Expectation-Maximization
Expectation-Maximization (EM) algorithm is an iterative method for
finding the maximum likelihood and maximum a posteriori estimates of
parameters in models that typically depend on hidden variables.
While serving as a clustering technique, EM is also used in
non-linear dimensionality reduction, missing value problems, and
other areas.
Details
Given a set 
of dimension
X
of n
feature vectors
of dimension p
, the problem is to find a maximum-likelihood estimate of
the parameters of the underlying distribution when the data is
incomplete or has missing values.Expectation-Maximization (EM) Algorithm in the General Form
Let 
depending on the parameters 
. Let 
be the latent or
missing data, so that 
is the complete data with
log-likelihood 
. The algorithm for solving the
problem in its general form is the following EM algorithm
([Dempster77],
[Hastie2009]):
X
be the observed data which has log-likelihood
depending on the parameters . Let be the latent or
missing data, so that is the complete data with
log-likelihood . The algorithm for solving the
problem in its general form is the following EM algorithm
([Dempster77],
[Hastie2009]):- Choose initial values of the parameters
. - Expectation step: in thej-th step, compute
as a
function of the dummy argument 
. - Maximization step: in thej-th step, calculate the new estimate
by maximizing 
over . - Repeat steps 2 and 3 until convergence.
EM algorithm for the Gaussian Mixture Model
Gaussian Mixture Model (GMM) is a mixture of k p-dimensional
multivariate Gaussian distributions represented as
where 
and 
.
and .The 
is the probability density function with
parameters 
, where 
the vector of means, and 
is the
variance-covariance matrix. The probability density function for a
is the probability density function with
parameters , where
the vector of means, and is the
variance-covariance matrix. The probability density function for a
p
-dimensional multivariate Gaussian distribution is defined as
follows:
Let 
be
the indicator function and 
.
be
the indicator function and .Computation
The EM algorithm for GMM includes the following steps:
Define the weights as follows:
for 
and 
.
and .- Choose initial values of the parameters:
- Expectation step: in thej-th step, compute the matrix
with the weights 
- Maximization step: in thej-th step, for all
compute:- The mixture weights
, where 
is the “amount” of the feature vectors that are assigned
to the r-th mixture component - Mean estimates
- Covariance estimate
of size 
with 
- Repeat steps 2 and 3 until any of these conditions is met:
, where the likelihood function is:
- The number of iterations exceeds the predefined level.
Initialization
The EM algorithm for GMM requires initialized vector of
weights, vectors of means, and variance-covariance
[Biernacki2003, Maitra2009].
The EM initialization algorithm for GMM includes the following
steps:
- Perform nTrials starts of the EM algorithm with nIterations iterations and start values:
- Initial means -kdifferent random observations from the input data set
- Initial weights - the values of
- Initial covariance matrices - the covariance of the input data
- Regard the result of the best EM algorithm in terms of the likelihood function values as the result of initialization
Initialization
The EM algorithm for GMM requires initialized vector of weights,
vectors of means, and variance-covariance. Skip the initialization
step if you already calculated initial weights, means, and
covariance matrices.
Batch Processing
Algorithm Input
The EM for GMM initialization algorithm accepts the input
described below. Pass the
Input ID
as a parameter to the methods
that provide input for your algorithm.Input ID | Input |
|---|---|
data | Pointer to the  numeric table with the data to which the EM
initialization algorithm is applied. The input can be an object of any
class derived from NumericTable. |
Algorithm Parameters
The EM for GMM initialization algorithm has the following parameters:
Parameter | Default Value | Description |
|---|---|---|
algorithmFPType | float | The floating-point type that the algorithm uses for intermediate computations. Can be float or double . |
method | defaultDense | Performance-oriented computation method, the only method supported by the algorithm. |
nComponents | Not applicable | The number of components in the Gaussian Mixture Model, a required parameter. |
nTrials | 20 | The number of starts of the EM algorithm. |
nIterations | 10 | The maximal number of iterations in each start of the EM algorithm. |
accuracyThreshold | 1.0e-04 | The threshold for termination of the algorithm. |
covarianceStorage | full | Covariance matrix storage scheme in the Gaussian Mixture Model:
|
engine | SharePtr< engines:: mt19937:: Batch>() | Pointer to the random number generator engine that is used internally to get the initial means in each EM start. |
.
Only diagonal elements of the matrix are updated during the
processing, and the rest are assumed to be zero.Algorithm Output
The EM for GMM initialization algorithm calculates the results described below.
