ESTIMATION OF LIKELIHOOD RATIO in VS .NET

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ESTIMATION OF LIKELIHOOD RATIO
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kernel (14.3) used for f ( | ), is given by Dc = f (y1 | , 2 )f ( | )d exp
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1 1 2 (1 zi )2 exp
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Similarly the second term, Ds , in the denominator, with the biweight kernel (14.3) used for f ( | ), is given by Ds = = f (y2 | , 2 )f ( | )d ns 16 m 2 15
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The numerator, Ncs , is given by Ncs = f (y1 y2 ) = 1 f (w | )f ( | )d 15 1 (y1 y2 )2 2 2 12 16 m 3 2 1 {w (xi + h zi )}2 dzi 2 2 3
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It can be shown that the likelihood ratio is given by the ratio of Ncs to the product of Dc and Ds . Numerical evaluation of the likelihood ratio may then be made with the substitution of by sw , by sb , and h by its optimal value (70/m)1/5 x. There is a boundary effect when an (xi , i = 1, . . . , m) is within h of zero. For these xi , the kernel expression 1 xi h
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2 2 2 = 1 zi 2
2 has to be adjusted with the factor ( 2 1 z)/( 0 2 1 ), where zi = ( x i )/(h ) and 0 , 1 , 2 are as in (14.4), to give
( 2 1 zi ) 2 1 zi 2 ( 0 2 1 )
APPLICATIONS V: FORENSIC GLASS ANALYSIS
which can be written as
2 (a bzi ) 1 zi
2 2 where a = 2 /( 0 2 1 ) and b = 1 /( 0 2 1 ). De ne an indicator function (zi ) such that
(zi ) = 1 if xi > h = (a bzi ) if xi < h Then the likelihood ratio Ncs /(Dc Ds ) can be adapted to account for boundary effects to give a value for the evidence of 1 exp
m 1 1
15 1 (y1 y2 )2 2 2 12 16 m 3 2 1 {w (xi + h zi )}2 dzi 2 2 3
2 (zi )(1 zi )2 exp
divided by the product of nc 16 m 2 15 and ns 16 m 2 15
m i=1 1 1 2 (zi )(1 zi )2 exp m i=1 1 1 2 (zi )(1 zi )2 exp
nc (y1 (xi + h zi ))2 dzi 2 2
ns (y2 (xi + h zi ))2 dzi 2 2
Adaptive Kernel
The value of the evidence, when the between-group distribution is taken to be nonnormal and is estimated by a normal kernel function as described in [1], equation (10.12), is adapted to allow for the correlation between the control and recovered data y1 and y2 if they are assumed, as in the numerator, to come from the same source. This expression is then extended to an adaptive kernel, where the smoothing parameter is dependent on xi and is thus denoted hi . The numerator is 1 (2 ) 1 m
n c + ns nc ns
1/2
2 + nc + ns
2 1/2
1/2
(h2 2 ) 1/2 i
2 + nc + ns
(h2 2 ) 1 i
ESTIMATION OF LIKELIHOOD RATIO
1 exp (y 1 y 2 )2 2
nc + ns nc ns
1 2 exp (w x i )2 2 + + h2 2 i 2 nc + ns
The rst term in the denominator is 1 2 (2 ) 1/2 2 + m nc
m 1/2
(hi 2 2 ) 1/2
1/2
2 +
2 nc
+ (h2 2 ) 1 i
1 2 exp (y 1 x i )2 2 + + h2 2 i 2 nc
The second term in the denominator is 1 2 (2 ) 1/2 2 + m ns
m 1/2
(h2 2 ) 1/2 i
1/2
2 + ns
(h2 2 ) 1 i
1 2 exp (y 2 x i )2 2 + + h2 2 i 2 ns
The constant term in the ratio is then m nc 2 (h2 + 1) + 2 i
ns 2 (h2 + 1) + 2 i
(nc + ns ) 2 (h2 + 1) + 2 i
The remaining term, that involving y 1 , y 2 and x i , is the ratio of 1 exp (y 1 y 2 )2 2 2
1 1 + nc ns
1 2 exp (w x i )2 2 + + h2 2 i 2 nc + ns
APPLICATIONS V: FORENSIC GLASS ANALYSIS
m i=1
1 2 exp (y 1 x i )2 2 + + h2 2 i 2 nc
1 2 exp (y 2 x i )2 2 + + h2 2 i 2 ns
14.2.4.1 Adaptive Smoothing Parameter The adaptive smoothing parameter hi is estimated using the procedure outlined in [331]. First, nd a pilot estimate f (x) that satis es f (xi ) > 0 for all i. This is achieved by standard kernel density estimation with Gaussian kernels [331]. Then de ne the smoothing parameter hi by
hi =
f (xi ) g
where g is the geometric mean of the f (xi ): log g = m 1 log f (xi )
and is a sensitivity parameter, a number satisfying 0 1.
APPLICATION
In order to evaluate the feature evaluation methods and LR estimators, a forensic dataset was obtained from the Forensic Research Institute in Krakow, Poland. An overview of the overall system can be found in Figure 14.1. One large piece of glass from each of 200 glass objects was selected. Each of these 200 pieces was wrapped in a sheet of gray paper and further fragmented. The fragments from each piece were placed in a plastic Petri dish. Four glass fragments, of linear dimension less than 0.5 mm with surfaces as smooth and at as possible, were selected for examination with the use of an SMXX Carl Zeiss (Jena, Germany) optical microscope (magni cation 100 ).