AN ABSTRACT OF THE THESIS OF
Stanley Claud Elliott Jr.
(Name)
in
Chemistry,
(Major)
for the
Master of Science
(Degree)
presented on
Title: A QUADRATIC PROGRAMMING ROUTINE FOR THE STATISTICAL ANALYSIS OF GAS CHROMATOGRAPHIC DATA
Abstract approved:
Redacted for Privacy
Stephen J. Hawkes
A quadratic programming method was developed for the statisti-
cal analysis of gas chromatographic data. The method was applied to
three types of flavor extracts, peppermint, hops and brewed tea. The
brewed tea data had to be abandoned because of the singularity of its
coveriance matrix. In all cases tried the method was able to pick the
exact origin of a pure hop or peppermint oil. The method was able to
classify blends of hop oils with less than five percent absolute error
for hypothetical blended values in proportions from zero to 100 percent of the whole.
The results for blends of peppermint oils were not
as straightforward. Some blends of peppermint were correctly
classified with less than a ten percent absolute error over a greater
composition range than others. When one three component blend and
one four component blend of the peppermint oil were tried the method
classified them correctly. The method can be applied to any problem
in which more than one parameter is needed to produce a classification.
A Quadratic Programming Routine for the
Statistical Analysis of Gas
Chromatographic Data
by
Stanley Claud Elliott. Jr.
A THESIS
submitted to
Oregon State University
in partial fulfilment of
the requirements for the
degree of
Master of Science
June 1971
APPROVED:
Redacted for Privacy
Associate professor of Chemistry
in charge of major
Redacted for Privacy
Assistant Professor of Siyistics
Redacted for Privacy
Head of Department of Chemistry
Redacted for Privacy
Dean of Graduate School
Date thesis is presented
100 "- Mak.
11
Typed by Barbara Eby for Stanley Claud Elliott Jr.
ACKNOWLEDGEMENTS
The author wishes to thank Drs. Stephen J. Hawkes and Norbert
A. Hartman for their assistance and guidance during this research.
Thanks are also due to David G. Niess and Billy Chow of the Computer
Center for their assistance.
TABLE OF CONTENTS
INTRODUCTION
1
STATISTICAL CONSIDERATIONS
2
III,
METHODOLOGY
5
IV,
RESULTS AND DISCUSSION
I.
II.
Mentha Arvensis Oils
Mentha Piperita Oils
V.
16
16
18
CONCLUSION
31
BIBLIOGRAPHY
32
APPENDIX
34
LIST OF TABLES
Page
Table
1
Mint data.
6
2
Mean vectors for mint oils.
7
3
Inverted
4
Hop oil data.
5
Tea data as received.
10
6
Reduced tea data.
12
7
Mean vectors for hop oils.
13
8
Inverted covariance for hop oils.
14
9
Hop oils.
17
10
Blends of mentha piperita with mentha arvensis oils.
19
11
Blends of mentha piperita origins one and four.
24
12
Other blends of mentha piperita.
26
13
Blends of origin one and four with five percent
S
matrix for mint.
8
9
seven.
28
A-1
Sample input file for MSUMT routine.
36
A-2
Symbols.
38
A QUADRATIC PROGRAMMING ROUTINE FOR THE
STATISTICAL ANALYSIS OF GAS
CHROMATOGRAPHIC DATA
INTRODUCTION
The question often asked of analysts is not one related to the de-
termination of a single component of a mixture, but one related to all
the components of a mixture. Such questions are origin of an oil,
price of an oil, variety of plant which produced the oil, percent composition of similar oils in blends and many others.
What is needed is a good general method that will provide this
kind of information for all such problems. The purpose of this study
was to investigate such a method.
2
STATISTICAL CONSIDERATIONS
The problems of interest are ones that can be handled by multi-
variate statistics. The density function of a multivariate normal distribution of the type of interest may be written in n dimensional space
as:
1
(270-2n
1
exp -
(1)
(X
(note that in one dimension this is the familiar Gaussian distribution)
where x is the observation vector (i. e. lxn matrix of observed peak areas
for one oil
the covariance matrix of the form:
10
1
Co.0
1
and
-1
.
U2
.1
ts inverse. An inverse is a matrix such that
its
-1 = I
(2) where I is a identity matrix. A matrix is invertable if and only if
it is nxn and nonsingular (the determinant is not zero) p, is the vec-
tor of means of the parameters of interest (i. e. relative areas).
the mean vector could be written as
Now
a. p.. where ai is the proportion
of the ith oil and Ili its mean vector. Then what is needed in our
type of problem are estimates of the ats.
3
The problem can therefore be stated in a mathematical model
similar to that of Hartmann and Hawkes (3):
min (x
subject to
(21
< 1 for all i
a.< 1 and 0 < a.i-1
1
Estimates of
pure components.
.
can be determined from data on the
and
Substituting the estimators back into Equation (1)
we obtain:
min (x subject to
a.-5c- .1'S
1
1
(x -
(3)
1
1
1
a. < 1 and 0 < a. < 1 when the matrix multiplication
in Equation (3) is carried out we produce an objective equation, in
matrix notation of the form:
-1
x'S lx - 2x'S
a .3-c
)+
a.x.)S
1 1
-
1
(4)
or in algebraic notation:
2
a. + B a.a.j + C a. + D
(5)
where A, B, C and D are constants calculated from the -X.'s, 5-1
and the observation vector x
This objective function along with the constraints can be handled
by a non-linear programming method to solve for the als. The 0. S. U.
