random number and random variate generation

Random Number and Random Variate Generation

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How does the computer generate observations from various distributions specified after input analysis?

There are two main components to the generation of observations from probability distributions.

1. Random number generation.2. Random variate generation.

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Random Number and Random Variate Generation

Random number generation – The generation of U(0,1) random variates (observations from Uniform (0,1) distribution).◦ This serves as the foundation for the generation of

observations from other distributions, which is called random variate generation.

Random Number Generator is the term used to describe the procedure and parameters used to generate the U(0,1) observations.

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Random Number Generation

Since the “stream” of random numbers generated is reproducible, random number generation procedures are also referred to as pseudo random number generators.

The stream or sequence of numbers produced by a generator should pass statistical tests for randomness.◦ An outside observer should not be able to tell the difference

(statistically) between a stream of pseudo random numbers and an actual random number stream.

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Pseudorandomness A pseudorandom process appears random, but isn’t Pseudorandom sequences exhibit statistical

randomness ◦ but generated by a deterministic process

Pseudorandom sequences are easier to produce than a genuine random sequences

Pseudorandom sequences can reproduce exactly the same numbers◦ useful for testing and fixing software.

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Pseudorandom Generator Seeds Random number generators typically compute the next

number in the sequence from the previous number The first number in a sequence is called the seed

◦ to get a new sequence, supply a new seed(current machine time is useful)

◦ to repeat a sequence, repeat the seed

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Desirable Attributes:

◦ Uniformity

◦ Independence

◦ Efficiency

◦ Replicability

◦ Long Cycle Length

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Random Number Generation (cont.)

Each random number Rt is an independent sample drawn from a continuous uniform distribution between 0 and 1

ì1 , 0 £ x £ 1pdf: f(x) = í

î0 , otherwise

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0

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REdxxRV

xdxxRE

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Random Number Generation (cont.)

One early method – the midsquare method (von Neumann and Metropolis 1940)◦ Start with a four digit positive integer Z0.◦ Square Z0 to get an integer with up to eight digits (append

zeros if less than eight).◦ Take the middle four digits as the next four digit integer Z1.◦ Place a decimal point to the left of Z1 to form the first

“U(0,1)” observation.◦ Repeat

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MidSquareExample:

X0 = 7182 (seed) = 51581124

==> R1 = 0.5811 = (5811) 2 = 33767721

==> R2 = 0.7677etc.

20X

20X

Random Number Generation - MidSquare

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Note: Cannot choose a seed that guarantees that the sequence will not degenerate and will have a long period. Also, zeros, once they appear, are carried in subsequent numbers.Ex1: X0 = 5197 (seed) = 27008809

==> R1 = 0.0088 = 00007744==> R2 = 0.0077Ex2: X0 = 4500 (seed) = 20250000==> R1 = 0.2500 = 06250000==> R2 = 0.2500

20X21X

20X21X

Random Number Generation- MidSquare

The prior method does not work well.◦ Degenerates to zero.

What are good methods?◦ Linear Congruential Generators (LCGs).◦ Composite generators.◦ Tausworthe generators.

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Random Number Generation - LCG

Linear Congruential Generators (LCGs). A LCG generates a sequence of integers Z1, Z2, Z3,…

using the following recursive formula,

mod m is short for modulo m or the remainder when divided by m.

mcaZZ ii mod)( 1

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Since the mod m operation is used, all Zi’s will be between 0 and m-1.

To get the “U(0,1)” random observations each Zi

generated is divided by m.

So are the Ui’s really U(0,1) random observations ?

,, 22

11 m

ZUmZU

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Let m=63, a=22, c=4 and Z0 =19.◦Generate the first five “U(0,1)” observations.

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In-class Exercise

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i 22*Zi-1+ 4 Zi Ui i 22*Zi-1+ 4 Zi Ui i 22*Zi-1+ 4 Zi Ui i 22*Zi-1+ 4 Zi Ui

0 19 16 356 41 0.6508 32 840 21 0.3333 48 400 22 0.34921 422 44 0.6984 17 906 24 0.3810 33 466 25 0.3968 49 488 47 0.74602 972 27 0.4286 18 532 28 0.4444 34 554 50 0.7937 50 1038 30 0.47623 598 31 0.4921 19 620 53 0.8413 35 1104 33 0.5238 51 664 34 0.53974 686 56 0.8889 20 1170 36 0.5714 36 730 37 0.5873 52 752 59 0.93655 1236 39 0.6190 21 796 40 0.6349 37 818 62 0.9841 53 1302 42 0.66676 862 43 0.6825 22 884 2 0.0317 38 1368 45 0.7143 54 928 46 0.73027 950 5 0.0794 23 48 48 0.7619 39 994 49 0.7778 55 1016 8 0.12708 114 51 0.8095 24 1060 52 0.8254 40 1082 11 0.1746 56 180 54 0.85719 1126 55 0.8730 25 1148 14 0.2222 41 246 57 0.9048 57 1192 58 0.920610 1214 17 0.2698 26 312 60 0.9524 42 1258 61 0.9683 58 1280 20 0.317511 378 0 0.0000 27 1324 1 0.0159 43 1346 23 0.3651 59 444 3 0.047612 4 4 0.0635 28 26 26 0.4127 44 510 6 0.0952 60 70 7 0.111113 92 29 0.4603 29 576 9 0.1429 45 136 10 0.1587 61 158 32 0.507914 642 12 0.1905 30 202 13 0.2063 46 224 35 0.5556 62 708 15 0.238115 268 16 0.2540 31 290 38 0.6032 47 774 18 0.2857 63 334 19 0.3016

What will happen after the 63rd number is generated?

m, a, and c are the parameters of the random number generator.◦ There can be an infinite number of different implementations

of a LCG. The values used for m, a, and c determine whether the

generator is good or bad.

