This work is devoted to functional ARMA(p, q) processes and approximating vector models based on functional PCA in the context of prediction. After deriving sufficient conditions for the existence of a stationary solution to both the functional and the vector model equations, the structure of the approximating vector model is investigated. The stationary vector process is used to predict the functional process. A bound for the difference between vector and functional best linear predictor is derived. The paper concludes by applying functional ARMA processes for the modeling and prediction of highway traffic data.

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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Quantum computing promises to overcome computational limitations with better and faster solutions for optimization, simulation, and machine learning problems. Europe and Germany are in the process of successfully establishing research and funding programs with the objective to advance the technology’s ecosystem and industrialization, thereby ensuring digital sovereignty, security, and competitiveness. Such an ecosystem comprises hardware/software solution providers, system integrators, and users from research institutions, start-ups, and industry. The vision of the Quantum Technology and Application Consortium (QUTAC) is to establish and advance the quantum computing ecosystem, supporting the ambitious goals of the German government and various research programs. QUTAC is comprised of ten members representing different industries, in particular automotive manufacturing, chemical and pharmaceutical production, insurance, and technology. In this paper, we survey the current state of quantum computing in these sectors as well as the aerospace industry and identify the contributions of QUTAC to the ecosystem. We propose an application-centric approach for the industrialization of the technology based on proven business impact. This paper identifies 24 different use cases. By formalizing high-value use cases into well-described reference problems and benchmarks, we will guide technological progress and eventually commercialization. Our results will be beneficial to all ecosystem participants, including suppliers, system integrators, software developers, users, policymakers, funding program managers, and investors.

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