fuzzy relation

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Phytoplankton assemblages, environmental influences
and their seasonal changes measured
using weighted averages and fuzzy set theory
IAGLR 2005
Ann Arbor, MI
May 23 - 27
Small data set
High phytoplankton diversity
Multiple environmental variables
Use canonical correspondence analysis
(CCA)
How are individual taxa affected by
environmental variables and each other?
Use fuzzy sets and fuzzy relations
•
Surface sample collection-- near Port Huron, Lake Huron
•
Temperature taken at time of sample collection
•
Nutrient measurements determined in the laboratory
•
Global: Statistical data analysis using CCA and partial CCA:
121 taxa, total
6 surface samples, 3 for June and 3 for August
6 environmental variables-- SiO2, NO3, TSP, NH3, Cl-, and temperature
•
Local: Fuzzy data analysis using fuzzy relations
axis 2
j1
a2
a3
TSP / NH3
Cltemp
j2
axis 1
NO3
SiO2
j3
j1, j2, j3 = June
a1, a2, a3 = August
a1
CCA
Monte Carlo permutation test-null model, 99 permutations-Test of first axis: F-ratio = 1.04
P-value = 0.04
Trace:
F-ratio = 1.96
P-value = 0.08
eigenvalues:
axis 1
axis 2
axis 3
axis 4
0.576
0.182
0.159
0.086
Sum of all unconstrained eigenvalues: 1.131
Sum of all constrained eigenvalues: 1.003
Residual unconstrained variation: 0.128
(89% of species variation explained)
intraset
correlations:
axis 1
SiO2
-0.834
NO3
0.994
TSP
-0.326
NH3
-0.326
Cl
0.968
temp
-0.984
1st Partial CCA-- Effects of Cl- and NO3 on taxon abundance:
eigenvalues:
axis 1
0.176
axis 2
0.129
Sum of unconstrained eigenvalues = 0.304
Sum of constrained eigenvalues = 0.176
intraset
correlations:
axis 1
NO3
-0.958
Cl
-0.932
2nd Partial CCA-- Effects of SiO2 and temperature on taxon abundance:
eigenvalues:
axis 1
0.180
axis 2
0.107
Sum of unconstrained eigenvalues = 0.415
Sum of constrained eigenvalues = 0.287
intraset
correlations:
axis 1
SiO2
-0.966
temp
-0.553
Fuzzy Set Theory:
Let X be a non-empty set that is defined as the universe of discourse, and
the elements of X are x1, x2, …, xn. A subset of X defined as a fuzzy set is
A˜  x, A˜ x , x  X 
where the fuzzy set is a grade of membership on the interval [0, 1].
That is,

 : A˜  0,1
is a mapping where each element x is assigned a degree of membership

0  A˜ x   1
Set theoretic operations on two fuzzy sets include intersection and union
and are defined, respectively as

x  X, A˜  B˜ x   minA˜ x , B˜ x 
x  X, A˜  B˜ x   maxA˜ x , B˜ x 
Similar to fuzzy sets, a fuzzy relation on X and Y is

˜
R  x, y, R˜ x, y 


x, y  X Y 

where the fuzzy relation is in the Cartesian product space X x Y. The fuzzy
relation is a grade of membership of ordered pairs on the interval [0, 1]. That is,

R˜ : X Y  0,1
is a mapping where elements x and y are assigned a degree of membership


0  R˜ x, y 1
For n-ary fuzzy relations
R˜1x, y,x, y  X Y
and
R˜ 2 y,z, y,z  Y  Z
The max-min composition is





˜
˜
R1 R2  x,z,max min R˜1 x, y , R˜ 2 y,z  x  X, y  Y,z  Z 
y



 


Two fuzzy relations:
R˜1x, y  and R˜ 2 w, x  or R˜1x, y  and R˜ 3 y,z
z1 z2
R˜1 :
x1
x2
x3
x4

y1 y
2 y3 y4 y5 y6 y7 y8 y9 y10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0

R˜ 2 :
x 1 x2 x3 x4
w1
w2
w3
w4
w5
w6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
R˜ 3 :
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

w represents environmental variables
x represents constrained axes influenced by all environmental variables
y represents taxa (n-dimensional fuzzified weighted averages) from June and August
z represents axes influences by Cl-and NO3 or SiO2 and temperature (from partial CCAs)
Aggregation operations
1st projection of a fuzzy relation:

