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Eigenvectors
Eigenvalues
Let I be a unit Matrix, we can then obtain the following
relationship:
When applied to spectral decomposition, this gives
It is important to understand that through there is a clear
relationship between Cholesky and spectral decomposition, both
approaches Cannot be mixed.
I.e. The basis A of each matrix is indeed different because Cholesky
decomposition is carried out on the Upper(Lower) half of the Matrix,
whereas spectral decomposition takes into account the full matrix
The main advantage of spectral decomposition lies in the handling of
the matrix rank r. Which allows us to define the Threshold used to
eliminate undesirable variance (either negative, negative or null
eigenvalues or eigenvalues below a certain target level). Alternatively
we can define a percentage of the distribution that must be retained and
eliminate the components that have little or no impact on the
multivariate distribution.
Hence we can apply the same as above, but retain the eigenvalues
(eigenvectors) up to the Rank r.
with
From this we can then apply the square root of our matrix to the a
series of random variates W.
In matrix notation and general form where delta W is your vector of
independent random variates.
or as individual elements
Finally when extending the Mean Zero Framework to Incorporate a mean
expected drift, we add the expected mean to the samples
The Framework presented above is general and applies to normal or
normal mixture models.
Student T Implementation
Moving away from Normality
If instead of normal variates W, we apply a series of p independant
Student-T variates T* , the vector
will have a multivariate Student T distribution with mean zero and
covariance matrix
The probability density function of the multivariate T- distribution
with covariance matrix sigma
is
In order to generate a multivariate Student T distribution from the
Methodology presented above,
we must therefore:
| Multivariate
Normal to Multivariate Student T Distribution |
| |
| 1 |
Scale
the Covariance Matrix  |
| 2 |
Decompose the scaled covariance matrix into eigenvalues and
eigenvectors up to rank r. |
| 3 |
Draw n T independent random variates T*.
Independent random T variates can be computed either from a Gamma/Chi
Square Normal distribution or through uniform polar coordinates. (see
StudentTVariates below) |
| 4 |
Apply the Student T variates to the general spectral
decomposition
framework |
Student T Variates
Two Different Implementations
The first implementation is based on the polar method of Box/Muller
for generating Normal variates is adapted to the Student-t distribution.
This implementation allows
for negative degrees of freedom.
The two generated variates are not independent and the expected number
of uniforms variates per Student T variate is 2.5464.
Ref: R Bailey, Polar generation of random variates with the
t-distribution, Mathematics of Computation 62, 779-781, 1994.
U is a uniform random variable [0,1]
with w >1
The second Implementation is based on the work of Milton Abramowitz
and Irene Stegun,Handbook of Mathematical Functions, US Department of
Commerce, 1964 and is based on the Normal Variates with mean 0 and
Variance df/(df-2) divided by the Chi Square Variates adjusted by df/(df-2)
This implementation does NOT
allows for negative degrees of freedom.
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