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HurstKolmogorov dynamics and uncertainty
Demetris Koutsoyiannis
Professor and Head, Department of Water Resources and Environmental Engineering, Faculty of Civil
Engineering, National Technical University of Athens, Heroon Polytechneiou 5, GR 157 80
Zographou, Greece, Tel. +30 210 772 2831, Fax +30 210 772 2832; dk@itia.ntua.gr;
http://www.itia.ntua.gr/dk
Abstract. The nonstatic, ever changing hydroclimatic processes are often described as
nonstationary. However, revisiting the notions of stationarity and nonstationarity, defined
within stochastics, suggests that claims of nonstationarity cannot stand unless the evolution in
time of the statistical characteristics of the process is known in deterministic terms,
particularly for the future. In reality, longterm deterministic predictions are difficult or
impossible. Thus, change is not synonymous with nonstationarity, and even prominent change
at a multitude of time scales, small and large, can be described satisfactorily by a stochastic
approach admitting stationarity. This “novel” description does not depart from the 60 to 70
year old pioneering works of Hurst on natural processes and of Kolmogorov on turbulence.
Contrasting stationary with nonstationary has important implications in engineering and
management. The stationary description with HurstKolmogorov (HK) stochastic dynamics
demonstrates that nonstationary and classical stationary descriptions underestimate the
uncertainty. This is illustrated using examples of hydrometeorological time series, which
show the consistency of the HK approach with reality. One example demonstrates the
implementation of this framework in the planning and management of the water supply
system of Athens, Greece, also in comparison with alternative nonstationary approaches,
including a trendbased and a climatemodelbased approach.
Key terms (MODELING) stochastic models, uncertainty analysis, simulation; (CLIMATE)
climate variability/change; (HYDROLOGY) meteorology, streamflow; (WATER
RESOURCES MANAGEMENT) planning, water supply.
Introduction
«Αρχή σοφίας ονοµάτων επίσκεψις» (Αντισθένης)
“The start of wisdom is the visit (study) of names” (Antisthenes; ~445365 BC)
Perhaps the most significant contribution of the intensifying climatic research is the
accumulation of evidence that climate has never in the history of Earth been static. Rather, it
2
has been ever changing at all time scales. This fact, however, has been hard, even for
scientists, to accept, as displayed by the redundant (and thus non scientific) term “climate
change”. The excessive use of this term reflects a belief, or expectation, that climate would
normally be static, and that its change is something extraordinary which to denote we need a
special term (“climate change”) and which to explain we need to invoke a special agent (e.g.
anthropogenic influence). Examples indicating this problem abound, e.g., “climate change is
real” (Tol, 2006) or “there is no doubt that climate change is happening and that we should
be taking action to address it now” (Institute of Physics, 2010). More recently the scientific
term “nonstationarity”, contrasted to “stationarity”, has also been recruited to express similar,
or identical ideas to “climate change”. Sometimes their use has been dramatized, perhaps to
communicate better a nonscientific message, as in the recent popular title of a paper in
Science: “Stationarity is Dead” (Milly et al., 2008). We will try to show below (in section
“Visiting names: stationarity and nonstationarity”), that such use of these terms is in fact a
diversion and misuse of the real scientific meaning of the terms.
Insisting on the proper use of the scientific terms “stationarity” and “nonstationarity” is not
just a matter of semantics and of rigorous use of scientific terminology. Rather, it has
important implications in engineering and management. As we demonstrate below,
nonstationary descriptions of natural processes use deterministic functions of time to predict
their future evolution, thus explaining part of the variability and eventually reducing future
uncertainty. This is consistent with reality only if the produced deterministic functions are
indeed deterministic, i.e., exact and applicable in future times. As this is hardly the case as far
as future applicability is concerned (according to a saying attributed to Niels Bohr or to Mark
Twain, “prediction is difficult, especially of the future”), the uncertainty reduction is a
delusion and results in a misleading perception and underestimation of risk.
In contrast, proper stationary descriptions, which, in addition to annual (or subannual)
variability, also describe the interannual climatic fluctuations, provide more faithful
representations of natural processes and help us characterize the future uncertainty in
probabilistic terms. Such representations are based on the HurstKolmogorov (HK) stochastic
dynamics (section “Change under stationarity and the HurstKolmogorov dynamics”), which
has essential differences from typical random processes. The HK representations may be
essential for water resources planning and management, which demand long time horizons
and can have no other rational scientific basis than probability (or its complement, reliability).
3
It is thus essential to illustrate the ideas discussed in this paper and the importance of rigorous
use of scientific concepts through a realworld case study of water resources management.
The case study we have chosen for this purpose is the complex water supply system of
Athens. While Athens is a very small part of Greece (about 0.4% of the total area), it hosts
about 40% of its population. The fact that Athens is a dry place (annual rainfall of about 400
mm) triggered the construction of water transfer works from the early stages of the long
history of the city (Koutsoyiannis et al., 2008b) . The modern water supply system transfers
water from four rivers at distances exceeding 200 km.
0
100
200
300
400
1900 1920 1940 1960 1980
Year
Runoff (mm)
Annual runoff ''Trend''
200
400
600
800
1000
1200
1900 1920 1940 1960 1980
Year
Rainfall (mm)
Annual rainfall ''Trend''
0
100
200
1988 1989 1990 1991 1992 1993 1994
Year
Runoff (mm)
Annual runoff Average 198894 Average 190887
Fig. 1 Time series of runoff (upper) and rainfall (middle) in the Boeoticos Kephisos River basin from the
beginning of observations to 1987, with focus of the runoff during the severe, 7year (198894) drought period
(lower).
4
Fig. 1 (upper panel) shows the evolution of the runoff of one of these rivers, the Boeoticos
Kephisos River (in units of equivalent depth over its about 2 000 km
2
catchment) from the
beginning of observations to 1987. A substantial falling trend is clearly seen in the time
series. The middle panel of Fig. 1 shows the time series of rainfall in a raingauge in the basin
(Aliartos), where a trend is evident and explains (to a large extent) the trend in runoff. Most
interesting is the runoff in the following seven years, 19881994, shown in the last panel of
Fig. 1, which is consistently below average, thus manifesting a longlasting and severe
drought that shocked Athens during that period. The average flow during these seven years is
only 44% of the average of the previous years. A typical interpretation of such time series
would be to claim nonstationarity, perhaps attributing it to anthropogenic global warming, etc.
