October 18, 2005 --- Class 12 --- More Random Walks, Hist, Biased Estimators 

Activities:

	More Random walks

	We continued the discussion started in the previous class of statistics
	of random walks.  Using Mathematica, it takes some time to create
	our random walks.

	We contrasted the time it took to generate these walks with the time
	to generate the random walks using a C program.  The C progam was much
	faster, but there is more coding required.  The C program is 
	~sg/rwalk_fsr.c
	The code was explained in detail.  It was then modified so it would
	only print the last point of the walk, not the entire walk.  When
	we printed the whole walk, we used awk to find the final value of
	each walk and this was slower than only printing the final value
	from the C program.

	A simple histogram program hist was introduced.  The code is in my
	src directory and it is called hist.c.  You are welcome to copy
	this code and use it to make histgrams for the rest of your life.
	We also used my blockerr program to find the mean and
	error in the mean from a file with numbers.  This code is also in
	src and you are welcome to copy and use it as you like.

	The options for hist are described in the comments at the top of hist.c:

/* Make  histogram of a list of data                *
*    option -n specifies number of bins            *
*    option -s specifies size of bin                *
*    option -x specifies lower and upper limits of histogram    *
*    option -g specifies output in form suitable for "graph"    *
*            or "axis"                *
*          if -g 1 we get a full bar for each bin    *
*          if -g 2 the bars don't go down to the origin    *
*    option -m specifies the line type for use with "axis"   *

	hist takes it input from the stardard input.  It will also take
	10 bins by default.  So, if you have file of number called file,

	hist <file
	
	will produce a histogram with 10 bins and to make a graph with 20 bin
	just give this command:

	hist -g -n 20 <file |axis |xplot

	blockerr is a simple program for blocking numbers together and
	reporting the mean value and its error.  Being able to block the
	data is useful for correlated data.  We will deal with this 
	complication later.  For now, just use a blocksize of 1.
	blockerr reports the error in the mean value, if your sample is
	large and you want the variance of the distribution, multiple the
	error reported by blockerr by sqrt(sample size).

	Biased Esimators Using Mathematica

	We returned to the problem posed in section H of the handout.
	We started by teaching Mathematica to do multiple integrals
	analytically and then having it get a numerical result as described
	in the handout.  In the next class, we will make a table of values
	from both analytic and numerical integration.

	We trained Mathematica to do multiple integrals analytically 
	for the problem we are considering.  Without
	waiting too long, we were able to get up to a sample size of 6;
	however, this result still shows significant bias.  In 
	~sg/oct19_2004.nb you will find a mathematica notebook showing how to
	time the command for doing a four-dimensional integral and the analytic
	and numerical results.

	Bias reduction

	We considered the jackknife approach to statistical analysis and
	how it can be used to reduce sample size bias.  In the next class,
	we will apply this method.

	Assuming that error goes like 1/n, where n is the sample size,

	S_n = A +e/n

	where A is the limit of n -> infinity and e determines the size of
	the error.  Considering samples sizes n and n-1, it is easy to
	solve for A.

	A = n S_n - (n-1) S_{n-1}