Curve fitting through orthogonal wavelets

German J Pliego, Purdue University

Abstract

The theory of wavelets provides the necessary infrastructure to generate a sequence of nested subspaces of $L\sp2$ which contain increasingly more accurate approximations of a function f in $L\sp2$. The subspaces are less complicated than $L\sp2$. This work concentrates on the identification and description of properties of $\varphi\sb{2N}$, Daubechies' orthogonal scaling functions, which generate the subspaces. These are used to understand and make use of the corresponding approximations for estimating a function f based on noisy observations at equally spaced intervals. It is shown that there is a constant $k\sb{2N}$, which depends upon the Daubechies orthogonal scaling function chosen, such that it is reasonable to assume that the observations are taken at the integers shifted by $k\sb{2N}$, rather than at the integers or half integers. Assuming that $\varphi\sb4(k\sb4$ + 1) = 1, it is then shown that the initial continuous approximation interpolates f at the observations. Here the scaling function coefficients of f in each subspace are described as the discrete representation of f. It is shown that the discrete representation of f reproduces polynomials of degree N when N equals 2 or 4. A boundary convention for finite data is examined. The following software was developed so that we could carry out an empirical curve fitting making use of the above Daubechies' scaling function properties $\varphi\sb{2N}$: (1) A modified version of a Fast Wavelet Transform (FWT) algorithm for orthogonal wavelets (STRANG) was designed. It carries out the decomposition of a function f based on a finite number of values into scaling and wavelet coefficients which provide discrete representations of f. (2) A complementary routine to the FWT algorithm that provides continuous representations of f was developed. (3) Two versions of the software were developed: One to carry out simulations assuming the function f known and including it in the output for comparison purposes, and the other to analyze real data cases where f is unknown. A series of simulations and examples are provided.

Degree

Ph.D.

Advisors

Bock, Purdue University.

Subject Area

Statistics

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