phytoplankton assemblages, environmental influences and their seasonal changes measured using...
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Phytoplankton assemblages, environmental influences and their seasonal changes measured
using weighted averages and fuzzy set theory
IAGLR 2005Ann Arbor, MI
May 23 - 27
Small data setHigh phytoplankton diversityMultiple environmental variables
How are individual taxa affected by environmental variables and each other?
Use canonical correspondence analysis (CCA)
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
j1, j2, j3 = Junea1, a2, a3 = August
Monte Carlo permutation test-- null model, 99 permutations--Test of first axis: F-ratio = 1.04
P-value = 0.04Trace: F-ratio = 1.96
P-value = 0.08
CCA
axis 1
axis
2
j1
a1
a2
a3
j2
j3
SiO2
NO3
Cl-
temp
TSP / NH3
intrasetcorrelations:
axis 1SiO2 -0.834NO3 0.994TSP -0.326NH3 -0.326Cl- 0.968temp -0.984
eigenvalues:
axis 1 0.576axis 2 0.182axis 3 0.159axis 4 0.086
Sum of all unconstrained eigenvalues: 1.131Sum of all constrained eigenvalues: 1.003Residual unconstrained variation: 0.128
(89% of species variation explained)
1st Partial CCA-- Effects of Cl- and NO3 on taxon abundance:
eigenvalues:
axis 1 0.176axis 2 0.129Sum of unconstrained eigenvalues = 0.304Sum of constrained eigenvalues = 0.176
intrasetcorrelations:
axis 1NO3 -0.958Cl- -0.932
2nd Partial CCA-- Effects of SiO2 and temperature on taxon abundance:
eigenvalues:
axis 1 0.180axis 2 0.107Sum of unconstrained eigenvalues = 0.415Sum of constrained eigenvalues = 0.287
intrasetcorrelations:
axis 1SiO2 -0.966temp -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,μ ˜ A x( )( ), x ∈ X{ }
where the fuzzy set is a grade of membership on the interval [0, 1]. That is,
€
μ : ˜ A → 0,1[ ]
is a mapping where each element x is assigned a degree of membership
€
0 ≤ μ ˜ A x( ) ≤1
Set theoretic operations on two fuzzy sets include intersection and unionand are defined, respectively as
€
∀x ∈ X, μ ˜ A ∩ ˜ B x( ) = min μ ˜ A
x( ),μ ˜ B x( )( )
∀x ∈ X, μ ˜ A ∪ ˜ B x( ) = max μ ˜ A
x( ),μ ˜ B x( )( )
€
˜ R = x,y( ),μ ˜ 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,
Similar to fuzzy sets, a fuzzy relation on X and Y is
€
μ ˜ R : X ×Y → 0,1[ ]
€
0 ≤ μ ˜ R x, y( ) ≤1
is a mapping where elements x and y are assigned a degree of membership
€
˜ R 1 o ˜ R 2 = x,z( ),maxy
min μ ˜ R 1x,y( ),μ ˜ R 2
y,z( ){ }{ } ⎡ ⎣ ⎢
⎤ ⎦ ⎥
⎧ ⎨ ⎩
x ∈ X,y ∈ Y,z ∈ Z ⎫ ⎬ ⎭
€
˜ R 2 y,z( ), y,z( )∈ Y × Z
€
˜ R 1 x, y( ), x, y( )∈ X ×Y and
The max-min composition is
For n-ary fuzzy relations
Two fuzzy relations:
€
˜ R 1 x, y( ) and
€
˜ R 3 y,z( )
€
˜ R 1 :0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0
y1 y2 y3 y4 y5 y6 y7 y8 y9 y10
x1
x2
x3
x4
€
˜ R 3 :
0 00 00 00 00 00 00 00 00 00 0
y1 y2 y3 y4 y5
y6 y7 y8
y9
y10
z1 z2
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)
€
˜ R 1 x, y( ) and
€
˜ R 2 w,x( ) or
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
w1
w2
w3
w4
w5
w6
x1 x2 x3 x4
€
˜ R 2 :
1st projection of a fuzzy relation:
€
˜ R (1) = x,maxy
μ ˜ R x, y( )
⎛ ⎝ ⎜ ⎞
⎠ ⎟ x, y( )∈ X ×Y
⎧ ⎨ ⎩
⎫ ⎬ ⎭
€
˜ R T( ) = maxx
maxy
μ ˜ R x, y( ) x, y( )∈ X ×Y
⎧ ⎨ ⎩
⎫ ⎬ ⎭
2nd projection of a fuzzy relation:
€
˜ R (2) = y,maxx
μ ˜ R x,y( )( ) x,y( )∈ X ×Y
⎧ ⎨ ⎩
⎫ ⎬ ⎭
Total projection of a fuzzy relation:
Aggregation operations
5 dominant taxa from each sampling month were used:
June-- Asterionella formosa August-- Achnanthidium minutissimum Fragilaria capucina Cyclotella #6 F. crotonensis C. comensis Urosolenia eriensis C. michiganiana Tabellaria fenestrata C. pseudostelligera
Fuzzification of species scores or intraset correlation coefficients:
€
fwa =w − axmin( )
axmax − axmin( ) Normalization of weighted averages or intraset correlation coefficients per axis.
