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Classical and Fuzzy Principal Component Analysis of Some Environmental Samples Concerning Pollution with Heavy Metals
COSTEL SÂRBU
Department of Chemsitry, Babeş-Bolyai University Cluj-Napoca ROMANIA
Principal Component Analysis
Principal component analysis (PCA) is a favorite tool in chemometrics for data compression and information extraction. PCA finds linear combinations of the original measurement variables that describe the significant variations in the data. However, it is well-known that PCA, as with any other multivariate statistical method, is sensitive to outliers, missing data, and poor linear correlation between variables due to poorly distributed variables. As a result data transformations have a large impact upon PCA. In this regard one of the most powerful approach to improve PCA appears to be the fuzzification of the matrix data, thus diminishing the influence of the outliers. Hier, we discuss and apply two robust fuzzy PCA algorithms (FPCA-1 and FPCA-o)
Soft Computing Methods
SoftComputing
Fuzzy LogicFuzzy Sets
PCA, PCR, PLS, ANN
Genetic Algorithms
Rough Sets
Chaos Theory
ApproximateReasoning
What is Soft ComputingWhat is Soft Computing ? ?
Soft Computing is a collection of methodologies (working synergistically, not competitively) which, in one form or another, reflect its guiding principle:
Exploit the tolerance for imprecision, uncertainty, approximate reasoning and partial truth to achieve tractability, robustness, and close resemblance with human like decision making.
Provides flexible information processing capability for representation and evaluation of various real life ambiguous and uncertain situations. Real World Computing
It may be argued that it is soft computing rather than hard computing that should be viewed as the foundation for Artificial Intelligence (AI).
Soft Computing vs Hard Computing
Hard computing requires programs to be written; soft computing can evolve its own programs
Hard computing uses two-valued logic; soft computing can use multivalued or fuzzy logic
Hard computing is deterministic; soft computing incorporates stochasticity
Hard computing requires exact input data; soft computing can deal with ambiguous and noisy data
Hard computing is strictly sequential; soft computing allows parallel computations
Hard computing produces precise answers; soft computing can yield approximate answers
In 1965* Zadeh published his seminal work "Fuzzy Sets" which described the mathematics of Fuzzy Set Theory, and by extension Fuzzy Logic.
It deals with the uncertainty and fuzziness arising from interrelated humanistic types of phenomena such subjectivity, thinking, reasoning, cognition, and perception. This type of uncertainty is characterized by structure that lack sharp boundaries. This approach provides a way to translate a linguistic model of the human thinking process into a mathematical framework for developing the computer algorithms for computerized decision-making processes.
*L. A. ZADEH, Fuzzy Sets, Information Control, 1965, 8, 338-353.
Fuzzy Sets and Fuzzy Logic
Fuzzy Sets Theory
A Fuzzy Set is a generalized set to which objects can belongs with various degrees (grades) of memberships over the interval [0,1].
Fuzzy systems are processes that are too complex to be modeled by using conventional mathematical methods.
In general, fuzziness describes objects or processes that are not amenable to precise definition or precise measurement. Thus, fuzzy processes can be defined as processes that are vaguely defined and have some uncertainty in their description. The data arising from fuzzy systems are in general, soft, with no precise boundaries.
Lotfi A. Zadeh betwen Orient and Occident
The Impact of Application of Fuzzy Sets Theory in Science and Technical Fields
“In 1999, Japan exported products at a total of $35 billion that use Fuzzy Logic or NeuroFuzzy. The remarkable fact that an emerging key technology in Asia and Europe went unnoticed by the U.S. public until recently, combined with its unusual name and revolutionary concept has led to a controversial discussion among engineers.”
Constantine von Altrock Inform Software Corp., Germany
Reasoning Styles in China and West
China WestPrinciple of ChangeReality is a dynamical, constantly-changing process. The concepts that reflect reality must be subjective, active, flexible.
Law of IdentityEverything is what it is. Thus it is a necessary fact that A equals A, no matter what A is.
Principle of ContradictionReality is full of contradictions and never clear-cut or precise. Opposites coexist in harmony with one another, opposed but connected
Law of NoncontradictionNo statement can be both true and false.
Principle of RelationshipTo know something completely, it is necessary to know its relations, what it affects and what affects it.
Law of the Excluded MiddleEvery statement is either true or false. There is no middle term.
School of Athens
Fuzziness in Everyday World
John is tall; Temperature is hot; Mr. B. G. is young (the paradox of Mr. B.G.); The girl next door is prettty; The Romanian Leu is getting relatively strong; The people living close to Bucharest; My car is slow, your car is fast;
Fuzziness in Chemistry
Water is an acid; Germanium is a metal; Those drugs are very effective; Varying peaks in chromatograms; Varying signal heights in spectra from the
same substance; Varying patterns in QSAR pattern recognition
studies;
Fuzziness in Everyday World(Orient versus Occident)
Fuzziness in Everyday World(Fuzzy girl-students in chemsitry)
Characteristic Function in the Case of Crisp Sets and Fuzzy Sets Respectively
P: X {0,1}
P(x) = 1 if x X
P(x) = 0 if x X
A : X [0,1]
A = {X, A(x)} if x X
Girl-Student Membership Function for “Young”
xif
xifx
xif
xS
400
402515
40251
Mr. B. G. Membership Function for “Young”
xif
xifx
xif
xB
700
704030
70401
Generalized Fuzzy c-Means Algorithm
njci
xA
xxA
L
LxdLxd
xCxA
LxdxALPJ
n
j
ji
n
j
jji
i
c
kkj
ij
jj
i
c
i
n
j
ijji
,...,1 ;,...,1
)(
)(
;
),(),(
)()(
),())((),(
1
2
1
2
12
2
1 1
22
Fuzzy 1-Line Regression Algorithm
njci
xA
xxA
uvLLxd
xA
xALxdxALPJ
n
j
ji
n
j
jji
j
ji
c
i
n
j
n
j
jijji
..., ,1 ; ..., ,1
)(
)(
),( ;),(
1
1)(
1.))((),())((),,(
1
2
1
2
2
1 1 1
222
Fuzzy Principal Component Analysis Algorithm
1. Determine the best value of . For this, loop with between 0 and 1. For
each iterative value of minimize the objective function above, and,
with the optimal membership degrees A(xj), compute the largest
eigenvalue of the matrix C given below. Select the optimal value of α
according to the maximal eigenvalue.
