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From last four decades of research it is well-established that all electrophysiological signals are nonlinear, irregular and aperiodic. Since those signals are used in everyday clinical practice as diagnostic tools (EMG, ECG, EEG), a huge progress in using it in making diagnostic more precise and
Self organization in electrochemical systems iSpringer
This document provides definitions and concepts related to nonlinear dynamics and self-organization. It discusses three key conditions for self-organization: an irreversible process, nonlinear dynamics, and feedback loops. It also defines important terms like bifurcation points, where a small change in a control parameter causes a qualitative change in behavior, such as the emergence of oscillations. Bifurcations can lead to phenomena like hysteresis, bistability, and multistability. Stability is also a crucial concept, as only stable states can be observed; unstable states will not survive fluctuations.
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Stochastic hydrology uses statistical models to represent hydrologic time series, such as rainfall and river flow records. Time series analysis involves extracting information from these records to understand the underlying physical system. Key aspects of time series include trends, periodic components, and stochastic/random components. Autocorrelation and partial autocorrelation functions are used to identify patterns in the data and select appropriate time series models, such as autoregressive (AR) or moving average (MA) models, for forecasting future values.
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This document describes a two-state regime switching autoregressive model and its application to river flow analysis. The model allows the autoregressive coefficients and innovation distributions to change between two regimes - an "ascending" regime where the process receives only positive shocks from a gamma distribution, and a "descending" regime where it follows a Gaussian autoregression. The durations of each regime are governed by independent negative binomial and geometric distributions, allowing for non-Markovian switching. Estimation is performed using Markov chain Monte Carlo methods, either treating the latent state values or change points as auxiliary parameters. The model is applied to daily river flow data and shown to reproduce important features like skewed margins and asymmetric hydrographs.
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This document discusses concepts from physics, chemistry, economics, and statistics as they relate to inertia and disruptive change. It covers topics like Lorentz force, Le Chatelier's principle, Samuelson's Correspondence Principle, the universe of clocks, statistical implications of change, and Benford's law. The overall theme appears to be examining how systems resist change through the lens of different disciplines and considering when resistance may be artificial.
From last four decades of research it is well-established that all electrophysiological signals are nonlinear, irregular and aperiodic. Since those signals are used in everyday clinical practice as diagnostic tools (EMG, ECG, EEG), a huge progress in using it in making diagnostic more precise and
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This document provides definitions and concepts related to nonlinear dynamics and self-organization. It discusses three key conditions for self-organization: an irreversible process, nonlinear dynamics, and feedback loops. It also defines important terms like bifurcation points, where a small change in a control parameter causes a qualitative change in behavior, such as the emergence of oscillations. Bifurcations can lead to phenomena like hysteresis, bistability, and multistability. Stability is also a crucial concept, as only stable states can be observed; unstable states will not survive fluctuations.
The document reviews thixotropy, a phenomenon where the apparent viscosity of a material decreases over time under constant shear and recovers when shear is removed. It discusses the evolution of the concept, experimental methods used to study thixotropy, and classification of materials that exhibit thixotropic behavior. Specifically, it provides a generalized definition of thixotropy, considers various experimental techniques like step changes in shear rate/stress and hysteresis loops, and groups thixotropic materials into classes based on origin or application.
Stochastic hydrology uses statistical models to represent hydrologic time series, such as rainfall and river flow records. Time series analysis involves extracting information from these records to understand the underlying physical system. Key aspects of time series include trends, periodic components, and stochastic/random components. Autocorrelation and partial autocorrelation functions are used to identify patterns in the data and select appropriate time series models, such as autoregressive (AR) or moving average (MA) models, for forecasting future values.
This document discusses time-dependent perturbation theory. It begins by introducing the concept of applying a time-dependent perturbation to a quantum system to induce transitions between its energy eigenstates. It then describes how the interaction picture can be used to focus on the slow evolution induced by the perturbation, without considering the rapid oscillation from the unperturbed Hamiltonian. The interaction picture defines a transformed state vector and operators such that the perturbation Hamiltonian governs the evolution operator in a Schrodinger equation.
This document describes a two-state regime switching autoregressive model and its application to river flow analysis. The model allows the autoregressive coefficients and innovation distributions to change between two regimes - an "ascending" regime where the process receives only positive shocks from a gamma distribution, and a "descending" regime where it follows a Gaussian autoregression. The durations of each regime are governed by independent negative binomial and geometric distributions, allowing for non-Markovian switching. Estimation is performed using Markov chain Monte Carlo methods, either treating the latent state values or change points as auxiliary parameters. The model is applied to daily river flow data and shown to reproduce important features like skewed margins and asymmetric hydrographs.
This document discusses detecting trigger points and irreversible thresholds in shock and trauma patients during catastrophic events when clinical infrastructure may be limited. It proposes that unstable recurrent patterns in physiological parameters could serve as early indicators of critical conditions. The document reviews using models like the Kuramoto-Sivashinsky equation to identify recurrent patterns in dissipative systems and associates these patterns with medical conditions to aid triage and forecasting needs under adverse conditions with sparse data. Further work is needed to determine relevant physiological parameters and associate recurrent patterns in those parameters with medical outcomes.
