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The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail - both as as a deep example and an important class of locally stationary processes. In the…
Machine Learning tools are nowadays widely applied extensively to the prediction of the properties of molecular materials, using datasets extracted from high-throughput computational models. In several cases of scientific and technological…
We establish the various properties as well as diverse relations of the ascent and descent spectra for bounded linear operators. We specially focus on the theory of subspectrum. Furthermore, we construct a new concept of convergence for…
Multivariate processes with long-range dependent properties are found in a large number of applications including finance, geophysics and neuroscience. For real data applications, the correlation between time series is crucial. Usual…
Multivariate spatial phenomena are ubiquitous, spanning domains such as climate, pandemics, air quality, and social economy. Cross-correlation between different quantities of interest at different locations is asymmetric in general. This…
In the 1940s, Wiener introduced a linear predictor, where the future prediction is computed by linearly combining the past data. A transformer generalizes this idea: it is a nonlinear predictor where the next-token prediction is computed by…
Layered stable (multivariate) distributions and processes are defined and studied. A layered stable process combines stable trends of two different indices, one of them possibly Gaussian. More precisely, in short time, it is close to a…
This paper revisits the definition of linear time-invariant (LTI) stochastic process within a behavioral systems framework. Building on [Willems, 2013], we derive a canonical representation of an LTI stochastic process and a physically…
Traditionally stationarity refers to shift invariance of the distribution of a stochastic process. In this paper, we rediscover stationarity as a path property instead of a distributional property. More precisely, we characterize a set of…
Univariate marked Hawkes processes are used to model a range of real-world phenomena including earthquake aftershock sequences, contagious disease spread, content diffusion on social media platforms, and order book dynamics. This paper…
A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in R^d and in some cases provide a full characterisation…
We consider a stationary process (with either discrete or continuous time) and find an adaptive approximating stationary process combining approximation quality and supplementary good properties that can be interpreted as additional…
Gaussian processes occupy one of the leading places in modern statistics and probability theory due to their importance and a wealth of strong results. The common use of Gaussian processes is in connection with problems related to…
We consider the junction of multiple one-dimensional systems and study how conserved currents transport at the junction. To characterize the transport process, we introduce reflection/transmission coefficients by applying boundary conformal…
Learning meaningful representations of data is an important aspect of machine learning and has recently been successfully applied to many domains like language understanding or computer vision. Instead of training a model for one specific…
This paper considers regression tasks involving high-dimensional multivariate processes whose structure is dependent on some {known} graph topology. We put forth a new definition of time-vertex wide-sense stationarity, or joint stationarity…
The paper studies properties of continuous time processes with spectrum degeneracy at a single point where their Fourier transforms vanish with a certain rate. It appears that these processes are linearly predictable in some weak sense,…
Linearity allows several versions of reality to simultaneously exist in the state vector. But it implies that there is no interaction between versions, and that there will never be perception of more than one version. It also implies, in…
Historical sciences like evolutionary biology reconstruct past events by using the traces that the past has bequeathed to the present. The Markov Chain Convergence Theorem and the Data Processing Inequality describe how the mutual…
In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent…