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Covariance matrix estimation
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Meva provides two methods to estimate the covariance matrix of log asset
returns: one, `cov_pca` based on pricipal component analysis (PCA) and one,
`cov_fa` based on factor analysis. Both functions return :math:`B` and
:math:`d` such that the covariance matrix :math:`V` is given by

.. math:: V = \mbox{cov}(y) = BB' + \mbox{diag}(d),

where :math:`B` is a :math:`n\times k` matrix mapping :math:`k` factors to
:math:`n` assets, :math:`d` is...

`cov_pca` (PCA-based)
=====================

The `cov_pca` function uses principal component analysis to fit a factor-model
decomposition of market variability. This model decomposes the market return
:math:`y` into two parts:

.. math:: y = Bx + w.

Here, :math:`B` is a :math:`n\times k` matrix mapping :math:`k` factors to
:math:`n` assets, :math:`x` is a draw from a :math:`k` dimensional iid standard
normal distribution, and :math:`w` is a draw from an independent normal
distribution in which the variance of the :math:`j`-th component is given by
:math:`d_j`.

The `cov_pca` function handles missing data, coded as `nan`.

`cov_fa` (factor-analysis-based)
================================

The `cov_fa` function fits the same factor model to the market returns. But
instead of using principal component analyisis to decompose the variance,
this factor analysis function uses the EM (expectation-maximization) algorithm
to iteratively find a maximum-likelihood fit.

For background on the factor analysis model and its EM solution see `Andrew
Ng's freely available machine-leaning notes
<http://see.stanford.edu/materials/aimlcs229/cs229-notes9.pdf>`_.

The only difference between our algorithm and his is that we use the matrix
identities

.. math:: \big(E - FH^{-1}G\big)^{-1} FH^{-1} = 
          E^{-1}F \big(H - GE^{-1}F\big)^{-1},

and

.. math:: \big(E - FH^{-1}G\big)^{-1} = 
          E^{-1} + E^{-1}F\big(H - GE^{-1}F\big)^{-1}GE^{-1}

to reduce :math:`n\times n` matrix inversions to :math:`k \times k` matrix
inversions. These identities result form standard blockwise inversion
teckniques, for instance on wikipedia
`here <http://en.wikipedia.org/wiki/Invertible_matrix#Blockwise_inversion>`_.

We apply them with :math:`E=I_{n\times n}`, :math:`F=B'`, :math:`G=-B`, and
:math:`H=\mbox{diag}(d)` during the E-step in order to avoid :math:`n\times n`
inversions.

The `cov_fa` function cannot handle missing data.

Examples
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