Stochastic Simulation Random number generation Bo Friis Nielsen - - PowerPoint PPT Presentation

Stochastic Simulation Random number generation

Bo Friis Nielsen

Applied Mathematics and Computer Science Technical University of Denmark 2800 Kgs. Lyngby – Denmark Email: bfn@imm.dtu.dk

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Random number generation Random number generation

• Uniform distribution
• Number theory
• Testing of random numbers
• Recommendations of random number generators

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Summary Summary

• We talk about generating pseudorandom numbers
• There exist a large number of RNG’s
• ... of varying quality
• Don’t implement your own, except for fun or as a research

project.

• Built-in RNG’s should be checked before use
• ... at least in general-purpose development environments.
• Scientific computing environments typically have state-of-the-art

RNG’s that can be trusted.

• Any RNG will fail, if the circumstances are extreme enough.

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History/background History/background

• The need for random numbers evident
• Tables
• Physical generators. Lottery machines
• Need for computer generated numbers

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Definition Definition

• Uniform distribution [0; 1].
• Randomness (independence).
• One basic problem is computers do not work in R Random

numbers: A sequence of independent random variable, Ui, uniformly distributed on ]0, 1[

1 1

• Generate a sequence of independently and identically distributed

U(0, 1) numbers.

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Random generation Random generation

Mechanics devices:

• Coin (head or tail)
• Dice (1-6)
• Monte-Carlo (Roulette) wheel
• Wheel of fortune
• Deck of cards
• Lotteries (Dansk tipstjeneste)

Other devices:

• electronic noise in a diode or resi-

stor

• tables of random numbers

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Definition of a RNG Definition of a RNG

An RNG is a computer algorithm that outputs a sequence of reals

• r integers, which appear to be
• Uniformly distributed on [0; 1] or {0, . . . , N − 1}
• Statistically independent.

Caveats: Caveats:

• “Appear to be” means: The sequence must have the same

relevant statistical properties as I.I.D. uniformly distributed random variables

• With any finite precision format such as double, uniform on

[0; 1] can never be achieved.

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• 1. Four digit integer

(output divide by 10000)

• 2. square it.
• 3. Take the middle four digits
• 4. repeat

i Zi Ui Z2

i

7182 0.7182 51,581,124 1 5811 0.5811 33,767,721 2 7677 0.7677 58,936,329 3 9363 0.9363 87,665,769 4 6657 0.6657 44,315,649 5 3156 0.3156 09,960,336 . . . . . . . . . . . . Might seem plausible - but rather dubious

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Fibonacci Fibonacci

Leonardo of Pisa (pseudonym: Fibonacci) dealt in the book ”Liber Abaci”(1202) with the integer sequence defined by: xi = xi−1 + xi−2 i ≥ 2 x0 = 1 x1 = 1

Fibonacci generator. Also called an additive congruential method.

xi = mod( xi−1 + xi−2, M ) Ui = xi M where x = mod( y, M ) is the modulus after division ie. y − nM where n = ⌊y/M⌋ Notice xi ∈ [0, M −1]. Consequently, there is M 2 −1 possible starting values. Maximal length of period is M 2 − 1 which is only achieved for M = 2, 3.

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Congruential Generator Congruential Generator

The generator Ui = mod( aUi−1, 1 ) Ui ∈ [0, 1] illustrates the principle provided a is large, the last digits are retained. Can be implemented as (xi is an integer) xi = mod( axi−1, M ) Ui = xi M Examples are a = 23 and M = 108 + 1.

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Mid conclusion Mid conclusion

• Initial state determine the whole sequence
• How many different cycles
• Length of each cycle

If xi can take N values, then the maximum length of a cycle is N.

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Properties for a Random generator Properties for a Random generator

• Cycle length
• Randomness
• Speed
• Reproducible
• Portable

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Linear Congruential Generator Linear Congruential Generator

LCG are defined as xi = mod( axi−1 + c, M ) Ui = xi M for a multiplier a, shift c and modulus M. We will take a, c and x0 such xi lies in (0, 1, ... , M − 1) and it looks random. Example: M = 16, a = 5, c = 1 With x0 = 3: 0 1 6 15 12 13 2 11 8 9 14 7 4 5 10 3

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Theorem 1 Theorem 1

Maximum cycle length The LCG has full length if (and only

if)

• M and c are relative prime.
• For each prime factor p of M, mod(a, p) = 1.
• if 4 is a factor of M, then mod(a, 4) = 1. Notice, If M is a

prime, full period is attained only if a = 1.

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Shuffling Shuffling

• eg. XOR between several generators.
• To enlarge period
• Improve randomness
• But not well understood
• LCGs widespread use, generally to be recommended

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Mersenne Twister Mersenne Twister

Matsumoto and Nishimura, 1998

• A large structured linear feedback shift register
• Uses 19,937 bits of memory
• Has maximum period, i.e. 219937 − 1
• Has right distribution
• ... also joint distribution of 623 subsequent numbers
• Probably the best PRNG so far for stochastic simulation (not for

cryptography).

