loss
Classification loss for generalized additive model (GAM)
Description
returns the Classification Loss (L
= loss(Mdl
,Tbl
,ResponseVarName
)L
),
a scalar representing how well the generalized additive model Mdl
classifies the predictor data in Tbl
compared to the true class labels
in Tbl.ResponseVarName
.
The interpretation of L
depends on the loss function
('LossFun'
) and weighting scheme ('Weights'
). In
general, better classifiers yield smaller classification loss values. The default
'LossFun'
value is 'classiferror'
(misclassification rate in decimal).
specifies options using one or more name-value arguments in addition to any of the input
argument combinations in previous syntaxes. For example,
L
= loss(___,Name,Value
)'LossFun','mincost'
sets the loss function to the minimal expected
misclassification cost function.
Examples
Determine Test Sample Classification Loss
Determine the test sample classification error (loss) of a generalized additive model. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.
Load the ionosphere
data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b'
) or good ('g'
).
load ionosphere
Randomly partition observations into a training set and a test set with stratification, using the class information in Y
. Specify a 30% holdout sample for testing.
rng('default') % For reproducibility cv = cvpartition(Y,'HoldOut',0.30);
Extract the training and test indices.
trainInds = training(cv); testInds = test(cv);
Specify the training and test data sets.
XTrain = X(trainInds,:); YTrain = Y(trainInds); XTest = X(testInds,:); YTest = Y(testInds);
Train a GAM using the predictors XTrain
and class labels YTrain
. A recommended practice is to specify the class names.
Mdl = fitcgam(XTrain,YTrain,'ClassNames',{'b','g'});
Mdl
is a ClassificationGAM
model object.
Determine how well the algorithm generalizes by estimating the test sample classification error. By default, the loss
function of ClassificationGAM
estimates classification error by using the 'classiferror'
loss (misclassification rate in decimal).
L = loss(Mdl,XTest,YTest)
L = 0.1052
The trained classifier misclassifies approximately 11% of the test sample.
Compare GAMs by Examining Classification Loss
Train a generalized additive model (GAM) that contains both linear and interaction terms for predictors, and estimate the classification loss with and without interaction terms. Specify whether to include interaction terms when estimating the classification loss for training and test data.
Load the ionosphere
data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b'
) or good ('g'
).
load ionosphere
Partition the data set into two sets: one containing training data, and the other containing new, unobserved test data. Reserve 50 observations for the new test data set.
rng('default') % For reproducibility n = size(X,1); newInds = randsample(n,50); inds = ~ismember(1:n,newInds); XNew = X(newInds,:); YNew = Y(newInds);
Train a GAM using the predictors X
and class labels Y
. A recommended practice is to specify the class names. Specify to include the 10 most important interaction terms.
Mdl = fitcgam(X(inds,:),Y(inds),'ClassNames',{'b','g'},'Interactions',10)
Mdl = ClassificationGAM ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'b' 'g'} ScoreTransform: 'logit' Intercept: 2.0026 Interactions: [10x2 double] NumObservations: 301 Properties, Methods
Mdl
is a ClassificationGAM
model object.
Compute the resubstitution classification loss both with and without interaction terms in Mdl
. To exclude interaction terms, specify 'IncludeInteractions',false
.
resubl = resubLoss(Mdl)
resubl = 0
resubl_nointeraction = resubLoss(Mdl,'IncludeInteractions',false)
resubl_nointeraction = 0
Estimate the classification loss both with and without interaction terms in Mdl
.
l = loss(Mdl,XNew,YNew)
l = 0.0615
l_nointeraction = loss(Mdl,XNew,YNew,'IncludeInteractions',false)
l_nointeraction = 0.0615
Including interaction terms does not change the classification loss for Mdl
. The trained model classifies all training samples correctly and misclassifies approximately 6% of the test samples.
Input Arguments
Mdl
— Generalized additive model
ClassificationGAM
model object | CompactClassificationGAM
model object
Generalized additive model, specified as a ClassificationGAM
or CompactClassificationGAM
model object.
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. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.
Tbl
must contain all the predictors used to train
Mdl
. Optionally, Tbl
can contain a column
for the response variable and a column for the observation weights.
The response variable must have the same data type as
Mdl.Y
. (The software treats string arrays as cell arrays of character vectors.) If the response variable inTbl
has the same name as the response variable used to trainMdl
, then you do not need to specifyResponseVarName
.The weight values must be a numeric vector. You must specify the observation weights in
Tbl
by using'Weights'
.
If you trained Mdl
using sample data contained in a table, then the input data for loss
must also be in a table.
Data Types: table
ResponseVarName
— Response variable name
name of variable in Tbl
Response variable name, specified as a character vector or string scalar containing the name
of the response variable in Tbl
. For example, if the response
variable Y
is stored in Tbl.Y
, then specify it as
'Y'
.
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, a logical or
numeric vector, or a cell array of character vectors. Each row of Y
represents the classification of the corresponding row of X
or
Tbl
.
Y
must have the same data type as Mdl.Y
. (The software treats string arrays as cell arrays of character
vectors.)
Data Types: single
| double
| categorical
| logical
| char
| string
| cell
X
— Predictor data
numeric matrix
Predictor data, specified as a numeric matrix. Each row of X
corresponds to one observation, and each column corresponds to one predictor variable.
If you trained Mdl
using sample data contained in a matrix, then the input data for loss
must also be in a matrix.
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: 'IncludeInteractions',false,'Weights',w
specifies to exclude
interaction terms from the model and to use the observation weights
w
.
IncludeInteractions
— Flag to include interaction terms
true
| false
Flag to include interaction terms of the model, specified as true
or
false
.
The default 'IncludeInteractions'
value is true
if Mdl
contains interaction terms. The value must be false
if the model does not contain interaction terms.
Example: 'IncludeInteractions',false
Data Types: logical
LossFun
— Loss function
'binodeviance'
| 'classifcost'
| 'classiferror'
| 'exponential'
| 'hinge'
| 'logit'
| 'mincost'
| 'quadratic'
| function handle
Loss function, specified as a built-in loss function name or a function handle.
The default value is 'mincost'
if the
ScoreTransform
property of the input model object
(
) is
Mdl
.ScoreTransform'logit'
; otherwise, the default value is
'classiferror'
.
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 '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','binodeviance'
Data Types: char
| string
| function_handle
Weights
— Observation weights
ones(size(X,1),1)
(default) | vector of scalar values | name of variable in Tbl
Observation weights, specified as a vector of scalar values or the name of a variable in Tbl
. The software weights the observations in each row of X
or Tbl
with the corresponding value in Weights
. The size of Weights
must equal the number of rows 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 in Tbl.W
, then
specify it as 'W'
.
loss
normalizes the weights in each class to add up to the value of the prior probability of 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.
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.
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 for the LossFun
name-value argument when a GAM object uses
posterior probabilities for classification scores, that is, when the
ScoreTransform
property of the input model object is
'logit'
. Otherwise, the function uses the
"classiferror"
option (misclassified rate in decimal) by default. In
previous releases, the default value was always "classiferror"
.
You do not need to make any changes to your code if you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification). The "mincost"
option is equivalent to the "classiferror"
option for the default cost matrix.
See Also
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