FUZZY SETS THEORY AND ITS APPLICATIONS IN ANALYTICAL …
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Transcript FUZZY SETS THEORY AND ITS APPLICATIONS IN ANALYTICAL …
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
[email protected]
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
Approximate
Reasoning
Fuzzy Logic
Fuzzy Sets
PCA, PCR,
PLS, ANN
Soft
Computing
Genetic
Algorithms
Chaos Theory
Rough Sets
What 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
Fuzzy Sets and Fuzzy Logic
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 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
West
Principle of Change
Law of Identity
Reality is a dynamical, constantly-changing
process. The concepts that reflect reality
must be subjective, active, flexible.
Everything is what it is. Thus it is a
necessary fact that A equals A, no matter
what A is.
Principle of Contradiction
Law of Noncontradiction
Reality is full of contradictions and never
clear-cut or precise. Opposites coexist in
harmony with one another, opposed but
connected
No statement can be both true and false.
Principle of Relationship
Law of the Excluded Middle
To know something completely, it is
necessary to know its relations, what it
affects and what affects it.
Every 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”
if
x 25
1
40 x
Sx
if 25 x 40
15
if
40 x
0
Mr. B. G. Membership Function for “Young”
x 40
1 if
70 x
Bx
if 40 x 70
30
70 x
0 if
Generalized Fuzzy c-Means Algorithm
J (P, L)
c
n
i 1
j 1
( Ai ( x j )) 2 d 2 ( x j , Li )
n
Ai ( x )
j
j
c
k 1
i 1,..., c ;
C (x )
;
2
j
i
d (x , L )
2
j
k
d (x , L )
j 1,..., n
L
i
j 1
n
Ai ( x j ) 2 x j
j 1
j
Ai ( x )
2
Fuzzy 1-Line Regression Algorithm
J ( P, L, ) ( Ai ( x )) d ( x , L ) ( A ( x )) .
1
i 1 j 1
j 1
c
n
n
j
Ai ( x j )
1
d 2 ( x j , L)
1
i 1, ..., c ;
j 1, ..., n
2
2
j
i
j
2
n
;
L ( v, u )
j 2 j
A
(
x
i )x
j 1
n
j 2
A
(
x
i )
j 1
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
a c c o r d i n g
t o
t h e
m
a x i m
a l
e i g e n
A ( x ) ( x x )( x x )
n
C
kl
v a l u e .
j 1
j
2
i
jk
k
jl
A ( x )
n
j 1
j
i
2
l
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
PCA
FPCA-1
FPCA-o
PCs Eigen- Prop.
value
%
Cum.
Prop.
%
Eigenvalue
Prop.
%
Cum.
Prop.
%
Eigen- Prop. Cum.
value
%
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
84.44
0.299
4.56
86.82
0.240
4.13 94.34
6.16
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)
