Optimization of Chromatogram Alignment Using A Class Separability Criterion

Transcription

1 Optimization of Chromatogram Alignment Using A Class Separability Criterion Gopal Yalla Department of Mathematics and Computer Science Department of Chemistry College of the Holy Cross April 28, 2015 Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

2 Outline 1 Introduction to Chromatography 2 Theory and Techniques 3 Experimental Data 4 Data Preprocessing 5 Results 6 Extended Results 7 Acknowledgements Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

3 Gas Chromatography The gas chromatograph (GC)) is the main instrument used for separating the components of a mixture. Two Phases: Mobile Phase and Stationary phase Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

4 Mass Spectrometry The mass spectrometer (MS) identifies the amount and type of chemicals present in a sample. Components are ionized and separated according mass. The mass spectrum is a definite pattern of the number of ions present at each mass level Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

5 Chromatograms GC + MS produces chromatograms. x-axis displays retention time in the GC column y-azis displays molecular abundance in sample Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

6 Chromatograms GC + MS produces chromatograms. x-axis displays retention time in the GC column y-azis displays molecular abundance in sample Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

7 Chromatographic Data Analysis Peak Area Extraction æ Judgement of number and type of chemical components must be made by the user. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

8 Chromatographic Data Analysis Peak Area Extraction æ Judgement of number and type of chemical components must be made by the user. æ Straightforward, but time consuming. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

9 Chromatographic Data Analysis Peak Area Extraction æ Judgement of number and type of chemical components must be made by the user. æ Straightforward, but time consuming. æ Sacrifice interesting trends. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

10 Chromatographic Data Analysis Peak Area Extraction æ Judgement of number and type of chemical components must be made by the user. æ Straightforward, but time consuming. æ Sacrifice interesting trends. æ Di cult with complex data... Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

11 Peak Area Extraction (Con t) Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

12 Alignment Issue When dealing with multiple samples, fluctuations in peak height and peak location occur. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

13 Alignment Issue When dealing with multiple samples, fluctuations in peak height and peak location occur. Without peak location alignment, trends determined by chemometric methods will be skewed or meaningless. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

14 Alignment Techniques. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

15 Alignment Techniques Correlation Optimized Warping (COW): Given two parameters segment size (m) andmax warp (t), a chromatogram P is aligned to a target chromatogram T. Dynamic Programming: Solves combinatorial optimization problems. COW uses two matrices, F and U of size (S + 1) (L + 1). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

16 COW Algorithm Correlation Optimized Warping (COW): Given two parameters segment size (m) andmax warp (t), a chromatogram P is aligned to a target chromatogram T. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

17 COW Algorithm Correlation Optimized Warping (COW): Given two parameters segment size (m) andmax warp (t), a chromatogram P is aligned to a target chromatogram T. Choice of target chromatogram is based on similarity index, NŸ SI j = r(x j, x n ). n=1 Where r(, ) represents Pearson s correlation coe cient. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

18 COW Algorithm Correlation Optimized Warping (COW): Given two parameters segment size (m) andmax warp (t), a chromatogram P is aligned to a target chromatogram T. What is the optimal choice of COW parameters?. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

19 Nomenclature and Terminology a = scalars a = column vector A = data matrices Row index n corresponds to sample chromatogram Column index m corresponds to retention time M total retention times N total chromatogram N k total chromatograms in the kth class K total classes x (Q) kn is the nth chromatogram in the kth class processed with correction method Q. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

20 Alignment Metrics: Warping E ect Warping E ect = Simplicity + Peak Factor Simplicity ([0, 1]): How close is data to rank 1 matrix Q Q ˆ RR Rÿ ıÿ simplicity = asvd ax/ Ù K ÿn k Mÿ bb r=1 xknm 2 k=1 n=1 m=1 4 Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

21 Alignment Metrics: Warping E ect Warping E ect = Simplicity + Peak Factor Simplicity ([0, 1]): How close is data to rank 1 matrix Q Q ˆ RR Rÿ ıÿ simplicity = asvd ax/ Ù K ÿn k Mÿ bb r=1 xknm 2 k=1 n=1 m=1 Peak Factor ([0, 1]): How much the shape and peak area of chromatograms have been changed by warping peak factor = 1 Kÿ ÿn k (1 min(c kn, 1) 2 ) N k=1 n=1 Î x (COW) kn Î Îx kn Î where c kn = represents a relative error between - Î x kn Î - aligned and unaligned chromatogram. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38 4

22 Alignment Metric: Hotelling Trace Criterion Hotelling Trace Criterion HTC Incorporates both within class and between class variation in the data set. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

23 Alignment Metric: Hotelling Trace Criterion Hotelling Trace Criterion HTC Incorporates both within class and between class variation in the data set. ø HTC Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

