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Forensic Analysis: Weighing Bullet Lead Evidence (2004)

Chapter: Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?

« Previous: Appendix G: Data Analysis of Table 1, Randich et al.
Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

H
Principal Components Analysis: How Many Elements Should Be Measured?

The number of elements in bullet lead that have been measured has ranged from three to seven, and sometimes the concentration of a measured element is so small as to be undetectable. The optimal number of elements to measure is unclear. An unambiguous way to determine it is to calculate, using two-sample equivalence t tests, the probability of a false match on the 1,837-bullet data set as described in Chapter 3. Recall that the equivalence t test requires specification of a value δ/RE where RE = relative error and a value α denoting the expected probability of a false match. Each simulation run would use a different combination of the elements: there are 35 possible subsets of three of the seven elements, 35 possible subsets of four of the seven elements, 21 possible subsets of five of the seven elements, seven possible subsets of six of the seven elements, and one simulation run corresponding to using all seven elements. Among the three-element subsets, the subset with the lowest false match probability would be selected, and a similar process would occur for the four-, five-, and six-element subsets. One could then plot the false match probability as a function of δ/RE for various choices of δ/RE and determine the reduction in false match probability in moving from three to seven elements for testing purposes. Such a calculation may well differ if applied to the full (71,000-bullet) data set.

An alternative, easier to apply but less direct approach is to characterize the variability among the bullets using all seven elements. To avoid the problem of many missing values of elemental concentrations in the 1,837-bullet dataset, we will use the 1,373-bullet subset, for which all 7 elemental calculations exist (after inputing some values for Cd). The variability can then be compared with the variability obtained using all possible three-, four-, five-, and six-element subsets. It is likely that the false match probability will be higher in subsets that

Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

comprise lesser amounts of the total variability and lower in subsets that comprise nearly all of the variability in the data set. Variability can be characterized by using principal components analysis (PCA).

Consider, for example, a PCA using the first three elements (As, Sb, and Sn—elements “123”), which yields 104.564 as the total variation in the data. PCA provides the three linear combinations that decompose this variation of 104.564 into three linear combinations of the three elements in a sequential fashion: the first linear combination explains the most variation (76.892); the second, independent of the first, explains the next-most (19.512), and the third accounts for the remainder (8.16). The total variation in all seven elements is 136.944. Thus, this three-element subset accounts for (104.564/136.944) × 100%, or 76.3% of the total variation. The results of PCA on all 35 3-element subsets are shown Table H.1; they illustrate that subset “237” (Sb, Sn, and Cd) appears to be best for characterizing the total variability in the set, accounting for (114.503/136.944) × 100% = 83.6% of the variability. Subset “137” (As, Sn, and Cd) is almost as good at (113.274/136.944) × 100% = 83.0%.

PCA is then applied to all 35 possible four-element subsets; the one that accounts for the most variation, (131.562/136.944) × 100% = 96.1%, is subset “1237” (As, Sb, Sn, and Cd). Among the five-element subsets, subset “12357” (As, Sn, Sb, Cu, and Cd) explains the greatest proportion of the variance: (134.419/136.944) × 100% = 98.2%, or about 2.1% more than the subset without Cu. The five-element subset containing Bi instead of Cu is nearly as efficient: (133.554/136.944) × 100% = 97.5%. Finally, among the six-element subsets, “123457” (all but Ag) comes very close to explaining the variation using all seven elements: (136.411/136.944) × 100% = 99.6%. Measuring all elements except Bi is nearly as efficient, explaining (134.951/136.944) × 100% = 98.5% of total variation. The values obtained for each three-, four-, five-, six-, and seven-element subset PCA are found in Tables H.1, H.3, H.5, H.7, and H.9 below. The corresponding variances in order of increasing percentages are found in Tables H.2, H.4, H.6, and H.8.

This calculation may not directly relate to results obtained by simulating the false match probability as described above, but it does give some indication of the contribution of the different elements, and the results appear to be consistent with the impressions of the scientists who have been measuring bullets and making comparisons (Ref. 1-3).

Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

TABLE H.1 Principal Components Analysis on All Three-Element Subsets of 1,373-Bullet Subset. Elements 1, 2, 3, 4, 5, 6, and 7 are As, Sb, Sn, Cu, Bi, Ag, and Cd, respectively. Row labels 1, 2, and 3 represent the first principal components through third, and the rows show the total variation due to each successive element included in the subset.

 

123

124

125

126

127

134

135

136

137

1

76.892

26.838

27.477

26.829

28.109

73.801

73.957

73.786

74.254

2

96.404

35.383

36.032

35.373

53.809

86.312

86.730

86.294

100.820

3

104.564

37.340

38.204

35.879

62.344

88.269

89.133

86.808

113.274

 

145

146

147

156

157

167

234

235

236

1

17.553

17.110

27.027

17.534

27.071

27.027

71.675

71.838

71.661

2

20.218

19.223

44.089

19.991

44.535

44.074

87.537

88.137

87.529

3

21.909

19.584

46.049

20.448

46.914

44.589

89.498

90.362

88.037

 

237

245

246

247

256

257

267

345

346

1

72.186

18.941

18.335

27.146

18.938

27.216

27.146

69.371

69.243

2

98.651

21.493

20.457

45.309

21.220

45.926

45.308

72.353

71.377

3

114.503

23.138

20.813

47.278

21.677

48.143

45.818

74.067

71.742

 

347

356

357

367

456

457

467

567

 

1

69.771

69.357

69.891

69.758

3.272

27.030

26.998

27.030

 

2

96.234

72.149

96.367

96.221

5.039

30.136

29.156

29.929

 

3

98.208

72.606

99.072

96.747

5.382

31.847

29.522

30.387

 

TABLE H.2 Total Variance (Compare with 136.944 Total Variance) for Three-Component Subsets, in Order of Increasing Variance.

456

146

156

246

256

145

245

467

567

457

5.382

19.584

20.448

20.813

21.677

21.909

23.138

29.522

30.387

31.847

126

124

125

167

267

147

157

247

257

127

35.879

37.340

38.204

44.589

45.818

46.049

46.914

47.278

48.143

62.344

346

356

345

136

236

134

135

234

235

367

71.742

72.606

74.067

86.808

88.037

88.269

89.133

89.498

90.362

96.747

347

357

123

137

237

 

 

 

 

 

98.208

99.072

104.564

113.274

114.503

 

 

 

 

 

Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

TABLE H.3 Principal Components Analysis on All Four-Element Subsets of 1,373-Bullet Subset. Elements 1, 2, 3, 4, 5, 6, and 7 are As, Sb, Sn, Cu, Bi, Ag, and Cd, respectively. Row labels 1, 2, 3, and 4 represent the first principal component through fourth, and the rows show the total variation due to each successive element included in the subset.

 

1234

1235

1236

1237

1245

1246

1247

1256

1257

1

76.918

77.133

76.903

77.362

27.517

26.865

28.126

27.506

28.599

2

96.441

97.085

96.430

103.955

36.072

35.410

53.844

36.061

54.501

3

104.603

105.249

104.590

123.430

38.556

37.516

62.380

38.279

63.047

4

106.557

107.421

105.096

131.562

40.197

37.872

64.337

38.736

65.202

 

1267

1345

1346

1347

1356

1357

1367

1456

1457

1

28.122

73.982

73.810

74.278

73.966

74.440

74.263

17.575

27.071

2

53.835

86.772

86.330

100.843

86.751

101.012

100.828

20.366

44.575

3

62.371

89.436

88.440

113.309

89.208

113.752

113.291

22.099

47.221

4

62.877

91.126

88.801

115.267

89.665

116.131

113.806

22.441

48.906

 

