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(Pseudo) Random Number
Generation
Properties of pseudo-random numbers
• Continuous numbers between 0 and 1
• Probability of selecting a number in interval
(a,b) ~ (b-a) – i.e. Uniformly distributed
• Numbers are statistically independent
• Can’t really generate random numbers
∞ information – finite algorithm or table
Example: XL spreadsheet function =RAND()
• Also, want fast and repeatable...
Random Number Generation
How to generate random numbers
– Table look-up
– Computer generation: these values cannot be
truly random and a computer cannot express a
number to an infinite number of decimal places
Pseudorandom numbers
Desirable Properties of Random
Number Generators• Fast
• Should not require much memory
• Long cycle or period
• Numbers should come close to:
– Uniformity (or known distribution)
– Independence
Historical Generator
Midsquare method:
1. Start with an initial seed (e.g. a 4-digit integer).
2. Square the number.
3. Take the middle 4 digits.
4. This value becomes the new seed. Divide the number by 10,000. This becomes the random number. Go to 2.
Midsquare Method, example
x0 = 5497
x1: 54972 = 30217009 x1 = 2170, R1 = 0.2170
x2: 21702 = 04708900 x2 = 7089, R2 = 0.7089
x3: 70892 = 50253921 x3 = 2539, R3 = 0.2539
Drawback: Hard to state conditions for picking initial seed that will generate a “good” sequence.
Midsquare Generator, examples
“Bad” sequences:
• x0 = 5197x1: 51972 = 27008809 x1 = 0088, R1 = 0.0088x2: 00882 = 00007744 x2 = 0077, R2 = 0.0077x3: 00772 = 00005929 x3 = 0059, R3 = 0.0059
• xi = 6500xi+1: 65002=42250000 xi+1=2500, Ri+1= 0.0088xi+2: 25002=06250000 xi+2=2500, Ri+1= 0.0088
Linear Congruential Generator
(LCG) GeneratorStart with random seed
Z0 < m = largest possible integer on machine
Recursively generate integers between 0 and M
Zi = (a Zi-1 + c) mod m
Use U = Z/m for pseudo-random number get
(avoid 0 and 1)
When c = 0 Called Multiplicative Congruential Generator
When c > 0 Mixed LCG
Linear Congruential Generator (LCG) (Lehmer 1951)
Let Zi be the ith number (integer) in the sequence
Zi = (aZi-1+c)mod(m) Zi{0,1,2,…,m-1}
where Z0 = seed
a = multiplier
c = increment
m = modulus
Define Ui = Zi /m (to obtain U(0,1) value)
LCG, example
16-bit machine
a = 1217 c = 0 Z0 = 23 m = 215-1 = 32767
Z1 = (1217*23) mod 32767 = 27991
U1 = 27991/32767 = 0.85424
Z2 = (1217*27991) mod 32767 = 20134
U2 = 20134/32767 = 0.61446
An LCG can be expressed as a function of the seed Z0
THEOREM:
Zi = [aiZ0+c(ai-1)/(a-1)] mod(m)
Proof: By induction on i
i=0 Z0 = [a0Z0+c(a0-1)/(a-1)] mod(m)
Assume for i. Show that expression holds for i+1
Zi+1 = [aZi+c] mod(m)
= [a {[aiZ0+c(ai-1)/(a-1)] mod(m)} +c] mod(m)
= [ai+1Z0+ac(ai-1)/(a-1) +c] mod(m)
= [ai+1Z0+c(ai+1-1)/(a-1) ] mod(m)
Examples:
Zi = (69069Zi-1+1) mod 232 Ui = Zi /232
Zi = (65539Zi-1+76) mod 231 Ui = Zi/231
Zi = (630360016Zi-1) mod (231-1) Ui = Zi/231
Zi = 1313Zi-1 mod 259 Ui = Zi/259
What makes one LCG better than another?
A full period (full cycle) LCG generates all m values
before it cycles.
Consider Zi = (3Zi-1+2) mod(9) with Z0 =7
Then Z1 = 5 Z2 = 8 Z3 = 8 Zj = 8 j = 3,4,5,6,…
On the other hand Zi = (4Zi-1+2) mod(9) has full period.
Why?
