reductions of hidden information sources in all but special circumstances measurements of time dependent processes reflect internal structures and correlations only indirectly building predictive models of such hidden information sources requires discovering in some way the internal states and mechanisms unfortunately there are often many possible models that are observationally equivalent here we show that the situation is not as arbitrary as one would think we show that generators of hidden stochastic processes can be reduced to a minimal form and compare this reduced representation to that provided by computational mechanics the e machine on the way to developing deeper measure theoretic foundations for the latter we introduce a new two step reduction process the first step internal event reduction produces the smallest observationally equivalent algebra and the second internal state reduction removes algebra components that are redundant for optimal prediction for several classes of stochastic dynamical systems these reductions produce representations that are equivalent to e machines

dominated and pointwise ergodic theorems with weighted averages for bounded lamperti representations of amenable groups abstract group representations by bounded lamperti operators in the spaces l 1 form a wide class of representations including representations by bounded positive operators and when 2 representations by isometric operators the dominated and the pointwise ergodic theorems det and pet for cesaro averages for the bounded lamperti representations of amenable compact locally compact groups in l 1 were proved by a tempelman in proc amer math soc 143 2015 4989 5004 by using a completely different functional analytical method developed by a shulman in his phd thesis in 1988 we generalize this result to weighted averages of such representations and discuss various conditions on the weights under which the det and the pet hold we conclude with applications of the general results to the bounded lamperti representations of groups of polynomial growth and of the groups r m and z m

application of computational mechanics to the analysis of natural data an example in geomagnetism we discuss how the ideal formalism of computational mechanics can be adapted to apply to a noninfinite series of corrupted and correlated data that is typical of most observed natural time series specifically a simple filter that removes the corruption that creates rare unphysical causal states is demonstrated and the concept of effective soficity is introduced we believe that computational mechanics cannot be applied to a noisy and finite data series without invoking an argument based upon effective soficity a related distinction between noise and unresolved structure is also defined noise can only be eliminated by increasing the length of the time series whereas the resolution of previously unresolved structure only requires the finite memory of the analysis to be increased the benefits of these concepts are demonstrated in a simulated times series by a the effective elimination of white noise corruption from a periodic signal using the expletive filter and b the appearance of an effectively sofic region in the statistical complexity of a biased poisson switch time series that is insensitive to changes in the word length memory used in the analysis the new algorithm is then applied to an analysis of a real geomagnetic time series measured at halley antarctica two principal components in the structure are detected that are interpreted as the diurnal variation due to the rotation of the earth based station under an electrical current pattern that is fixed with respect to the sun earth axis and the random occurrence of a signature likely to be that of the magnetic substorm in conclusion some useful terminology for the discussion of model construction in general is introduced

variable length markov chains we study estimation in the class of stationary variable length markov chains vlmc on a finite space the processes in this class are still markovian of high order but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high order markov chains from an algorithmic view the vlmc model class has attracted interest in information theory and machine learning but statistical properties have not yet been explored provided that good estimation is available the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest the latter is exemplified with some dna data a version of the tree structured context algorithm proposed by rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of vlmcs this remains true even when the underlying model increases in dimensionality fur thermore consistent estimation of minimal state spaces and mixing properties of fitted models are given we also propose a new bootstrap scheme based on fitted vlmcs we show its validity for quite general stationary categorical time series and for a broad range of statistical procedures

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

predictive rate distortion for infinite order markov processes predictive rate distortion analysis suffers from the curse of dimensionality clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length the challenge is compounded for infinite order markov processes since conditioning on finite sequences cannot capture all of their past dependencies spectral arguments confirm a popular intuition algorithms that cluster finite length sequences fail dramatically when the underlying process has long range temporal correlations and can fail even for processes generated by finite memory hidden markov models we circumvent the curse of dimensionality in rate distortion analysis of finite and infinite order processes by casting predictive rate distortion objective functions in terms of the forward and reverse time causal states of computational mechanics examples demonstrate that the resulting algorithms yield substantial improvements

doubly power bounded operators on lp 2 p 1 abstract we show the existence of a doubly power bounded t on l p 1 p p 2 such that t is spectral of scalar type hence polynomially bounded t is not similar to a lamperti operator hence is not similar to an isometry none of the powers of t is similar to a lamperti operator none of the powers is similar to a positive operator and for some f l p the averages 1 n k 1 n t k f or the averages along the primes or the squares fail to be a e convergent

observable operator models for discrete stochastic time series a widely used class of models for stochastic systems is hidden markov models systems that can be modeled by hidden markov models are a proper subclass of linearly dependent processes a class of s