Fuzzy importance matrices(from CCA):
0.00 0.43 0.48 0.170.08 0.00 1.00 0.270.33 1.00 0.00 1.000.33 1.00 0.00 1.000.99 0.59 0.97 0.001.00 0.51 0.60 0.13
temperature
species axis 1 species axis 2 species axis 3 species axis 4
SiO2
TSP
NH3
Cl-
NO3
0.31 0.56 0.50 0.400.89 0.46 0.31 0.400.07 0.49 0.31 0.310.16 0.50 0.40 0.410.12 0.51 0.29 0.410.03 0.52 0.29 0.510.84 0.63 0.55 0.430.96 0.25 0.08 0.400.88 0.50 0.37 0.420.84 0.55 0.44 0.41
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
temperature SiO2 TSP NH3 Cl- NO3
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
0.48 0.50 0.56 0.56 0.56 0.510.43 0.31 0.46 0.46 0.89 0.890.43 0.31 0.49 0.49 0.49 0.490.43 0.40 0.50 0.50 0.50 0.500.43 0.29 0.51 0.51 0.51 0.510.43 0.29 0.52 0.52 0.52 0.510.48 0.55 0.63 0.63 0.84 0.840.25 0.27 0.40 0.40 0.96 0.960.43 0.37 0.50 0.50 0.88 0.880.44 0.44 0.55 0.55 0.84 0.84
Max-min composition of fuzzy relation between fuzzy importance matrices:
1st projection:
2nd projection:
Total projection = 0.96
Achnanthidium minutissimum 0.56 Asterionella formosa 0.89 Cyclotella #6 0.49 C. comensis 0.50 C. michiganiana 0.51 C. pseudostelligera 0.52 Fragilaria capucina 0.84 Fragilaria crotonensis 0.96 Tabellaria fenestrata 0.84 Urosolenia eriensis 0.88
temperature 0.48 SiO2 0.55 TSP 0.63 NH3 0.63 Cl- 0.96 NO3 0.96
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
C. comensis
C. michiganiana
C. pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Tabellaria fenestrata
Urosolenia eriensis
Max-min composition of fuzzy relations --> NO3 and Cl- (from partial CCA):
Achn
anth
idiu
m m
inut
issim
um
Aste
rione
lla fo
rmos
a
Cyclo
tella
#6
Cyclo
tella
com
ensis
Cyclo
tella
mic
higa
nian
a
Cyclo
tella
pse
udos
telli
gera
Frag
ilaria
capu
cina
Frag
ilaria
crot
onen
sis
Uro
sole
nia
erie
nsis
Tabe
llaria
fene
strat
a
0.36 0.49 0.38 0.37 0.40 0.38 0.56 0.19 0.45 0.56
0.53 0.46 0.40 0.46 0.40 0.41 0.55 0.19 0.45 0.46
0.36 0.49 0.38 0.37 0.40 0.38 0.49 0.19 0.45 0.49
0.36 0.49 0.38 0.37 0.40 0.38 0.50 0.19 0.45 0.50
0.36 0.49 0.38 0.37 0.40 0.38 0.51 0.19 0.45 0.51
0.36 0.49 0.38 0.37 0.40 0.38 0.52 0.19 0.45 0.52
0.53 0.49 0.40 0.46 0.40 0.41 0.60 0.19 0.45 0.57
0.53 0.32 0.40 0.46 0.40 0.41 0.55 0.19 0.37 0.44
0.53 0.49 0.40 0.46 0.40 0.41 0.55 0.19 0.45 0.50
0.53 0.49 0.40 0.46 0.40 0.41 0.55 0.19 0.45 0.55
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
C. comensis
C. michiganiana
C. pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Tabellaria fenestrata
Urosolenia eriensis
Max-min composition of fuzzy relations --> SiO2 and temperature
Achn
anth
idiu
m m
inut
issim
um
Aste
rione
lla fo
rmos
a
Cyclo
tella
#6
Cyclo
tella
com
ensis
Cyclo
tella
mic
higa
nian
a
Cyclo
tella
pse
udos
telli
gera
Frag
ilaria
capu
cina
Frag
ilaria
crot
onen
sis
Uro
sole
nia
erie
nsis
Tabe
llaria
fene
strat
a
0.35 0.49 0.38 0.37 0.40 0.38 0.56 0.31 0.45 0.56
