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# Contents

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# Context

#### Learning | Clustering | Data Clustering

Data Clustering is an unsupervised learning algorithm that creates a new variable, [Factor_i]. The states of this variable represent the induced clusters.

Data Clustering can be used for different purposes:

1. For finding observations that look the same;
2. For finding observations that behave the same;
3. For representing an unobserved dimension;
4. For summarizing a set of variables;
5. For compactly representing the joint probability distribution.

From a technical perspective, the segments should be:

1. Homogeneous/pure;
2. Have clear differences with the other segments;
3. Stable.

From a functional perspective, the segments should be:

1. Easy to understand;
2. Operational;
3. A fair representation of the data.

Data Clustering with Bayesian networks is typically based on a Naive structure in which the newly created variable [Factor_i] is the parent of the variables that are used for clustering (usually called the Manifest variables).

This variable being hidden, i.e. with 100% of missing values, the marginal probability distribution of [Factor_i] and the conditional probability distributions of the Manifest variables are initialized with random distributions. Thus, an Expectation-Maximization (EM) algorithm is used to fit these distributions with the data:

1. Expectation: the network is used with its current distributions for computing the posterior probabilities of [Factor_i], for the entire set of observations described in the data set; These probabilities are used for soft imputing [Factor_i];
2. Maximization: based on these imputations, the distributions of the network are updated via Maximum-Likelihood. The algorithm goes back to Expectation, until no significant changes occur to the distributions.

# History

Data Clustering has been updated in versions 5.1 and 5.2.

# New Feature: Meta-Clustering

This new feature has been added for improving the stability of the induced solution (3rd technical quality). It consists in creating a data set made of a subset of the Factors that have been created while searching for the best segmentation, and using Data Clustering on these new variables. The final solution is thus a summary of the best solutions that have been found (4th purpose).

The five Manifest variables (bottom of the graph) are used in the data set for describing the observations.

The Factor variables  [Factor_1],  [Factor_2], and [Factor_3] have been induced with Data Clustering. They are then imputed to create a new data set.

In this example, three Factor variables are used for creating the final solution [Factor_4].

Example

Let's use a data set that contains house sale prices for King County, which includes Seattle. It describes homes sold between May 2014 and May 2015. More precisely, we have extracted the 94 houses that are more than 100 years old, that have been renovated, and come with a basement. For the sake of simplicity, we are just describing the houses here with the 5 Manifest variables below, discretized into 2 bins.

• sqrt_above: Square footage of house apart from basement
• sqft_living15: Living room area in 2015
• sqft_lot: Square footage of the lot
• lat: Latitude coordinate

The wizard below describes the setting used for segmenting these houses:

After 100 steps, the best solution consists in segmenting the houses into 4 groups. The mapping below (Analysis | Report | Target | Relationship with Target Node | Mapping) shows the created states/segments:

• the size of each segment is proportional to its marginal probability (i.e. how many houses belong to this segment),
• the intensity of the blue is proportional to the purity of the associated cluster (1st technical quality), and
• the layout reflects the neighborhood.

This radar chart (Analysis | Report | Target | Posterior Mean Analysis | Radar Chart) allows interpreting the generated segments. As we can see, they are easily distinguishable (2nd technical quality).

This solution with 4 segments thus satisfies the first two technical qualities listed above. However, what about the 3rd one, the stability? Below are the scores of the 10 best solutions that have been generated while learning:

Even though the best solution is made of 4 segments, this is the only solution with 4 clusters, all the other ones having pretty much the same score, but with 3 clusters. Thus, we can assume that a solution with 3 clusters would be more stable.

Using Meta-Clustering on the 10 best solutions (10%) indeed generates a final solution made of 3 clusters.

This mapping juxtaposes the mapping of the initial solution with 4 segments (lower opacity) and of the one corresponding to the meta-clustering solution.

The relationships between the final and initial segments are as follows:

• C1 groups C4 and C2 (the main difference between C4 and C2 was the Square footage of the lot),
• C2 corresponds to C3
• C3 corresponds to C1

# New Feature: Multinet

As stated in the Context,  Data Clustering with Bayesian network is typically done with Expectation-Maximization (EM) on a Naive structure. Thus, this is based on the hypothesis that the Manifests variables are conditionally independent of each others given [Factor_i]. Therefore, the Naive structure is well suited for finding observations that look the same (1st purpose), but not so good for finding observations that behave similarly (2nd purpose). The behavior should be represented with direct relationships between the Manifests.

