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SubspaceLDA

SubspaceLDA

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

From MLJ, the type can be imported using

SubspaceLDA = @load SubspaceLDA pkg=MultivariateStats

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

Multiclass subspace linear discriminant analysis (LDA) is a variation on ordinary LDA suitable for high dimensional data, as it avoids storing scatter matrices. For details, refer the MultivariateStats.jl documentation.

In addition to dimension reduction (using transform) probabilistic classification is provided (using predict). In the case of classification, the class probability for a new observation reflects the proximity of that observation to training observations associated with that class, and how far away the observation is from observations associated with other classes. Specifically, the distances, in the transformed (projected) space, of a new observation, from the centroid of each target class, is computed; the resulting vector of distances, multiplied by minus one, is passed to a softmax function to obtain a class probability prediction. Here "distance" is computed using a user-specified distance function.

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

  • normalize=true: Option to normalize the between class variance for the number of observations in each class, one of true or false.
  • outdim: the ouput dimension, automatically set to min(indim, nclasses-1) if equal to 0. If a non-zero outdim is passed, then the actual output dimension used is min(rank, outdim) where rank is the rank of the within-class covariance matrix.
  • dist=Distances.SqEuclidean(): The distance metric to use when performing classification (to compare the distance between a new point and centroids in the transformed space); must be a subtype of Distances.SemiMetric from Distances.jl, e.g., Distances.CosineDist.

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 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).

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.)
  • explained_variance_ratio: The ratio of explained variance to total variance. Each dimension corresponds to an eigenvalue.

Examples

using MLJ

SubspaceLDA = @load SubspaceLDA pkg=MultivariateStats

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

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

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

See also LDA, BayesianLDA, BayesianSubspaceLDA