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. Monte-Carlo October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang October 30th, 2012 Hyun Min Kang and Monte Carlo Methods Random Numbers Biostatistics 615/815 Lecture 15: . . Summary . 1 / 32 . Complex Distribution Random sampling Using PRG Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Summary October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang just based on the observations. . . Pseudo random numbers . atmospheric noise . . True random numbers . Random Numbers . . Monte-Carlo . . . . . . . . . . . . Random Numbers Using PRG Complex Distribution Random sampling 2 / 32 . . . . . . . . . . . . . . . . . . . . . . • Truly random, non-determinstric numbers • Easy to imagine conceptually • Very hard to generate one or test its randomness • For example, http://www.random.org generates randomness via • A deterministic sequence of random numbers (or bits) from a seed • Good random numbers should be very hard to guess the next number

. Complex Distribution October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang Usage of random numbers in statistical methods Summary . Monte-Carlo . 3 / 32 Random sampling . Using PRG Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Resampling procedure • Permutation • Boostrapping • Simulation of data for evaluating a statistical method. • Stochatic processes • Markov-Chain Monte-Carlo (MCMC) methods

. Complex Distribution October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang seed can also be deciphered. encrypting the seed to a sequence of random bits function Usage of random numbers in other areas Summary . Monte-Carlo . 4 / 32 . . . . . . . . . . . . Random Numbers Random sampling Using PRG . . . . . . . . . . . . . . . . . . . . . . • Hashing • Good hash function uniformly distribute the keys to the hash spcae • Good pseudo-random number generators can replace a good hash • Cryptography • Generating pseudo-random numbers given a seed is equivalent to • If the pattern of pseudo-random numbers can be predicted, the original

. Random sampling October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang True random numbers Summary . Monte-Carlo . Complex Distribution Using PRG Random Numbers . . . . . . . . . . . . 5 / 32 . . . . . . . . . . . . . . . . . . . . . . • Generate only through physical process • Hard to generate automatically • Very hard to provde true randomness

. Random sampling October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang Pseudo-random numbers : Example code . . Monte-Carlo Complex Distribution Summary 6 / 32 . Using PRG . Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #include <iostream> #include <cstdlib> int main(int argc, char** argv) { int n = (argc > 1) ? atoi(argv[1]) : 1; int seed = (argc > 2 ) ? atoi(argv[2]) : 0; srand(seed); // set seed -- same seed, same pseudo-random numbers for(int i=0; i < n; ++i) { std::cout << (double)rand()/(RAND_MAX+1.) << std::endl; // generate value between 0 and 1 } return 0; }

. Random sampling October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang Pseudo-random numbers : Example run . . Monte-Carlo Complex Distribution Summary 7 / 32 . Using PRG . Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . user@host:~/$ ./randExample 3 0 0.242578 0.0134696 0.383139 user@host:~/$ ./randExample 3 0 0.242578 0.0134696 0.383139 user@host:~/$ ./randExample 3 10 7.82637e-05 0.315378 0.556053

. Properties of pseudo-random numbers October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang weakness to generate random numbers with a skwed distribution. guess . . Irregularity and unpredictablility without knowing the seed . generate identical sequence of random numbers . . Deterministic given the seed . . Summary . . . . . . . . . . . . . Random Numbers Using PRG Complex Distribution Random sampling Monte-Carlo 8 / 32 . . . . . . . . . . . . . . . . . . . . . . • Given a fixed random seed, the pseudo-random numbers should • Deterministic feature is useful for debugging a code • Without knowning the seed, the random numbers should be hard to • If you can guess it better than random, it is possible to exploit the

. . October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang pseudo-random numbers function in PHP Good vs. bad random numbers Summary . Monte-Carlo Complex Distribution Random sampling Using PRG Random Numbers . . . . . . . . . . . . 9 / 32 . . . . . . . . . . . . . . . . . . . . . . • Images using true random numbers from random.org vs. rand() • Visible patterns suggest that rand() gives predictable sequence of

. Complex Distribution October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang . Generating uniform random numbers - example in R Summary . Monte-Carlo 10 / 32 Random sampling Using PRG Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . > x <- runif(10) # x is size 10 vector uniformly distributed from 0 to 1 > x <- runif(10,0,10) # x ranges 0 to 0 > x <- as.integer(runif(10,0,10)) > x [1] 6 0 7 4 4 8 1 4 3 4 > set.seed(3429248) # set an arbitrary seed > x <- as.integer(runif(10,0,10)) > x [1] 7 6 3 4 6 7 4 9 2 1 > set.seed(3429248) # setting the same seed > x <- as.integer(runif(10,0,10)) # reproduce the same random variables > x [1] 7 6 3 4 6 7 4 9 2 1

. Complex Distribution October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang . Generating uniform random numbers in C++ Summary . Monte-Carlo 11 / 32 Random sampling Using PRG Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . #include <iostream> #include <boost/random/uniform_int.hpp> #include <boost/random/uniform_real.hpp> #include <boost/random/variate_generator.hpp> #include <boost/random/mersenne_twister.hpp> int main(int argc, char** argv) { typedef boost::mt19937 prgType; // Mersenne-twister : a widely used prgType rng; // lightweight pseudo-random-number-generator boost::uniform_int<> six(1,6); // uniform distribution from 1 to 6 boost::variate_generator<prgType&, boost::uniform_int<> > die(rng,six); // die maps random numbers from rng to uniform distribution 1..6 int x = die(); // generate a random integer between 1 and 6 std::cout << "Rolled die : " << x << std::endl; boost::uniform_real<> uni_dist(0,1); boost::variate_generator<prgType&, boost::uniform_real<> > uni(rng,uni_dist); double y = uni(); // generate a random number between 0 and 1 std::cout << "Uniform real : " << y << std::endl; return 0; }

. Complex Distribution October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang The random number does not vary (unlike R) Running Example Summary . . Monte-Carlo Random sampling . . . . . . . . . . . . 12 / 32 Using PRG Random Numbers . . . . . . . . . . . . . . . . . . . . . . user@host:~/$ ./randExample Rolled die : 5 Uniform real : 0.135477 user@host:~/$ ./randExample Rolled die : 5 Uniform real : 0.135477

. Random sampling October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang Specifying the seed Summary . Monte-Carlo Complex Distribution . 13 / 32 . Using PRG . Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . int main(int argc, char** argv) { typedef boost::mt19937 prgType; prgType rng; if ( argc > 1 ) rng.seed(atoi(argv[1])); // set seed if argument is specified boost::uniform_int<> six(1,6); // ... same as before }

. Random sampling October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang Running Example . . Monte-Carlo Complex Distribution Summary 14 / 32 . Using PRG . Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . user@host:~/$ ./randExample Rolled die : 5 Uniform real : 0.135477 user@host:~/$ ./randExample 1 Rolled die : 3 Uniform real : 0.997185 user@host:~/$ ./randExample 3 Rolled die : 4 Uniform real : 0.0707249 user@host:~/$ ./randExample 3 Rolled die : 4 Uniform real : 0.0707249

. Random sampling October 30th, 2012 Biostatistics 615/815 - Lecture 15 Hyun Min Kang . Summary . Monte-Carlo Complex Distribution If we don’t want the reproducibility 15 / 32 . Using PRG . Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . // include other headers as before #include <ctime> int main(int argc, char** argv) { typedef boost::mt19937 prgType; prgType rng; if ( argc > 1 ) rng.seed(atoi(argv[1])); // set seed if argument is specified else rng.seed(std::time(0)); // otherwise, use current time to pick arbitrary seed to start boost::uniform_int<> six(1,6); // ... same as before }