Random Number Generation
Mixed congruential generator is full period if
1. m = 2B (B is often # bits in word) fast
2. c and m relatively prime (g.c.d. = 1)
3. If 4 divides m, then 4 divides a – 1
(e.g., a = 1, 5, 9, 13,…)
The period of an LCG is m (full period or full cycle) if and only if
— If q is a prime that divides m, then q divides a-1
— The only positive integer that divides both m and c is 1
— If 4 divides m, then 4 divides a-1.
Examples
Zi+1 = (16807Zi+3) mod (451605),
where 16807 =75, 16806 =(2)(3)(2801), 451605 =(3)(5)(7)(11)(17)(23)
This LCG does not satisfy the first two conditions.
Zi+1 = (16807Zi+5) mod (635493681)
where 16807 =75, 16806 = (2)(3)(2801), 635493681 = (34)(28012)
This LCG satisfies all three conditions.
- m = 2B where B = # bits in the machine is often a good choice
to maximize the period.
- If c = 0, we have a power residue or multiplicative generator.
Note that Zn = (aZn-1) mod(m) Zn = (anZ0) mod(m).
If m = 2B, where B = # bits in the machine, the longest period is m/4
(best one can do) if and only if
— Z0 is odd
— a = 8k+ 3, kZ+ (5,11,13,19,21,27,…)
Random Number Generation
Other kinds of generators
• Quadratic Congruential Generator
– Snew = (a1 Sold2 + a2 Sold
2 + b) mod L
• Combination of Generators
– Shuffling – L’Ecuyer – Wichman/Hill
• Tausworthe Generator
– Generates sequence of random bits
Feedback Shift Generators
• Tausworthe, Math of Computing 1965
• If {ak} is a sequence of binary digits (0 or 1)
defined by
ak = (c1ak-1 + c2ak-2 + … + cpak-p)mod 2
and the c’s are relatively prime, then {ak}
has period 2p-1
IBM - Randu
If c = 0 power residue generator
(multiplicative generator)
un = anu0 mod m
un = a un-1 mod m (homework)
NOTES
— Never “invent” your own LCG. It will probably not be “good.”
— All simulation languages and many software packages have
their own PRN generator. Most use some variation of a linear
congruential generator.
— Power residue generators are the most common.
Tests of RNG, cont’d
• Theoretical tests
– Prove sample moments over entire cycle are
correct
– Lattice structure of LCGs
• “random numbers fall mainly in the planes”
(Marsaglia)
• Spacing hyperplanes: the smaller, the better
Tests of Random Number
Generators• Empirical tests
– Uniformity
• Compute sample moments
• Goodness of fit
– Independence
• Gap Test
• Runs Test
• Poker Test
• Spectral Test
• Autocorrelation Test
Testing Random Number Generators
Desirable Properties:
• Mean and Variance
Theorem: E 1/2 and V 1/12 as m+.
Proof:
For a full period LCG, every integer value from 0 to m-1 is
represented. Therefore
E = (0+1+…+(m-1))/m2 = ((m-1)(m)/2)/m2 = (1/2)-(1/2m)
V = ((02+12+22+…+(m-1)2)/m3) - E2
= [(m)(m-1)(2m-1)/6]/m3 - [(1/2) - (1/2m)]2
= [(1/12) - (1/12m2)]
• Uniformity
2 Goodness of Fit Test
— Divide n observations into k (equal) intervals
— Do a frequency count fi, i=1,2,…,k
— Compute X2 = i (fi -n/k)2 / (n/k)
= i (fi -npi)2 / npi,
where pi = 1/k, i=1,2,…,k.
• • •
f1 f2 fk-1 fk
01
k
2
k
k 2
k
k 1
k
e1 e2 ek-1 ek
ei = expected number of observations in interval i
= n pi = n / k, i = 1, 2, …, k
Data Classification
1
NOTE
— (fi -npi)/(npi)1/2 is the N(0,1) approximation of
a multinomial distribution for pi small, where
E[fi] = npi and Var [fi] = npi (1-pi)).
— For n large, X2 is distributed 2 with k-1 degrees of freedom
—Reject randomness assumption X2 > 2
1k
NOTE: if X2 is too close to zero, it may be because the
numbers have been “fudged.”
