Table of Contents

Class QuadraticDiscriminantAnalysis<T>

Namespace
AiDotNet.Classification.DiscriminantAnalysis
Assembly
AiDotNet.dll

Quadratic Discriminant Analysis classifier.

public class QuadraticDiscriminantAnalysis<T> : ProbabilisticClassifierBase<T>, IProbabilisticClassifier<T>, IClassifier<T>, IFullModel<T, Matrix<T>, Vector<T>>, IModel<Matrix<T>, Vector<T>, ModelMetadata<T>>, IModelSerializer, ICheckpointableModel, IParameterizable<T, Matrix<T>, Vector<T>>, IFeatureAware, IFeatureImportance<T>, ICloneable<IFullModel<T, Matrix<T>, Vector<T>>>, IGradientComputable<T, Matrix<T>, Vector<T>>, IJitCompilable<T>

Type Parameters

T

The numeric data type used for calculations (e.g., float, double).

Inheritance
QuadraticDiscriminantAnalysis<T>
Implements
IFullModel<T, Matrix<T>, Vector<T>>
IModel<Matrix<T>, Vector<T>, ModelMetadata<T>>
IParameterizable<T, Matrix<T>, Vector<T>>
ICloneable<IFullModel<T, Matrix<T>, Vector<T>>>
IGradientComputable<T, Matrix<T>, Vector<T>>
Inherited Members
Extension Methods

Remarks

QDA is similar to LDA but allows each class to have its own covariance matrix, which creates quadratic (curved) decision boundaries.

For Beginners: Quadratic Discriminant Analysis (QDA) is a classification technique that:

  1. Models each class with its own Gaussian distribution
  2. Each class has its own covariance matrix (unlike LDA)
  3. Decision boundaries are quadratic (curved) instead of linear

When to use QDA over LDA:

  • When classes have different covariance structures
  • When you have enough samples per class to estimate covariance reliably
  • When decision boundaries are naturally curved

Trade-offs:

  • More flexible than LDA (can capture curved boundaries)
  • Needs more parameters (separate covariance per class)
  • More prone to overfitting with small datasets
  • Computationally more expensive than LDA

Constructors

QuadraticDiscriminantAnalysis(DiscriminantAnalysisOptions<T>?, IRegularization<T, Matrix<T>, Vector<T>>?)

Initializes a new instance of the QuadraticDiscriminantAnalysis class.

public QuadraticDiscriminantAnalysis(DiscriminantAnalysisOptions<T>? options = null, IRegularization<T, Matrix<T>, Vector<T>>? regularization = null)

Parameters

options DiscriminantAnalysisOptions<T>

Configuration options for the classifier.

regularization IRegularization<T, Matrix<T>, Vector<T>>

Optional regularization strategy.

Properties

Options

Gets the QDA-specific options.

protected DiscriminantAnalysisOptions<T> Options { get; }

Property Value

DiscriminantAnalysisOptions<T>

Methods

ApplyGradients(Vector<T>, T)

Applies pre-computed gradients to update the model parameters.

public override void ApplyGradients(Vector<T> gradients, T learningRate)

Parameters

gradients Vector<T>

The gradient vector to apply.

learningRate T

The learning rate for the update.

Remarks

Updates parameters using: θ = θ - learningRate * gradients

For Beginners: After computing gradients (seeing which direction to move), this method actually moves the model in that direction. The learning rate controls how big of a step to take.

Distributed Training: In DDP/ZeRO-2, this applies the synchronized (averaged) gradients after communication across workers. Each worker applies the same averaged gradients to keep parameters consistent.

Clone()

Creates a clone of the classifier model.

public override IFullModel<T, Matrix<T>, Vector<T>> Clone()

Returns

IFullModel<T, Matrix<T>, Vector<T>>

A new instance of the model with the same parameters and options.

ComputeGradients(Matrix<T>, Vector<T>, ILossFunction<T>?)

Computes gradients of the loss function with respect to model parameters for the given data, WITHOUT updating the model parameters.

public override Vector<T> ComputeGradients(Matrix<T> input, Vector<T> target, ILossFunction<T>? lossFunction = null)

Parameters

input Matrix<T>

The input data.

target Vector<T>

The target/expected output.

lossFunction ILossFunction<T>

The loss function to use for gradient computation. If null, uses the model's default loss function.

Returns

Vector<T>

A vector containing gradients with respect to all model parameters.

Remarks

This method performs a forward pass, computes the loss, and back-propagates to compute gradients, but does NOT update the model's parameters. The parameters remain unchanged after this call.

Distributed Training: In DDP/ZeRO-2, each worker calls this to compute local gradients on its data batch. These gradients are then synchronized (averaged) across workers before applying updates. This ensures all workers compute the same parameter updates despite having different data.

For Meta-Learning: After adapting a model on a support set, you can use this method to compute gradients on the query set. These gradients become the meta-gradients for updating the meta-parameters.