0.40 0.46 0.55 0.43 0.50 0.43 0.46 0.48 0.45 0.46
0.35 0.49 0.38 0.37 0.40 0.38 0.49 0.19 0.45 0.49
0.35 0.49 0.38 0.37 0.40 0.38 0.50 0.19 0.45 0.50
0.35 0.49 0.38 0.37 0.40 0.38 0.51 0.19 0.45 0.51
0.35 0.49 0.38 0.37 0.40 0.38 0.52 0.19 0.45 0.52
0.40 0.49 0.55 0.43 0.50 0.43 0.60 0.48 0.45 0.57
0.40 0.44 0.55 0.43 0.50 0.43 0.36 0.48 0.41 0.41
0.40 0.49 0.55 0.43 0.50 0.43 0.50 0.48 0.45 0.50
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 0.56 Asterionella formosa 0.55 Cyclotella #6 0.49 C. comensis 0.50 C. michiganiana 0.51 C. pseudostelligera 0.52 Fragilaria capucina 0.60 Fragilaria crotonensis 0.55 Tabellaria fenestrata 0.55 Urosolenia eriensis 0.55
2nd projection: (columns)
NO 3
and
Cl-
SiO 2
and
tem
pera
ture
Total projection = 0.60
Achnanthidium minutissimum 0.53 0.40 Asterionella formosa 0.49 0.49 Cyclotella #6 0.40 0.55 C. comensis 0.46 0.43 C. michiganiana 0.40 0.50 C. pseudostelligera 0.41 0.43 Fragilaria capucina 0.60 0.60 Fragilaria crotonensis 0.19 0.48 Tabellaria fenestrata 0.57 0.57 Urosolenia eriensis 0.45 0.45
Achn
anth
idiu
m m
inut
issim
um
Aste
rione
lla fo
rmos
a
Cyclo
tella
#6
Cyclo
tella
com
ensis
Cyclo
tella
mic
higa
nian
a
Cyclo
tella
pse
udos
telli
gera
Frag
ilaria
capu
cina
Frag
ilaria
crot
onen
sis
Uro
sole
nia
erie
nsis
Tabe
llaria
fene
strat
a
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
NO3 and Cl- influence
All
env
iron
men
tal i
nflu
ence
s
Achn
anth
idiu
m m
inut
issim
um
Aste
rione
lla fo
rmos
a
Cyclo
tella
#6
Cyclo
tella
com
ensis
Cyclo
tella
mic
higa
nian
a
Cyclo
tella
pse
udos
telli
gera
Frag
ilaria
capu
cina
Frag
ilaria
crot
onen
sis
Uro
sole
nia
erie
nsis
Tabe
llaria
fene
strat
a
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Fragilaria crotonensis
Urosolenia eriensis
Tabellaria fenestrata
SiO2 and temperature influence
All
env
iron
me n
tal i
nflu
e nc e
s
Fragilaria crotonensis
Achnanthidium minutissimum
Asterionella formosa
Cyclotella #6
Cyclotella comensis
Cyclotella michiganiana
Cyclotella pseudostelligera
Fragilaria capucina
Urosolenia eriensis
Tabellaria fenestrata
Augus
t tax
a (SiO 2
and t
empe
ratur
e inf
luenc
e)
June
taxa
(NO 3
and C
l- influ
ence
)
2nd projections:
>> 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.
Summary
2. Composition of fuzzy relations--10 taxa X 10 taxa (from partial CCAs):
Linguistic equivalent of 2nd projections as degree of influence* June August Taxa NO3 and Cl- SiO2 and temperature Fragilaria capucina Tabellaria fenestrata most highly most highly Asterionella formosa more highly more highly
Urosolenia eriensis highly highly
Achnanthidium minutissimum most highly less highly
Cyclotella comensis more highly less highly
Cyclotella #6 lesser most highly
Cyclotella michiganiana much lesser more highly
Cyclotella pseudostelligera much lesser less highly
Fragilaria crotonensis 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