4
computer center has two possible routines *FLEXI (4) and MSUMT
(5, 6,
7).
MSUMT being the most desirable in that it uses 1/6 of time
that *FLEXI would use to reach a solution, (5 sec vs 30 sec).
5
METHODOLOGY
In order to evaluate the power of such a method, three different
types of flavor extract were chosen because of their availability, though
crude oil data should also be amenable to this treatment as should any
complex mixture.
They were mint oil (Smith et al. (8)), hop oil
(Buttery et al, (9)) and extract of brewed tea (Vuataz et al. (10)). In
order to simplify our discussion the individual samples were given
numbers indicating their origin or variety which will be called oil
numbers (i. e, in the mint data all of the U. S. Mid-West oil samples
have the number one in common).
The mint oil data was the same as that used by Hartmann and
Hawkes (3) from Smith and Levi (8). Smith and Levi's analyses re-
ported fifteen peaks of which we used twelve excluding two because the
concentrations were too low and a third being the pure menthol peak to
eliminate the effect of dementholation in the original data (Table 1).
All peaks in the mint oils were calculated as percents of the totally
dementholated oil. The mean vectors (Table 2) for this data were ob-
tained from Hawkes (11), and the inverted covariance matrix (Table 3)
calculated by use of a statistical library program (12). The hop
(Table 4) and tea data (Table 5) (the tea data first being converted to
percent total relative area units) contain too many peaks even after
eliminating zero peaks to be used in the program to calculate the x.
6
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1.52
2.70
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'Table 2. Mean vectors for mint oils.
1.'1
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9.11
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6.98
14.68 26.81 5.79
10.81 30.95
6.94
6,.11 10.78
7.56
12.84 30.72 5.96
47.78 17.49
0
42.00 10.12
0
78.5q 5.16
0
44.00 13.76
(1
5.73
7.52
7.13
8.97
6.14 14.24
5.91
9.09
7.94 12.14
7.72 12.46
6.111 10.9?
4.57 12.78
7.63
1.95
7.47
2.07
1.14 1.48
3.14 4.28
3.34 1.96
5.69 3.6E
3.28 3.85
4.97 4.25
1.68
6.07
2.50 11.62
2.21 13.28
5.11 6.17
ELLIOTT TA0L7 7 T,IV73TE7_",
,4IRIC FOR MINT
31.480105-i.2734778 3.4094883 3.0087701-4.7054149 3.5473667
-0.2734773 4.7361271 2.3491466 2.8805958 2.6255618 2.7886828
7.40948F7 ?.8491466 5.6352223 4.2319336 3.4038498 4.1973786
3.0087701 2.8805958 4.2319336 4.3528205-1.2656533 4.1947795
-4.70541.49 2.6255618 3.4038498-1.267653392.2085714-1.4957146
3.5473667 2.7886828 4.1973786 4.1947795-1.4957146 4.6743787
4.2594104 2.8359551 4.3521256 3.8269526 .2564766 3.8888537
4.1675117 3.0445312 4.0503020 3.4645649 1.7941364 3.4762435
6.1198324 2.Q319207 5.2265262 4.6575051-3.3813058 4.2125206
3.73969tE,2 2,7591360 4.1906676 3.6613008 1.5010860 3.5552245
5.3976555 2.5575960 4.7272743 3.7156917 2.1471857 3.5373687
5.9284557 3.0417566 5.6747973 4.3600708 4.9209476 4.3047505
4.2594104
2.8359551
4.3521256
3.8269526
.2564766
3.8888537
4.0280890
3.6739475
4.3235243
3.8313681
3.9695174
4.6740528
4.1675817 6.1198324
3.0446312 2.9319207
4.0503020 5.2265262
3.4645649 4.6575051
1.7941364-3.3813058
3.4762435 4.2125206
3.6739475 4.3235243
4.5956997 3.3688709
3.3688709 9.0531316
3.6279751 3.7166574
3.6536814 5.4058959
4.3551736 5.5828131
3.7396962
2.7591360
4.1906676
3.6613008
1.5010860
3.5552245
3.8313681
3.6279751
3.7166574
3.9063816
3.7849337
4.5318278
5.3976555
2.5575960
4.7272743
3.7156917
2.1471857
3.5373687
3.9695874
3.6536814
5.4058959
3.7849337
5.5894870
4.9244563
5.9284657
3.0417566
5.6747973
4.3600708
4.9209476
4.3047505
4.6740528
4.3551736
5.5828131
4.5318278
4.9244563
6.4458754
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C
.42
0
0
.28
.14
.08
0
1.50
0
.04
.02
.47
.09
0
.23
.07
.40
.66
.03
.11
.03
.50
0
0
.09
.03
.02
.38
.10
.38
0
0
2.00
.08
.03
.05
.05
.01
.10
.12
.08
.03
.04
.93
0
.28
.09
.14
.28
.01
C
0
0
0
.28
.02
0
0
0
.39
.18
0
C
.08
.05
0
0
0
.20
(1
.35
.01
0
0
.25
.15
.07
0
.80
.16
.30
.65
.07
.40
1.10
0
.13
.95
.02
.32
1.20
.03
.18
.30
.05
.25
.20
.03
.13
.27
.01
.33
.46
.12
.09
.53
.34
.31
.72
.12
.06
.52
.11
0
.67
.16
.23
.75
.31
.40
.32
.70
.11
.27
.20
.20
.33
.10
.81
.20
.22
.43
.13
.38
.95
.25
.39
.54
.19
.17
.42
.15
.27
.74
.38
.42
1.40
.15
.40
.09
.30
.18
.05
.52
.38
.10
.42
.23
.06
.36
.2i
.15
.57
.15
.12
.83
.45
1.20
4.00
.15
.61
4.40
0
0
6.20
.45
.16
.14
.40
.15
.35
0
0
.30
.20
.60
0
0
.05
.20
.03
.35
.21
.03
.31
.25
.01
.48
6.50
.18
.45
3.90
.02
.28
6.70
.01
0
0
0
.42
.14
.20
.27
.13
.12
.71
.35
.20
.20
.15
.09
.27
.35
.30
RELATIVO 1127A
i'L._ "'Pt-' IIIT1 Tv- Sn 1- T40 LOITERS WERE GIVEN THE SAME OIL 1,UMEWR STARTING WITH
NUMBERS 2-r FOLLOWED IN CONSECUTIVE 9ROER.