Al
Pb
Cd
Zn
Zn
Ca
Mn
Fe
Al
Mg
K
Cr
Ni
Cu Co
CdPb
Fe
K
Ca
Co
Ni
Cr
Mn
Mg
Cu
Loading Plot PC1-PC2-PC3
(PCA and FPCA-o)
Pb
Cd
K
Zn
Ca
Mn
Fe
Al
Cr NiCu
Mg
Fe
Ca
Mg
K
Cr
Ni
Cu Co
Pb
Cd
CoMn
Zn
Al
Score Plot PC1-PC2
(PCA and FPCA-1)
4
8
7
6
i
b
4
2
i
i i
i iii
i i
3
m
ii
m
iiii
iii i i m
m
m
m i iiiii ii i im
m
i
bb
m
i iiiibiiim m
i iim
mm
b
b
bbb b b
m mb
m
mmm im
b b
bm bb m
b
m
mbbm
b
m
m
m
mmb b m b bb b
b b b
2
1
m
i
1
-1
b
b
PC 2:14.96%
PC 2: 14.05%
0
0
b
bb
bb
-2
-3
-1
-2
i
i
i
i ii
i
i
i
5
i i
i
i
i
ii i iiiiii iiiii i
i i i i ii i i
iiiii i ii
m
b
m
mmi i m
m
bbm
m mmm mm m
m bmb
m
m
mb m mm m
bm
m
b bmm
m
b
m
bb b b
m
m
bm b b
bbb
b b bb b
bm
b
b b
b
b
m
-6
-4
-2
0
2
PC 1: 43.38%
4
6
8
10
12
-4
-4
m
b
bbb
b
b
-3
-4
-5
-8
i
3
-2
0
2
PC 1:48.15%
4
6
b
8
Score Plot PC1-PC3
(PCA and FPCA-1)
3
10
b
8
2
6
b
i
1
4
2
PC 3: 10.79%
0
-2
b
b
b
bbb
0
PC 3: 10.71%
i
i
m ibm i
i
m
mm
bbimibi
bbm
b
m
b
mm
m
m
b
b
b mimmi bm
bb b b m
i i bm
mmm
b m
m m
bb b m
i imiiii ibiii m
b
m
m
bb
b
m
ii i iiiiiii ii ibi m
ii i
ii
ii i
m m
i
i
i i i
i
i i
b
b
ii i i b
ii
b iib b
b
i b b i
b bm
b
b
i
i ibiii
bi i
bi
ib
i bi mm
m
i
bb mm
i m
bbmmbmm
m
m
m
m
i
m
i mm
m
m
m
m
m
m
b
mm
m i
mb
m
m
mb
m
i
m
-1
i
ii
-2
-4
-6
-8
-6
-4
-2
0
2
PC 1: 43.38%
4
6
8
10
12
-3
-4
-2
bi
bb
i ib
ii
i
i
i
b
m
m
b
m
b
bbb
i
b
i
0
2
PC 1: 48.15%
4
6
8
Score Plot PC1-PC4
(PCA and FPCA-1)
8
6
6
5
4
i
b
mmb ii ii ibbb
mm
i
bbm
m
m
mm
m
ii iim mm
bibm
mm
im
i im
im
mm m
ii
i
i
im
i
i
i
b
i
i
i
m
b
i
m
bbmm
i iiibi b m
i bbm
i ib ibbbi i b b
b
b bb m
b
i
0
-2
i
i
i
4
iii
ii
2
b
m m
3
bbbb
b
2
1
PC 4: 8.44%
PC 4: 10.06%
b
0
-1
i
i i
i
i
m
i
i
m
b
m m b mi b
m
m
m
i
m m bm ibbm mi iibi ibi bb b b
m
bm mb b bm mbb i b b
bm
b m
mb m
m
mm
i i bi
mbm
b
b
b
i
bi
mi mi ii i
m i
i bi i
iii iii ii
ii
i ii
m
b
-4
-6
i
i
m
m
m
m
b
bbbbb b
-2
-8
-10
-8
-6
-4
-2
0
2
PC 1: 43.38%
4
6
8
10
12
-3
-4
-2
0
2
PC 1: 48.15%
4
6
8
Score Plot PC2-PC3
(PCA and FPCA-1)
3
10
b
8
i
2
6
b
b
i
1
4
i
2
PC 3: 10.79%
0
-2
i
i
i
i
ii
i
m
ii
0
i
PC 3:10.71%
bb b
bb m m
i
bbm
m
bbmbm
b
mm
bbm
m
b
b
b
m
b
b
mbm
bbm
bb i m i i
bmm
m
bbbbbm
m i mm m
mm
m
i ii
bimbbmibiim
m
i
iim
b
iii iiii i ii i i
i
ii
mi
-1
b
b
bb
b
-2
-4
-6
-5
-4
-3
-2
-1
0
1
2
PC 2: 14.05%
3
4
5
6
7
8
-3
-4
-3
i
ii i
b b
i
b b bb
i ii i i i
b
i
bb bmbb b b
i iii i
b
i
b
ii i
mb m
ib
m
b
bbmm mmmm i
bm
m
m
mm
m
i i
m mm
mm m
bm
m mi
bm
m
m