n
j
j
i
n
jljlkjk
j
i
kl
xA
xxxxxAC
1
2
1
2
)(
))(()(
Fuzzy Approaches
Fuzzy divisive hierarchical clustering;
Fuzzy horizontal clustering;
Fuzzy cross-clustering;
Fuzzy robust regression;
Fuzzy robust estimation of mean and spread
Data Set 1
The data collection was performed in the northern part of Romanian Carpathians Mountains : the western part of Bistriţa Mountains (b), the south-western part of Maramureş Mountains (m) and the north-western part of Igniş-Oaş Mountains (i), according to standardized methods for sampling, sample preparation and analysis. Thirteen different soil ion concentration were checked: lead, copper, manganese, zinc, nickel, cobalt, chromium, cadmium, calcium, magnesium, potassium, iron and aluminum
Eigenvalue and Proportion Considering the First Five Principal Components for PCA and FPCA
PCs
PCA FPCA-1 FPCA-o
Eigen-value
Prop.%
Cum.Prop.
%
Eigen-value
Prop.%
Cum.Prop.
%
Eigen-value
Prop.%
Cum.Prop.
%
1 5.639 43.37 43.37 3.161 48.15 48.15 3.161 62.78 62.78
2 1.826 14.04 57.42 0.982 14.96 63.11 0.724 14.38 77.14
3 1.403 10.79 68.22 0.703 10.71 73.82 0.417 8.28 85.44
4 1.308 10.06 78.28 0.554 8.44 82.26 0.208 4.77 89.57
5 0.801 6.16 84.44 0.299 4.56 86.82 0.240 4.13 94.34
Eigenvectors Corresponding to the First Four Principal Components for PCA and FPCA
PCA FPCA-1 FPCA-o
PC1 PC2 PC3 PC4 FPC1 FPC2 FPC3 FPC4 FPC1 FPC2 FPC3 FPC4
Pb -0.065 0.451 0.539 -0.165 -0.019 0.045 0.131 0.403 -0.019 -0.025 -0.589 -0.089
Cu 0.277 0.030 -0.004 -0.457 0.391 -0.415 0.419 0.046 0.391 0.341 -0.086 -0.416
Mn 0.265 0.251 -0.340 0.206 0.409 0.260 -0.477 -0.144 0.409 -0.205 0.127 0.481
Zn 0.311 0.372 -0.124 -0.119 0.470 0.196 0.114 0.186 0.470 -0.179 -0.164 -0.081
Ni 0.402 -0.105 0.111 -0.046 0.300 -0.221 0.035 0.019 0.299 0.222 -0.006 -0.090
Co 0.397 0.091 -0.139 0.078 0.404 0.079 -0.112 -0.086 0.404 -0.061 0.090 0.094
Cr 0.362 -0.159 0.206 -0.097 0.240 -0.341 0.022 0.043 0.240 0.317 -0.003 -0.100
Cd -0.058 0.585 0.345 0.032 0.013 0.296 0.034 0.809 0.013 -0.234 -0.743 0.094
Ca 0.175 0.066 0.088 0.609 0.127 0.041 -0.519 0.058 0.127 0.058 -0.041 0.607
Mg 0.380 -0.095 0.201 0.136 0.255 -0.183 -0.190 0.124 0.255 0.230 -0.059 0.148
K 0.311 -0.245 0.309 0.072 0.049 -0.228 -0.007 0.043 0.049 0.219 -0.016 -0.044
Fe 0.101 -0.063 -0.095 -0.541 0.111 -0.072 0.170 -0.038 0.111 0.012 0.014 -0.177
Al 0.121 0.359 -0.481 -0.027 0.226 0.607 0.463 -0.302 0.226 -0.704 0.192 -0.349
Loading Plot PC1-PC2-PC3(PCA and FPCA-1)
Co
NiM gCr
ZnK
M n
CuCa
A lFe
CdP b
Z n
M n
Co
Cu
N i
M g
A l
Cr
Ca
F e
K
CdP b
Loading Plot PC1-PC2-PC3(PCA and FPCA-o)
Zn
M nCu CoNiM g
Cr
A lCaFeK
Cd
P b
Co
NiM gCr
ZnK
M n
CuCa
A lFe
CdP b
Score Plot PC1-PC2(PCA and FPCA-1)
mmmmmm m
mm
mmmm
m
mm
mmm
m
m
m
m
m
mm
mm
mm
mmmm
mm
i
i
i
iii
i
i
iiii
i
iii i
i
i i
i
iiiii
i
iiii
i
iiii
i
i
i ii
iii
i
i
iiii bbbb
bbbbb
bbb bbb bb
bbb b
bbb b bb
bb b b
b bb
b
b bb
b
- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2
P C 1 : 4 3 .3 8 %
- 5
- 4
- 3
- 2
- 1
0
1
2
3
4
5
6
7
8
PC 2: 14.05%
m
mm
mmm
m mm
m m m m m
m
mmm m m
m mm
mm
m
m
m
m mmmmm
m mi
ii iii
i ii iii
ii i
i i i
i i
i ii iii
i i
iii
iiii i
i
ii i i
iii
ii
ii iibbbb
bbb
b
b
b
b
b bbb
b
bbb b b
b
bbb
b b
bb
b b b b b
b
b
b
b
b
- 4 - 2 0 2 4 6 8
P C 1 :4 8 .1 5 %
- 4
- 3
- 2
- 1
0
1
2
3
4
PC 2:14.96%
Score Plot PC1-PC3(PCA and FPCA-1)
mmmmmm m mm
m mmm m m
m
mmm m
m m
m
mm m
mm
m m
mmmm m
m
i
i
i
iii
i
i
iiii
i
iii i
i
i i
i
iiiii
i
iii i
iiiii
i
ii ii
iii
ii
iiii bbb
b
bbbbb
bbb bbb b
bbbb
bb
bbb
bb
bb
bb
b bb
b
b b b b
- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2
P C 1 : 4 3 .3 8 %
