Advanced Introduction to C++, Scientific Computing and Machine Learning

Claudius Gros, SS 2024

Institut für theoretische Physik
Goethe-University Frankfurt a.M.

NM : Monte Carlo & Metropolis Sampling

statistical evaluation of integrals

evaluation of multi-dimensional integrals
e.g. $1000$ trapezoids per dimension
not feasible for large $N$

Monte Carlo sampling

evaluate the integral by sampling statistically the integrand $\ f(x_1,x_x,\dots,x_N)$

Metropolis algorithm

evaluate the integral by sampling statistically the
most important terms of the integrand $\ f(x_1,x_x,\dots,x_N)$
(importance sampling)

integration as averaging

$\hspace{10ex} \displaystyle I \ =\ \int_a^b \, f(x)\, dx \ =\ \sum_{i=1}^N f(x_i)\,\Delta x $

box width $\ \Delta x = \frac{b-a}{N}$

$\hspace{10ex} \displaystyle I \ =\ (b-a) \left(\frac{1}{N} \sum_i\, f(x_i) \right) \ =\ (b-a)\, \langle f\rangle $

expectation value (average) $\ \ \ \langle f\rangle$
of integrand

$$ I\ =\ V\, \langle f\rangle$$

volume $\ V$
expectation values may be evaluated statistically

area integration

$\hspace{10ex} \displaystyle I\ = \ \int_{-a}^a f(x,y)\,dx\,dy $

with

$\hspace{10ex} \displaystyle f(x,y)\ =\ \left\{ \begin{array}{ccl} 1 && \mbox{inside}\\ 0 && \mbox{outside} \end{array} \right. $

evaluation of $\ \pi$

/** Evaluating \pi with simple Monte Carlo evaluation.
  */
#include <iostream>
#include <stdio.h>     // for printf
#include <stdlib.h>    // srand, rand
#include <math.h>      // for pow, etc.
#include <fstream>     // file streams
using namespace std;
 
double myFunction(double x, double y, double x0, double x1, double y0, double y1) 
{
 if ( (x<x0)||(x>x1)||(y<y0)||(y>y1) )
    {
    printf("# myFunction(): x or y value out of range \n");
    return 0.0;
    }
 if (y*y<=1.0-x*x)     // x*x+y*y=1 is the circle equation
   return 1.0;
 else
   return 0.0;
  } // end of myFunction()
 
// ---
// --- main
// ---
int main() 
{
 const double x0 = 0.0, x1 = 1.0;    // x-bounds
 const double y0 = 0.0, y1 = 1.0;    // y-bounds
//
 int N = 1000000;                           // number of draws
 int deltaN = N/10;                         // printout intervals
 double x, y, sum = 0;
 srand(time(NULL));                         // seed for random()
// --- loop over all random evaluations 
 printf("%8s %8s %8s\n","# steps","pi", "error");
 for (int i=0; i<N; i++)
   {
   x = (random()/(double)RAND_MAX);     // random draws
   y = (random()/(double)RAND_MAX);     // random draws
   sum += myFunction(x,y,x0,x1,y0,y1);
   double result = 4.0*sum/(i+1.0);
   if ((i+1)%deltaN==0)
     printf("%8d %8.5f %8.5f\n",i+1,result,(M_PI-result)/M_PI);
   }
 printf("%s\n","-----------------");
     printf("%8s %8.5f\n","infinity",M_PI);
 return 1;
 }  // end of main()

/** Evaluating \pi with simple Monte Carlo evaluation.
  */
#include <iostream>
#include <stdio.h>     // for printf
#include <stdlib.h>    // srand, rand
#include <math.h>      // for pow, etc.
#include <fstream>     // file streams
using namespace std;
 
double myFunction(double x, double y, double x0, double x1, double y0, double y1) 
{
 if ( (x<x0)||(x>x1)||(y<y0)||(y>y1) )
    {
    printf("# myFunction(): x or y value out of range \n");
    return 0.0;
    }
 if (y*y<=1.0-x*x)     // x*x+y*y=1 is the circle equation
   return 1.0;
 else
   return 0.0;
  } // end of myFunction()
 
