General Principles in Random Variates Generation

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1 General Principles in Random Variates Generation E. Moulines Ecole Nationale Supérieure des Télécommunications 20 mai 2007

2 Why use random numbers? Random numbers lie at the heart of many scientific and financial computations. Many important analytically intractable problems can be solved using Monte Carlo techniques. Many problems that can be solved by other computational methods, can be solved much more efficiently using Monte Carlo techniques

3 Why use random numbers Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed U(0, 1) random variables These random numbers are then transformed as needed to simulation random variables from different probability distributions, such as the normal, exponential, Poisson, etc...

4 Real source of randomness To draw a winning number for several million dollars lottery, people would generally not trust a computer! They would rather prefer a simple physical system that they understand well, such as drawing balls from a container to select successive digits. Even this requires many precautions : the balls must have identical weights and sizes, be well mixed, and changed regularly to reduce the chances that some numbers come out more frequently in the long run... Such procedure is clearly impractical for computer simulations, which often requires millions of random numbers!

5 Real source of randomness on a computer User movements (keyboard strikes, mouse). hard disk / network activity. Processor load, Cursor position, the row currently displayed. Some kernel parameters ( /dev/random )

6 Why use pseudorandom numbers rather than real RN s? As of today, the most convenient and most reliable way of generating the random numbers for stochastic simulations appears to be via deterministic algorithms with a solid mathematical basis. These algorithms produce a sequence of numbers which are in fact not random at all, but seem to behave like independent random numbers ; that is, like a realization of a sequence of IID U(0, 1) random variables Much more convenient to use pseudorandom numbers. Don t need radiation sources or lava lamps. Pseudorandom numbers can be generated on the fly and the sequences are reproducible.

7 Pseudo-Random generator A (pseudo)random number generator is a structure G = (S, s 0, T, U, G), where S is a finite set of states, s 0 S is the initial state (or seed), the mapping T : S S is the transition function, U is a finite set of output symbols, G : S U is the output function.

8 Basic Algorithm The state of the generator is initially s 0 and evolves according to the recurrence s n = T (s n 1 ), for n = 1, 2, 3,.... At step n, the generator outputs the number u n = G(s n ). The u n, n > 0 are the observations, and are also called the random numbers produced by the generator. Clearly, the sequence of states s n is eventually periodic, since the state space S is finite. Indeed, the generator must eventually revisit a state previously seen ; that is, s j = s i for some j > i > 0. From then on, one must have s j+n = s i+n and u j+n = u i+n for all n > 0.

9 Period The period length is the smallest integer ρ > 0 such that for some integer τ 0 and for all n τ, s ρ+n = s n. The smallest τ with this property is called the transient. Often, τ = 0 and the sequence is then called purely periodic. Note that the period length cannot exceed S, the cardinality of the state space. Good generators typically have their ρ very close to S (otherwise, there is a waste of computer memory).

10 Congruential Generators Linear Congruential Generator How to build a good random generator? How can one build a deterministic generator whose output looks totally random? Perhaps a first idea is to write a computer program more or less at random that can also modify its own code in an unpredictable way... However, experience shows that random number generators should not be built at random (see Knuth for more discussion on this). Building a good random number generator may look easy on the surface, but it is not. It requires a good understanding of heavy mathematics.

11 Congruential Generators Linear Congruential Generator A Linear Congruential Generator The best-known and (still) most widely used types of generators are the simple linear congruential generators (LCGs). The state at step n is an integer x n and the transition function T is defined by the recurrence x n = (ax n 1 + c) mod m, where m > 0, a > 0, and c are integers called the modulus, the multiplier, and the additive constant. One can identify s n with x n and the state space S is the set {0,..., m 1}. To produce values in the interval [0, 1], one can simply define the output function G by u n = G(x n ) = x n /m.

12 Congruential Generators Linear Congruential Generator Multiplicative Linear Congruential Generator When c = 0, this generator is called a multiplicative linear congruential generator (MLCG). It is cheaper to implement because it eliminates the addition operation. The maximal period length for the LCG is m in general. For the MLCG it cannot exceed m 1, since 0 is an absorbing state that must be avoided. Two popular values of m are n = and m = In the latter case, the modulo operation may be carried out efficiently by shift operations (or by extracting bits or words if these operations are carried out using unsigned integer arithmetics). However, these values may be too small for the requirements of today s simulations.