2nd projection of a fuzzy relation:

Total projection of a fuzzy relation:



(1)
˜
R  x,maxR˜ x, y

 y

x, y  X Y 


(2)
˜
R   y,maxR˜ x, y
x


x, y  X Y 




T 
˜
R  max maxR˜ x, y 
x
y 

x, y   X  Y 

5 dominant taxa from each sampling month were used:
June-- Asterionella formosa
Fragilaria capucina
F. crotonensis
Urosolenia eriensis
Tabellaria fenestrata
August-- Achnanthidium minutissimum
Cyclotella #6
C. comensis
C. michiganiana
C. pseudostelligera
Fuzzification of species scores or intraset correlation coefficients:
fwa 

w  ax min 
ax max  ax min 
 Normalization of weighted averages or
intraset correlation coefficients per axis.
Fuzzy importance matrices
(from CCA):
species axis 1 species axis 2 species axis 3 species axis 4
temperature
SiO2
TSP
NH3
ClNO3
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
0 .0 0
0 .0 8
0 .3 3
0 .3 3
0 .9 9
1 .0 0
0 .4 3
0 .0 0
1 .0 0
1 .0 0
0 .5 9
0 .5 1
0 .4 8
1 .0 0
0 .0 0
0 .0 0
0 .9 7
0 .6 0
0 .1 7
0 .2 7
1 .0 0
1 .0 0
0 .0 0
0 .1 3
0 .3 1
0 .8 9
0 .0 7
0 .1 6
0 .1 2
0 .0 3
0 .8 4
0 .9 6
0 .8 8
0 .8 4
0 .5 6
0 .4 6
0 .4 9
0 .5 0
0 .5 1
0 .5 2
0 .6 3
0 .2 5
0 .5 0
0 .5 5
0 .5 0
0 .3 1
0 .3 1
0 .4 0
0 .2 9
0 .2 9
0 .5 5
0 .0 8
0 .3 7
0 .4 4
0 .4 0
0 .4 0
0 .3 1
0 .4 1
0 .4 1
0 .5 1
0 .4 3
0 .4 0
0 .4 2
0 .4 1
Max-min composition of fuzzy relation between fuzzy importance matrices:
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
0 .4 8
0 .4 3
0 .4 3
0 .4 3
0 .4 3
0 .4 3
0 .4 8
0 .2 5
0 .4 3
0 .4 4
SiO2
TSP
NH3
Cl-
0 .5 0
0 .3 1
0.31
0 .4 0
0.29
0.29
0 .5 5
0 .2 7
0 .3 7
0 .4 4
0 .5 6
0 .4 6
0 .4 9
0 .5 0
0 .5 1
0 .5 2
0 .6 3
0 .4 0
0 .5 0
0 .5 5
0 .5 6
0 .4 6
0 .4 9
0 .5 0
0 .5 1
0 .5 2
0 .6 3
0 .4 0
0 .5 0
0 .5 5
0 .5 6
0.89
0 .4 9
0 .5 0
0 .5 1
0 .5 2
0.84
0.96
0.88
0.84
NO3
0 .5 1
0.89
0 .4 9
0 .5 0
0 .5 1
0 .5 1
0.84
0.96
0.88
0.84
1st projection:
2nd projection:
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
C. comensis
C. michiganiana
C. pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Tabellaria fenestrata
Urosolenia eriensis
temperature
SiO 2
TSP
NH3
ClNO3
Total projection = 0.96
0.56
0.89
0.49
0.50
0.51
0.52
0.84
0.96
0.84
0.88
0.48
0.55
0.63
0.63
0.96
0.96
Max-min composition of fuzzy relations --> NO3 and Cl- (from partial CCA):
Achnanthidium minutissimum
0.36 0.49 0.38 0.37 0.40 0.38 0.56 0.19 0.45 0.56
Asterionella formosa
0.53 0.46 0.40 0.46 0.40 0.41 0.55 0.19 0.45 0.46
Cyclotella #6
0.36 0.49 0.38 0.37 0.40 0.38 0.49 0.19 0.45 0.49
C. comensis
0.36 0.49 0.38 0.37 0.40 0.38 0.50 0.19 0.45 0.50
C. michiganiana
0.36 0.49 0.38 0.37 0.40 0.38 0.51 0.19 0.45 0.51
C. pseudostelligera
0.36 0.49 0.38 0.37 0.40 0.38 0.52 0.19 0.45 0.52
Fragilaria capuci na
0.53 0.49 0.40 0.46 0.40 0.41 0.60 0.19 0.45 0.57
Fragilaria crotonensis
0.53 0.32 0.40 0.46 0.40 0.41 0.55 0.19 0.37 0.44
Tabellaria fenestrata
0.53 0.49 0.40 0.46 0.40 0.41 0.55 0.19 0.45 0.50
Urosolenia eriensis
0.53 0.49 0.40 0.46 0.40 0.41 0.55 0.19 0.45 0.55
Max-min composition of fuzzy relations --> SiO2 and temperature
Achnanthidium minutissimum
0.35 0.49 0.38 0.37 0.40 0.38 0.56 0.31 0.45 0.56
Asterionella formosa
0.40 0.46 0.55 0.43 0.50 0.43 0.46 0.48 0.45 0.46
Cyclotella #6
0.35 0.49 0.38 0.37 0.40 0.38 0.49 0.19 0.45 0.49
