Lecture 20. Randomness and Monte Carlo. J. Chaudhry. Department of Mathematics and Statistics University of New Mexico

Transcription

1 Lecture 20 Randomness and Monte Carlo J. Chaudhry Department of Mathematics and Statistics University of New Mexico J. Chaudhry (UNM) CS / 40

2 What we ll do: Random number generators Monte-Carlo integration Monte-Carlo simulation Our testing and our methods often rely on randomness and the impact of finite precision Random input Random perturbations Random sequences J. Chaudhry (UNM) CS / 40

3 Randomness Randomness unpredictability One view: a sequence is random if it has no shorter description Physical processes, such as flipping a coin or tossing dice, are deterministic with enough information about the governing equations and initial conditions. But even for deterministic systems, sensitivity to the initial conditions can render the behavior practically unpredictable. we need random simulation methods J. Chaudhry (UNM) CS / 40

4 Quasi-Random Sequences For some applications, reasonable uniform coverage of the sample is more important than the randomness True random samples often exhibit clumping Perfectly uniform samples uses a uniform grid, but does not scale well at high dimensions quasi-random sequences attempt randomness while maintaining coverage J. Chaudhry (UNM) CS / 40

5 Quasi-Random Sequences quasi random sequences are not random, but give random appearance by design, the points avoid each other, resulting in no clumping J. Chaudhry (UNM) CS / 40

6 Randomness is easy, right? In May, 2008, Debian announced a vulnerability with OpenSSL: the OpenSSL pseudo-random number generator the seeding process was compromised (2 years) only 32,767 possible keys seeding based on process ID (this is not entropy!) all SSL and SSH keys from 9/2006-5/2008 regenerated all certificates recertified Cryptographically secure pseudorandom number generator (CSPRNG) are necessary for some apps Other apps rely less on true randomness J. Chaudhry (UNM) CS / 40

7 Repeatability With unpredictability, true randomness is not repeatable...but lack of repeatability makes testing/debugging difficult So we want repeatability, but also independence of the trials 1 >> rand( seed,1234) 2 >> rand(10,1) J. Chaudhry (UNM) CS / 40

8 Pseudorandom Numbers Computer algorithms for random number generations are deterministic...but may have long periodicity (a long time until an apparent pattern emerges) These sequences are labeled pseudorandom Pseudorandom sequences are predictable and reproducible (this is mostly good) J. Chaudhry (UNM) CS / 40

9 Random Number Generators Properties of a good random number generator: Random pattern: passes statistical tests of randomness Long period: long time before repeating Efficiency: executes rapidly and with low storage Repeatability: same sequence is generated using same initial states Portability: same sequences are generated on different architectures J. Chaudhry (UNM) CS / 40

10 Random Number Generators Early attempts relied on complexity to ensure randomness midsquare method: square each member of a sequence and take the middle portion of the results as the next member of the sequence...simple methods with a statistical basis are preferable J. Chaudhry (UNM) CS / 40

11 Linear Congruential Generators Congruential random number generators are of the form: x k = (ax k 1 + c) ( mod m) where a and c are integers given as input. x 0 is called the seed Integer m is the largest integer representable (e.g ) Quality depends on a and c. The period will be at most m. Example Let a = 13, c = 0, m = 31, and x 0 = 1. 1, 13, 14, 27, 10, 6,... This is a permutation of integers from 1,..., 30, so the period is m 1. J. Chaudhry (UNM) CS / 40

12 History From C. Moler, emphncm IBM used Scientific Subroutine Package (SSP) in the 1960 s the mainframes. Their random generator, rnd used a = 65539, c = 0, and m = arithmetic mod 2 31 is done quickly with 32 bit words. multiplication can be done quickly with a = with a shift and short add. Notice (mod m): x k+2 = 6x k+1 9x k...strong correlation among three successive integers J. Chaudhry (UNM) CS / 40

13 History From C. Moler, NCM Matlab used a = 7 5, c = 0, and m = for a while period is m 1. this is no longer sufficient J. Chaudhry (UNM) CS / 40

