# ClassificationDiscriminant class

Superclasses: CompactClassificationDiscriminant

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.

## Construction

```obj = fitcdiscr(x,Y)``` creates a discriminant classification object based on the input variables (also known as predictors, features, or attributes) `x` and output (response) `Y`. For syntax details, see `fitcdiscr`.

`obj = fitcdiscr(x,Y,Name,Value)` creates a classifier with additional options specified by one or more `Name,Value` pair arguments. If you use one of the following five options, `obj` is of class `ClassificationPartitionedModel`: `'CrossVal'`, `'KFold'`, `'Holdout'`, `'Leaveout'`, or `'CVPartition'`. Otherwise, `obj` is of class `ClassificationDiscriminant`.

### Input Arguments

 `x` Matrix of numeric predictor values. Each column of `x` represents one variable, and each row represents one observation. `NaN` values in `x` are considered missing values. Observations with missing values for `x` are not used in the fit. `y` A categorical array, cell array of strings, character array, logical vector, or a numeric vector with the same number of rows as `x`. Each row of `Y` represents the classification of the corresponding row of `x`. `NaN` values in `Y` are considered missing values. Observations with missing values for `Y` are not used in the fit.

## Properties

 `BetweenSigma` `p`-by-`p` matrix, the between-class covariance, where `p` is the number of predictors. `CategoricalPredictors` List of categorical predictors, which is always empty (`[]`) for SVM and discriminant analysis classifiers. `ClassNames` List of the elements in the training data `Y` with duplicates removed. `ClassNames` can be a categorical array, cell array of strings, character array, logical vector, or a numeric vector. `ClassNames` has the same data type as the data in the argument `Y`. `Coeffs` `k`-by-`k` structure of coefficient matrices, where `k` is the number of classes. `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 scalar`Linear` — A vector with `p` components, where `p` is the number of columns in `X``Quadratic` — `p`-by-`p` matrix, exists for quadratic `DiscrimType` 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`. If `fitcdiscr` had the `FillCoeffs` name-value pair set to `'off'` when constructing the classifier, `Coeffs` is empty (`[]`). `Cost` Square matrix, where `Cost(i,j)` is the cost of classifying a point into class `j` if its true class is `i` (i.e., 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```. `Delta` Value of the Delta threshold for a linear discriminant model, 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```. `DeltaPredictor` Row vector of length equal to 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`. `DiscrimType` String specifying the discriminant type. One of: `'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. `Gamma` Value of the Gamma regularization parameter, a scalar from `0` to `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` and `1` 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` (for `DiscrimType` `'quadratic'`) or `1` (for `DiscrimType` `'diagQuadratic'`). `LogDetSigma` Logarithm of the determinant of the within-class covariance matrix. The type of `LogDetSigma` depends on the discriminant type: Scalar for linear discriminant analysisVector of length `K` for quadratic discriminant analysis, where `K` is the number of classes `MinGamma` Nonnegative scalar, the minimal value of the Gamma parameter so that the correlation matrix is invertible. If the correlation matrix is not singular, `MinGamma` is `0`. `ModelParameters` Parameters used in training `obj`. `Mu` Class means, specified as a `K`-by-`p` matrix of scalar values class means of size. `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. `NumObservations` Number of observations in the training data, a numeric scalar. `NumObservations` can be less than the number of rows of input data `X` when there are missing values in `X` or response `Y`. `PredictorNames ` Cell array of names for the predictor variables, in the order in which they appear in the training data `X`. `Prior` Numeric vector of prior probabilities for each class. 