m
b m m mm i
m
m b b ii i
i
bb
i
ii i
i
b
i i
i
-2
-1
0
PC 2: 14.96%
1
i
i ii
i
i
i
i
i
2
3
4
Score Plot PC2-PC4
(PCA and FPCA-1)
8
6
6
5
b
4
4
iii i
i
b
i
b bbmbb bbib iim
m
i i
m
m
bbbbm
m
m
i
m
m
i
m
i
m
mm
ii i imimi iimi m
im
mim
bmm
m
mi b
m
m
b
i
i
m
i
i
m
i
m
bbmbm
b
i
i
b
i
i i
i
bbmbbm
i
i
i
ii
b
b bbbbb m
bb
i
2
0
-2
i
3
m
2
1
PC 4:8.44%
PC 4: 10.06%
-4
b
-6
i
i
i
0
b b bbb b
-1
i i
ii
i
i
i
m m
mm
i
m
m
b
i
b bmmm m
m m
i
m
i
mm i i i i i
mmm
bmbb bm
bbbm
b bb
m
bmm
m
bb bb bm
b
m
b
b
m
m
b
mb
b bbb mbmm
ii i
m iiiiii i i iii
b
i i i
ii
i
i
i
i i
ii
-2
-8
-10
-5
-4
-3
-2
-1
0
1
2
PC 2: 14.05%
3
4
5
6
7
8
-3
-4
-3
-2
-1
0
PC 2:14.96%
1
2
3
4
Score Plot PC3-PC4
(PCA and FPCA-1)
8
6
6
5
4
3
b
i
i
1
PC 4:8.44%
PC 4: 10.06%
-4
b
0
-1
ii
m
2
i
-2
-6
i
i
i
i
b
i mb m
ii iiibibb
bbb
b
mbmb
mm
bm
m
m
i iiiimiimimm
m
m
b i
mim
bm
m
ii i ii iiim
mimm
m
bim
bbbi i
bm
im
m
m
b
b
m
b
ii i
i
b
i
b
b bbbbb bb
m
b b
0
i
4
ii i
i
2
b
i
i
i
i
m mm
i
m
i
m
m
b
i
mmbmb
m
m
m
bbb b
m im i i
mbb m bb mb ib bbbb bi
i i i bb m m
m
m
m
m
bm
bi bbb b m
bbb bb i i
m
m
m
b
mbm
b
m
b i b i
i
i
i
m
m
i
m
i
i
i
i
i
i i
i
i i
i i i i
i
i
i ii
b
-2
-8
-10
-6
-4
-2
0
2
PC 3: 10.79%
4
6
8
10
-3
-3
-2
-1
0
PC 3:10.71%
1
2
3
Score Plot PC1-PC2
(FPCA-1 and FPCA-o)
4
4
i
3
2
1
0
-2
i
3
i
i ii
i
i
2
i i
i
i
i
ii iiiiiiii iiiii i
i i i i ii i i
iiiii i i i
m
b
m
mmi i
m
m
bbm
m mmm mm m
m bmb
m
m
mb m mm m
b
m
m
b bmm
m
b
m
bb b
m
m
b
bm b b
bbb
b b bb b
bm
b
b b
b
b
1
m
bbb
b
b
-2
0
2
PC 1:48.15%
0
b
-3
-4
-4
m
PC 2:14.38%
PC 2:14.96%
-1
b
b
bb
b b
i
4
6
b
-1
b
b b bm
b
b
bb b
m b
bbb
bbbm b b
bb m
b m b
b
b
m
bb
m
m mm
bm
mb m m
m
m m
m m mmm mmbm m
bb
m
m
mi
i
m
im
i
iiiiii i i
i i i
ii i i i i
i ii i ii
i i
iii i ii i i i
i
b
-2
i
-3
8
-4
-4
-2
0
b
m
m
m
m
i
i i
i
ii
i
i
2
PC 1:62.78%
4
6
8
Score Plot PC1-PC3
(FPCA-1 and FPCA-o)
3
2
b
2
1
0
-2
-3
-4
i
i i i
ii i
b
b
ii i i bii
b iib b
b
i b b i
b bm
b
b
i
i ibiii
b
i
i
bi
i bi mmm i b
m
i
bb m
bmmbmm
i m
b
m
m
ii
mm mm
m
m
m
m
mb
m
m
mm
m i
mb
m
m
mb
m
i
m
bi
ii
m
bb
i i
b
i
i
i
i
i
ii
i
-2
0
2
PC 1: 48.15%
0
b
-1
-2
i
b
bbbb b
b
m
m
i i i
i
i
-3
m
b
PC 3:8.28%
PC 3: 10.71%
-1
1
i
i ii
i i i ii i
i
i
i
i m
mi mm i ii i
bb
bm
bbbim b bi b
m
bm i bi bib mm
m
m
m
m
m
b
b
m
m
b
i
b
mm
bbbm mi i bi bi bb ib i b
mm
m m
i
i
m
m b mi
m
m
m b
i b
i
m
m
i
i
m
m
b
bbb
b
-4
-5
i
b
-6
4
6
8
i
i
-7
-4
-2
0
2
PC 1:62.78%
4
6
8