14 what s used? Two popular methods: 1. Method of Marsaglia (period ). 1 Initialize x 0,..., x 3 and c to random values given a seed 2 3 Let s = x n x n x n x n 1 + c 4 5 Compute x n = s mod c = floor(s/2 32 ) 2. rand() in Unix uses a = , c = 12345, m = In general, the digits in random numbers are not themselves random...some patterns reoccur much more often. J. Chaudhry (UNM) CS / 40

15 Linear Congruential Generators sensitive to a and c be careful with supplied random functions on your system period is m standard division is necessary if generating floating points in [0, 1). J. Chaudhry (UNM) CS / 40

16 Fibonacci produce floating-point random numbers directly using differences, sums, or products. Typical subtractive generator: x k = x k 17 x k 5 with lags of 17 and 5. Lags much be chosen very carefully negative results need fixing more storage needed than congruential generators no division needed very very good statistical properties long periods since repetition does not imply a period J. Chaudhry (UNM) CS / 40

17 Discrete random variables random variable x values: x 0, x 1,..., x n probabilities p 0, p 1,..., p n with n i=0 p i = 1 throwing a die (1-based index) values: x 1 = 1, x 2 = 2,..., x 6 = 6 probabilities p i = 1/6 J. Chaudhry (UNM) CS / 40

18 Expected value and variance expected value: average value of the variable E[x] = n x j p j j=1 variance: variation from the average σ 2 [x] = E[(x E[x]) 2 ] = E[x 2 ] E[x] 2 throwing a die expected value: E[x] = ( )/6 = 3.5 variance: σ 2 [x] = J. Chaudhry (UNM) CS / 40

19 estimated E[x] to estimate the expected value, choose a set of random values based on the probability and average the results N E[x] = 1 N j=1 x i bigger N gives better estimates throwing a die 3 rolls: 3, 1, 6 E[x] ( )/3 = rolls: 3, 1, 6, 2, 5, 3, 4, 6, 2 E[x] ( )/9 = 3.51 J. Chaudhry (UNM) CS / 40

20 Law of large numbers by taking N to, the error between the estimate and the expected value is statistically zero. That is, the estimate will converge to the correct value P(E[x] = lim N 1 N N x i ) = 1 This is the basic idea behind Monte Carlo estimation i=1 J. Chaudhry (UNM) CS / 40

21 Continuous random variable and density function random variable: x values: x [a, b] probability: density function ρ(x) with b a ρ(x) dx = 1 probability that the variable has value between (c, d) [a, b]: d c ρ(x) J. Chaudhry (UNM) CS / 40

22 Uniformly distributed random variable ρ(x) is constant b a ρ(x) dx = 1 means ρ(x) = 1/(b a) J. Chaudhry (UNM) CS / 40

23 Continuous extensions to expectation and variance expected value E[x] = E[g(x)] = variance σ 2 [x] = σ 2 [g(x)] = b a b a b a b a xρ(x) dx g(x)ρ(x) dx (x E[x]) 2 ρ(x) dx = E[x 2 ] (E[x]) 2 (g(x) E[g(x)]) 2 p(x) dx = E[g(x) 2 ] (E[g(x)]) 2 estimating the expected value E[g(x)] 1 N N g(x i ) J. Chaudhry (UNM) CS / 40 i=1

24 multidementional extensions difficult domains (complex geometries) expected value E[g(x)] = g(x)ρ(x) da x x D J. Chaudhry (UNM) CS / 40

25 Sampling over intervals Almost all systems provide a uniform random number generator on (0, 1) Matlab: rand If we need a uniform distribution over (a, b), then we modify x k on (0, 1) by (b a)x k + a Matlab: Generate 100 uniformly distributed values between (a, b) r = a + (b-a).*rand(100,1) Uniformly distributed integers: randi Random permutations: randperm J. Chaudhry (UNM) CS / 40

26 Non-uniform distributions sampling nonuniform distributions is much more difficult if the cumulative distribution function is invertible, then we can generate the non-uniform sample from the uniform. Consider probability density function ρ(t) = λe λt, t > 0 giving the cumulative distribution function as x x F(x) = λe λt dt = (e λt ) = (1 e λx ) thus inverting the CDF where y k is uniform in (0, 1)....not so easy in general 0 x k = log(1 y k )/λ 0 J. Chaudhry (UNM) CS / 40

