1. Phytoplankton assemblages, environmental influences and their seasonal changes measured using weighted averages and fuzzy set theory IAGLR 2005 Ann Arbor, MI May 23 - 27

2. Small data set High phytoplankton diversity Multiple environmental variables How are individual taxa affected by environmental variables and each other? Use canonical correspondence analysis (CCA) Use fuzzy sets and fuzzy relations

3. 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-- SiO 2, NO 3, TSP, NH 3, Cl -, and temperature Local: Fuzzy data analysis using fuzzy relations

4. j1, j2, j3 = June a1, a2, a3 = August 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 CCA axis 1 axis 2 j1 a1 a2 a3 j2 j3 SiO 2 NO 3 Cl - temp TSP/ NH 3

5. intraset correlations: axis 1 SiO 2 -0.834 NO 3 0.994 TSP-0.326 NH 3 -0.326 Cl - 0.968 temp-0.984 eigenvalues: axis 10.576 axis 20.182 axis 30.159 axis 40.086 Sum of all unconstrained eigenvalues: 1.131 Sum of all constrained eigenvalues: 1.003 Residual unconstrained variation: 0.128 (89% of species variation explained)

6. 1st Partial CCA-- Effects of Cl - and NO 3 on taxon abundance: eigenvalues: axis 10.176 axis 20.129 Sum of unconstrained eigenvalues = 0.304 Sum of constrained eigenvalues = 0.176 intraset correlations: axis 1 NO 3 -0.958 Cl - -0.932 2nd Partial CCA-- Effects of SiO 2 and temperature on taxon abundance: eigenvalues: axis 10.180 axis 20.107 Sum of unconstrained eigenvalues = 0.415 Sum of constrained eigenvalues = 0.287 intraset correlations: axis 1 SiO 2 -0.966 temp-0.553

7. Fuzzy Set Theory: Let X be a non-empty set that is defined as the universe of discourse, and the elements of X are x 1, x 2, …, x n. A subset of X defined as a fuzzy set is where the fuzzy set is a grade of membership on the interval [0, 1]. That is, is a mapping where each element x is assigned a degree of membership Set theoretic operations on two fuzzy sets include intersection and union and are defined, respectively as

8. 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 is a mapping where elements x and y are assigned a degree of membership

9. and The max-min composition is For n-ary fuzzy relations

10. Two fuzzy relations:and y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 x1x2x3x4x1x2x3x4 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 z 1 z 2 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 NO 3 or SiO 2 and temperature (from partial CCAs) andor x 1 x 2 x 3 x 4

11. 1st projection of a fuzzy relation: 2nd projection of a fuzzy relation: Total projection of a fuzzy relation: Aggregation operations

12. 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:  Normalization of weighted averages or intraset correlation coefficients per axis.

13. Fuzzy importance matrices (from CCA): temperature species axis 1 species axis 2 species axis 3 species axis 4 SiO 2 TSP NH 3 Cl - NO 3 Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata

14. temperature SiO 2 TSPNH 3 Cl - NO 3 Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata Max-min composition of fuzzy relation between fuzzy importance matrices:

15. 1st projection: 2nd projection: Total projection = 0.96

16. Max-min composition of fuzzy relations --> NO 3 and Cl - (from partial CCA): Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata

17. Max-min composition of fuzzy relations --> SiO 2 and temperature Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata

18. 1st projection: (Effects of all environmental variables => across rows) Degree of Cl - and NO 3 influence = degree of SiO 2 and temperature influence

19. 2nd projection: (columns) NO 3 and Cl - SiO 2 and temperature Total projection = 0.60

20. Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata NO 3 and Cl - influence All environmental influences

21. Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Fragilaria crotonensis Urosolenia eriensis Tabellaria fenestrata SiO 2 and temperature influence All environmental influences

22. Fragilaria crotonensis Achnanthidium minutissimum Asterionella formosa Cyclotella #6 Cyclotella comensis Cyclotella michiganiana Cyclotella pseudostelligera Fragilaria capucina Urosolenia eriensis Tabellaria fenestrata August taxa (SiO 2 and temperature influence) June taxa (NO 3 and Cl - influence) 2nd projections:

23. >> From CCA: NO 3 and Cl - influenced June taxa; SiO 2 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 NO 3 to a greater degree than August taxa by SiO 2 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 SiO 2. Summary

24. 2.Composition of fuzzy relations--10 taxa X 10 taxa (from partial CCAs): Linguistic equivalent of 2nd projections as degree of influence* *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.

25. 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