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RNGs in common environments RNGs in common environments

R: The Mersenne Twister is the default, many others can be chosen. Python: Mersenne Twister chosen. S-plus: XOR-shuffling between a congruential generator and a (Tausworthe) feedback shift register generator. The period is about 262 ≈ 4 · 1018, but seed dependent (!). Matlab 7.4 and higher: By default, the Mersenne Twister. Also

• ne other available.

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Characteristics Characteristics

Definition: A sequence of pseudo-random numbers Ui is a deterministic sequence of numbers in ]0, 1[ having the same relevant statistical properties as a sequence of random numbers. The question is what are relevant statistical properties.

• Distribution type
• Randomness (independence, whiteness)

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Theoretical tests/properties Theoretical tests/properties

• Test of global behaviour (entire cycles)
• Mathematical theorems
• Typically investigates multidimensional uniformity

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Testing random number generators Testing random number generators

• Test for distribution type

⋄ Visual tests/plots ⋄ χ2 test ⋄ Kolmogorov Smirnov test

• Test for independence

⋄ Visual tests/plots ⋄ Run test up/down ⋄ Run test length of runs ⋄ Test of correlation coefficients

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Significance test Significance test

• We assume (known) model - The hypothesis
• We identify a certain characterising random variable - The test

statistic

• We reject the hypothesis if the test statistic is an abnormal
• bservation under the hypothesis

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Key terms Key terms

• Hypothesis/Alternative
• Test statistic
• Significance level
• Accept/Critical area
• Power
• p-value

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Multinomial distribution Multinomial distribution

• n items
• k classes
• each item falls in class j with probabibility pj
• Xj is the (random) number of items in class j
• We write X = (X1, . . . , X2) ∼ Mul(n, p1, . . . , pk)

Thus Xj ∼ Bin(n, pj) E(Xj) = npj, Var(Xj) = npj(1 − pj) And E

• Xj−npj

√

npj(1−pj)

• = 0 Var
• Xj−npj

√

npj(1−pj)

• = 1

Thus

Xj−npj

√

npj(1−pj) n→∞

∼ N(0, 1)

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Test statistic for k − 2 Test statistic for k − 2

Recall

Xj−npj

√

npj(1−pj) n→∞

∼ N(0, 1) thus

• Xj−npj

√

npj(1−pj)

2 = (Xj−npj)2

npj(1−pj) asymp

∼ χ2(1) Consider now the case k = 2

(X1−np1)2 np1(1−p1) = (X1−np1)2(p1+1−p1) np1(1−p1)

= (X1−np1)2

np1

+ (X1−np1)2

n(1−p1)

= (X1−np1)2

np1

+ (X1−n−n(p1−1))2

n(1−p1)

= (X1−np1)2

np1

+ (−X2+np2)2

np2

= (X1−np1)2

np1

+ (X2−np2)2

np2

• the χ2 statistic
• the proof can be completed by induction

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Test for distribution type χ2 test Test for distribution type χ2 test

The general form of the test statistic is T =

nclasses

• i=1

(nobserved,i − nexpected,i)2 nexpected,i

• The test statistic is to be evaluated with a χ2 distribution with

d f degrees of freedom. d f is generally nclasses − 1 − m where m is the number of estimated parameters.

• It is recommend to choose all groups such that nexpected,i ≥ 5

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Test for distribution type Kolmogorov Smirnov test Test for distribution type Kolmogorov Smirnov test

• Compare empirical distribution function Fn(x) with hypothesized

distribution F(x).

• For known parameters the test statistic does not depend on

F(x)

• Better power than the χ2 test
• No grouping considerations needed
• Works only for completely specified distributions in the original

version

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Empirical distribution Empirical distribution

20 N(0, 1) variates (sorted):

• 2.20, -1.68, -1.43, -0.77, -0.76, -0.12, 0.30, 0.39, 0.41, 0.44, 0.44,

0.71, 0.85, 0.87, 1.15, 1.37, 1.41, 1.81, 2.65, 3.69 Xi iid random variables with F(x) = P(X ≤ x) Each leads to a (simple) random function Fe,i(x) = 1{Xi≤x} leading to Fe(x) = 1

n

n

i=1 Fe,i(x) = 1 n

n

i=1 1{Xi≤x}

E (Fe(x)) = E 1

n

n

i=1 1{Xi≤x}

• = 1

n

n

i=1 E

• 1{Xi≤x}
• = F(x)