24 Alignment Metric: Hotelling Trace Criterion Hotelling Trace Criterion HTC Incorporates both within class and between class variation in the data set. HTC Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

25 Hotelling Trace Criterion Define the sample mean vector and sample covariance matrix for the kth class as: x k = 1 N k ÿn k n=1 x kn, S k = 1 N k 1 ÿn k n=1 (x kn x k )(x kn x k ) t. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

26 Hotelling Trace Criterion Define the sample mean vector and sample covariance matrix for the kth class as: x k = 1 N k ÿn k n=1 x kn, S k = 1 N k 1 ÿn k n=1 (x kn x k )(x kn x k ) t. Let P k = N k /N be the probability of occurrence of class k. Thegrand mean vector is given by: Kÿ x = P k x k. k=1 Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

27 Hotelling Trace Criterion Define the sample mean vector and sample covariance matrix for the kth class as: x k = 1 N k ÿn k n=1 x kn, S k = 1 N k 1 ÿn k n=1 (x kn x k )(x kn x k ) t. Let P k = N k /N be the probability of occurrence of class k. Thegrand mean vector is given by: Kÿ x = P k x k. k=1 The within-class scatter matrix and between-class scatter matrix is defined as: Kÿ Kÿ S wc = P k S k, S bc = P k ( x k x)( x k x) t. k=1 k=1 Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

28 Hotelling Trace Criterion (Con t) The HTC is defined as: J = tr! S 1 " wc S bc Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

29 Hotelling Trace Criterion (Con t) The HTC is defined as: J = tr! S 1 " wc S bc When K = 2, HTC reduces to the Mahalanobis distance J =( x 1 x 2 ) t S 1 ( x 1 x 2 ) Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

30 Hotelling Trace Criterion (Con t) The HTC is defined as: J = tr! S 1 " wc S bc When K = 2, HTC reduces to the Mahalanobis distance J =( x 1 x 2 ) t S 1 ( x 1 x 2 ) When K = 2andM = 1, HTC reduces to the square of a t-statistic 1 2 J = t 2 2 N Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

31 Experimental Data 5 Classes of Biodiesel: Soy (6 di erent samples) Canola (3 di erent samples) Tallow (3 di erent samples) Waste Grease (2 di erent samples) Hybrid (1 sample) } Each sample tested 3di erentruns 45 Total Chromatograms Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

32 Experimental Data 5 Classes of Biodiesel: Soy (6 di erent samples) Canola (3 di erent samples) Tallow (3 di erent samples) Waste Grease (2 di erent samples) Hybrid (1 sample) Chemical Structure: FAMEs (Fatty acid methyl ester) } Each sample tested 3di erentruns 45 Total Chromatograms Variable length of carbon chain and number of double bonds. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

33 Experimental Data 5 Classes of Biodiesel: Soy (6 di erent samples) Canola (3 di erent samples) Tallow (3 di erent samples) Waste Grease (2 di erent samples) Hybrid (1 sample) Chemical Structure: FAMEs (Fatty acid methyl ester) } Each sample tested 3di erentruns 45 Total Chromatograms Variable length of carbon chain and number of double bonds. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

34 Experimental Data 5 Classes of Biodiesel: Soy (6 di erent samples) Canola (3 di erent samples) Tallow (3 di erent samples) Waste Grease (2 di erent samples) Hybrid (1 sample) Reaction Process: } Each sample tested 3di erentruns 45 Total Chromatograms Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

35 Experimental Data 5 Classes of Biodiesel: Soy (6 di erent samples) Canola (3 di erent samples) Tallow (3 di erent samples) Waste Grease (2 di erent samples) Hybrid (1 sample) Sample Chromatogram: } Each sample tested 3di erentruns 45 Total Chromatograms Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

36 Data Preprocessing: Timeline 1 Baseline Correction 2 COW Alignment 3 Normalization & Mean Centering 4 Principal Component Transformation 5 Computed Metrics Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

37 Baseline Problem Need to correct for non-linear increase in baseline caused from: Gradual increase in oven temperature Column Bleeding Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

38 Baseline Correction Use asymmetric least squares smoothing to determine baseline vector b Õ that minimizes f (b Õ )=Îw t (b Õ x kn )Î 2 + ÎDb Õ Î 2 w is a vector of weights is a relaxation parameter D is a second di erence matrix Î Î is the Euclidean norm Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

39 Baseline Correction Use asymmetric least squares smoothing to determine baseline vector b Õ that minimizes f (b Õ )=Îw t (b Õ x kn )Î 2 + ÎDb Õ Î 2 w is a vector of weights is a relaxation parameter D is a second di erence matrix Î Î is the Euclidean norm Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

