Clustering module (WIP)

Provide clustering methods and missing values imputation / collaborative filtering / reccomendation systems using clustering methods as backend.

The module provides the following functions. Use ?[function] to access their full signature and detailed documentation:

• initRepresentatives(X,K;initStrategy,Z₀): Initialisation strategies for Kmean and Kmedoids
• `kmeans(X,K;dist,initStrategy,Z₀)`: Classical KMean algorithm
• `kmedoids(X,K;dist,initStrategy,Z₀)`: Kmedoids algorithm
• `em(X,K;p₀,mixtures,tol,verbosity,minVariance,minCovariance,initStrategy)`: EM algorithm over GMM
• `predictMissing(X,K;p₀,mixtures,tol,verbosity,minVariance,minCovariance)`: Fill mixing values / collaborative filtering using EM as backbone

{Spherical|Diagonal|Full}Gaussian mixtures for em / predictMissing are already provided. User defined mixtures can be used defining a struct as subtype of Mixture and implementing for that mixture the following functions:

• initMixtures!(mixtures, X; minVariance, minCovariance, initStrategy)
• lpdf(m,x,mask) (for the e-step)
• updateParameters!(mixtures, X, pₙₖ; minVariance, minCovariance) (the m-step)

em(X,K;p₀,mixtures,tol,verbosity,minVariance,minCovariance,initStrategy)

Compute Expectation-Maximisation algorithm to identify K clusters of X data, i.e. employ a Generative Mixture Model as the underlying probabilistic model.

X can contain missing values in some or all of its dimensions. In such case the learning is done only with the available data. Implemented in the log-domain for better numerical accuracy with many dimensions.

Parameters:

• X : A (n x d) data to clusterise
• K : Number of cluster wanted
• p₀ : Initial probabilities of the categorical distribution (K x 1) [default: nothing]
• mixtures: An array (of length K) of the mixture to employ (see notes) [def: [DiagonalGaussian() for i in 1:K]]
• tol: Tolerance to stop the algorithm [default: 10^(-6)]
• verbosity: A verbosity parameter regulating the information messages frequency [def: STD]
• minVariance: Minimum variance for the mixtures [default: 0.05]
• minCovariance: Minimum covariance for the mixtures with full covariance matrix [default: 0]. This should be set different than minVariance (see notes).
• initStrategy: Mixture initialisation algorithm [def: grid]

Returns:

• A named touple of:
  • pⱼₓ: Matrix of size (N x K) of the probabilities of each point i to belong to cluster j
  • pⱼ : Probabilities of the categorical distribution (K x 1)
  • μ : Means (K x d) of the Gaussian
  • σ² : Variance of the gaussian (K x 1). We assume here that the gaussian has the same variance across all the dimensions
  • ϵ : Vector of the discrepancy (matrix norm) between pⱼₓ and the lagged pⱼₓ at each iteration
  • lL : The log-likelihood (without considering the last mixture optimisation)
  • BIC : The Bayesian Information Criterion (lower is better)
  • AIC : The Akaike Information Criterion (lower is better)

Notes:

• The mixtures currently implemented are SphericalGaussian(μ,σ²),DiagonalGaussian(μ,σ²) and FullGaussian(μ,σ²)
• Reasonable choices for the minVariance/Covariance depends on the mixture. For example 0.25 seems a reasonable value for the SphericalGaussian, 0.05 seems better for the DiagonalGaussian, and FullGaussian seems to prefer either very low values of variance/covariance (e.g. (0.05,0.05) ) or very big but similar ones (e.g. (100,100) ).
• For initStrategy, look at the documentation of initMixtures! for the mixture you want. The provided gaussian mixtures support either grid or kmeans. grid is faster (expecially if X contains missing values), but kmeans often provides better results.

Resources:

• Paper describing EM with missing values
• Class notes from MITx 6.86x (Sec 15.9)
• Limitations of EM

Example:

julia> clusters = em([1 10.5;1.5 0; 1.8 8; 1.7 15; 3.2 40; 0 0; 3.3 38; 0 -2.3; 5.2 -2.4],3,verbosity=HIGH)

initMixtures!(mixtures::Array{T,1}, X; minVariance=0.25, minCovariance=0.0, initStrategy="grid")

TThe parameter initStrategy can be either grid or kmeans. If the data contains missing values, a first run of predictMissing is done under init=grid to impute the missing values just to allow the kmeans algorithm. Then the em algorithm is used with the output of kmean as init values.

initRepresentatives(X,K;initStrategy,Z₀)

Initialisate the representatives for a K-Mean or K-Medoids algorithm

Parameters:

• X: a (N x D) data to clusterise
• K: Number of cluster wonted
• initStrategy: Wheter to select the initial representative vectors:
  • random: randomly in the X space
  • grid: using a grid approach [default]
  • shuffle: selecting randomly within the available points
  • given: using a provided set of initial representatives provided in the Z₀ parameter
• Z₀: Provided (K x D) matrix of initial representatives (used only together with the given initStrategy) [default: nothing]

Returns:

• A (K x D) matrix of initial representatives

Example:

julia> Z₀ = initRepresentatives([1 10.5;1.5 10.8; 1.8 8; 1.7 15; 3.2 40; 3.6 32; 3.6 38],2,initStrategy="given",Z₀=[1.7 15; 3.6 40])

kmeans(X,K;dist,initStrategy,Z₀)

Compute K-Mean algorithm to identify K clusters of X using Euclidean distance

Parameters:

• X: a (N x D) data to clusterise
• K: Number of cluster wonted
• dist: Function to employ as distance (see notes). Default to Euclidean distance.
• initStrategy: Wheter to select the initial representative vectors:
  • random: randomly in the X space
  • grid: using a grid approach [default]
  • shuffle: selecting randomly within the available points
  • given: using a provided set of initial representatives provided in the Z₀ parameter
• Z₀: Provided (K x D) matrix of initial representatives (used only together with the given initStrategy) [default: nothing]

Returns:

• A tuple of two items, the first one being a vector of size N of ids of the clusters associated to each point and the second one the (K x D) matrix of representatives

Notes:

• Some returned clusters could be empty
• The dist parameter can be:
  • Any user defined function accepting two vectors and returning a scalar
  • An anonymous function with the same characteristics (e.g. dist = (x,y) -> norm(x-y)^2)
  • One of the above predefined distances: l1_distance, l2_distance, l2²_distance, cosine_distance

Example:

julia> (clIdx,Z) = kmeans([1 10.5;1.5 10.8; 1.8 8; 1.7 15; 3.2 40; 3.6 32; 3.3 38; 5.1 -2.3; 5.2 -2.4],3)

kmedoids(X,K;dist,initStrategy,Z₀)

Compute K-Medoids algorithm to identify K clusters of X using distance definition dist

Parameters:

• X: a (n x d) data to clusterise
• K: Number of cluster wonted
• dist: Function to employ as distance (see notes). Default to Euclidean distance.
• initStrategy: Wheter to select the initial representative vectors:
  • random: randomly in the X space
  • grid: using a grid approach
  • shuffle: selecting randomly within the available points [default]
  • given: using a provided set of initial representatives provided in the Z₀ parameter
• Z₀: Provided (K x D) matrix of initial representatives (used only together with the given initStrategy) [default: nothing]

Returns:

• A tuple of two items, the first one being a vector of size N of ids of the clusters associated to each point and the second one the (K x D) matrix of representatives

Notes:

• Some returned clusters could be empty
• The dist parameter can be:
  • Any user defined function accepting two vectors and returning a scalar
  • An anonymous function with the same characteristics (e.g. dist = (x,y) -> norm(x-y)^2)
  • One of the above predefined distances: l1_distance, l2_distance, l2²_distance, cosine_distance

Example:

julia> (clIdx,Z) = kmedoids([1 10.5;1.5 10.8; 1.8 8; 1.7 15; 3.2 40; 3.6 32; 3.3 38; 5.1 -2.3; 5.2 -2.4],3,initStrategy="grid")

lpdf(m::DiagonalGaussian,x,mask) - Log PDF of the mixture given the observation x

lpdf(m::FullGaussian,x,mask) - Log PDF of the mixture given the observation x

lpdf(m::SphericalGaussian,x,mask) - Log PDF of the mixture given the observation x

predictMissing(X,K;p₀,mixtures,tol,verbosity,minVariance,minCovariance)

Fill missing entries in a sparse matrix assuming an underlying Gaussian Mixture probabilistic Model (GMM) and implementing an Expectation-Maximisation algorithm.

While the name of the function is predictMissing, the function can be used also for system reccomendation / collaborative filtering and GMM-based regressions.

Implemented in the log-domain for better numerical accuracy with many dimensions.

Parameters:

• X : A (N x D) sparse matrix of data to fill according to a GMM model
• K : Number of mixtures desired
• p₀ : Initial probabilities of the categorical distribution (K x 1) [default: nothing]
• mixtures: An array (of length K) of the mixture to employ (see notes) [def: [DiagonalGaussian() for i in 1:K]]
• tol: Tolerance to stop the algorithm [default: 10^(-6)]
• verbosity: A verbosity parameter regulating the information messages frequency [def: STD]
• minVariance: Minimum variance for the mixtures [default: 0.05]
• minCovariance: Minimum covariance for the mixtures with full covariance matrix [default: 0]. This should be set different than minVariance (see notes).
• initStrategy: Mixture initialisation algorithm [def: grid]

Returns:

• A named touple of:

  • ̂X̂ : The Filled Matrix of size (N x D)
  • nFill: The number of items filled
  • lL : The log-likelihood (without considering the last mixture optimisation)
  • BIC : The Bayesian Information Criterion (lower is better)
  • AIC : The Akaike Information Criterion (lower is better)

  Notes:

  • The mixtures currently implemented are SphericalGaussian(μ,σ²),DiagonalGaussian(μ,σ²) and FullGaussian(μ,σ²)
  • For initStrategy, look at the documentation of initMixtures! for the mixture you want. The provided gaussian mixtures support either grid or kmeans. grid is faster, but kmeans often provides better results.

Example:

julia>  cFOut = predictMissing([1 10.5;1.5 missing; 1.8 8; 1.7 15; 3.2 40; missing missing; 3.3 38; missing -2.3; 5.2 -2.4],3)