ASSI,NE) THE N11M3ER 1.
-01 F-10 -r' 'FP)
0
0
.25
.65
.07
.11
.34
.39
.15
.02
.14
.17
.23
.53
0
0
.32
.22
0
.12
.50
3.60
.15
.67
8.40
.12
8.80
.04
.15
.20
.03
.40 21.00
.04
0
0
.85
.15
9.80
.09
.29
.42
.02
.09
.37 11.00
.02
.30
.02
.45
.01
.15
.45 12.50
.05.43
.42
.01
.25
.76 24.00
.06
.88
.05
.76
0
.09
.50 10.30
.57
0
.01
.55 11.00
0
0
.37
.64
.0i
.26
.08
.20
.30
.01
.29
0
.10
1.10
5.10
.25
.97
8.90
.50
.33
.30
4.50
.11
.26
8.20
.24
.18
6.20
.05
.27
8.20
.14
T
. _ _.
_ . . _
, .
.
05-
0-9'
.64
_. _ _ . . . _ , . _
6.20
0
.30
.08
0
.20 12.00
.03
0
.04
.60
.08
.20
.09
0
.04
.09
7.20
0
.44.
0
.03
.30 30.00
.22
.25
.01
.74
03
0
.20 22.00
.20
.04.5a
.03
0
.50 26.00
.01
.38
.04
.85
TABLES TEA DATA
io
AS RECEI VED
L.0,001
1.
1.0002
11741.
60138.
6742.
1.0,003
L.0,004
L.0,005
L.0,006
1.0,007
L.0,008
L.0,009
L.0,010
L.0,011
L.0,012
1.0,013
L.0,014
1.01015
1.
/1388.
122676.
14939.
1. ,
20896.
115408.
17011.
1..
17492.
97977.
4503.
46732.
651.
23107.
127287,
7897.
O.
O.
16555.
92510.
7358.
64824.
263305.
23338.
O.
O.
18727.
88601.
1978.
69774.
259565.
19262.
O.
O.
5728.
63925.
3489.
25616.
195516.
13960.
0 3.
47.84iH+64411.1.1.417
1.0,018
L.0,019
L.0,020
L.0,021
1.0,022
L.0,023
L.0,024
1.0,025
1.0,026
1.0,027
1.0,028
L.0.029
L.0,030
L.0,031
47203.
456004.
25330,
2,
52374.
360022.
17306.
2,
49902.
437537.
5369.
2,
56587.
594902.
162743.
25660.
17003.
10763.
115638.
18701.
4605.
24232.
171848.
18617.
1249.
14593.
201628.
25409.
L.0037
1.0,038
L.0,039
L.01040
1.0,041
1.0,042
1.0,043
1.01044
L.01045
L.0,046
1.0,047
L.0,048
L.0050
1.0,051
L.0,052
L.0,053
1.0,054
L.0,055
L.0,056
L.0,057
L.0,058
1.01059
L.0060
1.0461
16970.
9315.
23490.
10770.
13086.
16842.
70774.
8009.
19097.
17087.
2286.
58029.
422986.
48735.
7089
O.
O.
20916.
89345.
11428
23965.
278853.
49598.
48170.
576470.
62072.
O.
O.
69402.
2178.
0.
590.
O.
O.
O.
O.
O.
122662.
62136.
55692.
210892.
56422.
70138.
74072.
O.
O.
O.
O.
193658.
60510.
23431.
4,
O.
95994.
2143444.
24922.
567451.
107192.
62912.
4.
0.
644704.
122180.
124972.
M.0,072 READY.1708/08/17/70.
O.
O.
O.
14744.
1607792.
19458.
64974.
25557.
22187.
312
243728.
29032.
O.
O.
O.
14033
37948.
818062.
14092.
4.
O.
O.
O.
O.
113842.
2275936.
L.0064
50404.
ri..0,070 DONE
L.0,062
1.01063
O.
O.
0,
324260.
60712.
16038.
O.
3147.
67834.
30886.
O.
41138.
1456450.
26220.
114979.
31702.
4148.
12209.
O.
3,
of
502.
5172.
112231.
28916.
83459.
358003.
40941,
69820.
1010426.
49612.
3,
170'7:
0.
O.
O.
O.
O.
11110.
38866.
O.
4g06:;.
52419.
22398.
18303.
1400.
81485.
35677.
20614.
1429.
91014.
32375.
495306.
74230.
29526.
6004.
360856.
46302.
30260.
57320.
1721464.
147076.
O.
O.
O.
O.
O.
22710.
983840.
98916.
