ArticlePDF Available

Abstract and Figures

In the scientific vocabulary, the term “process” is used to denote change in time. Even a stationary process describes a system changing in time, rather than a static one which keeps a constant state all the time. However, this is often missed, which has led to misusing the term “nonstationarity” as a synonym of “change”. A simple rule to avoid such misuse is to answer the question: can the change be predicted in deterministic terms? Only if the answer is positive it is legitimate to invoke nonstationarity. In addition, we should have in mind that models are made to simulate the future rather than to describe the past; the past is rather characterized by observations (data). Usually future changes are not deterministically predictable and thus the models should, on the one hand, be stationary and, on the other hand, describe in stochastic terms the full variability, originating from all agents of change. Even if the past evolution of the process of interest contains changes explainable in deterministic terms (e.g. urbanization), again it is better to describe the future conditions in stationary terms, after “stationarizing” the past observations, i.e. adapting them to represent the future conditions.
Content may be subject to copyright.
Negligent killing of scientific concepts:
the stationarity case
Demetris Koutsoyiannis1 and Alberto Montanari2
1Department of Water Resources and Environmental Engineering, School of Civil Engineering,
National Technical University of Athens, Greece (
2Department DICAM, University of Bologna
Abstract In the scientific vocabulary, the term “process” is used to denote change in time. Even a
stationary process describes a system changing in time, rather than a static one which keeps a constant
state all the time. However, this is often missed, which has led to misusing the term “nonstationarity”
as a synonym of “change”. A simple rule to avoid such misuse is to answer the question: can the
change be predicted in deterministic terms? Only if the answer is positive it is legitimate to invoke
nonstationarity. In addition, we should have in mind that models are made to simulate the future rather
than to describe the past; the past is rather characterized by observations (data). Usually future changes
are not deterministically predictable and thus the models should, on the one hand, be stationary and,
on the other hand, describe in stochastic terms the full variability, originating from all agents of
change. Even if the past evolution of the process of interest contains changes explainable in
deterministic terms (e.g. urbanization), again it is better to describe the future conditions in stationary
terms, after “stationarizing” the past observations, i.e. adapting them to represent the future conditions.
While a phrase like “there is a lot of heat in the kitchen” is perfectly understandable in everyday
language, the same expression is unacceptable for a scientific text, or even in oral communications
among scientists. One would expect from a scientist to say “there is a lot of thermal energy in the
kitchen” or reformulate it as “the temperature in the kitchen is high”. Interestingly, despite such
scientific concepts like heat, (thermal) energy and temperature are centuries old, misconceptions and
misuses are still common, even among university students of thermal physics (Georgiou and Sharma,
2012) and may hinder learning and understanding. Perhaps the difficulties stem from the fact that
thermal physics concepts are au fond statistical concepts, while our education systems favour
deterministic approaches and give much less importance to probability, statistics and stochastic
processes, areas that are collectively described by the term stochastics.
While stochastics offers a much more powerful approach to model physical systems
(Koutsoyiannis 2010) including hydrological systems (Montanari and Koutsoyiannis 2012), it also
requires a different type of understanding for the concepts it introduces. Lack of such understanding
may result in misuse of the terms and the related concepts.
Take for instance the terms stationarity and nonstationarity. Interestingly, while the adjective
stationary is contained in English dictionaries with the meaning not moving or not changing, the nouns
stationarity and nonstationarity do not appear in common dictionaries like the Cambridge
(, the (, the Merriam-Webster
( and the Oxford ( The reason
is that they are only scientific terms, not words used in everyday language. Thus, in contrast to heat
and temperature which may be used colloquially in a loose manner, misuse of the stationarity and
nonstationarity can be more easily detected and hopefully avoided as there is no ambiguity owing to
difference in everyday language and scientific meaning.
However, for a wide variety of reasons, including its stochastic origin, ideologico-political
influences and a clustering effect, misuse of the terms stationarity and nonstationarity has been
common. Illustration of the clustering effect in the term’s misuse is offered by the paper by Milly et al.
(2008) with the bold title “Stationarity is dead”. A Google Scholar search reveals that up to now
(March 2014) 1350 papers contain the phrase “Stationarity is dead”. This can be interpreted as an
indication that more or less those papers have adopted the misuse of the term in the original paper.
Interestingly, only four papers query “Is stationarity dead?”, not a single paper says “Stationarity is not
dead” and only one paper says “Stationarity is alive”. Other searches, again using Google Scholar,
reveal that (as of March 2014) 660 papers speak about “Nonstationary world”, 551 about
“Nonstationary climate”, 15 about “Nonstationary catchment(s)” and 14 700 papers about
“Nonstationary data”. It can be conjectured that most of them misuse the term nonstationary, as will
be justified below.
In an attempt to combat such misuse, the World Meteorological Organization (Commission for
Hydrology 2013) has issued a statement with clarifications about the terms. This note has exactly the
same purpose.
Semantics and historical review
In this section we will try to reconstruct a possible history of the concept of stationarity based on some
key references including historical ones. The term stationary is originally a classical mathematical
term: given a function x(t), a stationary point thereof is a point in which the derivative is zero, i.e. x΄(t)
= 0. In dynamical systems, the notion of a stationary point has a richer meaning. A dynamical system
described by its state x(t) is typically characterized by a differential equation of the form (t) = g(x(t)).
On the other hand, it is common to express the dynamical system in terms of a transformation St of its
initial state x(0) (at time 0) to its current state x(t) (at time t), that is, x(t) = St(x(0)) (Lasota and Mackey
1994). In this case, if x is a stationary point, i.e. a solution of (t) = 0, it also satisfies x = St(x), that is,
it remains invariant under the transformation St. In classical algebra a point x satisfying x = St(x) is
called a fixed point or an invariant point and is distinguished from the stationary point. However in
differential equations books (e.g. Conrad, 2003) and dynamical systems books (e.g. Wiggins, 2003) all
these terms (and other ones, e.g., “equilibrium point”, “rest point”) are used interchangeably.