// ---
// --- main
// ---
int main() 
{
 const double x0 = 0.0, x1 = 1.0;    // x-bounds
 const double y0 = 0.0, y1 = 1.0;    // y-bounds
//
 int N = 1000000;                           // number of draws
 int deltaN = N/10;                         // printout intervals
 double x, y, sum = 0;
 srand(time(NULL));                         // seed for random()
// --- loop over all random evaluations 
 printf("%8s %8s %8s\n","# steps","pi", "error");
 for (int i=0; i<N; i++)
   {
   x = (random()/(double)RAND_MAX);     // random draws
   y = (random()/(double)RAND_MAX);     // random draws
   sum += myFunction(x,y,x0,x1,y0,y1);
   double result = 4.0*sum/(i+1.0);
   if ((i+1)%deltaN==0)
     printf("%8d %8.5f %8.5f\n",i+1,result,(M_PI-result)/M_PI);
   }
 printf("%s\n","-----------------");
     printf("%8s %8.5f\n","infinity",M_PI);
 return 1;
 }  // end of main()

basic statistics

statistics

mean (expecatation value)
variance $\ \sigma^2$
standard deviation $\ \sigma$
note: $\ \sigma\ $ is not the standard deviation for the process,
not the an estimate for the accuracy of $\ \langle f\rangle$
question: how do we estimate the accuracy of the measurement: $\ \langle f\rangle$

Monte Carlo - convergence

data binning

divide the $\ N\ $ measurements into $\ k=1,\dots,K\ $ bins
containing each $\ N_K=N/K\ $ data points
example: $\ N_k=3$
the accuracy of $\ \langle f\rangle\ $ is given
by the standard deviation of $\ \langle f\rangle_k $

$$ \Delta \langle f\rangle \ =\ \sqrt{\frac{1}{K}\sum_{k=1}^K \Big(\langle f\rangle_k-\langle f\rangle\Big)^2} $$

error estimate: $\ \Delta \langle f\rangle$
usually $\ K\sim 10,\, 20$
$N_k\ $ large

law of large numbers

$$ \Delta \langle f\rangle \ \sim\ \frac{1}{\sqrt{N}} $$

convergence very slow

Monte Carlo with error estimation

evaluation of $\ \pi$ (again)

/** Evaluating \pi with simple Monte Carlo evaluation,
  * calculating variance and standard deviation (error).
  * On input: number of measurements N.
  */
#include <iostream>
#include <stdio.h>     // for printf
#include <stdlib.h>    // srand, rand
#include <math.h>      // for pow, etc.
#include <fstream>     // file streams
 
using namespace std;
 
double myFunction(double x, double y, double x0, double x1, double y0, double y1) 
{
 if ( (x<x0)||(x>x1)||(y<y0)||(y>y1) )
    {
    printf("# myFunction(): x or y value out of range \n");
    return 0.0;
    }
 if (y*y<=1.0-x*x)     // x*x+y*y=1 is the circle equation
   return 1.0;
 else
   return 0.0;
  } // end of myFunction()
 
// ---
// --- main
// ---
int main(int argLength, char* argValues[])   // parsing command line arguments
{
 const double x0 = 0.0, x1 = 1.0;    // x-bounds
 const double y0 = 0.0, y1 = 1.0;    // y-bounds
//
 int N = 0;                          // number of MC steps
 if (argLength==2)
   N = atof(argValues[1]);
 else
   {
   printf("on input N: number of MC steps\n");
   return 0;
   }
//
// --- other parameters and variables
//
 const int K = 20;                                // number of bins
 int nK = N/K;                                    // number of data per bin
 double* data = new double[N];                    // storing all measurements
 double* meanBin = new double[K];                 // means of bins
 double mean = 0.0, sigma = 0.0;                  // result: mean and error
 double x, y;
//
// --- loop over all random evaluations 
//
 srand(time(NULL));                         // seed for random()
 for (int i=0; i<N; i++)
   {
   x = (random()/(double)RAND_MAX);            // random draws
   y = (random()/(double)RAND_MAX);            // random draws
   data[i] = 4.0*myFunction(x,y,x0,x1,y0,y1);      // storing measurement
   }  
//
// --- evaluate the overall mean and the mean of the bins
//
 for (int i=0; i<N; i++) 
    mean += data[i];
 mean = mean/N;
//
 for (int k=0; k<K; k++)
   {
   meanBin[k] = 0.0;       
   for (int n=0; n<nK; n++)                     // nK data points in each bin
      meanBin[k] += data[k*nK+n];
   meanBin[k] = meanBin[k]/nK;
   printf("# (mean of bin) - mean : %8.5f\n",meanBin[k]-mean);
   }  // end of loop over all bins
//
// --- evaluate the error: variance of means of bins
//
 for (int k=0; k<K; k++)
   sigma += (meanBin[k]-mean)*(meanBin[k]-mean);
 sigma = sqrt(sigma/K);
//
// --- printout of result
//
 printf(" number of measurements : %d\n",N);
 printf(" number of bins         : %d\n",K);
 printf(" data points per bin    : %d\n",nK);
 printf(" estimate for pi        : %8.5f\n",mean);
 printf(" estimate for accuracy  : %8.5f\n",sigma);
 printf(" true error             : %8.5f\n",M_PI-mean);
// 
 return 1;
 }  // end of main()

