DOC fix typos in manifold learning user guide
@@ -234,7 +234,7 @@ The MLLE algorithm comprises three stages:
to that of standard LLE. The second term has to do with constructing the
weight matrix from multiple weights. In practice, the added cost of
constructing the MLLE weight matrix is relatively small compared to the
- cost of steps 1 and 3.
+ cost of stages 1 and 3.
3. **Partial Eigenvalue Decomposition**. Same as standard LLE
@@ -402,8 +402,8 @@ Multi-dimensional Scaling (MDS)
representation of the data in which the distances respect well the
distances in the original high-dimensional space.
-In general, is a technique used for analyzing similarity or
-dissimilarity data. :class:`MDS` attempts to model similarity or dissimilarity data as
+In general, :class:`MDS` is a technique used for analyzing similarity or
+dissimilarity data. It 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.
@@ -426,7 +426,7 @@ 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 objective, called the
-stress, is then defined by :math:`sum_{i < j} d_{ij}(X) - \hat{d}_{ij}(X)`
+stress, is then defined by :math:`\sum_{i < j} d_{ij}(X) - \hat{d}_{ij}(X)`
Metric MDS
@@ -443,7 +443,7 @@ Nonmetric MDS
-------------
Non metric :class:`MDS` focuses on the ordination of the data. If
-:math:`S_{ij} < S_{kl}`, then the embedding should enforce :math:`d_{ij} <
+:math:`S_{ij} < S_{jk}`, then the embedding should enforce :math:`d_{ij} <
d_{jk}`. A simple algorithm to enforce that is to use a monotonic regression
of :math:`d_{ij}` on :math:`S_{ij}`, yielding disparities :math:`\hat{d}_{ij}`
in the same order as :math:`S_{ij}`.