As an example consider the logistic differential equation
x΄(t) = g(x(t)) = a x(t) (1 x(t)/b)
Solving the differential equation for x(t) = St(x(0)) with initial condition x(0) x we find that the
transformation St(x) is
    
It can be readily seen that there are two stationary points, x = 0 and x = b, which satisfy both = g(x)
= 0 and x = St(x).
Now, a more effective description of dynamical systems is achieved when, instead of studying
trajectories of points (i.e. the St(x) for varying t), we study the evolution in time t of probability
densities f(x) (Lasota and Mackey, 1994). In this case, the description through St(x) is replaced by that
based on Kt f(x), where Kt is a transformation operator. Accordingly, a stationary density is defined to
be a probability density function f which satisfies f = Kt f thus remaining unchanged in the course of
time. For a simple account about why evolution of probability densities provide a better description of
a system than trajectories of points and for more details about transformation operators, the reader is
referred to Koutsoyiannis (2010).
In this context, Kolmogorov (1931) used the term stationary to describe a probability density
function that is unchanged in time. In the same work, Kolmogorov introduced the term stochastic
process although he cited Bachelier (1900) as having already used stochastic processes in continuous
time; however, it appears that Bachelier did not use the term stochastic process. A more formal
definition of a stochastic process and stationarity was given a few years later by Khinchin (Khintchine
1934; notice the different transliteration in the English and the German literature of author’s name,
originally appearing in the Cyrillic alphabet). A concise presentation thereof has been given by
Kolmogorov (1938) as follows:
[…] a stationary stochastic process in the sense of Khinchin […] is a set of random variables xt
depending on the parameter t, −∞ < t < +, such that the distributions of the systems
(xt1, xt2, …, xtn) and (xt1 + τ, xt2 + τ, …, xtn + τ)
coincide for any n, t1, t2, … ,tn, and τ.
Kolmogorov (1947) has also defined wide-sense stationarity in terms of independence on time t of the
expectation E[xt + τ xt].
These authors were Russian (then Soviet) and their papers were published mostly in German.
The concepts were tranplanted in the English literature, unfortunately with some loss of clarity.
Perhaps the first who introduced them into the English literature was Doob (1934), from whom we
quote the following definition:
A stochastic process is defined by Khintchine [(1934)] to be a one parameter set of chance
variables: x(t), –∞ < t < ∞. It is supposed that if t1, ..., tn is any finite set of values of t, and aj < x
< bj, j = 1, . . .,n any set of intervals, the probability that
aj < x(tj) < bj, j = 1, …, n
is defined. If the probability that [(4)] is true is independent of translations of the t-axis, the
process is called stationary.
By comparing with the original paper (in German) or with Kolmogorov’s definition in (3) it can be
seen that Doob distorts somewhat Khinchin’s definition. Furthermore, the famous book by Kendall
and Stuart (1966, p. 404), while essentially keeping the Khinchin-Kolmogorov definition, speaks
about a stationary series instead of a stationary process. It further states that stationary time-series are
a particular case of the theory of stochastic processes. However, the definition of a time series given in
the book does not support the latter statement, while a definition of a stochastic process is missing.
Specifically the definition of a time series given by Kendall and Stuart (1966, p. 342) is the following:
Observations on a phenomenon which is moving through time generate an ordered set known as
a time series. The values assumed by a variable at time t may or may not embody an element of
random variation, but in the majority of cases with which we shall be concerned some such
element is present, if only as an error of observation.
This seems to recognize a time series as a series of observations, which could be a series of
realizations of the random variables that constitute the stochastic process (else known as a sample
function of the stochastic process). Also, it could be a series of values not necessarily associated with a
stochastic process. However, in the definition of the stationarity, the concept of a time series looks to
be treated as identical to (or subcase of) that of a stochastic process.
Indeed, in the English literature the concept of a time series is ambiguous, sometimes denoting a
realization of a stochastic process and other times denoting the stochastic process per se (with the
specification that the index set defining the stochastic process denotes time; e.g. Parzen 1957).
Nevertheless, there are books in the English literature characterized by perfect clarity, of which we
mention Papoulis (1991; first edition 1965). Such books follow the Khinchin-Kolmogorov definition
of stochastic processes and stationarity.
Some preliminary conclusions that we can draw from this historical review are: (a) stationarity
refers to stochastic processes; (b) stochastic processes are families of random variables usually (but
not necessarily) indexed by time; and (c) random variables are variables associated with a probability
distribution or density function. It follows that any attempt to conceptualize stationarity without
reference to the notion of a stochastic process will be inconsistent with the theory.
The concept of nonstationarity
Inevitably, the negation of stationarity, i.e. nonstationarity, needs to be defined and conceived again
within the notion of stochastic processes. The negation could be conceived with reference to Eqn. (3),
meaning that in a nonstationary process the probability density f(xt1+τ, xt2+τ, …, xtn+τ) for some (or all) τ
should not equal f(xt1, xt2, …, xtn). In turn, this means that the mathematical expression of f(xt1+τ, xt2+τ, …,
xtn + τ) should explicitly contain the time shift τ, or else that it should be a deterministic function of τ.
Often, in looser terms, a stationary process is thought of as a process whose statistical properties
do not change with time, while, by negation, a nonstationary process is thought of as a process whose
statistical properties do change with time. The latter statement, if combined with a loose perception of
what statistical properties are, and perhaps dismissal of “statistical” from “properties” has led to very
widespread misconceptions. Two common examples of such misconceptions are (a) that random
changes in some properties indicate nonstationarity, and (b) that it is not necessary to have
deterministic functions of time in order to claim nonstationarity.