13 Congruential Generators Linear Congruential Generator Period of Multiplicative MLCG If the modulus m is a prime number, the maximal period ρ = m 1 results if the multiplier a is a primitive root of m, i.e. a m = 1 mod m and a q 1 mod m for 0 < q < m 1. For this restriction of m and a, each positive integer in {0,..., M} appears exactly once in any sequence of M 1 consecutive numbers produced by this generator.

14 Congruential Generators Linear Congruential Generator Period of LCG Simple conditions are available ensuring that the generator has full period - i.e. that the number of distinct values generated from any seed x 0 is m 1. If c 0, these conditions are c and m are relatively prime every prime number that divides m divides a 1, a 1 is divisible by 4 if m is. As a simple consequence, we observe that if m is a power of two, the generator has full period if c is odd a = 4n + 1 for some integer n.

15 Congruential Generators Linear Congruential Generator A running example For a concrete illustration, let m = = , c = 0, and a = These parameters were originally proposed by Lewis. Take x 0 = Then x 1 = 16807x12345 mod m = u 1 = /m = , x 2 = 16807x mod m = u 2 = /m = , x 3 = 16807x mod m = u 3 = /m =

16 Congruential Generators Multiple Recursive Generators Principle Consider the linear recurrence x n = (a l x n a k x n k ) mod m, where the order k and the modulus m are positive integers, while the coefficients a 1,..., a k are integers in the range { (m 1),..., m 1}. Define Z m as the set {0, 1,..., m 1} on which operations are performed modulo m. The state at step n of the multiple recursive generator (MRG) is the vector s n = (x n,..., x n+k 1 ) Z k m. The output function can be defined simply by u n = G(s n ) = x n /m, which gives a value in [0, 1], or by a more refined transformation if a better resolution than 1/m is required.

17 Congruential Generators Multiple Recursive Generators Properties The characteristic polynomial P of the MRG is defined by P (z) = z k a 1 z k 1 a k z. The maximal period length of the MRG is ρ = m k 1 (for all initial seeds), reached if and only if m is prime and P is a primitive polynomial over Z m, identified here as the finite field with m elements.

18 Congruential Generators Multiple Recursive Generators Primitive roots Suppose that m is prime and let r = (m k 1)/(m 1). The polynomial P is primitive over Z m if and only if it satisfies the following conditions, where everything is assumed to be modulo m 1. [( 1) k+l a k ] (m l)/q = 1 for each prime factor q of m 1 2. z r mod P (z) = ( 1) k+l a k 3. z r/q mod P (z) has degree > 0 for each prime factor q of r, 1 < q < r.

19 Congruential Generators Multiple Recursive Generators Properties If m is not prime, the period length of the MRG has an upper bound typically much lower than m k 1. For k = 1 and m = 2 e, e > 4, the maximum period length is 2 e 2, which is reached if a 1 = 3 or 5( mod 8) and x 0 is odd. Otherwise, if m = p e for p prime and e > 1, and k > 1 or p > 2, the upper bound is (p k 1)p e 1.

20 Congruential Generators Multiple Recursive Generators Choice of m Clearly, p = 2 is very convenient from the implementation point of view, because the modulo operation then amounts to chopping-off the higher-order bits. So to compute ax mod m in that case, for example with e = 32 on a 32-bit computer, just make sure that the overflow-checking option or the compiler is turned off, and compute the product ax using unsigned integers while ignoring the overflow. However, taking m = 2 e imposes a big sacrifice on the period length, especially for k > 1. For example, if k = 7 and in m = (a prime), the maximal period length is (2 31 1) But for m = 2 31 and the same value of k, the upper bound becomes ρ < (2 7 1) < 2 37, which is more than times shorter.

21 Congruential Generators Multiple Recursive Generators Implementation for prime m For k > 1 and prime m, for the characteristic polynomial P to be primitive, it is necessary that a k and at least another coefficient a j be nonzero. From the implementation point of view, it is best to have only two nonzero coefficients ; that is, a recurrence of the form x n = (a r x n r + a k x n k ) mod m with characteristic trinomial P defined by P (z) = z k a r z k r a k.

