2 research outputs found
Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables
Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety
of engineering and scientific fields. Dynamic mode decomposition (DMD), which
is a numerical algorithm for the spectral analysis of Koopman operators, has
been attracting attention as a way of obtaining global modal descriptions of
NLDSs without requiring explicit prior knowledge. However, since existing DMD
algorithms are in principle formulated based on the concatenation of scalar
observables, it is not directly applicable to data with dependent structures
among observables, which take, for example, the form of a sequence of graphs.
In this paper, we formulate Koopman spectral analysis for NLDSs with structures
among observables and propose an estimation algorithm for this problem. This
method can extract and visualize the underlying low-dimensional global dynamics
of NLDSs with structures among observables from data, which can be useful in
understanding the underlying dynamics of such NLDSs. To this end, we first
formulate the problem of estimating spectra of the Koopman operator defined in
vector-valued reproducing kernel Hilbert spaces, and then develop an estimation
procedure for this problem by reformulating tensor-based DMD. As a special case
of our method, we propose the method named as Graph DMD, which is a numerical
algorithm for Koopman spectral analysis of graph dynamical systems, using a
sequence of adjacency matrices. We investigate the empirical performance of our
method by using synthetic and real-world data.Comment: 34 pages with 4 figures, Published in Neural Networks, 201
A Koopman Operator-Based Prediction Algorithm and its Application to COVID-19 Pandemic
The problem of prediction of behavior of dynamical systems has undergone a
paradigm shift in the second half of the 20th century with the discovery of the
possibility of chaotic dynamics in simple, physical, dynamical systems for
which the laws of evolution do not change in time. The essence of the paradigm
is the long term exponential divergence of trajectories. However, that paradigm
does not account for another type of unpredictability: the ``Black Swan" event.
It also does not account for the fact that short-term prediction is often
possible even in systems with exponential divergence. In our framework, the
Black Swan type dynamics occurs when an underlying dynamical system suddenly
shifts between dynamics of different types. A learning and prediction system
should be capable of recognizing the shift in behavior, exemplified by
``confidence loss". In this paradigm, the predictive power is assessed
dynamically and confidence level is used to switch between long term prediction
and local-in-time prediction. Here we explore the problem of prediction in
systems that exhibit such behavior. The mathematical underpinnings of our
theory and algorithms are based on an operator-theoretic approach in which the
dynamics of the system are embedded into an infinite-dimensional space. We
apply the algorithm to a number of case studies including prediction of
influenza cases and the COVID-19 pandemic. The results show that the predictive
algorithm is robust to perturbations of the available data, induced for example
by delays in reporting or sudden increase in cases due to increase in testing
capability. This is achieved in an entirely data-driven fashion, with no
underlying mathematical model of the disease