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
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m
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m
m
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m
m
i
i
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i
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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
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m
m
m
m
m
m
m
m
m
m
m
i
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b
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b
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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
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b
b
b
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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
-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
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m
m
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m
m
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m
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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
m
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m
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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
m
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m
m
m
m
m
m
m
i
i
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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
-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
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m
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m
m
m
m
m
m
m
m
i
i
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i
b
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b
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b
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b
b
b
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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
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m
m
i
i
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i
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i
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i
i
b
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
-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
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m
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m
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m
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m
m
m
m
m
m
m
m
m
m
i
i
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i
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i
i
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i
i
b
b
b
b
b
b
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b
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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
-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
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m
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m
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m
i
i
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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
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i
i
i
i
b
b
b
b
b
b
b
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
-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
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m
m
m
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m
m
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m
i
i
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i
i
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i
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i
i
i
i
i
b
b
b
b
b
b
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b
b
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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
-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
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i
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i
i
i
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i
i
i
i
i
i
i
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i
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i
i
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i
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i
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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 1:62.78%
-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
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m
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m
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m
m
m
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m
m
m
m
i
i
i
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i
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i
i
i
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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
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
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m
m
m
m
m
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m
m
m
m
m
m
i
i
i
i
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i
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i
i
i
i
i
i
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
-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
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m
m
m
m
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m
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m
m
m
m
i
i
i
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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
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
-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
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i
i
i
i
i
i
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
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
-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
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m
m
m
m
i
i
i
i
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i
i
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i
i
i
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i
i
i
i
i
i
i
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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
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
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
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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
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i
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i
i
i
i
i
i
i
b
b
b
b
b
b
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
-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
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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
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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
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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
-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
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i
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i
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i
i
i
i
i
i
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
-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
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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
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