Score Plot PC1-PC4
(FPCA-1 and FPCA-o)
6
4
b
i
5
i
i
i
3
ii
i
i
4
2
3
i
i
m
2
-1
m
m
m
0
b
PC 4:4.77%
PC 4: 8.44%
0
1
m
i
i
m
b
m m b mi b
m
m
m
i
m
i
m
i
i
b
m m bm bb
m
bm mb i bibmi mbbibbb ib b bb
m
bm
b m
m
mm
i i bi mbm i b
b
bmb
bi
mi mi ii i
m i
i bi i
iii iii ii
ii
i ii
m
1
i
i i
i
i
bbbbb b
i
i
i ib
ibb
i
m
b
m
i
mb
m
m
m m
mm
im
bb
b m i
mmm m
m
mm
ii m m
mbmmm
mi
i
m
i
m
i
m
bi i m b i
bb bim
i
m
m
i
b i
b mii i i
ii
i
m
b bi i biiibi b
i b
ib i b
b
i
bi
b
b bb b
i
b
m
bb m
bb
b
b
-2
-2
-3
-4
-1
i
b
-2
0
2
PC 1: 48.15%
4
6
8
-3
-4
-2
0
2
PC 1:62.78%
4
6
8
Score Plot PC2-PC3
(FPCA-1 and FPCA-o)
2
3
ii
b
i
2
b
1
0
b
b
bb
b
-2
-3
-4
-3
i
ii i
b b
i
b b bb
i ii i i i
b
i
bb bmbb b b
i iii i
b
b
i ii i
ib
mb mm
b
bbmm mmmm i
bm
m
m
mm
m
i i
m mm
m
m
m
m i
b m
m
b m m
m
b mm
m mm i
m
m b b ii i
i
bb
i
ii i
i
b
i i
i
-2
-1
0
PC 2: 14.96%
1
i
i ii
i
i
0
-1
i
i iiii ii i ii
i ii m
m mb
i
b
b
i
bb
bmm
mbb m
mm
b b
bbbbbmbm bbbbb
i i i i mm
b
m
m
b
b
b
m
b
i i i
i
b
i
m bm
i m mm
i
b
m m m
m
mm
b b
i
i
m
m m
m
i
m
i
i i
i
i
b bb b b
b
i i
i
ii
-2
i
i
-3
PC 3:8.28%
PC 3:10.71%
-1
1
i
-4
i
i
-5
i
b
-6
2
3
4
-7
-4
-3
-2
-1
0
PC 2:14.38%
1
2
3
4
Score Plot PC2-PC4
(FPCA-1 and FPCA-o)
6
4
b
i
5
i
i
i
i
i i
i
3
4
i
i i
i
b
i
i i i bb b
mb
m
mm
i
mb
m
m mmm
bm bb
i m
mm
mm m
i i m m
m
m
mbm
m
i
i
m
m
i i i m m
bb
mm
i ii i i
m bm b
b
i iii i
b
iii
b b bm
iii i
b
b
bb
b
b
i i
b bb b
i
2
3
m
2
1
b b bbb b
-1
i
i
i
m m
mm
i
mm
b
i
b bmmm m
m m
i
mmm i i
mmm
bmbb bm
i i ii
bbbm
b bb
m
bmm
bb bb bm
b
m
bm
b
b
m
i
b
m
b
b
b
m
b b
iii i ii i
m
m
m
i
iii i iii
b
ii
i
1
i
i
0
PC 4:4.77%
PC 4:8.44%
0
i i
ii
i
i
i i
ii
iii
i
-1
i
i
b
b
b
b
-2
-2
-3
-4
i
bb
b
-3
-2
-1
0
PC 2:14.96%
1
2
3
4
-3
-4
-3
-2
-1
0
PC 2:14.38
1
2
3
4
Score Plot PC3-PC4
(FPCA-1 and FPCA-o)
6
5
4
b
i
i
i
i
i
4
i
i
2
3
i
-1
i
i
m
m
m
m
1
i
b
i
m mm
i
m
i
m m bm
i
mm b
m
m
m b ii
bbb i b b
m
b ib b i
mbim
b m m
i
i
m
m
m
m
bbmm bb m
bi bbb b m
bbb bbbbb bi i
m
m
mb mbm
b
m
b i b i
i
m
i i
mi i m i
i
i ii
ii
i i
i i i i
i
i
i ii
m
0
PC 4:4.77%
PC 4:8.44%
0
ii
m
2
1
b
i
i
ii
i
i
i
-1
i
i
i
i
i
bb i
bb ii
m
b
bb i
bm
bb m
mm
m i m mbmi
mm
bbi i
m
m mmmmm
m
m
bm
m
i
m
i i bm
bmmmi
bmb bbm
m i
ib
i i i ii
m
bi bbbi i i i
i
bb iibb
ib b
bbbb
b
ii
-2
-2
-3
-3
i
ii
3
b
-3
-2
-1
0
PC 3:10.71%
1
2
3
-7
-6
-5
-4
-3
-2
PC 3:8.28%
-1
0
1
2
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
PCA
FPCA-1
Cum.