This document discusses early-warning signals that may indicate when complex systems are approaching critical thresholds or tipping points. It summarizes:
1) As systems approach tipping points, they exhibit "critical slowing down" - they take longer to recover from small perturbations. This can increase autocorrelation and variance in fluctuations.
2) Other potential early-warning signals include increased skewness in fluctuations, as the system spends relatively more time near unstable thresholds, and "flickering" between alternative stable states under stochastic forcing close to bifurcation points.
3) These signals, including slower recovery rates, higher autocorrelation, and increased variance and skewness, are expected to occur generically across diverse
This document provides an introduction to complexity science and key concepts related to complex adaptive systems. It discusses how complex systems are made up of relatively simple parts whose interactions give rise to new, unpredictable behaviors at a higher level. Emergence refers to behaviors that arise from interactions between components that cannot be predicted from studying the components individually. Complex adaptive systems can adapt based on feedback and are more robust than non-complex systems. The document outlines some basic terms like agents, feedback, and emergence, and provides historical context on the development of complexity science.
What does it mean for something to be a dynamical system What is .pdfvikasbajajhissar
A dynamical system is a mathematical model of a system whose state changes over time. Dynamical systems are described using differential or difference equations. An example of a dynamical system that is sensitive to initial conditions is one that exhibits the butterfly effect, where small changes to the initial state can lead to large changes in the system's behavior over time.
Simple Exponential Observer Design for the Generalized Liu Chaotic Systemijtsrd
In this paper, the generalized Liu chaotic system is firstly introduced and the state observation problem of such a system is investigated. Based on the time-domain approach with differential and integral equalities, a novel state observer for the generalized Liu chaotic system is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be precisely calculated. Finally, numerical simulations are presented to exhibit the effectiveness and feasibility of the obtained results. Yeong-Jeu Sun"Simple Exponential Observer Design for the Generalized Liu Chaotic System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd7126.pdf http://www.ijtsrd.com/engineering/engineering-maths/7126/simple-exponential-observer-design-for-the-generalized-liu-chaotic-system/yeong-jeu-sun
Chaos theory proposes that seemingly random events may actually arise from deterministic systems and can be predicted. It views organizations as complex systems with nonlinear relationships between variables. Applying chaos theory to organizational crises suggests that: 1) small changes can have large effects; 2) long-term predictions are impossible but short-term are feasible; 3) crises may arise from bifurcation points where outcomes oscillate. Chaos theory provides an alternative lens for analyzing unpredictable crisis events in organizations.
Using a theory of nematic liquid crystals to model swimming microorganismsNigel Mottram
This document summarizes research on modeling swimming microorganisms using theories of nematic liquid crystals. It discusses how flocking behavior can emerge from simple interaction rules and similarities to liquid crystal models. The document then analyzes a simplified 1D active nematic model to explore spontaneous flow, flow induced by shear, and backflow coupling. It finds different stable states can form depending on the activity parameter and initial conditions. Finally, it poses questions about including additional factors like density and polar order, using continuum models for large organisms, and effects on mixing.
This document provides an overview of statistical thermodynamics. It discusses key concepts like macroscopic vs microscopic scales, equilibrium states, extensive vs intensive quantities, and reversible vs irreversible processes. The document also defines important thermodynamic concepts like open and closed systems, adiabatic processes, and temperature. The goal of statistical thermodynamics is to use microscopic approaches to calculate macroscopic properties and relate them using both thermodynamic and statistical physics methods.
1. The document discusses statistical thermodynamics, which deals with systems of large numbers of particles at equilibrium. It examines thermodynamic systems and processes from both a macroscopic and microscopic perspective.
2. Key concepts discussed include extensive and intensive quantities, equilibrium, quasi-static processes, temperature, equations of state, and thermodynamic coefficients like compressibility and thermal expansivity.
3. The ideal gas law is examined as the simplest equation of state, relating pressure, volume, temperature, and number of particles or moles of gas. Temperature scales and their relation to empirical measurements are also summarized.
The document provides an index for a course on transport phenomena, outlining topics over 12 weeks that cover concepts like Newton's law of viscosity, shell momentum balance, boundary layers, and mass transfer. Key aspects of transport phenomena are discussed, including the governing equations for momentum, heat, and mass transfer as well as the boundary layer concept. Dimensionless groups and their importance in understanding similarity between different transport processes are also highlighted.
This document discusses the incompatibility between classical mechanics and electromagnetism. It shows that under a Galilean transformation, the wave equation governing electromagnetic waves takes on a different form in different reference frames, violating Galilean invariance. This means that the laws of electromagnetism depend on the choice of reference frame. As such, classical mechanics and electromagnetism cannot be unified without modifications to account for this issue.
The document discusses several ideas for improving stochastic simulation algorithms based on a literature review. It proposes using tau-leaping to facilitate parallelization of multi-compartment models. It also suggests using molecule volumes to efficiently simulate cell growth and division dynamics. Additionally, it recommends visualizing reaction topology and propensity information to aid understanding of simulation results.
Utility of chaos theory in product developmentTapani Taskinen
Chaos theory relates to product development in several ways:
1) Product development involves complex systems influenced by many internal and external factors, making outcomes difficult to predict and causing uncertainty similar to chaotic systems.