/** Evaluating \pi with simple Monte Carlo evaluation,
  * calculating variance and standard deviation (error).
  * On input: number of measurements N.
  */
#include <iostream>
#include <stdio.h>     // for printf
#include <stdlib.h>    // srand, rand
#include <math.h>      // for pow, etc.
#include <fstream>     // file streams
 
using namespace std;
 
double myFunction(double x, double y, double x0, double x1, double y0, double y1) 
{
 if ( (x<x0)||(x>x1)||(y<y0)||(y>y1) )
    {
    printf("# myFunction(): x or y value out of range \n");
    return 0.0;
    }
 if (y*y<=1.0-x*x)     // x*x+y*y=1 is the circle equation
   return 1.0;
 else
   return 0.0;
  } // end of myFunction()
 
// ---
// --- main
// ---
int main(int argLength, char* argValues[])   // parsing command line arguments
{
 const double x0 = 0.0, x1 = 1.0;    // x-bounds
 const double y0 = 0.0, y1 = 1.0;    // y-bounds
//
 int N = 0;                          // number of MC steps
 if (argLength==2)
   N = atof(argValues[1]);
 else
   {
   printf("on input N: number of MC steps\n");
   return 0;
   }
//
// --- other parameters and variables
//
 const int K = 20;                                // number of bins
 int nK = N/K;                                    // number of data per bin
 double* data = new double[N];                    // storing all measurements
 double* meanBin = new double[K];                 // means of bins
 double mean = 0.0, sigma = 0.0;                  // result: mean and error
 double x, y;
//
// --- loop over all random evaluations 
//
 srand(time(NULL));                         // seed for random()
 for (int i=0; i<N; i++)
   {
   x = (random()/(double)RAND_MAX);            // random draws
   y = (random()/(double)RAND_MAX);            // random draws
   data[i] = 4.0*myFunction(x,y,x0,x1,y0,y1);      // storing measurement
   }  
//
// --- evaluate the overall mean and the mean of the bins
//
 for (int i=0; i<N; i++) 
    mean += data[i];
 mean = mean/N;
//
 for (int k=0; k<K; k++)
   {
   meanBin[k] = 0.0;       
   for (int n=0; n<nK; n++)                     // nK data points in each bin
      meanBin[k] += data[k*nK+n];
   meanBin[k] = meanBin[k]/nK;
   printf("# (mean of bin) - mean : %8.5f\n",meanBin[k]-mean);
   }  // end of loop over all bins
//
// --- evaluate the error: variance of means of bins
//
 for (int k=0; k<K; k++)
   sigma += (meanBin[k]-mean)*(meanBin[k]-mean);
 sigma = sqrt(sigma/K);
//
// --- printout of result
//
 printf(" number of measurements : %d\n",N);
 printf(" number of bins         : %d\n",K);
 printf(" data points per bin    : %d\n",nK);
 printf(" estimate for pi        : %8.5f\n",mean);
 printf(" estimate for accuracy  : %8.5f\n",sigma);
 printf(" true error             : %8.5f\n",M_PI-mean);
// 
 return 1;
 }  // end of main()

integrals over distribution functions

probability distribution function $\ \rho(x)$
example: Boltzman factor $\ e^{\beta E_\lambda}$
probability that a state with energy $\ E_\lambda$
is thermally occupied
inverse temperature $\ \beta = 1/(k_B T)$
partition function $\ Z$

importance sampling

the integral can be dominate by small regions of
phase space with large values for $\ \rho(x)$

