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# Expectation Maximization

The Expectation Maximization(EM) algorithm estimates the parameters of the multivariate probability density function in the form of a Gaussian mixture distribution with a specified number of mixtures.

Consider the set of the N feature vectors { x_1, x_2,...,x_{N} } from a d-dimensional Euclidean space drawn from a Gaussian mixture:

p(x;a_k,S_k, \pi _k) = \sum _{k=1}^{m} \pi _kp_k(x), \quad \pi _k \geq 0, \quad \sum _{k=1}^{m} \pi _k=1,

p_k(x)= \varphi (x;a_k,S_k)= \frac{1}{(2\pi)^{d/2}\mid{S_k}\mid^{1/2}} exp \left \{ - \frac{1}{2} (x-a_k)^TS_k^{-1}(x-a_k) \right \} ,

where m is the number of mixtures, p_k is the normal distribution density with the mean a_k and covariance matrix S_k, \pi_k is the weight of the k-th mixture. Given the number of mixtures M and the samples x_i, i=1..N the algorithm finds the maximum-likelihood estimates (MLE) of all the mixture parameters, that is, a_k, S_k and \pi_k :

L(x, \theta )=logp(x, \theta )= \sum _{i=1}^{N}log \left ( \sum _{k=1}^{m} \pi _kp_k(x) \right ) \to \max _{ \theta \in \Theta },

\Theta = \left \{ (a_k,S_k, \pi _k): a_k \in \mathbbm{R} ^d,S_k=S_k^T>0,S_k \in \mathbbm{R} ^{d \times d}, \pi _k \geq 0, \sum _{k=1}^{m} \pi _k=1 \right \} .

The EM algorithm is an iterative procedure. Each iteration includes two steps. At the first step (Expectation step or E-step), you find a probability p_{i,k} (denoted \alpha_{i,k} in the formula below) of sample `i` to belong to mixture `k` using the currently available mixture parameter estimates:

\alpha _{ki} = \frac{\pi_k\varphi(x;a_k,S_k)}{\sum\limits_{j=1}^{m}\pi_j\varphi(x;a_j,S_j)} .

At the second step (Maximization step or M-step), the mixture parameter estimates are refined using the computed probabilities:

\pi _k= \frac{1}{N} \sum _{i=1}^{N} \alpha _{ki}, \quad a_k= \frac{\sum\limits_{i=1}^{N}\alpha_{ki}x_i}{\sum\limits_{i=1}^{N}\alpha_{ki}} , \quad S_k= \frac{\sum\limits_{i=1}^{N}\alpha_{ki}(x_i-a_k)(x_i-a_k)^T}{\sum\limits_{i=1}^{N}\alpha_{ki}}

Alternatively, the algorithm may start with the M-step when the initial values for p_{i,k} can be provided. Another alternative when p_{i,k} are unknown is to use a simpler clustering algorithm to pre-cluster the input samples and thus obtain initial p_{i,k} . Often (including machine learning) the `kmeans()` algorithm is used for that purpose.

One of the main problems of the EM algorithm is a large number of parameters to estimate. The majority of the parameters reside in covariance matrices, which are d \times d elements each where d is the feature space dimensionality. However, in many practical problems, the covariance matrices are close to diagonal or even to \mu_k*I , where I is an identity matrix and \mu_k is a mixture-dependent “scale” parameter. So, a robust computation scheme could start with harder constraints on the covariance matrices and then use the estimated parameters as an input for a less constrained optimization problem (often a diagonal covariance matrix is already a good enough approximation).

References:

- Bilmes98 J. A. Bilmes. A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models. Technical Report TR-97-021, International Computer Science Institute and Computer Science Division, University of California at Berkeley, April 1998.

## EM

class `EM` : public `Algorithm`

The class implements the EM algorithm as described in the beginning of this section. It is inherited from `Algorithm`.

## EM::EM

The constructor of the class

C++:`` `EM::``EM`(int nclusters=EM::DEFAULT_NCLUSTERS, int covMatType=EM::COV_MAT_DIAGONAL, const TermCriteria& termCrit=TermCriteria(TermCriteria::COUNT+TermCriteria::EPS, EM::DEFAULT_MAX_ITERS, FLT_EPSILON) )

Python:`` `cv2.``EM`([nclusters[, covMatType[, termCrit]]]) → <EM object>

  Parameters:

  - nclusters – The number of mixture components in the Gaussian mixture model. Default value of the parameter is `EM::DEFAULT_NCLUSTERS=5`. Some of EM implementation could determine the optimal number of mixtures within a specified value range, but that is not the case in ML yet.
  - covMatType –

    Constraint on covariance matrices which defines type of matrices. Possible values are:

    - EM::COV_MAT_SPHERICAL A scaled identity matrix \mu_k * I. There is the only parameter \mu_k to be estimated for each matrix. The option may be used in special cases, when the constraint is relevant, or as a first step in the optimization (for example in case when the data is preprocessed with PCA). The results of such preliminary estimation may be passed again to the optimization procedure, this time with `covMatType=EM::COV_MAT_DIAGONAL`.
    - EM::COV_MAT_DIAGONAL A diagonal matrix with positive diagonal elements. The number of free parameters is `d` for each matrix. This is most commonly used option yielding good estimation results.
    - EM::COV_MAT_GENERIC A symmetric positively defined matrix. The number of free parameters in each matrix is about d^2/2. It is not recommended to use this option, unless there is pretty accurate initial estimation of the parameters and/or a huge number of training samples.
  - termCrit – The termination criteria of the EM algorithm. The EM algorithm can be terminated by the number of iterations `termCrit.maxCount` (number of M-steps) or when relative change of likelihood logarithm is less than `termCrit.epsilon`. Default maximum number of iterations is `EM::DEFAULT_MAX_ITERS=100`.