2) Characteristics of chaotic systems like sensitive dependence on initial conditions, feedback loops, and short term predictability but long term unpredictability reflect aspects of product development processes.
3) Understanding chaos theory may help improve the effectiveness of product development by providing insights into how to manage uncertainty and make better decisions in complex, unpredictable environments.
This document discusses preliminary work towards developing auto-resilient systems. It defines resilience as the ability of a system to maintain its identity and optimal behavior despite changes. The document outlines three key aspects of resilience - perception, apperception, and entelechism. It proposes frameworks to evaluate and compare systems' resilience capabilities. Finally, it describes a proposed "handshake mechanism" where systems declare their resilience and environments state requirements, enabling scenarios to autonomously enhance resilience through collaboration. The goal is to pave the way for future systems that can independently reason about and optimize their own architectures to best guarantee persistence under varying conditions.
On the Unification of Physic and the Elimination of Unbound Quantitiesijrap
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This document defines key concepts in systems modeling and simulation. It discusses:
1) What a system is and how systems can be categorized as discrete or continuous.
2) The difference between experimenting with an actual system versus a model of the system. Physical models are sometimes used but mathematical models are more common.
3) How simulation is used when analytical solutions for complex mathematical models are not possible. Simulation numerically exercises the model to observe outputs.
4) Three dimensions for classifying simulation models: static vs dynamic; deterministic vs stochastic; and continuous vs discrete. Static models represent a system at a single time while dynamic models evolve over time. Deterministic models don't include probabilities while stochastic models include random inputs.
The document provides an overview of time series econometrics concepts including:
1) Time series econometrics analyzes the dynamic structure and interrelationships over time in economic data. It examines stationary and non-stationary stochastic processes.
2) A time series is stationary if its mean, variance, and autocovariance remain constant over time. A random walk process is a type of non-stationary process where the variable fluctuates around a stochastic trend.
3) The document discusses key time series econometrics models and techniques including unit root tests, vector autoregressive models, causality tests, cointegration, and error correction models.
The document discusses thermodynamics from both macroscopic and microscopic viewpoints. It defines key concepts like system, surroundings, open and closed systems, intensive and extensive properties, state, equilibrium, processes, cycles, work, heat transfer, and different types of thermodynamic processes. Specific processes discussed include isobaric, isochoric, isothermal, and polytropic processes. The document also explains the zeroth law of thermodynamics and its importance for temperature measurement.
The document discusses key concepts in scientific methodology including:
1. The scientific method is a systematic process of inquiry using observation, experimentation, and repetition to characterize phenomena and test hypotheses.
2. A hypothesis is a proposed explanation for a phenomenon that can be tested, while a theory is a well-substantiated explanation backed by evidence.
3. Measurement is the process of assigning a number to a property of an object using a unit, with the understanding that all measurements have some uncertainty.
Dr. Cleber Gomes has extensive experience in electronics engineering and research. He has a Ph.D from Tokyo University of Technology and Agriculture and has lived and worked in both Tokyo and Israel for several years, holding positions in research and development at major technology companies. The presentation will discuss dynamical complex systems, using the stock market as an example, and will propose a system for forecasting stock market behavior.
Sen, Z. (1974) Propiedades de muestras pequeñas de modelos estocásticos estac...SandroSnchezZamora
This document provides an abstract for a thesis submitted for the degree of Doctor of Philosophy in the University of London in November 1974.
The thesis examines the small sample properties of parameters for stationary stochastic hydrological models. It aims to improve techniques for simulating observed streamflow sequences and substitute computationally expensive Monte Carlo techniques with exact or approximate analytical expressions.
Specifically, it analytically derives the small sample expectations of parameters like variance, standard deviation, and serial correlation coefficients for normal stationary processes. It also derives expectations for variables related to Hurst's phenomenon, including the range and resealed range of cumulative departures from the sample mean. A new model called the white Markov process is proposed for modeling Hurst's law
Molz, FJ, Rajaram, H. y Lu, S. (2004). Modelos de heterogeneidad basados en...SandroSnchezZamora
This document summarizes research on using fractal-based stochastic models to characterize heterogeneity in subsurface hydrology. It discusses how modern measurement techniques have revealed highly irregular and nonstationary property distributions in porous media. Researchers have explored using nonstationary stochastic processes with stationary increments to model these, which relates to the mathematical theory of stochastic fractals. The document reviews applications of Gaussian and Levy stochastic fractals and multifractals to model properties like porosity and hydraulic conductivity. It also discusses limitations like models based on increments being less predictive than stationary models, and issues around measuring media that exhibit variations on all scales. The conclusion is that additional data is needed to advance fractal-based theories, especially for 3D fracture networks.
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2) Other potential early-warning signals include increased skewness in fluctuations, as the system spends relatively more time near unstable thresholds, and "flickering" between alternative stable states under stochastic forcing close to bifurcation points.
