to a different one. Persistence is an important characteristic of any system,
which describes part of its statistical physics behaviour. The study of
complex systems' persistence involves different definitions and uses different
techniques, depending on whether we consider short-term or long-term
persistence. Short-term persistence has been described by considering
definitions focused on measuring the average time spent at a given state, or
the probability of changing to another one. Markov chain methods or auto-
regressive models have been applied, among others, to describe the
characteristics of systems with short-term persistence. On the other hand,
long-term persistence is usually defined based on the autocorrelation function
of the time series describing the complex system at hand. A long-range
persistent process exhibits a power-law scaling of its autocorrelation
function. Long-term persistence can also be defined in terms of the power
spectral density of the time series (which forms a Fourier par with its
autocorrelation function). A process can be defined as long-range persistent
if its power spectral density scales asymptotically as a power law for
frequencies close to the origin. Some methods used to describe the
characteristics of long-term persistent systems have been described in this
article, such as Hurst R/S analysis, DFA or methods based on Wavelets.

Concepts related to persistence in arbitrary systems appear in a wide spectrum
and diversity of the scientific literature. In this review we provided a
structured literature review in this work, highlighting the most important
fields where persistence of complex systems has been studied. Without doubt,
the persistence of systems in Earth and atmospheric sciences has been
profusely studied over the last fifty years. Short-term and long-term
persistence of atmospheric processes such as precipitation, global
temperature, droughts and heat waves, soil moisture, climate models
performance, sea level, air pollution and different hydrologic-related
processes have been reviewed. In connection with atmospheric and climate, the
persistence of renewable energy resources has been studied in different works,
which have been discussed in this paper. Wind speed and solar persistence are
the two renewable resources which have attracted the most attention in the
last years. Note that solar resource is in general much more persistent than
wind, as discussed in the literature review carried out. We have also revised
some works dealing with the persistence of systems in Earth science, such as
geophysics and seismology. We have also dealt with the review of persistence
in complex networks, where a number works dealing with network dynamics and
persistence, or link persistence, among others, have been discussed. The
research on persistence in Economics has been intense in the last years. We
have reviewed here a number of works dealing with persistence in inflation,
assets prices, stock market indices or exchange rates. We have next revised
different works on persistence of non-equilibrium thermodynamics systems, most
of them characterized by probabilities of system's state change which decay as
power law of the form $P_{o}(t)\sim t^{-\theta}$. We have also revised
different techniques to construct synthetic time series with long-term
persistence properties, some of them specific for hydrology applications. The
last large block of works discussed in this paper is related to persistence in
optimization and planning. In this type of problems, persistence is referred
to the optimal solution found, which should maintain a certain degree of
performance versus changes in the original definition of the problem. We have
closed this review of existing works about persistence in complex systems by
reviewing some previous studies in biomedical applications and sport science.

Specific case studies and applications have been described in the last part of
this paper: The study of time scales in DFA over different time series, how
persistence-based methods can be accurate in solar radiation prediction
problems, an study of soil moisture persistence and its analysis, how to
exploit directional persistence of wind to reduce the computational cost in
turbine layout problems for wind farms design and the effect of persistence in
ML algorithms when dealing with data streams, are the problems exhaustively
discussed in this work, to show the importance of studying persistence of real
complex systems.

The study of complex systems' persistence is expected to continue and be a
major topic of research interest in the near future. We anticipate a cross-
fertilization of the field, and suggest here some prospective uses and
applications of persistence analysis for complex systems:

  * 1.

Merging with ML algorithms. ML is currently the paradigm of computational
techniques, applied to the study of almost every known system or open problem.
The development of ML is being huge in the last years, specially with the
raise of deep learning as the last and maybe most powerful tool for knowledge
discovering and analysis from data. However, in the study of different highly
complex real systems, sometimes the solutions obtained by ML techniques
(exclusively working on data) do not respect the most elementary laws of
physics, like mass or energy conservation [468]. Some researchers have pointed
out the necessity of mixing ML algorithms with numerical models dealing with
the physical equations of the phenomenon under study, so the physical meaning
of ML solution is ensured [469]. The inclusion of the system's persistence as
one of the rules that ML algorithms must keep within their outcome could be
basic in order to obtain solutions with physical meaning out of ML approaches.

  * 2.