However, we will present a different interpretation of the observed behavior and its
implications on water resources planning and management (section “Implications in
engineering design and water resources management”). For Athens, these implications were
particularly important even after the end of the persistent drought, because it was then
preparing for the Olympic games—and these would not be possible in water shortage
conditions. Evidently, good planning and management demand a strong theoretical basis and
the proper application of fundamental (but perhaps forgotten or abused) notions.
Visiting names: stationarity and nonstationarity
Finding invariant properties within motion and change is essential to science. Newton’s laws
are eminent examples. The first law asserts that, in the absence of an external force, the
position x of a body may change in time t but the velocity u := dx/dt is constant. The second
law is a generalization of the first for the case that a constant force F is present, whence the
velocity changes but the acceleration a = du/dt is constant and equal to F/m, where m is the
mass of the body. In turn, Newton’s law of gravitation is a further generalization, in which the
attractive force F (weight) exerted, due to gravitation, by a mass M on a body of mass m
located at a distance r is no longer constant. In this case, the quantity G = F r
2
/(m M) is
constant, whereas in the application of the law for planetary motion another constant emerges,
i.e., the angular momentum per unit mass, (dθ/dt) r
2
, where θ denotes angle.
However, whilst those laws give elegant solutions (e.g., analytical descriptions of trajectories)
for simple systems comprising two bodies and their interaction, they can hardly describe the
irregular trajectories of complex systems. Complex natural systems consisting of very many
elements are impossible to describe in full detail nor their future evolution can be predicted in
detail and with precision. Here, the great scientific achievement is the materialization of
5
macroscopic descriptions rather than modeling the details. This is essentially done using
probability theory (laws of large numbers, central limit theorem, principle of maximum
entropy). Here lies the essence and usefulness of the stationarity concept, which seeks
invariant properties in complex systems.
According to the definitions quoted from Papoulis (1991), “A stochastic process x(t) is called
strictsense stationary … if its statistical properties are invariant to a shift of the origin” and
“… is called widesense stationary if its mean is constant (E[x(t)] = η) and its autocorrelation
depends only on [time difference] τ…, (E[x(t + τ) x(t)] = R(τ)]”. We can thus note that the
definition of stationarity applies to stochastic processes (rather than to time series; see also
Koutsoyiannis, 2006b). Processes that are not stationary are called nonstationary and in this
case some of their statistical properties are deterministic functions of time.
Abstract representation
Model
Ensemble: mental copies of
natural system
Stochastic process
Abstract representation
Model
Ensemble: mental copies of
natural system
Stochastic process
Real world
Natural
system
Unique
evolution
Time series
Real world
Natural
system
Unique
evolution
Time series
Many different models
can be constructed
Mental copies depend
on model constructed
Can generate arbitrar
ily many time series
Stationarity and
nonstationarity
apply here
Fig. 2 Schematic for the clarification of the notions of stationarity and nonstationarity.
Fig. 2 helps us to further clarify the definition. The left part of this graphic symbolizes the real
world. Any natural system we study has a unique evolution (a unique trajectory in time), and
if we observe this evolution, we obtain a time series. The right part of the graphic symbolizes
the abstract world, the models. Of course, we can build many different models of the natural
system, any one of which can give us an ensemble, i.e., mental copies of the realworld
system. The idea of mental copies is due to Gibbs, known from statistical thermodynamics.
An ensemble can also be viewed as multiple realizations of a stochastic process, from which
we can generate synthetic time series. Clearly, the notions of stationarity and nonstationarity
6
apply here, to the abstract objects—not to the realworld objects. In this respect, profound
conclusions such as that “hydroclimatic processes are nonstationary” or “stationarity is dead”
may be pointless.
To illustrate further the notion of stationarity we use an example of a synthetic time series,
shown in Fig. 3, whose generating model will be unveiled below, along with some indication
that it could be a plausible representation of a complex natural system. The upper panel of the
figure depicts the first 50 terms of the time series. Looking at the details of this irregular
trajectory, one could hardly identify any property that is constant. However, in a
macroscopic—i.e., statistical—description one could assume that this time series comes from
a stochastic process with a mean constant in time (E[x
i
] = µ, where E denotes expected value,
i denotes discrete time, x
i
is the time series and x
i
is the stochastic process). In a similar
manner, one can assume that the process has a standard deviation σ constant in time (i.e.,
E[(x
i
– µ)
2
] = σ
2
) and so on. Both µ and σ are not material properties of the process (that can
be measured by a certain device), but, rather, abstract statistical properties.
The middle panel of Fig. 3 depicts 100 terms of the time series. One could easily identify two
periods, i < 70 with a local time average m
1
= 1.8 and i ≥ 70 with a local time average m
2
=
3.5. One could then be tempted to use a nonstationary description, assuming a “change” or
“shift” of the mean at time i = 70. But this is just a temptation (if one follows the conventional
views of natural phenomena as either “clockwork” or “dice throwing”; see Koutsoyiannis,
2010); it does not reflect any objective scientific truth and it is not the only option. Rather, a
stationary description may be possible.
In fact, as is more evident from the lower panel of Fig. 3, a stationary model was used to
generate the time series. This model consists of the superposition of: (a) a stochastic process,
with values m
j
derived from the normal distribution N(2, 0.5), each lasting a period τ
j
exponentially distributed with E[τ
j
] = 50 (the thick line with consecutive plateaus); and (b)
white noise, with normal distribution N(0, 0.2). Nothing in this model is nonstationary and,
clearly, the process of our example is stationary. In fact, shifting mean models such as the one
above have been suggested in the water literature by several researches (e.g. Klemes, 1974;
Salas and Boes, 1980; Sveinsson et al, 2003).
