April 5, 2012 --- Class 24 --- Pseudorandom Numbers Generators,
Serial Correlation Tests
Activities:
Properties of Pseudo Random Number Generators
We looked at some properties of Linear Congruential PRNGs.
x_{n+1} = (a x_n + c) mod m
a is the multiplier
c is the increment
m is the modulus.
If c=0, this is a multiplicative generator, otherwise, it is mixed.
We start the sequence with a seed x_0.
m determines the maximum possible length of the sequence. It is
necessary to pick a and c carefully to get the maximum length
sequence.
Here is a theorem: The length of the period is m if the following
three conditions all hold.
a) c is relatively prime to m
b) (a-1) is a multiple of p for every prime factor of m
c) if m is a multiple of 4, (a-1) is a multiple of 4.
We looked at the case, m=32, a=3, c=4. This does not obey the
theorem and there is no sequence longer than 8. There is even a
sequence of length one. So much for randomness! To explore this
generator use the Mathematica definition
myrand[x_] := Mod[ 3x+4, 32]
The NestList command can be used to study a sequence:
seed = 3
NestList[myrand, seed, 32]
Adjusting a and c, we can easily find sequences of length 32;
however, that is not enough to ensure that the sequence looks random.
Serial Correlation Tests
In addition to having a uniform probability density, a pseudo random
number generator should have zero autocorrelation. There are tests
based on looking at pairs or triplets of numbers. For linear
congruential random number generators, there is a well known
problem that like the rain in Spain they tend to fall mainly in
the planes. This can be demonstrated by showing a two dimensional
scatter plot of (r_n, r_{n+1}). If you would like to see this, there
is an executable in ~sg/bin called random_lc:
random_lc | axis m 0 | xplot
will show you the correlations. The code is in ~sg/src and this
makes use of a poor linear congruential random number generator. You
would not see the same pattern with the fsr random number generator.
We looked in detail at another test of correlations based on
triplets of numbers. If we divide the interval (0,1) into L bins,
and look at triplets in the resulting L^3 cells, we derived that
the probability of having a cell remain unoccupied is exponentially
falling. Specifially, if we examine the occupancy after tL^3
triplets have been generated, the number of unoccupied cells should
be L^3 Exp[-t].
The box_rand.c code in ~sg/src implements this code for L=10.
To compile the code use the command:
cc -o box_rand box_rand.c new_fsr.o
This assumes that the new_fsr.c code has already been complied with
cc -c new_fsr.c
Running box_rand
We ran a few tests with the fsr random number generator and saw that
the number of unoccupied cells was decreasing. Then using a
while shell command, we ran the code 50 times, putting the reults
in a file. Using awk and aver, it was easy to get the average of
the 50 trials and plot the result with a semilog scale using axis.
The result clearly shows exponential decay of the number of unoccupied
boxes.
In our next class, we will look at the Feedback Shift Register
Random Number Generator code.