Var (Fe(x)) =

1 n2nF(x)(1 − F(x)) = F(x)G(x) n

Fe(x)

n→∞

∼ N

• F(x), F(x)G(x)

n

• In the limit (n → ∞) we have a random continuous function of x -

a stochastic process, more particularly a Brownian bridge

Empirical distribution Empirical distribution

20 N(0, 1) variates (sorted):

• 2.20, -1.68, -1.43, -0.77, -0.76, -0.12, 0.30, 0.39, 0.41, 0.44, 0.44,

0.71, 0.85, 0.87, 1.15, 1.37, 1.41, 1.81, 2.65, 3.69 Dn = sup

x {|Fn(x) − F(x)|}

the test statistic follows Kolmogorovs distribution

Test statistic and significance levels Test statistic and significance levels

DTU

Level of significance (1 − α) Case Adjusted test statistic 0.850 0.900 0.950 0.975 0.990 All parameters known √n + 0.12 + 0.11

√n

• Dn

1.138 1.224 1.358 1.480 1.628 N( ¯ X(n), S2(n)) √n − 0.01 + 0.85

√n

• Dn

0.775 0.819 0.895 0.955 1.035 exp( ¯ X(n)) √n + 0.26 + 0.5

√n

Dn − 0.2

n

• 0.926

0.990 1.094 1.190 1.308

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Test for correlation - Visual tests Test for correlation - Visual tests

• Plot of Ui+1 versus Ui

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Random numbers U_i against U_{i+1}, X_{i+1} = (5 X_i + 1)(mod 16) ’ranplot.lst’ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Random numbers U_i against U_{i+1}, X_{i+1} = (129 X_i + 26461)(mod 65536) ’ranplot2.lst’

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Indepedence test: Test for multidimensional uniformity Indepedence test: Test for multidimensional uniformity

• In the two dimensional version test for uniformity of (U2i−1, U2i)
• Typically χ2 test
• The number of groups increases drastically with dimension

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Run test I Run test I

Above/below

• The run test given in Conradsen, can be used by e.g. comparing

with the median.

• The number of runs (above/below the median) is (asymptotically)

distributed as N

• 2 n1n2

n1 + n2 + 1, 2 n1n2(2n1n2 − n1 − n2) (n1 + n2)2(n1 + n2 − 1)

• where n1 is the number of samples above and n2 is the number

below.

• The test statistic is the total number of runs T = Ra + Rb with Ra

(runs above) and Rb (runs below)

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Run tests II Run tests II

Up/Down from Knuth

A test specifically designed for testing random number generators is the following UP/DOWN run test, see e.g. Donald E. Knuth, The Art

• f Computer Programming Volume 2, 1998, pp. 66-.

The sequence: 0.54, 0.67, |0.13, 0.89, |0.33, 0.45, 0.90, |0.01, 0.45, 0.76, 0.82, |0.24, |0.17 has runs of length 2,2,3,4,1, ... i.e. runs of consecutively increa- sing numbers.

Run test II Run test II

Generate n random numbers.The observed number of runs of length 1, . . . , 5 and ≥6 are recorded in the vector R. The test statistic is calculated by: Z = 1 n − 6(R − nB)TA(R − nB)

A =              4529.4 9044.9 13568 18091 22615 27892 9044.9 18097 27139 36187 45234 55789 13568 27139 40721 54281 67852 83685 18091 36187 54281 72414 90470 111580 22615 45234 67852 90470 113262 139476 27892 55789 83685 111580 139476 172860              B =             

1 6 5 24 11 120 19 720 29 5040 1 840

             The test statistic is compared with a χ2(6) distribution. One should have n > 4000

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Run test III Run test III

The-Up-and-Down Test This test is described in Rubinstein

81 “Simulation and the Monte Carlo Method” and Iversen 07 (in Danish). The sequence: 0.54, 0.67, 0.13, 0.89, 0.33, 0.45, 0.90, 0.01, 0.45, 0.76, 0.82, 0.24, 0.17 is converted to <, >, <, >, <, <, >, <, <, <, >, > giving in total 8 runs of length 1, 1, 1, 1, 2, 1, 3, 2

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Run test III Run test III

The expected number of runs of length k is n+1

12 , 11n−4 12

for runs of length 1 and 2 respectively, and 2[(k2 + 3k + 1)n − (k3 + 3k2 − k − 4)] (k + 3)! for runs of length k < N − 1. Define X to be the total number of runs, then Z = X − 2n−1

3

• 16n−29

90

is asymptotically N(0,1).

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Correlation coefficients Correlation coefficients

• the estimated correlation

ch = 1 n − h

n−h

• i=1

UiUi+h ∼ N

• 0.25,

7 144n

Exercise 1 Exercise 1

In today’s exercise you should implement everything including the tests yourself, including the chi-square and KS tests. Later, when your code is working you are free to use builtin functions.

• Write a program generating 10.000 (pseudo-) random

numbers and present these numbers in a histogramme (e.g. 10 classes). ⋄ First implement the LCG yourself by experimenting with different values of “a”, “b” and “c”. ⋄ Evaluate the quality of the generators by graphical descriptive statistices (histogrammes, scatter plots) and statistical tests (χ2,Kolmogorov-Smirnov, run-tests, and correlation test. ⋄ Then apply a system available generator and perform the various statistical tests for this generator too.