ClassificationDiscriminant
Discriminant analysis classification
Description
A ClassificationDiscriminant
object
encapsulates a discriminant analysis classifier, which is a Gaussian mixture model for
data generation. A ClassificationDiscriminant
object
can predict responses for new data using the predict
method. The object contains the data used for training, so can
compute resubstitution predictions.
Creation
Create a ClassificationDiscriminant
object by using fitcdiscr
.
Properties
BetweenSigma
— Betweenclass covariance
square matrix
This property is readonly.
Betweenclass covariance, specified as a p
byp
matrix, where p
is the number of predictors.
Data Types: double
CategoricalPredictors
— Categorical predictor indices
[]
This property is readonly.
Categorical predictor indices, which is always empty ([]
).
ClassNames
— Class names in training data Y
categorical array  cell array of character vectors  character array  string array  logical vector  numeric vector
This property is readonly.
Class names in the training data Y
with duplicates removed.
ClassNames
has the same data type as the data in the argument
Y
in the training data. ClassNames
can have
the following data types:
Categorical array
Cell array of character vectors
Character array
Logical vector
Numeric vector
(The software treats string arrays as cell arrays of character vectors.)
Data Types: single
 double
 logical
 char
 string
 cell
 categorical
Coeffs
— Coefficient matrices
k
byk
structure  []
This property is readonly.
Coefficient matrices, specified as a k
byk
structure, where k
is the number of classes. If fitcdiscr
had the FillCoeffs
namevalue pair set to
'off'
when constructing the classifier, Coeffs
is empty ([]
).
Coeffs(i,j)
contains coefficients of the linear or quadratic
boundaries between classes i
and j
. Fields in
Coeffs(i,j)
:
DiscrimType
Class1
—ClassNames
(i)
Class2
—ClassNames
(j)
Const
— A scalarLinear
— A vector withp
components, wherep
is the number of columns inX
Quadratic
—p
byp
matrix, exists for quadraticDiscrimType
The equation of the boundary between class i
and class
j
is
Const
+ Linear
* x
+ x'
* Quadratic
* x
=
0
,
where x
is a column vector of length p
.
Data Types: struct
Cost
— Cost of classifying a point
square matrix
Cost of classifying a point, specified as a square matrix.
Cost(i,j)
is the cost of classifying a point into class
j
if its true class is i
(the rows correspond
to the true class and the columns correspond to the predicted class). The order of the
rows and columns of Cost
corresponds to the order of the classes in
ClassNames
. The number of rows and columns in
Cost
is the number of unique classes in the response.
Change a Cost
matrix using dot notation: obj.Cost =
costMatrix
.
Data Types: double
Delta
— Delta threshold for a linear discriminant model
nonnegative scalar
Value of the Delta threshold for a linear discriminant model, specified as a
nonnegative scalar. If a coefficient of obj
has magnitude smaller
than Delta
, obj
sets this coefficient to
0
, and so you can eliminate the corresponding predictor from the
model. Set Delta
to a higher value to eliminate more
predictors.
Delta
must be 0
for quadratic discriminant
models.
Change Delta
using dot notation: obj.Delta =
newDelta
.
Data Types: double
DeltaPredictor
— Minimum value of Delta coefficient for predictor to be in model
row vector of length p
This property is readonly.
Minimum value of Delta coefficient for predictor to be in model, specified as a row
vector of length p
, where p
is the number of
predictors in obj
. If
DeltaPredictor(i) < Delta
then coefficient
i
of the model is 0
.
If obj
is a quadratic discriminant model, all elements of
DeltaPredictor
are 0
.
Data Types: double
DiscrimType
— Discriminant type
character vector
Discriminant type, specified as a character vector or string. Available values:
'linear'
'quadratic'
'diagLinear'
'diagQuadratic'
'pseudoLinear'
'pseudoQuadratic'
Change DiscrimType
using dot notation: obj.DiscrimType =
newDiscrimType
. You can change between linear types, or between quadratic
types, but cannot change between linear and quadratic types.