1467

1567

2345

2346

2347

2356

2357

2367

2456

1

27.027

27.071

71.861

71.683

72.209

71.847

72.378

72.195

18.969

2

44.108

44.556

88.174

87.562

98.673

88.164

98.855

98.660

21.650

3

46.221

46.989

90.710

89.674

114.534

90.437

115.149

114.526

23.328

4

46.581

47.446

92.355

90.030

116.495

90.894

117.360

115.035

23.670

 

2457

2467

2567

3456

3457

3467

3567

4567

 

1

27.217

27.146

27.217

69.378

69.911

69.777

69.898

27.031

 

2

45.955

45.333

45.952

72.492

96.387

96.241

96.374

30.276

 

3

48.496

47.454

48.218

74.257

99.355

98.375

99.147

32.037

 

4

50.135

47.810

48.675

74.599

101.065

98.740

99.604

32.380

 

TABLE H.4 Total Variance (Compare with 136.944 Total Variance) for Four-Component Subsets, in Order of Increasing Variance.

1456

2456

4567

1246

1256

1245

1467

1567

2467

2567

22.441

23.670

32.380

37.872

38.736

40.197

46.581

47.446

47.810

48.675

1457

2457

1267

1247

1257

3456

1346

1356

2346

2356

48.906

50.135

62.877

64.337

65.202

74.599

88.801

89.665

90.030

90.894

1345

2345

3467

3567

3457

1236

1234

1235

1367

2367

91.126

92.355

98.740

99.604

101.065

105.096

106.557

107.421

113.806

115.035

1347

1357

2347

2357

1237

 

 

 

 

 

115.267

116.131

116.495

117.360

131.562

 

 

 

 

 

Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

TABLE H.5 Principal Components Analysis on All Five-Element Subsets of 1,373-Bullet Subset. Elements 1, 2, 3, 4, 5, 6, and 7 are As, Sb, Sn, Cu, Bi, Ag, and Cd, respectively. Row labels 1, 2, 3, 4, and 5 represent the first principal components through fifth, and the rows show the total variation due to each successive element included in the subset.

 

12345

12346

12347

12356

12357

12367

12456

12457

12467

1

77160

76.930

77.388

77.144

77.608

77.373

27.547

28.624

28.140

2

97.127

96.468

103.981

97.114

104.205

103.966

36.103

54.541

53.871

3

105.292

104.630

123.467

105.278

124.130

123.456

38.716

63.088

62.408

4

107.775

106.733

131.600

107.496

132.265

131.588

40.387

65.560

64.514

5

109.414

107.089

133.554

107.953

134.419

132.094

40.729

67.194

64.869

 

12567

13456

13457

13467

13567

14567

23456

23457

23467

1

28.617

73.991

74.464

74.286

74.448

27.072

71.870

72.401

72.217

2

54.530

86.795

101.037

100.852

101.021

44.598

88.203

98.878

98.682

3

63.076

89.584

113.794

113.328

113.773

47.372

90.867

115.186

114.559

4

65.277

91.316

116.440

115.438

116.206

49.096

92.546

117.714

116.617

5

65.734

91.658

118.124

115.799

116.663

49.439

92.887

119.353

117.028

 

23567

24567

34567

 

 

 

 

 

 

1

72.387

27.218

69.918

 

 

 

 

 

 

2

98.864

45.984

96.394

 

 

 

 

 

 

3

115.177

48.655

99.495

 

 

 

 

 

 

4

117.435

50.326

101.254

 

 

 

 

 

 

5

117.892

50.667

101.597

 

 

 

 

 

 

TABLE H.6 Total Variance (Compare with 136.944 Total Variance) for Five-Component Subsets, in Order of Increasing Variance.