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This document summarizes research on modeling swimming microorganisms using theories of nematic liquid crystals. It discusses how flocking behavior can emerge from simple interaction rules and similarities to liquid crystal models. The document then analyzes a simplified 1D active nematic model to explore spontaneous flow, flow induced by shear, and backflow coupling. It finds different stable states can form depending on the activity parameter and initial conditions. Finally, it poses questions about including additional factors like density and polar order, using continuum models for large organisms, and effects on mixing.
This document provides an overview of statistical thermodynamics. It discusses key concepts like macroscopic vs microscopic scales, equilibrium states, extensive vs intensive quantities, and reversible vs irreversible processes. The document also defines important thermodynamic concepts like open and closed systems, adiabatic processes, and temperature. The goal of statistical thermodynamics is to use microscopic approaches to calculate macroscopic properties and relate them using both thermodynamic and statistical physics methods.
1. The document discusses statistical thermodynamics, which deals with systems of large numbers of particles at equilibrium. It examines thermodynamic systems and processes from both a macroscopic and microscopic perspective.
2. Key concepts discussed include extensive and intensive quantities, equilibrium, quasi-static processes, temperature, equations of state, and thermodynamic coefficients like compressibility and thermal expansivity.
3. The ideal gas law is examined as the simplest equation of state, relating pressure, volume, temperature, and number of particles or moles of gas. Temperature scales and their relation to empirical measurements are also summarized.
The document provides an index for a course on transport phenomena, outlining topics over 12 weeks that cover concepts like Newton's law of viscosity, shell momentum balance, boundary layers, and mass transfer. Key aspects of transport phenomena are discussed, including the governing equations for momentum, heat, and mass transfer as well as the boundary layer concept. Dimensionless groups and their importance in understanding similarity between different transport processes are also highlighted.
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Chaos theory relates to product development in several ways:
1) Product development involves complex systems influenced by many internal and external factors, making outcomes difficult to predict and causing uncertainty similar to chaotic systems.
2) Characteristics of chaotic systems like sensitive dependence on initial conditions, feedback loops, and short term predictability but long term unpredictability reflect aspects of product development processes.
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3) How simulation is used when analytical solutions for complex mathematical models are not possible. Simulation numerically exercises the model to observe outputs.
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1. The scientific method is a systematic process of inquiry using observation, experimentation, and repetition to characterize phenomena and test hypotheses.
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3. Measurement is the process of assigning a number to a property of an object using a unit, with the understanding that all measurements have some uncertainty.
Dr. Cleber Gomes has extensive experience in electronics engineering and research. He has a Ph.D from Tokyo University of Technology and Agriculture and has lived and worked in both Tokyo and Israel for several years, holding positions in research and development at major technology companies. The presentation will discuss dynamical complex systems, using the stock market as an example, and will propose a system for forecasting stock market behavior.
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Specifically, it analytically derives the small sample expectations of parameters like variance, standard deviation, and serial correlation coefficients for normal stationary processes. It also derives expectations for variables related to Hurst's phenomenon, including the range and resealed range of cumulative departures from the sample mean. A new model called the white Markov process is proposed for modeling Hurst's law
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Stochastic Models in Hydrology- Adrían E. Schei.pdf
1. VOL.6, NO.3 WATER
RESOURCES
RESEARCH $UNE1970
Stochastic
Models
inHydrology
ADRIAN E. SCIIEIDEGGER
U.$. Geological
Survey,University
ofIllinois,Urbana,
Illinois61801
Abstract.The stochastic
models
that canbe used
to represent
growth
andsteady
state
phenomena
in hydro!ogy
are reviewed.
It is shown
that thereare essentially
two typesof
growthmodelspossible;
the cyclicgrowthmodeland the randomconfiguration
model.For
steady
statephenomena
(timeseries)
wearegenerally
restricted
to a Gaussian
typeof model
with or withoutautocorrelation.
Self-similarity
models(fractionalBrownfan
motion)leadto
physically
absurd
conditions
if theyareextrapolated
to highfrequencies.
INTRODUCTION
While investigating
the causes
of hydrologi-
cal processes,
a completelynovel approach
emergedduring the past five years. Whereas
suchcauses
were soughtin entirely determin-
isticphenomena
in the past,LeopoldandLa•g-
bein [1962] recognized
that while on a micro-
scopicscaleeach minute event affectingthe
evolutionof for examplea landscape,
a river
course, or • drainage basin is certainly de-
terministic, the combinedeffects of all the in-
dividualmicroscopic
eventscanbestbe treated
by modelsthat are analogous
to thoseusedin
statisticalthermodynamics.
The two main approaches
to statisticalhy-
drology,just as in statisticalthermodynamics,
are by two types of modelswhich might be
calledtime evolutionaryand ensemble
statisti'-
cal. In identical statistical populationsthe
ergodic hypothesisprovides a connectionbe-
tweenthe two approaches.
However,depending
on the generation
of the stochastic
models,
the
populationsin a particular evolutionarymodel
and a particular ensemble
statisticalmodel can-
not be assumed
a priori to be the same.Some
startlingdifferences
in basicpopulations
have
been noted in the statistical treatments of a
phenomenon,
althoughsuperficially
thesepop-
ulationswouldbe thoughtof asbeingidentical.