46228.
114544.
663662.
62120.
47968.
51914.
475770.
40286.
,3,
5591.
44081.
10160.
3126.
5813.
46916.
13374.
4049.
16528.
57533.
11920.
22038.
O.
49
30254.
1033920.
51032.
0.
O.
O.
O.
O.
O.
O.
42819.
284038.
31995.
L 01
1.0.034
L.0,035
L.0,036
332779.
48255.
G.
3054.
24Cc8.
5629.
9429.
7154.
52902.
13501.
21154.
760408.
82112.
O.
5360.
536642.
102534.
O.
92573.
1168454.
154978.
O.
145690.
1248258.
190748.
O.
6702.
92512.
8418.
0.
O.
O.
0.
0.
0.
53196.
4436.
17-814.
O.
O.
0.
150950.
10668.
24570.
O.
O.
194914.
18565.
O.
9208.
202920.
29586.
O.
O.
O.
O.
4977.
247.
4418.
10678.
O.
8245.
O.
O.
O.
O.
O.
11785.
139661.
1709.
15568.
O.
O.
O.
10166.
114593.
4481.
18469.
O.
O.
O.
5561.
57926.
O.
O.
O.
43693.
O.
O.
4710.
64539.
O.
O.
O.
85757.
O.
O.
3157.
12285.
119409.
O.
20945.
12311.
163493.
O.
17994.
11138.
106093.
3929.
24393.
13663.
147275.
O.
34770.
20443.
I
72622.
O.
O.
O.
I
1344.
37892.
311200.
O.
172416.
114260.
108474.
113752.
29576.
12516.
122552.
557792.
O.
O.
O.
174424.
52656.
43398.
73856.
8196.
32244.
217676.
0.
106286.
43920.
42288.
147282.
116810.
55198.
O.
84775.
84956.
68727.
0.
O.
O.
O.
O.
47980.
O.
0.
73566.
O.
O.
O.
84951.
0.
0.
0.
O.
O.
8202.
0.
11806.
290164.
22400.
O.
294306.
86888.
110382.
195300.
9962.
88186.
O.
60807.
O.
O.
O.
O.
10770.
5733.
6326.
32226.
248756.
54882.
96588.
O.
O.
0.
O.
3572.
14246.
203566.
8882.
23214.
263752.
O.
17283.
18633.
321836.
0.
166916.
23894.
O.
37572.
39098.
475816.
420630.
0.
11
vector and the S-1 matrix. This program (12) will handle a maximum
of fifteen variables. So it was necessary to transform the peaks into
data values that represent groups of peaks, with the same significance
in the decision making process as the original peaks.
This process was accomplished by use of the large factor analy-
sis program (13) of the computer center. This produced ten data
values for the tea data (Table 6) and nine data values for the hop data.
(It should be noted that in practice the factor matrix needs to be calcu.
lated only once for any type of problem.)
It was then discovered that the S matrix for the transformed tea
data is singular (the determinant is zero within the limits of the computer)
and so the tea data could not be handled by the method described in this paper.
The hop data was satisfactory and the mean vectors and the in-
verted covariance matrix were calculated with the same program (12)
that was used for the tea data (Table 7, hop mean vectors; Table 8,
covariance matrix).
In order to verify the power of the method the composition of
hypothetical blends were calculated. For example if we want to make
a 50-50 blend we pick one oil from one subgroup and one from another
then halve each of the data points (i. e. half of area for each peak) for
each oil and then add the corresponding data points together to make
an observation vector. In the case of the hop data such a vector would
be reduced by multiplying by the calculated factor matrix for this data.
TABLE 4 REDUCED TEA DATA
.724770
-2.901911
-4.507309
- 6.067580
-25.150159
1.860656
.936775
- 3.435898
-6.953154
-6.347849
22.089978
2.169451
2.279165
-9.406037
-0.298111
.452826
-0.729572
3.817869
-7.246936
-0.197614
2.120247
2.104660
-4.742289
-0.308625
1.049949
9.399353
-0.675245
3.357576
-0.178915
-0.787956
6.373845
2.300667
-1.307709
1.564157
9.536407
-0.955015
.845551
12.399354
-0.118713
14.776254
.830575
.163616
10.512995
2.639004
-0.830044
13.287252
.665297
-0.8590)5
-0.742580
4.685494
-0.710683
6.204428
-0.276078
-3.346172
-2.015724
-0.680521
-0.777901
1.944273
-4.185336
-3.073995
.047788
4.371610
-1.833799
.389198
-1.438081
-9.647770
-2.679509
-9.225943
-1.677126
2.902930
-0.465663
-0.920227
4.033642
.056755
3.573343
5.033695
5.232794
4.955489
4.181537
-3.915865
7.145320
35.504127
10.977323
30.097012
5.131405
-6.372471
5.174237
.661515
-11.749833
-7.530247
-11.324330
-11.569881
-12.2,9565
-11.379362
-11.597407
-10.907342
1.160770
-0.360420
1.253766
-0.977375
-0.036990
1.012522
.827466
-0.012083
.012131
-0.910095
1.219349
.989582
-0.185685
-1.882137
.041569
-2.152370
-5.451285
-9.382530
-7.070983
-6.458045
-1.965020
-0.593902
-3.633859
-1.027811
4.192809
3.403608
-0.637600
1.724101
6.308172
12.243880
4.262161
4.086302
-2.841453
-9.801327
-3.572815
-11.466847
3.617703
7.448194
4.570957
3.265826
1.821903
-2.490160
5.842081
4.215699
.863928
-3.361956
4.198613
-2.310345
-4.235477
-10.781620
-5.476995
-10.375544
-0.588240
2.074272
-1.110969
.288649
3.821339
1.621658
3.904055
3.835448
3.510591
2.944020
4.717086
5.851729
13
Table 7. Mean vectors for hop oils.
