w7b

CASE STUDY 7

Plastic explosives detection.

Data Set: Training Set: Pex23 training Testing Set: Pex23 testing

The data comes from a study for the detection of plastic explosives in
suitcases using X-ray signals.

The 23 variables are the discrete xcomponents of the xray absorption spectrum.

The response is the last variable in the dataset. It takes two values:

0: There is explosive

1: There is not.

The objective is to detect the suitcases with explosives.

Discriminant Analysis Examples

Suppose we have k distinct populations or groups and suppose we observe x from one of them and we wish to know which population came from.

Pop	Pop₂	. . .	Pop_k	Populations
p₁	p₂	. . .	p_k	Pror Probs
m₁, S₁	m₂, S₂	. . .	m_k, S_k	Pop params

Sample 1	Sample 2	. . .	Sample k	Sample
₁, S₁	₂, S₂,	. . .	_k, S_k	sample estimators
L(x)₁	L(x)₂	. . .	L(x)_k	Likelihood
p₁L(x)₁	p₂L(x)₂	. . .	p_kL(x)_k	Posterior Probs

2
D (x|j) prop to -2 Log(p_jL_j(x))

   2         _   _       -1 _   _
D (i|j) = (X - X )' COV   (X - X ) + ln |COV | - 2 ln prior
              i   j      j    i   j           j              j

The posterior probabilities are

                        2                    2
     Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
                        j        k           k

data crops;
   title 'Discriminant Analysis of
Remote Sensing Data on Five Crops';
   input crop $ 1-10 x1-x4 xvalues $ 11-21;
   cards;
CORN      16 27 31 33
CORN      15 23 30 30
CORN      16 27 27 26
CORN      18 20 25 23
CORN      15 15 31 32
CORN      15 32 32 15
CORN      12 15 16 73
SOYBEANS 20 23 23 25
SOYBEANS 24 24 25 32
SOYBEANS 21 25 23 24
SOYBEANS 27 45 24 12
SOYBEANS 12 13 15 42
SOYBEANS 22 32 31 43
COTTON    31 32 33 34
COTTON    29 24 26 28
COTTON    34 32 28 45
COTTON    26 25 23 24
COTTON    53 48 75 26
COTTON    34 35 25 78
SUGARBEETS22 23 25 42
SUGARBEETS25 25 24 26
SUGARBEETS34 25 16 52
SUGARBEETS54 23 21 54
SUGARBEETS25 43 32 15
SUGARBEETS26 54 2 54
CLOVER    12 45 32 54
CLOVER    24 58 25 34
CLOVER    87 54 61 21
CLOVER    51 31 31 16
CLOVER    96 48 54 62
CLOVER    31 31 11 11
CLOVER    56 13 13 71
CLOVER    32 13 27 32
CLOVER    36 26 54 32
CLOVER    53 08 06 54
CLOVER    32 32 62 16
;

proc discrim data=crops
             method=normal pool=no
             crossvalidate;
   class crop;
   priors prop;
   id xvalues;
   var x1-x4;
   title2 'Using Quadratic Discriminant Function';
run;

OUTPUT

Discriminant Analysis

           36 Observations        35 DF Total
            4 Variables           31 DF Within Classes
            5 Classes              4 DF Between Classes

Class Level Information

Prior
CROP Frequency Weight Proportion Probability

CLOVER              11     11.0000     0.305556      0.305556
CORN                 7      7.0000     0.194444      0.194444
COTTON               6      6.0000     0.166667      0.166667
SOYBEANS             6      6.0000     0.166667      0.166667
SUGARBEETS           6      6.0000     0.166667      0.166667

Covariance Natural Log of the Determinant
CROP Matrix Rank of the Covariance Matrix

CLOVER               4                     23.64618
CORN                 4                     11.13472
COTTON               4                     13.23569
SOYBEANS             4                     12.45263
SUGARBEETS           4                     17.76293

Pairwise Generalized Squared Distances Between Groups

   2         _   _       -1 _   _
D (i|j) = (X - X )' COV   (X - X ) + ln |COV | - 2 ln PRIOR
              i   j      j    i   j           j              j

Generalized Squared Distance to CROP

From CROP CLOVER CORN COTTON SOYBEANS

CLOVER         26.01743         1320    104.18297    194.10546
CORN           27.73809     14.40994    150.50763     38.36252
COTTON         26.38544    588.86232     16.81921     52.03266
SOYBEANS       27.07134     46.42131     41.01631     16.03615
SUGARBEETS     26.80188    332.11563     43.98280    107.95676

Generalized Squared Distance to CROP

From CROP SUGARBEETS

CLOVER         31.40816
CORN           25.55421
COTTON         37.15560
SOYBEANS       23.15920
SUGARBEETS     21.34645

Classification Summary for Calibration Data: WORK.CROPS

Resubstitution Summary using Quadratic Discriminant Function

Generalized Squared Distance Function:

      2         _       -1   _
     D (X) = (X-X )' COV (X-X ) + ln |COV | - 2 ln PRIOR
      j          j      j     j           j              j

Posterior Probability of Membership in each CROP:

                        2                    2
     Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
                        j        k           k

Number of Observations and Percent Classified into CROP:

From CROP CLOVER CORN COTTON

CLOVER 9 0 0
81.82 0.00 0.00

CORN 0 7 0
0.00 100.00 0.00

COTTON 0 0 6
0.00 0.00 100.00

SOYBEANS 0 0 0
0.00 0.00 0.00

SUGARBEETS 0 0 1
0.00 0.00 16.67

Total 9 7 7
Percent 25.00 19.44 19.44

Priors 0.3056 0.1944 0.1667

Classification Summary for Calibration Data: WORK.CROPS

Resubstitution Summary using Quadratic Discriminant Function

Number of Observations and Percent Classified into CROP:

From CROP SOYBEANS SUGARBEETS Total

CLOVER 0 2 11
0.00 18.18 100.00

CORN 0 0 7
0.00 0.00 100.00

COTTON 0 0 6
0.00 0.00 100.00

SOYBEANS 6 0 6
100.00 0.00 100.00

SUGARBEETS 1 4 6
16.67 66.67 100.00

Total 7 6 36
Percent 19.44 16.67 100.00

Priors 0.1667 0.1667

Error Count Estimates for CROP:

CLOVER CORN COTTON SOYBEANS SUGARBEETS Total

Rate 0.1818 0.0000 0.0000 0.0000 0.3333 0.1111

Priors 0.3056 0.1944 0.1667 0.1667 0.1667
;