Data Types: char
 string
Gamma
— Gamma regularization parameter
scalar from 0
through 1
Value of the Gamma regularization parameter, specified as a scalar from
0
through 1
. Change Gamma
using dot notation: obj.Gamma = newGamma
.
If you set
1
for linear discriminant, the discriminant sets its type to'diagLinear'
.If you set a value between
MinGamma
and1
for linear discriminant, the discriminant sets its type to'linear'
.You cannot set values below the value of the
MinGamma
property.For quadratic discriminant, you can set either
0
(forDiscrimType
'quadratic'
) or1
(forDiscrimType
'diagQuadratic'
).
Data Types: double
HyperparameterOptimizationResults
— Description of crossvalidation optimization of hyperparameters
BayesianOptimization
object  table of hyperparameters and associated values
This property is readonly.
Description of the crossvalidation optimization of hyperparameters, returned as a
BayesianOptimization
object or a table of
hyperparameters and associated values. Nonempty when the
OptimizeHyperparameters
namevalue pair is nonempty at creation.
Value depends on the setting of the HyperparameterOptimizationOptions
namevalue pair at creation:
'bayesopt'
(default) — Object of classBayesianOptimization
'gridsearch'
or'randomsearch'
— Table of hyperparameters used, observed objective function values (crossvalidation loss), and rank of observations from lowest (best) to highest (worst)
LogDetSigma
— Logarithm of determinant of withinclass covariance matrix
scalar  vector
This property is readonly.
Logarithm of the determinant of the withinclass covariance matrix, returned as a
scalar or vector. The type of LogDetSigma
depends on the discriminant
type:
Scalar for linear discriminant analysis
Vector of length
K
for quadratic discriminant analysis, whereK
is the number of classes
Data Types: double
MinGamma
— Minimal value of the Gamma parameter so that the correlation matrix is invertible
nonnegative scalar
This property is readonly.
Minimal value of the Gamma parameter so that the correlation matrix is
invertible, specified as a nonnegative scalar. If the correlation matrix is
not singular, MinGamma
is 0
.
Data Types: double
ModelParameters
— Parameters used in training the model
DiscriminantParams
object
This property is readonly.
Parameters used in training the model, returned as a
DiscriminantParams
object. The returned parameters
have the following properties.
Property  Value 

DiscrimType 

Gamma  scalar from 0 through
1 
Delta  nonnegative scalar 
FillCoeffs  logical scalar 
SaveMemory  logical scalar 
Version  scalar 
Method  'Discriminant' 
Type  'classification' 
Mu
— Class means
real K
byp
matrix
This property is readonly.
Class means, specified as a K
byp
matrix of
real values. K
is the number of classes, and p
is
the number of predictors. Each row of Mu
represents the mean of the
multivariate normal distribution of the corresponding class. The class indices are in
the ClassNames
attribute.
Data Types: double
NumObservations
— Number of observations in the training data
positive integer
This property is readonly.
Number of observations in the training data, returned as a positive integer.
NumObservations
can be less than the number of rows of input data
when there are missing values in the input data or response data.
Data Types: double
PredictorNames
— Names of predictor variables
cell array
This property is readonly.
Names of predictor variables, returned as a cell array. The names are in the order in
which they appear in the training data X
.
Data Types: cell
Prior
— Prior probabilities for each class
numeric vector
Prior probabilities for each class, returned as a numeric vector. The order of the
elements of Prior
corresponds to the order of the classes in
ClassNames
.
Add or change a Prior
vector using dot notation: obj.Prior
= priorVector
.
Data Types: double
ResponseName
— Name of the response variable Y
character vector
This property is readonly.
Name of the response variable Y
, returned as a character
vector.
Data Types: char
 string
RowsUsed
— Rows of the original predictor data X
used for fitting
logical vector
This property is readonly.
Rows of the original predictor data X
used for fitting, returned as an
n
element logical vector, where n
is the
number of rows of X
. If the software uses all rows of
X
for constructing the object, then RowsUsed
is an empty array ([]
).
Data Types: logical
ScoreTransform
— Score transformation function
name of a builtin function  function handle  'none'
Score transformation function, specified as a character vector or string representing
a builtin transformation function, or as a function handle for transforming scores.
