Likelihood Ratio Estimation in Visual Studio .NET

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14.4.2 Likelihood Ratio Estimation
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There are 200 within-group comparisons of control and recovered data and 200 199/2 = 19, 900 between-group comparisons For the 200 within-group comparisons, the likelihood ratio should be greater than 1 and for the 19,900 between-group comparisons, the likelihood ratio should be less than 1. Results are recorded for a normal kernel estimation (nn), an exponential kernel estimation (exp), an adaptive kernel estimation for = 0, 0.1, 0.2, and 0.5 and a biweight kernel estimation (b).
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TABLE 14.1 Ranking of Features Element FRFS IG OneR O 0.003929 0.267 52.50 Na 0.025909 0.597 55.75 Mg 0.025426 0.718 52.13 Al 0.025996 0.843 57.63 Si 0.009267 0.129 41.38 K 0.003118 0.825 55.75 Ca 0.008341 0.511 53.25 Fe 0.022455 0.191 45.50
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FuzEnt 1.839151 1.581143 1.658684 1.647498 1.801163 1.582525 1.600595 1.741598
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GR 0.211 0.391 0.367 0.275 0.127 0.338 0.312 0.145
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Relief-F 0.0364 0.0894 0.1234 0.0661 0.0309 0.0928 0.0881 0.0453
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2 368.4882 968.2267 1010.738 1167.625 150.6826 1578.554 686.8860 174.0253
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SU 0.1615 0.3345 0.3589 0.3301 0.0846 0.3682 0.2774 0.1136
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EXPERIMENTATION
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TABLE 14.2
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Summary of Feature Ranking Feature Ranking Al > Na > Mg > Fe > Si > Ca > O > K Al > K > Mg > Na > Ca > O > Si > Fe Al > Na = K > Ca > O > Mg > Fe > Si Na > K > Ca > Al > Mg > Fe > Si > O Na > Mg > K > Ca > Al > O > Fe > Si Mg > K > Na > Ca > Al > Fe > O > Si K > Al > Mg > Na > Ca > O > Fe > Si K > Mg > Na > Al > Ca > O > Fe > Si
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Selection Method FRFS IG OneR FuzEnt GR Relief-F 2 SU
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Note: A > B indicates that A is a more informative feature than B, and A = B indicates identical ranking.
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The results for the aluminum data (selected by FRFS, IG, and OneR) are shown in Table 14.3, magnesium data (selected by Relief-F) in Table 14.4, potassium data (selected by 2 and SU) in Table 14.5, and sodium data (selected by FuzEnt and GR) in Table 14.6. It can be seen that, overall, the results produced using the aluminum data are superior to those generated using the other considered elements. The within-group estimation of likelihood ratio is as good as or better than the others, and the between-group estimation is more accurate. The results also indicate that magnesium is a very informative feature for likelihood ratio estimation. The level of false positives (important from a forensic viewpoint) is rather high. This could be the result of several factors. The SEM-EDX method delivers information about the main elemental content of glass; thus differences between glass objects are very small. Moreover soda-lime-silica glass (the most common) has a strict composition, and the production recipes, used in many factories, are very similar. Also the data are created mostly from glass used for car windows and building windows, which have extremely similar elemental concentrations. Sodium results are relatively bad as the concentration of Na is restricted by the de nition of Na Ca Si glass. Potassium results are also unsatisfactory because it is only present in glass objects that have special optical properties, and so is absent or undetectable in many objects (this may explain the poor performance of the exponential model for this element). Aluminum and magnesium results are relatively superior as these elements are added to glass to create desired features of glass objects, and thus their content varies with glass properties. For this dataset the feature ranking methods FRFS, IG, and OneR have selected the best feature for likelihood estimation. This is in keeping with other investigations that have shown the utility of FRFS, in particular, for this purpose [168]. Although Relief-F selected an alternative feature, the results achieved were almost as accurate. The results show that the normal kernel estimation procedure is inadequate at modeling the underlying data distributions in general. Excepting the case of sodium, it be seen that even an exponential distribution is a better model as the resulting likelihood ratios are more accurate.
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