We can clarify these misconceptions, by considering Eqn. (3) with just one variable, xt1, also
assuming t1 = 0, so that we have to compare statistical properties of xτ, with those of x0 (notice that
underlined symbols denote random variables and stochastic processesthe Dutch convention). For
misconception (a), let us assume that xτ can be decomposed in two parts as follows:
xτ = dτ + vτ
where vt is a stationary stochastic process. If the component dτ is a constant (dτ d, where d is a
number), then obviously xt is also a stationary stochastic process. If dτ is a deterministic function of
time (dτ d(τ)), then xτ will be a nonstationary stochastic process as its mean will be E[xτ] = d(τ) +
E[vτ], i.e., a function of time. Now if dτ changes in time, but not according to a deterministic function,
then it should be modelled as a stochastic process itself (dτ dτ). If the latter process is a stationary one
(see further clarification about this in section “Stationarity under change) then the process xτ is
stationary (because E[xτ] = E[dτ] + E[vτ] = constant)even though many would regard it as
As an example to analyse misconception (b), let us consider as statistical properties of xτ its
moments E[
] for varying q = 0, 1, …. Such moments are not random variables, and thus they can be
either constants (E[
] mq) or deterministic functions of τ (E[
] mq(τ)). If we assume that all
moments E[
] for all q are not functions of τ then, by virtue of the moment theorem (Papoulis 1991,
p. 116) the probability density function f(xτ) is uniquely determined by the moments and will not be a
function of τ per se. Therefore, it is necessary that at least one moment E[
] for a certain q be a
deterministic function mq(τ) in order to have a time dependent f(xτ).
The above example is for first-order stationarity, because we used only one xτ. A process can be
first-order stationary yet being nonstationary, if joint statistical properties of variables corresponding
to different times are deterministic functions of time (e.g., if the joint moment E[xτ xτ + t] is a function
of both t and τ, the process is nonstationary). However, the above result can be generalized for two or
more variables as the moment theorem holds for more than one variable (e.g., Papoulis 1991, p. 160).
Hence, if a stochastic process is nonstationary, then at least one of its moments, marginal or joint,
should be a deterministic function of time.
In conclusion, as the meaning of change in everyday language is quite general and certainly
includes random change, statements like a nonstationary process is a process whose statistical proper-
ties change with time may be misunderstood. In order to be strictly valid, such statements should be
accompanied by the clarification that change here is meant as a deterministic function of time.
Philosophical reflections
While misconceptions about stationarity and nonstationarity can be relieved because the concept is
only 80 years old, the discourse in the Introduction (in particular, the typical phrases that involve
nonstationarity) reveals that there may be a more fundamental misconception about a notion that is 2.5
thousand years old. Specifically, this is related to confusing the material world with the world of
models, which usually are abstract (mathematical) entities. The distinction between the two and the
relevant discussion are at least as old as Plato’s metaphysical theory.
Figure 1 A schematic to illustrate Plato’s metaphysical theory (upper row) in comparison to a view more
compatible to modern science (lower row). The left column depicts the real world, which in Plato’s theory is the
world of the archetypes, while the material world is composed by imperfect “shadows” of the archetypes. By
rotating the schematic corresponding to Plato’s theory by 180º we identify the material world with the real world
and we replace the ideal archetypes with imperfect models of reality. In both cases, however, the intellectual
structures are distinguished from the material ones (image of pyramid from
Real/perfect Imperfect (approximation)
Real/perfect Imperfect (shadow)
According to Plato’s theory, the real world is a world of ideal or perfect forms (αρχέτυπα,
archetypes). It is unchanging and unseen and it can only be perceived by reason (νοούμενα,
nooumena). The physical world is an imperfect image of the world of archetypes. Physical objects and
events are “shadows” of their ideal forms, are subject to change and can be perceived by senses
(φαινόμενα, phenomena).
By turning upside-down Plato’s theory (or making a 180º turn, as illustrated in Fig. 1) we obtain
a view that is more consistent with modern science. According to the latter, the physical world is the
real world. It is perfect and it is perpetually changing. Abstract representations or models of the real
world are imperfect but can be useful to describe the real world. It should be noted though that, while
Plato’s archetypes and modern models have in common the fact that are abstract concepts perceived
by reason, the two notions are not identical.
Whether one accepts the original or the upside-down version of Plato’s theory, it is important
that both make the distinctions: physical world ≠ models (or forms) and phenomena ≠ nooumena.
Also, both recognize that in the physical world change is the rule, a fact that had been earlier and aptly
expressed by Heraclitus as “Πάντα ῥεῖ” (Panta rhei, Everything flows). The latter aphorism has now
become the emblem of the IAHS initiative for the current decade (Montanari et al. 2013).
With respect to the discussion of the earlier section, it is quite easy to understand that
probability and all concepts based on it are abstract concepts, and not properties of the physical world.
Attempts to define probability in terms of experience from the physical world resulted in logical
problems (e.g. circular logic), which were resolved only when Kolmogorov (1933) introduced the
axiomatic foundation of probability. The abstract character of probability is more evident in the
Bayesian interpretation thereof. Likewise, the stochastic processes, as well as the random variables
they consist of and their probability density functions, are models and not physical objects. This means
that stationarity and nonstationarity are properties belonging to the world of models. In contrast, the
objects and processes of the real world are neither stationary nor nonstationary; they are just
perpetually changing.
Stationarity under change
Stationarity does not contradict change. Rather it offers a powerful way to model change. Without
change, we would not need the concept of a stationary process, not even that of a process. Note that a
process is defined in common English dictionaries, such as those mentioned above, as a series of
changesor actions. In addition, in the scientific literature the term process has been introduced as
synonymous to change, as evident in Kolmogorov’s (1931) pioneering paper, which starts as A
physical process [is] a change of a certain physical system.
It is very common in science to try to identify invariant properties within change (Koutsoyiannis
2011). For example, in the absence of an external force, the position of a body in motion changes in
time but the velocity is unchanged (Newton’s first law). If a constant force is present, then the velocity
changes but the acceleration is constant (Newton’s second law). If the force changes, e.g. the
gravitational force with changing distance in planetary motion, the acceleration is no longer constant,
but other invariant properties emerge, e.g. the angular momentum (Newton’s law of gravitation; see
also Koutsoyiannis 2011).
