Principal Components Analysis (PCA)

Principal Component Analysis (PCA) is an algorithm for exploratory data analysis and dimensionality reduction. PCA transforms a set of feature vectors of possibly correlated features to a new set of uncorrelated features, called principal components. Principal components are the directions of the largest variance, that is, the directions where the data is mostly spread out.

Operation

Computational methods

Programming Interface

Training

Covariance

SVD

train(…)

train_input

train_result

Inference

Covariance

SVD

infer(…)

infer_input

infer_result

Mathematical formulation

Refer to Developer Guide: Principal Components Analysis.

Programming Interface

All types and functions in this section are declared in the oneapi::dal::pca namespace and be available via inclusion of the oneapi/dal/algo/pca.hpp header file.

Descriptor

template<typename Float = float, typename Method = method::by_default, typename Task = task::by_default>
class descriptor
Template Parameters
  • Float – The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

  • Method – Tag-type that specifies an implementation of algorithm. Can be method::cov or method::svd.

  • Task – Tag-type that specifies type of the problem to solve. Can be task::dim_reduction.

Constructors

descriptor(std::int64_t component_count = 0)

Creates a new instance of the class with the given component_count property value.

Properties

std::int64_t component_count

The number of principal components \(r\). If it is zero, the algorithm computes the eigenvectors for all features, \(r = p\). Default value: 0.

Getter & Setter
std::int64_t get_component_count() const
auto & set_component_count(int64_t value)
Invariants
bool deterministic

Specifies whether the algorithm applies the sign-flip technique. If it is true, the directions of the eigenvectors must be deterministic. Default value: true.

Getter & Setter
bool get_deterministic() const
auto & set_deterministic(bool value)

Method tags

struct cov

Tag-type that denotes Covariance computational method.

struct svd

Tag-type that denotes SVD computational method.

using by_default = cov

Alias tag-type for Covariance computational method.

Task tags

struct dim_reduction

Tag-type that parameterizes entities used for solving dimensionality reduction problem.

using by_default = dim_reduction

Alias tag-type for dimensionality reduction task.

Model

template<typename Task = task::by_default>
class model
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::dim_reduction.

Constructors

model()

Creates a new instance of the class with the default property values.

Properties

const table &eigenvectors

An \(r \times p\) table with the eigenvectors. Each row contains one eigenvector. Default value: table{}.

Getter & Setter
const table & get_eigenvectors() const
auto & set_eigenvectors(const table &value)

Training train(...)

Input

template<typename Task = task::by_default>
class train_input
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::dim_reduction.

Constructors

train_input(const table &data)

Creates a new instance of the class with the given data property value.

Properties

const table &data

An \(n \times p\) table with the training data, where each row stores one feature vector. Default value: table{}.

Getter & Setter
const table & get_data() const
auto & set_data(const table &data)

Result

template<typename Task = task::by_default>
class train_result
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::dim_reduction.

Constructors

train_result()

Creates a new instance of the class with the default property values.

Public Methods

const table &get_eigenvectors() const

An \(r \times p\) table with the eigenvectors. Each row contains one eigenvector.

Properties

const table &variances

A \(1 \times r\) table that contains the variances for the first r features. Default value: table{}.

Getter & Setter
const table & get_variances() const
auto & set_variances(const table &value)
const table &eigenvalues

A \(1 \times r\) table that contains the eigenvalues for for the first r features. Default value: table{}.

Getter & Setter
const table & get_eigenvalues() const
auto & set_eigenvalues(const table &value)
const table &means

A \(1 \times r\) table that contains the mean values for the first r features. Default value: table{}.

Getter & Setter
const table & get_means() const
auto & set_means(const table &value)
const model<Task> &model

The trained PCA model. Default value: model<Task>{}.

Getter & Setter
const model< Task > & get_model() const
auto & set_model(const model< Task > &value)

Operation

template<typename Descriptor>
pca::train_result train(const Descriptor &desc, const pca::train_input &input)
Parameters
  • desc – PCA algorithm descriptor pca::descriptor

  • input – Input data for the training operation

Preconditions
input.data.has_data == true
input.data.column_count >= desc.component_count
Postconditions
result.means.row_count == 1
result.means.column_count == desc.component_count
result.variances.row_count == 1
result.variances.column_count == desc.component_count
result.variances[i] >= 0.0
result.eigenvalues.row_count == 1
result.eigenvalues.column_count == desc.component_count
result.model.eigenvectors.row_count == 1
result.model.eigenvectors.column_count == desc.component_count

Inference infer(...)

Input

template<typename Task = task::by_default>
class infer_input
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::dim_reduction.

Constructors

infer_input(const model<Task> &trained_model, const table &data)

Creates a new instance of the class with the given model and data property values.

Properties

const table &data

The dataset for inference \(X'\). Default value: table{}.

Getter & Setter
const table & get_data() const
auto & set_data(const table &value)
const model<Task> &model

The trained PCA model. Default value: model<Task>{}.

Getter & Setter
const model< Task > & get_model() const
auto & set_model(const model< Task > &value)

Result

template<typename Task = task::by_default>
class infer_result
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::dim_reduction.

Constructors

infer_result()

Creates a new instance of the class with the default property values.

Properties

const table &transformed_data

An \(n \times r\) table that contains data projected to the r principal components. Default value: table{}.

Getter & Setter
const table & get_transformed_data() const
auto & set_transformed_data(const table &value)

Operation

template<typename Descriptor>
pca::infer_result infer(const Descriptor &desc, const pca::infer_input &input)
Parameters
  • desc – PCA algorithm descriptor pca::descriptor

  • input – Input data for the inference operation

Preconditions
input.data.has_data == true
input.model.eigenvectors.row_count == desc.component_count
input.model.eigenvectors.column_count == input.data.column_count
Postconditions
result.transformed_data.row_count == input.data.row_count
result.transformed_data.column_count == desc.component_count

Usage example

Training

pca::model<> run_training(const table& data) {
   const auto pca_desc = pca::descriptor<float>{}
      .set_component_count(5)
      .set_deterministic(true);

   const auto result = train(pca_desc, data);

   print_table("means", result.get_means());
   print_table("variances", result.get_variances());
   print_table("eigenvalues", result.get_eigenvalues());
   print_table("eigenvectors", result.get_eigenvectors());

   return result.get_model();
}

Inference

table run_inference(const pca::model<>& model,
                  const table& new_data) {
   const auto pca_desc = pca::descriptor<float>{}
      .set_component_count(model.get_component_count());

   const auto result = infer(pca_desc, model, new_data);

   print_table("labels", result.get_transformed_data());
}

Examples

Batch Processing:

Batch Processing:

Batch Processing: