Regularized discriminant analysis in Julia.
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October 2013

Discriminant Analysis

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DiscriminantAnalysis.jl is a Julia package for multiple linear and quadratic regularized discriminant analysis (LDA & QDA respectively). LDA and QDA are distribution-based classifiers with the underlying assumption that data follows a multivariate normal distribution. LDA differs from QDA in the assumption about the class variability; LDA assumes that all classes share the same within-class covariance matrix whereas QDA relaxes that constraint and allows for distinct within-class covariance matrices. This results in LDA being a linear classifier and QDA being a quadratic classifier.

The package is currently a work in progress work in progress - see issue #12 for the package status.

Getting Started

A bare-bones implementation of LDA is currently available but is not exported. Calls to the solver must be prefixed with DiscriminantAnalysis after running using DiscriminantAnalysis. Below is a brief overview of the API:

  • lda(X, y; kwargs...): construct a Linear Discriminant Analysis model.
    • X: the matrix of predictors (design matrix). Data may be per-column or per-row; this is specified by the dims keyword argument.
    • y: the vector of class indices. For c classes, the values must range from 1 to c.
    • dims=1: the dimension along which observations are stored. Use 1 for row-per-observation and 2 for column-per-observation.
    • canonical=false: compute the canonical coordinates if true. For c classes, the data is mapped to a c-1 dimensional space for prediction.
    • compute_covariance=false: compute the full class covariance matrix if true. Data is whitened prior to compute discriminant values, so generally the covariance is not computed unless specified.
    • centroids=nothing: matrix of pre-computed class centroids. This can be used if the class centroids are known a priori. Otherwise, the centroids are estimated from the data. The centroid matrix must have the same orientation as specified by the dims argument.
    • priors=nothing: vector of pre-computed class prior probabilities. This can be used if the class prior probabilities are known a priori. Otherwise, the priors are estimated from the class frequencies.
    • gamma=nothing: real value between 0 and 1. Gamma is a regularization parameter that is used to shrink the covariance matrix towards an identity matrix scaled by the average eigenvalue of the covariance matrix. A value of 0.2 retains 80% of the original covariance matrix.
  • posteriors(LDA, Z): compute the class posterior probabilities on a new matrix of predictors Z. This matrix must have the same dims orientation as the original design matrix X.
  • classify(LDA, Z): compute the class label predictions on a new matrix of predictors Z. This matrix must have the same dims orientation as the original design matrix X.

The script below demonstrates how to fit an LDA model to some synthetic data using the interface described above:

using DiscriminantAnalysis
using Random

const DA = DiscriminantAnalysis

# Generate two sets of 100 samples of a 5-dimensional random normal 
# variable offset by +1/-1
X = [randn(250,5) .- 1;
     randn(250,5) .+ 1];

# Generate class labels for the two samples
#   NOTE: classes must be indexed by integers from 1 to the number of 
#         classes (2 in this case)
y = repeat(1:2, inner=250);

# Construct the LDA model
model = DA.lda(X, y; dims=1, canonical=true, priors=[0.5; 0.5])

# Generate some new data
Z = rand(10,5) .- 0.5

# Get the posterior probabilities for new data
Z_prob = DA.posteriors(model, Z)

# Get the class predictions
Z_class = DA.classify(model, Z)