The reasonis that evolutionaryprocesses
nat-
urally suggest
the propertyof cyclicfry,
whereas
the ensemble
approachdoesnot. It is therefore
not the ergodichypothesisthat is infringed
upon,but the factthat the generation
of a pop-
ulationby a cyclicevolutionary
process
and the
750
populationobtainedby regarding
all possible
configurations
of systems
are generallyquite
different from one another.
The authorof this paper intendsto investi-
gate the stochasticmodels that can be used to
representgrowthand steadystate phenomena
in hydrology.
ENSEMBLES, OBSERVABLES,
AND
EXPECTATION VALUES
We begin by discussing
some fundamental
conceptsnecessaryfor the stochastictreatment
of geomorphology.
These concepts
are well
known from their applicationin statistical
physics.
For a generaldefinitionof the perti-
nent terms, the reader is referred to $ommer-
feld's[1964] standardtext.
We are concerned
with a system(landscape,
river net, etc.) in which certain observables
(elevation,bifurcationratio, etc.) are to. be
measured.The prob•.em
is to predict the nu-
merical values that will be obtained if these
observables
are measured
undercertainspeci-
fied conditions.
In deterministic
theory,thecorrectprediction
will simply state the value of the observable
that will be obtained.
In • stochastic
theory,
however,only an expectationvalue can be
specified.
This expectation
value is alwaysde-
fined as the average value of the observable
over an ensemble
of statesof the system.This
ensemble
of statesmustbe clearlyspecified
in
eachparticularcase.The reasons
why a prob-
abilistic approach rather than a deterministic
2. Stochastic Models 751
one may be indicated usually lies in the great
mechanicalcomplexityof the systemunder con-
sideration,
whichprecludes
the fo!lowing
up of
eachmicroscopic
eventin detail. Thus the single
processunder consideration
is replacedby an
ensembleof like processes
that are all equiva-
lent within the limits of one'sknowledge.
As an example,we consider
time evolutionary
processes.
A systemis subjectto influences
fluc-
tuating in time (they may actually be deter-
ministie,but causedby somany influences
that
they can be treated as fluctuating) according
to someprobabilistic
law, and we are interested
in the expectationvalue of some observable.
This expectation
valueisthenthe average
value
of the observab!e
in questionover the ensemble
of statesthat is obtainedif the process
isstarted
over and over againfrom the sameinitial con-
dition,but subjectto the fluctuating
influences
mentioned above.
Under certain specificcircumstancesthe time
average can be substituted for the ensemble
average (ergodieityof the system).In this case,
the ensemble over which the observable is av-
eragedare the statesof the systemat a series
of time instants.One of the necessary,
but by no
meanssufficient,conditionsfor this type of av-
eragingto be possibleis that the systembe in
a steadystate.
The general proceduresoutlined above are
standardpractice in statisticalphysicsand are
widely appliedin all fieldswherethe statements
are of a probabilistic
nature.Thus the scheme
doesnot only apply to statisticalphysicsas
such [Sommer]eid,1964], but also,to quantum
theory [Dirac, 1947].
It wouldbe conceivable,
givenan ensemble
of
states and an observable (which has a certain
value in each state), to define its expectation
value differentlyfrom the mannerin whichthis
was done above. For instance,we could de-
termine the most probable state of the system
and term that value of the observable which
it attains for this most probable state as its
expectationvalue [Shreve, 1966]. However this
procedure,while mathematically well defined,
is physicallyunsound.Measurementvaluesare
generally either time averageson a fluctuating
systemor the result of averagingvaluesfrom
repeated measurements
on similar systems.In
either ease,the averageof the observableover
the ensembleis the appropriate theoretical ex-
pression,and not someother mathematically
possible
expression.
CYCLIC MODELS
Let us considera time evolutionary process
as discussed
as an examplein the sectionon en-
sembles.Actually the time evolution of a sys-
tem represents
the most important application
of stochasticideasto hydrology.
Thus the growth of a system is followed
sequentiallythrough subsequent
stages.These
stagesdeve!opthroughthe actionof influences
that fluctuate accordingto some probabilistic
law. At any one time instant t, the ensemble
for calculatingexpectationvalues of a desired
observableis obtained by imagining that the
growthprocess
is startedmanytimesoveragain
from the initial condition to t, but subject to
differingrandom influences
and subjectto the
sameprobabilitylaw asmentionedabove.Evi-
dently this leads to an ensembleof possible
statesof the systemat eachstageof its develop-
ment specified
by the giventime t,. Expectation
valuesof any observable
at that time t, are then
calculated by the usual procedure of taking
averages
overthe ensemble.
In the procedure outlined above there is no
intrinsic specificationas to how the ensemble
at any particular time t, will look. This
dependsentirely on the probabilisticlaw re-
ferringto the fluctuatinginfluences.
However it is often difficult to specify this
law accurately,and the alternate path is then
to postulate dimefly (ab hypothesi) certain
propertiesfor the sequentialensembles
(at t•,
t2,..., t•,...).
The simplestsuchpostulateis that the stages
are cyclic regarding some time interval At,
(this may vary with i), and that each cycle
taken at its proper sea!eis similar to. the pre-
vious ones.Under such conditions,the growth
of the systemissaidto be eyelieandself-similar.
