.. _library_lda_projection:

`lda_projection`

Linear Discriminant Analysis projection for labeled continuous datasets. Supports continuous attributes only.

The library implements the dimension_reducer_protocol defined in the dimension_reduction_protocols library and learns discriminant directions by centering the training data, optionally standardizing continuous attributes, building regularized within-class and between-class scatter matrices, whitening the Fisher criterion using a Cholesky factorization, and extracting discriminant directions using deterministic power iteration with deflation.

Requires a dataset implementing supervised_dimension_reduction_dataset_protocol and therefore uses class labels during training.

API documentation

Open the `../../apis/library_index.html#lda_projection <../../apis/library_index.html#lda_projection>`__ link in a web browser.

Loading

To load this library, load the loader.lgt file:

| ?- logtalk_load(lda_projection(loader)).

Testing

To test this library predicates, load the tester.lgt file:

| ?- logtalk_load(lda_projection(tester)).

Features

Supervised Continuous Datasets: Accepts labeled datasets implementing supervised_dimension_reduction_dataset_protocol and containing only continuous attributes.
Centering and Optional Scaling: Centers all attributes and optionally standardizes them before estimating class scatter matrices.
Portable Fisher Eigensolver: Uses regularized within-class scatter, Cholesky whitening, and deterministic power iteration instead of backend-specific linear algebra libraries.
Projection API: Transforms a new instance into a list of component_N-Value pairs.
Model Export: Learned reducers can be exported as predicate clauses or written to a file.
Missing Values: Missing or nonnumeric values are rejected.

Options

The learn/3 predicate accepts the following options:

n_components/1: Maximum number of discriminant components to extract. Requests that exceed the minimum of the number of features and number_of_classes - 1 raise domain_error(component_count, Requested-Maximum). The default is 2.
feature_scaling/1: Whether to standardize continuous attributes before estimating scatter matrices. Options: true (default) or false.
maximum_iterations/1: Maximum number of power-iteration steps used when estimating each discriminant direction. The default is 1000.
tolerance/1: Positive convergence tolerance used both for power-iteration stopping and for deciding when additional discriminant directions are numerically negligible. The default is 1.0e-8.
regularization/1: Positive diagonal regularization added to the within-class scatter matrix before whitening. The default is 1.0e-6.

Usage

The following examples use the sample labeled dataset shipped with the dimension_reduction_protocols library:

| ?- logtalk_load(dimension_reduction_protocols('test_datasets/labeled_measurements')).

Learning a reducer ~~~~~~~~~~~~~~~~~~

| ?- lda_projection::learn(labeled_measurements, DimensionReducer).

| ?- lda_projection::learn(labeled_measurements, DimensionReducer, [n_components(1), feature_scaling(false), maximum_iterations(250), tolerance(1.0e-7), regularization(1.0e-5)]).

Transforming new instances ~~~~~~~~~~~~~~~~~~~~~~~~~~

| ?- lda_projection::learn(labeled_measurements, DimensionReducer), lda_projection::transform(DimensionReducer, [length-5.1, width-3.5, height-1.4, weight-0.2], ReducedInstance).

| ?- lda_projection::learn(labeled_measurements, DimensionReducer, [n_components(1)]), lda_projection::transform(DimensionReducer, [length-6.2, width-3.4, height-5.4, weight-2.3], ReducedInstance).

Exporting and reusing the reducer ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

| ?- lda_projection::learn(labeled_measurements, DimensionReducer, [n_components(1)]), lda_projection::export_to_file(labeled_measurements, DimensionReducer, reducer, 'lda_projection_reducer.pl').

| ?- logtalk_load('lda_projection_reducer.pl'), reducer(Reducer), lda_projection::transform(Reducer, [length-5.1, width-3.5, height-1.4, weight-0.2], ReducedInstance).

Dimension reducer representation

The learned dimension reducer is represented by a compound term with the functor chosen by the implementation and arity 4. For example:

lda_projection_reducer(Encoders, Components, ClassValues, Diagnostics)

Where:

Encoders: List of continuous attribute encoders storing attribute name, mean, and scale.
Components: List of learned discriminant vectors in component order.
ClassValues: Ordered list of class labels used during training.
Diagnostics: Learned reducer metadata including the effective training options and model details.
When exported using export_to_clauses/4 or export_to_file/4, this reducer term is serialized directly as the single argument of the generated predicate clause so that the exported model can be loaded and reused as-is.

References

Fisher, R. A. (1936) - "The use of multiple measurements in taxonomic problems".
Rao, C. R. (1948) - "The utilization of multiple measurements in problems of biological classification".