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The example LCG demonstrates cycling in the prior table.◦ Since m=63, it can generate at most 63 numbers before it

repeats the same sequence. This small random number generator has full period

since it generates all possible (m=63) numbers before cycling.◦ A long period (full if possible) is desirable since more

observations can be generated before cycling.◦ No “gaps”.

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The example generator has full period but bad statistical properties (next slide).

A good random number generator will have values for m, a, and c such that full or close to full period is obtained, as well as good statistical properties.◦Crystal Ball m = 231 – 1 a = 62089911 c = 0 Period = 231 – 2

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Theorem (Hull and Dobell 1962)◦ The LCG Zi = (aZi-1 + c) mod m has full period if and

only if the following three conditions hold.1. The only positive integer that exactly divides both m and c

is 1.2. If q is a prime number that divides m, then q divides a-1.3. If 4 divides m, then 4 divides a-1.

The parameters of the LCG dictate the period length of the LCG as well as other properties of the numbers generated.

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A generator that has the maximum possible period is called a full-period generator.

Lower autocorrelations between successive numbers are preferable.

Both generators have the same full period, but the first one has a correlation of 0.25 between xn-1 and xn, whereas the second one has a negligible correlation of less than 2-18.

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Period vs. Autocorrelation

Types of LCGs◦ When c = 0, the LCG is called a multiplicative generator.◦ When c ≠ 0, the LCG is called a mixed generator.

Most LCGs implemented are multiplicative◦ Can’t have full period.

How is m selected.◦ A large period is desired → m=231 (based on a 32 bit word

size). With m=231 it has been proven that the period can be at

most 229 (25% of the values are cycled and gaps may be present).

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Multiplicative LCG: c=0

Two types:

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Multiplicative LCG

mxax ii mod1

k

k

m

m

2

2

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Example:Using the multiplicative congruential method, find the period of the generator for a = 13, m = 26, and X0 = 1, 2, 3, and 4. The solution is given in next slide. When the seed is 1 and 3, the sequence has period 16. However, a period of length eight is achieved when the seed is 2 and a period of length four occurs when the seed is 4.

Multiplicative LCG

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Period Determination Using Various seedsi Xi Xi Xi Xi

0 1 2 3 41 13 26 39 522 41 18 59 363 21 42 63 204 17 34 51 45 29 58 236 57 50 437 37 10 478 33 2 359 45 7

10 9 27 11 53 31 12 49 19 13 61 55 14 25 11 15 5 15 16 1 3

Multiplicative LCG

Maximum possible period 2k-2

Period achieved if multiplier a is of the form 8i± 3, and the initial seed is an odd integer One-fourth the maximum possible may not be too small Low order bits of random numbers obtained using multiplicative LCG's with m=2k have a cyclic

pattern.

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Multiplicative LCG with m=2k

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Examining Bits of a Multiplicative LCG

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Examining Bits of a Mixed LCG

When the modulus m is a prime number and a >1, the maximum period is m-1, no matter whether c=0 or not.

The maximum period m-1 is obtained if and only if ‘a’ is a primitive element modulo m.

If m is prime then ‘a’ is a primitive element modulo m (or primitive root of m) if and only if an mod m ≠ 1 for n=1, 2, 3, …,m-2.

Recommended. (Prime moduli are best in terms of sequence randomness.)

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Multiplicative LCG with m≠2k

Example:

Starting with a seed of x0=1: 1, 3, 9, 27, 19, 26, 16, 17, 20, 29, 25, 13, 8, 24, 10, 30,

28, 22, 4, 12, 5, 15, 14, 11, 2, 6, 18, 23, 7, 21, 1, … The period is 30 ⇒ 3 is a primitive root of 31 With a multiplier of a = 5: 1, 5, 25, 1,… The period is only 3 5 is not a primitive root of 31⇒ Primitive roots of 31= 3, …????????.