The notion of self-similarityis essentialfor the
definitionof a cycle.
It shouldbe emphasizedonce more that as-
suming a system's growth is eyelie and self-
similaris in lieu of, and thereforeequivalentto,
assuminga specificprobabilisticlaw regarding
the fluctuating influencesaffecting the growth
of the system.Such an assumption,therefore,
must be justified upon somephysicalor other
grounds.
3. 752 ADRIAN E. SCI-IEIDEGGER
The notionof cyclicityand self-similarityhas
bee
n tied up abovewith certainspecified,
but
not.necessarily constant, time intervals
The notionof cyclicitycanthen be generalized,
or rather extrapolated,to infinitely small or in-
finitely largeintervalsAt,. We shalldiscuss
such
extrapolations
laterin greater
detail,but let it
be statedhere that an observedcyclicityin
giventime intervalt• < t, < re,whereperhaps
self-similarityis empiricallyfound for a series
of finiteAt, -- a(t,) At,+• witha some
func-
tion, impliesnothingat all for t < t• or t > re.
Carrying self-similarity to infinitely small or
infinitelylargevaluesof t canleadto suchab-
surd
•notions
asa nonintegrable
length
ofa river
or to totally unjustifiedextrapolationso.ft.ime
series.
If:•a systemis cyclicallyself-similar,then it
isppSSible
to regard
each
cycle
asa compo-
nent:Cell.
If thequantitythat characterizes
each
cycleis reduced
by the correspo.nding
similarity
factorsto somecommonscale,then we end up
with a seriesof cells,in eachof whichthere is
(statistically) an equalamountof the quantity
mentionedabove. A systemo.f this type is
anMogous
in gasdynamics
to onein whichthere
is an equipartitionof energyin the individual
phase cells.Therefore there is an obviousanal-
ogyin suchsystems
with thetemperature
in gas
dynamics.The mean value o.fthe quantity in
questionis, therefore,canonicallydistributedin
the cells.The distributionof the quantity in
question
is exactly
canonical,
provided
it can
be further assumed that the deviations from the
mean are Gaussian. Since the statistical behavior
of the systemis generallydue to the action of
many linearly superposed
independentrandom
influences,
the central limit theoremapplicable
in this caseensures
that this is usuallythe case.
It is possible,
but not veryinstructive,
to put
the argumentsof this sectiongiven in words
into an abstract mathematical formalism. The
concepts
becomeclearwhen givensomespecific
examples.
River nets. It is well known that river nets
followHorton's
lawof stream
numbers;
i.e.,the
numbersn, of streamsegments
of (Strahler or
Horton) order i form (on the average) a geo-
metric sequence.
Horton [1945,p. 337] already
attempted a hydrophysicalexplanation of this
law in terms of a self-similarcyclic growth
model,an ideawhichwasformalizedby Wolden-
berg[1966].Accordingly,
ea.ch
newstreamorder
corresponds
to a cyclein the drainage
basinde-
velopment.In this modelit is easilypossible
to
definea temperature
analog;thisis simplythe
bifurcationratio for a givenchange(equalto
+ 1 in oneStrahlerorder for example) [Schei-
degger,
1968a].Unfortunately,
naturedoesnot
produce
river netsthat followtheabovegrowth
model[Ranalli (rndScheidegger,
1968].
Meanders. River meanders can also be re-
gardedin terms of a cyclicgrowthmodel.New
meanderloops are addedagain and again to
thealreadyexisting
ones.
In principle,a certainfixedlengthof river
segmentcan be regardedas generatingcells
(adjoining
segments,
numbered
by i) ofthesys-
tem; the stochasticvariable is the deviation
angle•, (change
of direction
angle)overeach
segment
i. Since• canbe positiveor negative,
it cannotdirectlybe relatedto a temperature
analog.
For this analog
we haveto take •,•'
[Scheidegger,
1967],andfor thisquantity•'
wehaveanequipartition
theorem
whichappears
to be satisfiedin nature [Thakur and Schei-
degger,
1968].
RANDON C0Nr•UR•ON •OD•rS
Aswasnotedin thelastsection,
the assump-
tion that the outcome
of a fluctuating
growth
processin a system is that the latter becomes
cyclically self-similar is in lieu of assum-
ing a specificprobabilitylaw for the fluctua-
tions.Thereis nothinginherentlysacred
about
makingsuchanhypothesis,
except
the apparent
simplicity of the resultingstructure of the
system.
Theoretically,
any otheroutcome
might
have been chosen.
In this view, an equallysimpleassumption
is that the probabilitylaw is suchthat the out-
comeis an ensemble
of systems
in whichevery
possible
internal configuration
within any one
systemis likely. We call sucha modela random
configuration
model. In gas dynamicsit cor-
respondsto assuminga microcanonical
proba-
bility distribution.
It is nowno longerpossible
to,splitthe sys-
temintocomponent
cells.
A temperature
analog
still exists,but this is with the microcanonical
temperaturein gasdynamics.