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

mixing ergodic and nonergodic processes with rapidly growing information between blocks we construct mixing processes over an infinite alphabet and ergodic processes over a finite alphabet for which shannon mutual information between adjacent blocks of length n grows as n where 0 1 the processes are a modification of nonergodic santa fe processes which were introduced in the context of natural language modeling the rates of mutual information for the latter processes are alike and also established in this paper as an auxiliary result it is shown that infinite direct products of mixing processes are also mixing

a partial ordering of sets making mean entropy monotone consider a state of a system with several subsystems the entropies of the reduced state on different subsystems obey certain inequalities provided there is an equivalence relation and a function measuring volumes or weights of subsystems the entropy per unit volume or unit weight the mean entropy then decreases with respect to an order relation of the subsystems defined in this paper in the context of statistical mechanics a lattice system is studied in detail and a decrease of mean energy is deduced for blow up sequences of regular and irregular octagons examples referring to quantum information with strict relations for mean entropies of subsystems in multipartite systems are also presented

inferring statistical complexity in the dripping faucet experiment abstract we have used topological e machine reconstruction to calculate statistical complexity associated to period five orbits in the dripping faucet experiment basic structures related to the periodic movement and to peripheral tracks in the corresponding directed graphs were identified

characterizing the dynamics of stochastic bistable systems by measures of complexity the dynamics of noisy bistable systems is analyzed by means of lyapunov exponents and measures of complexity we consider both the classical kramers problem with additive white noise and the case when the barrier fluctuates due to additional external colored noise in the case of additive noise we calculate the lyapunov exponents and all measures of complexity analytically as functions of the noise intensity or the mean escape time respectively for the problem of a fluctuating barrier the usual description of the dynamics with the mean escape time is not sufficient the application of the concept of measures of complexity allows us to describe the structures of motion in more detail most complexity measures indicate the value of the correlation time at which the phenomenon of resonant activation occurs with an extremum

analysis of the phase transition in the two dimensional ising ferromagnet using a lempel ziv string parsing scheme and black box data compression utilities in this work we consider information theoretic observables to analyze short symbolic sequences comprising time series that represent the orientation of a single spin in a two dimensional 2d ising ferromagnet on a square lattice of size l 2 128 2 for different system temperatures t the latter were chosen from an interval enclosing the critical point t mathrm c of the model at small temperatures the sequences are thus very regular at high temperatures they are maximally random in the vicinity of the critical point nontrivial long range correlations appear here we implement estimators for the entropy rate excess entropy i e complexity and multi information first we implement a lempel ziv string parsing scheme providing seemingly elaborate entropy rate and multi information estimates and an approximate estimator for the excess entropy furthermore we apply easy to use black box data compression utilities providing approximate estimators only for comparison and to yield results for benchmarking purposes we implement the information theoretic observables also based on the well established m block shannon entropy which is more tedious to apply compared to the first two algorithmic entropy estimation procedures to test how well one can exploit the potential of such data compression techniques we aim at detecting the critical point of the 2d ising ferromagnet among the above observables the multi information which is known to exhibit an isolated peak at the critical point is very easy to replicate by means of both efficient algorithmic entropy estimation procedures finally we assess how good the various algorithmic entropy estimates compare to the more conventional block entropy estimates and illustrate a simple modification that yields enhanced results

correlated biased random walk with latency in one and two dimensions asserting patterned and unpredictable movement the correlated biased random walk with latency in one and two dimensions is discussed with regard to the portion of irreducible random movement and structured movement it is shown how a quantitative analysis can be carried out by using computational mechanics the stochastic matrix for both dynamics are reported latency introduces new states in the finite state machine description of the system in both dimensions allowing for a full nearest neighbor coordination in the two dimensional case complexity analysis is used to characterize the movement independently of the set of control parameters making it suitable for the discussion of other random walk models the complexity map of the system dynamics is reported for the two dimensional case

information trimming sufficient statistics mutual information and predictability from effective channel states one of the most basic characterizations of the relationship between two random variables x and y is the value of their mutual information unfortunately calculating it analytically and estimating it empirically are often stymied by the extremely large dimension of the variables one might hope to replace such a high dimensional variable by a smaller one that preserves its relationship with the other it is well known that either x or y can be replaced by its minimal sufficient statistic about y or x while preserving the mutual information while intuitively reasonable it is not obvious or straightforward that both variables can be replaced simultaneously we demonstrate that this is in fact possible the information x s minimal sufficient statistic preserves about y is exactly the information that y s minimal sufficient statistic preserves about x we call this procedure information trimming as an important corollary we consider the case where one variable is a stochastic process past and the other its future in this case the mutual information is the channel transmission rate between the channel s effective states that is the past future mutual information the excess entropy is the amount of information about the future that can be predicted using the past translating our result about minimal sufficient statistics this is equivalent to the mutual information between the forward and reverse time causal states of computational mechanics we close by discussing multivariate extensions to this use of minimal sufficient statistics

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

stabilization for equations of one dimensional viscous compressible heat conducting media with nonmonotone equation of state abstract we consider the navier stokes system describing motions of viscous compressible heat conducting and self gravitating media we use the state function of the form p u p 0 u p 1 u linear with respect to the temperature but we admit rather general nonmonotone functions p 0 and p 1 of u which allows us to treat various physical models of nuclear fluids for which p and u are the pressure and the specific volume or thermoviscoelastic solids for solutions to an associated initial boundary value problem with fixed free boundary conditions and arbitrarily large data we prove a collection of estimates independent of time interval including uniform two sided bounds for u and describe asymptotic behavior as t namely we establish the stabilization pointwisely and in l q for u in l 2 for and in l q for v the velocity for any q 2 for completeness the proof of the corresponding global existence theorem is also included

fractal renewal processes generate 1 f noise 1 mathit f mathit d noise occurs in an impressive variety of physical systems and numerous complex theories have been proposed to explain it we construct two relatively simple renewal processes whose power spectral densities vary as 1 mathit f mathit d i a standard renewal point process with 0d1 and ii a finite valued alternating renewal process with 0d2 the resulting event number statistics coincidence rates minimal coverings and autocorrelation functions are shown also to follow power law forms these fractal characteristics derive from interevent time probability density functions which themselves decay in a power law fashion a number of applications are considered trapping in amorphous semiconductors electronic burst noise movement in systems with fractal boundaries the digital generation of 1 mathit f mathit d noise and ionic currents in cell membranes

recursion theory on the reals and continuous time computation abstract we define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers corresponding to a conceptual analog computer that operates in continuous time this class turns out to be surprisingly large and includes many functions which are uncomputable in the traditional sense we stratify this class of functions into a hierarchy according to the number of uses of the zero finding operator at the lowest level are continuous functions that are differentially algebraic and computable by shannon s general purpose analog computer at higher levels are increasingly discontinuous and complex functions we relate this hierarchy to the arithmetical and analytical hierarchies of classical recursion theory

robust linear regression with broad distributions of errors abstract we consider the problem of linear fitting of noisy data in the case of broad say stable distributions of random impacts noise which can lack even the first moment this situation common in statistical physics of small systems in earth sciences in network science or in econophysics does not allow for application of conventional gaussian maximum likelihood estimators resulting in usual least squares fits such fits lead to large deviations of fitted parameters from their true values due to the presence of outliers the approaches discussed here aim onto the minimization of the width of the distribution of residua the corresponding width of the distribution can either be defined via the interquantile distance of the corresponding distributions or via the scale parameter in its characteristic function the methods provide the robust regression even in the case of short samples with large outliers and are equivalent to the normal least squares fit for the gaussian noises our discussion is illustrated by numerical examples