-6
-4
-2
0
2
4
6
8
1 0
PC 3: 10.79%
m
mm
mmm
mm
mm
m m m m
m
m
mm
m mmmmm
m m
m
mm
m
mm mm m m
i
ii iii
i i
ii
ii
ii i
ii
i
ii
i ii iii
ii i
i
i
i
ii ii
i
i
i i
i
iiii
i
i
iii b
bbb
bbbb
b
b
b
bbbbb
bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b b
b
b
b
b b
-4 -2 0 2 4 6 8
P C 1 : 4 8 .1 5 %
-3
-2
-1
0
1
2
3
PC 3: 10.71%
Score Plot PC1-PC4(PCA and FPCA-1)
mmm
mmm m mmmmmm m
m
mmmm mm m
mmmm m
m m mmmmm m m
iiiiiii i
i iiii i i i i ii i
i i i iii
i i iii
i iiii
i
ii iiiiii
iiiii bbbb bb bbb
b
bb bbb b
bbb bb
b
bb b
b
b
b
b
b
b b bb
b
b
bb
b
- 8 -6 -4 -2 0 2 4 6 8 1 0 1 2
P C 1 : 4 3 .3 8 %
-1 0
-8
-6
-4
-2
0
2
4
6
8
PC 4: 10.06%
mmmmm
m
mm
m
mm
m m mm
mmm
mm
m
m
m
m m m
mm
m m
m
m
m
m
m
m
i
i
i
iii
i
i
i
iii
i
i
i i i
i
i i
i
i
i
iii
i
i ii
i
i
ii i i
i
i
i ii
iii
i
iii i
ibbbb
bbb bbb
b
b bbb b
b
bb b bb
bb b b b
b
bb
b b bb
b
b b b b
-4 -2 0 2 4 6 8
P C 1 : 4 8 .1 5 %
-3
-2
-1
0
1
2
3
4
5
6
PC 4: 8.44%
Score Plot PC2-PC3(PCA and FPCA-1)
mmmmm
mm mm
m mmm m m
m
m mm m
m m
m
mmm
mm
m m
mm mm m
m
i
i
i
ii i
i
i
iiii
i
iiii
i
ii
i
ii
iii
i
iiii
iiiii
i
iii i
iii
ii
iiiibbb
b
bbb
bb
b bbbbbbb
bbbb
b
bbbb
b
bbbb
bbb
b
bb b b
- 5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
P C 2 : 1 4 .0 5 %
-6
-4
-2
0
2
4
6
8
1 0
PC 3: 10.79%
m
mm
mmm
mm
mmmmmm
m
m
mmmm m
mmm
mm
m
mmm
mm mmmm
i
iiiii
ii
ii
i i
iii
ii
i
ii
i iiiii
ii i
i
i
i
ii ii
i
i
ii
i
iii
i
i
i
iiib
bbb
bbb
bb
b
b
bb bb
b
bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b b
b
b
b
bb
- 4 -3 -2 -1 0 1 2 3 4
P C 2 : 1 4 .9 6 %
-3
-2
-1
0
1
2
3
PC 3:10.71%
Score Plot PC2-PC4(PCA and FPCA-1)
mmmmm mmmm
mmmm m
m
mm mm m m m
mmm
m mm m m
mm mm m m
iiiii i ii
iiii iii iii
iiiiiiii
ii ii
i
iiiiii
iii iiii i
iiii ibbbbbbb bb
b
bbbbbb
bbb bb
b
bbb
b
b
b
b
b
b bb b
b
b
bb
b
- 5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
P C 2 : 1 4 .0 5 %
-1 0
-8
-6
-4
-2
0
2
4
6
8
PC 4: 10.06%
m mmm m
m
mm
m
mmmmm
m
mmm
mm
m
m
m
m mm
mm
mm
m
m
m
m
m
m
i
i
i
ii i
i
i
i
ii i
i
i
i ii
i
ii
i
i
i
iii
i
i ii
i
i
ii ii
i
i
iii
iii
i
iii i
ibbbb
bbb bbb
b
bb bbb
b
bbbbb
bbbbb
b
bb
bbb b
b
bbbb
- 4 -3 -2 -1 0 1 2 3 4
P C 2 :1 4 .9 6 %
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
Score Plot PC3-PC4(PCA and FPCA-1)
mmmmmmmm mmmmmm
m
m mmmmmm
mmmmmmmm
mmmmmm
iiiiii ii
iiii iiiiii
iiiiiiii
iiii
i
iiiii
i
iii iiii i
iiii ibbbb bbbb b
b
bbbbbb
bbbbb
b
bb b
b
b
b
b
b
bbbb
b
b
bbb
-6 -4 -2 0 2 4 6 8 1 0
P C 3 : 1 0 .7 9 %
-1 0
-8
-6
-4
-2
0
2
4
6
8
PC 4: 10.06%
mm mmm
m
mm
m
mmmmm
m
mmm
mm
m
m
m
m mm
mm
m m
m
m
m
m
m
m
i
i
i
ii i
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i
i
i i i
i
i
iii
i
ii
i
i
i
ii i
i
i ii
i
i
iiii
i
i
iii
iii
i
iiii
ibbb b
bb b b bb
b
b b bb b
b
b b b bb
bb b b b
b
bb
b bbb
b
b b bb
- 3 -2 -1 0 1 2 3
P C 3 :1 0 .7 1 %
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
Score Plot PC1-PC2(FPCA-1 and FPCA-o)
m
mmmmm
m mm
m m m mm
m
mmm m m
m mm
mm
mm
m
m mmmmm m m
i
ii iiii i
i iii
ii i
i i ii i
iii iii
ii
ii
i
iiii i
i
ii i i
iiiii
ii ii
bbbb
bbb
b
b
b
bb b
bbb
bbb b bbb
bb b bb
b
b b b bb
b
b
b
b
b
- 4 -2 0 2 4 6 8
P C 1 :6 2 .7 8 %
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.38%m
mm
mmm
m mm
m m m m m
m
mmm m m
m mm
mm
m
m
m
m mmmmm
m mi
ii iii
i ii iii
ii i
i i i
i i
i ii iii
i i
iii
iiii i
i
ii ii
iii
ii
ii iibbbb
bbb
b
b
b
b
b bbb
b
bbb b b
b
bbb
b b
bb
b b b b b
b
b
b
b
b
-4 -2 0 2 4 6 8
P C 1 :4 8 .1 5 %
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.96%
Score Plot PC1-PC3(FPCA-1 and FPCA-o)
m
mm
mmm
mm
mm
m m m m
m
m
mm
m mmmmm
m m
m
mm
m
mm mm m m
i
ii iii
i i
ii
ii
ii i
ii
i
ii
