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### Random Numbers

URGE to Compute

What is a random number generator?

- Most random number generators generate a sequence of integers by a recurrence (linear congruent generator):
x0 = given

xn+1 = P1 xn + P2 (mod N) n=0,1,2,...

divide by N to get a number in [0,1]

A sample sequence

- x0 =79, N = 100, P1 = 263, and P2 = 71
- x1 = 79*263 + 71 (mod 100) = 20848 (mod 100) = 48,
- x2 = 48*263 + 71 (mod 100) = 12695 (mod 100) = 95,
- x3 = 95*263 + 71 (mod 100) = 25056 (mod 100) = 56,
- x4 = 56*263 + 71 (mod 100) = 14799 (mod 100) = 99,
- Subsequent numbers are: 8, 75, 96, 68, 36, 39, 28, 35, 76, 59, 88, 15, 16, 79, 48.
- The sequence then repeats

Sequences

- P1, P2, and N determine the characteristics of the random number generator
- The choice of x0 (the seed ) determines the particular sequence of random numbers that is generated.

What makes a good random number generator?

- A sequence is good if it passes several well established statistical tests.
- Or, it's good if it gives good results in particular applications (where the meaning of "good results" is heavily dependent upon the context).

One Test – plot pairs

- 100 points (xi, xi+1).
- It's clear that there are only 20 points; 100 were drawn, so five lie on top of each other
- The dots appear to lie along six slanted lines.
- Not very ‘random’

Second plot

- P1 = 16807,
P2 = 0,

N= 231 -1 = 2147483647

- Distinctly better

Linear generators

- All sequences generated by a linear congruent formula will eventually enter a cycle which repeats itself endlessly; a good generator will produce a long sequence of numbers before repeating. The max. length of course is N.
- A linear congruent formula will generate a sequence of maximum length if and only if the following conditions are met (See Knuth:
- i) P2 is relatively prime to N;
- ii) B = (P1 - 1) is a multiple of p, for every prime p dividing N;
- iii) B = (P1 - 1) is a multiple of 4, if N is a multiple of 4.

THE MTH$RANDOM ALGORITHM

- The VMS Run-Time Library provides a random number generator routine called MTH$RANDOM.
SEED = (69069*SEED + 1) MOD 2**32

X = SEED/2**32

- Note MTH$RANDOM satisfies the conditions above: i) 1 is relatively prime to 2**32 since 1 is relatively prime to all numbers.
- ii) 69068 is a multiple of 2, which is the only prime dividing 2**32.
- iii) 69068 is a multiple of 4.

THE MTH$RANDOM ALGORITHM

- Note for the MTH$RANDOM function if SEED is initially an ODD value then the new value of SEED will always be an even value. And if SEED is an EVEN value, then the new value of SEED will be an ODD value. Thus if the algorithm is repeatedly called, the value of SEED will alternate between EVEN and ODD values.

THE MTH$RANDOM ALGORITHM

- More important than starting the MTH$RANDOM generator to get one random sequence is the problem of restarting the generator to get several different sequences. You may wish to run a simulation several times and use a different random sequence each time.

THE MTH$RANDOM ALGORITHM

- To come up with a good "random" initial SEED value to start the generator, use the generator itself to produce a random SEED value to start our random number generator with! Select an initial nonrandom SEED value, then run the random number generator a few cycles to generate a random SEED value. We then restart our random number generator with the new random SEED value, and the output is then a properly initialized random sequence.

THE RANDU ALGORITHM

- The VMS FORTRAN Run-Time Library contains a random number generator RANDU, first introduced by IBM IN 1963. This turned out to be a poor random number generator, but nonetheless it has been widely spread.
- INTEGER*4 SEED
- INTEGER*2 W(2)
- EQUIVALENCE( SEED, W(1) )
- R = FOR$IRAN( W(1), W(2) )
- R is the return value between [0,1). W(1) and W(2) together is the seed value for the generator. This goes back to the PDP-11 days of 16 bit integers. SEED is really a 32 bit integer, but it was represented as two 16 bit integers.

THE RANDU ALGORITHM

SEED = (65539*SEED) MOD 2**31

X = SEED/2**31

- Note if SEED is initially an odd value, the new SEED generated will also be an odd value. Similarly, if SEED is initially an even value. Thus there are at least two disjoint cycles for the RANDU generator.

THE RANDU ALGORITHM

Actually, the situation is even worse than that.

- For odd SEED values there are two separate disjoint cycles, one generated by the SEED value 1, and one generated by the SEED value 5.
- The cycles <1> and <5> each contain 536,870,912 values. Together they account for all of the (2**31)/2 possible odd SEED values.

THE RANDU ALGORITHM

- There are 30 different disjoint cycles using even SEED values.
- TABLE 5.2.2 RANDU WITH EVEN VALUES OF SEED
CYCLE LENGTH OF CYCLE

<2> 268435456 <4> 134217728

<8> 67108864 <10> 268435456

<16> 33554432 <20> 134217728

<32> 16777216 <40> 67108864

<64> 8388608 <80> 33554432

<128> 4194304 <160> 16777216

<256> 2097152 <320> 8388608

<512> 1048576 <640> 4194304

<1024> 524288 <1280> 2097152

<2048> 262144 <2560> 1048576

<4096> 131072 <5120> 524288

<8192> 65536 <10240> 262144

<16384> 32768 <20480> 131072

<32768> 16384 <40960> 65536

<81920> 32768 <163840> 16384

THE RANDU ALGORITHM

- There are a total of (2**31)/2 = 1,073,741,824 possible even SEED values; we've accounted for 1,073,709,056 of them. The remaining 32768 SEED values are ones for which the 31 bit binary representation of them has the lower 16 bits set to 0. These SEED values are treated by RANDU as if the SEED value were 1, and they result in the cycle <1>.

Tests

The 1-D TEST is a frequency test. Imagine a number line stretching from 0 to 1. Use the random number generator to plot random points on this line. First divide the line into a number of "bins". 1 2 3 4

|---------|---------|---------|---------|

0 .25 .50 .75 1.0

See how randomly the random number generator fills our bins. If the bins are filled too unevenly, the Chi-Square test will give a value that's high, indicating the points do not appear random.

Tests

In 2D, divide the plane into squares. You can think of similar tests in higher dimensions also.

Define: N = number of trials

k = number of possible outcomes of the chance experiment

f(ZETA_i) = number of occurrences of ZETA_i in N trials

E(ZETA_i) = The expected number of occurrences of ZETA_i in N trials. E(ZETA_i) = N*Pr(ZETA_i).

i=k [ f(ZETA_i) - E(ZETA_i) ]**2

CHISQ = SUM -------------------------------------

i=1 E(ZETA_i)

Tests

MTH$RANDOM

SEED = (69069*SEED + 1) mod 2**32

X = SEED/2**32 returns real in range [0,1)

RATING:

Fails 1-D above 350,000 bpd (bins per dimension)

Fails 2-D above 600 bpd

Fails 3-D above 100 bpd

Fails 4-D above 27 bpd

Comments: This generator is also used by the VAX FORTRAN intrinsic function RAN, and by the VAX BASIC function RND.

Tests

RANDU

SEED = (65539*SEED) mod 2**31

X = SEED/2**31 returns real in range [0,1)

RATING:

Fails 1-D above 200,000 bpd

Fails 2-D above 400 bpd

Fails 3-D above 3 bpd

Fails 4-D above 6 bpd

Comments: Note the extremely poor performance for dimensions 3 and above. This generator is obsolete.

Shuffling

A simple way to greatly improve any random number generator is to shuffle the output.

- Start with an array of dimension around 100 (exact size is not important.)
- Initialize the array by filling it with random numbers from your generator.
- When the program wants a random number, randomly choose one from the array and output it to the program.
- Replace the number chosen in the array with a new random number from the random number generator.
- Note that this shuffling method uses two numbers from the random number generator for each random number output to the calling program.

