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Added paragraphs on metric and nonmetric MDS, explaining the difference

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1 parent 5d80149 commit 096e7927fc0d51a99efc22cc4afc1acf8ce509fa @NelleV NelleV committed with GaelVaroquaux May 3, 2012
Showing with 31 additions and 4 deletions.
  1. +31 −4 doc/modules/manifold.rst
@@ -347,19 +347,46 @@ The overall complexity of standard LTSA is
Multi-dimensional Scaling (MDS)
===============================
-Multidimensional scaling is a technique used for analysis similarity or
+Multidimensional scaling is a technique used for analyzing similarity or
dissimilarity data. MDS attempts to model similarity or dissimilarity data as
distances in a geometric spaces. The data can be ratings of similarity between
objects, interaction frequencies of molecules, or trade indices between
countries.
There exists two types of MDS algorithm: metric and non metric. In Metric MDS,
-the distances between two points are set to be as closed as possible as the
-similarity or dissimilarity data. In non metric, the algorithms will try to
-preserve the order of the distances, and hence seek for a monotonic
+the distances between two points are set to be as close as possible as the
+similarity or dissimilarity data. In non metric vision, the algorithms will
+try to preserve the order of the distances, and hence seek for a monotonic
relationship between the distances in the embedded space and the
similarities/dissimilarities.
+Let :math:`S` be the similarity matrix, and :math:`X` the coordinates of the
+:math:`n` input points. Disparities :math:`\hat{d}_{ij}` are transformation of
+the similarities chosen in some optimal ways.
+The stress is then defined by :math:`sum_{i < j} d_{ij}(X) - \hat{d}_{ij}(X)}`
+
+Metric MDS
+----------
+
+The simplest metric MDS model, called absolute MDS, disparities are defined by
+:math:`\hat{d}_{ij} = S_{ij}`. With absolute MDS, the value :math:`S_{ij}`
+should then correspond exactly to the distance between point :math:`i` and
+:math:`j` in the embedding point.
+
+Most commonly, disparities are set to :math:`\hat{d}_ij = b S_{ij}`.
+
+Nonmetric MDS
+-------------
+
+Non metric MDS focuses on the ordination of the data. If :math:`S_{ij} <
+S_{kl}`, then the embedding should enforce :math:`d_{ij} < d_{jk}`. A simple
+algorithm to enforce that is to use a monotonic regression is computed of
+:math:`d_{ij}` on :math:`S_{ij}`, yielding disparities :math:`\hat{d}_{ij}` in
+the same order as `S_{ij}`.
+A trivial solution to this problem is to set all the points on the origin. In
+order to avoid that, the disparities :math:`\hat{d}_ij` are normalized.
+
+
.. figure:: ../auto_examples/manifold/images/plot_mds.png
:target: ../auto_examples/manifold/plot_mds.html
:align: center

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