Concerning Pollution with Heavy Metals

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

c
ostel
srb@
yahoo.co
.
uk

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
algorithm
s

(
FPCA
-
1
and
FPCA
-
o
)





Soft
C
omputing

Methods

Soft

Computing

Fuzzy Logic

Fuzzy Sets

PCA, PCR,

PLS, ANN

Genetic

Algorithms

Rough Sets

Chaos Theory

Approximate

Reasoning





What is Soft Computing

?

Aim :

To exploit the tolerance for imprecision uncertainty, approximate reasoning and partial truth
to achieve
tractability, robustness, low solution cost,
and

close resemblance
with human

like decision making


To find an approximate solution to an imprecisely/precisely formulated problem.




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

t
ractability
,

r
obustness
,

and

close

r
esemblance


with

human

like

decision

making
.





Provides

f
lexible

i
nformation

p
rocessing

c
apability

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
C
omputing

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

humanist
i
c

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
S
tyles
in
China

and W
est



China

West

Principle of Change

Reality is a dynamical, constantly
-
changing
process. The concepts that reflect reality
must
b
e subjective, active, flexible.


Law of Identity

Everything is what it is. Thus it is a
necessary fact that A equals A, no matter
what A is
.

Principle of Contradiction

Reality is full of contradictions and never
clear
-
cut or precise. Opposites coexist in
harmony with one another, opposed but
connected

Law of Noncontradiction

No statement can be both true and false.

Principle of Relationship

To know something completely, it is
necessary to know its relations, what it
affects and what affects it.

Law of the Excluded Middle

Every statement is either true or false. There
is no middle term.


School of Athens

Fuzziness

in
E
veryday
W
orld


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
C
hemistry


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”



















x
if
x
if
x
x
if
x
S
40
0
40
25
15
40
25
1
Mr. B. G. Membership Function for “Young”
















x
if
x
if
x
x
if
x
B
70
0
70
40
30
70
40
1
Generalized Fuzzy c
-
Means
Algorithm



n
j
c
i
x
A
x
x
A
L
L
x
d
L
x
d
x
C
x
A
L
x
d
x
A
L
P
J
n
j
j
i
n
j
j
j
i
i
c
k
k
j
i
j
j
j
i
c
i
n
j
i
j
j
i
,...,
1

;
,...,
1
)
(
)
(

;
)
,
(
)
,
(
)
(
)
(
)
,
(
))
(
(
)
,
(
1
2
1
2
1
2
2
1
1
2
2















Fuzzy 1
-
Line Regression
Algorithm

n
j
c
i
x
A
x
x
A
u
v
L
L
x
d
x
A
x
A
L
x
d
x
A
L
P
J
n
j
j
i
n
j
j
j
i
j
j
i
c
i
n
j
n
j
j
i
j
j
i
...,

,
1

;

...,

,
1
)
(
)
(
)
,
(


;
)
,
(
1
1
)
(
1
.
))
(
(
)
,
(
))
(
(
)
,
,
(
1
2
1
2
2
1
1
1
2
2
2



























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
(
x
j
), 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 v a l u e.














n
j
j
i
n
j
l
jl
k
jk
j
i
kl
x
A
x
x
x
x
x
A
C
1
2
1
2
)
(
)
)(
(
)
(



Fuzzy
Approaches


Fuzzy divisive hierarchical clustering
;


F
uzzy horizontal clustering
;


F
uzzy cross
-
clustering
;


F
uzzy 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

Bistri
ţa

Mountains

(
b
),

the

south
-
western

part

of

Maramureş

Mountains

(
m
)

and

the

north
-
western

part

of

Igni
ş
-
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

8
5
.
4
4

4

1.308

10.06

78.28

0.554

8.44

82.26

0.208

4.77

8
9.
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
Ni
Mg
Cr
Zn
K
Mn
Cu
Ca
Al
Fe
Cd
Pb
Zn
Mn
Co
Cu
Ni
Mg
Al
Cr
Ca
Fe
K
Cd
Pb
Loading Plot PC1
-
PC2
-
PC3

(
PCA
and

FPCA
-
o
)



Zn
Mn
Cu
Co
Ni
Mg
Cr
Al
Ca
Fe
K
Cd
Pb
Co
Ni
Mg
Cr
Zn
K
Mn
Cu
Ca
Al
Fe
Cd
Pb
Score Plot PC1
-
PC2

(
PCA
and

FPCA
-
1
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-8
-6
-4
-2
0
2
4
6
8
10
12
PC 1: 43.38%
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
PC 2: 14.05%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
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
PC 1:48.15%
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.96%
Score Plot PC1
-
PC3

(
PCA
and

FPCA
-
1
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-8
-6
-4
-2
0
2
4
6
8
10
12
PC 1: 43.38%
-6
-4
-2
0
2
4
6
8
10
PC 3: 10.79%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
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i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
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
PC 1: 48.15%
-3
-2
-1
0
1
2
3
PC 3: 10.71%
Score Plot PC1
-
PC4

(
PCA
and
FPCA
-
1
)



m
m
m
m
m
m
m
m
m
m
m
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m
m
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m
m
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m
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m
m
i
i
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i
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i
b
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b
b
b
b
b
b
b
b
-8
-6
-4
-2
0
2
4
6
8
10
12
PC 1: 43.38%
-10
-8
-6
-4
-2
0
2
4
6
8
PC 4: 10.06%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
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m
i
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b
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b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-2
0
2
4
6
8
PC 1: 48.15%
-3
-2
-1
0
1
2
3
4
5
6
PC 4: 8.44%
Score Plot PC2
-
PC3

(
PCA
and

FPCA
-
1
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
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m
m
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m
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m
m
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m
m
m
m
i
i
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i
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i
i
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i
b
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b
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b
b
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b
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b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
PC 2: 14.05%
-6
-4
-2
0
2
4
6
8
10
PC 3: 10.79%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
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m
m
m
m
m
m
m
m
m
m
m
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m
m
i
i
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i
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i
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i
i
i
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i
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i
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i
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i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-3
-2
-1
0
1
2
3
4
PC 2: 14.96%
-3
-2
-1
0
1
2
3
PC 3:10.71%
Score Plot PC2
-
PC4

(
PCA
and

FPCA
-
1
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
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i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
PC 2: 14.05%
-10
-8
-6
-4
-2
0
2
4
6
8
PC 4: 10.06%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
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i
i
i
i
i
i
i
i
i
i
i
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i
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i
i
i
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i
i
i
i
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i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.96%
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
Score Plot PC3
-
PC4

(
PCA
and

FPCA
-
1
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
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i
i
i
i
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i
i
i
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i
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i
i
i
i
i
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i
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i
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i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-6
-4
-2
0
2
4
6
8
10
PC 3: 10.79%
-10
-8
-6
-4
-2
0
2
4
6
8
PC 4: 10.06%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
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m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
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i
i
i
i
i
i
i
i
i
i
i
i
i
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i
i
i
i
i
i
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i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-3
-2
-1
0
1
2
3
PC 3:10.71%
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
Score Plot PC1
-
PC2

(
FPCA
-
1

and

FPCA
-
o
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
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i
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i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
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
PC 1:62.78%
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.38%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
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i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
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i
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i
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i
i
i
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i
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i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
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
PC 1:48.15%
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.96%
Score Plot PC1
-
PC3

(
FPCA
-
1

and

FPCA
-
o
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
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i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
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i
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i
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i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
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b
b
b
b
b
b
b
b
b
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
PC 1: 48.15%
-3
-2
-1
0
1
2
3
PC 3: 10.71%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
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i
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i
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i
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i
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i
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i
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i
i
i
i
i
i
i
i
b
b
b
b
b
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b
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b
b
b
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b
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b
b
b
b
b
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
PC 1:62.78%
-7
-6
-5
-4
-3
-2
-1
0
1
2
PC 3:8.28%
Score Plot PC1
-
PC4

(
FPCA
-
1

and

FPCA
-
o
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
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i
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i
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i
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i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
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b
b
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b
b
b
b
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b
b
b
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b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-2
0
2
4
6
8
PC 1: 48.15%
-3
-2
-1
0
1
2
3
4
5
6
PC 4: 8.44%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
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i
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i
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i
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i
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i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
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
PC 1:62.78%
-3
-2
-1
0
1
2
3
4
PC 4:4.77%
Score Plot PC2
-
PC3

(
FPCA
-
1

and

FPCA
-
o
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
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i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
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b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-3
-2
-1
0
1
2
3
4
PC 2: 14.96%
-3
-2
-1
0
1
2
3
PC 3:10.71%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.38%
-7
-6
-5
-4
-3
-2
-1
0
1
2
PC 3:8.28%
Score Plot PC2
-
PC4

