/**
@mainpage
Apache MADlib is an open-source library for scalable
in-database analytics. It provides data-parallel implementations of
mathematical, statistical, graph and machine learning methods for structured
and unstructured data.

Useful links:
<ul>
<li><a href="http://madlib.apache.org">MADlib web site</a></li>
<li><a href="https://cwiki.apache.org/confluence/display/MADLIB">MADlib wiki</a></li>
<li><a href="https://issues.apache.org/jira/browse/MADLIB/">JIRAs for reporting bugs and reviewing backlog</a></li>
<li><a href="https://mail-archives.apache.org/mod_mbox/madlib-user/">User mailing list</a></li>
<li><a href="https://mail-archives.apache.org/mod_mbox/madlib-dev/">Dev mailing list</a></li>
<li>User documentation for earlier releases:
    <a href="../v1.16/index.html">v1.16</a>,
    <a href="../v1.15.1/index.html">v1.15.1</a>,
    <a href="../v1.15/index.html">v1.15</a>,
    <a href="../v1.14/index.html">v1.14</a>,
    <a href="../v1.13/index.html">v1.13</a>
</li>
</ul>

Please refer to the
<a href="https://github.com/apache/madlib/blob/master/README.md">ReadMe</a>
file for information about incorporated third-party material. License information
regarding MADlib and included third-party libraries can be found in the
<a href="https://github.com/apache/madlib/blob/master/LICENSE">
License</a> directory.

@defgroup grp_datatrans Data Types and Transformations
@details Data types and operations that transform and shape data.
    @defgroup grp_arraysmatrix Arrays and Matrices
    @ingroup grp_datatrans
    @brief Mathematical operations for arrays and matrices.
    @details
These modules provide basic mathematical operations to be run on array and matrices.

For a distributed system, a matrix cannot simply be represented as a 2D array of numbers in memory.
<b>We provide two forms of distributed representation of a matrix</b>:

- Dense: The matrix is represented as a distributed collection of 1-D arrays.
        An example 3x10 matrix would be the below table:
<pre>
 row_id |         row_vec
--------+-------------------------
   1    | {9,6,5,8,5,6,6,3,10,8}
   2    | {8,2,2,6,6,10,2,1,9,9}
   3    | {3,9,9,9,8,6,3,9,5,6}
</pre>
- Sparse: The matrix is represented using the row and column indices for each
non-zero entry of the matrix. Example:
<pre>
 row_id | col_id | value
--------+--------+-------
      1 |      1 |     9
      1 |      5 |     6
      1 |      6 |     6
      2 |      1 |     8
      3 |      1 |     3
      3 |      2 |     9
      4 |      7 |     0
(6 rows)
</pre>
&nbsp;
All matrix operations work with either form of representation.

In many cases, a matrix function can be <b>decomposed to vector operations
applied independently on each row of a matrix (or corresponding rows of two
matrices)</b>. We have also provided access to these internal vector operations
(\ref grp_array) for greater flexibility. Matrix operations like
<em>matrix_add</em> use the corresponding vector operation (<em>array_add</em>)
and also include additional validation and formating. Other functions like
<em>matrix_mult</em> are complex and use a combination of such vector operations
and other SQL operations.

<b>It's important to note</b> that these array functions are only available for the
dense format representation of the matrix. In general, the scope of a single
array function invocation is limited to only an array (1-dimensional or
2-dimensional) that fits in memory. When such function is executed on a table of
arrays, the function is called multiple times - once for each  array (or pair of
arrays). On contrary, scope of a single matrix function invocation  is the
complete matrix stored as a distributed table.
    @{
        @defgroup grp_array Array Operations
        @defgroup grp_matrix Matrix Operations

        @defgroup grp_matrix_factorization Matrix Factorization
        @brief Linear algebra methods that factorize a matrix into a product of matrices.
        @details Linear algebra methods that factorize a matrix into a product of matrices.
        @{
            @defgroup grp_lmf Low-Rank Matrix Factorization
            @defgroup grp_svd Singular Value Decomposition
        @}