Prop.
%
Eigenvalue
Prop.
%
FPCA-o
PCs Eigenvalue
Prop.
%
Cum.
Prop.
%
Eigenvalue
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
Prop. Cum.
%
Prop.%
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)
dT
pPb
DR
DR
pPb
sPb
dT
sPb
Loading Plot PC1-PC2-PC3
(FPCA-1 and FPCA-o)
sPb
dT
DR
DR
pPb
pPb
dT
sPb
Score Plot PC1-PC2
(PCA and FPCA-1)
3
3
2
11
1
1
2
1
2
2
1
0
-1
4
-2
-3
-7
-6
-5
3
11 111111
2 3 2122 1111 2 1111
1 1 1
4 3
12113 1 2 2 111 1 1
4
2
2
2
1 1121 12 1 11
42 3
2 2 21 11 22111 1 21 1111
111111
1
11111
2
1
111
2 3
1
1
1
4
1
1 1 1111111
11
2
21111 111
1
2
2 4
1
1 1 1
11111
3 32
11
1
3
1
2
1
11111
11 1 111111111111
22
2 111 1
21
4
2
4
2 1
3 1
11
2
11
11
11
2
-4
-3
-2
PC 1: 46.96%
-1
0
1
2
4 4
0
3
1
2
4 4
3
2
22
2
3 21
2
11
11
11
2 1
1 111
21111111
1111111
11
1
11
1111
1
11
11111111111
21
11
2
1
3
24
11
3 1 1
1
2
112111 1
11
2
2
1 1 111
4
2241212113 1 1 1 11111
433
12 1
1 4
1 1111
2
2 11 2111211 1 11
1 111
2 1
21
2
2
1
23
1
2 1111
1
222 11
3
11
21111111
11
1
1
32
1 212 11111
111
1
PC 2:28.10%
PC 2;24.47%
4
2
1 1 11
22 111111
-1
2
2
-2
-3
-5
-4
-3
-2
-1
PC 1:50.75%
0
1
2
3
Score Plot PC1-PC3
(PCA and FPCA-1)
6
2
4
1
4
4
4
4
2
0
0
4
-2
2
1
-1
-2
PC 3:13.20%
PC 3:17.04%
3 3
4
3
3 23 323 2 2222221 12121121112
22
2 212 1 1 1
2 1221221111 111111111111111111211112111111111111111111 1 1111
3
2 2 211111 1 121 1111111 111111 11111111
22
1 1 11 1
3
2 1 1111 111 11111111111 1
1
1
23
1
1
11
22 22 1
1
1 1
2
1
4
44
4
1
-4
-6
-7
1
2
2
12 12 1 111
31 2121111 1
3 23 3 2 22222 111111111 11111121
4
11 1 12111111 111 1
321 232 21 1111111121 1 21 1111111 1
4
1
3 11 2 2 1 11 11121 1 1111111
4
2
2 1111 1111
121 11 1
4
11 11111 1111 11111
1
22
11
2 2 12 1
11111
3
1
11
1
2 2
2
3
4
1
1
1
4
1
2
2
2
3
1
2
42
4
1
-3
1
1
-4
-6
-5
-4
-3
-2
PC 1:46.96%
-1
0
1
2
3
-5
-5
-4
-3
-2
-1
PC 1:50.75%
0
1
2
3
Score Plot PC1-PC4
(PCA and FPCA-1)
2
4
1
2
2
2
11 1
1 1 111
2 111111 1
2 2 2 1 11 111
3 32 3 11121211221 111111111111111
1 211 212 2 2 111 1111111111111
2
1
2 32
2222 1111
2 3 2 1 111 11 1 111
1
2 2 22
1
22 21 1111111111
2
3
2
21 1111111111111 1
2
1
2 42 1 3 1
2 11 11111
4 3
3
2
11
4
3
4
111
11111
1
1
1
1
1
4 44
1
0
-1
-2
11
1 11
2
222121 111 1111111111 1111111 11111111111111 11 1 11 1
2
2 2
22 1111111111111111 1111111111111111111111111111111111
2
222222 222222222222 1222 1 221 121 1 11111 1 111111111
33
2 2
1
2
3 3 33 3