time series information and unsupervised learning of representations numerous control and learning problems face the situation where sequences of high dimensional highly dependent data are available but no or little feedback is provided to the learner which makes any inference rather challenging to address this challenge we formulate the following problem given a series of observations x 0 ldots x n coming from a large high dimensional space mathcal x find a representation function f mapping mathcal x to a finite space mathcal y such that the series f x 0 ldots f x n preserves as much information as possible about the original time series dependence in x 0 ldots x n we show that for stationary time series the function f can be selected as the one maximizing a certain information criterion that we call time series information some properties of this functions are investigated including its uniqueness and consistency of its empirical estimates implications for the problem of optimal control are presented

practical unitary simulator for non markovian complex processes stochastic processes are as ubiquitous throughout the quantitative sciences as they are notorious for being difficult to simulate and predict in this letter we propose a unitary quantum simulator for discrete time stochastic processes which requires less internal memory than any classical analogue throughout the simulation the simulator s internal memory requirements equal those of the best previous quantum models however in contrast to previous models it only requires a small finite dimensional hilbert space moreover since the simulator operates unitarily throughout it avoids any unnecessary information loss we provide a stepwise construction for simulators for a large class of stochastic processes hence directly opening the possibility for experimental implementations with current platforms for quantum computation the results are illustrated for an example process

an almost machine independent theory of program length complexity sophistication and induction abstract the purpose of this paper is to use a variant of program length complexity to formally define the structure of a binary string where the structure of an object is taken to mean the aggregate of its projectible properties

minimum description length induction bayesianism and kolmogorov complexity the relationship between the bayesian approach and the minimum description length approach is established we sharpen and clarify the general modeling principles minimum description length mdl and minimum message length mml abstracted as the ideal mdl principle and defined from bayes s rule by means of kolmogorov complexity the basic condition under which the ideal principle should be applied is encapsulated as the fundamental inequality which in broad terms states that the principle is valid when the data are random relative to every contemplated hypothesis and also these hypotheses are random relative to the universal prior the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized if we restrict the model class to finite sets then application of the ideal principle turns into kolmogorov s minimal sufficient statistic in general we show that data compression is almost always the best strategy both in model selection and prediction

informational and causal architecture of continuous time renewal processes we introduce the minimal maximally predictive models epsilon text machines of processes generated by certain hidden semi markov models their causal states are either discrete mixed or continuous random variables and causal state transitions are described by partial differential equations as an application we present a complete analysis of the epsilon text machines of continuous time renewal processes this leads to closed form expressions for their entropy rate statistical complexity excess entropy and differential information anatomy rates

compression of two dimensional data distortion free compressibility of individual pictures i e two dimensional arrays of data by finite state encoders is investigated for every individual infinite picture i a quantity rho i is defined called the compressibility of i which is shown to be the asymptotically attainable lower bound on the compression ratio that can be achieved for i by any finite state information lossless encoder this is demonstrated by means of a constructive coding theorem and its converse that apart from their asymptotic significance might also provide useful criteria for finite and practical data compression tasks the proposed picture compressibility is also shown to possess the properties that one would expect and require of a suitably defined concept of two dimensional entropy for arbitrary probabilistic ensembles of infinite pictures while the definition of rho i allows the use of different machines for different pictures the constructive coding theorem leads to a universal compression scheme that is asymptotically optimal for every picture the results are readily extendable to data arrays of any finite dimension

computational mechanics of cellular automata an example abstract we illustrate and extend the techniques of computational mechanics in explicating the structures that emerge in the space time behavior of elementary one dimensional cellular automaton rule 54 the dominant regular domain of the cellular automation is identified and a domain filter is constructed to locate and classify defects in the domain the primary particles are identified and a range of interparticle interactions is studied the deterministic equation of motion of the filtered space time behavior is derived filters of increasing sophistication are constructed for the efficient gathering of particle statistics and for the identification of higher level defects particle interactions and secondary domains we define the emergence time at which the space time behavior condenses into configurations consisting only of domains particles and particle interactions taken together these techniques serve as the basis for the investigation of pattern evolution and self organization in this representative system

identifiability of hidden markov information sources and their minimum degrees of freedom if only a function of the state in a finite state markov chain is observed then the stochastic process is no longer markovian in general this type of information source is found widely and the basic problem of its identifiability remains open that is the problem of showing when two different markov chains generate the same stochastic process the identifiability problem is completely solved by linear algebra where a block structure of a markov transition matrix plays a fundamental role and from which the minimum degree of freedom for a source is revealed

