# Discriminant Analysis

Linear Discriminant Analysis (LDA) is a technique for multi-class classification
and dimensionality reduction.
It is based on the assumption that the observations in each class or group
are distributed as a multivariate Gaussian distribution, and that all groups
have the same covariance matrix. (A variation where groups can have different
covariance matrices is called *quadratic discriminant analysis* (QDA).

When used for dimensionality reduction, the features are projected onto the directions that most separate the classes. When used for classification, it is sufficient to consider the distance to the group centroids in the projected space.

Discriminant analysis models are implemented by the LinearDiscriminantAnalysis class.

## Constructing Linear Discriminant Analysis Models

The LinearDiscriminantAnalysis class has four constructors.

The first constructor takes two arguments. The first is a ICategoricalVector that represents the dependent variable. The second is a parameter array of Vector<T> objects that represent the independent variables.

```
var dependent = Vector.CreateCategorical(yData);
var independent1 = Vector.Create(x1Data);
var independent2 = Vector.Create(x2Data);
var model1 = new LinearDiscriminantAnalysis(dependent, independent1, independent2);
```

The second constructor takes 3 arguments. The first argument is a IDataFrame (a DataFrame<R, C> or Matrix<T>) that contains the variables to be used in the analysis. The second argument is a string containing the name of the dependent variable. The third argument is a parameter array of strings containing the names of the independent variables. All the names must exist in the column index of the data frame specified by the first parameter.

```
var dataFrame = DataFrame.FromColumns(
( "y", dependent ),
( "x1", independent1 ),
( "x2", independent2 ));
var model2 = new LinearDiscriminantAnalysis(dataFrame, "y", "x1", "x2");
```

The third constructor takes 2 arguments. The first argument is once again a IDataFrame (a DataFrame<R, C> or Matrix<T>) that contains the variables to be used in the analysis. The second argument is a string containing a formula that specifies the dependent and independent variables. See the section on Defining models using formulas for details of formula syntax.

`var model3 = new LinearDiscriminantAnalysis(dataFrame, "y ~ x1 + x2");`

## Computing the Analysis

The Compute method performs the actual analysis. Most properties and methods throw an exception when they are accessed before the Compute method is called. You can verify that the model has been calculated by inspecting the Computed property.

`model1.Fit();`

The Predictions property returns a CategoricalVector<T> that contains the values of the dependent variable as predicted by the model. The PredictedProbabilities property returns a Matrix<T> that gives the probability of each outcome for each observation. A related property, PredictedLogProbabilities property returns the natural logarithm of the predicted probabilities. The ProbabilityResiduals property returns a matrix containing the difference between the actual (0 or 1) and the predicted probabilities.

## Discriminant Functions

The result of a discriminant analysis is a set of *discriminant functions*.
These are linear functions of the variables in the direction that most separates
the observations in each group from those in other groups.
The LinearDiscriminantAnalysis
class' DiscriminantFunctions
property returns a collection of
LinearDiscriminantFunction
object that represent the discriminant functions.
The discriminant functions are constructed by the model. You cannot create them directly.

Discriminant functions perform a role similar to factors in Factor Analysis or components in Principal Component Analysis (PCA). Discriminant functions are based on a generalized eigenvalue decomposition. The corresponding eigenvalue and eigenvector can be accessed through the Eigenvalue and Eigenvector properties. The EigenvalueDifference property returns the difference between the eigenvalues of the discriminant function and the next most significant discriminant function. The ProportionOfVariance and CumulativeProportionOfVariance properties give the contribution of the discriminant function to the variation in the data in relative terms.

The CanonicalCorrelation
property returns the canonical correlation between the discriminant function and the groups.
A higher coefficient indicates a higher relevance.
Similarly, WilksLambda
returns *Wilks' lambda*, which is a statistic that is based on the
canonical correlations.

The significance of a discriminant function can be quantified further using a hypothesis test based on Wilks' lambda. This is an F test that is exact when the number of groups is less than 3 or when the number of included functions is 1 or 2. The GetFTest() method returns this test.

In the example below, these properties are printed for the discriminant functions from the earlier example:

```
Console.WriteLine(" # Eigenvalue Difference Contribution Contrib. % Can.Corr F stat. df1 df2");
for (int i = 0; i < model1.DiscriminantFunctions.Count; i++)
{
var fn = model1.DiscriminantFunctions[i];
var f = fn.GetFTest();
Console.WriteLine("{0,2}{1,12:F4}{1,11:F4}{2,14:F3}%{3,10:F3}%{4,9:F4}{5,9:F3}{6,9:F3}{7,4}{8,4}",
i, fn.Eigenvalue, fn.EigenvalueDifference,
100 * fn.ProportionOfVariance,
100 * fn.CumulativeProportionOfVariance,
fn.CanonicalCorrelation,
fn.WilksLambda,
f.Statistic,
f.NumeratorDegreesOfFreedom,
f.DenominatorDegreesOfFreedom);
}
```

## Classification

When using a linear discriminant analysis for classification, the probability that an observation belongs to each class is computed. The class with the highest probability is selected.

The Predict method that takes a vector or data frame and produces the model's prediction for the supplied data. When a single observation is supplied (as a vector), the method returns an integer that is the level index of the predicted class. When multiple observations are supplied, the method returns a vector of level indexes.

Similarly, the PredictProbabilities method returns the predicted probabilities for each class. When a single observation is supplied (as a vector), the method returns a vector that contains the probabilities that the observation belongs to each of the classes. When multiple observations are supplied, the method returns a matrix, where each row contains the probabilities for the corresponding observation.

```
var index = model1.Predict(Vector.Create(1.2, 3.0));
var input = Matrix.CreateRandom(10, 2);
var predictions = model1.Predict(input);
```

## Dimensionality Reduction

When used for dimensionality reduction, the observations are projected onto the direction that most separate the classes. The LinearDiscriminantAnalysis class implements the ITransformationModel interface to support this operation. The Transform(Matrix<Double>) method performs this operation. It takes one argument: a matrix whose rows contain the observations. It returns a matrix whose columns are the projections of the original features on the discriminant directions.

`var transformed = model1.Transform(dataFrame.ToMatrix<double>());`