Note that 'a', 'b', 'c' and 'X0' are
parameters that we have to determine in the file "Data.m", as constants
or random variables following a certain distribution.
The above files are copied to "Model"
folder. Then, open with
Mathematica
the file "DependentPC.nb" and execute the cell in the notebook sections:
1.- Generation of the auxiliary system
of differential equations.
In this section we generate the auxiliary system of differential
equations symbolically and we write it in the file "SEDO" inside the
folder "Output". Also, we store the list of symbolic inner products to
be carried out in the file "listaProductos" inside the folder "Output".
Moreover, we calculate the symbolic mean and standard deviation of the
solution of the auxiliary system and store them in the file "Media-DT"
inside the folder "Output".
Note that, in this section, we built the solutions symbolically, only
manipulating expressions. Any computation has been done yet.
2.- Manipulation of the distributions
and generation of the inner products.
In this section we generate the inner product using the joint
probability distributions. The result is written in the file
"Inner_Product" inside the
folder "Output". As before, any computation has been done yet.
At this point, we should say that if we could simplify the expression
of this inner product, further computations will be faster. Therefore,
we recommend the user to take advantage of his/her knowledge of
Mathematica in order to find
simpler forms of the inner product using the ideas of the Section 3.1
of the paper.
3.- Computation of the inner products.
Here, we start with real computations. Despite these computations are
parallelized, this is a crucial point and usually consumes most of the
procedure computation time. In this section, we compute the inner
products from their symbolic expressions stored in the file
"Output/listaProductos". It usually takes very long time, the more time
the more dependent random variable together (in the same subgroup)
because the inner product has more nested integrals.
4.- Resolution of the auxiliary system
of differential equations.
The last step of the procedure is to substitute the results of the
inner products into the symbolic auxiliary system of differential
equations and solve it. We should take into account that, even though
our objective has been to work symbolically as much as possible, here
we use the
Mathematica
command
NDSolve[], that
solves numerically the differential equations. Furthermore, we also
have to realise that the auxiliary system uses to have a large number
of equations what is an additional difficulty when computing the
solution.
5.- Drawing the solutions.
At this point, the procedure has finished. Our last step is to draw the
mean and the standard deviation. Some commands appear in the section to
plot the results.
More examples
In the following, links to the "ModelData.m" and "Data.m" files of the
remainder paper examples are presented. Remember that these files have
to be copied inside the "Model" folder. Then, repeat the execution
process using the
Mathematica
notebook "DependentPC.nb".
REFERENCES
