November 27, 2007 --- Class 25 --- 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 a the Feedback Shift Register 
	Random Number Generator code.