'none'
means no transformation; equivalently,
'none'
means @(x)x
. For a list of builtin
transformation functions and the syntax of custom transformation functions, see
fitcdiscr
.
Implement dot notation to add or change a ScoreTransform
function
using one of the following:
cobj.ScoreTransform = '
function
'cobj.ScoreTransform = @
function
Data Types: char
 string
 function_handle
Sigma
— Withinclass covariance
numeric array
This property is readonly.
Withinclass covariance, returned as a numeric array. The dimensions depend on
DiscrimType
:
'linear'
(default) — Matrix of sizep
byp
, wherep
is the number of predictors'quadratic'
— Array of sizep
byp
byK
, whereK
is the number of classes'diagLinear'
— Row vector of lengthp
'diagQuadratic'
— Array of size1
byp
byK
'pseudoLinear'
— Matrix of sizep
byp
'pseudoQuadratic'
— Array of sizep
byp
byK
Data Types: double
W
— Scaled observation weights
numeric vector of length n
This property is readonly.
Scaled observation weights
, returned as a numeric
vector of length n
, where n
is the
number of rows in X
.
Data Types: double
X
— Predictor values
real matrix
This property is readonly.
Predictor values, returned as a real matrix. Each column of
X
represents one predictor (variable), and each row
represents one observation.
Data Types: single
 double
Xcentered
— X
data with class means subtracted
real matrix
This property is readonly.
X
data with class means subtracted, returned as a real
matrix. If Y(i)
is of class j
,
Xcentered(i,:)
= X(i,:)
–
Mu(j,:)
,
where Mu
is the class mean property.
Data Types: single
 double
Y
— Row classifications
classification variables
This property is readonly.
Row classifications, returned as a categorical array, cell array of
character vectors, character array, logical vector, or numeric vector with
the same number of rows as X
. Each row of
Y
represents the classification of the corresponding
row of X
.
Data Types: single
 double
 logical
 char
 string
 cell
 categorical
Object Functions
compact  Reduce size of discriminant analysis classifier 
compareHoldout  Compare accuracies of two classification models using new data 
crossval  Crossvalidate machine learning model 
cvshrink  Crossvalidate regularization of linear discriminant 
edge  Classification edge for discriminant analysis classifier 
lime  Local interpretable modelagnostic explanations (LIME) 
logp  Log unconditional probability density for discriminant analysis classifier 
loss  Classification loss for discriminant analysis classifier 
mahal  Mahalanobis distance to class means of discriminant analysis classifier 
margin  Classification margins for discriminant analysis classifier 
nLinearCoeffs  Number of nonzero linear coefficients in discriminant analysis classifier 
partialDependence  Compute partial dependence 
plotPartialDependence  Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots 
predict  Predict labels using discriminant analysis classifier 
resubEdge  Resubstitution classification edge for discriminant analysis classifier 
resubLoss  Resubstitution classification loss for discriminant analysis classifier 
resubMargin  Resubstitution classification margins for discriminant analysis classifier 
resubPredict  Classify observations in discriminant analysis classifier by resubstitution 
shapley  Shapley values 
testckfold  Compare accuracies of two classification models by repeated crossvalidation 
Examples
Train Discriminant Analysis Model
Load Fisher's iris data set.
load fisheriris
Train a discriminant analysis model using the entire data set.
Mdl = fitcdiscr(meas,species)
Mdl = ClassificationDiscriminant ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 150 DiscrimType: 'linear' Mu: [3x4 double] Coeffs: [3x3 struct]
Mdl
is a ClassificationDiscriminant
model. To access its properties, use dot notation. For example, display the group means for each predictor.
Mdl.Mu
ans = 3×4
5.0060 3.4280 1.4620 0.2460
5.9360 2.7700 4.2600 1.3260
6.5880 2.9740 5.5520 2.0260
To predict labels for new observations, pass Mdl
and predictor data to predict
.
