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