How to calculate integral, numerically, in Rcpp

Question

I've searched for an hour for the methods doing numerical integration. I'm new to Rcpp and rewriting my old programs now. What I have done in R was:

   x=smpl.x(n,theta.true)   
   joint=function(theta){# the joint dist for
                         #all random variable
     d=c()
     for(i in 1:n){
       d[i]=den(x[i],theta)
     }
     return(prod(d)*dbeta(theta,a,b))   }  
 joint.vec=Vectorize(joint)##vectorize the function, as required when
                           ##using integrate()   
margin=integrate(joint.vec,0,1)$value # the
                                ##normalizeing constant at the donominator  
 area=integrate(joint.vec,0,theta.true)$value # the values at the
                                              ## numeritor

• The integrate() function in R will be slow, and since I am doing the integration for a posterior distribution of a sample of size n, the value of the integration will be huge with large error.
• I am trying to rewrite my code with the help of Rcpp, but I　don't know how to deal with the integrate. Should I include a c++ h file? Or any suggestions?

Answer 1

You can code your function in C and call it, for instance, via the sourceCpp function and then integrate it in R. In alternative, you can call the integrate function of R within your C code by using the Function macro of Rcpp. See Dirk's book (Seamless R and C++ Integration with Rcpp) on page 56 for an example of how to call R functions from C. Another alternative (which I believe is the best for most cases) is to integrate your function written in C , directly in C, using the RcppGSL package.

As about the huge normalizing constant, sometimes it is better to scale the function at the mode before integrating it (you can find modes with, e.g., nlminb, optim, etc.). Then, you integrate the rescaled function and to recover the original nroming constant multiply the resulting normalizing constant by the rescaling factor. Hope this may help!

Answer 2

after reading your @utobi advice, I felt programming by my own maybe easier. I simply use Simpson formula to approximate the integral:

// [[Rcpp::export]]
double den_cpp (double x, double theta){
return(2*x/theta*(x<=theta)+2*(1-x)/(1-theta)*(theta<x));
}
// [[Rcpp::export]]
double joint_cpp ( double theta,int n,NumericVector x, double a, double b){
double val = 1.0;
NumericVector d(n);
for (int i = 0; i < n; i++){
double tmp = den_cpp(x[i],theta);
val = val*tmp;
}
val=val*R::dbeta(theta,a,b,0);
return(val);
}
// [[Rcpp::export]]
List Cov_rate_raw ( double theta_true,  int n, double a, double b,NumericVector x){
//This function is used to test, not used in the fanal one
int steps = 1000;
double s = 0;
double start = 1.0e-4;
std::cout<<start<<" ";
double end = 1-start;
std::cout<<end<<" ";
double h = (end-start)/steps;
std::cout<<"1st h ="<<h<<" ";
double area = 0;
double margin = 0;
for (int i = 0; i < steps ; i++){
double at_x = start+h*i;
double f_val = (joint_cpp(at_x,n,x,a,b)+4*joint_cpp(at_x+h/2,n,x,a,b)+joint_cpp(at_x+h,n,x,a,b))/6;
s = s + f_val;
}
margin = h*s;
s=0;
h=(theta_true-start)/steps;
std::cout<<"2nd h ="<<h<<" ";
for (int i = 0; i < steps ; i++){
double at_x = start+h*i;
double f_val = (joint_cpp(at_x,n,x,a,b)+4*joint_cpp(at_x+h/2,n,x,a,b)+joint_cpp(at_x+h,n,x,a,b))/6;
s = s + f_val;
}
area = h * s;
double r = area/margin;
int cover = (r>=0.025)&&(r<=0.975);
List ret;
ret["s"] = s;
ret["margin"] = margin;
ret["area"] = area;
ret["ratio"] = r;
ret["if_cover"] = cover;
return(ret);
}

I'm not that good at c++, so the two for loops like kind of silly.

It generally works, but there are still several potential problems:

1. I don't really know how to choose the steps, or how many sub intervals do I need to approximate the integrals. I've taken numerical analysis when I was an undergraduate, I think maybe I need to check my book about the expression of the error term, to decide the step length.
2. I compared my results with those from R. the integrate() function in R can take care of the integral over the interval [0,1]. That helps me because my function is undefined at 0 or 1, which takes infinite value. In my C++ code, I can only make my interval from [1e-4, 1-1e-4]. I tried different values like 1e-7, 1e-10, however, 1e-4 was the one most close to R's results....What should I do with it?