Legendre polynomials as a recommended basis for numerical differentiation in the presence of stochastic white noise

Abstract

Abstract. In this paper, we consider the problem of estimating the derivative of a function from its noisy version contaminated by a stochastic white noise and argue that in certain relevant cases the reconstruction of by the derivatives of the partial sums of Fourier–Legendre series of has advantage over some standard approaches. One of the interesting observations made in the paper is that in a Hilbert scale generated by the system of Legendre polynomials the stochastic white noise does not increase, as it might be expected, the loss of accuracy compared to the deterministic noise of the same intensity. We discuss the accuracy of the considered method in the spaces L2 and C and provide a guideline for an adaptive choice of the number of terms in differentiated partial sums (note that this number is playing the role of a regularization parameter). Moreover, we discuss the relation of the considered numerical differentiation scheme with the well-known Savitzky–Golay derivative filters, as well as possible applications in diabetes technology.

Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

“…Secondly, we avoid the use of least square optimization altogether, using a summability method. We show that, by employing recent results [18] together with the modifications, we derive a method that yields lower noise propagation error than in other approach considered in [18].…”

“…We demonstrate numerically that these modifications lead to a performance superior to the Savitzky-Golay method as modified in [18] on a number of numerical examples.…”

“…However, in addition to the intrinsic ill-conditioning of numerical differentiation, the solution of the least square problem as posed above involves a system of linear equations with the Hilbert matrix of order n, which is notoriously illconditioned. Therefore, it is proposed in [18] to use Legendre polynomials rather than the monomials as the basis for the space of polynomials of degree n. A procedure to choose n is given in [18], together with error bounds in terms of n and δ which are optimal up to a constant factor for the method in the sense of the oracle inequality.…”

“…In this paper, we propose two modifications of the approach [18]. Firstly, we propose the use of judiciously selected weights in the least square method as in (2) except for the use of Legendre basis.…”

“…In the paper [18], the authors considered the following modification of (11) for approximating f ′ (t)…”