A slight generalizationof the abovemodelis
that the probabilitydistributionof the possible
configurations
of the systemis not constantbut
4. Stochastic Models 753
Gaussianregardingsomecharacterizing
param-
eters.It shouldbe noted that a constant
•proba-
bility distributionis a specialcaseof a Gaus-
sianone(infinitedispersion).
Thesebasicconcepts
areillustratedin the fol-
lowingexample.
We noted above that as far as river nets are
concerned,
nature doesnot supportthe cyclic
growth model.We will thereforetry to use a
randomconfiguration
modelof the type which
isthe subjectof the presentsection.
This model implies in the caseof river nets
that a particularriver networkis a realization
of a particular graph (arborescence)
in the en-
semble.
of all possiblegraphs (arborescences),
whereinthe probabilitydistribution
is suchthat
all possible
arborescences
(with a givennumber
of free ends) are equally likely. This corre-
spondsto. a microcanonical
probability distri-
bution of the individualpossible
statesof the
system.We can then set up a microcanonical
temperature
analogfor river nets[Scheidegger,
1968b].
OTHER MODELS
We have discussed
above growth modelsin
which the growth is a stochastic
process.
What
hasbeen investigatedis the probability of sys-
temsthat are the outcomeof the growthprocess.
Two typesof possible
outcomes
havebeencon-
sidered. The resulting systems can or cannot
be subdivided into. individual cells. The latter
casein somefashion can be t•ken as a special
case of the former. Only one single cell en-
compasses
the wholesystem.
The questionarisesas to whether there are
any other types of outcome that might rea-
sonablybe expected?Dependingon the prob-
abilisticlawsgoverningthe growth process,
the
outcomemight be anything; i.e., any arbitrary
probability distribution for the configurations
that the systemmight have at time t{ couldbe
the outcome.However, we have to assumethat
the growthprocess
is the outcomeof the effect
of many independent
individuallinearilysuper-
posedinfluences,
and under suchconditionsthe
central limit theorem imposessomelimitations.
Usually the particular individual possible
configurations
of a systemat a certain time t,,
which in their tota•lity representthe ensemble,
will be characterizedby certain parameters.Be-
causethe influencesproducingthese configura-
tions are usuallyassumed
to be great in num-
ber, additiveand independent,
it will be per-
missibleto apply the central limit theorem,
indicatingthat the distributionof probabilities
regarding
theseparameters
or configurations
is
Gaussian.Thus barring queer processes,
the
mostgeneraloutcome
for the probabilitydis-
tributions in each cell will be that the latter is
Gaussianaround somemean. A specialcaseof
a Gaussian
probabilitydistribution
isthat every
configuration
is equallylikely (infinitedisper-
sion).
Apart from queerprocesses,
the following
possibilities
existfor theoutcome
of the prob-
ability distributions:
1. Systemdivisibleinto cells (cycles).In
each cell there is a Gaussian (which includesa
constant)probabilitydistributionof the pos-
sibleconfigurations.
2. Systemnot divisibleinto cells(consisting
of a single
cell).Thedistribution
of probabilities
for the configurations
is Gaussian(which in-
cludesa constant).
These are in essencethe casesdiscussedabove.
Except for queerprocesses,
no othersare pos-
sible.
THE STEADY STATE
Up to now we have discussed
modelsof
stochasticgrowth processes.
However, an im-
portant class
of hydrologic
systems
is not grow-
ingbut is (presumably)
fluctuating
in a steady
state.
A systemis givenhere which,in a random
fluctuatingfashion,passes
in time through a
variety of states.The statesthat are formedat
timestl, t2, "', t{, "' form an ensemble.
The
expectation
value of any observable
is formed
by takingits average
valueoverthis ensemble
(time average).
Alternatively, we can consider all possible
states of the systemat a given time t,. These
statesalsoform an ensemble
with a givenprob-
ability distribution.Under generalassumptions
(ergodie hypothesis)averaging over this en-
semblecan be substitutedfor the time average.
In otherwords,the systempasses
in time (arbi-
trarily closely)througheachpossible
state,sub-
ject to a certain probability distribution.
In hydrologythe problem usually arisesto
predictthe stochastic
behaviorof someobserv-
5. 754 •DRX• r.
able probabilities,
suchas its mean,its dis-
persion,probabilitiesof high or low seriesof
values,
etc.Theproblem
canbesolved
by simple
averagingover the ensemble.
The di•culty is
that the ensemble
is only imperfectlyknown
from a short interval of measurements. There-
fore modelswill have to be set up for the
extrapolation from the measurementsto other
time ranges.
To establish such models we must look for
physicalreasons
for choosing
them. A randomly
fluctuatingsystemis subjectto certain influ-
ences.If theseinfluences
are independent,and
their effects
are additivewith regardto a certain
observableX(t), •hen the distribution of •he
latter will be Gaussian,and •he dispersionof
•he cumulativevariablex(t) • •X(t)dt will be
proportional •o the time t (BrownJan condi-
tion)
=
wherewe have assumed
that X(t) = 0 and
angiebracketsdenotethe average.
A modification of the above model is obtained
if it is assumed that there is a correlation be-
tween subsequent
influences.