7
1
1.5
2
2.5
0 10 20 30 40 50
Time,
i
Time series
Local average
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 10 20 30 40 50 60 70 80 90 100
Time, i
Time series
Local average
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 100 200 300 400 500 600 700 800 900 1000
Time, i
Time series
Local average
Fig. 3 A synthetic time series for the clarification of the notions of stationarity and nonstationarity (see text);
(upper) the first 50 terms; (middle) the first 100 terms; (lower) 1000 terms.
8
In this example, distinguishing stationarity from nonstationarity is a matter of answering a
simple question: Does the thick line of plateaus in Fig. 3 represent a known (deterministic)
function or an unknown (random) function? In the first case (deterministic function), we
should adopt a nonstationary description, while in the second case (random function, which
could be assumed to be a realization of a stationary stochastic process), we should use a
stationary description. As stated above, contrasting stationary with nonstationary descriptions
has important implications in engineering and management. To see this we have copied in
Fig. 4 the lower panel of Fig. 3, now in comparison to two “mental copies” of it. For the
construction of the middle panel of Fig. 4 we assumed nonstationarity, which implies that the
sequence of consecutive plateaus is a deterministic function of time. Thus, the thick lines of
plateaus is exactly the same as in the original time series of the upper panel. The uncertainty,
expressed as the unexplained variance, i.e., the variance of differences between the thick line
of plateaus and the rough line, is (by construction of the process) 0.2
2
= 0.04. The mental
copy shown in the lower panel of Fig. 4 was constructed assuming stationarity. This copy has
a different random realization of the line of plateaus. As a result, the total variance (that of the
“nondecomposed” time series) is unexplained, and this is calculated to be 0.38, i.e., almost
10 times greater than in the nonstationary description. Thus, a nonstationary description
reduces uncertainty, because it explains part of the variability. This is consistent with reality
only if the produced deterministic functions are indeed deterministic, i.e., exact and applicable
in future times. As this is hardly the case, as far as future applicability is concerned, the
uncertainty reduction is a illusion and results in a misleading perception and underestimation
of risk.
In summary, the referred example illustrates that (a) stationary is not synonymous with static;
(b) nonstationary is not synonymous with changing; (c) in a nonstationary process the change
is described by a deterministic function; (d) nonstationarity reduces uncertainty (because it
explains part of variability); and (e) unjustified/inappropriate claim of nonstationarity results
in underestimation of variability, uncertainty and risk. In contrast, a claim of nonstationarity is
justified and, indeed, reduces uncertainty, if the deterministic function of time is constructed
by deduction (the Aristoteleian apodeixis), and not by induction (direct use of data). Thus, to
claim nonstationarity, we must: (a) establish a causative relationship; (b) construct a
quantitative model describing the change as a deterministic function of time; and (c) ensure
applicability of the deterministic model into the future.
9
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 100 200 300 400 500 600 700 800 900 1000
Time, i
Time series
Local average
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 100 200 300 400 500 600 700 800 900 1000
Time, i
Time series
Local average
0
0.5
1
1.5
2
2.5
3
3.5
4
0 100 200 300 400 500 600 700 800 900 1000
Time,
i
Time series
Local average
Fig. 4 The time series of Fig. 3 (upper) along with mental copies of it assuming that the local average is a
deterministic function and thus identical with that of the upper panel (middle) or assuming that the local average
is a random function, i.e. a realization of the stochastic process described in text, different from that of the upper
panel (lower).
10
Because the inflationary use of the term “nonstationarity” in hydrology has recently been
closely related to “climate change”, it is useful to examine whether the terms justifying a
nonstationary description of climate hold true or not. The central question is: Do climate
models (also known as general circulation models—GCMs) enable a nonstationary approach?
More specific versions of this question are: Do GCMs provide credible deterministic
predictions of future climate evolution? Do GCMs provide good predictions for temperature
and somewhat less good for precipitation (as often thought)? Do GCMs provide good
predictions at global and continental scales and, after downscaling, at local scales? Do GCMs
provide good predictions for the distant future (albeit less good for the nearer future, e.g., for
the next 1020 years—or for the next season or year)? In the author’s opinion, the answers to
all these questions should be categorically negative. Not only are GCMs unable to provide
credible climatic predictions for the future, but they also fail to reproduce the known past and
even the past statistical characteristics of climate (see Koutsoyiannis et al., 2008a;
Anagnostopoulos et al., 2010). An additional, very relevant question is: Is climate predictable
in deterministic terms? Again, the author’s answer is negative (Koutsoyiannis, 2006a; 2010).
Only stochastic climatic predictions could be scientifically meaningful. In principle, these
could also include nonstationary descriptions wherever causative relationships of climate with
its forcings are established. But until such a stochastic theory of climate, which includes
nonstationary components, could be shaped, there is room for developing a stationary theory
that characterizes future uncertainty as faithfully as possible; the main characteristics of such
a theory are outlined in section “Change under stationarity and the HurstKolmogorov
dynamics” (see also Koutsoyiannis et al., 2007).
While a nonstationary description of climate is difficult to establish or possibly even
infeasible, in cases related to water resources it may be much more meaningful. For example,
in modeling streamflow downstream of a dam, we would use a nonstationary model with a
shift in the statistical characteristics before and after the construction of the dam. Gradual
changes in the flow regime, e.g., due to urbanization that evolves in time, could also justify a
nonstationary description, provided that solid information or knowledge (as opposed to
ignorance) of the agents affecting a hydrological process is available. Even in such cases, as
far as modeling of future conditions is concerned, a stationary model of the future is sought
most frequently. A procedure that could be called “stationarization” is then necessary to adapt
the past observations to future conditions. For example, the flow data prior to the construction
of the dam could be properly adapted, by deterministic modeling, so as to determine what the
11
flow would be if the dam existed. Also, the flow data at a certain phase of urbanization could
be adapted so as to represent the future conditions of urbanization. Such adaptations enable
the building of a stationary model of the future.