(
FPCA
-
1

and

FPCA
-
o
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.96%
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-4
-3
-2
-1
0
1
2
3
4
PC 2:14.38
-3
-2
-1
0
1
2
3
4
PC 4:4.77%
Score Plot PC3
-
PC4

(
FPCA
-
1

and

FPCA
-
o
)



m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-3
-2
-1
0
1
2
3
PC 3:10.71%
-3
-2
-1
0
1
2
3
4
5
6
PC 4:8.44%
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
-7
-6
-5
-4
-3
-2
-1
0
1
2
PC 3:8.28%
-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
)



DR
dT
sPb
pPb
DR
dT
pPb
sPb
Loading Plot PC1
-
PC2
-
PC3

(
FPCA
-
1

and

FPCA
-
o
)



DR
pPb
dT
sPb
DR
dT
pPb
sPb
Score Plot PC1
-
PC2

(
PCA
and

FPCA
-
1
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
PC 1: 46.96%
-3
-2
-1
0
1
2
3
PC 2;24.47%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:50.75%
-3
-2
-1
0
1
2
3
PC 2:28.10%
Score Plot PC1
-
PC3

(
PCA
and

FPCA
-
1
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
PC 1:46.96%
-6
-4
-2
0
2
4
6
PC 3:17.04%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:50.75%
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
Score Plot PC1
-
PC4

(
PCA
and

FPCA
-
1
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:50.75%
-10
-8
-6
-4
-2
0
2
4
PC 4:7.95%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
PC 1:46.96%
-4
-3
-2
-1
0
1
2
PC 4:11.51%
Score Plot PC2
-
PC3

(
PCA
and

FPCA
-
1
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:28.10%
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:24.47%
-6
-4
-2
0
2
4
6
PC 3:17.04%
Score Plot PC2
-
PC4

(
PCA
and

FPCA
-
1
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:28.10%
-10
-8
-6
-4
-2
0
2
4
PC 4:7.95%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:24.47%
-4
-3
-2
-1
0
1
2
PC 4:11.51%
Score Plot PC3
-
PC4

(
PCA
and

FPCA
-
1
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-6
-4
-2
0
2
4
6
PC 3:17.04%
-4
-3
-2
-1
0
1
2
PC 4:11.51%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
-10
-8
-6
-4
-2
0
2
4
PC 4:7.95%
Score Plot PC1
-
PC2

(
FPCA
-
1

and

FPCA
-
o
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:50.75%
-3
-2
-1
0
1
2
3
PC 2:28.10%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:53.57%
-3
-2
-1
0
1
2
3
PC 2:27.71%
Score Plot PC1
-
PC3

(
FPCA
-
1

and

FPCA
-
o
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:50.75%
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:53.57%
-3
-2
-1
0
1
2
3
4
5
PC 3:13.89%
Score Plot PC1
-
PC4

(
FPCA
-
1

and

FPCA
-
o
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:50.75%
-10
-8
-6
-4
-2
0
2
4
PC 4:7.95%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
3
PC 1:53.57%
-2
-1
0
1
2
3
4
5
6
7
8
PC 4:4.83
Score Plot PC2
-
PC3

(
FPCA
-
1

and

FPCA
-
o
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:28.10%
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:27.71%
-3
-2
-1
0
1
2
3
4
5
PC 3:13.89%
Score Plot PC2
-
PC4

(
FPCA
-
1

and

FPCA
-
o
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:28.10%
-10
-8
-6
-4
-2
0
2
4
PC 4:7.95%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
PC 2:27.71%
-2
-1
0
1
2
3
4
5
6
7
8
PC 4:4.83%
Score Plot PC3
-
PC4

(
FPCA
-
1

and

FPCA
-
o
)



1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-5
-4
-3
-2
-1
0
1
2
PC 3:13.20%
-10
-8
-6
-4
-2
0
2
4
PC 4:7.95%
1
1
1
1
1
1
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
3
1
1
1
1
1
1
3
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
3
1
1
1
2
2
1
1
1
1
1
2
1
1
1
1
1
4
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
2
2
1
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
2
3
4
1
1
1
1
1
2
1
1
1
3
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
2
2
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
3
2
1
1
1
1
3
1
1
1
2
2
1
4
2
1
1
2
4
2
3
2
1
4
3
1
4
2
2
1
1
2
2
2
-3
-2
-1
0
1
2
3
4
5
PC 3:13.89%
-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

F
uzzy

Soft

Computing

could

be

one

of

the

best

s
olution

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

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

The Bright Future of
C
hemometrics