Also, in motion of fluids we speak about steady flow. This is not a contradiction. Obviously,
there is change because of the flow (change and flow are tightly connected), but in a steady flow the
velocity does not change in time.
Likewise, the theory of dynamical systems is a theory describing change. Amazingly, most of
dynamical systems used to model natural processes are autonomous systems, that is, they are
expressed by autonomous differential equations, e.g., (t) = g(x(t)), which do not explicitly depend on
time. In the more general case of non-autonomous systems, i.e. those expressed as (t) = g(x(t); t), the
dynamics is not the same on the intervals [0, t΄] and [t, t + t΄]. However, this is rarely the case because
the laws of nature which hold now are identical to those holding for any time in the past or future.
Even if change in the laws happens to be the case (e.g. in a macroscopic description of a complex
system) again the non-autonomous system is converted into an autonomous one. This is a rather easy
task as it only needs the definition of new dependent variables (Mackey 1992).
Now, coming to the notion of a stochastic process, we note that it was invented to describe the
irregular changes of complex natural systems, which are impossible to model deterministically in full
detail or predict their future evolution in detail and with precision. Here, the great scientific
achievement is the invention of macroscopic descriptions instead of modelling the details. This is
essentially done using stochastics. Here lies the essence and usefulness of the stationarity concept,
which seeks invariant properties in complex systems (Koutsoyiannis, 2011). As in the case of
converting non-autonomous to autonomous systems, again even if a process is nonstationary, it should
be converted into a stationary one, which can be effectively studied and used in predictions.
As the stochastic model is a mathematical construction, the use of a stationary or a
nonstationary model is a modelling option rather than a property of the observational data of the real
world. Of course any model, whether deterministic or stochastic, stationary or nonstationary, should
be consistent with the data. But the data alone do not suffice to make a correct modelling choice.
This is illustrated in Figure 2, which depicts 1000 terms of a synthetic time series; the present
time, as designated in the figure, divides the time axis into two parts, the past (with 150 observations)
and the future. Considering the 150 observations of the past, one sees two step changes: a large rise at
time 70 and a smaller drop at time 120. According to common statistical practice, one would describe
such changes as nonstationarities and would construct a nonstationary model with different statistical
properties for each of the three periods designated by the two step changes. But such a description is
totally useless if prediction of the future is of interest and, at the same time, it contradicts the fact that
the stochastic process which generated the time series is in fact a stationary process, which is
described in detail in Koutsoyiannis (2011).
Figure 2 Schematic for a prediction problem based on a time series.
0 100 200 300 400 500 600 700 800 900 1000
Time, i
Time series
Local average
Global average
It is recalled that nonstationarity is determined based on ensemble means or, more generally,
ensemble statistical properties of a stochastic process. The ensemble mean is defined as
E[x] = 
while the temporal mean is
 
 
It is seen that the temporal mean is a stochastic process per se, that is, a random function of time. The
ensemble mean is not a random quantity, but in a stationary process it is a single number. As explained
in section The concept of nonstationarity”, if the process is nonstationary, E[x] can vary in time,
being a deterministic function of time (i.e., precisely known and perfectly predictable). But, as the
graphical example shows, this cannot be inferred from data only, because the data can give us only the
temporal mean (actually a realization thereof) and not the ensemble mean. Rather, the data can
mislead us to adopt a nonstationary description, while the underlining process is stationary. To
establish a deterministic function of time, as required in order to claim nonstationarity, we need at
least both of the following conditions to hold: (a) deductive reasoning in order to establish the
deterministic function of time; (b) validation of the deterministic function by data which were not used
in the model construction.
It seems the frequent invocation of nonstationarity, as well as deterministic trends, shifts and
cycles, is a remnant of deterministic thinking, while using stochastic descriptions. The actual
stochastic concept that should replace all these, within a stationary stochastic setting is dependence,
possibly a long-range one. The handling of dependence, and eventually its utilization in prediction, is
made through conditioning on known information, e.g. observations of the past. For example,
assuming that x0, x1, …, xn, are observations at n past times t0, t1, …, tn 1, and at the present time tn,
the conditional average of the process x(t) for any future time t > tn will be a function of time, h(t):
E[x(t)|x0, x1, …, xn] = h(t) E[x(t)]
Likewise, the conditional variance is also a (non-decreasing) function of time and every statistical
characteristic, conditional on observations, can be a function of time. All these functions of time are
deterministic functions, derived by deductive reasoning (using stochastic processes algebra). Hence,
conditioning results in statistical characteristics that vary in time, even though overall the model is
Figure 2 further illustrates the effect of conditioning on the past, within stationarity, in order to
predict the future. While the past values can be known from observation, the evolution of the process
in future times is unknown. Ignorance of the future does not concern only the precise value at each
time. The local averages, shown as dashed lines in the figure are also unknown (had they been known,
the model would be nonstationary). Let us suppose we are required to make a prediction for the future.
If the prediction horizon is long, then we will use the global (i.e. the true or ensemble) average and the
global variance for our prediction. However, if the prediction horizon is short, then we will use the
local average at the present time and a reduced variance. More generally, by utilizing the dependence
structure of the model, which can be inferred from the observations, assuming that their length is
sufficient, we are able to make predictions conditioning on the observations, i.e. formally evaluating
Eqn. (8) for the assumed stationary stochastic model and for the required prediction horizon t.
Justified use of nonstationary models
As clarified above, in a nonstationary process, to describe change we use both stochastic variations
and deterministic functions. By definition, a deterministic function should be constructed by deduction
(the Aristotelian apodeixis), not by induction (the Aristotelian epagoge) which makes direct use of the
data (see also next section). Because it explains in deterministic terms part of the variability, a
nonstationary description is associated with reduced uncertainty. Hence unjustified or inappropriate
claim of nonstationarity results in underestimation of variability, uncertainty and risk.