12456

14567

24567

12467

12567

12457

13456

23456

34567

12346

40.73

49.44

50.67

64.87

65.73

67.19

91.66

92.89

101.60

107.09

%29.74

36.10

37.00

47.37

48.00

49.07

66.93

67.83

74.19

78.20

12356

12345

13467

12567

23467

23567

13457

23457

12367

12347

107.95

109.41

115.80

116.66

117.03

117.89

118.12

119.35

132.09

133.55

78.83

79.90

84.56

85.19

85.46

86.09

8 6.26

87.15

96.46

97.53

12357

 

134.42

 

98.16

 

Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

TABLE H.7 Principal Components Analysis on All Six-Element Subsets of 1,373-Bullet Subset. Elements 1, 2, 3, 4, 5, 6, and 7 are As, Sb, Sn, Cu, Bi, Ag, and Cd, respectively. Row labels 1, 2, 3, 4, 5, and 6 represent the first principal component through sixth, and the rows show the total variation due to each successive element included in the subset.

 

123456

123457

123467

123567

124567

134567

234567

1

77.172

77.635

77.399

77.620

28.643

74.472

72.411

2

97.157

104.232

103.993

104.216

54.571

101.046

98.887

3

105.322

124.172

123.494

124.159

63.118

113.817

115.215

4

107.934

132.307

131.628

132.294

65.721

116.590

117.872

5

109.605

134.779

133.731

134.494

67.385

118.314

119.543

6

109.946

136.411

134.087

134.951

67.726

118.656

119.885

TABLE H.8 Total Variance (Compare with 136.944 Total Variance) for Six-Component Subsets, in Order of Increasing Variance

124567

123456

134567

234567

123467

123567

123457

67.726

109.946

118.656

119.885

134.087

134.951

136.411

49.45%

80.28%

86.65%

87.54%

97.91%

98.54%

99.61%

TABLE H.9 Principal Components Analysis on all Seven-Element Subsets of 1,373-Bullet Subset. Elements 1, 2, 3, 4, 5, 6, and 7 are As, Sb, Sn, Cu, Bi, Ag, and Cd, respectively. Row labels 1, 2, 3, 4, 5, and 6 represent the first principal component through sixth, and the rows show the total variation due to each successive element included in the subset.

 

1234567

1

77.64703

2

104.24395

3

124.20241

4

132.33795

5

134.94053

6

136.60234

7

136.94360

Summary:

 

3 elements: 237

(83.6% of total variance)

4 elements: 1237

(96.07% of total variance)

5 elements: 12357

(98.16% of total variance) or 12347 (97.52%)

6 elements: 123567

(99.61% of total variance) or 123457 (98.54%) (Bi-Ag correlation)

7 elements: 1234567

(100.00% of total variance)

REFERENCES

1. Koons, R. D. and Grant, D. M. J. Foren.. Sci. 2002, 47(5), 950.

2. Randich, E.; Duerfeldt, W.; McLendon, W.; and Tobin, W. Foren. Sci. Int. 2002, 127, 174–191.

3. Peele, E. R.; Havekost, D. G.; Peters, C. A.; and Riley, J. P. USDOJ (ISBN 0-932115-12-8), 57, 1991.

Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 157
Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 158
Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 159
Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 160
Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 161
Suggested Citation:"Appendix H: Principal Components Analysis: How Many Elements Should Be Measured?." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 162
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Since the 1960s, testimony by representatives of the Federal Bureau of Investigation in thousands of criminal cases has relied on evidence from Compositional Analysis of Bullet Lead (CABL), a forensic technique that compares the elemental composition of bullets found at a crime scene to the elemental composition of bullets found in a suspect’s possession. Different from ballistics techniques that compare striations on the barrel of a gun to those on a recovered bullet, CABL is used when no gun is recovered or when bullets are too small or mangled to observe striations. Forensic Analysis: Weighing Bullet Lead Evidence assesses the scientific validity of CABL, finding that the FBI should use a different statistical analysis for the technique and that, given variations in bullet manufacturing processes, expert witnesses should make clear the very limited conclusions that CABL results can support. The report also recommends that the FBI take additional measures to ensure the validity of CABL results, which include improving documentation, publishing details, and improving on training and oversight.

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