This correlationis
givenby the correlationfunctionC(s)
C(s)
=•••••X(t)X(t
+s)dt
if the individual measurements X are taken at
the t•es t. The integralis to be taken over all
times; since,in practiceonly a finite numberof
measurements
can be made, the integral is an
approximationto reality. If there is • correla-
tion betweensubsequent
influences,
the auto-
correlation
timeL, is [Pai, 19'57]
Lt = C(8)ds
It is •plicitly assumed
that the integralcon-
verges.
If it doesconverge,
thenat leastasymp-
totically for longtimes (for t >>L•)
(x(t)) = x(t) at t
This is characteristic of Brownian behavior. For
shorttimes(• <<L•) wehave
t
SCYIEIDEGGER
sothat the behaviorof (x• (t)) is bracketed
be-
tween proportionalityto t"-and proportionality
to t.
The idea of a finite correlation time Lt is
basedon the assumption
that if we wait long
enough,
the influences
producing
the X(t) are
independent
and additive,so that in the long
run the processcan be consideredas Gaussian.
This reflectsitself in the integrabilitycondi-
tion for C(s). It is possible
to conceive
of
processes where the corre!ation time is not
finite,but infinite; i.e., C(s) is not integrable.
The behavior of (x2) must then lie between t
and t2 as demonstrated
above,but this is also
for infinitetimes(the limiting(x-ø)
• t isnever
reached).
It is evenpossible
to formally
setup
a mathematical
modelfor a stochastic
process
that always(for shortand longtime ranges)
hasa behaviorfor (x•) ~ t•, with 2H a con-
stant betweenI and 2, independently
of the
timerangeunderconsideration.
By postulating
a certain type of self-shnilarity[Mandelbrot,
1965], such mathematical modelscan be made
soasto bedetermined
by the soleparameter
H.
It shouldbe emphasized,
however,
that the
physicalreasons
for postulatingsuchself-simi-
lar models are not clear. In the modified Gauss
process,we have assumedan initial correlation
anda longrunindependence
of theeffects
giv-
ingriseto thetimeseries
X(t•) orx(t•). Physi-
callythisseems
to be • very satisfactory
pro-
cedure.
On the otherhand,whenpostulating
• self-similar
process
with H • • (fractional
Brownian motion), we really make a com-
pletely arbitrary extrapolationfrom short to
very long time ranges.Actuallypostulating
a certain
H -v•• simply
amounts
to postulating
• prioria certainasymptotic
behavior
for X(t)
for longtimes.Suchan assumption
is of course
also
madein theusualGauss-Markov
model(ul-
timate independence
of the events),but there
isat leasta physical
reason
for doingso.
Usingself-similarity
(withH v•:•) to ex-
trapolate the correlated behavior from a finite
time spanto.an asymptotically
infiniteoneis
physically
completely
unjustified.
Furthermore,
usingself-similarity
to intrapolate
to a very
short
timespan,
anultraviolet
catastrophe
arises.
Since
thecycles
aresupposed
tobeself-similar,
it canbe seen
that the phenomenon
becomes
so
highlyoscillatory
for smallfrequencies
asto be
nolonger
definable.
Thisisphysically
absurd.
6. Stochastic Models 755
Any physicaltime seriesconsistsof a finite
number of measurements that have been taken
over a finite total time range.Empiricallyit is
foundthat within this rangethe behaviorof the
mean-root square variance of the cumulative
series is describedby proportionalityto t
whereH is betweenx/• and 1. This is exactlyas
predictedby the usual Gauss-Markovcorrela-
tion model with a suitable choiceof Lt. To ex-
trapolate beyond the time range over which
measurements
havebeenmade,or to intrapolate
into the intervals between the individual meas-
urements,we haveto proceedfrom a reasonable
physicalassumption.
To simply postulateself-
similarity for the seriesand extrapolateto in-
finitelylongandto intrapo•ateto infinitelyshort
time intervalsis physically unsound,as evi-
denced
by the ultravioletcatastrophe
mentioned
above. There is therefore a physicalreasonto
reject the fractional Brownian motion model at
the high frequencyend.Why then shouldthere
be a reasonto acceptit for the low frequency
end? If the correlation between events is such
that the latter cannot be assumed to become in-
dependentand additive for the contemplated
extrapolation
time range,thereis no possibility
of predictingwhat this correlationwill be like.
This correlation has to be measured or some
physicalreason
mustbe givento makean edu-
cated guess.Self-similaritymay be a mathe-
maticallypretty idea•but unless
a physicalrea-
son is given for its realization in nature it is
not clear why the asymptoticbehaviorof the
correlationengendered
by it should
be any more
likely in realitythan any other.
The modelof hydrologicphenomenathat
physically most reasonableis therefore one of
additiverandomeventscausing
the fluctuation
of someobservable.
Theseeventsmay be cor-
relatedby a seriesof effects,eachwith a cer-
tain correlationintensity and time range
Unless
some
physical
reasons
aregiven,extrapo-
lationsin • steadystate can thereforebe made
only basedon empiricism.
The value of H (i.e.,
the correlationbehavior) for thetime rangeT is
found under investigation, and an educated
guessis madethat this H doesnot changevery
much for additional time intervals of the size
of maybe T/2.
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(ManuscriptreceivedJanuary 13, 1970.)