By Achim Zielesny
This profitable booklet offers in its moment variation an interactive and illustrative advisor from two-dimensional curve becoming to multidimensional clustering and computer studying with neural networks or aid vector machines. alongside the way in which issues like mathematical optimization or evolutionary algorithms are touched. All options and ideas are defined in a transparent reduce demeanour with graphically depicted plausibility arguments and a bit effortless mathematics.
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Extra resources for From Curve Fitting to Machine Learning: An Illustrative Guide to Scientific Data Analysis and Computational Intelligence
Example text
G. g. see [Hamilton 1964], [Barlow 1989], [Bevington 2002], [Brandt 2002] or [Press 2007]). Again, note that the term linear model function denotes a function that is linear in its parameters only. It does not necessarily mean a linear dependence of the function value y on the argument x. This subtle difference often causes some misunderstandings in scientific practice as far as non-linear fits are concerned. g. y = f (x) = a1 ea2 x To recognize the non-linearity in parameters of the example function a power series expansion is helpful (in this case around x = 0 with a display up to the 4th power): Series[Subscript[a, 1]*Exp[Subscript[a, 2]*x],{x,0,4}] a1 + a1 a2 x + 12 a1 a22 x2 + 16 a1 a32 x3 + 1 4 4 5 24 a1 a2 x + O[x] The cross terms like a1 a2 or a1 a22 and the higher powers of a2 like a22 , a32 , a42 etc.
E. 0)} ]; The grid points are calculated with nested Do loops in the xy plane. 4]]; functionGraphics3D=CIP‘Graphics‘Plot3dPointsWithFunction[ gridPoints3D,pureFunction,labels, GraphicsOptionArgument1Range3D -> xRange, GraphicsOptionArgument2Range3D -> yRange, GraphicsOptionViewPoint3D -> viewPoint3D, GraphicsOptionPlotStyle3D -> plotStyle3D]; Show[functionGraphics3D,arrowGraphics3D] The winner grid point of the global grid search does only approximate the global optimum with an error corresponding to the defined grid spacing.
And it becomes clear that any a priori knowledge about the location of an optimum from theoretical considerations or practical experience may play a crucial role. Throughout the later chapters a number of standard problems are discussed and strategies for their circumvention are described. 2 Optimization 17 <