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BayesianLDA

BayesianLDA

A model type for constructing a Bayesian LDA model, based on MultivariateStats.jl, and implementing the MLJ model interface.

From MLJ, the type can be imported using

BayesianLDA = @load BayesianLDA pkg=MultivariateStats

Do model = BayesianLDA() to construct an instance with default hyper-parameters. Provide keyword arguments to override hyper-parameter defaults, as in BayesianLDA(method=...).

The Bayesian multiclass LDA algorithm learns a projection matrix as described in ordinary LDA. Predicted class posterior probability distributions are derived by applying Bayes' rule with a multivariate Gaussian class-conditional distribution. A prior class distribution can be specified by the user or inferred from training data class frequency.

See also the package documentation. For more information about the algorithm, see Li, Zhu and Ogihara (2006): Using Discriminant Analysis for Multi-class Classification: An Experimental Investigation.

Training data

In MLJ or MLJBase, bind an instance model to data with

mach = machine(model, X, y)

Here:

  • X is any table of input features (eg, a DataFrame) whose columns are of scitype Continuous; check column scitypes with schema(X).
  • y is the target, which can be any AbstractVector whose element scitype is OrderedFactor or Multiclass; check the scitype with scitype(y)

Train the machine using fit!(mach, rows=...).

Hyper-parameters

  • method::Symbol=:gevd: choice of solver, one of :gevd or :whiten methods.
  • cov_w::StatsBase.SimpleCovariance(): An estimator for the within-class covariance (used in computing the within-class scatter matrix, Sw). Any robust estimator from CovarianceEstimation.jl can be used.
  • cov_b::StatsBase.SimpleCovariance(): The same as cov_w but for the between-class covariance (used in computing the between-class scatter matrix, Sb).
  • outdim::Int=0: The output dimension, i.e., dimension of the transformed space, automatically set to min(indim, nclasses-1) if equal to 0.
  • regcoef::Float64=1e-6: The regularization coefficient. A positive value regcoef*eigmax(Sw) where Sw is the within-class scatter matrix, is added to the diagonal of Sw to improve numerical stability. This can be useful if using the standard covariance estimator.
  • priors::Union{Nothing, UnivariateFinite{<:Any, <:Any, <:Any, <:Real}, Dict{<:Any, <:Real}} = nothing: For use in prediction with Bayes rule. If priors = nothing then priors are estimated from the class proportions in the training data. Otherwise it requires a Dict or UnivariateFinite object specifying the classes with non-zero probabilities in the training target.

Operations

  • transform(mach, Xnew): Return a lower dimensional projection of the input Xnew, which should have the same scitype as X above.
  • predict(mach, Xnew): Return predictions of the target given features Xnew, which should have the same scitype as X above. Predictions are probabilistic but uncalibrated.
  • predict_mode(mach, Xnew): Return the modes of the probabilistic predictions returned above.

Fitted parameters

The fields of fitted_params(mach) are:

  • classes: The classes seen during model fitting.
  • projection_matrix: The learned projection matrix, of size (indim, outdim), where indim and outdim are the input and output dimensions respectively (See Report section below).
  • priors: The class priors for classification. As inferred from training target y, if not user-specified. A UnivariateFinite object with levels consistent with levels(y).

Report

The fields of report(mach) are:

  • indim: The dimension of the input space i.e the number of training features.
  • outdim: The dimension of the transformed space the model is projected to.
  • mean: The mean of the untransformed training data. A vector of length indim.
  • nclasses: The number of classes directly observed in the training data (which can be less than the total number of classes in the class pool).
  • class_means: The class-specific means of the training data. A matrix of size (indim, nclasses) with the ith column being the class-mean of the ith class in classes (See fitted params section above).
  • class_weights: The weights (class counts) of each class. A vector of length nclasses with the ith element being the class weight of the ith class in classes. (See fitted params section above.)
  • Sb: The between class scatter matrix.
  • Sw: The within class scatter matrix.

Examples

using MLJ

BayesianLDA = @load BayesianLDA pkg=MultivariateStats

X, y = @load_iris ## a table and a vector

model = BayesianLDA()
mach = machine(model, X, y) |> fit!

Xproj = transform(mach, X)
y_hat = predict(mach, X)
labels = predict_mode(mach, X)

See also LDA, SubspaceLDA, BayesianSubspaceLDA