However, there are cases where use of a nonstationary description is justified. Sometimes we
undertake modelling of the past, in order to explain observed behaviours. Changes in hydrological
behaviours happen all the time, as a result of changes in quantifiable characteristics of catchments and
conceptual parameters of models. If we know the evolution of these characteristics and parameters
(e.g. in addition to hydrological observations, we have information about how the percent of urban
area changed in time), then we can build a nonstationary model. The specific information has reduced
uncertainty thus enabling a nonstationary description. In contrast, if we see a changing behaviour but
we do not have this quantitative information, then we may treat the catchment characteristics and
parameters as random variables. In this case we build stationary models entailing larger uncertainty.
It is important to distinguish explanation of observed phenomena in the past from modelling
that is made for the future. Except for trivial cases, the future is not easy to predict in deterministic
terms. If changes in the recent past are foreseen to endure in the future (e.g. urbanization, hydraulic
infrastructures), then the model of the future should be adapted to the most recent past. This may
imply a stationary model of the future that is different from that of the distant past (prior to the
change). It may also require “stationarizing” of the past observations, i.e. adapting them to represent
the future conditions (e.g. the flow data prior to the construction of the dam could be adapted to
determine what the flow would be if the dam existed).
In the case of planned and controllable future changes (e.g. catchment modification by hydraulic
infrastructures, water abstractions), which indeed allow prediction in deterministic terms or at least
condition our predictions, nonstationary models are justified.
Stationarity, ergodicity and inductive inference
It is important to note that stationarity is also related to ergodicity, which in turn is a prerequisite to
make inference from data, that is, induction. In dynamical systems, by definition (e.g. Mackey, 1992,
p. 48), ergodicity is the property of a system whose all invariant sets under the dynamic transformation
are trivial (have zero probability). In other words, in an ergodic transformation starting from any point,
the trajectory of the system state will visit all other points, without being trapped to a certain subset.
The ergodic theorem (Birkhoff 1931; see also Mackey 1992 p. 54) allows redefining ergodicity within
the stochastic processes domain (Papoulis 1991 p. 427; Koutsoyiannis 2010) in the following manner:
A stochastic process xt is ergodic if the time average of any (integrable) function g(xt), as time tends to
infinity, equals the true (ensemble) expectation E[g(xt)]. Thus, with reference to Eqn. (7), in an ergodic
system the time average, as t΄ → ∞, will tend to the ensemble average given in (6). This allows the
estimation (i.e. approximate calculation) of the true but unknown property E[xt] from the , that is
from the available data. Without ergodicity inference from data would not be possible.
Now, if the system that is modelled in a stochastic framework has deterministic dynamics
(meaning that a system input will give a single system response, as happens for example in most
hydrological models) then a theorem applies (Mackey 1992, theorem 4.5 p. 52), according to which a
dynamical system with dynamics St(x) has a stationary probability density if and only if it is ergodic.
Therefore, a stationary system is also ergodic and vice versa, and a nonstationary system is also non-
ergodic and vice versa. Here we note that even if a system has deterministic dynamics, again it is
legitimate to use a stochastic description, replacing the study of the evolution of system states St(x)
with the evolution of probability densities of states f(x) as already mentioned in the section “Semantics
and historical review”; one reason to prefer the stochastic description over the pure deterministic
description is that the former incudes quantification of uncertainty, whereas the deterministic
dynamics does not eliminate uncertainty (Koutsoyiannis 2010). Furthermore, we clarify that the
deterministic description through the transformation St(x) is fully compatible with a stochastic
description that is stationary and ergodic, according to theorem stated above: while the system state is
changing in time t according to the transformation St(x), its statistical properties (and the probability
density f(x)) can be constant in time.
If the system dynamics is stochastic (a single input could result in multiple outputs), then
ergodicity and stationarity do not necessarily coincide. However, recalling that a stochastic process is
a model and not part of the real world, we can always conveniently device a stochastic process that is
ergodic, provided that we have excluded nonstationarity. To clarify this idea we give the following
example. A well-known stochastic process xt which is stationary and non-ergodic is (Papoulis 1991, p.
xt a ut
where a is a random variable and ut is a stationary and ergodic stochastic process. A single realization
of xt does not enable the estimation of its true statistics because all its history corresponds to a single
realization a of the random variable a. However, the single realization allows the estimation of the
statistics of the process x΄t a ut xt|a = a, which is stationary and ergodic. In most practical
problems corresponding to physical systems, which actually do not allow different realizations but
rather have a unique evolution, the properties of the ergodic process x΄t rather than those of the non-
ergodic xt are actually of interest.
In conclusion, from a practical point of view ergodicity can always be assumed when there is
stationarity, while this assumption if fully justified by the theory if the system dynamics is
deterministic. Conversely, if nonstationarity is assumed, then ergodicity cannot hold, which forbids
inference from data. This contradicts the basic premise in geosciences, where data are the only reliable
information in building models and making inference and prediction.
Change is Nature’s style and occurs at all times and all time scales (i.e. Πάντα ῥεῖ). The frequent use
of the term nonstationarity lately indicates that it is confused with change. However, change is not
synonymous to nonstationarity: While change is a general notion applicable everywhere, including to
the real (material) world, stationarity and nonstationarity apply only to models, not to the real world,
and are defined within stochastics. The confusion about these terms extends also to the world of
models, as several properties related to stochastic dependence of processes are often interpreted as
nonstationarity. Nonstationary descriptions are justified only if the future can be predicted in
deterministic terms and are associated with reduction of uncertainty, while misuse of nonstationarity
results in underestimation of variability, uncertainty and risk. In absence of credible predictions of the
future, admitting stationarity (and larger uncertainty) provides a more consistent and more effective
modelling option.
Acknowledgment The present work was developed within the framework of the Panta Rhei Research
Initiative of the International Association of Hydrological Sciences (IAHS). We thank two anonymous
reviewers, the eponymous reviewer Tim Cohn and the Guest Editor Guillaume Thirel for their
comments which resulted in expansion and improvement of the paper.