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

time dependent implicit evolution equations on presente des theoremes d existence et d unicite pour des problemes de cauchy correspondant aux equations abstraites de la forme d b t u t dt a t u t f t ou b t est une famille dependante du temps d operateurs lineaires

a polynomially bounded operator on hilbert space which is not similar to a contraction let e 0 we prove that there exists an operator te e2 2 such that for any polynomial p we have iip te ii not similar to a contraction i e there does not exist an invertible operator s f2 e f2 such that jis 1tesii tributed to halmos after his well known 1970 paper ten problems in hilbert space we also give some related finite dimensional estimates department of mathematics texas a m university college station texas 77843 universite paris vi equipe d7analyse case 186 75252 paris cedex 05 france e mail address gipqccr jussieu fr this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 52 47 utc all use subject to http about jstor org terms

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

finite statistical complexity for sofic systems we propose a measure of complexity for symbolic sequences which is based on conditional probabilities and captures computational aspects of complexity without the explicit construction of minimal deterministic finite automata dfa moreover if the sequence is obtained from a dynamical system through a suitable encoding and its equations of motion are known we show how to estimate the regions of phase space that correspond to computational states with statistically equivalent futures causal states

extending sdl and lmc complexity measures to quantum states an extension of sdl shiner davison landsberg and lmc lopez ruiz mancini calbet complexity measures is proposed for the quantum information context considering that von neumann entropy is a natural disorder quantifier for quantum states as a first example of application the simple qubit was studied presenting results similar to that obtained by applying sdl and lmc measures to a classical probability distribution then for the werner state a mixture of bell states sdl and lmc measures were calculated depending on the mixing factor providing some conjectures concerning quantum systems

subsequence ergodic theorems for contractions in this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations or dunford schwartz operators are extended to operators which are positive contractions on l p for p fixed

resolvent conditions and growth of powers of operators abstract following bermudez et al 5 we study the rate of growth of the norms of the powers of a linear operator under various resolvent conditions or cesaro boundedness assumptions in hilbert spaces we prove that if t satisfies the kreiss condition t n o n log n if t is absolutely cesaro bounded t n o n 1 2 e for some e 0 which depends on t if t is strongly kreiss bounded then t n o log n for some 0 we show that a kreiss bounded operator on a reflexive space is abel ergodic and its cesaro means of order converge strongly when 1

optimal classical simulation of state independent quantum contextuality simulating quantum contextuality with classical systems requires memory a fundamental yet open question is what is the minimum memory needed and therefore the precise sense in which quantum systems outperform classical ones here we make rigorous the notion of classically simulating quantum state independent contextuality qsic in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite qsic set we obtain the minimum memory needed to simulate arbitrary qsic sets via classical systems under the assumption that the simulation should not contain any oracular information in particular we show that while classically simulating two qubits tested with the peres mermin set requires log 2 24 4 585 bits simulating a single qutrit tested with the yu oh set requires at least 5 740 bits

asymptotic behaviour and changes of phase in one dimensional nonlinear viscoelasticity in this paper we study the asymptotic behaviour as t co of solutions u x t to the nonlinear partial differential equation ut m4 a o o 1 1 with initial conditions u 0 q x u x u x o 0 1 3a

many roads to synchrony natural time scales and their algorithms we consider two important time scales the markov and cryptic orders that monitor how an observer synchronizes to a finitary stochastic process we show how to compute these orders exactly and that they are most efficiently calculated from the e machine a process s minimal unifilar model surprisingly though the markov order is a basic concept from stochastic process theory it is not a probabilistic property of a process rather it is a topological property and moreover it is not computable from any finite state model other than the e machine via an exhaustive survey we close by demonstrating that infinite markov and infinite cryptic orders are a dominant feature in the space of finite memory processes we draw out the roles played in statistical mechanical spin systems by these two complementary length scales

minimum memory for generating rare events we classify the rare events of structured memoryful stochastic processes and use this to analyze sequential and parallel generators for these events given a stochastic process we introduce a method to construct a process whose typical realizations are a given process rare events this leads to an expression for the minimum memory required to generate rare events we then show that the recently discovered classical quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations