Developer Guide and Reference

  • 2021.4
  • 09/27/2021
  • Public Content
Contents

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.

Mathematical formulation

Training
Given the training set LaTex Math image. of
p
-dimensional feature vectors and the number of principal components
r
, the problem is to compute
r
principal directions (
p
-dimensional eigenvectors [Lang87]) for the training set. The eigenvectors can be grouped into the LaTex Math image. matrix
T
that contains one eigenvector in each row.
Training method:
Covariance
This method uses eigenvalue decomposition of the covariance matrix to compute the principal components of the datasets. The method relies on the following steps:
  1. Computation of the covariance matrix
  2. Computation of the eigenvectors and eigenvalues
  3. Formation of the matrices storing the results
Covariance matrix computation is performed in the following way:
  1. Compute the vector-column of sums LaTex Math image..
  2. Compute the cross-product LaTex Math image..
  3. Compute the covariance matrix LaTex Math image..
To compute eigenvalues LaTex Math image. and eigenvectors LaTex Math image., the implementer can choose an arbitrary method such as [Ping14].
The final step is to sort the set of pairs LaTex Math image. in the descending order by LaTex Math image. and form the resulting matrix LaTex Math image.. Additionally, the means and variances of the initial dataset are returned.
Training method:
SVD
This method uses singular value decomposition of the dataset to compute its principal components. The method relies on the following steps:
  1. Computation of the singular values and singular vectors
  2. Formation of the matrices storing the results
To compute singular values LaTex Math image. and singular vectors LaTex Math image. and LaTex Math image., the implementer can choose an arbitrary method such as [Demmel90].
The final step is to sort the set of pairs LaTex Math image. in the descending order by LaTex Math image. and form the resulting matrix LaTex Math image.. Additionally, the means and variances of the initial dataset are returned.
Sign-flip technique
Eigenvectors computed by some eigenvalue solvers are not uniquely defined due to sign ambiguity. To get the deterministic result, a sign-flip technique should be applied. One of the sign-flip techniques proposed in [Bro07] requires the following modification of matrix
T
:
LaTex Math image.
where LaTex Math image. is
i
-th row, LaTex Math image. is the element in the
i
-th row and
j
-th column, LaTex Math image. is the signum function,
LaTex Math image.
Inference
Given the inference set LaTex Math image. of
p
-dimensional feature vectors and the LaTex Math image. matrix
T
produced at the training stage, the problem is to transform LaTex Math image. to the set LaTex Math image., where LaTex Math image. is an
r
-dimensional feature vector, LaTex Math image..
The feature vector LaTex Math image. is computed through applying linear transformation [Lang87] defined by the matrix
T
to the feature vector LaTex Math image.,
LaTex Math image.
Inference methods:
Covariance
and
SVD
Covariance and SVD inference methods compute LaTex Math image. according to (1).

Programming Interface

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

oneAPI DPC++
Batch Processing:
oneAPI C++
Batch Processing:
Python* with DPC++ support
Batch Processing:

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.