22 Congruential Generators Multiple Recursive Generators Overflow When m is not a power of 2, computing and adding the products modulo m is not necessarily straightforward, using ordinary integer arithmetic, because of the possibility of overflow. For example, if m = and a l = 16807, then x n 1 can be as large as , so the product a 1 x n 1 can easily exceed 2 31.

23 Congruential Generators Multiple Recursive Generators Floating-point arithmetic In this case, it is appropriate to represent all the numbers and perform all the arithmetic modulo m in double-precision floating point. This works provided that the multipliers a i are small enough so that the integers a i x n i and their sum are always represented exactly by the floating-point values. A sufficient condition is that the floating-point numbers are represented with at least [log 2 ((m 1)(a l + + a k ))] bits of precision in their mantissa, where [x] denotes the smallest integer larger or equal to x. On computers with good 64-bit floating-point hardware (most computers nowadays), this approach usually gives by far the fastest implementation.

24 Objectives Define the error in a meaningful. Devise a procedure for choosing which of several candidate generators with, say, the same modulus, is best according to the specified error criterion. Reduce the error achievable by a particular congruential genera- generator by modifying its structure in a way that preserves its computability, execution time, reproducibility property, and ease of implementation.

25 Structural / Statistical Methods The techniques used to evaluate the quality of random number generators can be partitioned into two main classes : The structural analysis methods (sometimes called theoretical tests) : studies the mathematical structure underlying the successive values produced by the generator, most often over its entire period length the statistical methods (also called empirical tests). It observes the output and applies a statistical test of hypothesis to catch up significant statistical defects. An unlimited number of such tests can be designed.

26 Structural methods Lattice A lattice of dimension t in the t-dimensional real space R t, is a set of the form t L def = V = z j V j, z j Z, (1) j=1 where {V 1,..., V t } is a basis of R t. The lattice L is thus the set of all integer linear combinations of the vectors V 1,..., V t ; ese vectors are called a lattice basis of L. The basis {W 1,..., W t } which satisfies V i, W j = δ i,j is the dual basis, and the lattice generated by this dual basis is called the dual lattice to L.

27 Structural methods Lattice Consider the set T t T t def = {u n = (u n,..., u n+t 1 ), s 0 = (x 0,..., x k 1 ) Z k m} of all overlapping t-uples of successive values produced by the MRG, with u n = x n /m, from all possible initial seeds. This set T t is the intersection of a lattice L t with the t-dimnensional unit hypercube I t = [0, 1] t.

28 Structural methods Lattice Property of an MRG For t k, each vector (x 0,..., x t 1 ) Z t m can be taken as s 0, so T t = Z t m/m = (Z t /m) I t ; In dimension t > k, the set T t contains only m k points, while Z t m/m contains m t points. Therefore, for large t, T t contains only a small fraction of the t-dimensional vectors whose coordinates are multiples of 1/m.

29 Structural methods (a) a = (b) a = (c) a = 51

30 Structural methods Spectral Distance When t k, the points also lie on a lattice L t composed of parallel hyperplanes. The spectral distance d t is the minimal distance between two successive hyperplanes of any family of parallel hyperplanes covering the lattice ; this distance may be shown to be the minimal norm of a non-zero vector belonging to the dual lattice. The shorter the distance d t, the better, because a large d t means thick empty slices. This spectral distance can be computed by solving a quadratic programming problem in integer random variables (which can be solved using branch and bound algorithms for example). Appropriate values of the multiplier have often been found by minimizing this spectral distance (for several values of t).

31 Structural methods Spectral distance The spectral distance is lower bounded by d t d t = 1 γ t m k/t, where γ t is a constant depending only on t. As an example, for m = t = t = 3 d t t = t = t = 6 (2) showing that, even for moderate value of t, the space between the hyperplanes can be pretty large!

