Clustering is an unsupervised learning problem whereby we aim to group subsets of entities with one another based on some notion of similarity. Clustering is often used for exploratory analysis and/or as a component of a hierarchical supervised learning pipeline (in which distinct classifiers or regression models are trained for each cluster).

The `spark.mllib`

package supports the following models:

- Table of contents {:toc}

## K-means

K-means is one of the most commonly used clustering algorithms that clusters the data points into a predefined number of clusters. The `spark.mllib`

implementation includes a parallelized variant of the k-means++ method called kmeans||. The implementation in `spark.mllib`

has the following parameters:

*k*is the number of desired clusters. Note that it is possible for fewer than k clusters to be returned, for example, if there are fewer than k distinct points to cluster.*maxIterations*is the maximum number of iterations to run.*initializationMode*specifies either random initialization or initialization via k-means||.*runs*This param has no effect since Spark 2.0.0.*initializationSteps*determines the number of steps in the k-means|| algorithm.*epsilon*determines the distance threshold within which we consider k-means to have converged.*initialModel*is an optional set of cluster centers used for initialization. If this parameter is supplied, only one run is performed.

**Examples**

## Gaussian mixture

A Gaussian Mixture Model represents a composite distribution whereby points are drawn from one of *k* Gaussian sub-distributions, each with its own probability. The `spark.mllib`

implementation uses the expectation-maximization algorithm to induce the maximum-likelihood model given a set of samples. The implementation has the following parameters:

*k*is the number of desired clusters.*convergenceTol*is the maximum change in log-likelihood at which we consider convergence achieved.*maxIterations*is the maximum number of iterations to perform without reaching convergence.*initialModel*is an optional starting point from which to start the EM algorithm. If this parameter is omitted, a random starting point will be constructed from the data.

**Examples**

## Power iteration clustering (PIC)

Power iteration clustering (PIC) is a scalable and efficient algorithm for clustering vertices of a graph given pairwise similarities as edge properties, described in Lin and Cohen, Power Iteration Clustering. It computes a pseudo-eigenvector of the normalized affinity matrix of the graph via power iteration and uses it to cluster vertices. `spark.mllib`

includes an implementation of PIC using GraphX as its backend. It takes an `RDD`

of `(srcId, dstId, similarity)`

tuples and outputs a model with the clustering assignments. The similarities must be nonnegative. PIC assumes that the similarity measure is symmetric. A pair `(srcId, dstId)`

regardless of the ordering should appear at most once in the input data. If a pair is missing from input, their similarity is treated as zero. `spark.mllib`

's PIC implementation takes the following (hyper-)parameters:

`k`

: number of clusters`maxIterations`

: maximum number of power iterations`initializationMode`

: initialization model. This can be either "random", which is the default, to use a random vector as vertex properties, or "degree" to use normalized sum similarities.

**Examples**

In the following, we show code snippets to demonstrate how to use PIC in `spark.mllib`

.

## Latent Dirichlet allocation (LDA)

Latent Dirichlet allocation (LDA) is a topic model which infers topics from a collection of text documents. LDA can be thought of as a clustering algorithm as follows:

- Topics correspond to cluster centers, and documents correspond to examples (rows) in a dataset.
- Topics and documents both exist in a feature space, where feature vectors are vectors of word counts (bag of words).
- Rather than estimating a clustering using a traditional distance, LDA uses a function based on a statistical model of how text documents are generated.

LDA supports different inference algorithms via `setOptimizer`

function. `EMLDAOptimizer`

learns clustering using expectation-maximization on the likelihood function and yields comprehensive results, while `OnlineLDAOptimizer`

uses iterative mini-batch sampling for online variational inference and is generally memory friendly.

LDA takes in a collection of documents as vectors of word counts and the following parameters (set using the builder pattern):

`k`

: Number of topics (i.e., cluster centers)`optimizer`

: Optimizer to use for learning the LDA model, either`EMLDAOptimizer`

or`OnlineLDAOptimizer`

`docConcentration`

: Dirichlet parameter for prior over documents' distributions over topics. Larger values encourage smoother inferred distributions.`topicConcentration`

: Dirichlet parameter for prior over topics' distributions over terms (words). Larger values encourage smoother inferred distributions.`maxIterations`

: Limit on the number of iterations.`checkpointInterval`

: If using checkpointing (set in the Spark configuration), this parameter specifies the frequency with which checkpoints will be created. If`maxIterations`

is large, using checkpointing can help reduce shuffle file sizes on disk and help with failure recovery.

All of `spark.mllib`

's LDA models support:

`describeTopics`

: Returns topics as arrays of most important terms and term weights`topicsMatrix`

: Returns a`vocabSize`

by`k`

matrix where each column is a topic

*Note*: LDA is still an experimental feature under active development. As a result, certain features are only available in one of the two optimizers / models generated by the optimizer. Currently, a distributed model can be converted into a local model, but not vice-versa.

The following discussion will describe each optimizer/model pair separately.

**Expectation Maximization**

Implemented in `EMLDAOptimizer`

and `DistributedLDAModel`

.

For the parameters provided to `LDA`

:

`docConcentration`

: Only symmetric priors are supported, so all values in the provided`k`

-dimensional vector must be identical. All values must also be $> 1.0$. Providing`Vector(-1)`

results in default behavior (uniform`k`

dimensional vector with value $(50 / k) + 1$`topicConcentration`

: Only symmetric priors supported. Values must be $> 1.0$. Providing`-1`

results in defaulting to a value of $0.1 + 1$.`maxIterations`

: The maximum number of EM iterations.