$\hspace{6ex}\displaystyle \rho(x) \ =\ \frac{1}{y_n}\frac{x^n}{x+a}, \quad\quad y_n\ =\ \int_0^1\, \frac{x^n}{x+a}\, dx $

importance sampling
Monte Carlo sampling of phase space with probability $\rho(x)$ to sample $\ x$

random walk through configuration space

random walk
probability to visit a given phase space $\ x$
construct walk with correct limiting behaviour
we then have
and

$$ I\ =\ \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N g(x_n) $$

this is the basis of importance sampling:
the important parts of phase space are visited (sampled) often
the important parts of phase space seldomly
by the random walk $\ \{x_n\}$

Metropolis algorithm

method to generate a random walk $\ \{x_n\}$
with the correct limiting behviour
Markov chain
random walk where the probility to select $\ x_{n+1}\ $
depends only on the $\ x_n$
(and not on $\ x_{n-1},\ x_{n-2},\ \dots\ $: no memory)

transition rules

select candidate for $\ \tilde x_{n+1}\ $ randomly
draw a random number $\ r\in[0,1]$
repeat
transition probability from state $\ x\ $ to state $\ y$

detailed balance

Markov chain $\ \{x_n\}\ $ with transition probability
two states $\ x\ $ and $\ y\ $ phase space
with $\ \rho(y)>\rho(x)\ $
(relative) number of transtions $\ x\to y$ $$ T(x\to y) \ =\ P_N(x)\,P(x\to y) \ =\ P_N(x) $$ since $\ \rho(y)/\rho(x)>1$
(relative) number of transtions $\ y\to x$ $$ T(y\to x) \ =\ P_N(y)\,P(y\to x) \ =\ P_N(y)\rho(x)/\rho(y) $$ since $\ \rho(x)/\rho(y)<1$

detailed balance

for a large number of walkers (random walks)
of for $\ N\to\infty$
the probability $\ P_N(x)\ $ will be determined
by the steady-state condition (detailed balance) $$ T(y\to x) \ =\ T(x\to y) $$ there would be otherwise an infinite accumulation
leading to
the Metropolis algorithm hence correctly yields

evaluation of relative properties

with the Metropolis algorithm only relativ quantities can be evaluated
use statistics (error estimate) from Monte Carlo program

/** Evaluating y_{n+1}/y_n, with y_n = \int_0^1 x^n/(a+x) dx
  * with Monte Carlo (Metropolis) algorithm.
  * On input: number of Monte Carlo steps N.
  */
#include <iostream>
#include <stdio.h>     // for printf
#include <stdlib.h>    // srand, rand
#include <math.h>      // for pow, etc.
#include <fstream>     // file streams
 
using namespace std;
 
/** The proability density.
  */
double myDensity(double x, double nPow, double aPar) {
 return pow(x,nPow)/(x+aPar);
  } 
 
// ---
// --- main
// ---
int main(int argLength, char* argValues[])   // parsing command line arguments
{
 const double aPar = 10.0;                 // parameter
 const double nPow = 3.0;                  // power
 srand(time(NULL));                  // seed for random()
//
 int N = 0;                          // number of MC steps
 if (argLength==2)
   N = atof(argValues[1]);
 else
   {
   printf("on input N: number of MC steps\n");
   return 0;
   }
//
// --- other parameters and variables
//
 const int K = 20;                                // number of bins
 int nK = N/K;                                    // number of data per bin
 double* data = new double[N];                    // storing all measurements
 double* meanBin = new double[K];                 // means of bins
 double mean = 0.0, sigma = 0.0;                  // result: mean and error
 double x = random()/(double)RAND_MAX;            // random starting value
 double y, ratio;                                 // variables
 data[0] = x;                                     // first measurment
//
// --- loop over all Monte Carlo steps
//
 for (int i=1; i<N; i++)
  {
  y = random()/(double)RAND_MAX;
  ratio = myDensity(y,nPow,aPar)/ myDensity(x,nPow,aPar);
//
  double probability = random()/(double)RAND_MAX; // of acceptance
  if (ratio>probability)
    x = y;                                        // accept candidate
  data[i] = x;                                    // storing measurement
  }  // end of loop over all Monte Carlo draws
//
// --- evaluate the overall mean and the mean of the bins
//
 for (int i=0; i<N; i++) 
    mean += data[i];
 mean = mean/N;
//
 for (int k=0; k<K; k++)
   {
   meanBin[k] = 0.0;       
   for (int n=0; n<nK; n++)                     // nK data points in each bin
      meanBin[k] += data[k*nK+n];
   meanBin[k] = meanBin[k]/nK;
   }  // end of loop over all bins
//
// --- evaluate the error: variance of means of bins
//
 for (int k=0; k<K; k++)
   sigma += (meanBin[k]-mean)*(meanBin[k]-mean);
 sigma = sqrt(sigma/K);
//
// --- printout of result
//
 printf(" a  in  y^n/(x+a)       : %8.4f\n",aPar);
 printf(" n  in  y^n/(x+a)       : %8.4f\n",nPow);
 printf(" number of measurements : %d\n",N);
 printf(" number of bins         : %d\n",K);
 printf(" data points per bin    : %d\n",nK);
 printf(" y_{n+1}/y_n  estimate  : %8.4f\n",mean);
 printf(" estimate for accuracy  : %8.4f\n",sigma);
 printf("\n");
 printf(" ratio y_4/y_3 (exact)  : %8.4f\n",0.797490089);
// 
 return 1;
 }  // end of main()

