Numerical differentiation procedures for non-exact data

Abstract

Abstract. The numerical differentiation of data divides naturally into two distinct problems:(i) the differentiation of exact data, and (ii) the differentiation of non-exact (experimental) data. In this paper, we examine the latter. Because methods developed for exact data axe based on abstract formalisms which are independent of the structure within the data, they prove, except for the regulaxization procedure of Cullum, to be unsatisfactory for non-exact data. We therefore adopt the point of view that satisfactory methods for non-exact data must take the structure within the data into account in some natural way, and use the concepts of regression and spectrum analysis as a basis for the development of such methods. The regression procedure is used when either the structure within the non-exact data is known on independent grounds, or the assumptions which underlie the spectrum analysis procedure [viz., stationaxity of the (detrended) data] do not apply. In this latter case, the data could be modelled using splines. The spectrum analysis procedure is used when the structure within the nonexact data (or a suitable transformation of it, where the transformation can be differentiated exactly) behaves as if it were generated by a stationary stochastic process. By proving that the regulaxization procedure of Cullum is equivalent to a certain spectrum analysis procedure, we derive a fast Fourier transform implementation for regularization (based on this equivalence) in which an acceptable value of the regulaxization parameter is estimated directly from a time series formulation based on this equivalence. Compared with the regularization procedure, which involves 0 (n 8) operations (where n is the number of data points), the fast Fourier transform implementation only involves O (n log n).

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