LINEAR ALGEBRA (added 23-Oct)

STATISTICAL DISTANCE (added 23-Nov)

HYPOTHESIS TESTING

TABLE.   Joint Distribution of   X  and  Y  in  P0  and P1 

    Probability function        Probability function      Likelihood ratio,

     under P0,  fo(x,y)           under P1, f1(x,y)        f1(x,y) / fo(x,y) 

    -------------------           ----------------       ----------------------

                  y                       y                       y

         |   9  10  11   12         9  10  11  12            9   10    11    12   

 _______ | _________________      _______________        ______________________ 

         |                                         

     64  | .04 .08 .12  .16       .01 .02 .03 .04        0.25  0.25  0.25  0.25

 x   66  | .03 .06 .09  .12       .02 .02 .04 .02        0.67  0.33  0.44  0.17 

     68  | .02 .04 .06  .08       .15 .18 .04 .03        7.50  4.50  0.67  0.38 

     70  | .01 .02 .03  .04       .12 .25 .02 .01       12.00 12.50  0.67  0.25  

(a)Consider the test that has rejection region{(x,y):x> 68 and y < 10}.What is the level of this test ?

(b) What is the power of this test ?

(c) What is the rejection region of the optimal .02 level test of H₀vs. H₁?

(d)What is the power of this test ?

(e)What is the rejection region of the optimal .05 level test of H₀vs. H₁?

(f) What is the power of this test ?

(g) What is the rejection region of the Bayes test when the prior probabilities are .8, .2 ?

(h) What is the level of this test ?

(i) What is the power of this test ?

(j)What is the rejection region of the Bayes test when the prior probabilities are equal?

(k)What is the level of this test ?

(l)What is the power of this test ?

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CLASSIFICATION

C1. Negative Exponential Distributions

DISCRIMINATION

Consider classification with two multinormal distributions.Assuming prior probabilitiesp₁ = .8,p₂ = .2,

the constants in the two classification functionsare0.5and0.6.

What will be the constants if the prior probabilities are changed top₁ = .9,p₂ = .1 ?

D2.Form of Classification Functions

The form of the classification functions in the bivariate normal case ( p = 2 ) with equal covariance matricesis

C₁(x,y)=b₁₀ + b₁₁x + b₁₂ yandC₂(x,y)=b₂₀ + b₂₁x + b₂₂ y.

What is the form of the classification functions in the bivariate normal case withunequal covariance matrices?

D3.Means the Same, Covariance Matrices Different

In Population 1, X is multinormal with mean vector 0 and covariance matrix 4I; in Population 2, it is multinormal with mean vector 0 and covariance matrix 9I.  The two populations have equal prior probabilities.  Find the discriminant function.

D4.Height within Sexes.If the mean height for males is 69",   the mean height for females is 64" and the standard deviations are 3" and 2.5", respectively, and the prior probabilities are .5 and .5, find the classification regions.Hint:Find the roots of the associated quadratic.

D5.Height and Weight within Sexes.If the mean height for males is 69", the mean height for females is 64" and the standard deviations are 3" and 2.5", the mean weight for males is 160 lbs. with a standard deviation of 20 lbs., the mean weight for females is 125 lbs. with a standard deviation of 15 lbs., and the correlation between height and weight is +0.6 in both groups, find the classification regions. (Probably it's enough to think about the boundary (or boundaries, as the case may be) between the classification regions.)

D6.   Show that, in multinormal classification with different covariance matrices, the MAP procedure is to classify  x  into population k*,  where

D7. (continuation)  The prior probability term can be ignored if the prior probabilities are equal .   Then one is looking at  D²(x,m;S) + ln|S|  for the different populations.  Is this a form of minimum-distance classification; that is, is  [ D²(x,m;S) + ln|S| ]^1/2  a distance function?

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HISTOGRAMS

H1.     The number of days ill in a year is given for a sample of n = 50 coal miners in the dataset http://www.uic.edu/classes/ids472/data/daysill.txt . Make a series of histograms with various bin widths to illustrate the shape of the distribution, as follows.

One often reasonable rule for the number of bins is that if  n is about   2^k,  use  k+1  bins.  Here n = 50, which is between 32 and 64, the fifth and sixth powers of 2.    The rule would suggest six or seven bins.  Using Excel or Minitab, make histograms with 5, 7 and 9 bins. Compare them.  Which seems to show the overall pattern best ?  Making a succession of histograms with different bin widths can be help in formulating a mixture model for the data.

H2.Work out the SIC model selection criterion for histograms.Hint:multinomial distribution.

H2.  Stock Rates of Return:  Crude Oil

3.1.  Given a price series  {P(t)},  the rate of return is  [P(t+1) - P(t)]/P(t), which is approximately   ln P(t+1)/P(t),  or
ln P(t+1) - ln P(t) .  This is often multiplied by 100 to correspond to percent.  This logarithmic approximation to the percentage RORs of the crude-oil nearest month futures prices is in the worksheet  in the column 100*DIFF.   Make histograms of 100*DIFF using bin widths of say, 1.5, 2.0 and 2.5, or whatever best exhibits the information in the data.

3.2.  Do a K-Means clustering of these data, using a number of clusters and starting points suggested by the histograms.  (Use Minitab or other software.)

CLUSTERING