/** Evaluating y_{n+1}/y_n, with y_n = \int_0^1 x^n/(a+x) dx
  * with Monte Carlo (Metropolis) algorithm.
  * On input: number of Monte Carlo steps N.
  */
#include <iostream>
#include <stdio.h>     // for printf
#include <stdlib.h>    // srand, rand
#include <math.h>      // for pow, etc.
#include <fstream>     // file streams
 
using namespace std;
 
/** The proability density.
  */
double myDensity(double x, double nPow, double aPar) {
 return pow(x,nPow)/(x+aPar);
  } 
 
// ---
// --- main
// ---
int main(int argLength, char* argValues[])   // parsing command line arguments
{
 const double aPar = 10.0;                 // parameter
 const double nPow = 3.0;                  // power
 srand(time(NULL));                  // seed for random()
//
 int N = 0;                          // number of MC steps
 if (argLength==2)
   N = atof(argValues[1]);
 else
   {
   printf("on input N: number of MC steps\n");
   return 0;
   }
//
// --- other parameters and variables
//
 const int K = 20;                                // number of bins
 int nK = N/K;                                    // number of data per bin
 double* data = new double[N];                    // storing all measurements
 double* meanBin = new double[K];                 // means of bins
 double mean = 0.0, sigma = 0.0;                  // result: mean and error
 double x = random()/(double)RAND_MAX;            // random starting value
 double y, ratio;                                 // variables
 data[0] = x;                                     // first measurment
//
// --- loop over all Monte Carlo steps
//
 for (int i=1; i<N; i++)
  {
  y = random()/(double)RAND_MAX;
  ratio = myDensity(y,nPow,aPar)/ myDensity(x,nPow,aPar);
//
  double probability = random()/(double)RAND_MAX; // of acceptance
  if (ratio>probability)
    x = y;                                        // accept candidate
  data[i] = x;                                    // storing measurement
  }  // end of loop over all Monte Carlo draws
//
// --- evaluate the overall mean and the mean of the bins
//
 for (int i=0; i<N; i++) 
    mean += data[i];
 mean = mean/N;
//
 for (int k=0; k<K; k++)
   {
   meanBin[k] = 0.0;       
   for (int n=0; n<nK; n++)                     // nK data points in each bin
      meanBin[k] += data[k*nK+n];
   meanBin[k] = meanBin[k]/nK;
   }  // end of loop over all bins
//
// --- evaluate the error: variance of means of bins
//
 for (int k=0; k<K; k++)
   sigma += (meanBin[k]-mean)*(meanBin[k]-mean);
 sigma = sqrt(sigma/K);
//
// --- printout of result
//
 printf(" a  in  y^n/(x+a)       : %8.4f\n",aPar);
 printf(" n  in  y^n/(x+a)       : %8.4f\n",nPow);
 printf(" number of measurements : %d\n",N);
 printf(" number of bins         : %d\n",K);
 printf(" data points per bin    : %d\n",nK);
 printf(" y_{n+1}/y_n  estimate  : %8.4f\n",mean);
 printf(" estimate for accuracy  : %8.4f\n",sigma);
 printf("\n");
 printf(" ratio y_4/y_3 (exact)  : %8.4f\n",0.797490089);
// 
 return 1;
 }  // end of main()