27 Monte Carlo Central to the PageRank (and many many other applications in fincance, science, informatics, etc) is that we randomly process something what we want to know is on average what is likely to happen what would happen if we have an infinite number of samples? let s take a look at integral (a discrete limit in a sense) J. Chaudhry (UNM) CS / 40

28 (deterministic) numerical integration split domain into set of fixed segments sum function values with size of segments (Riemann!) J. Chaudhry (UNM) CS / 40

29 Monte Carlo Integration We have for a random sequence x 1,..., x n 1 0 f (x) dx 1 n n f (x i ) i=1 1 n=100; 2 x=rand(n,1); 3 a=f(x); 4 s=sum(a)/n; J. Chaudhry (UNM) CS / 40

30 Monte Carlo over random domains We have for a random sequence x 1,..., x n in Ω I = Ω f (x) dx Q n = V 1 n where V is the volume given by V = Ω dx n f (x i ) Law of large numbers gurantees that Q n I as n Works in any dimension! One dimension: b a f (x) dx (b a) 1 n i=1 n f (x i ) i=1 where x 1,..., x n are uniformly distributed in (a, b) J. Chaudhry (UNM) CS / 40

31 2d example: computing π Use the unit square [0, 1] 2 with a quarter-circle { 1 (x, y) circle f (x, y) = 0 else A quarter circle = f (x, y) dxdy = pi 4 J. Chaudhry (UNM) CS / 40

32 2d example: computing π Estimate the area of the circle by randomly evaluating f (x, y) A quarter circle 1 N N f (x i, y i ) i=1 J. Chaudhry (UNM) CS / 40

33 2d example: computing π By definition A quarter circle = π/4 so π 4 N N f (x i, y i ) i=1 J. Chaudhry (UNM) CS / 40

34 2d example: computing π algorithm 1 input N 2 call rand in 2d 3 for i=1:n 4 sum = sum + f (x i, y i ) 5 end 6 sum = 4 sum/n J. Chaudhry (UNM) CS / 40

35 2d example: computing π algorithm The expected value of the error is O( 1 N ) convergence does not depend on dimension deterministic integration is very hard in higher dimensions deterministic integration is very hard for complicated domains Trapezoid rule in dimension d: O( 1 N 2/d ) J. Chaudhry (UNM) CS / 40

36 Error in Monte Carlo Estimation The distribution of an average is close to being normal, even when the distribution from which the average is computed is not normal. What? Let x 1,..., x n be some independent random variables from any PDF Each x i has expected value µ and standard deviation σ Consider the Average A n = (x x n )/n The expected value of A n is µ and the standard deviation is σ n This follows from Central Limit Theorem What? The sample mean has an error of σ/ n J. Chaudhry (UNM) CS / 40

37 Stochastic Simulation Stochastic simulation mimics physical behavior through random simulations...also known as Monte Carlo method (no single MC method!) Usefulness: nondeterministic processes deterministic models that are too complicated to model deterministic models with high dimensionality J. Chaudhry (UNM) CS / 40

38 Stochastic Simulation Two requirements for MC: knowing which probability distributions are needed generating sufficient random numbers The probability distribution depends on the problem (theoretical or empirical evidence) The probability distribution can be approximated well by simulating a large number of trials J. Chaudhry (UNM) CS / 40

39 Stochastic Simulation: The Monte Hall Problem Question You are on a gameshow. There are three doors that you have the option of opening. There is a prize behind one of the doors, and there are goats behind the other two. You select one of the doors. One of the remaining two doors is opened to reveal a goat. You are then given the option of opening your door or the other remaining unopened door. Should you switch. Determine the probability that you will win the prize if you switch. Answer: Yes, always switch. Probability of success is 2/3. Can show using conditinal probability J. Chaudhry (UNM) CS / 40

40 Stochastic Simulation: The Monte Hall Problem Solution via random simuation 1 input N 2 wins = 0 3 for i=1:n 4 Generate random permutation of goats and prize 5 You randomly choose a door 6 if the door had a goat, host always chooses the other door with the goat 7 else if the door had a prize, host randomly chooses one of the doors with the goat 8 end 9 You switch your choice 0 if the switched door had the prize 1 wins = wins end 3 4 end 5 probability of success = wins/n J. Chaudhry (UNM) CS / 40