24.3449
88.502
49.4387
-28.9843
35.3283
-11.(155?
7.73C12
-11.6440
-1 .??147
-3G).9993
-1.6,22C
7.76071
3.0366
-1.7595
4.9553
4.0082
-8.6573
-4.2(1'34
-2.821?
1.4726
-4.r,737
-7.7776)
-4.593?
-0.1461
4.9920
3.0376
-1.0?82
-7.9376
-2.6841
-2.948?
77.ii477
11 .14
?4.0959
-6.16?7
-5.274
3.181?
-.794r;
-7.774C
-7.0702
-5.0508
14
TARLE 8 INVERTED COVARIANCE
FOR HOP OILS
ROW
ROW
1
3.45369
-9.71951
19.42561
5.19423
-20.01592
5.23144
9.26335
5.64690
20.38168
5.19423
11.74983
-41.97387
10.23245
15.78350
4.11720
22.61597
10.23245
-47.73705
13.57705
21.56361
8.49748
28.61422
2
- 17.88107
- ?8.85229
ROW
3
5.23144
-16.87996
- 39.23864
ROW
ROW
ROW
ROW
4
5.64590
4.11720
- 12.72788
- 35.95000
- 26.31707
8.49748
14.92093
16.16514
45.37401
-9.71961
31.84700
59.75190
-17.88107
65.05141
-16.87906
-26.38276
-12.72788
-20.01992
55.00141
1?9.90248
-41.97387
191.81190
-47.71705
-75.43127
- 117.76941
9.26775
15.78750
-75.48127
21.56361
49.33841
14.92393
55.35416
22.61597
-117.76939
28.6142?
55.35416
45.37401
164.56656
-28.85229
129.90248
-35.23854
-56.14813
-36.55000
-130.97840
7.45759
-2.71951
-19.42561
5.19427
-23.01592
5.23144
9.26335
5.64690
20.38168
5.194?
-17.83107
-23.85179
11.74983
-41.97337
10.23245
15.78350
4.11720
22.61597
5.73144
-15.37906
Y3.77964
10.23245
-47.78705
17.57705
21.56361
8.49748
28.51422
9.54521
12.7279
4.11720
-26.31707
1.4974R
14.92093
16.16514
45.37401
-2.71251
71.14721
55.76190
-17.88107
65.03141
-16.87906
-26.11276
-12.72788
-50.29630
71.01517
-41.97387
191.81190
-47.73705
- 26.31708
- 75,43127
-117.76941
15.78350
-75.48127
21.56361
49.73141
14.92091
55.35416
2"..78161
-51.21r)30
174'1
22.61597
-117.76939
28.61422
55.35415
45.77401
164.56656
-1'1.4?551
-28.352?9
95,7511
129.(3Q248
-75.23854
-f).14313
-76.55190
-130.93940
5
- 50.29630
6
-26.31708
7
- 1_6.78275
-66.14317
Row
8
20.73168
- 90.29530
-131.97840
ROW
9
- 13.42551
55.76191
131.57625
ROW
ROW
POW
ROW
1
?
3
4
- 75.55021
ROW
ROW
5
6
6r;.00141
1.?9.90?41
ROW
7
4.26!-35
5.78779
-55.1481_7
ROW
ROW
1
131.576'75
15
Data of pure oils and of blends were then used to calculate objective
equations and subject to the minimization routine.
16
RESULTS AND DISCUSSION
The results for the hop oils (Table 9) were done by two routines
*FLEXI and MSUMT. The method was able to exactly pick which
group the pure oil belonged to. The MSUMT routine was able to class-
ify the mixture within five percent of the hypothetical blended value for
the nine blends tried ranging in values from ten to 90 percent of the
whole.
The mint data can be divided into two groups, those numbered
one through six (Mentha piperita oils) which contain no octanol but do
contain menthofuran and those numbered seven through ten (Mentha
arvensis oils) which contain octanol but no menthofuran. The routine
MSUMT was used for all mint classifications.
Mentha Arvensis Oils (Seven-Ten)
An oil from group seven (Brazilian) and one from group nine
(Japanese) were chosen at random and the routine classified them
unambiguously, estimating the fraction of the oil coming from other
origins at less than 0. 01 percent in the Brazilian and less than two
percent in the Japanese.
The routine was then used to find the concentration of these oils
in hypothetical blends with each other and with oils from other origins,
17
Table 9. Hop oils.
Oil Numbers
2
1
4
3
5
*FLEXI
1. 00
1. 0000
.
50
.50
.
4338
.
5021
.
90
.
10
.
8790
.
0985
.
90
0021
.
MSUMT
10
.
0049
.0895
.20
9047
.
80
.
8093
30
.
70
2849
.
7134
.
60
.
40
3825
.
6169
.
50
.
50
5181
.
40
.
.
1871
.
4601
.
.
40
5610
.
70
6617
.
0002
.
0031
0002
.
0016
0002
.
0008
.90
8684
4218
.30
.
.
3247
.20
80
7703
.
0028
.
.
0001
.2271
.10
.
1295
.
0639
0019
18
with up to three other oils in any one blend.