40 Baseline Correction Use asymmetric least squares smoothing to determine baseline vector b Õ that minimizes f (b Õ )=Îw t (b Õ x kn )Î 2 + ÎDb Õ Î 2 w is a vector of weights is a relaxation parameter D is a second di erence matrix Î Î is the Euclidean norm Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

41 Baseline Correction Use asymmetric least squares smoothing to determine baseline vector b Õ that minimizes f (b Õ )=Îw t (b Õ x kn )Î 2 + ÎDb Õ Î 2 w is a vector of weights is a relaxation parameter D is a second di erence matrix Î Î is the Euclidean norm Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

42 Baseline Correction: Finding Peaks Let x kn = s + b + where s is true peak height, b is true smooth basline, and is normal random error with small deviation. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

43 Baseline Correction: Finding Peaks Let x kn = s + b + where s is true peak height, b is true smooth basline, and is normal random error with small deviation. Let m i be median vector of points in x kn over an appropriate window centered at time index i. æ m b æ x kn b + æ median ( x kn m ). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

44 Baseline Correction: Finding Peaks Let x kn = s + b + where s is true peak height, b is true smooth basline, and is normal random error with small deviation. Let m i be median vector of points in x kn over an appropriate window centered at time index i. Y _] 0 if x kni > m i ± 2 w i = _[ 1 if x kni Æm i ± 2 Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

45 Baseline Correction: Results Using b Õ to estimate b gives, x (BC) kn = x kn b Õ s + Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

46 Baseline Correction: Results Using b Õ to estimate b gives, x (BC) kn = x kn b Õ s + Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

47 Normalization and Mean Centering Each chromatogram x (BC,COW) kn should be normalized to account for variations in injection volume. x (BC,COW,NORM) kn = Ā x (BC,COW) kn A kn where A kn represents total area of each chromatogram, and Ā is average total area of all chromatograms. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

48 Normalization and Mean Centering Each chromatogram x (BC,COW) kn should be normalized to account for variations in injection volume. x (BC,COW,NORM) kn = Ā x (BC,COW) kn A kn where A kn represents total area of each chromatogram, and Ā is average total area of all chromatograms. Each chromatogram should be mean centered to the origin. x (BC,COW,NORM,MC) kn = x (BC,COW,NORM) kn x (BC,COW,NORM) where x (BC,COW,NORM,MC) kn is the sample mean chromatogram. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

49 Principal Component Analysis HTC was evaluated on the principal component transformed data. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

50 Principal Component Analysis HTC was evaluated on the principal component transformed data. Let S represent the the sample covariance matrix of the entire set of preprocessed data, with eigenvalue decomposition: S = U U t Then y kn, the vector of PC s, is computed via the transformation y kn = U t x (BC,COW,NORM,MC) kn Eigenvalues correspond to how much variation is explained in each PC. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

51 Principal Component Analysis HTC was evaluated on the principal component transformed data. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

52 HTC Evaluated on PCs Let z kn =(y kn1, y kn2,, y knl ) t denote the L 1 vector corresponding to the first L PCs of y kn.thesample mean vector and sample covariance matrix for the kth class are given respectively by z k = 1 N k ÿn k n=1 z kn, S k = The grand mean vector is given by z = 1 N k 1 ÿn k n=1 Kÿ P k z k. k=1 (z kn z k )(z kn z k ) t. The within-class scatter matrix and between-class scatter matrix is defined as: Kÿ Kÿ S wc = P k S k, S bc = P k ( z k z)( z k z) t. HTC is given by, k=1 k=1 J = tr (S 1 wc S bc ) Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

53 Computed Metrics Density Plots for Warp E ect & HTC: Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

54 Computed Metrics Density Plots for Warp E ect & HTC: Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

55 Computed Metrics Density Plots for Warp E ect & HTC: Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

56 Computed Metrics Density Plots for Warp E ect & HTC: Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

57 Results: PC1 vs. PC2 Max Warp E ect: (26,15) Max HTC (1 PC): (64,3) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

58 Results: PC1 vs. PC2 Max Warp E ect: (26,15) Max HTC (2 PC): (55,8) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

59 Results: PC1 vs. PC2 Max Warp E ect: (26,15) Max HTC (3 PC): (70,6) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

60 Results: PC1 vs. PC3 Max Warp E ect: (26,15) Max HTC (1 PC): (64,3) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

61 Results: PC1 vs. PC3 Max Warp E ect: (26,15) Max HTC (2 PC): (55,8) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

62 Results: PC1 vs. PC3 Max Warp E ect: (26,15) Max HTC (3 PC): (70,6) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