Change under stationarity and the HurstKolmogorov dynamics
It was asserted earlier that nonstationarity is not synonymous with change. Even in the
simplest stationary process, the white noise, there is change all the time. But, as this case is
characterized by independence in time, the change is only shortterm. There is no change in
longterm time averages. However, a process with dependence in time exhibits longerterm
changes. Thus, change is tightly linked to dependence and longterm change to longrange
dependence. Hence, stochastic concepts that have been devised to study dependence also help
us to study change.
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90 100
Lag
Autocorrelation
Empirical
Markov
Fig. 5 Empirical autocorrelogram of the time series of Fig. 3 in comparison to the theoretical autocorrelogram of
a Markovian process with lag one autocorrelation equal to the empirical.
Here, we are reminded of three such concepts, or stochastic tools, stressing that all are
meaningful only for stationary processes (albeit this is sometimes missed). The
autocorrelogram, which is a plot of the autocorrelation coefficient vs. lag time, provides a
very useful characterization and visualization of dependence. Fig. 5 depicts the empirical
autocorrelogram estimated from the 1000 items of the time series of Fig. 3. The fact that the
autocorrelation is positive even for lags as high as 100 is an indication of longrange
dependence. The popular Markovian (AR(1)) dependence would give much lower
autocorrelation coefficients, as also shown in Fig. 5, whereas a white noise process would
give zero autocorrelations, except for lag 0, which is always 1 irrespectively of the process.
12
We recall that the process in our example involves no “memory” mechanism; it just involves
change in two characteristic scales, 1 (the white noise components) and 50 (the average length
of the plateaus). Thus, interpretation of longrange dependence as “long memory”, despite
being very common (e.g. Beran, 1994), may be misleading; it is more insightful to interpret
longrange dependence as longterm change. This has been first pointed out—or implied—by
Klemes, 1974, who wrote “… the Hurst phenomenon is not necessarily an indicator of infinite
memory of a process”. The term “memory” should better refer to systems transforming inputs
to outputs (cf. the definition of memoryless systems in Papoulis, 1991), rather than to a single
stochastic process.
Slope = 1
0.001
0.01
0.1
1
10
100
0.001 0.01 0.1 1
Frequency
Spectral density
Empirical
Power law approximation
Fig. 6 Empirical power spectrum of the time series of Fig. 3.
The power spectrum, which is the inverse finite Fourier transform of the autocorrelogram, is
another stochastic tool for the characterization of change with respect to frequency. The
power spectrum of our example is shown in Fig. 6, where a rough line appears, which has an
overall slope of about –1. This negative slope, which indicates the importance of variation at
lower frequencies relative to the higher ones, provides a hint of longrange dependence.
However, the high roughness and scattering of the power spectrum does not allow accurate
estimations. A better depiction is provided in Fig. 7 by the climacogram (from the Greek
climax, i.e., scale), which provides a multiscale stochastic characterization of the process.
Based on the process x
i
at scale 1, we define a process x
i(k)
at any scale k ≥ 1 as:
13
∑
+−=
=
ik
kil
l
k
i
x
k
x
1)1(
)(
1
:
(1)
A key multiscale characteristic is the standard deviation σ
(k)
of x
i(k)
. The climacogram is a
plot (typically double logarithmic) of σ
(k)
as a function of the scale k ≥ 1. While the power
spectrum and the autocorrelogram are related to each other through a Fourier transform, the
climacogram is related to the autocorrelogram by a simpler transformation, i.e.,
k
k
k
α
σ
σ
=
)(
,
∑
−
=
−+=
1
1
121
k
j
jk
k
j
ρα
↔
11
2
1
2
1
−+
−
+−
+
=
jjjj
j
j
j
αααρ
(2)
To estimate the climacogram, the standard deviation σ
(k)
could be calculated either from the
autocorrelogram by means of (2) or directly from time series x
i(k)
aggregated by (1). It is
readily verified (actually this is the most classical statistical law) that in a process with
independence in time (white noise), σ
(k)
= σ/k, which implies a slope of –1/2 in the
climacogram. Positively autocorrelated processes yield higher σ
(k)
and perhaps milder slopes
of the climacogram. Fig. 7 illustrates the constant slope of –1/2 of a whitenoise process,
which is also asymptotically the slope of a Markovian process, while the process of our
example suggests a slope of –0.25 for scales k near 100.
Slope = 0.5
Slope = 0.25
0.1
1
1 10 100
Scale, k
Standard deviation, σ
(k)
Empirical
Markov
White noise
Fig. 7 Empirical climacogram of the time series of Fig. 3 in comparison to the theoretical climacograms of a
whitenoise and a Markovian process.
Recalling that our example involves two time scales of change (1 and 50), we can imagine a
process with additional time scales of change. The simplest case of such a process (which
14
assumes theoretically infinite time scales of fluctuation, although practically, three such scales
suffice; Koutsoyiannis, 2002), is the one whose climacogram has a constant slope H – 1, i.e.
σ
(k)
= k
H – 1
σ (3)
This simple process, which is essentially defined by (3), has been termed the Hurst
Kolmogorov (HK) process (after Hurst, 1951, who first analyzed statistically the longterm
behavior of geophysical time series, and Kolmogorov, 1940, who, in studying turbulence, had
proposed the mathematical form of the process), and is also known as simple scaling
stochastic model or fractional Gaussian noise (cf. Mandelbrot and Wallis, 1968). The constant
H is called the Hurst coefficient and in positivelydependent processes ranges between 0.5
and 1. The elementary statistical properties of the HK process are shown in Table 1, where it
can be seen that all properties appear to be power laws of scale, lag and frequency.
Table 1 Elementary statistical properties of the HK process.