i ii iii
ii i
i
i
i
ii ii
i
i
i i
i
iiii
i
i
iii b
bbb
bbbb
b
b
b
bbbbb
bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b b
b
b
b
b b
-4 -2 0 2 4 6 8
P C 1 : 4 8 .1 5 %
-3
-2
-1
0
1
2
3
PC 3: 10.71%
mmmmm
mm
m
m
mm
m m mm
mmm
mm
m
m
m
m m m
mm
m m
m
m
m
m
mm
i
i
i
iii
i
i
i
iii
i
i
i i i
i
ii
i
i
i
iii
i
i ii
i
i
ii i i
i
i
i ii
iii
i
iii i
ibbbb bbb bbb
b
b bbb b
b
bb b b bb
b b b b
b
bb b b b b
b
b b b b
-4 -2 0 2 4 6 8
P C 1 :6 2 .7 8 %
-7
-6
-5
-4
-3
-2
-1
0
1
2
PC 3:8.28%
Score Plot PC1-PC4(FPCA-1 and FPCA-o)
mmmmm
m
mm
m
mm
m m mm
mmm
mm
m
m
m
m m m
mm
m m
m
m
m
m
m
m
i
i
i
iii
i
i
i
iii
i
i
i i i
i
ii
i
i
i
iii
i
ii
ii
i
ii i i
i
i
i ii
iii
i
ii
i i
ibbbb
bbb bbb
b
b bbb b
b
bb b bb
bb b b b
b
bb
b b bb
b
b b b b
-4 -2 0 2 4 6 8
P C 1 : 4 8 .1 5 %
-3
-2
-1
0
1
2
3
4
5
6
PC 4: 8.44%
m
mm
mmm
mm
mmm m m m
m
m
mmm m
mm
mm
mm
m
m mm
mm mm m m
i
iiiii
ii
ii
ii
ii iii
iii
i ii iii
i i i
ii
iii ii
i
ii i
i
iii
i
i
i
iii bbb
bbbb b
b
b
b
bbbb
b
bbb
b
b
b
b
b
b
b
b
b
b
b
bb
b b
b
b
b
bb
-4 -2 0 2 4 6 8
P C 1 :6 2 .7 8 %
-3
-2
-1
0
1
2
3
4
PC 4:4.77%
Score Plot PC2-PC3(FPCA-1 and FPCA-o)
m
mm
mmm
mm
mmmmmm
m
m
mmmm m
mmm
mm
m
mmm
mm mmmm
i
iiiii
ii
ii
i i
iii
ii
i
ii
i iiiii
ii i
i
i
i
ii ii
i
i
ii
i
iii
i
i
i
iiib
bbb
bbb
bb
b
b
bb bb
b
bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b b
b
b
b
bb
- 4 -3 -2 -1 0 1 2 3 4
P C 2 : 1 4 .9 6 %
-3
-2
-1
0
1
2
3
PC 3:10.71%
mmmmm
mmm
m
mmmmm
m
mm
m
mm
m
m
m
mm m
m m
mm
m
m
m
m
mm
i
i
i
iii
i
i
i
iii
i
i
iii
i
ii
i
i
i
i ii
i
iii
i
i
iiii
i
i
iii
iii
i
iiii
ibb bb bbbb bb
b
bbbb b
b
bbbb bb
b bbb
b
bbbbb
b
b
b b b b
- 4 -3 -2 -1 0 1 2 3 4
P C 2 :1 4 .3 8 %
-7
-6
-5
-4
-3
-2
-1
0
1
2
PC 3:8.28%
Score Plot PC2-PC4(FPCA-1 and FPCA-o)
m mmm m
m
mm
m
mmmmm
m
mmm
mm
m
m
m
m mm
mm
mm
m
m
m
m
m
m
i
i
i
ii i
i
i
i
ii i
i
i
i ii
i
ii
i
i
i
iii
i
i ii
i
i
ii ii
i
i
iii
iii
i
iii i
ibbbb
bbb bbb
b
bb bbb
b
bbbbb
bbbbb
b
bb
bbb b
b
bbbb
- 4 -3 -2 -1 0 1 2 3 4
P C 2 :1 4 .9 6 %
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
m
mm
mmm
mm
mmmmmm
m
m
mmmm
mm
mm
mm
m
mmm
mmm mmm
i
ii
i ii
ii
ii
ii
i iiiii
ii
iiii ii
iii
ii
iiii
ii
iii
i
iii
i
i
i
ii i bb b
bbbbb
b
b
b
bbbbb
bbbb
b
b
b
b
b
b
b
b
b
b
bbbb
b
b
b
bb
-4 -3 -2 -1 0 1 2 3 4
P C 2 :1 4 .3 8
-3
-2
-1
0
1
2
3
4
PC 4:4.77%
Score Plot PC3-PC4(FPCA-1 and FPCA-o)
mm mmm
m
mm
m
mmmmm
m
mmm
mm
m
m
m
m mm
mm
m m
m
m
m
m
m
m
i
i
i
ii i
i
i
i
i i i
i
i
iii
i
ii
i
i
i
ii i
i
i ii
i
i
iiii
i
i
iii
iii
i
iiii
ibbb b
bb b b bb
b
b b bb b
b
b b b bb
bb b b b
b
bb
b bbb
b
b b bb
- 3 -2 -1 0 1 2 3
P C 3 :1 0 .7 1 %
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
m
mm
mmm
mm
m mm mmm
m
m
mmmm
mm
mm
mm
m
mmm
m mm mm m
i
ii
i ii
ii
ii
i i
i i iii
iii
i ii i i i
i i i
ii
ii
i ii
i
ii i
i
iii
i
i
i
iii bbb
bbbbb
b
b
b
bbbbb
b bbb
b
b
b
b
b
b
b
b
b
b
bbbb
b
b
b
bb
- 7 -6 -5 -4 -3 -2 -1 0 1 2
P C 3 :8 .2 8 %
-3
-2
-1
0
1
2
3
4
PC 4:4.77%
Data Set 2
The data set consists of 234 differently polluted sampling locations (East Germany) characterized by four variables: soil lead content (sPb), plant lead content (pPb), traffic density (tD), and distance from the road (dR). As an additional feature a classification number resulting from the a-priori knowledge of the loading situation at the particular sampling location according to the following list is given:
Loading situation Class number Samples number
Unpolluted 1 175
Moderately polluted 2 40
Polluted 3 10
Extremely polluted 4 9
Eigenvalue and Proportion Considering the First Five Principal Components for PCA and FPCA
PCs
PCA FPCA-1 FPCA-o
Eigen-value
Prop.%
Cum.Prop.