        @defgroup grp_linalg Norms and Distance Functions
        @defgroup grp_svec Sparse Vectors
    @}

        @defgroup grp_encode_categorical Encoding Categorical Variables
        @ingroup grp_datatrans

        @defgroup grp_path Path
        @ingroup grp_datatrans

        @defgroup grp_pivot Pivot
        @ingroup grp_datatrans

        @defgroup grp_sessionize Sessionize
        @ingroup grp_datatrans

        @defgroup grp_stemmer Stemming
        @ingroup grp_datatrans

@defgroup grp_graph Graph
@brief Graph algorithms and measures associated with graphs.
@details Graph algorithms and measures associated with graphs.
@{
    @defgroup grp_apsp All Pairs Shortest Path
    @defgroup grp_bfs Breadth-First Search
    @defgroup grp_hits HITS

    @defgroup grp_graph_measures Measures
    @brief A collection of metrics computed on a graph.
    @details A collection of metrics computed on a graph.
    @{
        @defgroup grp_graph_avg_path_length Average Path Length
        @defgroup grp_graph_closeness Closeness
        @defgroup grp_graph_diameter Graph Diameter
        @defgroup grp_graph_vertex_degrees In-Out Degree
    @}

    @defgroup grp_pagerank PageRank
    @defgroup grp_sssp Single Source Shortest Path
    @defgroup grp_wcc Weakly Connected Components
@}

@defgroup grp_mdl Model Selection
@brief Functions for model selection and model evaluation.
@details Functions for model selection and model evaluation.
@{
    @defgroup grp_validation Cross Validation
    @ingroup grp_mdl
    @defgroup grp_pred Prediction Metrics
    @ingroup grp_mdl
    @defgroup grp_train_test_split Train-Test Split
    @ingroup grp_mdl
@}

@defgroup grp_sampling Sampling
@brief A collection of methods for sampling from a population.
@details A collection of methods for sampling from a population.
@{
    @defgroup grp_balance_sampling Balanced Sampling
    @defgroup grp_strs Stratified Sampling
@}

@defgroup grp_stats Statistics
@brief A collection of probability and statistics modules.
@details A collection of probability and statistics modules.
@{
    @defgroup grp_desc_stats Descriptive Statistics
    @brief Methods to compute descriptive statistics of a dataset.
    @details Methods to compute descriptive statistics of a dataset.
    @{
        @defgroup grp_sketches Cardinality Estimators
        @brief Methods to estimate the number of unique values contained in data.
        @{
            @defgroup grp_countmin CountMin (Cormode-Muthukrishnan)
            @defgroup grp_fmsketch FM (Flajolet-Martin)
            @defgroup grp_mfvsketch MFV (Most Frequent Values)
        @}

        @defgroup grp_correlation Covariance and Correlation
        @defgroup grp_summary Summary
    @}

    @defgroup grp_inf_stats Inferential Statistics
    @brief Methods to compute inferential statistics of a dataset.
    @details Methods to compute inferential statistics of a dataset.
    @{
        @defgroup grp_stats_tests Hypothesis Tests
    @}

    @defgroup grp_prob Probability Functions
@}

@defgroup grp_super Supervised Learning
@brief Methods to perform a variety of supervised learning tasks.
@details Methods to perform a variety of supervised learning tasks.
@{
    @defgroup grp_crf Conditional Random Field
    @defgroup grp_knn k-Nearest Neighbors
    @defgroup grp_nn Neural Network
    @defgroup grp_regml Regression Models
    @brief A collection of methods for modeling conditional expectation of a response variable.
    @details A collection of methods for modeling conditional expectation of a response variable.
    @{
        @defgroup grp_clustered_errors Clustered Variance
        @defgroup grp_cox_prop_hazards Cox-Proportional Hazards Regression
        @defgroup grp_elasticnet Elastic Net Regularization
        @defgroup grp_glm Generalized Linear Models
        @defgroup grp_linreg Linear Regression
        @defgroup grp_logreg Logistic Regression
        @defgroup grp_marginal Marginal Effects
        @defgroup grp_multinom Multinomial Regression
        @defgroup grp_ordinal Ordinal Regression
        @defgroup grp_robust Robust Variance
    @}

