Fourier shape analysis and shape group determination by principal component analysis and fuzzy measure theory of <italic>Asterionella</italic> hassall (Heterokontophyta, Bacillariophyceae) from the Great Lakes.

Abstract

Species separation of character-poor <italic>Asterionella</italic>, an abundant Great Lakes diatom, is difficult to accomplish by visual inspection alone. <italic>Asterionella</italic> shape, which is an ontogenic property, was studied quantitatively including novel application of fuzzy measure theory. To quantify shape, Fourier coefficients were calculated from arc lengths and tangent angles around the valve periphery of 96 <italic>Asterionella</italic> from Lakes Superior, Huron, and Michigan. Polar Fourier coefficients resulted from orthogonal polynomial regression of F&ast;t</fen> =a0+n=1 NAn cos nt+an</fen> , where a₀ is the zeroth Fourier coefficient, A_n is the nth amplitude, and alpha_n is the nth phase angle for N coefficients. 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 and visual inspection produced seven <italic>Asterionella</italic> shape groups. Classification integration and fuzzy measures were used to determine degree of shape group overlap and degree that specimens belonged to their 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=<?Eqn TeX style=display> Hx</fen>&j0; g&bull;</fen> ,<?Eqn TeX endstyle=display> where E is the evaluation of h(x) (partial evidence) and g&bull;</fen> (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 cross-over point was 0.5. Two exceptions, shape groups II--III and shape groups IV--V at E = 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 &ge; 0.6. Most northern Great Lakes <italic>Asterionella</italic> were members of shape groups I--III. Lake Michigan <italic>Asterionella</italic> were mostly in shape groups IV--VII. Reproductive isolation is supported by additional evidence with regard to Great Lakes water movement patterns and isozyme analysis of Lakes Superior and Michigan <italic>Asterionella</italic>.