Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Meth.pdf

Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization SK Mishra Dept. of Economics NEHU, Shillong (India) Introduction: The Gielis superformula 1 1 2 3 ( ) 1 1 4 4( ) ( ). cos( ) sin( ) ( ). ( ) ; 0 nn nm m a br f f g mθ θ θ θ θ θ ??
= + =    … (1) describes almost any closed curve in terms of the deformed circle (or ellipse), ( ),g θ and another function, ( ),f θ and their parameters (Gielis, 2003). The function ( )f θ may be considered as a modifier of the Gielis function, ( )g θ . Estimation of Gielis Parameters: For a scientific purpose, Gielis parameters need to be estimated from empirical data. Presently, we are concerned with the possibilities of the same. Let the n true points be [ ( , ); 1,2,..., ]i i iz x y i n= = , of which the corresponding observed values are ( , )i iz x y′ ′ ′= , possibly with errors of measurement and displacement of origin by ( , )x yc c , unknown to us. Let ( , )x yc c

be the approximate or assumed values of ( , ).x yc c Let us denote by ( , ) ( , ).i i i i x i yz x y x c y c′ ′= = ? ?




From these values we obtain 2 2( )i i ir x y= +


. We also obtain 1tan ( / )i i iy xθ ?=


. On the other hand, we obtain 1 2 3? ( , , , , , , ). ( ),i ir g a b m n n n fθ θ=







where (.)g is the Gielis super-formula defined in (4) and ( )f θ is variously defined. The wavy bar on the arguments of (.)g and (.)f indicates that all parameters have taken on some arbitrary values, which may not be the correct values. The deviation of assumed values of parameters from their true values gives rise to ?( )i i id abs r r= ?
and consequently 2 2 1 0. n i i S d = = ≥ Only if the assumed values of parameters are the true values, 2S can be zero, but smaller it is, closer are the assumed values of the parameters from their true values (assuming empirical uniqueness of the parameters to a given set of data). Thus we