Page 245 - Special Topic Session (STS) - Volume 1

STS426 Asis K.C.
log 1 (,) > log 2 (1|2) ………….Rule1
2 (,)
1 (1|2)

else, classify to group 2, where is the prior probability for the i th group, i =

1,2. If we assume = = 1/2 , then the above rule R1 classifies (x,y) to Group
2
1
1 if
(, )
1
log > log ( ) − log ( )

2

1
(, )
2
For the estimation of unknown parameters involved in the classification
rule, they have used the maximum likelihood estimates.
The total cost of misclassification (TCM) according to Rule 1 corresponding
to three forms of bivariate Gamma distribution and two sets of choices of
classification cost are shown in the following table-1. The analysis was carried
out on the basis of above mentioned data set on the galaxy NGC5128 and
starting with two groups obtained by k-means clustering with k=2.

Table 1: TCM for NGC 5128 data set
Gamma distribution TCM ( =0.9, =0.3) TCM ( =0.2, =0.8)
2
1
2
1
First form 0.3409 0.2657
Second form 0.1365 0.0182
Third form 0.1501 0.0265

The authors have proposed under the above set up, a classification rule by
including the knowledge of missing proportion in the construction of
classification rule, as described below.
Assume few observations are missing in a data set containing n
observations. If l is the cost per observation, the total cost is nl when no
observation is missing. But for m missing observations in the data, total cost

reduces to ( − ). Now ( − ) = (1 − ). If p is the proportion of

missing observations in the data, then = . Hence, ( − ) = (1 −

) ∝ (1 − ) , which indicates that it will be reasonable to take the loss as
(1 − ), 0 < < 1, in such a situation.
Consequently, if we have two groups to classify, we redefine the loss due
to misclassification as (1|2) = (1 − ) and (2|1) = (1 − ), where

2
2
1
are known constants, is the proportion of missing observations in the i th

group ,i = 1,2. It can be noted that when there is no missing value in the data,
i.e., = = 0, then (1|2) = and (2|1) = . So, the proposed loss due
2
1
1
2
to misclassification becomes equal to that without missing observations.
Hence, one can apply earlier adopted Rule 1 after discarding the missing
observations (i.e. marginalization) in the data or after substituting the missing
observations (i.e. imputation). But under marginalization there would be loss
of information and when the proportion of missing observation is quite
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