Statistical property At scale k = 1 (e.g. annual) At any scale k
Standard deviation σ ≡ σ
(1)
σ
(k)
= k
H – 1
σ
Autocorrelation
function (for lag j) ρ
j
≡ ρ
(1)
j
=ρ
(k)
j
≈ H (2 H – 1) j

2H – 2
Power spectrum (for
frequency ω)
s(ω) ≡ s
(1)
(ω) ≈
4 (1 – H) σ
2
(2 ω)
1 – 2 H
s
(k)
(ω) ≈
4(1 – H) σ
2
k
2H – 2
(2 ω)
1 – 2 H
Fluctuations at multiple temporal or spatial scales, which may suggest HK stochastic
dynamics, are common in Nature, as seen for example in turbulent flows, in large scale
meteorological systems, and even in humanrelated processes. We owe the most characteristic
example of a large spatialscale phenomenon that exhibits HK temporal dynamics to the
Nilometer time series, the longest available instrumental record. Fig. 8 shows the record of the
Nile minimum water level from the 7th to the 13th century AD (663 observations, published
by Beran, 1994 and available online from http://lib.stat.cmu.edu/S/beran, here converted into
meters). Comparing this Nilometer time series with synthetically generated white noise, also
shown in Fig. 8 (lower panel), we clearly see a big difference on the 30year scale. The
fluctuations in the realworld process are much more intense and frequent than the stable
curve of the 30year average in the white noise process.
15
0
2
4
6
8
600 700 800 900 1000 1100 1200 1300
Year AD
Minimum water level (m)
Annual
30year average
0
2
4
6
8
600 700 800 900 1000 1100 1200 1300
"Year"
Minimum roulette wheel outcome
"Annual"
30"year" average
Fig. 8 The annual minimum water level of the Nile River from the Nilometer (upper) and, for comparison, a
synthetic series, each value of which is the minimum of 36 outcomes of a roulette wheel (lower); both time
series have equal length (663) and standard deviation (about 1.0).
The climacogram of the Nilometer series, shown in Fig. 9, suggests that the HK model is a
good representation of reality. To construct this climacogram, the annual time series of 663
observations, was aggregated (averaged) into time scales of 2, 3, …, 66 years, each one
having, respectively, 331, 221, …, 10 data points. The sample standard deviations s
(k)
(actually their logarithms) are plotted in Fig. 9. Their plot departs substantially from those
corresponding to the white noise process as well as the Markovian (AR(1)) processes, whose
theoretical climacograms are also plotted in Fig. 9. The former is a straight line with slope
–0.5 while the second is a curve (whose analytical expression is given in Koutsoyiannis,
2002), but for scales > 10
0.5
it becomes again a straight line with slope –0.5. On the other
16
hand, the HK model with a Hurst coefficient is H = 0.89 seems to be consistent with the
empirical points. The theoretical climacogram of this HK model is plotted in Fig. 9 as a
straight line with slope –0.11. However, the empirical sample standard deviations s
(k)
are not
directly comparable to the theoretical σ
(k)
of the HK model, because, as will be detailed below
the HK behavior implies substantial bias in the estimates of variance and standard deviation.
For this reason, another curve, labelled “HurstKolmogorov adapted for bias” is also plotted
in the figure, in which the bias (also predicted by the HK model as shown in Table 2) was
subtracted from the theoretical model. The latter curve agrees well with the empirical points.
The value of the Hurst coefficient H = 0.89 was estimated by the LSSD (least squares based
on standard deviation) algorithm (Koutsoyiannis, 2003; Tyralis and Koutsoyiannis, 2010).
Interestingly, a similar value (H = 0.85) is estimated (by the same algorithm, Koutsoyiannis et
al., 2008) from the modern record (131 years) of the Nile flows at Aswan (although due to
high uncertainty implied be HK, estimates by other algorithms may differ; see Montanari et
al., 2000).
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0 0.5 1 1.5 2
Log(scale in years)
Log(standard deviation in m)
Empirical
White noise ("Roulette")
Markov
HurstKolmogorov, theoretical
HurstKolmogorov adapted for bias
Fig. 9 Climacogram of the Nilometer time series of Fig. 8.
The same behavior can be verified in several geophysical time series; examples are given in
most related publications referenced herein. Two additional examples are depicted in Fig. 10,
which refers to the monthly lower tropospheric temperature, and in Fig. 11, which refers to
17
the monthly Atlantic Multidecadal Oscillation (AMO) index. Both examples suggest
consistency with HK behavior with a very high Hurst coefficient, H = 0.99.
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
2009
Temperature ( °C)
Monthly
12month running average
3year running average
All data average
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Log (scale in months)
Log (standard deviation in °C)
Empirical
White noise
HurstKolmogorov, theoretical
HurstKolmogorov, adapted for bias
Bias
Fig. 10 Monthly time series (upper) and climacogram (lower) of the global lower tropospheric temperature (data
for 19792009, from http://vortex.nsstc.uah.edu/public/msu/t2lt/tltglhmam_5.2).
18
0.6
0.4
0.2
0
0.2
0.4
0.6
1850 1870 1890 1910 1930 1950 1970 1990 2010
Year
AMO index
Monthly
5year average
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5
Log(scale in months)
Log(Standard deviation)
Empirical
White noise
HurstKolmogorov, theoretical
HurstKolmogorov adapted for bias
Fig. 11 Monthly time series (upper) and climacogram (lower) of the Atlantic Multidecadal Oscillation (AMO)
index (data for 18562009, from NOAA, http://www.esrl.noaa.gov/psd/data/timeseries/AMO/).
One of the most prominent implications of the HK behavior concerns the typical statistical
estimation. The HK dynamics implies dramatically higher intervals in the estimation of
location statistical parameters (e.g., mean) and highly negative bias in the estimation of
dispersion parameters (e.g., standard deviation). The HK framework allows calculating the
statistical measures of bias and uncertainty of statistical parameters (Koutsoyiannis, 2003;
Koutsoyiannis and Montanari, 2007), as well as those of future predictions (Koutsoyiannis et
al., 2007). It is surprising, therefore, that in most of the recent literature the HK behavior is
19
totally neglected, despite the fact that books such as those by Salas et al. (1980), Bras and
RodriguezIturbe (1985), and Hipel and McLeod (1994) have devoted a significant attention
to the Hurst findings and methods to account for it. Even studies recognizing the presence of
HK dynamics usually do not account for the implications in statistical estimation and testing.