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Multiplicative LCG with m≠2k

31mod3 1 ii xx


LCGs are a special case of the form Zi = g(Zi-1, Zi-2, ...) (mod m), Ui = Zi /m, for some function g

Examples: g(Zi-1) = aZi-1 + c LCG g(Zi-1, Zi-2, ..., Zi-q) = a1Zi-1 + a2Zi-2 + ... + aqZi-q

multiple recursive generator g(Zi-1) = a'Z 2

i-1 + aZi-1 + c quadratic CG g(Zi-1, Zi-2) = Zi-1 + Zi-2 Fibonacci (bad)

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Composite GeneratorsCombine two (or more) individual generators in some way. Differencing LCGs

◦ Z1i and Z2i from LCGs with different moduli◦ Let Zi = (Z1i – Z2i ) (mod m); Ui = Zi / m◦ Very good statistical properties◦ Very portable (micros, different languages)

Wichmann/Hill◦ Use three LCGs to get U1i, U2i, and U3i sequences◦ Let Ui = fractional part of U1i + U2i + U3i

◦ Long period, good statistics, portability

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Originated in cryptography Can achieve very long periods Theoretical appeal: for properly chosen parameters, can prove

that over a cycle,◦ mean 1/2 (as for true U(0,1))◦ Variance 1/12 (as for true U(0,1))◦ Autocorrelation 0 (as for true IID sequence)

Define a sequence of binary digits B1,B2, . . ., by

where cj = 0 or 1.

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Tausworthe Generator

2mod1

q

jjiji bcb

Looks a bit like a generalization of LCG’s. Let D = delay operator such that Db(n)=b(n+1)

or

Since in mod 2 arithmetic subtraction is equivalent to addition, the preceding equation is equivalent to

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2mod)()()()( 02

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1 qibcqibDcqibDcqibD qq

qq

q

2mod02

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1 cDcDcD qq

qq

q

2mod002

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1

cDcDcD q

qq

qq

2mod002

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1

cDcDcD q

qq

qq

The polynomial on the left-hand side of this equation is called a characteristic polynomial and is traditionally

written using x in place of D

The period of a Tausworthe generator depends upon the characteristic polynomial. In particular, the period is the smallest positive integer n for which xn - 1 is divisible by the characteristic polynomial. The maximum possible period with a polynomial of order q is 2q - 1. The polynomials that give this period are called primitive polynomials.

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02

21

1 cxcxcx qq

qq

q

Example: Consider the following polynomial:x7 + x3 + 1

Using the D operator in place of x, we get

oror using the XOR operator

or

Substituting n-7 for n, we get

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2mod0)()()( 37 nbnbDnbD2mod037 nnn bbb

,2,1,0037 nbbb nnn

,2,1,037 nbbb nnn

,9,8,774 nbbb nnn

Starting with b0 = b1 = ... = b6 = 1, we get the following bit sequence:

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110011011011011

4711

3610

259

148

037

bbbbbb

bbbbbbbbb

The complete sequence is:1111111 0000111 0111100 1011001 0010000 00100010011000 1011101 0110110 0000110 0110101 00111001111011 0100001 0101011 1110100 1010001 10111000111111 1000011 1000000. Period = 127 or 27-1 bits ⇒ The polynomial x7+x3+1 is a primitive polynomial.

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A Tausworthe sequence can be easily generated in hardware using Linear-Feedback Shift Registers (LFSRs).

For example, the polynomial x5 + x3 + 1 results in the generator bn = bn-2 bn-5. This can be implemented using the LFSR shown in the Figure presented next slide.

The circuit consists of six registers, each holding one bit. On every clock cycle, each register’s content is shifted out, and the new content is determined by the input to the register.

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Linear Feedback Shift Register:x5+x3+1 b⇒ n= bn-2 ⊕ bn-5

This can be easily implemented using shift registers:

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Generating U(0,1): Divide the sequence into successive groups of s bits and

use the first l bits of each group as a binary fraction:xn = 0.bsnbsn+1bsn+2bsn+3 ...bsn+l-1

Here, s is a constant greater than or equal to l and is relatively prime to 2q-1.

s ≥ l x⇒ n and xj for n ≠ j have no bits in common. Relative prime-ness guarantees a full period 2q-1 for xn.

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Example: bn = bn-4 ⊕ bn-7

The period 27-1=127 l=8, s=8:

x0 = 0.111111102 = 0.9921910

x1 = 0.000111012 = 0.1132810

x2 = 0.111001012 = 0.8945310

x3 = 0.100100102 = 0.2968810

x4 = 0.000001002 = 0.3632810

x5 = 0.010011002 = 0.4218810

….

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List of Primitive Trinomialsx2 + x + 1 x3 + x + 1 x4 + x + 1 x5 + x2 + 1x6 + x + 1 x7 + x + 1 x7 + x3 + 1 x9 + x4 + 1x10 + x3 + 1 x11 + x2 + 1 x15 + x + 1 x15 + x4 + 1x15 + x7 + 1 x17 + x3 + 1 x17 + x5 + 1 x17 + x6 + 1x18 + x7 + 1 x20 + x3 + 1 x21 + x2 + 1 x22 + x + 1x23 + x5 + 1 x23 + x9 + 1 x25 + x3 + 1 x25 + x7 + 1x28 + x3 + 1 x28 + x9 + 1 x28 + x13 + 1 x29 + x2 + 1x31 + x3 + 1 x31 + x6 + 1 x31 + x7 + 1 x31 + x13 +1If xq + xr + 1 is listed, then xq + xq-r +1 is also primitive.

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Homework: Generate random numbers using the primitive

polynomial x5+x2+1. (use l=4) Generate the same sequence using LFSR.

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random number and random variate generation

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