Prediction of functional ARMA processes with an application to traffic data

Klepsch, Johannes; Klüppelberg, Claudia; Wei, Tong

doi:10.1016/j.ecosta.2016.10.009

Cited by 65 publications

(49 citation statements)

References 17 publications

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“…where (ν i ) i∈N is some ONB of H, and consider the projection of a functional random element on A D . In [2] and [18] the authors consider the projection of a FAR process (X n ) n∈Z on A D , where ν 1 , . .…”

Section: Prediction Based On a Finite Dimensional Projectionmentioning

confidence: 99%

“…More practical results are given for example in [1], where prediction is performed nonparametrically with a functional kernel regression technique, or in [2], [16] and [18], where the dimensionality of the prediction problem is reduced via functional principal component analysis. In a multivariate setting, the Innovations Algorithm proposed in [8] gives a robust prediction method for linear processes.…”

Section: Introductionmentioning

confidence: 99%

“…To achieve convergence in (1.1) we work with finite dimensional projections of the functional process, similarly as in [2] and [18]. We start with a representation of the functional linear model in terms of an arbitrary orthonormal basis of the Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

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An innovations algorithm for the prediction of functional linear processes

Klepsch

Klüppelberg

2017

Journal of Multivariate Analysis

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When observations are curves over some natural time interval, the field of functional data analysis comes into play. Functional linear processes account for temporal dependence in the data. The prediction problem for functional linear processes has been solved theoretically, but the focus for applications has been on functional autoregressive processes. We propose a new computationally tractable linear predictor for functional linear processes. It is based on an application of the Multivariate Innovations Algorithm to finite-dimensional subprocesses of increasing dimension of the infinite-dimensional functional linear process. We investigate the behavior of the predictor for increasing sample size. We show that, depending on the decay rate of the eigenvalues of the covariance and the spectral density operator, the resulting predictor converges with a certain rate to the theoretically best linear predictor.

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Section: Prediction Based On a Finite Dimensional Projectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An innovations algorithm for the prediction of functional linear processes

Klepsch

Klüppelberg

2017

Journal of Multivariate Analysis

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“…Hays et al , and daily vehicle traffic curves, e.g. Klepsch et al . Other examples are given in the books by Horváth and Kokoszka and Kokoszka and Reimherr .…”

Section: Introductionmentioning

confidence: 99%

Principal Components Analysis of Periodically Correlated Functional Time Series

Kidziński

Kokoszka

Jouzdani

2018

Journal Time Series Analysis

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Within the framework of functional data analysis, we develop principal component analysis for periodically correlated time series of functions. We define the components of the above analysis including periodic operator‐valued filters, score processes, and the inversion formulas. We show that these objects are defined via a convergent series under a simple condition requiring summability of the Hilbert–Schmidt norms of the filter coefficients and that they possess optimality properties. We explain how the Hilbert space theory reduces to an approximate finite‐dimensional setting which is implemented in a custom‐build |R| package. A data example and a simulation study show that the new methodology is superior to existing tools if the functional time series exhibits periodic characteristics.

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“…Klepsch and Klüppelberg (2016) proposed a functional moving average (FMA) process and introduced an innovations algorithm to ensure that the best linear predictor could be determined. Klepsch, Klüppelberg, and Wei (2017) extended the FAR and FMA processes to a functional autoregressive moving average (FARMA). Recently, Li, Robinson, and Shang (2017) considered long-range dependent functional time series, and developed a functional autoregressive integrated moving average process.…”

Section: Introductionmentioning

confidence: 99%

Forecasting intraday S&P 500 index returns: A functional time series approach

Shang

2017

Journal of Forecasting

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Financial data often take the form of a collection of curves that can be observed sequentially over time; for example, intraday stock price curves and intraday volatility curves. These curves can be viewed as a time series of functions that can be observed on equally spaced and dense grids. Owing to the so‐called curse of dimensionality, the nature of high‐dimensional data poses challenges from a statistical perspective; however, it also provides opportunities to analyze a rich source of information, so that the dynamic changes of short time intervals can be better understood. In this paper, we consider forecasting a time series of functions and propose a number of statistical methods that can be used to forecast 1‐day‐ahead intraday stock returns. As we sequentially observe new data, we also consider the use of dynamic updating in updating point and interval forecasts for achieving improved accuracy. The forecasting methods were validated through an empirical study of 5‐minute intraday S&P 500 index returns.

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Prediction of functional ARMA processes with an application to traffic data

Cited by 65 publications

References 17 publications

An innovations algorithm for the prediction of functional linear processes

An innovations algorithm for the prediction of functional linear processes

Principal Components Analysis of Periodically Correlated Functional Time Series

Forecasting intraday S&P 500 index returns: A functional time series approach

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