%
Eigen-value
Prop.%
Cum.Prop.
%
Eigen-value
Prop.%
Cum.Prop.%
1 1.8792 46.98 46.98 1.3269 50.75 50.75 1.3269 53.57 53.57
2 0.9788 24.47 71.45 0.7349 28.10 78.85 0.6862 27.71 81.28
3 0.6817 17.04 88.49 0.3452 13.20 92.05 0.3441 13.89 95.17
4 0.4604 11.51 100.00 0.2078 7.95 100.00 0.1195 4.83 100.00
Eigenvectors Corresponding to the First Three Principal Components for PCA and FPCA
PCA FPCA-1 FPCA-o
PC1 PC2 PC3 PC4 FPC1 FPC2 FPC3 FPC4 FPC1 FPC2 FPC3 FPC4
pPb -0.560 -0.153 0.609 -0.540 -0.356 0.085 -0.106 -0.924 -0.356 -0.101 -0.126 0.920
sPb -0.528 0.195 -0.749 -0.350 -0.425 0.078 -0.860 0.269 -0.425 -0.045 0.903 -0.046
dT -0.399 -0.772 -0.141 0.474 -0.356 0.862 0.310 0.181 -0.356 -0.868 -0.225 -0.264
dR 0.497 -0.586 -0.223 -0.600 0.752 0.493 -0.390 -0.200 0.752 -0.485 0.344 0.285
Loading Plot PC1-PC2-PC3(PCA and FPCA-1)
D R
d Ts P b
p P bD R
d T
p P b
s P b
Loading Plot PC1-PC2-PC3(FPCA-1 and FPCA-o)
D R
p P b
d T
s P b
D R
d T
p P b
s P b
Score Plot PC1-PC2(PCA and FPCA-1)
11
1
1
11
1
1
1
1
1
1
1
1
2
1
1
1
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
31
1
11
1
1
31
1
1
1222
2
1
1
1
1
1
1
1
3 1
1
1
22
1
1
1
1
1
2 1
1
1
1142
1
1
11
1
1
1
1
1
2 1
1
1
1
1
1
22
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
14 1
1
1
1
1
1
1
1
21
1
1 1
1
1
2
341
1
1
1
1
2
1
1
1323
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
41
1
22
1
1
2
1
1
1
1
1
2
1
1
1
1
1
1
2
1
2
1
1
1
4
2
1
1
32
1
1
1
1
3 1
1
1
2
2
14
2
1
1
2
4
2
3
2
1
4 3
1
422
1
1
2
2
2
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
P C 1 : 4 6 .9 6 %
-3
-2
-1
0
1
2
3
PC 2;24.47%
11
1
1
11
1
1
1
1
1
1
1
1
21 1
1
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
21
1
31
1
11
1
1
3 1
1
1
1222
2
1
1
1
1
1
1
1
3 1
1
1
221
1
1
1
1
2 1
1
1
11
42
1
1
1 1
1
1
1 1
1
2 1
1
1
11
1
221
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
41
1
11
1
1
1
1
2
1
1
1 1
1
1
23
41
1
11
1
21
1
13
23
2
2
21
1
11
1
11
1
1
1
1
1
1
1
41
1
22
1
1
21
1
11
1
2
1
1
11
1
1
21
2
1
1
1
421
1
3
2
1
1
1
1
31
1
1
22
1
4
2
1
1
2
4232
1
43
1
422
11 22
2
-5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 0 .7 5 %
-3
-2
-1
0
1
2
3
PC 2:28.10%
Score Plot PC1-PC3(PCA and FPCA-1)
11 11
1
1 11
1 11
1 11
2
1
11
2 211 111 1
1
11
11 1
11
1 11 1 11 1 1
2
1 1
3
11
111
1
3
11
1
1222 2
11 1
11 1
1
31 1
12
2
1 1
11
1
21
11
1142
11
11 1
1
1
11
2 11
11
11
22 1 1
1
1 11
1 1
111
1
11 1
11
4
1
111
11 1 1
2
111 1 11
2
34
1
11
112
11
1
3
23
2 2
2
11
1
11
1
1 11 11
1 11
4
11
2
2
11
2
11
1
11
2 11
11 112
1 21 11
4
2
11
3
2 11 1 1
31 1
1
2
2
1
4
2 11
2
4 2
32
1
4
3
1
4
22
1
1
2
2
2
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
P C 1 :4 6 .9 6 %
-6
-4
-2
0
2
4
6
PC 3:17.04%
11
1
1
1
1 1
1
11
1
11
1
2
1
1
1
2 2
1
11
11 1
1
11
1
11
1
1
11
11
1
11
1
2 1
1
31
1
111
1
31
1
1
1
222 2
1
1 1
1
1 1
13
11
1
2
2
11
11
1
2 11
1
11
4
2
1
1
1
11
1
1
1
1
21
1
1
1
1
1
2
2
1
11
11
1
1 1
1
11
1
1
11
11
41
11
1
11
1
1
2
11
1
1 1
1
2
3
4
1
1
1
1
1
2
1
11
3
23 2
2
2
1
1
1
1
1
1
1
11
1
1
11
1
41
1
2
2
1
1
2
1
1
1
1
12
1
1
11
1
1
2
1
2
1 1
1
4
2
1
1
3
21
1 1
13
11
1
2
2
14
21
1
2
4
2
3
2
14
31
4
22
1
1
2
2
2
- 5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 0 .7 5 %
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
Score Plot PC1-PC4(PCA and FPCA-1)
11 11
1 11
11 1 11 1 1
2
1
1 12
2 1
1 1 11 111 11
11 11 1
11 1 11 1
12
11
3
1 111 1 1
3
1 1 11222 2
11 1 11 1 1
3
1 1 122 1 11 1
121 1 111
4
2
11
1 1 1 1
1
1 1211 1
11 122
11
1 11 11
111
11 11 1
11
4
1
11 1
11 112
11
1 1 1 123
4
1
11 1 12 1
1
1
32
32
2
2 1 11
1 11 1 11 1 1
1 1 1
4
1 12
2 1 12 1 1
1
1 12
1 111 1 121
2
1 1 1
4
2
1 13
2 11 1
1
3
1 1 12
21
4
2 1 12
4
232 1
4
3
1
4
22
1
12
2 2