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```. `ResponseName` String describing the response variable `Y`. `ScoreTransform` Function handle for transforming scores, or string representing a built-in transformation function. `'none'` means no transformation; equivalently, `'none'` means `@(x)x`. For a list of built-in 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` `Sigma` Within-class covariance matrix or matrices. The dimensions depend on `DiscrimType`: `'linear'` (default) — Matrix of size `p`-by-`p`, where `p` is the number of predictors`'quadratic'` — Array of size `p`-by-`p`-by-`K`, where `K` is the number of classes`'diagLinear'` — Row vector of length `p``'diagQuadratic'` — Array of size `1`-by-`p`-by-`K``'pseudoLinear'` — Matrix of size `p`-by-`p``'pseudoQuadratic'` — Array of size `p`-by-`p`-by-`K` `W` Scaled `weights`, a vector with length `n`, the number of rows in `X`. `X` Matrix of predictor values. Each column of `X` represents one predictor (variable), and each row represents one observation. `Xcentered` `X` data with class means subtracted. If `Y(i)` is of class `j`, `Xcentered(i,:)` = `X(i,:)` – `Mu(j,:)`, where `Mu` is the class mean property. `Y` A categorical array, cell array of strings, character array, logical vector, or a numeric vector with the same number of rows as `X`. Each row of `Y` represents the classification of the corresponding row of `X`.

## Methods

 compact Compact discriminant analysis classifier crossval Cross-validated discriminant analysis classifier cvshrink Cross-validate regularization of linear discriminant resubEdge Classification edge by resubstitution resubLoss Classification error by resubstitution resubMargin Classification margins by resubstitution resubPredict Predict resubstitution response of classifier

### Inherited Methods

 compareHoldout Compare accuracies of two classification models using new data edge Classification edge logP Log unconditional probability density for discriminant analysis classifier loss Classification error mahal Mahalanobis distance to class means margin Classification margins nLinearCoeffs Number of nonzero linear coefficients predict Predict classification

## Definitions

### Discriminant Classification

The model for discriminant analysis is:

• Each class (`Y`) generates data (`X`) using a multivariate normal distribution. That is, the model assumes `X` 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:

`$\stackrel{^}{y}=\underset{y=1,...,K}{\mathrm{arg}\mathrm{min}}\sum _{k=1}^{K}\stackrel{^}{P}\left(k|x\right)C\left(y|k\right),$`

where

• $\stackrel{^}{y}$ is the predicted classification.

• K is the number of classes.

• $\stackrel{^}{P}\left(k|x\right)$ is the posterior probability of class k for observation x.

• $C\left(y|k\right)$ is the cost of classifying an observation as y when its true class is k.

For details, see How the predict Method Classifies.

### 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 $\stackrel{^}{X}$ be the centered data (the data X minus the mean by class). Define

`$D=\text{diag}\left({\stackrel{^}{X}}^{T}*\stackrel{^}{X}\right).$`

The regularized covariance matrix $\stackrel{˜}{\Sigma }$ is

`$\stackrel{˜}{\Sigma }=\left(1-\gamma \right)\Sigma +\gamma D.$`

Whenever γ ≥ `MinGamma`, $\stackrel{˜}{\Sigma }$ is nonsingular.

Let μk be the mean vector for those elements of X in class k, and let μ0 be the global mean vector (the mean of the rows of X). Let C be the correlation matrix of the data X, and let $\stackrel{˜}{C}$ be the regularized correlation matrix:

`$\stackrel{˜}{C}=\left(1-\gamma \right)C+\gamma I,$`

where I is the identity matrix.

The linear term in the regularized discriminant analysis classifier for a data point x is

`${\left(x-{\mu }_{0}\right)}^{T}{\stackrel{˜}{\Sigma }}^{-1}\left({\mu }_{k}-{\mu }_{0}\right)=\left[{\left(x-{\mu }_{0}\right)}^{T}{D}^{-1/2}\right]\left[{\stackrel{˜}{C}}^{-1}{D}^{-1/2}\left({\mu }_{k}-{\mu }_{0}\right)\right].$`

The parameter δ enters into this equation as a threshold on the final term in square brackets. Each component of the vector $\left[{\stackrel{˜}{C}}^{-1}{D}^{-1/2}\left({\mu }_{k}-{\mu }_{0}\right)\right]$ 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

`$|{\stackrel{˜}{C}}^{-1}{D}^{-1/2}\left({\mu }_{k}-{\mu }_{0}\right)|\le \delta .$`

Therefore, when δ ≥ `DeltaPredictor(i)`, the regularized classifier does not use predictor `i`.

## Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB® documentation.

## Examples

Create a discriminant analysis classifier for the Fisher iris data:

```load fisheriris obj = fitcdiscr(meas,species) obj = ClassificationDiscriminant: PredictorNames: {'x1' 'x2' 'x3' 'x4'} ResponseName: 'Y' ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 150 DiscrimType: 'linear' Mu: [3x4 double] Coeffs: [3x3 struct]```

## 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.