    @defgroup grp_svm Support Vector Machines
    @defgroup grp_tree Tree Methods
    @brief A collection of recursive partitioning (tree) methods.
    @details A collection of recursive partitioning (tree) methods.
    @{
        @defgroup grp_decision_tree Decision Tree
        @defgroup grp_random_forest Random Forest
    @}
@}

@defgroup grp_tsa Time Series Analysis
@brief A collection of methods to analyze time series data.
@details A collection of methods to analyze time series data.
@{
    @defgroup grp_arima ARIMA
@}

@defgroup grp_unsupervised Unsupervised Learning
@brief A collection of methods for unsupervised learning tasks.
@details A collection of methods for unsupervised learning tasks.
@{
    @defgroup grp_association_rules Association Rules
    @brief Methods used to discover patterns in transactional datasets.
    @details Methods used to discover patterns in transactional datasets.
    @{
        @defgroup grp_assoc_rules Apriori Algorithm
    @}

    @defgroup grp_clustering Clustering
    @brief Methods for clustering data.
    @details Methods for clustering data.
    @{
        @defgroup grp_kmeans k-Means Clustering
    @}

    @defgroup grp_pca Dimensionality Reduction
    @brief Methods for reducing the number of variables in a dataset to obtain a set of principle variables.
    @details Methods for reducing the number of variables in a dataset to obtain a set of principle variables.
    @{
        @defgroup grp_pca_train Principal Component Analysis
        @defgroup grp_pca_project Principal Component Projection
    @}

    @defgroup grp_topic_modelling Topic Modelling
    @brief A collection of methods to uncover abstract topics in a document corpus.
    @details A collection of methods to uncover abstract topics in a document corpus.
    @{
        @defgroup grp_lda Latent Dirichlet Allocation
    @}
@}


@defgroup grp_other_functions Utilities
@details Useful utilities for data science workflows.
@{
    @defgroup grp_cols2vec Columns to Vector
    @defgroup @grp_utilities Database Functions

    @defgroup grp_linear_solver Linear Solvers
    @brief Methods that implement solutions for systems of consistent linear equations.
    @details Methods that implement solutions for systems of consistent linear equations.
    @{
        @defgroup grp_dense_linear_solver Dense Linear Systems
        @defgroup grp_sparse_linear_solver Sparse Linear Systems
    @}

    @defgroup grp_minibatch_preprocessing Mini-Batch Preprocessor
    @defgroup grp_pmml PMML Export
    @defgroup grp_text_utilities Term Frequency
    @defgroup grp_vec2cols Vector to Columns
@}

@defgroup grp_early_stage Early Stage Development
@details Implementations which are in an early stage of development.
Interface and implementation are subject to change.
@{
    @defgroup grp_cg Conjugate Gradient
    @defgroup grp_dl Deep Learning
    @brief A collection of modules for deep learning.
    @details A collection of modules for deep learning.
    @{
        @defgroup grp_gpu_configuration GPU Configuration
        @defgroup grp_keras Keras
        @defgroup grp_keras_model_arch Load Models
        @defgroup grp_model_selection Model Selection for DL
        @brief Train multiple deep learning models at the same time for model architecture search and hyperparameter selection.
        @details Train multiple deep learning models at the same time for model architecture search and hyperparameter selection.
        @{
            @defgroup grp_keras_run_model_selection Run Model Selection
            @defgroup grp_keras_setup_model_selection Setup Model Selection
        @}
        @defgroup grp_input_preprocessor_dl Preprocessor for Images
    @}
    @defgroup grp_bayes Naive Bayes Classification
    @defgroup grp_sample Random Sampling
@}

@defgroup grp_deprecated Deprecated Modules
Deprecated modules that will be removed in the
next major version (2.0).  There are newer MADlib modules
that have replaced these functions.
@{
    @defgroup grp_indicator Create Indicator Variables
    @defgroup grp_mlogreg Multinomial Logistic Regression
@}

*/