Of the 27 hypothetical
two component blends and one three component blend of these oils with
other oils ranging in value from 30 to 95 percent of the whole for oil
number seven and 45 to 95 percent of the whole for oil number nine,
the absolute error was less than two percent. In the case of the four
component blend (which contained two piperita oils which are discussed
in the next section) the absolute error was less than two percent for oil
nine and less than four percent for oil seven (Table 10).
Mentha Piperita Oils (One-Six)
The classifications between oils from origins one through six
were not as good. The results seem to depend on which pair is used
to make up the mixture. For instance in the classification between
origins one (U. S. Mid-West) and origin seven (Brazilian) the absolute
errors range from five percent when the hypothetical percentage for
component one was five percent of the whole to less than three percent
when the hypothetical value was 70 percent.
In the case of ten blends of origin one and four (Italian) the
routine was able to classify this pair with absolute errors of less than
seven percent when the hypothetical percentage of oil four was 40
percent of the whole to less than three percent when the hypothetical
percentage was 95 percent. The absolute error for origin one varied
from 12 percent when the hypothetical percentage of the oil was 20 percent
Blends of mentha piperita with mentha arvensis oils.
Table 10.
Oil Numbers
2
1
4
3
.05
.0004
.
0291
.10
.0587
.
0222
.
0183
.
0120
15
.1124
.20
.1650
0036
.25
.2173
.
6
7
1. 00
1. 000
0
.
5
0080
30
.2695
.35
.3228
.40
.3761
0109
.
0071
0034
.
0053
.
95
.
9705
.
90
.
9190
.
85
.
8692
.
80
.
8193
.
75
.
7694
.
70
.
7196
.
65
.
6700
.
60
.
6205
8
9
10
Table 10 continued.
Oil Numbers
1
2
4
3
5
7
6
.
55
.4290
.
5760
.50
.
50
.
5213
.
45
45
.
.
4787
.55
.
5282
. 4717
.
60
.
40
.
4217
.5778
.
65
.
35
.
6274
.
3726
.
30
.
3230
.70
.
6770
9
1. 00
0
0
.
0132
.
0
9868
.95
.05
0
.
.
0502
.
0120
.
1040
.
0069
.
9376
.90
10
.
8891
10
Table 10 continued.
Oil Numbers
1
2
4
3
5
6
8
7
15
.
0
1576
.
0018
.
.20
.
0343
.
25
.
0981
. 1743
.
1601
.
1601
1618
.
1030
3535
8406
.
80
.
7914
.
75
.
7418
.55
.
5432
.50
.50
4939
.
4174
.
55
. 45
.
4813
.
.
0887
.
4443
1. 00
. 9868
0
.
85
.
6922
.45
.
.
.70
.30
.
9
05
0509
.
0120
.
95
.
9371
10
Table 10 continued.
Oil Numbers
1
2
4
3
5
6
.90
.10
.1053
0063
.15
1594
.
.20
.1705
.1010
.0088
.0156
85
.
8393
.
80
.6893
.1543
.80
.15
.0657
.25
.1242
.
.70
.30
.05
.1098
8882
.75
.7396
.25
.0899
0012
.
.7894
.1800
.1425
9
8
7
.8157
25
.
0493
.2467
.25
.2160
.25
.2627
10
23
to less than three percent when the hypothetical percentage was 60
percent (Table 11). Fouteen other hypothetical blends tried are shown
in (Table 12). Thirteen of these blends were from origin three
(Yakima Valley) and five (English). In these blends the sum of the
values picked by the routine from origin one and three is quite close
to the hypothetical value for origin three (.3100 when the hypothetical
value was 30 percent).
This can be expected since the plants that pro-
duced both of these oils were descended from Italian plants and the
weather in both areas is similar. However it should be noted that if
all U. S. origins were lumped together, the method would be able to
exactly classify such a category. The same problem arises in the
cases of the three component blend and the four component blend.
Note in the case of the four component, the absolute error for origin
five is less than one percent, while that for origin three is not as
good.
(Note lumping origin one and three together would produce an
absolute error of less than three percent for such a origin.) This
method is the first to the author's knowledge to classify three and four
component blends.
When blends of oils one and four with five percent of oil seven
blended in were tried, it was found that an absolute error in the value
for the number four oil in the 55 to 95 percent range was reduced to
within five percent (Table 13). This could mean that the value of some
oils could be improved by spiking in certain other oils. If this was
Blends of mentha piperita origins one and four.
Table 11.
Oil Numbers
2
1
4
3
6
5
1. 00
.
05
.20
.
0825
1323
.9295
.
0694
.95
.9257
.
0727
.80
.7682
.
0159
.30
.2024
.70
35
.65
.
.2630
.
40
3215
.
45
.
3793
.
.
.
.
1137
.
1112
.60
.5673
1103
.55
.5105
0972
.50
.4523
.50
.
4405
.
6757
62 32
.
0100
7
8
9
10
Table 11 continued.
Oil Numbers
4
1
2
.55
.5098
.0551
.45
.3911
.0439
0132
.40
.3300
.0776
.60
.5791
.
3
5
6
7
8
9
10
Table 12.
Other blends of mentha piperita.
Oil Numbers
1
2
3
0590
. 05
.
0697
.
0804
.
0912
.
10
. 15
.20
. 0999
.
1100
.
1250
1176
.
90
.
9163
7
8
9
10
. 85
.
9048
.
80
8650
0320
.
.
25
1335
. 75
.
6
. 95
. 9278
.