32 Structural methods Discrepancy : Definition Consider the N points u n = (u n,..., u n+t 1 ), for n = 0,..., N 1, in dimension t, formed by (overlapping) vectors of t successive output values of the generator. For any hyper-rectangular box aligned with the axes, of the form R = t j=1 [α j, β j [, with 0 α j < β j < 1, let I(R) be the number of points un falling into R, V (R) = t j=1 (β j α j ) be the volume of R. Let R be the set of all such regions R. The t-dimensional (extreme) discrepancy of the set of points {u 0,..., u N 1 } is given by D (t) N = max V (R) I(R)/N. R R If we impose α j = 0 for all j, then the corresponding quantity is called the star-discrepancy

33 Structural methods Why? A low discrepancy value means that the points are very evenly distributed in the unit hypercube. To get superuniformity of the sequence over its entire period, one might want to minimize the discrepancy D (t) N or D (t) N for t = 1, 2,... A major practical difficulty with discrepancy is that it can be computed only for very special cases. For LCGs, for example, it can be computed efficiently in dimension t = 2, but for larger t, the computing cost then increases as O(N t ). In most cases, only (upper and lower) bounds on the discrepancy are available. Often, these bounds are expressed as orders of magnitude as a function of N, are defined for N = ρ, and/or are averages over a large (specific) class of generators (e.g., over all full-period MLCGs with a given prime modulus)

34 Structural methods Super-Uniformity and Discrepancy We previously argued for superuniformity over the entire period, which means seeking the lowest possible discrepancy. When a subsequence of length N is used (for N < ρ), starting, say, at a random point along the entire sequence, the discrepancy of that subsequence should behave (viewed as a random variable) as the discrepancy of a sequence of IID U(0, 1) random variables. The latter is (roughly) of order O(N 1/2 ) for both the star and extreme discrepancies.

35 Structural methods Super-Uniformity and Discrepancy Niederreiter shows that the discrepancy of full-period MLCGs over their entire period (of length ρ = m 1), on the average over multipliers a, is of order O(m 1 (log m) t log log(m + 1)). This order is much smaller (for large m) than O(m 1/2 ), meaning superuniformity. Over small fractions of the period length, the available bounds on the discrepancy are more in accordance with the law of the iterated logarithm. This is yet another important justification for never using more than a negligible fraction of the period.

36 Statistical methods Example I Suppose that one generates n random numbers from a generator whose output is supposed to imitate IID U(0, 1) random variables. Let T be the number of values that turn out to be below 1/2, among those n. For large n, T should normally be not too far from n/2... In fact, one should expect T to behave like a binomial random variable with parameters (n, 1/2). So if one repeats this experiment several times (e.g., generating N values of T), the distribution of the values of T obtained should resemble that of the binomial distribution (and the normal distribution with mean n/2 and standard deviation n/2 for large n).

37 Statistical methods Example II If N = 100 and n = 10000, the mean and standard deviation are 5000 and 50, respectively. With these parameters, if one observes, for instance, that 12 values of T are less than 4800, or that 98 values of T out of 100 are less than 5000, one world readily conclude that something is wrong with the generator. On the other hand, if the values of T behave as expected, one may conclude that the generator seems to reproduce the correct behavior for this particular statistic T (and for this particular sample size). But nothing prevents other statistics than this T to behave wrongly!

38 Statistical methods Test Statistics Define the null hypothesis H 0 as : The generator s output is a sequence of IID U(O, 1) random variables. Formally, this hypothesis is false, since the sequence is periodic and usually deterministic (except perhaps for the seed). But if this cannot be detected by reasonable statistical tests, one may assume that H 0 holds anyway. In fact, what really counts in the end is that the statistics of interest in a given simulation have (sample) distributions close enough to their theoretical ones.

39 Statistical methods Tests Statistics The most usual procedure is to devise some statistic whose distribution under the null could be obtained without too much trouble. Then, if extreme values of this statistic were observed, the null hypothesis of randomness is rejected. A statistical test for H 0 can be defined by any function T of a finite number of U(0, 1) random variables, for which the distribution under H 0 is either known or can be approximated well enough. The random variable T is called the test statistic. The statistical test tries to find empirical evidence against H 0.

40 Statistical methods Single-Level procedures When applying a statistical test to a random number generator, a single-level procedure computes the value of T, say t, then computes the p-value δ = P H0 [T > t H 0 ] (for a one-sided test) A single-sided test will reject only of δ is too close to 0, or only if it is too close to 1.