3
4
3 3
4
44
4
4
4
-4
PC 4:7.95%
PC 4:11.51%
4
-4
-7
2 12
0
-2
-3
1
2
-6
4
-8
4
-10
-6
-5
-4
-3
-2
PC 1:46.96%
-1
0
1
2
3
-5
-4
-3
-2
-1
PC 1:50.75%
0
1
2
3
Score Plot PC2-PC3
(PCA and FPCA-1)
6
2
4
1
4
4
2
0
3 3
4
4
3
2 2224 12 2134 22 111 2212
33
2 3 2 222
1
2
21
2
1
1111111
12 14 12 111 1 11 1 111111121211111 1121111111
1
1
1
1
3
1
2
2
1
1
1
1
1
1
2
2 1 1 11 1
1 1 1 111111111 1111 21 11
1 11 11 11 1111111111121311111111 111 121 121112 11 1
1
1
4
121 311
21 1
1
1
221
2
2
1
1 1
2
2
-1
-2
PC 3:13.20%
PC 3:17.04%
-2
0
4
1
1
1
-4
-6
-3
1
2
1 1 22
1
2
1
11211 1 1122 11 111 13 12 1 2 2 31
1
3
1
1
2
1
1
1
2
4 1
1111 1 1 11 11221111 111311111112111
1 1 11
1 211 3 1 3221
1 4 11
1
2
1
1
1
11 2 2 1111121
1
1
1
3
1
2
1
1
4
4
11
1 1
1 1111 111111114111111121121
1 11111 2 1 1
1
1
1
2
1
2
1 21
1111
2111111 1221
3
1
11
1
2
2
23
41
1
4
1 1
2
3 21
42
2
4
-3
1
1
1
-4
-2
-1
0
PC 2:24.47%
1
2
3
-5
-3
1 2
1
-2
-1
0
PC 2:28.10%
1
2
3
Score Plot PC2-PC4
(PCA and FPCA-1)
2
4
1
2
1
1
1
21
1
11
1
2
2 1
1
1
1
11
2 2 1 1 1 111
2
1 11111 111 111
1
1
2
1
1
1
1
1
3
3
2 12
3
11 1 121 1 1 2 1 11 11111111
1
1
2 2221 21 21112111 2111 1 21
2
1
11
1
2
3
3
1 1 1 1 1
1
1 11 212 2121 21111
1
22 3 21111
22
111111
1
1
1
1
1
1 2
2 1 11 11 11111
1
1 1 111111 4
11 22 32 12
4
3
11
4 42
3
13
111111111
1
1 1
1
4 44
4
1
1
0
-1
-2
1
1
1 2
-4
PC 4:7.95%
PC 4:11.51%
-4
-3
1
1
0
-2
-3
1
1
2
2
1 11 1
1
1111
11 1121111111111111111111211211112111111211 111 1211111
1
1
1
1
1
1 1111 111111111 11111111 11 12 111211111111 111 11 112 21 1 2 111
1
2
2
1
1
1
2
1
1
2
1
2
1
1
1
2
2
1
1
3
1
2
1
1
2 2 2121 222 2 2 2
3222
2
22 2 2
1
2
3
3
3
3 34
3
3
3
4 44
4
4
4
4
2
-6
4
-8
4
-10
-2
-1
0
PC 2:24.47%
1
2
3
-3
-2
-1
0
PC 2:28.10%
1
2
3
Score Plot PC3-PC4
(PCA and FPCA-1)
2
1
0
4
1
1
-1
1
2
1 1111 2
1
11
1211211
1 1111111111111212111211233
12 1 111 232 2
1 21 111 212 111111111211111111122
12 1121 12212
2
2 113121111 1121 3
1 212
2
111111111212111111 2 223
1
11
2
1
1 221 1 11111111111 1 2 43
4 3
3 1 132
4 4
1
1
1
1
1
1 1111 1
1
4
44
2
1
1
2 2 1
1
1
1
1
22 1 22 11 111 212111111 1111111111111111111111111 11111 1 1
1
1
1
1
1
2 1 21 12 11111111111
1
1
1 1 111111
1321211121 11212111112211221111111212121122222112212 2
3
1 2 2222 223 33 2
3
3 3
4
33
44
4 4
4
4
4
1
0
-2
-4
PC 4:7.95%
PC 4:11.51%
4
-2
-3
-4
-6
1
-6
4
-8
4
-10
-4
-2
0
PC 3:17.04%
2
4
6
-5
-4
-3
-2
-1
PC 3:13.20%