-5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 0 .7 5 %
-1 0
-8
-6
-4
-2
0
2
4
PC 4:7.95%
11
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
2
2
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
21
1
3
1
1
11
1
1
3
1
1
1
1
2222
1
1
1
1
1
1
13
1
1
1
22
1
1
11
1
21
1
1
11
4
21
1
11
1
1
1
1
1
21
1
1
1 1
1
2
2
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
4
1 1
1 1
1
1 1
1
2
1
1
11
1
1
2
3
4
1 1
1
1
12
1
1
1
3
23 2
2
2
1
1
11
1
11
1
1 1
1
1
1
14
1
12
2
1
12
1
1
1 1
1
2 1
1
1
1
1
1
2 1
2
11
1
4
2
1
1
32
1
11
1
3
1
1
12
2
1
4
2 1
12
4
2
3
2
1
4
31
4
22 1
12
2
2
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
P C 1 :4 6 .9 6 %
-4
-3
-2
-1
0
1
2
PC 4:11.51%
Score Plot PC2-PC3(PCA and FPCA-1)
11
1
1
1
1 1
1
11
1
11
1
2
1
1
1
2 2
1
11
11 1
1
11
1
11
1
1
11
11
1
11
1
2 1
1
31
1
111
1
31
1
1
1
222 2
1
1 1
1
1 1
13
11
1
2
2
11
11
1
211
1
11
4
2
1
1
1
11
1
1
1
1
21
1
1
1
1
1
2
2
1
11
11
1
1 1
1
11
1
1
11
11
41
11
1
11
1
1
2
11
1
1 1
1
2
3
4
1
1
1
1
1
2
1
11
3
23 2
2
2
1
1
1
1
1
1
1
11
1
1
11
1
41
1
2
2
1
1
2
1
1
1
1
12
1
1
11
1
1
2
1
2
1 1
1
4
2
1
1
3
21
1 1
13
11
1
2
2
14
21
1
2
4
2
3
2
143
1
4
22
1
1
2
2
2
-3 -2 -1 0 1 2 3
P C 2 :2 8 .1 0 %
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
1111
1
111
111
111
2
1
11
221 111 111
11
111
11
11111111
2
11
3
11
111
1
3
11
1
12222
111
111
1
311
1 2
2
11
11
1
21
11
1142
11
111
1
1
11
211
11
11
2 211
1
111
11
1 11
1
111
1 1
4
1
1 11
1 111
2
11 1111
2
34
1
11
11 2
11
1
3
23
22
2
11
1
11
1
11 111
111
4
11
2
2
11
2
11
1
11
211
1 111 2
12 111
4
2
11
3
21111
311
1
2
2
1
4
211
2
42
32
1
4
3
1
4
22
1
1
2
2
2
-3 -2 -1 0 1 2 3
P C 2 :2 4 .4 7 %
-6
-4
-2
0
2
4
6
PC 3:17.04%
Score Plot PC2-PC4(PCA and FPCA-1)
11 11
111
11 1 11 1 12
1
1 12
2 1
1 1 11 111 11
11 11 1
11 1 11 1
12
11
3
1 111 1 1
3
1 1 112 22 2
11 1 11 1 1
3
1 1 122 1 11 1 1
21 1 111
4
2
11
11 1 1
1
1 1211 1
11 122
11
111 111
1111 1
1 1 11
4
11
1 111 112
11
11 1 12 3
4
1
11 1 12 1
1
1
323
22
2 1 11
1 11 1 11 1 11 1 1
4
1 122 1 12 1 1
1
1 12
1 111 1 121
2
1 1 1
4
2
1 13
2 11 1 1
3
1 1 12
21
4
2 1 12
4
232 1
4
3
1
4
22
1
12
2 2
- 3 -2 -1 0 1 2 3
P C 2 :2 8 .1 0 %
-1 0
-8
-6
-4
-2
0
2
4
PC 4:7.95%
11
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
2
2
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
21
1
3
1
1
11
1
1
3
1
1
1
1
2222
1
1
1
1
1
1
13
1
1
1
22
1
1
11
1
21
1
1
11
4
21
1
11
1
1
1
1
1
21
1
1
11
1
2
2
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
4
11
11
1
11
1
2
1
1
11
1
1
2
3
4
11
1
1
12
1
1
1
3
232
2
2
1
1
11
1
11
1
11
1
1
1
14
1
12
2
1
1 2
1
1
11
1
21
1
1
1
1
1
21
2
11
1
4
2
1
1
32
1
11
1
3
1
1
1 2
2
1
4
21
1 2
4
2
3
2
1
4
31
4
221
12
2
2
- 3 - 2 - 1 0 1 2 3
P C 2 :2 4 .4 7 %
- 4
- 3
- 2
- 1
0
1
2
PC 4:11.51%
Score Plot PC3-PC4(PCA and FPCA-1)
11
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
2
2
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
21
1
3
1
1
11
1
1
3
1
1
1
1
22 22
1
1
1
1
1
1
13
1
1
1
22
1
1
11
1
21
1
1
11
4
21
1
11
1
1
1
1
1
21
1
1
1 1
1
2
2
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
4
1 1
1 1
1
11
1
2
1
1
111
1
2
3
4
1 1
1
1
12
1
1
1
3
232
2
2
1
1
11
1
11
1
11
1
1
1
14
1
12
2
1
12
1
1
1 1
1
21
1
1
1
1
1
2 1
2
11
1
4
2
1
1
321
11
1
3
1
1
12
2
1
4
21
12
4
2
3
2
1
4
31
4
221
12
2
2
-6 -4 -2 0 2 4 6
P C 3 :1 7 .0 4 %
-4
-3
-2
-1
0
1
2
PC 4:11.51%
1111
1 11
1111 1112
1
11221
111 1 11 111
111 11
111111
12
11
3
11 1111
3
111 122 22
1111111
3
11122 11 111
2111 11
4
2
11
1 111
1
11 2 111
111 22
11
1 111 11
111 11
1111
4
111 111 11 2
111 111
23
4
1
11 112 1
1
1
3232
2
2 111
111 11 1 11
111
4
112
2 112 11
1
11211 1 1112
12
111
4
2
113
2 1111
3
1112
21
4
2 112
4
2321
4
3
1
4
22
1
12
22
-5 -4 -3 -2 -1 0 1 2
P C 3 :1 3 .2 0 %
-1 0
-8
-6
-4
-2
0
2
4
PC 4:7.95%
Score Plot PC1-PC2(FPCA-1 and FPCA-o)
11
1
1
11
1
1
1
1
1
1
1
1
21 1
1
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
21
1
31
1
11
1
1
3 1
1
1
1222
2
1
1
1
1
1
1
1
3 1
1
1
221
1
1
1
1
2 1
1
1
11
42
1
1
1 1
1
1
1 1
1
2 1
1
1
11
1
221
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
41
1
11
1
1
1
1
2
1
1
1 1
1
1
23
41
1
11
1
21
1
13
23
2
2
21
1
11
1
11
1
1
1
1
1
1
1
41
1
22
1
1
21
1
11
1
2
1
1
11
1
1
21
2
1
1
1
421
1
3
2
1
1
1
1
31
1
1
22
1
4
2
1
1
2
4232
1
43
1
422
11 22
2
-5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 0 .7 5 %
-3
-2
-1
0
1
2
3
PC 2:28.10%
11
1
1
1 1
1
1
1
1
1
1
1
1
211
1
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
21
1
31
1
11
1
1
3 1
1
1
1222
2
1
1
1
1
1
1
1
3 1
1
1
221
1
1
1
1
2 1
1
1
11
42
1
1
1 1
1
1
11
1
2 1
1
1
1
1
1
221
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
4 1
1
1
1
1
1
1
1
2
1
1
1 1
1
1
23
41
1
11
1
21
1
13
232
2
21
1
11
1
11
1
1
1
1
1
1
1
41
1
22
1
1
21
1
1
1
1
21
1
11
1
1
2
1
2
1
1
1
42 1
1
3
2
1
1
1
1
31
1
1
22
14
2
1
1
2
4 232
1
4 3
1
422
11 2
2
2
- 5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 3 .5 7 %
-3
-2
-1
0
1
2
3
PC 2:27.71%
Score Plot PC1-PC3(FPCA-1 and FPCA-o)
11
1
1
1
1 1
1
11
1
11
1
2
1
1
1
2 2
1
11
11 1
1
11
1
11
1
1
11
11
1
11
1
2 1
1
31
1
111
1
31
1
1
1
222 2
1
1 1
1
1 1
13
11
1
2
2
11
11
1
2 11
1
11
4
2
1
1
1
11
1
1
1
1
21
1
1
1
1
1
2
2
1
11
11
1
1 1
1
11
1
1
11
11
41
11
1
11
1
1
2
11
1
1 1
1
2
3
4
1
1
1
1
1
2
1
11
3
23 2
2
2
1
1
1
1
1
1
1
11
1
1
11
1
41
1
2
2
1
1
2
1
1
1
1
12
1
1
11
1
1
2
1
2
1 1
1
4
2
1
1
3
21
1 1
13
11
1
2
2
14
21
1
2
4
2
3
2
14
3
1
4
22
1
1
2
2
2
- 5 - 4 - 3 - 2 - 1 0 1 2 3
P C 1 :5 0 .7 5 %
- 5
- 4
- 3
- 2
- 1
0
1
2
PC 3:13.20%
11 1
1
1
1 1
1
11
1
11
1
2
1
1
1
2 2
1
11
11 1
1
11
1
11
1
1
11
1 11
11
1
21
1
3
1
1
111
1
3 11
1
12
22 2
1
1 1
1
1 1
1
31
1
1
2
2
11
11
1
2 11
1
11
4
2
1
1
1
11
1
1
11
21
1
1
1
1
1
2
2
11
1
11
1
1 1
1
11
1
1
11
11
4
1
11
1
111
1
2
111
1 11
2
34 1
1
1
1
1
2
1
1
13
232
2
2
1
1
1
1
1
1
111
1
1
1 11
41
12
2
1
1
2
1
1
1
1
12 1
1
111
12
12
1 1
14
2
1
1
3
21
1 11
31
1
1
2
2
1
4
21
1
2
4
2
3
2
1
43
14
22
1
1
2
2
2
-5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 3 .5 7 %
-3
-2
-1
0
1
2
3
4
5
PC 3:13.89%
Score Plot PC1-PC4(FPCA-1 and FPCA-o)
11 11
1 11
11 1 11 1 1
2
1
1 12
2 1
1 111 1
11 11
11 11 1
11 1 11 1
12
11
3
1 111 1 1
3
1 1 11222 2
11 1 11 1 1
3
1 1 122 1 11 1
121 1 111
4
2
11
1 1 1 1
1
1 1211 1
11 122
11
1 11 11
111
11 11 1
11
4
1
11 1
11 112
11
1 1 1 123
4
1
11 1 12 1
1
1
32
32
2
2 1 11
1 11 1
11 1 11 1 1
4
1 12
2 1 12 1 1
1
1 12
1 111 1 121
2
1 1 1
4
2
1 13
2 11 1
1
3
1 1 12
21
4
2 1 12
4
232 1
4
3
1
4
22
1
12
2 2
-5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 0 .7 5 %
-1 0
-8
-6
-4
-2
0
2
4
PC 4:7.95%
11 11
1 11
11
1 11
112
11
122 1
1 11
1 11
1 11
11
11 1
1
11
1
1 11
2
1
1
3
11
11 11
3
1 1 11
2222
11 1
1
11
1
3
1 112
21
11 1
121 1
111
4
2
1
1
1 1 11
1 112 1
11
1 112
2
1
1
1 11
111 11
11 1
11
11
4
11
1 11
1 1
12 11
11 1
12
3
4
11
1 11
2
11
1
3
23
22
2 11
1 11
1 11
1 11
1 11
4
11
221 121 1
1
11
21
111
1 12 1
2
1 11
4
2 11
3
2 1
1 11
3
11
122
1
4
2 1 12
4
23
21
4
3
1
4
22
11
2
2 2
-5 -4 -3 -2 -1 0 1 2 3
P C 1 :5 3 .5 7 %
-2
-1
0
1
2
3
4
5
6
7
8
PC 4:4.83
Score Plot PC2-PC3(FPCA-1 and FPCA-o)
11
1
1
1
1 1
1
11
1
11
1
2
1
1
1
2 2
1
11
11 1
1
11
1
11
1
1
11
11
1
11
1
2 1
1
31
1
111
1
31
1
1
1
222 2
1
1 1
1
1 1
13
11
1
2
2
11
11
1
211
1
11
4
2
1
1
1
11
1
1
1
1
21
1
1
1
1
1
2
2
1
11
11
1
1 1
1
11
1
1
11
11
41
11
1
11
1
1
2
11
1
1 1
1
2
3
4
1
1
1
1
1
2
1
11
3
23 2
2
2
1
1
1
1
1
1
1
11
1
1
11
1
41
1
2
2
1
1
2
1
1
1
1