*Note*: It is important to do enough iterations. In early iterations, EM often has useless topics, but those topics improve dramatically after more iterations. Using at least 20 and possibly 50-100 iterations is often reasonable, depending on your dataset.

`EMLDAOptimizer`

produces a `DistributedLDAModel`

, which stores not only the inferred topics but also the full training corpus and topic distributions for each document in the training corpus. A `DistributedLDAModel`

supports:

`topTopicsPerDocument`

: The top topics and their weights for each document in the training corpus`topDocumentsPerTopic`

: The top documents for each topic and the corresponding weight of the topic in the documents.`logPrior`

: log probability of the estimated topics and document-topic distributions given the hyperparameters`docConcentration`

and`topicConcentration`

`logLikelihood`

: log likelihood of the training corpus, given the inferred topics and document-topic distributions

**Online Variational Bayes**

Implemented in `OnlineLDAOptimizer`

and `LocalLDAModel`

.

For the parameters provided to `LDA`

:

`docConcentration`

: Asymmetric priors can be used by passing in a vector with values equal to the Dirichlet parameter in each of the`k`

dimensions. Values should be $>= 0$. Providing`Vector(-1)`

results in default behavior (uniform`k`

dimensional vector with value $(1.0 / k)$)`topicConcentration`

: Only symmetric priors supported. Values must be $>= 0$. Providing`-1`

results in defaulting to a value of $(1.0 / k)$.`maxIterations`

: Maximum number of minibatches to submit.

In addition, `OnlineLDAOptimizer`

accepts the following parameters:

`miniBatchFraction`

: Fraction of corpus sampled and used at each iteration`optimizeDocConcentration`

: If set to true, performs maximum-likelihood estimation of the hyperparameter`docConcentration`

(aka`alpha`

) after each minibatch and sets the optimized`docConcentration`

in the returned`LocalLDAModel`

`tau0`

and`kappa`

: Used for learning-rate decay, which is computed by $(\tau_0 + iter)^{-\kappa}$ where $iter$ is the current number of iterations.

`OnlineLDAOptimizer`

produces a `LocalLDAModel`

, which only stores the inferred topics. A `LocalLDAModel`

supports:

`logLikelihood(documents)`

: Calculates a lower bound on the provided`documents`

given the inferred topics.`logPerplexity(documents)`

: Calculates an upper bound on the perplexity of the provided`documents`

given the inferred topics.

**Examples**

In the following example, we load word count vectors representing a corpus of documents. We then use LDA to infer three topics from the documents. The number of desired clusters is passed to the algorithm. We then output the topics, represented as probability distributions over words.

## Bisecting k-means

Bisecting K-means can often be much faster than regular K-means, but it will generally produce a different clustering.

Bisecting k-means is a kind of hierarchical clustering. Hierarchical clustering is one of the most commonly used method of cluster analysis which seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two types:

- Agglomerative: This is a "bottom up" approach: each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy.
- Divisive: This is a "top down" approach: all observations start in one cluster, and splits are performed recursively as one moves down the hierarchy.

Bisecting k-means algorithm is a kind of divisive algorithms. The implementation in MLlib has the following parameters:

*k*: the desired number of leaf clusters (default: 4). The actual number could be smaller if there are no divisible leaf clusters.*maxIterations*: the max number of k-means iterations to split clusters (default: 20)*minDivisibleClusterSize*: the minimum number of points (if >= 1.0) or the minimum proportion of points (if < 1.0) of a divisible cluster (default: 1)*seed*: a random seed (default: hash value of the class name)

**Examples**

## Streaming k-means

When data arrive in a stream, we may want to estimate clusters dynamically, updating them as new data arrive. `spark.mllib`

provides support for streaming k-means clustering, with parameters to control the decay (or "forgetfulness") of the estimates. The algorithm uses a generalization of the mini-batch k-means update rule. For each batch of data, we assign all points to their nearest cluster, compute new cluster centers, then update each cluster using:

`\begin{equation} c_{t+1} = \frac{c_tn_t\alpha + x_tm_t}{n_t\alpha+m_t} \end{equation}`

`\begin{equation} n_{t+1} = n_t + m_t \end{equation}`

Where `$c_t$`

is the previous center for the cluster, `$n_t$`

is the number of points assigned to the cluster thus far, `$x_t$`

is the new cluster center from the current batch, and `$m_t$`

is the number of points added to the cluster in the current batch. The decay factor `$\alpha$`

can be used to ignore the past: with `$\alpha$=1`

all data will be used from the beginning; with `$\alpha$=0`

only the most recent data will be used. This is analogous to an exponentially-weighted moving average.

The decay can be specified using a `halfLife`

parameter, which determines the correct decay factor `a`

such that, for data acquired at time `t`

, its contribution by time `t + halfLife`

will have dropped to 0.5. The unit of time can be specified either as `batches`

or `points`

and the update rule will be adjusted accordingly.

**Examples**

This example shows how to estimate clusters on streaming data.

As you add new text files with data the cluster centers will update. Each training point should be formatted as `[x1, x2, x3]`

, and each test data point should be formatted as `(y, [x1, x2, x3])`

, where `y`

is some useful label or identifier (e.g. a true category assignment). Anytime a text file is placed in `/training/data/dir`

the model will update. Anytime a text file is placed in `/testing/data/dir`

you will see predictions. With new data, the cluster centers will change!