Naturally, the implications magnify as the strength of the HK behavior increases, i.e., as H
approaches 1. Table 2 provides in a tabulated form the equations (or simplified
approximations thereof) that determine the bias and uncertainty metrics for the three most
typical statistical estimators, i.e. of the mean, standard deviation and autocorrelation (which
are indicators of location, dispersion, and dependence, respectively). The reader interested to
see more detailed presentation of the equations including their derivations is referenced to
Beran (1994), Koutsoyiannis (2003) and Koutsoyiannis and Montanari (2007). In addition to
the theoretical equations, Table 2 provides, a numerical example for n
=
100 and H
=
0.90. Fig.
10 and Fig. 11 depict the huge bias in the standard deviation when H = 0.99. This bias
increases with increased time scale because the sample size for higher time scales becomes
smaller.
Table 2 Impacts on statistical estimation: HurstKolmogorov statistics (HKS) vs. classical statistics (CS)
(sources: Koutsoyiannis, 2003; Koutsoyiannis and Montanari, 2007).
True values →
Mean, µ Standard deviation, σ Autocorrelation ρ
l
for lag l
Standard
estimator x
– := 1
n ∑
i = 1
n
x
i
s := 1
n – 1 ∑
i = 1
n
(x
i
– x
–)
2
r
l
:= 1
(n – 1)s
2
·
∑
i = 1
n – l
(x
i
– x
–)(x
i + l
– x
–)
Relative bias of
estimation, CS 0 ≈ 0 ≈ 0
Relative bias of
estimation,
HKS
0 1 − 1
n΄/ 1 − 1
n − 1 ≈ − 1
2n΄
(–22%)
≈ – 1/ρ
l
− 1
n΄− 1 (–79%)
Standard
deviation of
estimator, CS
σ
n (10%) ≈ σ
2(n – 1) (7.1%)
Standard
deviation of
estimator, HKS
σ
n΄ (63%)
≈ σ (0.1 n + 0.8)
λ(H)
(1 –n
2H − 2
)
2(n – 1)
where λ(H) := 0.088 (4H
2
–
1)
2
(9.3%)
Notes (a) n΄ :=
σ
2
/Var[x
–] =
n
2 – 2H
is the “equivalent” or “effective” sample size: a sample with size n΄ in CS
results in the same uncertainty of the mean as a sample with size n in HKS; (b) the numbers in parentheses are
numerical examples for n
=
100, σ
=
1, H
=
0.90 (so that n΄
=
2.5) and l =
10.
20
Implications in engineering design and water resources management
Coming back to the Athens water supply system, it is interesting to estimate the return period
of the multiyear drought mentioned in the Introduction. Let us first assume that the annual
runoff in the Boeoticos Kephisos basin can be approximated by a Gaussian distribution (this
is fairly justified given that the coefficient of skewness is 0.35 at the annual scale and drops to
zero or below at the 3year scale and beyond) and that the multiyear standard deviation σ
(k)
at
scale (number of consecutive years) k is given by the classical statistical law, σ
(k)
= σ/k,
which assumes independence in time. We can then easily assign a theoretical return period to
the lowest (as well as to the highest) recorded value for each time scale. More specifically, the
theoretical return period of the lowest observed value x
L(k)
, for each time scale k, can be
determined as T
L
= k
δ / F(x
L(k)
), where δ = 1 year and F denotes the probability distribution
function. The latter is Gaussian with mean µ (independent of scale, estimated as the sample
average x
– at the annual scale) and standard deviation σ
(k)
(for scale k, determined as σ/k,
with σ estimated as the sample standard deviation s at the annual scale). Likewise, for the
highest value x
H(k)
the theoretical return period is T
H
= k
δ / (1 – F(x
H(k)
).
1
10
100
1000
10000
100000
0 2 4 6 8 10
Scale, k
Return period (years)
Loewest value, classical
Highest value, class ical
Lowest value, HK
Highest value, HK
Emprirically expected
Fig. 12 Return periods of the lowest and highest observed annual runoff, over time scale (or number of
consecutive years) k = 1 (annual scale) to 10 (decadal scale), of the Boeoticos Kephisos basin assuming normal
distribution (adapted from Koutsoyiannis et al., 2007).
21
Fig. 12 shows the assigned return periods of the lowest and highest values for time scales
(number of consecutive years) k = 1 to 10. Empirically, since the record length is about 100
years, we expect that the return period of lowest and highest values would be of the order of
100 years for all time scales. This turns out to be true for k = 1 to 2, but the return periods
reach 10 000 years at scale k = 5. Furthermore, the theoretical return period of the lowest
value at scale k = 10 (10yearlong drought) reaches 100 000 years!
0
100
200
300
400
1900 1920 1940 1960 1980 2000
Year
Runoff (mm)
Annual 30year average
19072002 average
1.5
1.6
1.7
1.8
1.9
0 0.2 0.4 0.6 0.8 1
Log(scale in years)
Log(standard deviation in mm)
Empirical
Classical statistics
HurstKolmogorov, theoretical
HurstKolmogorov, adapted for bias
Fig. 13 The entire annual time series (upper) and the climacogram (lower) of the Boeoticos Kephisos runoff.
Is this sufficient evidence that Athens experienced a very infrequent drought event, which
happens on the average once every 100 000 years, in our lifetime? In the initial phase of our
involvement in this case study we were inclined to believe that we witnessed an event that
extraordinary but, gradually, we understood that the event may not be that infrequent. History
22
is the key to the past, to the present, and to the future; and the longest available historical
record is that of the Nilometer (Fig. 8). This record offers a precious empirical basis of long
term changes. It suffices to compare the time series of the Beoticos Kephisos runoff (shown in
its entirety in Fig. 13) with that of the Nilometer series. We observe that a similar pattern had
appeared in the Nile flow between 680 and 780 AD: a 100year falling trend (which, notably,
reverses after 780 AD), with a clustering of very low water level around the end of this
period, between 760 and 780 AD. Such clustering of similar events was observed in several
geophysical time series by Hurst (1951), who stated: “Although in random events groups of
high or low values do occur, their tendency to occur in natural events is greater. This is the
main difference between natural and random events.”