0
1
2
Score Plot PC1-PC2
(FPCA-1 and FPCA-o)
3
2
3
2
11
4 4
1
2
4 4
3
2
22
2
3 21
2
11
11
11
2 1
1 111
21111111
1111111
11
1
11
1111
1
11
11111111111
21
2
11
2
1
3
24
11
3 1 1
1
2
112111 1
11
2
2
1 1 111
4
2241212113 1 1 1 11111
433
12 1
1 4
1 1111
2
2 11 2111211 1 11
1 111
2 1
21
2
2
1
23
1
2 1111
1
222 11
3
11
2111111
11
11
1
3
1 2 212 11111
111
1
0
1
0
PC 2:27.71%
PC 2:28.10%
-1
2
2
-2
-3
-5
-4
-3
-2
-1
PC 1:50.75%
0
1
2
1
1 212 111111
2
3
11
11
21111111
1
1
3 2 1
22 111
1
2 111
1
2 2 3 1221
11111111111
1
2
1 111
2 11 2111211 111
1 4
1 11
11
1 1 21
11
433 2212 2113 21 1 1 11111
1
1
4
4
1
2
2
11
2 111
1111
2
111111 1
111
1
1
24
11
3 1 1
1111111
3
11
1
2
2
11111
2
11
1 2111
2
21
3
21
22
2
2
42 4
11
3 1
11
1
2
1
11
2
1
3
-1
4 4
-2
-3
-5
-4
1
-3
-2
-1
PC 1:53.57%
0
1
2
3
Score Plot PC1-PC3
(FPCA-1 and FPCA-o)
5
2
1
2
2
12 12 1 111
31 2121111 1
3 23 3 2 22222 111111111 11111121
4
11 1 12111111 111 1
321 232 21 1111111121 1 21 1111111 1
4
31 11 2 2 1 11 11121 1 1111111
4
221 11
1 2 1111 1111
4
1 11111 1111 11111
22 1 1 1 1
1
1
2 2 12 1
11111
3
1
1
1
1
2 2
2
3
4
1
1
1
4
1
2
2
2
3
1
2
42
1
4
0
-1
-2
4
1
1
3
2
2
4
2
1
1
-4
-5
-5
4
PC 3:13.89%
PC 3:13.20%
1
-3
1
12
3 22
4
1
2 432
4
3
0
4
1
1
1
1
1 1
1
1
1
1
2
1
2
2 2 2 1 1 11 1
111 111 111111111
1
1
1
1
1
2 12
1
1 111 1 11 1
1 11 2
1 1111 1111121211 11111111111
1
1
1
2
3 2 112221112121111111111111111211211112 111
12 1 11 1
2
4
3 3 312331222122321211111111 1
2
21
2
2
4
-1
1
4
4
-2
-4
-3
-2
-1
PC 1:50.75%
0
1
2
3
-3
-5
-4
-3
-2
-1
PC 1:53.57%
0
1
2
3
Score Plot PC1-PC4
(FPCA-1 and FPCA-o)
8
4
1
2
7
2 12
11
1 11
2
222121 111 1111111111 1111111 1111111111111 11 1 111 1
2
2 2
22 111111111111111111 11111111111111111111111111111111
2
222222 222222222222 1222 1 221 1121 1 11111 1 111111111
33
2 2
1
2
3 3 3 33 3
4
3 3
4
44
0
-2
4
4
4
6
5
4
3
4
PC 4:7.95%
PC 4:4.83
-4
-6
4
4
2
4
1
0
4
2
-8
1
-1
-10
-5
4 4
4
-4
-3
-2
-1
PC 1:50.75%
0
1
2
3
-2
-5
-4
-3
4
3 3
3
3
1
3
32 32 23 2 2 2 2 2 2
1
111
2
2
2
2
1
22
1
2
1
2
2222222211 211 1 211 111111 1111111111111111 111
2222
111 111 1 1
1 1 1
2 12212 11 1111211 1111111111111111111111111111111111 1111 1111111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1111 111 1 1
1
3
3
2
12
-2
-1
PC 1:53.57%
0
1
2
3
Score Plot PC2-PC3
(FPCA-1 and FPCA-o)
5
2
1
0
-1
-2
-3
4
1 2
1
1
1
3
2
2
2
1
1
1
1
-4
-5
-3
1
PC 3:13.89%
PC 3:13.20%