Bachelier, M. L., 1900. Théorie de la spéculation, Ann. Ecole Norm. Super. 17, 21-86.
Birkhoff, G. D., 1931. Proof of the ergodic theorem, Proc. Nat. Acad. Sci., 17, 656660.
Commission for Hydrology, 2013, CHy Statement on the Terms Stationarity and Nonstationarity, World
Meteorological Organization,
Conrad, B. P., 2003, Differential Equations, A Systems Approach, Prentice Hall, Upper Saddle River, NJ, USA,
462 pp.
Doob, J.L., 1934. Stochastic processes and statistics. Proceedings of the National Academy of Sciences of the
United States of America, 20(6), p.376.
Georgiou, H., and Sharma, M. D. 2012, University students’ understanding of thermal physics in everyday
contexts, International Journal of Science and Mathematics Education, 10, 1119-1142.
Kendall, M. G. & Stuart, A. (1966) The Advanced Theory of Statistics, vol. 3: Design and Analysis, and Time-
series. 552 pp., Griffin, London, UK.
Khintchine, A., 1934. Korrelationstheorie der stationären stochastischen Prozesse. Mathematische Annalen, 109
(1), 604615.
Kolmogorov, A. N., 1931. Uber die analytischen Methoden in der Wahrscheinlichkcitsrechnung, Math. Ann.
104, 415-458. (English translation: On analytical methods in probability theory, In: Kolmogorov, A.N.,
1992. Selected Works of A. N. Kolmogorov - Volume 2, Probability Theory and Mathematical Statistics
A. N. Shiryayev, ed., Kluwer, Dordrecht, The Netherlands, pp. 62-108).
Kolmogorov, A. N., 1933. Grundbegriffe der Wahrscheinlichkeitsrechnung (Ergebnisse der Mathematik und
Ihrer Grenzgebiete), Berlin (2nd English Edition: Foundations of the Theory of Probability, 84 pp.
Chelsea Publishing Company, New York, 1956).
Kolmogorov, A.N., 1938, A simplified proof of the Birkhoff-Khinchin ergodic theorem, Uspekhi Mat. Nauk 5,
52-56. (English edition: Kolmogorov, A.N., 1991, Selected Works of A. N. Kolmogorov - Volume 1,
Mathematics and Mechanics, Tikhomirov, V. M. ed., Kluwer, Dordrecht, The Netherlands, pp. 271-276).
Kolmogorov, A.N., 1947, Statistical theory of oscillations with continuous spectrum, Collected papers on the
30th anniversary of the Great October Socialist Revolution, Vol. 1, Akad. Nauk SSSR, Moscow-
Leningrad, pp. 242-252. (English edition: Kolmogorov, A.N., 1992. Selected Works of A. N.
Kolmogorov - Volume 2, Probability Theory and Mathematical Statistics A. N. Shiryayev ed., Kluwer,
Dordrecht, The Netherlands, pp. 321-330).
Koutsoyiannis, D., 2010, A random walk on water, Hydrology and Earth System Sciences, 14, 585601.
Koutsoyiannis, D., 2011, Hurst-Kolmogorov dynamics and uncertainty, Journal of the American Water
Resources Association, 47 (3), 481495.
Lasota, A., and Mackey, M. C.: Chaos, Fractals and Noise, Springer-Verlag, 1994.
Mackey, M. C. 1992, Time’s Arrow: The Origins of Thermodynamic Behavior, Dover, Mineola, NY, USA, 175
Milly, P. C. D., J. Betancourt, M. Falkenmark, R. M. Hirsch, Z. W. Kundzewicz, D. P. Lettenmaier and R. J.
Stouffer, 2008, Stationarity is dead: whither water management?, Science, 319, 573-574.
Montanari, A., and D. Koutsoyiannis, 2012, A blueprint for process-based modeling of uncertain hydrological
systems, Water Resources Research, 48, W09555, doi:10.1029/2011WR011412.
Montanari, A., et al. 2013, “Panta Rhei Everything Flows”, Change in Hydrology and Society – The IAHS
Scientific Decade 2013-2022, Hydrological Sciences Journal, 58 (6), 12561275.
Papoulis, A., 1991, Probability, Random Variables, and Stochastic Processes, 3rd ed., McGraw-Hill, New York,
Parzen, E. 1957. On consistent estimates of the spectrum of a stationary time series, The Annals of Mathematical
Statistics 28 (2), 329-348.
Wiggins, S., 2003. Introduction to applied nonlinear dynamical systems and chaos, Ed. 2. Springer-Verlag, New
York, USA, 843 pp.
... Anthropogenic climate change (ACC) in uence has been noted unequivocally in many Earth systems 1,2 , with a growing human imprint from the start of the industrial revolution onward 14 . Over the past halfcentury, the ACC signals have been substantiated in various hydroclimatic and heat budget records, e.g., precipitation, global surface temperatures, sea surface temperature (SST), and Ocean Heat Content (OHC) 3,4 . ...
... The human-induced modi cations in water systems, land cover, and land use have triggered uncontrolled water resources exploitation [9][10][11] , and introduced the notion of "nonstationarity" in the hydrologic system 13 . The concept implies signi cant changes in the hydrologic systems' intra-annual and interannual variabilities with a time-variant mean, variance, and non-uniform density distribution 14,15 . ...
Full-text available
Anthropogenic climate change signature has been detected in several Earth’s systems as a manifestation of the warming atmosphere and oceans that led to significant changes in the global water cycle. As a result, the hydrological systems display substantial and intermittent shifts in their natural courses with time-variant intra-annual and interannual variability, i.e., nonstationarity. Via GRACE (Gravity Recovery and Climate Experiment) and GRACE-FO satellite gravimetry records, we assessed the regional stationarity across 39 hotspot hydrological systems in a global context. Three main nonstationarity tests were applied around the deterministic and periodic components. Distinctively, 38 percent of the tested hotspot regions are identified as nonstationary systems, where a combination of anthropogenic and climatic forcings is dominant. The nonstationary hydrologic systems would be more susceptible to hydrologic extremes with compromised capacities to timely cope, respond, and mitigate. Continuation of satellite gravity missions is crucial to evaluate the (non)stationary status in global hydrologic systems.