41 Statistical methods Two-levels procedures A two-level test obtains (say) N independent copies of T, denoted T 1,..., T N, and computes their empirical distribution ˆF N. This empirical distribution is then compared to the theoretical distribution of T under H 0, say F, via a standard goodness-of-fit test, such as the Kolmogorov-Smirnov (KS) or Anderson-Darling test. One version of the KS goodness-of-fit test uses the statistic D N = sup ˆF N (x) F (x), <x< for which an approximation of the distribution under H 0 is available, assuming that the distribution F is continuous. Once the value d N of the statistic D N is known, one computes the p-value of the test, defined as δ = P[D N > d N H 0 ] ; we reject H 0 if δ is too close to 0.

42 Statistical methods Which one is best? For a given test and a fixed computing budget, the question arises of what is best : to choose a small N (e.g., N = 1) and base the test statistic T on a large sample size, or the opposite? There is no universal winner. It depends on the test and on the alternative hypothesis. The rationale for two-level testing is to test the sequence not only globally, but also locally, by looking at the distribution of values of T over shorter subsequences. In most cases, when testing random number generators, N = 1 turns out to be the best choice because the same regularities or defects of the generators tend to repeat themselves over all long-enough subsequences. But it also happens for certain tests that the cost of computing T increases faster than linearly with the sample size, and this gives another argument for choosing N > 1.

43 Statistical methods Which are the best tests? Simply testing uniformity, or pair correlations, is far from enough. Good tests are designed to catch higher-order correlation properties or geometric patterns of the successive numbers. Such patterns can easily show up in certain classes of applications. If the generator is to be used to estimate the expectation of some random variable T by generating replicates of T, the best test would be the one based on T as a statistic. But this is impractical, since if one knew the distribution of T, one would not use simulation to estimate its mean.

44 Statistical methods Which are the best tests? Ideally, a good test for this kind of application should be based on a statistic T whose distribution is known and resembles that of T. But such a test is rarely easily available. Moreover, only the user can apply it. When designing a general purpose generator, one has no idea of what kind of random variable interests the user. So, the best the designer can do (after the generator has been properly designed) is to apply a wide variety of tests that tend to detect defects of different natures.

45 Statistical methods Batteries of tests The statistical tests described by Knuth have long been considered the standard tests for random number generators. A Fortran implementation of (roughly) this set of tests is given in the package TESTRAND. A newer battery of tests is DIEHARD, designed by Marsaglia. It contains more stringent tests, in the sense that more generators tend to fail some of the tests.

46 Statistical methods Die Hard test I Birthday Spacings : Choose random points on a large interval. The spacings between the points should be asymptotically Poisson distributed. The name is based on the birthday paradox. Overlapping Permutations : Analyze sequences of five consecutive random numbers. The 120 possible orderings should occur with statistically equal probability. Ranks of matrices : Select some number of bits from some number of random numbers to form a matrix over 0,1, then determine the rank of the matrix. Count the ranks. Monkey Tests : Treat sequences of some number of bits as words. Count the overlapping words in a stream. The number of words that don t appear should follow a known distribution. The name is based on the infinite monkey theorem.

47 Statistical methods Die Hard test II Count the 1 s : Count the 1 bits in each of either successive or chosen bytes. Convert the counts to letters, and count the occurrences of five-letter words. Parking Lot Test : Randomly place unit circles in a 100 x 100 square. If the circle overlaps an existing one, try again. After 12,000 tries, the number of successfully parked circles should follow a certain normal distribution. Minimum Distance Test : Randomly place 8,000 points in a 10,000 x 10,000 square, then find the minimum distance between the pairs. The square of this distance should be exponentially distributed with a certain mean. Random Spheres Test : Randomly choose 4,000 points in a cube of edge 1,000. Center a sphere on each point, whose radius is the minimum distance to another point. The smallest sphere s volume should be exponentially distributed with a certain mean.

48 Statistical methods Die Hard Test III The Squeeze Test : Multiply 2 31 by random floats on [0, 1[ until you reach 1. Repeat this 100, 000 times. The number of floats needed to reach 1 should follow a certain distribution. Overlapping Sums Test : Generate a long sequence of random floats on [0, 1[. Add sequences of 100 consecutive floats. The sums should be normally distributed with characteristic mean and sigma. Runs Test : Generate a long sequence of random floats on [0, 1]. Count ascending and descending runs. The counts should follow a certain distribution. The Craps Test : Play 200, 000 games of craps, counting the wins and the number of throws per game. Each count should follow a certain distribution.

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