Probabilistic Principal Component Analysis¶
Probabilistic Principal Component Analysis (PPCA) represents a constrained form of the Gaussian distribution in which the number of free parameters can be restricted while still allowing the model to capture the dominant correlations in a data set. It is expressed as the maximum likelihood solution of a probabilistic latent variable model [BSHP06].
This package defines a PPCA
type to represent a probabilistic PCA model, and provides a set of methods to access the properties.
Properties¶
Let M
be an instance of PPCA
, d
be the dimension of observations, and p
be the output dimension (i.e the dimension of the principal subspace)

indim
(M)¶ Get the input dimension
d
, i.e the dimension of the observation space.

outdim
(M)¶ Get the output dimension
p
, i.e the dimension of the principal subspace.

mean
(M)¶ Get the mean vector (of length
d
).

projection
(M)¶ Get the projection matrix (of size
(d, p)
). Each column of the projection matrix corresponds to a principal component.The principal components are arranged in descending order of the corresponding variances.

loadings
(M)¶ The factor loadings matrix (of size
(d, p)
).

var
(M)¶ The total residual variance.
Transformation and Construction¶
Given a probabilistic PCA model M
, one can use it to transform observations into latent variables, as
or use it to reconstruct (approximately) the observations from latent variables, as
Here, is the factor loadings or weight matrix.
The package provides methods to do so:

transform
(M, x)¶ Transform observations
x
into latent variables.Here,
x
can be either a vector of lengthd
or a matrix where each column is an observation.

reconstruct
(M, z)¶ Approximately reconstruct observations from the latent variable given in
z
.Here,
y
can be either a vector of lengthp
or a matrix where each column gives the latent variables for an observation.
Data Analysis¶
One can use the fit
method to perform PCA over a given dataset.

fit
(PPCA, X; ...)¶ Perform probabilistic PCA over the data given in a matrix
X
. Each column ofX
is an observation.This method returns an instance of
PCA
.Keyword arguments:
Let
(d, n) = size(X)
be respectively the input dimension and the number of observations:name description default method The choice of methods:
:ml
: use maximum likelihood version of probabilistic PCA:em
: use EM version of probabilistic PCA:bayes
: use Bayesian PCA
:ml
maxoutdim Maximum output dimension. d1
mean The mean vector, which can be either of:
0
: the input data has already been centralizednothing
: this function will compute the mean a precomputed mean vector
nothing
tol Convergence tolerance 1.0e6
tot Maximum number of iterations 1000
Notes:
 This function calls
ppcaml
,ppcaem
orbayespca
internally, depending on the choice of method.
Example:
using MultivariateStats
# suppose Xtr and Xte are training and testing data matrix,
# with each observation in a column
# train a PCA model
M = fit(PPCA, Xtr; maxoutdim=100)
# apply PCA model to testing set
Yte = transform(M, Xte)
# reconstruct testing observations (approximately)
Xr = reconstruct(M, Yte)
Core Algorithms¶
Three algorithms are implemented in this package: ppcaml
, ppcaem
, and bayespca
.

ppcaml
(Z, mean, tw; ...)¶ Compute probabilistic PCA using on maximum likelihood formulation for a centralized sample matrix
Z
.Parameters:  Z – provides centralized samples.
 mean – The mean vector of the original samples, which can be a vector of length
d
, or an empty vectorFloat64[]
indicating a zero mean.
Returns: The resultant PPCA model.
Note: This function accepts two keyword arguments:
maxoutdim
andtol
.

ppcaem
(S, mean, n; ...)¶ Compute probabilistic PCA based on expectationmaximization algorithm for a given sample covariance matrix
S
.Parameters:  S – The sample covariance matrix.
 mean – The mean vector of original samples, which can be a vector of length
d
, or an empty vectorFloat64[]
indicating a zero mean.  n – The number of observations.
Returns: The resultant PPCA model.
Note: This function accepts two keyword arguments:
maxoutdim
,tol
, andtot
.

bayespca
(S, mean, n; ...)¶ Compute probabilistic PCA based on Bayesian algorithm for a given sample covariance matrix
S
.Parameters:  S – The sample covariance matrix.
 mean – The mean vector of original samples, which can be a vector of length
d
, or an empty vectorFloat64[]
indicating a zero mean.  n – The number of observations.
Returns: The resultant PPCA model.
Note: This function accepts two keyword arguments:
maxoutdim
,tol
, andtot
.Additional notes:
 Function uses the
maxoutdim
parameter as an upper boundary when it automatically determines the latent space dimensionality.