October 16, 2007---Class 14---Jackknife Mean, Jackknife Method Via Mathematica,
		Introduction to Chaos and Logistic Map

Activities:

	Hist
	
	We reviewed the options to hist.
	
	Jackknife Mean
	
	In addition to using the jackknife method for bias reduction, you
	can use it to calculate the sample mean.  The sample mean is
	related to variance of the jackknife samples.  Because these
	samples are so correlated (only one variable changes in each
	sample), the variance of the jackknife values must be multiplied by
	(n-1):

	     (\sigma_J)^2 = (n-1)/n \sum (S'_i - S)^2

	where S'_i is the jackknife statistic with the i'th measurement
	removed and S is the statistic on the entire set of n measurements.

	Notice that in the simple case that the statistic is the mean value
	of the sample, the naive variance would be given by:

	\sigma^2 = 1/[n(n-1)] \sum (x_i - x_avg)^2

	In this case, it is easy to show S'_i - S = (x_avg - x_i)/(n-1).
	Here is a deriviation:

	(n-1) S'_i + x_i  = sum of all x_i = n(S) = n x_avg = (n-1)S + x_avg

	since S and x_avg are the same in this example.  Bring the terms
	involving S to the left, the terms involving x to the right and
	divide by (n-1) to complete the derivation.

	Jackknife Method Via Mathematica

	Mathematica can be used to generate samples of random numbers and
	to apply the jackknife technique.  This is a good opportunity to
	introduce a new mathematica construct the "module", which is like
	a subroutine.  In ~sg/jackknife/jackn.math  is the code to implement
	this method.

	First we need a way to create our set of random variables.  This is
	done via the newvars command:

	newvars[num_] := Do[z[i] = Random[], {i, num}]

	Once this is defined, you may create as many new random numbers
	in the vector z by calling newvars with the number you desire
	as the argument.

	A Module in mathematica takes two arguments.  The first is a list
	of variables local to the module, and the second is an expression,
	i.e., a sequence of statements.

	Using a module statement we define jackknife[x_, num_Integer],
	where x is vector of random numbers and num is the sample size.
	Here is the code:

jackknife[x_, num_Integer] := 
  Module[{mysum, mysumpr, stat, statpr, estimate, fctr, i}, 
   fctr = N[(num - 1)/num]; mysum = Sum[x[i], {i, 1, num}]; 
    stat = num/mysum; estimate = num*stat; 
    Do[mysumpr = mysum - x[i]; statpr = (num - 1)/mysumpr; 
      estimate -= fctr*statpr, {i, 1, num}]; {stat, estimate}]

	Successive calls to newvars and jackknife are equivalent to what
	was done in the C code in the last class.  I made a table from many
	calls of newvars and jackknife, from which a scatterplot was
	produced using ListPlot.

	Chaos and the Logistic Map

	We introduced the logistic map from 
	consideration of an insect population model.  

		P_{n+1} = P_n (A - B P_n)

	Upon scaling the population, the population is
	replaced by x which lies in the range [0,1].  The next generation
	is given by

	x <- 4 r x * (1-x) .