.30
. 1158
5
1. 00
. 9392
0
.
4
.
1842
.
35
. 7518
.70
6953
.
65
.2350
.6388
.40
.
60
.2866
.
5612
E.)
a,
Table 12 continued.
Oil Numbers
2
1
.45
.3342
.1154
.1184
.1264
4
3
.
.
0508
.50
.3684
0500
.55
.4045
.
0183
0652
.
0585
50
3926
.
1122
.
3075
.40
.
2168
1572
.50
.50
.4405
.4417
.
.45
.60
.1215
55
.4781
.
.
6
5
.
0971
.
4523
.
0101
7
8
9
10
Table 13. Blends of origin one and four with five percent seven.
Oil Numbers
2
1
. 95
0
0
.
0342
.
1145
.10
0080
.15
.0694
.
1722
.20
.1257
.
1613
1814
.30
.2351
.
.
1416
35
.2876
.40
.3413
.
1351
1296
8
10
9
.05
.
.
0038
05
. 0041
.05
0046
.
0057
. 0051
.
0041
.6537
.05
.0107
.
0039
.65
.05
.
0025
8032
.
80
.05
.7519
.
7026
.
05
.
6034
. 0174
.
60
.
.5521
.55
.
7
.
.
.70
.1501
6
8807
.75
.25
.
.
.85
.1831
5
.9620
.90
.05
.
4
3
.
4994
05
0244
.05
0298
. 0008
t
oo
Table 13 continued.
Oil Numbers
2
1
.45
.3916
.50
.4576
4
3
5
.50
.
1412
.
0981
. 3898
7
.05
4476
.
0335
.05
.45
.
6
.
0174
.
0370
30
proven to be true the routine could be used to determine which oils
and their percentage to spike in.
It should be noted that an error in classification does not mean
necessarily a wrong answer but the possibility of producing the sample
by the blend picked.
It should be noted that all of the oils numbered
one through six are botanically related being descended from English
plants so that some difficulty in their classification can be expected.
31
CONCLUSION
The quadratic minimization routine seems to be a promising
method to handle gas chromatographic data to answer such questions
as compositions of blends of oils. The whole data handling package
can be combined to produce rapid answers by interfacing a gas chromatograph with a computer. Greater accuracy could be obtained by obtain-
ing data with the accuracy needed for the desired problem.
The method could be applied to any problem that produces several
measurable parameters subject to natural variations.
32
BIBLIOGRAPHY
1.
Anderson, T. W. An introduction to multivariate statistical
analysis. New York, Wiley, 1958.
2.
Shields, P. C. Linear algebra. Mass, Addison-Wesley, 1964.
3.
Hartman,
N. and S. J. Hawkes. Statistical analysis of multivariate chromatographic data on natural mistures, with particular reference to peppermint oils. Journal of Chromatographic
Science 8, 610 (1970).
4.
Chow, B. 0. S. U. Computer Center. Personal communication.
July 1970.
5.
Chow, B. 0. S. U. Computer Center.
July 1970.
6.
Fiacco, A. V. and G. P, McCormick. Nonlinear programming:
sequential unconstrained minimization techniques. New York,
Personal communication.
Wiley, 1968.
A method for nonlinear constraints in minimization problems. In: Optimization ed. by R. Fletcher,
Academic Press London.
7,
Powell, M. J. D.
8.
Smith, M. D. and L. Levi. Treatment of compositional data for
the characterization of essential oils. Determination of Geographical origins of Peppermint Oil by Gas Chrographic Analysis. Journal of Agriculture and Food Chemistry 9, 238 (1961).
9.
Buttery, R. G. and L. C. Ling. Identification of hop varieties
by gas chromatographic analysis of their essential oils.
Capillary Gas Chromatography Patterns and Analyses of Hop
Oils from American Grown Varieties. Journal of Agriculture
and Food Chemistry. 15, 531 (1967).
10.
Vuataz and D. Reymond. Mathematical treatment of gas chromatographic data: application of tea quality evaluation. In: Advances
in Chromatagraphy ed. by A. Ziatkis, Chromatography Simposium
Dept. of Chemistry University of Houston, Texas also appeared
in Journal of Gas Chromatography 9, 168 (1971) Raw Data received in Personal communication August, 1970.
33
11.
Hawkes, S. J. Personal communication, Sept. 1970.
12.
12, 0. S. U. Statistical Analysis Program
Library. Oregon State University, Oregon, October 1967. Sec.
Revision, October 1969.
13.
Neiss, D. Large factor analysis program, 0. S. U. Computer
Center, Oregon State University.
Yates, T. L. 0. S. U.
APPENDIX
34
APPENDIX
Explanation of Sample Input
File-Appendix Table 1
The sample input file shown is for a mint blend. The file can be
divided into two parts. The first half which would not change for any
mint blend problem contains control parameter cards for MSUMT.
The inequality constraints are contained in this part of the file. The
second half of the file contains the A vector, the B matrix of Equation
(5) of the main text and a vector of zeros. The vector of zeros con-
tains one zero for each a. . The format for this half of the input file
is 8F10. 5. Several files of different problems like hop and mint oils
or several groups of origin problems can be combined, the advantage
being that only one loading operation need be carried out instead of
several.
The input files are set up as the example to take advantage of a
compiled program called *USERSUB that will generate the objective
functions and their gradients.
*USERSUB eliminates the need to write
input programs for MSUMT.
If one has only one type of problem (i. e. only origin of mint oil)
a program could be written that would only require that raw data be
entered to produce an output.
