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Correlation Distance Matrix
===========================

Given :math:`n` feature vectors :math:`x_1 = (x_{11}, \ldots, x_{1p}), \ldots x_n = (x_{n1}, \ldots, x_{np})`
of dimension :math:`p`,
the problem is to compute the symmetric :math:`n \times n` matrix :math:`D_{\text{cor}} = (d_{ij})`
of distances between feature vectors, where

.. math::

    d_{ij} = 1 - \frac
    {\sum_{k=1}^{p} (x_{ik} - \overline{x_i}) (x_{jk} - \overline{x_j})}
    {\sqrt{ \sum_{k=1}^{p} (x_{ik} - \overline{x_i})^2 }
    \sqrt{ \sum_{k=1}^{p} (x_{jk} - \overline{x_j})^2 }}

.. math::
    \overline{x_i} = \frac{1}{p} \sum_{k=1}^{p} x_{ik}

.. math::
    \overline{x_j} = \frac{1}{p} \sum_{k=1}^{p} x_{jk}

.. math::
    i = \overline{1, n}

.. math::
    j = \overline{1, n}


Batch Processing
****************

Algorithm Input
---------------

The correlation distance matrix algorithm accepts the input described below.
Pass the ``Input ID`` as a parameter to the methods that provide input for your algorithm.
For more details, see :ref:`algorithms`.

.. tabularcolumns::  |\Y{0.2}|\Y{0.8}|

.. list-table:: Algorithm Input for Correlation Distance Matrix (Batch Processing)
   :widths: 10 60
   :header-rows: 1

   * - Input ID
     - Input
   * - ``data``
     - Pointer to the :math:`n \times p` numeric table for which the distance is computed.

       The input can be an object of any class derived from ``NumericTable``.

Algorithm Parameters
--------------------

The correlation distance matrix algorithm has the following parameters:

.. tabularcolumns::  |\Y{0.15}|\Y{0.15}|\Y{0.7}|

.. list-table:: Algorithm Parameters for Correlation Distance Matrix (Batch Processing)
   :header-rows: 1
   :widths: 10 10 60
   :align: left
   :class: longtable

   * - Parameter
     - Default Value
     - Description
   * - ``algorithmFPType``
     - ``float``
     - The floating-point type that the algorithm uses for intermediate computations. Can be ``float`` or ``double``.
   * - ``method``
     - ``defaultDense``
     - Performance-oriented computation method, the only method supported by the algorithm.

Algorithm Output
----------------

The correlation distance matrix algorithm calculates the result described below.
Pass the ``Result ID`` as a parameter to the methods that access the results of your algorithm.
For more details, see :ref:`algorithms`.

.. tabularcolumns::  |\Y{0.2}|\Y{0.8}|

.. list-table:: Algorithm Output for Correlation Distance Matrix (Batch Processing)
   :widths: 10 60
   :header-rows: 1

   * - Result ID
     - Result
   * - ``correlationDistance``
     - Pointer to the numeric table that represents the :math:`n \times n` symmetric distance matrix :math:`D_\text{cor}`.

       By default, the result is an object of the ``PackedSymmetricMatrix`` class with the ``lowerPackedSymmetricMatrix`` layout.
       However, you can define the result as an object of any class derived from ``NumericTable`` except ``PackedTriangularMatrix`` and ``CSRNumericTable``.

Examples
********

.. tabs::

  .. tab:: C++ (CPU)

    Batch Processing:

    - :cpp_example:`cor_dist_dense_batch.cpp <distance/cor_dist_dense_batch.cpp>`

  .. tab:: Java*

    .. note:: There is no support for Java on GPU.

    Batch Processing:

    - :java_example:`CorDistDenseBatch.java <distance/CorDistDenseBatch.java>`

  .. tab:: Python*

    Batch Processing:

    - :daal4py_example:`correlation_distance_batch.py`

Performance Considerations
**************************

To get the best overall performance when computing the correlation distance matrix:

- If input data is homogeneous, provide the input data and store results in homogeneous numeric tables
  of the same type as specified in the ``algorithmFPType`` class template parameter.
- If input data is non-homogeneous, use :ref:`AOS <aos_layout>` layout rather than SOA layout.

.. include:: ../../../opt-notice.rst