... In geosciences, data are considered the only reliable information for building models, inferences, and predictions. It is worth noting that since a stochastic model is a mathematical construction, stationarity or nonstationarity is a modeling option rather than a property of the observed data (Dimitriadis & Koutsoyiannis, 2018;Koutsoyiannis & Montanari, 2015). Any model, irrespective of it being deterministic or stochastic, stationary, or nonstationary, should be consistent with the data. ...
... Any model, irrespective of it being deterministic or stochastic, stationary, or nonstationary, should be consistent with the data. Koutsoyiannis and Montanari (2015) reported that ergodicity can always be assumed when stationarity exists in practice. In other words, the sample function contains the characteristic information of the entire stochastic process. ...
Full-text available
Under the changing climate, the natural characteristics of hydrological processes are assumed to have been intensified, and the volatility of these processes to have increased significantly. However, the applicability of traditional time series analysis methods and the commonly used Gaussian GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) to streamflow modeling has been limited. Although SV‐type (stochastic volatility) models have appeared as competitive alternatives to GARCH‐type models owing to their flexibility in modeling volatilities, they have not been applied in hydrological studies. Therefore, this study assesses the applicability of SV models to streamflow modeling using daily streamflow series from 21 hydrological stations in the Yellow River basin. Considering the influence of different types of GARCH and residual distributions, 7 hybrid models with variance fluctuation (fractional autoregressive integrated moving average‐GARCH) based on 9 distributions are compared to determine the optimal model for each station. Then 4 SV‐type models are introduced and compared with the results of the optimal GARCH‐type model to verify their applicability. The results show that: (a) the Gaussian distribution is not applicable in both GARCH‐type and SV‐type models for modeling daily streamflow; (b) although the GARCH‐type models are shown to describe the volatility of streamflow processes and improve the modeling performance, large residuals have been observed in the results of the analysis during the peak flow period; and (c) SV‐type models can better describe the streamflow series with time‐varying variance and accurately capture the occurrence of peak streamflow. The findings of this study offer practical and promising time series analysis methods for daily streamflow modeling.
... TCEV : F Y (y) = e −e −y −Λ * e −y/θ * (8) and to identify Homogeneous Regions (HRs), where the theoretical skewness, and therefore the parameters Λ * and θ * for the TCEV, or the parameter b (with b = 0 in the case of EV1) for the GEV, can be assumed as constant [5] or they follow laws based on geomorphological covariates (altitude, hydrographic basin area, etc.). The developed methodology in EXTRASTAR (useful for either onsite or regional approaches) is based on the following steps: ...
... Moreover, it should be stated that the concept of "change" is not mutually exclusive with the term "stationarity" [8]. In fact, according to the examples with Newton's laws: (1) without an external force, the position of a body in motion changes in time, but the velocity is unchanged; (2) a constant force implies a constant acceleration and a changing velocity. ...
... Stationarity means that the curve obtained from the time series of research elements can continue the existing shape for the coming period. The change is not synonymous with non-stationarity, change is a general notion applicable everywhere, stationarity and non-stationarity apply only to models, and are defined within stochastics (Koutsoyiannis and Montanari, 2015). The air temperature can be simulated as a non-linear trend stationary process (Estrada et al., 2013) or the extreme value of standardized value can be stationarity (Parey et al., 2019). ...
Since 20th century, urban warming has become the major global change concern, affecting sustainability of cities. Based on observation data of air temperature from 1728 meteorological stations, we constructed 612 effective grid cells with a spatial resolution of 1°×1° by using the gridding method, analyzed their spatiotemporal variability and non-stationarity by using the temperature anomaly and autocorrelation test model. Results showed that China's accelerated warming trend, with an increase of 0.995℃ from 1959 to 2018. The air temperature changes of grid cells show significant spatial variability. Grid cells with air temperature rise more than 1℃, accounted for 42.81%, mainly distributed in the northwest, east coast, northeast areas. China's warming has become more worrying in the past 30 years, with an increase of 0.797℃. The annual average temperature increased by 1.124℃, 1.029℃, 1.048℃ respectively in Beijing-Tianjin-Hebei, Yangtze River Delta, Pearl River Delta Urban Agglomerations from 1959 to 2018, which were higher than the country increased value. The temperature autocorrelation coefficients of 49.35%-96.08% grid cells were greater than the critical value of 0.05 significance level within lag 1-13 periods. The 30-year (1959-1988 or 1989-2018) mean the temperature s of 98.04% grid cells were outside the 90% confidence threshold of 60-year mean temperature. Therefore, temperature changes exhibited non-stationarity over China, providing scientific supports for future temperature prediction, global change response planning, and sustainable development.
... In many cases, these anthropogenic changes are difficult to be predicted deterministically. Therefore, it is appropriate to consider the increase in uncertainty under a stationary hypothesis rather than accepting non-stationarity and changing the parameters, as mentioned by Koutsoyiannis and Montanari (2014) [13]. No warm-up period for calibration was provided. ...
Full-text available
It is a grand challenge to realize robust rainfall-runoff prediction for a changing climate through conceptual hydrological models. Although multi-model ensemble (MME) is considered useful in improving the robustness of hydrological prediction, it has yet to be thoroughly evaluated. We evaluated the robustness of MME by 44 conceptual hydrological models in 582 river basins. We found that MME was more accurate and robust than each individual model alone. Although the performance of MME degrades in the validation period, the extent of degradation is smaller for MME than for individual models, especially when the climatology of river discharge in the validation period is greatly different from that in the calibration period. This implies the robustness of MME to climate change. It was found to be difficult to quantify the robustness of MME when the number of basins and models is small, which implies the importance of the large number of models and watersheds to evaluate the robustness and uncertainty in hydrological prediction.
