An Innovations Algorithm for the prediction of functional linear processes

Estimating functional time series by moving average model fitting

“…Note that there are major differences to the multivariate prediction case. Due to the infinite dimensionality of function spaces, Xf n+1,k in (3.1) is not guaranteed to have a representation in terms of its past values and operators in L, see for instance Proposition 2.2 in Bosq [5] and the discussion in Section 3 of Klepsch and Klüppelberg [13]. A typical remedy in FDA is to resort to projections into principal directions and then to let the dimension d of the projection subspace grow to infinity.…”

“…This, however, has to be done with care, especially if sample versions of the predictors in (3.1) are considered. Even at the population level, the rate at which d tends to infinity has to be calibrated scrupulously to ensure that the inversions of matrices occurring, for example, in the multivariate Innovations Algorithm are meaningful and well defined (see Theorem 5.3 of Klepsch and Klüppelberg [13]).…”

“…where Xn−k,0 = 0. On a population level, it was shown in Klepsch and Klüppelberg [13] that the coefficients θ k,i with k, i ∈ N can be computed with the following algorithm.…”

“…This has led to an upsurge in contributions to the functional time series literature. The many recent works in this area include papers on time-domain methods such as Hörmann and Kokoszka [10], who introduced a framework to describe weakly stationary functional time series, and Aue et al [3] and Klepsch and Klüppelberg [13], who developed functional prediction methodology; as well as frequency domain methods such as Panaretos and Tavakoli [22], who utilized functional cumulants to justify their functional Fourier analysis, Hörmann et al [9], who defined the concept of dynamic functional principal components, and Aue and van Delft [1], who designed stationarity tests based on functional periodogram properties. This paper is concerned with functional moving average (FMA) processes as a building block to estimate potentially more complicated functional time series.…”

“…The operators in the FMA representation, a functional linear filter, are estimated using a functional Innovations Algorithm. This counterpart of the well-known univariate and multivariate Innovations Algorithms was recently introduced by Klepsch and Klüppelberg [13], where its properties were analyzed on a population level. These results are extended to the sample case and used as a first step in the estimation.…”