12
1
1
11
1
1
2
1
2
1 1
1
4
2
1
1
3
21
1 1
13
11
1
2
2
14
21
1
2
4
2
3
2
143
1
4
22
1
1
2
2
2
-3 -2 -1 0 1 2 3
P C 2 :2 8 .1 0 %
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
111
1
1
11
1
11
1
11
1
2
1
1
1
22
1
11
111
1
11
1
11
1
1
11
111
11
1
21
1
3
1
1
111
1
3 11
1
12222
1
11
1
11
1
31
1
1
2
2
11
11
1
211
1
11
4
2
1
1
1
11
1
1
11
21
1
1
1
1
1
2
2
11
1
11
1
11
1
11
1
1
11
1 1
4
1
1 1
1
1 11
1
2
11 1
111
2
341
1
1
1
1
2
1
1
13
232
2
2
1
1
1
1
1
1
11 1
1
1
111
41
12
2
1
1
2
1
1
1
1
121
1
1 11
12
12
11
1 4
2
1
1
3
21
111
31
1
1
2
2
1
4
21
1
2
4
2
3
2
1
43
14
22
1
1
2
2
2
-3 -2 -1 0 1 2 3
P C 2 :2 7 .7 1 %
-3
-2
-1
0
1
2
3
4
5
PC 3:13.89%
Score Plot PC2-PC4(FPCA-1 and FPCA-o)
11 11
111
11 1 11 1 12
1
1 12
2 1
1 1 11 111 11
11 11 1
11 1 11 1
12
11
3
1 111 1 1
3
1 1 112 22 2
11 1 11 1 1
3
1 1 122 1 11 1 1
21 1 111
4
2
11
11 1 1
1
1 1211 1
11 122
11
111 111
1111 1
1 1 11
4
11
1 111 112
11
11 1 12 3
4
1
11 1 12 1
1
1
323
22
2 1 11
1 11 1 11 1 11 1 1
4
1 122 1 12 1 1
1
1 12
1 111 1 121
2
1 1 1
4
2
1 13
2 11 1 1
3
1 1 12
21
4
2 1 12
4
232 1
4
3
1
4
22
1
12
2 2
- 3 -2 -1 0 1 2 3
P C 2 :2 8 .1 0 %
-1 0
-8
-6
-4
-2
0
2
4
PC 4:7.95%
1111
1 11
11
111
11 2
11
1 221
111
111
111
11
111
1
11
1
111
2
1
1
3
11
1111
3
1111222
21
111
11
1
3
111 2
21
111
1211
111
4
2
1
1
1111
111 21
11
111 2
2
1
1
111
1 111 1
111
11
11
4
11
111
11
1 211
111
12
3
4
11
111
2
11
1
3
23
22
211
111
111
111
111
4
11
2211 211
1
11
21
11 1
11 21
2
111
4
211
3
21
111
3
11
1 22
1
4
211 2
4
23
21
4
3
1
4
22
112
22
- 3 -2 -1 0 1 2 3
P C 2 :2 7 .7 1 %
-2
-1
0
1
2
3
4
5
6
7
8
PC 4:4.83%
Score Plot PC3-PC4(FPCA-1 and FPCA-o)
1111
1 11
1111 1112
1
11221
111 1 11 111
111 11
111111
12
11
3
11 1111
3
111 122 22
1111111
3
11122 11 111
2111 11
4
2
11
1 111
1
11 2 111
111 22
11
1 111 11
111 11
1111
4
111 111 11 2
111 111
23
4
1
11 112 1
1
1
3232
2
2 111
111 11 1 11
111
4
112
2 112 11
1
11211 1 1112
12
111
4
2
113
2 1111
3
1112
21
4
2 112
4
2321
4
3
1
4
22
1
12
22
-5 -4 -3 -2 -1 0 1 2
P C 3 :1 3 .2 0 %
-1 0
-8
-6
-4
-2
0
2
4
PC 4:7.95%
1 1 11
111
111 1
11
12
11
122 1
1 11
111
1 11
11
11 1
1
11
1
1 11
2
1
1
3
11
11 11
3
1 1 11
2 222 111
1
11
1
3
1 11 2
21
11 1
121 1
111
4
2
1
1
11 11
11121
11
11122
1
1
111
111 1 1
11 1
11
11
4
1111111
12 11
111
12
3
4
11
111
2
11
1
3
23
22
211
111
111
111
1 11
4
11
2 21 1 21 1
1
11
21
1111 1 21
2
111
4
211
3
21
111
3
11
1 22
1
4
21 1 2
4
23
21
4
3
1
4
2 2
11
2
2 2
-3 -2 -1 0 1 2 3 4 5
P C 3 :1 3 .8 9 %
-2
-1
0
1
2
3
4
5
6
7
8
PC 4:4.83%
Conclusions
FPCA algorithms achieved better results mainly because they are more compressible and robust than classical PCA
Applying FPCA algorithms it should be possible to explain some (many!) discrepancies, found in the literature, relating to PCA, PCR and PLS
Concluding Remark
“Are the Concepts of Chemistry all fuzzy?”
(The title of the Conference organized by Rouvray and Kirby, 1995)
If Yes, then Fuzzy Soft Computing could be one of the best solution for solving problems in chemistry!?
Chemistry
“In any branch of study of the natural world, the amount of actual science contained therein is directly proportional to the amount of mathematics used. Chemistry can under no circumstances be regarded as a science”
KANT
The responsibility for The responsibility for changechange … …lies within us. We must lies within us. We must begin with ourselves, begin with ourselves, teaching ourselves notteaching ourselves not to to close our minds close our minds prematurely to the novel, prematurely to the novel, the surprising, the the surprising, the seemingly radical.seemingly radical.
Alvin ToefflerAlvin Toeffler
The Bright Future of Chemometrics