... The assumption of non-stationarity is more appropriate for the current study because it is targeting climate variability/change. However, the stationarity or non-stationarity assumption is decided based on how well they represent statistical estimations drawn from a time series (Koutsoyiannis & Montanari 2015;Dimitriadis & Koutsoyiannis 2018). ...
Full-text available
Point precipitation frequency estimates (PPFEs) are used in the design of stormwater drainage infrastructures. The PPFEs for 1-year to 1,000-year recurrence intervals with durations ranging from 5 min to 60 days are published in the National Oceanic and Atmospheric Administration (NOAA) Atlas 14. NOAA's published PPFEs are accepted as a reliable resource for the design of urban drainage infrastructures, yet they are based on the stationary climate assumption. However, future climate change may affect the distribution, frequency, and intensity of precipitation events. To evaluate the potential impacts of climate change on the stationary climate assumption, a weather research and forecasting (WRF) model was developed based on the medium climate change emission scenario. Then, the PPFEs were determined for 2010–2015, 2030–2035, 2050–2055, and 2070–2075 at six different locations in the state of Arizona. The comparison of the published and simulated PPFEs revealed a varying trend in depth and frequency values dependent on location and climate. Additionally, depth values were diminished for <3-h events at a majority of the stations.
The software EXTRASTAR (EXTRemes Abacus for STAtistical Regionalization) is here proposed. It can represent a useful and quick tool for both regional and at-site statistical analyses of Annual Maxima (AM) time series. The innovative aspects (compared to other algorithms/softwares) are: 1) an easy comparison of sample skewness of several observed series with pre-determined Monte Carlo Prediction Intervals (depending on sample size) from hypothesized probability distributions, for obtaining a first important indication about possible clusters and regional values for some parameters, which can be afterward refined (if deemed as necessary) with specific and more complex algorithms/softwares proposed in literature; 2) for at-site analyses, the possibility to benefit from calibration of one function for some parameters of the others (thus significantly reducing the computational costs). EXTRASTAR was tested with the daily AM series for the Italian rain-gauge network, by implementing EV1, GEV and TCEV distributions.
Intensity-duration-frequency (IDF) curves – sometimes also called precipitation frequency estimates – are used to design urban drainage systems for stormwater control and, when combined with hydrologic modeling, to design flood control infrastructure and other structures. However, common approaches to estimating IDF curves need to be revised to account for non-stationarity from global climate change. This review synthesizes the current knowledge and on-going research regarding IDF curves under non-stationarity. In particular, this review briefly summarizes known and projected changes in extreme precipitation at the global scale, describes approaches for IDF curve estimation under non-stationarity (focusing on the covariate-based approach and including a brief overview of the specific challenges facing snow-dominated regions), addresses the topic of regionalization under non-stationarity (which has been overlooked in previous reviews), explores the challenges of design values and uncertainty in the context of non-stationarity, provides details on these topics in the context of the United States, and finishes by enumerating needed avenues of future research.
Through 2019, the Caspian Sea excluded, the majority (54–60%) of Earth’s irrigation-impacted endorheic lake and sea (ELS) areal extent has been lost in basins that contain as much as 20% of global irrigated agricultural land. Estimates of irrigated agriculture contribution to ELS desiccation based on a steady-state water balance equation for endorheic basins generally agree that this contribution is on the order of 70–90% at the global scale. However, large uncertainties or errors in attribution – as large as 100% – are observed with respect to particular ELS, suggesting that attributions based on a single irrigated agriculture dataset, should be treated cautiously. The observed areal contraction in ELS attributed to irrigated agriculture corresponds to an estimated one-third decrease in ELS volume, excluding the Caspian Sea. Such volumetric decrease is expected to at least double solute concentration in 40–47% of Earth’s ELS.
Full-text available
The new Scientific Decade 2013–2022 of IAHS, entitled “Panta Rhei—Everything Flows”, is dedicated to research activities on change in hydrology and society. The purpose of Panta Rhei is to reach an improved interpretation of the processes governing the water cycle by focusing on their changing dynamics in connection with rapidly changing human systems. The practical aim is to improve our capability to make predictions of water resources dynamics to support sustainable societal development in a changing environment. The concept implies a focus on hydrological systems as a changing interface between environment and society, whose dynamics are essential to determine water security, human safety and development, and to set priorities for environmental management. The Scientific Decade 2013–2022 will devise innovative theoretical blueprints for the representation of processes including change and will focus on advanced monitoring and data analysis techniques. Interdisciplinarity will be sought by increased efforts to connect with the socio-economic sciences and geosciences in general. This paper presents a summary of the Science Plan of Panta Rhei, its targets, research questions and expected outcomes.
Full-text available
We present a probability based theoretical scheme for building process-based models of uncertain hydrological systems, thereby unifying hydrological modeling and uncertainty assessment. Uncertainty for the model output is assessed by estimating the related probability distribution via simulation, thus shifting from one to many applications of the selected hydrological model. Each simulation is performed after stochastically perturbing input data, parameters and model output, this latter by adding random outcomes from the population of the model error, whose probability distribution is conditioned on input data and model parameters. Within this view randomness, and therefore uncertainty, is treated as an inherent property of hydrological systems. We discuss the related assumptions as well as the open research questions. The theoretical framework is illustrated by presenting real-world and synthetic applications. The relevant contribution of this study is related to proposing a statistically consistent simulation framework for uncertainty estimation which does not require model likelihood computation and simplification of the model structure. The results show that uncertainty is satisfactorily estimated although the impact of the assumptions could be significant in conditions of data scarcity.
We have focused our attention on solving ordinary differential equations that involve one dependent variable. However, many physical situations are modeled with more than one differential equation and involve more than independent variable. For example, if we want to determine the population of two interacting populations in a predator-prey relationship, such as foxes and rabbits, we have two dependent variables representing the two populations. In a basic model of the interaction of these two populations, these populations depend on one independent variable representing time. Situations similar to these lead to systems of differential equations.