For Beginners: Think of this as "dry run" training: - The model sees what direction it should move (the gradients) - But it doesn't actually move (parameters stay the same) - You get to decide what to do with this information (average with others, inspect, modify, etc.)

Exceptions

InvalidOperationException

If lossFunction is null and the model has no default loss function.

CreateNewInstance()

Creates a new instance of the same type as this classifier.

protected override IFullModel<T, Matrix<T>, Vector<T>> CreateNewInstance()

Returns

IFullModel<T, Matrix<T>, Vector<T>>

A new instance of the same classifier type.

GetModelMetadata()

Gets metadata about the model.

public override ModelMetadata<T> GetModelMetadata()

Returns

ModelMetadata<T>

A ModelMetadata object containing information about the model.

Remarks

This method returns metadata about the model, including its type, feature count, complexity, description, and additional information specific to classification.

For Beginners: Model metadata provides information about the model itself, rather than the predictions it makes. This includes details about the model's structure (like how many features it uses) and characteristics (like how many classes it can predict). This information can be useful for understanding and comparing different models.

GetModelType()

Returns the model type identifier for this classifier.

protected override ModelType GetModelType()

Returns

ModelType

GetParameters()

Gets all model parameters as a single vector.

public override Vector<T> GetParameters()

Returns

Vector<T>

A vector containing all model parameters.

Remarks

This method returns a vector containing all model parameters for use with optimization algorithms or model comparison.

For Beginners: This method packages all the model's parameters into a single collection. This is useful for optimization algorithms that need to work with all parameters at once.

Predict(Matrix<T>)

Predicts class labels for the given input data by taking the argmax of probabilities.

public override Vector<T> Predict(Matrix<T> input)

Parameters

input Matrix<T>

The input features matrix where each row is an example and each column is a feature.

Returns

Vector<T>

A vector of predicted class indices for each input example.

Remarks

This implementation uses the argmax of the probability distribution to determine the predicted class. For binary classification with a custom decision threshold, you may want to use PredictProbabilities() directly and apply your own threshold.

For Beginners: This method picks the class with the highest probability for each sample.

For example, if the probabilities are [0.1, 0.7, 0.2] for classes [A, B, C], this method returns class B (index 1) because it has the highest probability (0.7).

PredictLogProbabilities(Matrix<T>)

Predicts log-probabilities for each class.

public override Matrix<T> PredictLogProbabilities(Matrix<T> input)

Parameters

input Matrix<T>

The input features matrix where each row is a sample and each column is a feature.

Returns

Matrix<T>

A matrix where each row corresponds to an input sample and each column corresponds to a class. The values are the natural logarithm of the class probabilities.

Remarks

The default implementation computes log(PredictProbabilities(input)). Subclasses that compute log-probabilities directly (like Naive Bayes) should override this method for better numerical stability.

For Beginners: Log-probabilities are probabilities transformed by the natural logarithm. They're useful for numerical stability when working with very small probabilities.

For example:

  • Probability 0.9 → Log-probability -0.105
  • Probability 0.1 → Log-probability -2.303
  • Probability 0.001 → Log-probability -6.908

Log-probabilities are always negative (since probabilities are between 0 and 1). Higher (less negative) values mean higher probability.

PredictProbabilities(Matrix<T>)

Predicts class probabilities for each sample in the input.

public override Matrix<T> PredictProbabilities(Matrix<T> input)

Parameters

input Matrix<T>

The input features matrix where each row is a sample and each column is a feature.

Returns

Matrix<T>

A matrix where each row corresponds to an input sample and each column corresponds to a class. The values represent the probability of the sample belonging to each class.

Remarks

This abstract method must be implemented by derived classes to compute class probabilities. The output matrix should have shape [num_samples, num_classes], and each row should sum to 1.0.

For Beginners: This method computes the probability of each sample belonging to each class. Each row in the output represents one sample, and each column represents one class. The values in each row sum to 1.0 (100% total probability).

SetParameters(Vector<T>)

Sets the parameters for this model.

public override void SetParameters(Vector<T> parameters)

Parameters

parameters Vector<T>

A vector containing all model parameters.

Exceptions

ArgumentException

Thrown when the parameters vector has an incorrect length.

Train(Matrix<T>, Vector<T>)

Trains the QDA classifier on the provided data.

public override void Train(Matrix<T> x, Vector<T> y)

Parameters

x Matrix<T>
y Vector<T>

WithParameters(Vector<T>)

Creates a new instance of the model with specified parameters.

public override IFullModel<T, Matrix<T>, Vector<T>> WithParameters(Vector<T> parameters)

Parameters

parameters Vector<T>

A vector containing all model parameters.

Returns

IFullModel<T, Matrix<T>, Vector<T>>

A new model instance with the specified parameters.

Exceptions

ArgumentException

Thrown when the parameters vector has an incorrect length.

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