Pass the
Result ID
as a parameter to the methods that access the results of your algorithm.Result ID | Result |
|---|---|
weights | Pointer to the  numeric table with mixture weights.By default, this result is an object of the HomogenNumericTable class,
but you can define the result as an object of any class derived from NumericTable
except PackedTriangularMatrix , PackedSymmetricMatrix , and CSRNumericTable . |
means | Pointer to the  numeric table with each row containing the estimate
of the means for the i -th mixture component, where  .By default, this result is an object of the HomogenNumericTable class,
but you can define the result as an object of any class derived from NumericTable
except PackedTriangularMatrix , PackedSymmetricMatrix , and CSRNumericTable . |
covariances | Pointer to the DataCollection object that contains k numeric tables,
each with the  variance-covariance matrix for the i -th mixture
component of size:
By default, the collection contains objects of the HomogenNumericTable class,
but you can define them as objects of any class derived from NumericTable
except PackedTriangularMatrix and CSRNumericTable . |
Computation
Batch Processing
Algorithm Input
The EM for GMM algorithm accepts the input described below.
Pass the
Input ID
as a parameter to the methods that provide input for your algorithm.Input ID | Input |
|---|---|
data | Pointer to the numeric table with the data to which the EM
algorithm is applied. The input can be an object of any class derived
from NumericTable . |
inputWeights | Pointer to the numeric table with initial mixture weights. This input can be an object of any class derived from NumericTable. |
inputMeans | Pointer to a numeric table. Each row in this table contains the
initial value of the means for the i -th mixture component, where  .
This input can be an object of any class derived from NumericTable . |
inputCovariances | Pointer to the DataCollection object that contains k numeric tables,
each with the variance-covariance matrix for the i -th mixture component of size:
The collection can contain objects of any class derived from NumericTable. |
inputValues | Pointer to the result of the EM for GMM initialization algorithm. The
result of initialization contains weights, means, and a collection of
covariances. You can use this input to set the initial values for the EM
for GMM algorithm instead of explicitly specifying the weights, means,
and covariance collection. |
Algorithm Parameters
The EM for GMM algorithm has the following parameters:
Parameter | Default Value | Description |
|---|---|---|
algorithmFPType | float | The floating-point type that the algorithm uses for intermediate computations. Can be float or double . |
method | defaultDense | Performance-oriented computation method, the only method supported by the algorithm. |
nComponents | Not applicable | The number of components in the Gaussian Mixture Model, a required parameter. |
maxIterations | 10 | The maximal number of iterations in the algorithm. |
accuracyThreshold | 1.0e-04 | The threshold for termination of the algorithm. |
covariance | Pointer to an object of the BatchIface class | Pointer to the algorithm that computes the covariance matrix. By default, the respective oneDAL algorithm is used,
implemented in the class derived from BatchIface . |
regularizationFactor | 0.01 | Factor for covariance regularization in the case of ill-conditional data. |
covarianceStorage | full | Covariance matrix storage scheme in the Gaussian Mixture Model:
|
Algorithm Output
The EM for GMM algorithm calculates the results described below. Pass
the
Result ID
as a parameter to the methods that access the results
of your algorithm.Result ID | Result |
|---|---|
weights | Pointer to the numeric table with mixture weights.By default, this result is an object of the HomogenNumericTable class,
but you can define the result as an object of any class derived from NumericTable
except PackedTriangularMatrix , PackedSymmetricMatrix , and CSRNumericTable . |
means | Pointer to the numeric table with each row containing the estimate
of the means for the i -th mixture component, where .By default, this result is an object of the HomogenNumericTable class,
but you can define the result as an object of any class derived from NumericTable
except PackedTriangularMatrix , PackedSymmetricMatrix , and CSRNumericTable . |
covariances | Pointer to the DataCollection object that contains k numeric tables,
each with the variance-covariance matrix for the i -th mixture component of size:
By default, the collection contains objects of the HomogenNumericTable class,
but you can define them as objects of any class derived from NumericTable
except PackedTriangularMatrix and CSRNumericTable . |
goalFunction | Pointer to the  numeric table with the value of the logarithm of
the likelihood function after the last iteration.By default, this result is an object of the HomogenNumericTable class. |
nIterations | Pointer to the numeric table with the number of iterations
computed after completion of the algorithm.By default, this result is an object of the HomogenNumericTable class. |
Examples
C++ (CPU)
Batch Processing:
Java*
There is no support for Java on GPU.
Batch Processing:
Python*
Batch Processing:
Performance Considerations
To get the best overall performance of the expectation-maximization
algorithm at the initialization and computation stages:
- If input data is homogeneous, provide the input data and store results in homogeneous numeric tables of the same type as specified in the algorithmFPType class template parameter.
- If input data is non-homogeneous, use AOS layout rather than SOA layout.
Product and Performance Information |
|---|
Performance varies by use, configuration and other factors.
Learn more at www.Intel.com/PerformanceIndex. Notice revision #20201201 |