1
2
1 1 22
1
2
1
11211 1 1122 11 111 13 12 1 2 2 31
1
3
1
1
2
1
1
1
2
4 1
1111 1 1 11 11221111 111311111112111
1 1 11
1 211 3 1 3221
1 4 11
1
2
1
1
1
11 2 2 1111121
1
1
1
3
1
2
1
1
4
4
11
1
1
1 11111 111111114111111121121
1 11111 2 1 1
1
1
1
2
1
2
1 21
1111
2111111 1221
3
1
11
1
2
2
23
41
1
4
1 1
2
3 21
42
2
4
0
-1
2
11
14
2
2113
4
1
1
2
21 2
4
1
34 1 1 1
211 11112
1
3 1
1
2
2
2
1
1
1
11 1 1 1 1 1 111 111111
211111111111111 11111 1 1121 1 11 2 111111 1
1
1
2
1
1
1
1
1
1
12
4 111111111 11 1111121113 122 2111 2 211111
1111 1
1
1
1
1
1
1
2
1
1 42
21 11 1 121141132 2 111222 111 1311 11 3 1 1
2 1 2
2 11 3 31
1
2
3
2
2
1
4 2
4
-2
-2
-1
0
PC 2:28.10%
1
2
3
-3
-3
-2
-1
0
PC 2:27.71%
1
2
3
Score Plot PC2-PC4
(FPCA-1 and FPCA-o)
4
8
1
1
1
2
2
1
121111111 111111211111111211 1 2 11
1
11111
1
1
1
1
1 1111 1111111111 1111111111111111111211121112121111111 111111 11111112 121 1 2 111
1
1
2
2
1
1
1
2
1
1
2
1
2
22 2211111 222 32 21 1 11 1 2 3222
212
2
2
22 2 2
2
3
3
3
3 34
3
3
3
4 44
4
4
4
4
2
4
7
1
0
-2
6
1
1 2
5
4
444
3
4
-4
PC 4:4.83%
PC 4:7.95%
-6
4
2
-3
-2
-1
0
PC 2:28.10%
1
2
3
4
4
3
3
3
-1
3
3
3 21
2 22
3 23 2 3 3
2
2
1
2
2
11111122 2
2
2
1
2
1
1
2
2
1
2
1
1
1
1
2
2
2
2
2
1 1 11 2212 2 2111112111 11111 11 1 1111111 1 2 12 21 1 1111111111 111
11 2 111 21 1 12111122111111111111 111 1 11 1 1111111
1
111 1
1
1
11 12 1111111111111 11111 1 11
12
1
1
1
-2
-3
-2
1
0
-8
-10
4
2 1
1
-1
0
PC 2:27.71%
1
2
3
Score Plot PC3-PC4
(FPCA-1 and FPCA-o)
4
8
2
1
4
7
1
1
2 2 1
1
1
1
1
22 1 22 11 111 2121111111111111111111111111111111 11111 1 1
11111
1
1
1
1
1
2 1 21 12 11111111111
1
1
1
1
1
1
1
1
2
1
1
1
1
1321211121 121111222211112121211222122112212 2
3
1 2 2222 223 33 2
3
3 3
4
33
44
4 4
4
4
4
1
0
-2
6
5
4
4
4
4
3
44
4
-4
PC 4:4.83%
PC 4:7.95%
-6
4
4
334
3
3 3
1
3
3 322 2222222 211121213
22 2 22 22 1211 1111 21 1 2 1 22
22 12111 11 1
2 2 121111121212111111111111111111111111111121111111112211 1 22 1 12 1
1
1 1111 1111
11
22
11111111111111 111 1111 11 1
1
1
2
1
0
-8
-1
-10
-5
-4
-3
-2
-1
PC 3:13.20%
0
1
2
-2
-3
-2
-1
0
1
PC 3:13.89%
2
1
1
1
3
4
5
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 Bright Future of Chemometrics
The responsibility for
change …
lies within us. We must
begin with ourselves,
teaching ourselves not
to close our minds
prematurely to the novel,
the surprising, the
seemingly radical.
Alvin Toeffler