Lagged Fibonacci Generators

- The name lfg comes from the Fibonacci sequence 1, 1, 2, 3, 5, 8, ......Xn = Xn-1 + Xn-2.
- LFGs generate random numbers from the following iterative scheme: Xn = Xn-i + Xn-k (mod m) the lags i and k satisfy the conditions i > k > 0. i initial values X0, X1,.....,Xi-1 are needed. For most applications m is power of 2, and with proper choice of i, k, and the first i values of X, the period is (2i - 1)2(M-1). One problem with LFG is that i words of memory must be kept current, whereas LCG requires only that the last value of X be saved.

Hardware generators

- A hardware (true) random number generator is a piece of electronics that plugs into a computer and produces genuine random numbers - as opposed to the pseudo-random numbers that are produced by a computer program. A typical method is to amplify noise generated by a resistor or a semi-conductor diode and feed this to a comparator or Schmitt trigger. If you sample the output (not too quickly) you (hope to) get a series of bits which are statistically independent. These can be assembled into bytes, integers or floating point numbers and then, if necessary, into random numbers from other distributions using methods.

The Marsaglia CD-ROM

- George Marsaglia produced a CD-ROM containing 600 megabytes of random numbers. These were produced using the best pseudo-random number generators, but were then combined bytes from a variety of random sources or semi-random sources (such as rap music).
- Suppose X and Y are independent random bytes (integer values 0 to 255), and at least one of them is uniformly distributed over the values 0 to 255. Then both the bitwise exclusive-or of X and Y, and X+Y mod 256, are uniformly distributed over 0 to 255. In addition if both X and Y are approximately uniformly distributed, then the combination will be more closely uniformly distributed.
- In the Marsaglia CD-ROM the idea is to get the excellent properties of the pseudo-random number generator but to break up any remaining patterns with the random or semi-random generators.

Transformations

- We now have an idea of how to generate a uniform probability distribution, so that the probability of generating a number between x and x + dx, denoted p(x)dx, is given by
p(x)dx = dx 0 < x < 1

0 otherwise

Transformations

- Now suppose that we generate a uniform deviate x and then take some prescribed function of it, y(x).
- The probability distribution of y, denoted p(y)dy, is determined by the fundamental transformation law of probabilities, which is simply
- |p(y)dy| = |p(x)dx| or p(y) = p(x) dxdy

Exponential

- As an example, suppose that y(x) ≡ −ln(x), and that p(x) is as given by a uniform deviate. Then
- p(y)dy = |dx/dy| dy = e−y dy
which is distributed exponentially. This exponential distribution occurs frequently in real problems, usually as the distribution of waiting times between independent Poisson-random events, for example the radioactive decay of nuclei.

Gaussian

- Another example is the Box-Muller method for generating random deviates with a normal (Gaussian) distribution,
- p(y)dy =1/√2π e−y2/2 dy

Gaussian

Since this is the product of a function of y2 alone and a function of y1 alone, each y is independently distributed according to the normal distribution

Gaussian

One further trick is useful: suppose that, instead of picking uniform deviates x1 and x2 in the unit square, we instead pick v1 and v2 as the ordinate and abscissa of a random point inside the unit circle around the origin.

Then the sum of their squares, R2≡ v12 + v22 is a uniform deviate, which can be used for x1, while the angle that (v1, v2) defines with respect to the v1 axis can serve as the random angle 2πx2.

Van der Corput Sequence

- If the desire is to have a uniform sequence,
- The van der Corput ssequence is ideal.
- The sequence
1/2

1/4 3/4

1/8 5/8 3/8 7/8

1/16 9/16 5/16 13/16 3/16 11/16 7/16 15/16

Finding Pi

- Choose a random point in the unit square by finding two random numbers (x1, x2).
- If this point lies inside the unit circle, consider the choice a ‘hit’, if not, a ‘miss’.
- Compute the area of a circle by finding the ratio of hits to the total number of points chosen, N. Determine the relationship between the number of choices and the digits of accuracy

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