35
Following the table is a sample output. For further information
see the MSUMT write up available at 0. S. U. Computer Center.
36
APPENOIX TABLE 1 SAMPLE INPUT FILE FOR MSUMT ROUTINE
1=0 4=.95 7=.05
10
0
0
1.E-06
0.04
0.0001
1
-1
?
0.
0.
0.
-1.
-1.
21
20
1.E-08
0.0001
0.0001
-2
3
-1.
-1.
-1.
-1.
-1.
0
-3
-4
4
0.
0.
0.
-1.
0
0.
0.0001
0.0001
a
1.E-O5
0.15
5
-5
-1.
-1.
-1.
-1.
6
-6
0.0001
0.80
7
-7
8
-8
0.0001
-9
9
10 -10
0.
-1.
-1.
0.
0.
-1.
-1.
-1.
-1.
-1.
-1.
0.
-776.15161-717.14127-777.05P41-777.34254-786.85320-781.36755-787.13007-792.45889
-770.34967-765.53631
388.6591.4 391.75173 387.95145 388.06438 392.97513 390.11271 394.31980 396.94688
394.20601 382.56931 391.75873 395.43802 391.68891 391.85876 396.96721 393.94444
397.48854 400.21290 388.10949 386.33563 387.95145 391.68891 389.11895 388.55481
393.24581 790.75209 792.20948 394.76543 382.23936 380.23169 388.06438 391.85876
388.55481 311.62263 393.37093 390.63960 393.52376 396.19353 385.07285 382.61243
192.97513 796.86721 393.24581 393.37093 798.44937 395.46133 398.71562 401.42015
390.36981 117.62767 390.11278 393.94444 390.75209 390.63960 395.46133 392.81044
395.29719 197.96042 316.51080 384.04660 394.31963 397.48835 392.20926 393.52358
398.71542 395.29703 404.39774 407.45446 401.80281 397.07786 396.94672 400.21274
394.76525 396.19737 401.41997 397.96029 407.45446 410.15875 406.06243 400.54069
384.20448 788.79183 382.23752 385.07126 390.36805 386.49947 401.80271 406.06233
429.06952 411.84300 382.56824 386.33447 380.23040 382.61132 387.62643 384.04567
397.07778 400.5406? 411.14301 400.72465
0
0
0
0
0
0
0
0
0
THIS
TS
RUN FOR
SUMT -RWL
A
INEQUALITY Q0IsTRAINTS
1=0 4=.95 7=.05
0 EQUALITY CONSTRAINTS
LINEAR CONSTRAINTS
21, BOUNDS
20, EQUALITIES
SCALE FA0TOP IS
1.00000000E 00
0
THE STARTING 4EnToP. lc
4.00000011F-)0
1.000000990-)4
1.00000000E-14
1.00009000E-14
1.50000000E-01
1.00000000E-04
LINEAR CONSTRAINTS VALUES
4.00000109E-02 9.60000090E-01
9.99000000-01 1.00309900E-04
1.05000101:794 9.99900000E-11
FUNCTION COUNT =
1.00000000E-04
1.00000000E-04
9.99900000E-91
1.30.000000E-04
1.00000000E-04
9.99900000E-01
8.00000900E-01
9.99909000E-11
8.00000000E-01
1.50000000E-01
2.00000000E-01
1.00100000E-04
1.00000000E-04
8.50000000E-01
1.00000000E-04
9.99900900E-01
1.00000000E-04
9.99900000E-01
2.94091823E-03
n SUMT-ROWELL VALUE = -1.88522804E 02 UNSCALED FUNCTION VALUE = -3.88522804E 02 SUMT RK
=
1.00000000E 00
*
FINAL SOLUTION REPORTS
44
FUNCTION CQUHT =
?5
4
4
4 4 4 4 4 4 4
THE MINIMIZED OBJECTIVE VALUE =
4
4
4 4 4 4 4 4
44
4
-3.83720761E 02
4
4
SUMT RK
1.00000000E 00
TH0 FINAL X VALUES
-1.64P:191707-11
-?.9273500?E-11
1.41775361E-02
2.976091777-13
0 EQUALITY CONSTRAINTS
INEQUALITY C:0001STRATNIS
LINEAP CONSTRAINTS
2.05221196E-10
3.77096182F-03
?1, 00UNOS
20, EQUALITIES
9.61950904F-01 -2.16211932E-11 -2.31575557E-10 -3.80232679E-10
AT THE STAGE OF TERMINATION
0
2
LINEAR QONSTPAINT VALUES(INCLUDE BOUNDS)
-1.648991717-11 1.00990900E 00 3.41775361E-02 9.65822464E-01
1,R'f=0Q157';-07 -'.16?11172E-11
1.00001000E CO -2.31.570557F-10
-2.q22'5, Y)7-1C
1.000D01097. 00
2.97609137E-1j 1.0;01:00,000-00
CU"NLATTV7 '-Y7rflTT'" TI"F =
2.05721196E-10
1.00000000E-00
1.00009000E 00 -3.83232679E-10
.i.77096032F-63
9.'06229039E-01
5.100 SITCONOS
THESE ARE FINAL ANSWERS
9.61950004E-01
1.00000000E 00
1.01499259E-04
38
SYMBOLS
The proportion of the ith pure component
Vector of means of parameters of interest
The variance-covariance matrix
The inverse of the variance-covariance matrix
Summation symbol
x
The observation vector
Estimator of Ili
Estimator of