63 Results: PC2 vs. PC3 Max Warp E ect: (26,15) Max HTC (1 PC): (64,3) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

64 Results: PC2 vs. PC3 Max Warp E ect: (26,15) Max HTC (2 PC): (55,8) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

65 Results: PC2 vs. PC3 Max Warp E ect: (26,15) Max HTC (3 PC): (70,6) soy ( ), canola (ù), tallow ( ), waste grease (ú), hybrid (+). Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

66 Summary of Results Based on our data, HTC leads to better alignment than warping e ect æ Greater Euclidean Distance between class means Ratios for Segment Length/Max Warp (55,8) to (26,15) Class Soy Canola Tallow Waste Grease Soy Canola Tallow Waste Grease Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

67 Summary of Results Based on our data, HTC leads to better alignment than warping e ect æ Greater Euclidean Distance between class means Ratios for Segment Length/Max Warp (55,8) to (26,15) Class Soy Canola Tallow Waste Grease Soy Canola Tallow Waste Grease æ Smaller within-class variation. Ratios for Segment Length/Max Warp (55,8) to (26,15) Class 1st Major Axis 2nd Major Axis Soy Canola Tallow Waste Grease Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

68 Summary of Results Based on our data, HTC leads to better alignment than warping e ect æ Greater Euclidean Distance between class means Ratios for Segment Length/Max Warp (55,8) to (26,15) Class Soy Canola Tallow Waste Grease Soy Canola Tallow Waste Grease æ Smaller within-class variation. Ratios for Segment Length/Max Warp (55,8) to (26,15) Clear parametric distinction. Class 1st Major Axis 2nd Major Axis Soy Canola Tallow Waste Grease Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

69 Project Milestone! 1 Published Work in Journal of Chemometrics Soares Edward J., Yalla Gopal R., O Connor John B., Walsh Kevin A., and Hupp Amber M. (2015), Hotelling trace criterion as a figure of merit for the optimization of chromatogram alignment, J. Chemometrics, 29, pages Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

70 More Complex Data: Biodiesel-Diesel Blends 210 chromatograms with three di erent attributes æ Feedstock: Pure Diesel, Soy, Canola, IRE Tallow, Texas Tallow, Waste Grease æ Diesel Type: Flynn,Hess,Shell,Sunoco æ Blend Ratio: B2, B5, B10, B20 Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

71 Diesel Results Before Alignment: Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

72 Diesel Results After Alignment and Optimization: Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

73 Diesel Results After Alignment and Optimization: Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

74 Classification B10 Biodiesel Samples Shell Sunoco Texas Tallow 12 5( ) 5(*) IRE Tallow 12 5( ) 5(*) Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

75 Classification B10 Biodiesel Samples Shell Sunoco Texas Tallow 12 5( ) 5(*) IRE Tallow 12 5( ) 5(*) Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

76 Broader Impact 1 Determine chemical components that contribute the most to the energy content of fuel æ Create synthetic biomaterial with energy content? 2 Forensic / Environment Concerns æ Determine origins and consequence of oil spill Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

77 Future Work Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

78 Future Work 1 Algorithmic Development æ COW has very long computation time. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

79 Future Work 1 Algorithmic Development æ COW has very long computation time. æ No parametric pattern Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

80 Future Work 1 Algorithmic Development æ COW has very long computation time. æ No parametric pattern 2 Larger Sample Size for HTC Results Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

81 Acknowledgements Thank you for listening! Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

82 Acknowledgements Thank you for listening! Professor Amber Hupp Professor Kevin Walsh Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

83 Acknowledgements Thank you for listening! Professor Amber Hupp Professor Kevin Walsh Colette Houssan Mike Comiskey Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

84 Acknowledgements Thank you for listening! Professor Amber Hupp Professor Kevin Walsh Colette Houssan Mike Comiskey Department of Mathematics & Computer Science Department of Chemistry Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

85 Acknowledgements Journal of Chemometrics Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

86 Acknowledgements Journal of Chemometrics University Syringe Program Grant from Hamilton Company (AMH). Robert L. Ardizzone Fund for Junior Faculty Excellence (AMH). College of the Holy Cross. Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38

87 Iowa Renewable Energy Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38 Acknowledgements Journal of Chemometrics University Syringe Program Grant from Hamilton Company (AMH). Robert L. Ardizzone Fund for Junior Faculty Excellence (AMH). College of the Holy Cross. National Institute of Standards and Technology (NIST, Gaithersburg, MD) Western Dubuque Biodiesel ADM Company, Keystone Biofuels, TMT Biofuels, Texas Green Manufacturing

88 Thank you Professor Soares! Couldn t have done it without you Sauce! Gopal Yalla (Holy Cross) Analysis of Biofuels April 28, / 38