R help - Mar 2014 - Importance Sampling Monte Carlo method for bivariate integration

I have the following integral

           (x^-0.5)                                   ;in x=[0.01,1]
To solve this using Importance Sampling MC integration, one needs to select
an importance pdf that is approximately the same as the function plot

My R code to solve the same is this :

#function 1 - importance sampling
w <- function(x) dunif(x,0.01,1)/dbeta(x,0.7,1)
f <- function(x) x^(-0.5)
X <- rbeta(1000,0.7,1)
Y <- w(X)*f(X)
c(mean(Y),var(Y))
True integral value - 1.8
Using the Importance Sampling code above - 1.82 (where my importance PDF is
Beta(0.7,1)

which is quite alright so I'm assuming the code is correct.




---------------------------------------------
However I now have two bivariate functions that look like this in intervals
[x,y] in [-pi,pi] and [x,y] in [-5,5] respectively -

[image: bivariate function 1]


Could anyone guide on how to perform MC Importance Sampling for these
functions? I know I could select two independent distributions however, I
have no idea about how to choose the functions or coding it like the way
above.
-- 

Regards and Cheers,
Pooja Voladoddi
+91-90351 93110




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On March 6, 2014 3:56:13 PM PST, voladoddi <pooja.voladoddi at gmail.com>
wrote:
>I have the following integral
>
>           (x^-0.5)                                   ;in x=[0.01,1]
>To solve this using Importance Sampling MC integration, one needs to
>select
>an importance pdf that is approximately the same as the function plot
>
>My R code to solve the same is this :
>
>#function 1 - importance sampling
>w <- function(x) dunif(x,0.01,1)/dbeta(x,0.7,1)
>f <- function(x) x^(-0.5)
>X <- rbeta(1000,0.7,1)
>Y <- w(X)*f(X)
>c(mean(Y),var(Y))
>True integral value - 1.8
>Using the Importance Sampling code above - 1.82 (where my importance
>PDF is
>Beta(0.7,1)
>
>which is quite alright so I'm assuming the code is correct.
>
>
>
>
>---------------------------------------------
>However I now have two bivariate functions that look like this in
>intervals
>[x,y] in [-pi,pi] and [x,y] in [-5,5] respectively -
>
>[image: bivariate function 1]
>
>
>Could anyone guide on how to perform MC Importance Sampling for these
>functions? I know I could select two independent distributions however,
>I
>have no idea about how to choose the functions or coding it like the
>way
>above.