More About
Discriminant Classification
The model for discriminant analysis is:
Each class (
Y
) generates data (X
) using a multivariate normal distribution. That is, the model assumesX
has a Gaussian mixture distribution (gmdistribution
).For linear discriminant analysis, the model has the same covariance matrix for each class, only the means vary.
For quadratic discriminant analysis, both means and covariances of each class vary.
predict
classifies so as to minimize the expected
classification cost:
$$\widehat{y}=\underset{y=1,\mathrm{...},K}{\mathrm{arg}\mathrm{min}}{\displaystyle \sum _{k=1}^{K}\widehat{P}\left(kx\right)C\left(yk\right)},$$
where
$$\widehat{y}$$ is the predicted classification.
K is the number of classes.
$$\widehat{P}\left(kx\right)$$ is the posterior probability of class k for observation x.
$$C\left(yk\right)$$ is the cost of classifying an observation as y when its true class is k.
For details, see Prediction Using Discriminant Analysis Models.
Regularization
Regularization is the process of finding a small set of predictors
that yield an effective predictive model. For linear discriminant
analysis, there are two parameters, γ and δ,
that control regularization as follows. cvshrink
helps
you select appropriate values of the parameters.
Let Σ represent the covariance matrix of the data X, and let $$\widehat{X}$$ be the centered data (the data X minus the mean by class). Define
$$D=\text{diag}\left({\widehat{X}}^{T}*\widehat{X}\right).$$
The regularized covariance matrix $$\tilde{\Sigma}$$ is
$$\tilde{\Sigma}=\left(1\gamma \right)\Sigma +\gamma D.$$
Whenever γ ≥ MinGamma
, $$\tilde{\Sigma}$$ is nonsingular.
Let μ_{k} be the mean vector for those elements of X in class k, and let μ_{0} be the global mean vector (the mean of the rows of X). Let C be the correlation matrix of the data X, and let $$\tilde{C}$$ be the regularized correlation matrix:
$$\tilde{C}=\left(1\gamma \right)C+\gamma I,$$
where I is the identity matrix.
The linear term in the regularized discriminant analysis classifier for a data point x is
$${\left(x{\mu}_{0}\right)}^{T}{\tilde{\Sigma}}^{1}\left({\mu}_{k}{\mu}_{0}\right)=\left[{\left(x{\mu}_{0}\right)}^{T}{D}^{1/2}\right]\left[{\tilde{C}}^{1}{D}^{1/2}\left({\mu}_{k}{\mu}_{0}\right)\right].$$
The parameter δ enters into this equation as a threshold on the final term in square brackets. Each component of the vector $$\left[{\tilde{C}}^{1}{D}^{1/2}\left({\mu}_{k}{\mu}_{0}\right)\right]$$ is set to zero if it is smaller in magnitude than the threshold δ. Therefore, for class k, if component j is thresholded to zero, component j of x does not enter into the evaluation of the posterior probability.
The DeltaPredictor
property is a vector related
to this threshold. When δ ≥ DeltaPredictor(i)
, all classes k have
$$\left{\tilde{C}}^{1}{D}^{1/2}\left({\mu}_{k}{\mu}_{0}\right)\right\le \delta .$$
Therefore, when δ ≥ DeltaPredictor(i)
, the regularized
classifier does not use predictor i
.
References
[1] Guo, Y., T. Hastie, and R. Tibshirani. "Regularized linear discriminant analysis and its application in microarrays." Biostatistics, Vol. 8, No. 1, pp. 86–100, 2007.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
The
predict
function supports code generation.When you train a discriminant analysis model by using
fitcdiscr
or create a compact discriminant analysis model by usingmakecdiscr
, the value of the'ScoreTransform'
namevalue pair argument cannot be an anonymous function.
For more information, see Introduction to Code Generation.
Version History
Introduced in R2011bR2023b: Model stores observations with missing predictor values
Starting in R2023b, training observations with missing predictor values are
included in the X
, Xcentered
,
Y
, and W
data properties. The
RowsUsed
property indicates the training observations
stored in the model, rather than those used for training. Observations with missing
predictor values continue to be omitted from the model training process.
In previous releases, the software omitted training observations that contained missing predictor values from the data properties of the model.
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