0.99
0.95
0.8
0.5
0.2
0.05
0.01
0
100
200
300
400
500
3 2 1 0 1 2 3
Reduced normal variate
Distribution quantile (mm)
PE/classical
MCCL/classical
PE/HK
MCCL/HK
0.99
0.95
0.8
0.5
0.2
0.05
0.01
0
100
200
300
400
500
3 2 1 0 1 2 3
Reduced normal variate
Distribution quantile (mm)
PE/classical
MCCL/classical
PE/HK
MCCL/HK
0.01 0.05 0.2 0.5 0.8 0.95 0.99
0
100
200
300
400
500
Distribution quantile (mm)
PE
MCCL/classical
MCCL/HK
Probability of nonexceedence
0.01 0.05 0.2 0.5 0.8 0.95 0.99
0
100
200
300
400
500
Distribution quantile (mm)
PE
MCCL/classical
MCCL/HK
Probability of nonexceedence
Fig. 14 Point estimates (PE) and 95% Monte Carlo confidence limits (MCCL) of the distribution quantiles of the
Boeoticos Kephisos runoff at the annual (upper) and climatic (30year; lower) time scales, both for classical and
HK statistics (adapted from Koutsoyiannis et al., 2007).
23
Thus, the Athens story may prompt us to replace the classical statistical framework (i.e. that
assuming independence in time) with a HK framework. As shown in Fig. 13 (lower panel) the
Boeticos Kephisos runoff time series is consistent with the HK model, with a Hurst
coefficient H = 0.79. Redoing the calculations of return period, we find that the return period
for scale k reduces from the extraordinary value of 100 000 years to a humble value of 270
years. Also, the HK framework renders the observed downward trend a natural and usual
behavior (Koutsoyiannis, 2003). The Boeticos Kephisos runoff is another “naturally trendy”
process to use an expression coined by Cohn and Lins (2005).
Thus, the HK framework implies a perspective of natural phenomena that is very different
from that of classical (i.e. independencebased) statistics, particularly in aggregate scales. This
is further demonstrated in Fig. 14 (adapted from Koutsoyiannis et al., 2007, where additional
explanation on its construction is given), which depicts normal probability plots of the
distribution quantiles of the Boeoticos Kephisos runoff at the annual and the climatic, 30
year, time scale. At the annual time scale (k = 1) the classical and the HK statistics yield the
same point estimates of distribution quantiles (i.e. the same amount of uncertainty due to
variability), but the estimation (or parameter) uncertainty, here defined by the 95% confidence
limits constructed by a Monte Carlo method, is much greater according to the HK statistics.
The confidence band is narrow in classical statistics (shaded area in Fig. 14) and becomes
much wider in the HK case.
More interesting is the lower panel of Fig. 14, which refers to the typical climatic time scale
(k = 30). The low variability and uncertainty in the classical model is depicted as a narrow,
almost horizontal, band in the lower panel of Fig. 14. Here, the HK model, in addition to the
higher parameter uncertainty, results in uncertainty due to variability much wider than in the
classical model. As a result, while the total uncertainty (by convention defined as the
difference of the upper confidence limit at probability of exceedence 97.5% minus the lower
confidence limit at probability of exceedence 2.5%) is about 50% of the mean in the classical
model, in the HK case it becomes about 200% of the mean, or four times larger. Interestingly,
it happens that the total uncertainty of the classical model at the annual scale is 200% of the
mean. In other words, the total uncertainty (due to natural variability and parameter
estimation) at the annual level according to the classical model equals the total uncertainty at
the 30year scale according to HK model. This allows paraphrasing a common saying (which
sometimes has been used to clarify the definition of climate, e.g., NOAA Climate Prediction
Center, 2010) that “climate is what we expect, weather is what we get” in the following way:
24
“weather is what we get immediately, climate is what we get if you keep expecting for a long
time”.
For reasons that should be obvious from the above discussion, the current planning and
management of the Athens water supply system are based on the HK framework. Appropriate
multivariate stochastic simulation methods have been developed (Koutsoyiannis, 2000, 2001)
that are implemented within a general methodological framework termed parameterization
simulationoptimization (Nalbantis and Koutsoyiannis, 1997; Koutsoyiannis and Economou,
2003; Koutsoyiannis et al., 2002, 2003; Efstratiadis et al., 2004). The whole framework
assumes stationarity, but simulations always use the current initial conditions (in particular,
the current reservoir storages) and the recorded past conditions:, in a Markovian framework,
only the latest observations affect the future probabilities, but in the HK framework the entire
record of past observations should be taken into account to condition the simulations of future
(Koutsoyiannis, 2000).
Nonetheless, it is interesting to discuss two alternative methods that are more commonly used
than the methodology developed for Athens. The first alternative approach, which is
nonstationary, consists of the projection of the observed “trend” into the future. As shown in
Fig. 15, according to this approach the flow would disappear by 2050. Also, this approach
would lead to reduced uncertainty (because it assumes that the observed “trend” explains part
of variability); the initial standard deviation of 70 mm would decrease to 55 mm. Both these
implications are glaringly absurd.
0
100
200
300
400
1900 1950 2000 2050
Year
Runoff (mm)
Before time of modelling
After time of modelling
Trend
Fig. 15 Illustration of the alternative method of trend projection into the future for modeling of the Boeoticos
Kephisos runoff.
25
The second alternative, again admitting nonstationarity, is to use outputs of climate models
and to feed them into hydrological models to predict the future runoff. This approach is
illustrated in Fig. 16, also in comparison to the HK stationary approach and the classical
statistical approach. Outputs from three different GCMs (ECHAM4/OPYC3, CGCM2,
HadCM3), each one for two different scenarios, were used, thus shaping 6 combinations
shown in the legend of Fig. 16 (each line of which corresponds to each of the three models in
the order shown above; see more details in Koutsoyiannis et al., 2007). To smooth out the
annual variability, the depictions of Fig. 16 refer to the climatic (30year) scale. In fact,
outputs of the climate models exhibited huge departures from reality (highly negative
efficiencies at the annual time scale and above); thus, adjustments, also known as “statistical
downscaling”, were performed to make them match the most recent observed climatic value
(30year average).
