Affine combination of independent univariate random variables¶
Introduction¶
Let be the random vector defined as the
affine transform of
independent univariate random variables.
More precisely, consider:
(1)¶
where is a deterministic vector with
,
is a
deterministic matrix and
are
independent random variables.
In this case, it is possible to directly evaluate the distribution of
and then to ask
any request compatible
with a distribution: moments, probability and cumulative density
functions, quantiles (in dimension 1 only)…
In this document, we present a method using the Poisson summation
formula to evaluate the distribution of
.
Evaluation of the probability density function¶
Since, by hypothesis, the univariate random variables are independent, the
characteristic function of
, denoted
, is
easily defined from the characteristic function of
denoted
as follows:
(2)¶
for any .
Once
is evaluated, it is possible to evaluate the
probability density function of
, denoted
:
several techniques are possible, as the inversion of the Fourier
transformation, but this method is not easy to implement.
We can alternatively use the Poisson summation formula:
(3)¶
where and
is the complex imaginary number, i.e.
.
If
are close to zero, then:
and:
because of the decreasing properties of . Thus the nested sums of the left
term of (3) are reduced to the central term
: the left term is approximately equal to
.
Furthermore, the right term of (3) is a series which
converges very fast: few terms of the series are enough to get
machine-precision accuracy. Let us note that the factors
, which are expensive to
evaluate, do not depend on
and are evaluated once only.
It is also possible to greatly improve the performance of the
algorithm by noticing that the equation is linear between and
. We denote by
and
respectively
the density and the characteristic function of the multivariate normal
distribution with the same mean
and same covariance
matrix
as the affine combination. By applying this
multivariate normal distribution to the equation, we obtain by
subtraction:
(4)¶
where ,
and
.
In the case where
, using the limit central theorem,
the law of
tends to the normal distribution density
, which will drastically reduce
. The sum on
will become the most CPU-intensive part, because in the
general case we will have to keep more terms than the central one in
this sum, since the parameters
were
calibrated with respect to
and not
.
The parameters are calibrated using the
following formula:
where and
,
are respectively the number of standard
deviations covered by the marginal distribution (
by
default) and
the number of marginal deviations beyond
which the density is negligible (
by default).
The parameter
is dynamically calibrated: we start with
then we double
value until the total contribution
of the additional terms is negligible.
Evaluation of the moments¶
The relation (1) enables to evaluate all the moments of the affine combination, if mathematically defined. For example, we have:
Computation on a regular grid¶
We want to compute the density function on a regular grid and to get an approximation quickly. The regular grid is:
for all and
.
Denoting
:
for which the term is the most CPU
consuming. This term rewrites:
with:
The aim is to rewrite the previous expression as a - discrete
Fourier transform, in order to apply Fast Fourier Transform (FFT) for
its evaluation.
We set
and
and
. For convenience, we introduce
the functions:
We use instead of
in this function to simplify
expressions below.
We obtain:
For performance reasons, we want to use the discrete Fourier transform with the following convention in dimension 1:
which extension to dimensions 2 and 3 are respectively:
We decompose sums of on the interval into three parts:
(5)¶
If we compute for dimension
, then the
middle term in this sum is trivial.
To compute the last sum, we apply a change of variable :
This implies:
Thus:
To compute the first sum of equation, we apply a change of variable
:
This implies:
Thus:
To summarize:
In order to compute sum from
to
, we multiply by
and consider
In order to compute sum from
to
, we consider