y
is a character array, then each element must correspond to one row of the array.The length of
Y
must be equal to the number of observations inX
orTbl
.
Data Types: categorical
| char
| string
| logical
| single
| double
| cell
X
— Predictor data
numeric matrix
Predictor data, specified as a numeric matrix. By default,
loss
assumes that each row of X
corresponds to one observation, and each column corresponds to one predictor
variable.
Note
If you orient your predictor matrix so that observations correspond to columns and
specify 'ObservationsIn','columns'
, then you might experience a
significant reduction in computation time.
The length of Y
and the number of observations in X
must be equal.
If you set 'Standardize',true
in fitcnet
when training Mdl
, then the software standardizes the numeric columns of the predictor data using the corresponding means and standard deviations.
Data Types: single
| double
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: loss(Mdl,Tbl,"Response","LossFun","crossentropy")
specifies
to compute the cross-entropy loss for the model Mdl
.
LossFun
— Loss function
'mincost'
(default) | 'binodeviance'
| 'classifcost'
| 'classiferror'
| 'crossentropy'
| 'exponential'
| 'hinge'
| 'logit'
| 'quadratic'
| function handle
Loss 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 'classifcost'
Observed misclassification cost 'classiferror'
Misclassified rate in decimal 'crossentropy'
Cross-entropy loss (for neural networks only) 'exponential'
Exponential loss 'hinge'
Hinge loss 'logit'
Logistic loss 'mincost'
Minimal expected misclassification cost (for classification scores that are posterior probabilities) 'quadratic'
Quadratic loss For more details on loss functions, see Classification Loss.
To specify a custom loss function, use function handle notation. The function must have this form:
lossvalue =
lossfun
(C,S,W,Cost)The output argument
lossvalue
is a scalar.You specify the function name (
lossfun
).C
is ann
-by-K
logical matrix with rows indicating the class to which the corresponding observation belongs.n
is the number of observations inTbl
orX
, andK
is the number of distinct classes (numel(Mdl.ClassNames)
. The column order corresponds to the class order inMdl.ClassNames
. CreateC
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.Cost
is aK
-by-K
numeric matrix of misclassification costs. For example,Cost = ones(K) – eye(K)
specifies a cost of0
for correct classification and1
for misclassification.
Example: 'LossFun','crossentropy'
Data Types: char
| string
| function_handle
ObservationsIn
— Predictor data observation dimension
'rows'
(default) | 'columns'
Predictor data observation dimension, specified as 'rows'
or
'columns'
.
Note
If you orient your predictor matrix so that observations correspond to columns and
specify 'ObservationsIn','columns'
, then you might experience a
significant reduction in computation time. You cannot specify
'ObservationsIn','columns'
for predictor data in a
table.
Data Types: char
| string
Weights
— Observation weights
nonnegative numeric vector | name of variable in Tbl
Observation weights, specified as a nonnegative numeric vector or the name of a
variable in Tbl
. The software weights each observation in
X
or Tbl
with the corresponding value in
Weights
. The length of Weights
must equal
the number of observations in X
or
Tbl
.
If you specify the input data as a table Tbl
, then
Weights
can be the name of a variable in
Tbl
that contains a numeric vector. In this case, you must
specify Weights
as a character vector or string scalar. For
example, if the weights vector W
is stored as
Tbl.W
, then specify it as 'W'
.
By default, Weights
is ones(n,1)
, where
n
is the number of observations in X
or
Tbl
.
If you supply weights, then loss
computes the weighted
classification loss and normalizes weights to sum to the value of the prior
probability in the respective class.
Data Types: single
| double
| char
| string
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).
Version History
Introduced in R2021aR2022a: Default LossFun
value has changed
Behavior changed in R2022a
Starting in R2022a, the loss
function uses the
"mincost"
option (minimal expected misclassification cost) as the
default value for the LossFun
name-value argument. The
"mincost"
option is appropriate when classification scores are
posterior probabilities. In previous releases, the default value was
"classiferror"
.
You do not need to make any changes to your code for the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification).
R2022a: loss
can return NaN for predictor data with missing values
Behavior changed in R2022a
The loss
function no longer omits an observation with a
NaN score when computing the weighted average classification loss. Therefore,
loss
can now return NaN when the predictor data
X
or the predictor variables in Tbl
contain any missing values, and the name-value argument LossFun
is
not specified as "classifcost"
, "classiferror"
, or
"mincost"
. In most cases, if the test set observations do not
contain missing predictors, the loss
function does not
return NaN.
This change improves the automatic selection of a classification model when you use
fitcauto
.
Before this change, the software might select a model (expected to best classify new
data) with few non-NaN predictors.
If loss
in your code returns NaN, you can update your code
to avoid this result by doing one of the following:
Remove or replace the missing values by using
rmmissing
orfillmissing
, respectively.Specify the name-value argument
LossFun
as"classifcost"
,"classiferror"
, or"mincost"
.
The following table shows the classification models for which the
loss
object function might return NaN. For more details,
see the Compatibility Considerations for each loss
function.
Model Type | Full or Compact Model Object | loss Object
Function |
---|---|---|
Discriminant analysis classification model | ClassificationDiscriminant , CompactClassificationDiscriminant | loss |
Ensemble of learners for classification | ClassificationEnsemble , CompactClassificationEnsemble | loss |
Gaussian kernel classification model | ClassificationKernel | loss |
k-nearest neighbor classification model | ClassificationKNN | loss |
Linear classification model | ClassificationLinear | loss |
Neural network classification model | ClassificationNeuralNetwork , CompactClassificationNeuralNetwork | loss |
Support vector machine (SVM) classification model | loss |
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