0
100
200
300
400
1930 1960 1990 2020 2050
Year
Runoff (mm)
MP01GG01 MP01GS01
CCCma_A2 CCCma_B2
HADCM3_A2 HADCM3_B2
Observed Point forecast
MCCL/HK MCCL/classical
Fig. 16 Illustration the alternative GCMbased method for modeling of the Boeoticos Kephisos runoff, vs. the
uncertainty limits (Monte Carlo Confidence Limits—MCCL) estimated for classical and HK statistics; runoff is
given at climatic scale, i.e. runoff y at year x is the average runoff of a 30year period ending at year x (adapted
from Koutsoyiannis et al., 2007).
Fig. 16 shows plots of the GCMbased time series after the adjustments. For the past, despite
adjustments, the congruence of models with reality is poor (they do not capture the falling
trend, except one part reflecting the more intense water resources exploitation in recent years).
Even worse, the future runoff obtained by adapted GCM outputs is too stable. All different
26
model trajectories are crowded close to the most recent climatic value. Should one attempt to
estimate future uncertainty by enveloping the different model trajectories, this uncertainty
would be lower even from that produced by the classical statistical model. Hence, the GCM
based approach is too risky, as it predicts a future that is too stable, whereas the more
consistent HK framework entails a high future uncertainty (due to natural variability and
unknown parameters), which is also shown in Fig. 16. The planning and management of the
Athens water supply system is based on the latter uncertainty.
Some interesting questions were raised during the review phase of the paper and need to be
discussed: Isn’t there a danger in ignoring results from deterministic models? What if, unlike
in the Athens example, the GCM results were not contained within the uncertainty limits of
the HK statistics? In the author’s opinion, whether results from deterministic model should be
considered or ignored depends on whether the models results have been validated against
reality. In hydrology there is a long tradition in model building, assessing the prediction skill
of models, and evaluating the skill not only in the model calibration period, but also in a
separate validation period, whose data were not used in the calibration (the splitsample
technique, Klemeš, 1986). Models that have not passed such scrutiny, may not be provide
usable results regardless of whether these results are contained or not into confidence limits.
In the Athens case, as stated above, the outputs of the climate models exhibited huge
departures from reality. In contrast, the HK approach seems to have provided a better
alternative with a sound yet parsimonious theoretical basis and an appropriate empirical
support. Obviously, any modeling framework is never a perfect description of the real world
and can never provide solution to all problems over the globe—and this holds also for the HK
approach. Obviously, any model involves uncertainty in parameter estimation. In the HK
approach this uncertainty is amplified, as detailed above, and this amplification may even hide
the presence of the HK dynamics if observation records are short. On the other hand, as far as
longterm future predictions are concerned, a macroscopic—and thus stochastic—approach
may be more justified than deterministic modeling. This approach should be consistent with
the longterm statistical properties of hydroclimatic processes, like the HK behavior, as
observed from long instrumental and proxy time series, where available. Incorporating in such
a stochastic approach what is known about the driving causal mechanisms of hydroclimatic
processes could potentially provide a more promising scientific and technological direction
than the current deterministic GCM approach.
27
Additional remarks
Whilst this exposition has focused on climatic averages and low extremes (droughts), it may
be useful to note that change, which underlies HK dynamics, also affects high extremes such
as intense storms and floods. This concerns both the marginal distribution tail as well as the
timing of high intensity events. For example, Koutsoyiannis (2004) has shown that an
exponential distribution tail of rainfall may shift to a power tail if the scale parameter of the
former distribution changes in time; and it is well known that a power tail yields much higher
rainfall amounts in comparison to an exponential tail for high return periods. Also, Blöschl
and Montanari (2010) demonstrated that five of the six largest floods of the Danube River at
Vienna (100 000 km
2
catchment area) in the 19
th
century were grouped in its last two decades.
This is consistent with Hurst’s observation about grouping of similar events and should
properly be taken into account in flood management—rather than trying to speculate about
humaninduced climate effects. Likewise, Franks and Kuczera (2002) showed that the usual
assumption that annual maximum floods are identically and independently distributed is
inconsistent with the gauged flood evidence from 41 sites in Australia whereas Bunde et al.
(2005) found that the scaling behavior leads to pronounced clustering of extreme events and
demonstrated that this can be seen in long climate records.
Overall, the “new” HK approach presented herein is as old as Kolmogorov’s (1940) and
Hurst’s (1951) expositions. It is stationary (not nonstationary) and demonstrates how
stationarity can coexist with change at all time scales. It is linear (not nonlinear) thus
emphasizing the fact that stochastic dynamics need not be nonlinear to produce realistic
trajectories (while, in contrast, trajectories from linear deterministic dynamics are not
representative of the evolution of complex natural systems). The HK approach is simple,
parsimonious, and inexpensive (not complicated, inflationary and expensive) and is
transparent (not misleading) because it does not hide uncertainty and it does not pretend to
predict the distant future deterministically.
Conclusions
• Change is Nature’s style.
• Change occurs at all time scales.
• Change is not nonstationarity.
• HurstKolmogorov dynamics provides a useful key to perceive multiscale change and
model the implied uncertainty and risk.
28
• In general, the HurstKolmogorov approach can incorporate deterministic descriptions
of future changes, if available.
• In the absence of credible predictions of the future, HurstKolmogorov dynamics admits
stationarity.
Acknowledgment. I am grateful to J. D. Salas for his very detailed and constructive
suggestions and an additional anonymous reviewer for his very positive and encouraging
comments. I also thank a third anonymous reviewer for his criticism and disagreement, the
Associate Editor for the constructive attitude and the Editor Kenneth Lanfear for the positive
decision.
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