Models of memory and persistence with a biological inspiration. Quantifying
persistence in nonlinear systems requires nonlinear models of memory and
adaptation, and artificial neural networks excel into that. Neural nets have
found inspiration in biological processes in the brain system or parts
thereof, such as the visual, auditory or olfactory brain. Such bio-inspiration
has resulted in a great success in applied fields of engineering, such as
computer vision, speech or natural language processing. There is no reason why
this should not continue in the future. In order to maintain a persistent
activity, it has been argued that a positive feedback is strictly needed [31].
There are also some evidences that neuronal responses can instead be
maintained by a purely feedforward mechanism that maintains short-term memory
in which both feedforward and feedback processes interact to govern network
behavior. Such mechanisms are currently accounted explicitly in recurrent
neural networks, but one could gain in computational efficiency by accounting
for memory without feedbacks [470]. The framework of attractor neural networks
[471] could be extended to represent time-dependent stimuli in artificial
neural nets. Actually, persistent activity states (attractors) in biological
neurons is believed to be the basis of the working memory [472]. The existence
of persistent states was originally suggested in [473], and gave rise to
exciting developments in multiscale neural networks [46]. Yet, also relevant
computational principles of memory are concerned here [474]. After all, the
ability to store and use information is essential for a variety of adaptive
behaviors, including learning, generalization, prediction and inference, which
are familiar concepts in ML (and deep learning).

  * 3.

Extreme events. The persistence of extreme events has not been deeply studied
in the literature so far, but in some specific cases related to meteorology
and climate, such as droughts or heat waves. Even in these cases, the concept
is still far from being fully understood and modelled, and it seems that
persistence of these events is fully dependent on each specific case. It is
necessary a more intense research to characterize extreme events persistence,
which is difficult, due to the lack of observational data because the low rate
of occurrence of these events.

  * 4.

Persistence and physical drivers. In many works, persistence of a complex
system is described from a statistical physics point of view, i.e. they try to
characterize the statistics of the time series representing an event, without
a clear connection with the physical processes (in a broad sense) driving the
system. We have touched upon these topics in Section 3.1.5, where the
relationship of persistence in ML with the exogenous variables describing the
system has been discussed. In a more general framework, to study the
relationship between the persistence of a complex system, and the physical
variables of the phenomenon (exogenous variables) would be extremely
interesting and also challenging. This point clearly connects with topics such
as causal inference, observed physical processes, and persistence.

  * 5.

Persistence of new complex systems. Persistence analysis has proven to be a
very useful way of characterizing part of the statistical properties of
complex systems. The techniques revised in this work, and other alternatives
that may arise in the next years, could be applied to any new complex system
of interest, in any research area. Some specific new problems in which
persistence analysis is extremely interesting arise, for example, in mass
media research (why a given news item remains at newscasts during more time
than others? how is this process of news items importance decaying over time?
and in turn, how does the audience interest in a news item decay over time?),
social research and politics, sports science, further problems related to
persistence in optimization of systems, and many others.

The study of persistence in complex systems has reached a point of maturity in
theoretical developments and exciting applications. We reviewed the
literature, gave empirical evidence of performance in several case studies,
and outlined a collective agenda for the future research and developments. We
anticipate a wider adoption of the techniques by newcomers and experienced
researchers willing to make sound advances in persistence characterization.
And vice-versa, we expect to spark new ideas across fields. Our insights and
prospects held over this field pave a promising path plenty of opportunities
for persistence related studies, including a synergistic interplay with other
domains of knowledge.

## Acknowledgments

This research has been partially supported by the project PID2020-115454GB-C21
of the Spanish Ministry of Science and Innovation (MICINN). This research has
also been partially supported by Comunidad de Madrid, PROMINT-CM project
(grant ref: P2018/EMT-4366). J. Del Ser would like to thank the Basque
Government for its funding support through the EMAITEK and ELKARTEK programs
(3KIA project, KK-2020/00049), as well as the consolidated research group
MATHMODE (ref. T1294-19). GCV work is supported by the European Research
Council (ERC) under the ERC-CoG-2014 SEDAL Consolidator grant (grant agreement
647423) and the ERC Synergy Grant `Understanding and Modelling the Earth
System with Machine Learning' (USMILE) under the Horizon 2020 research and
innovation programme (Grant agreement No. 855187).

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