Functional estimation problem algorithm derivation

The remainder of this chapter is structured as follows. In Section II the problem of deriving an estimate of an unknown function from empirical data is posed and studied in a theoretical level. Then, following Vapnik s original work (Vapnik, 1982), the problem is formulated in mathematical terms and the sources of the error related to any proposed solution to the estimation problem are identified. Considerations on how to reduce these errors show the inadequacy of the NN solutions and lead in Section III to the formulation of the basic algorithm whose new element is the pointwise presentation of the data and the dynamic evolution of the solution itself. The algorithm is subsequently refined by incorporating the novel idea of structural adaptation guided by the use of the L" error measure. The need... 

All major NLP algorithms require estimation of first derivatives of the problem functions to obtain a solution and to evaluate the optimality conditions. If the values of the derivatives are computed inaccurately, the algorithm may progress very slowly, choose poor directions for movement, and terminate due to lack of progress or reaching the iteration limits at points far from the actual optimum, or, in extreme cases, actually declare optimality at nonoptimal points. 

We can approach performance modeling in different ways. One way is to formally derive the asymptotic behavior of the most time critical part of the program. The asymptotic behavior of an algorithm gives an estimate of the execution time as a function of problem size and of possibly other parameters. The notation that is commonly used is called big-oh [43]. For example, the statement that says that some method scales as means that there are positive constants c and... 

Also in chemistry artificial neural networks have found wide use. They have been used to fit spectroscopic data, to investigate quantitative structure-activity relationships (QSAR), to predict deposition rates in chemical vapor deposition, to predict binding sites of biomolecules, to derive pair potentials from diffraction data on liquids, " to solve the Schrodinger equation for simple model potentials like the harmonic oscillator, to estimate the fitness function in genetic algorithm optimizations, in experimental data analysis, to predict the secondary structure of proteins, to predict atomic energy levels, " and to solve classification problems from clinical chemistry, in particular the differentiation between diseases on the basis of characteristic laboratory data. ... 

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