## 13 Expectation Maximization

Learn how to use Expectation Maximization Clustering algorithm.

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Expectation Maximization (EM) estimation of mixture models is a popular probability density estimation technique that is used in a variety of applications. Oracle Data Mining uses EM to implement a distribution-based clustering algorithm (EM-clustering).

#### 13.1.1 Expectation Step and Maximization Step

Expectation Maximization is an iterative method. It starts with an initial parameter guess. The parameter values are used to compute the likelihood of the current model. This is the Expectation step. The parameter values are then recomputed to maximize the likelihood. This is the Maximization step. The new parameter estimates are used to compute a new expectation and then they are optimized again to maximize the likelihood. This iterative process continues until model convergence.

#### 13.1.2 Probability Density Estimation

In density estimation, the goal is to construct a density function that captures how a given population is distributed. In probability density estimation, the density estimate is based on observed data that represents a sample of the population. Areas of high data density in the model correspond to the peaks of the underlying distribution.

Density-based clustering is conceptually different from distance-based clustering (for example k-Means) where emphasis is placed on minimizing inter-cluster and maximizing the intra-cluster distances. Due to its probabilistic nature, density-based clustering can compute reliable probabilities in cluster assignment. It can also handle missing values automatically.

### 13.2 Algorithm Enhancements

Although Expectation Maximization (EM) is well established as a distribution-based clustering algorithm, it presents some challenges in its standard form. The Oracle Data Mining implementation includes significant enhancements, such as scalable processing of large volumes of data and automatic parameter initialization. The strategies that Oracle Data Mining uses to address the inherent limitations of EM clustering are described further in this section.

Note:

The EM abbreviation is used here to refer to EM-clustering.

Limitations of Standard Expectation Maximization:

• Scalability: EM has linear scalability with the number of records and attributes. The number of iterations to convergence tends to increase with growing data size (both rows and columns). EM convergence can be slow for complex problems and can place a significant load on computational resources.

• High dimensionality: EM has limited capacity for modeling high dimensional (wide) data. The presence of many attributes slows down model convergence, and the algorithm becomes less able to distinguish between meaningful attributes and noise. The algorithm is thus compromised in its ability to find correlations.

• Number of components: EM typically requires the user to specify the number of components. In most cases, this is not information that the user can know in advance.

• Parameter initialization: The choice of appropriate initial parameter values can have a significant effect on the quality of the model. Initialization strategies that have been used for EM have generally been computationally expensive.

• From components to clusters: EM model components are often treated as clusters. This approach can be misleading since cohesive clusters are often modeled by multiple components. Clusters that have a complex shape need to be modeled by multiple components.

#### 13.2.1 Scalability

Expectation Maximization (EM) in Oracle Data Mining, uses database parallel processing to achieve excellent scalability.

The Oracle Data Mining implementation of Expectation Maximization (EM) uses database parallel processing to achieve excellent scalability. EM computations naturally lend themselves to row parallel processing, and the partial results are easily aggregated. The parallel implementation efficiently distributes the computationally intensive work across slave processes and then combines the partial results to produce the final solution.

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#### 13.2.2 High Dimensionality

The Oracle Data Mining implementation of Expectation Maximization (EM) can efficiently process high-dimensional data with thousands of attributes. This is achieved through a two-fold process:

• The data space of single-column (not nested) attributes is analyzed for pair-wise correlations. Only attributes that are significantly correlated with other attributes are included in the EM mixture model. The algorithm can also be configured to restrict the dimensionality to the M most correlated attributes.

• High-dimensional (nested) numerical data that measures events of similar type is projected into a set of low-dimensional features that are modeled by EM. Some examples of high-dimensional, numerical data are: text, recommendations, gene expressions, and market basket data.

#### 13.2.3 Number of Components

Typical implementations of Expectation Maximization (EM) require the user to specify the number of model components. This is problematic because users do not generally know the correct number of components. Choosing too many or too few components can lead to over-fitting or under-fitting, respectively.