Our new Multinet clustering is an EM2 algorithm based both on a Naive structure (Look) and on a set of Maximum Weight Spanning Trees (MWST) (Behavior). Once the distributions of the Naive randomly set, the algorithm works as follows:

1. Expectation_Naive: the Naive network is used with its current distributions for computing the posterior probabilities of [Factor_i], for the entire set of observations described in the data set; These probabilities are used for hard imputing [Factor_i], i.e. choosing the state with the highest posterior probability;
2. Maximization_MWST[Factor_i] is used as a breakout variable. A MWST is learned on each subset of data.
3. Expectation_MWST: the joint probabilities of the observations are computed with each MWST and used for updating the imputation of [Factor_i].
4. Maximization_Naive: based on this updated imputation, the distributions of the Naive network are updated via Maximum-Likelihood. Then, the algorithm goes back to Expectation_Naive, until no significant changes occur to the distributions.

Two parameters allow changing the Look/Behavior equilibrium. They can be considered as probabilities to run the Naive and MWST steps at each iteration.

Setting a weight of 0 for Behavior defines a Data Clustering quite similar to the usual one, but based on hard imputation instead of soft imputation.

Example

Let's use the same data set that describes houses in Seattle.

The wizard below describes the setting we used for segmenting these houses:

After 100 steps, the best solution consists in segmenting the houses into three groups. The final network is a Naive Augmented Network, with a direct link between two Manifests variables (that are thus not independent given the segmentation, i.e. the Behavior part). Note that this dependency is valid for C3 only (conclusion drawn after some inference with the network).

The radar chart allows analyzing the Look of the segments.

# New Feature: Heterogeneity Weight

The assumption that the data is homogeneous, given all the Manifest Variables, can sometimes be unrealistic. There may be significant heterogeneity in the data across unobserved groups, and it can bias the machine learned Bayesian networks. This phenomena is known as Unobserved Heterogeneity, i.e. an unobserved variable in the data set.

Data Clustering represents a solution for searching for such hidden unobserved groups (3rd purpose). However, whereas the default scoring function in Data Clustering is based on the entropy of the data, finding heterogenous groups requires modifying the scoring function.

We thus defined an Heterogeneity Index :

,

with

where:

•  is the induced Factor,
•  are the states of the Factor, i.e. the segments used to split the data,
•  is the marginal probability of state , i.e. the relative size of the segment,
•  is the set of Manifest variables,
•  is the entire data set,
•  is the subset of the data that corresponds to the segment
•  is the Mutual Information between the Manifest variable  and the Target node  computed on the data set .

The Heterogeneity Weight allows setting a weight of the Heterogeneity Index in the score, which will thus bias the selection of the solutions toward segmentations that maximize the Mutual Information of the Manifest variables with the Target Node

Example

Let's use the entire data set that describes houses in Seattle, with this subset of Manifest variables:

• Renovated: indicates if the house has been renovated
• Age: Age of the house
• sqft_living15: Living room area in 2015
• long: Longitude coordinate
• lat: Latitude coordinate
• Price (K\$): Price of the house.

After setting Price (K\$) as a Target Node and selecting all the other variables, we use the following settings for Data Clustering:

This returns a solution with 2 segments, generating an Heterogeneity Index of 60%. This indicates thus that using [Factor_i] as a breakout variable would allow increasing by 60% the sum of the Mutual Informations of the Manifest variables with the Target Node.

The Multi-Quadrant below highlights the improvement of the Mutual Information. The points correspond to the Mutual Informations on the entire data set, and the vertical scales shows the variations of the Mutual Informations by splitting the data based on the values of [Factor_i].

The Heterogeneity Index is computed on the Manifest variables that are used during the segmentation only. In order to take into account other variables in the computation of the index, these variables have to be included in the segmentation, with a weight of 0 for preventing them to influence the segmentation.

# New Feature: Random Weights

By default, the weight associated with a variable is set to 1. Whereas a weight of 0 renders the variable purely illustrative, a weight of 2 is equivalent to duplicating the variable in the data set. The option Mutual Information Weight, introduced in version 5.1, allows to weight the variable by taking into account its Mutual Information with the Target node.

As of version 7.0, a new option, Random Weights, allows to randomly modify the weight values while trying the find the best segmentation. The amplitude of the randomness is inversely proportional to the current number of trials, starting thus with the maximum level of randomness and ending with almost no weight modification. This option can be useful for escaping from local minima.