IntegrateRandomVectorsProduct: Integral outer product of random vectors

In StableEstim: Estimate the Four Parameters of Stable Laws using Different Methods

Integral outer product of random vectors

Description

Computes the integral outer product of two possibly complex random vectors.

Usage

IntegrateRandomVectorsProduct(f_fct, X, g_fct, Y, s_min, s_max,
                              subdivisions = 50,
                              method = c("Uniform", "Simpson"),
                              randomIntegrationLaw = c("norm","unif"),
                              ...)

Arguments

f_fct	function object with signature f_fct=function(s,X) and returns a matrix ns \times nx where nx=length(X) and ns=length(s); s is the points where the integrand is evaluated.
X	random vector where the function f_fct is evaluated. See Details.
g_fct	function object with signature g_fct=function(s,Y) and returns a matrix ns \times ny where ny=length(Y) and ns=length(s); s is the points where the integrand is evaluated.
Y	random vector where the function g_fct is evaluated. See Details.
s_min,s_max	limits of integration. Should be finite.
subdivisions	maximum number of subintervals.
method	numerical integration rule, one of "uniform" (fast) or "Simpson" (more accurate quadratic rule).
randomIntegrationLaw	Random law pi(s) to be applied to the Random product vector, see Details. Choices are "unif" (uniform) and "norm" (normal distribution).
...	other arguments to pass to random integration law. Mainly, the mean (mu) and standard deviation (sd) of the normal law.

Details

The function computes the nx \times ny matrix C = \int_{s_{min}}^{s_{max}} f_s(X) g_s(Y) π(s) ds, such as the one used in the objective function of the Cgmm method. This is essentially an outer product with with multiplication replaced by integration.

There is no function in R to compute vectorial integration and computing C element by element using integrate may be very slow when length(X) (or length(y)) is large.

The function allows complex vectors as its integrands.

Value

an nx \times ny matrix C with elements:

c_{ij} = \int_{s_{min}}^{s_{max}} f_s(X_i) g_s(Y_j) π(s) ds .

Examples

## Define the integrand
f_fct <- function(s, x) {
    sapply(X = x, FUN = sampleComplexCFMoment, t = s, theta = theta)
}
f_bar_fct <- function(s, x) Conj(f_fct(s, x))

## Function specific arguments
theta <- c(1.5, 0.5, 1, 0)
set.seed(345)
X <- rstable(3, 1.5, 0.5, 1, 0)
s_min <- 0;
s_max <- 2
numberIntegrationPoints <- 10
randomIntegrationLaw <- "norm"

Estim_Simpson <-
    IntegrateRandomVectorsProduct(f_fct, X, f_bar_fct, X, s_min, s_max, 
                                  numberIntegrationPoints, 
                                  "Simpson", randomIntegrationLaw)
Estim_Simpson

StableEstim documentation built on Aug. 7, 2022, 5:17 p.m.