C. comensis
0.35 0.49 0.38 0.37 0.40 0.38 0.50 0.19 0.45 0.50
C. michiganiana
0.35 0.49 0.38 0.37 0.40 0.38 0.51 0.19 0.45 0.51
C. pseudostelligera
0.35 0.49 0.38 0.37 0.40 0.38 0.52 0.19 0.45 0.52
Fragilaria capuci na
0.40 0.49 0.55 0.43 0.50 0.43 0.60 0.48 0.45 0.57
Fragilaria crotonensis
0.40 0.44 0.55 0.43 0.50 0.43 0.36 0.48 0.41 0.41
Tabellaria fenestrata
0.40 0.49 0.55 0.43 0.50 0.43 0.50 0.48 0.45 0.50
Urosolenia eriensis
0.40 0.49 0.55 0.43 0.50 0.43 0.55 0.48 0.45 0.55
1st projection: (Effects of all environmental variables => across rows)
Degree of Cl- and NO3 influence = degree of SiO2 and temperature influence
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
C. comensis
C. michiganiana
C. pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Tabellaria fenestrata
Urosolenia eriensis
0.56
0.55
0.49
0.50
0.51
0.52
0.60
0.55
0.55
0.55
2nd projection: (columns)
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
C. comensis
C. michiganiana
C. pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Tabellaria fenestrata
Urosolenia eriensis
Total projection = 0.60
0.53
0.49
0.40
0.46
0.40
0.41
0.60
0.19
0.57
0.45
0.40
0.49
0.55
0.43
0.50
0.43
0.60
0.48
0.57
0.45
NO3 and Cl- influence
All environmental influences
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
SiO2 and temperature influence
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
2nd projections:
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
Summary
>> From CCA:
NO3 and Cl- influenced June taxa; SiO2 and temperature influenced August taxa
>> From composition of fuzzy relations:
1. Environmental influences X weighted averages (from CCA):
June taxa are affected by Cl and NO3 to a greater degree than
August taxa by SiO2 and temperature.
August taxa are approximately equally affected by all environmental variables with three
exceptions; Cyclotella #6, C. michiganiana and C. pseudostelligera are less affected by SiO2.
2.
Composition of fuzzy relations--10 taxa X 10 taxa (from partial CCAs):
Linguistic equivalent of 2nd projections as degree of influence*
Taxa
June
NO3 and Cl-
Augu st
SiO2 and temperature
Fragilaria capuci na
Tabellaria fenestrat a
Asterionella formos a
most highl y
more highl y
most highly
more highl y
Urosolenia eriensis
highly
highly
Achnan thidium minutissimum most highl y
less highly
Cyclotella comensis
more highly
less highly
Cyclotella #6
lesser
most highly
Cyclotella michiga niana
much lesser
more highly
Cyclotella pseudostelliger a
much lesser
less highly
Fragilaria crotonensi s
very least
more highly
*most highly > more highly > highly > less highly > much lesser > very least

If the five dominant taxa from June are present in the assemblage, they are more
influential than the five dominant August taxa in seasonal variation from June to
August.
From here…
• compare results to what is known about the ecological status of individual taxa
 fuzzy decision-making:
-
Devise a fuzzy truth table of results
-
Incorporate expert opinion(s) into decision-making
-
Combine results from multiple regions of a lake into decision-making
-
Devise linguistic solutions from results of fuzzy decision-making
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