BE WARY OF PRN WHICH LOOK TOO RANDOM
Do Not
Reject HOReject HO
k1,2
2 Goodness of Fit Test
- Repeat test m times with independent samples of size n
- If H0 is true, test will reject H0 m times (on average)
Trouble Spots
— Choosing the intervals evenly
— Choosing the intervals such that you would expect
each class to contain at least 5 or 10 observations
— pi should (ideally) be small (<.05)
Example
n = 1000
[0, .1) fi = 87 [.1, .2) fi = 93 [.2, .3) fi = 113
[.3, .4) fi = 106 [.4, .5) fi = 108 [.5, .6) fi = 99
[.6, .7) fi = 91 [.7, .8) fi = 95 [.8, .9) fi = 103
[.9, 1.] fi = 105
X2 = 628/100 = 6.28 Do not reject H0 : U(0,1).
9,.052
16.919
NOTE
— The 2 goodness of fit test is also used to fit distributions to
data, where
X2 = i (fi -ei)2 / ei
ei = expected number of observations in interval i.
Kolmogorov-Smirnov Goodness-of-fit Test
— Order n U[0,1] variates {x[i]}
— Construct an empirical CDF for the n variates {x[i]}
(i.e., F(x[i]) = i/n i = 1,2,…,n)
— Construct a hypothesized CDF for n uniform variates
(i.e., = x, 0x1)
— Compute D = max {D+, D-}, where
D+ = Max1<i<n [(i/n)-
D- = Max1<i<n [ -((i-1)/n)].
— Check tables
• Reject if D is too large, with a risk , which means that
we reject (uniformity) falsely with probability .
(x)F~
˜ F (x[i])]
)(xF~
[i]
1.00
.75
.50
.25
0 .1 .2 .3 .9 1.0
D+ = max {.15, .30, .45, .10}=.45
D- = max {.10, -.05, -.20, .15}=.15
D+
D-
Examples
— If {Ui} = {.1, .2, .3, .9}, then D = .45.
— If {Ui} = {.2, .6, .8, .9}, then D = .35.
— If {Ui} = {.25, .5, .75, 1.}, then D = .25.
NOTE: The minimum value that D can take on is 1/2n.
(How?)
Independence
— Sign Test
* Test Statistic: S = runs of numbers above or below median)
* For large N, S is distributed N( = 1+(N/2), 2 = N/2)
Example
N = 15, S = 7, distributed N( = 8.5, 2 = 15/2)
Maximum value for S: N (negative dependency)
Minimum value for S: 1 (positive dependency)
.87 .15 .23 .45 .69 .32 .30 .19 .24 .18 .65 .82 .93 .22 .81
+ - - - + - - - - - + + + - +
Normal Curve Rejection Regions
REJECT
(+ve)
REJECT
(-ve)Do
Not
REJECT
Reject H0 in favor of HA if
Z = (S - (1+(N/2))) / (N/2)1/2 Z/2 or Z Z/2
Z/2-Z/2
H0 : Independence
HA : Dependence
— Runs Up and Down Test
(runs of increasing and decreasing numbers)
• Assign + if xi <xi+1, assign - if xi>xi+1
• Test Statistic: S = number of runs up AND down
(sequence of + and -)
• E(S) = (2N-1)/3, V(S) = (16N-29)/90
• Use Normal approximation for N>30.
Example:
N = 15, S = 8, distributed N(µ = 29/3, 2 = 211/90)
Maximum value for S: N-1 (negative dependency)
Minimum value for S: 1 ?
.87 .15 .23 .45 .69 .32 .30 .19 .24 .18 .65 .82 .93 .22 .81
- + + + - - - + - + + + - +
Normal Curve Rejection Regions
REJECT (-ve)REJECT
Do Not
REJECT
H0 : Independence
HA : Dependence
Z/2-Z/2
Reject H0 in favor of HA if
Z = (S - (2N-1)/3) / (16N-29/90)1/2 Z/2 or Z Z/2
Test of Cycling
Floyd’s Test for Cycling
• Assume ui = G(ui-1)
• x0 = y0 = seeds
• xi = G(xi-1) yi = G(G(yi-1)), i.e. skip every
other one so y will go twice as fast as x.
Then check to see if there is some value of
n for which xn = yn.
• If xn = yn, cycling occurred.
Marsaglia’s Theorem
All N-tuples generated by a congruential generator
will fall in fewer than (N!m)1/N hyperplanes.
(Proc. Nat. Acad. Sci. 61, 1968 pp.25-28)
e.g. all 10-tuples fall in fewer than 13 9-dimensional
planes for m = 216. Randu in ONLY 15
PLANES in 3D cube.
(Solution: Make m bigger – limited by computer
word size.)
Plot of RNDi+1 vs RNDi using LCG in SIGMA
