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Fourier shape analysis and fuzzy measure shape group differentiation of Great Lakes Asterionella Hassall (Heterokontophyta, Bacillariophyceae)

Pappas, Janice; Stoermer, Eugene

Pappas, Janice; Stoermer, Eugene

2001

Citation:Proceedings 16th IDS, pp. 485-501 <http://hdl.handle.net/2027.42/94109>

Abstract: Species separation of character-poor Asterionella, an abundant Great Lakes diatom, is difficult to accomplish by visual inspection alone. Diatom shape is an inherited property which is measurable. Quantifiable variation is mostly the result of genetic influences, while qualitative changes in valve morphology are mostly the result of environmental influences. Asterionella valve shape and shape group separation was studied using principal components analysis (PCA) and fuzzy measure theory. To quantify shape, Fourier coefficients were calculated from arc lengths and tangent angles around the periphery of the valve outline. Polar Fourier coefficients resulted from an orthogonal polynomial N regression of Phi*(t) = a sub 0 + sum from n to N of A sub n times cos(nt +a sub n), where a sub 0 is the zero'th Fourier coefficient, A sub n is the nth amplitude, and a sub n is the n'th phase angle. One-hundred x,y coordinates were used to calculate 22 shape coefficients to get a best fit closed curve in a least squares sense. Standardized PCA of mean-corrected, square root transformed amplitudes produced seven Asterionella shape groups. Classification integration and fuzzy measures were used to determine degree of shape group overlap and degree that specimens belonged to an assigned shape group. Fuzzy measures were based on morphometry of head pole, foot pole, and mid-valve widths or a combination and scaled and ordered on the interval [0,1). Sugeno's or the fuzzy integral, E= fuzzy integral of h(x)o g(•),where E is the evaluation of h(x) (partial evidence) and g(•) (importance or possibility measure), was used. Partial evidence was fuzzy average overlap. Degree of shape group membership was evaluated as degree of certainty (partial evidence) and Sugeno's measure (importance measure). Complete overlap or specimen inclusion was equal to one, complete lack of overlap or specimen exclusion was equal to zero, and the crossover point was 0.5. Two exceptions, shape groups II-III and shape groups IV-V atE= 0.6 exhibited overlap. Two specimen assignments were slightly questionable at E= 0.49 and E = 0.57 for shape groups II and VI, respectively. All other specimen assignments were E > or = 0.6. Overall, shape group differences were evident and may indicate different species of Asterionella Hassal. ·