## EM::train

Estimates the Gaussian mixture parameters from a samples set.

C++:`` bool `EM::``train`(InputArray samples, OutputArray logLikelihoods=noArray(), OutputArray labels=noArray(), OutputArray probs=noArray())

C++:`` bool `EM::``trainE`(InputArray samples, InputArray means0, InputArray covs0=noArray(), InputArray weights0=noArray(), OutputArray logLikelihoods=noArray(), OutputArray labels=noArray(), OutputArray probs=noArray())

C++:`` bool `EM::``trainM`(InputArray samples, InputArray probs0, OutputArray logLikelihoods=noArray(), OutputArray labels=noArray(), OutputArray probs=noArray())

Python:`` `cv2.EM.``train`(samples[, logLikelihoods[, labels[, probs]]]) → retval, logLikelihoods, labels, probs

Python:`` `cv2.EM.``trainE`(samples, means0[, covs0[, weights0[, logLikelihoods[, labels[, probs]]]]]) → retval, logLikelihoods, labels, probs

Python:`` `cv2.EM.``trainM`(samples, probs0[, logLikelihoods[, labels[, probs]]]) → retval, logLikelihoods, labels, probs

  Parameters:

  - samples – Samples from which the Gaussian mixture model will be estimated. It should be a one-channel matrix, each row of which is a sample. If the matrix does not have `CV_64F` type it will be converted to the inner matrix of such type for the further computing.
  - means0 – Initial means a_k of mixture components. It is a one-channel matrix of nclusters \times dims size. If the matrix does not have `CV_64F` type it will be converted to the inner matrix of such type for the further computing.
  - covs0 – The vector of initial covariance matrices S_k of mixture components. Each of covariance matrices is a one-channel matrix of dims \times dims size. If the matrices do not have `CV_64F` type they will be converted to the inner matrices of such type for the further computing.
  - weights0 – Initial weights \pi_k of mixture components. It should be a one-channel floating-point matrix with 1 \times nclusters or nclusters \times 1 size.
  - probs0 – Initial probabilities p_{i,k} of sample i to belong to mixture component k. It is a one-channel floating-point matrix of nsamples \times nclusters size.
  - logLikelihoods – The optional output matrix that contains a likelihood logarithm value for each sample. It has nsamples \times 1 size and `CV_64FC1` type.
  - labels – The optional output “class label” for each sample: \texttt{labels}_i=\texttt{arg max}_k(p_{i,k}), i=1..N (indices of the most probable mixture component for each sample). It has nsamples \times 1 size and `CV_32SC1` type.
  - probs – The optional output matrix that contains posterior probabilities of each Gaussian mixture component given the each sample. It has nsamples \times nclusters size and `CV_64FC1` type.

Three versions of training method differ in the initialization of Gaussian mixture model parameters and start step:

- train - Starts with Expectation step. Initial values of the model parameters will be estimated by the k-means algorithm.
- trainE - Starts with Expectation step. You need to provide initial means a_k of mixture components. Optionally you can pass initial weights \pi_k and covariance matrices S_k of mixture components.
- trainM - Starts with Maximization step. You need to provide initial probabilities p_{i,k} to use this option.

The methods return `true` if the Gaussian mixture model was trained successfully, otherwise it returns `false`.

Unlike many of the ML models, EM is an unsupervised learning algorithm and it does not take responses (class labels or function values) as input. Instead, it computes the Maximum Likelihood Estimate of the Gaussian mixture parameters from an input sample set, stores all the parameters inside the structure: p_{i,k} in `probs`, a_k in `means` , S_k in `covs[k]`, \pi_k in `weights` , and optionally computes the output “class label” for each sample: \texttt{labels}_i=\texttt{arg max}_k(p_{i,k}), i=1..N (indices of the most probable mixture component for each sample).

The trained model can be used further for prediction, just like any other classifier. The trained model is similar to the `CvNormalBayesClassifier`.

## EM::predict

Returns a likelihood logarithm value and an index of the most probable mixture component for the given sample.

C++:`` Vec2d `EM::``predict`(InputArray sample, OutputArray probs=noArray())`const`

Python:`` `cv2.EM.``predict`(sample[, probs]) → retval, probs

  Parameters:

  - sample – A sample for classification. It should be a one-channel matrix of 1 \times dims or dims \times 1 size.
  - probs – Optional output matrix that contains posterior probabilities of each component given the sample. It has 1 \times nclusters size and `CV_64FC1` type.

The method returns a two-element `double` vector. Zero element is a likelihood logarithm value for the sample. First element is an index of the most probable mixture component for the given sample.

## CvEM::isTrained

Returns `true` if the Gaussian mixture model was trained.

C++:`` bool `EM::``isTrained`()`const`

Python:`` `cv2.EM.``isTrained`() → retval

## EM::read, EM::write

See `Algorithm::read()` and `Algorithm::write()`.

## EM::get, EM::set

See `Algorithm::get()` and `Algorithm::set()`. The following parameters are available:

- `"nclusters"`
- `"covMatType"`
- `"maxIters"`
- `"epsilon"`
- `"weights"` (read-only)
- `"means"` (read-only)
- `"covs"` (read-only)

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