When model search is enabled, the number of EM components is automatically determined. The algorithm uses a held-aside sample to determine the correct number of components, except in the cases of very small data sets when Bayesian Information Criterion (BIC) regularization is used.

#### 13.2.4 Parameter Initialization

Choosing appropriate initial parameter values can have a significant effect on the quality of the solution. Expectation Maximization (EM) is not guaranteed to converge to the global maximum of the likelihood function but may instead converge to a local maximum. Therefore different initial parameter values can lead to different model parameters and different model quality.

In the process of model search, the EM model is grown independently. As new components are added, their parameters are initialized to areas with poor distribution fit.

#### 13.2.5 From Components to Clusters

Expectation Maximization (EM) model components are often treated as clusters. However, this approach can be misleading. Cohesive clusters are often modeled by multiple components. The shape of the probability density function used in EM effectively predetermines the shape of the identified clusters. For example, Gaussian density functions can identify single peak symmetric clusters. Clusters of more complex shape need to be modeled by multiple components.

Ideally, high density areas of arbitrary shape must be interpreted as single clusters. To accomplish this, the Oracle Data Mining implementation of EM builds a component hierarchy that is based on the overlap of the individual components' distributions. Oracle Data Mining EM uses agglomerative hierarchical clustering. Component distribution overlap is measured using the Bhattacharyya distance function. Choosing an appropriate cutoff level in the hierarchy automatically determines the number of high-level clusters.

The Oracle Data Mining implementation of EM produces an assignment of the model components to high-level clusters. Statistics like means, variances, modes, histograms, and rules additionally describe the high-level clusters. The algorithm can be configured to either produce clustering assignments at the component level or at the cluster level.

### 13.3 Configuring the Algorithm

Configure Expectation Maximization (EM).

In Oracle Data Mining, Expectation Maximization (EM) can effectively model very large data sets (both rows and columns) without requiring the user to supply initialization parameters or specify the number of model components. While the algorithm offers reasonable defaults, it also offers flexibility.

The following list describes some of the configurable aspects of EM:

• Whether or not independent non-nested column attributes are included in the model. The choice is system-determined by default.

• Whether to use Bernoulli or Gaussian distribution for numerical attributes. By default, the algorithm chooses the most appropriate distribution, and individual attributes may use different distributions. When the distribution is user-specified, it is used for all numerical attributes.

• Whether the convergence criterion is based on a held-aside data set or on Bayesian Information Criterion (BIC). The convergence criterion is system-determined by default.

• The percentage improvement in the value of the log likelihood function that is required to add a new component to the model. The default percentage is 0.001.

• Whether to define clusters as individual components or groups of components. Clusters are associated to groups of components by default.

• The maximum number of components in the model. If model search is enabled, the algorithm determines the number of components based on improvements in the likelihood function or based on regularization (BIC), up to the specified maximum.

• Whether the linkage function for the agglomerative clustering step uses the nearest distance within the branch (single linkage), the average distance within the branch (average linkage), or the maximum distance within the branch (complete linkage). By default the algorithm uses single linkage.

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### 13.4 Data Preparation for Expectation Maximization

Learn how to prepare data for Expectation Maximization (EM).

If you use Automatic Data Preparation (ADP), you do not need to specify additional data preparation for Expectation Maximization. ADP normalizes numerical attributes (in non-nested columns) when they are modeled with Gaussian distributions. ADP applies a topN binning transformation to categorical attributes.

Missing value treatment is not needed since Oracle Data Mining algorithms handle missing values automatically. The Expectation Maximization algorithm replaces missing values with the mean in single-column numerical attributes that are modeled with Gaussian distributions. In other single-column attributes (categoricals and numericals modeled with Bernoulli distributions), NULLs are not replaced; they are treated as a distinct value with its own frequency count. In nested columns, missing values are treated as zeros.

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