Showing papers on "Function approximation published in 1983"

Calculator Function Approximation

Abstract: (1983). Calculator Function Approximation. The American Mathematical Monthly: Vol. 90, No. 5, pp. 317-325.

Function approximations in the mathematical laboratory

Abstract: The article treats the subject of function approximations for a computer's built‐in library as a typical topic for a school mathematical laboratory. Following a brief description of how such a laboratory is envisaged as part of mathematical education, approximation by interpolation is discussed at the pre‐calculus level. Special attention is given to the removal of ‘black box’ procedures, and inherent concepts such as quality of approximation, relative error and computational efficiency are examined. Suitable examples are given, demonstrating the actual construction of library functions.

On the problem of function approximation by sample set processing for an incompletely determined model

Abstract: In mathematical statistics, the problem of function approximation is usually considered in the following way. One is given a finite system of functions, {+ i (x ) } , such that an unknown function to be approximated can be expressed as a linear combination of the functions 4i(x) plus a random error. The problem of estimating the unknown coefficients of the linear combination using a sample set has been well studied for different cases. However, in many practical problems, the system of functions (+,(x)} is not known completely, but only partially. We shall call such models incompletely determined. The model for predicting solar flares4 and some models for medical diagnosis are typical examples. As one considers such models, one finds some new problems that do not exist for completely determined models. Among them is the question, Should one use all available functions 4i(x) in the model or discard some of them? If the latter is preferable (and we shall see that it is) then what is the criterion of selection? If it is possible to use the complete model, is that always preferable to discarding some functions? We shall show that the use of an incompletely determined model sometimes leads to better results. The influence of the precision of the model and the experimental data on the selection criterion is also considered. The case with errors in both dependent and independent variables is especially discussed. A study of such problems was made by the author for a simplified and, in a sense, artificial model of function approximation in Reference 5. In this paper, the study is performed for the method of least squares, which is often used for function approximation in practice.

Restoration of noisy images using a raised cosine function approximation

Abstract: In order to implement image restoration techniques on a digital computer an accurate representation of the image in discrete form is required. A new set of functions called raised cosine functions is suggested for this purpose. These functions have properties similar to those of the spline functions which are useful for image representation and inversion of the degrading phenomenon. The singular value decomposition in the raised cosine function domain is used for restoration of noisy degraded images.

On the problem of function system selection for function approximation based on the use of a sample set with defects

Abstract: The problem of function approximation is considered in mathematical statistics as follows. One knows a finite system of functions, {di(x)], such that an unknown function to be approximated can be expressed as a linear combination of the functions {di(x)] plus a random error, the coefficients of the linear combination being determined by a sample set with the help of a well-known procedure, such as the method of least squares. But, under real conditions, the investigator rarely knows the system of functions { d i ( x ) ) precisely. Even if he is certain about some of them, the others may well be considered only candidates that should be tried beforehand. In any case, he almost never knows the whole system { d i ( x ) ] . That is why such a model, the incompletely determined model, was considered in Reference 1, highlighting the problem of the selection of the functions, di(x), to be used. If the coefficients of the model are determined with the help of the method of least squares, the main result of the study of the orthonormalized system of functions, { d i ( x ) } , may be formulated as follows.* If the modulus of the coefficient of the proposed function is less than a threshold value, the function must be discarded; otherwise, it must be included in the model. The threshold value has the form

Projection Pursuit Regression: Some Mathematics.

Abstract: : We present some mathematical analysis for a class of curve fitting algorithms labeled 'projection pursuit' algorithms. These algorithms approximate a general function of p variables by a sum of non-linear functions of linear combinations. The approximation is computationally feasible and performs well in examples of nonparametric regression with noisy data, high dimensional density estimation, and multidimensional spline approximation. This note treats the algorithms from the point of view of approximation theory. It is easy to show that approximation is always possible.