thermodynamic entropy and chaos in a discrete hydrodynamical system we show that the thermodynamic entropy density is proportional to the largest lyapunov exponent lle of a discrete hydrodynamical system a deterministic two dimensional lattice gas automaton the definition of the lle for cellular automata is based on the concept of boolean derivatives and is formally equivalent to that of continuous dynamical systems this relation is justified using a markovian model in an irreversible process with an initial density difference between both halves of the system we find that boltzmann s h function is linearly related to the expansion factor of the lle although the latter is more sensitive to the presence of traveling waves

restoration of monotonicity respecting in dynamic regression abstractdynamic regression models including the quantile regression model and aalen s additive hazards model are widely adopted to investigate evolving covariate effects yet lack of monotonicity respecting with standard estimation procedures remains an outstanding issue advances have recently been made but none provides a complete resolution in this article we propose a novel adaptive interpolation method to restore monotonicity respecting by successively identifying and then interpolating nearest monotonicity respecting points of an original estimator under mild regularity conditions the resulting regression coefficient estimator is shown to be asymptotically equivalent to the original our numerical studies have demonstrated that the proposed estimator is much more smooth and may have better finite sample efficiency than the original as well as when available as only in special cases other competing monotonicity respecting estimators illustration with a clinical study is provided

lyapunov exponents for products of rectangular real complex and quaternionic ginibre matrices we study the joint density of eigenvalues for products of independent rectangular real complex and quaternionic ginibre matrices in the limit where the number of matrices tends to infinity it is shown that the joint probability density function for the eigenvalues forms a permanental point process for all three classes the moduli of the eigenvalues become uncorrelated and log normal distributed while the distribution for the phases of the eigenvalues depends on whether real complex or quaternionic ginibre matrices are considered in the derivation for a product of real matrices we explicitly use the fact that all eigenvalues become real when the number of matrices tends to infinity finally we compare our results with known results for the lyapunov exponents as well as numerical simulations

on the defect of additivity of fuzzy measures it is well known that fuzzy measures are non additive in this paper we introduce the concept of defect which gives us the degree of non additivity of fuzzy measures for large classes of fuzzy measures this defect is calculated and estimated three kinds of applications are given to the approximative calculation of some fuzzy integrals to the best approximation of a fuzzy measure by classical measures and to the introduction of an invariant at translations distance on the set of fuzzy measures

false discovery rate control for high dimensional networks of quantile associations conditioning on covariates motivated by gene coexpression pattern analysis we propose a novel sample quantile contingency squac statistic to infer quantile associations conditioning on covariates it features enhanced flexibility in handling variables with both arbitrary distributions and complex association patterns conditioning on covariates we first derive its asymptotic null distribution and then develop a multiple testing procedure based on the squac statistic to test simultaneously the independence between one pair of variables conditioning on covariates for all p p 1 2 pairs here p is the length of the outcomes and could exceed the sample size the testing procedure does not require resampling or perturbation and thus is computationally efficient we prove by theory and numerical experiments that this testing method asymptotically controls the false discovery rate it outperforms all alternative methods when the complex association patterns exist applied to a gastric cancer data set this testing method successfully inferred the gene coexpression networks of early and late stage patients it identified more changes in the networks which are associated with cancer survivals we extend our method to the case that both the length of the outcomes and the length of covariates exceed the sample size and show that the asymptotic theory still holds

two integrals and some modified versions critical remarks the aim of this paper is to discuss different constructions of integrals sections 3 and 4 based on decomposable measures section 1 according to the classification of the continuous t conorms in essentially two types namely v and archimedean t conorms there exist mainly two types of integrals namely the constructions of sugeno section 3 and of weber section 4 further constructions corresponding to the archimedean case result to be special cases or not well defined section 4 in all cases a crucial property is some restricted distribution law for the pair with an appropriate operation section 2 some applications shall illustrate the use of the two integrals section 5

nonparametric estimation and inference on conditional quantile processes this paper presents estimation methods and asymptotic theory for the analysis of a nonparametrically specified conditional quantile process two estimators based on local linear regressions are proposed the first estimator applies simple inequality constraints while the second uses rearrangement to maintain quantile monotonicity the bandwidth parameter is allowed to vary across quantiles to adapt to data sparsity for inference the paper first establishes a uniform bahadur representation and then shows that the two estimators converge weakly to the same limiting gaussian process as an empirical illustration the paper considers a dataset from project star and delivers two new findings

liapunov spectra for infinite chains of nonlinear oscillators we argue that the spectrum of liapunov exponents for long chains of nonlinear oscillators at large energy per mode may be well approximated by the liapunov exponents of products of independent random matrices if in addition statistical mechanics applies to the system the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone under a certain isotropy hypothesis which is not always satisfied we argue that the liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix this construction uses an integral equation first derived by newman we then derive and discuss a method to compute the spectrum of a typical random matrix putting the pieces together we see that the liapunov spectrum can be computed from the potential between the oscillators

the largest liapunov exponent for random matrices and directed polymers in a random environment we study the largest liapunov exponent for products of random matrices the two classes of matrices considered are discrete d dimensional laplacians with random entries and symplectic matrices that arise in the study ofd dimensional lattices of coupled nonlinear oscillators we derive bounds on this exponent for all dimensions d and we show that ifd 3 and the randomness is not too strong one can obtain an explicit formula for the largest exponent in the thermodynamic limit our method is based on an equivalence between this problem and the problem of directed polymers in a random environment

scaling law and asymptotic distribution of lyapunov exponents in conservative dynamical systems with many degrees of freedom the authors study by numerical means the infinite product of 2n 2n conservative random matrices which mimics the chaotic behaviour of hamiltonian systems with n 1 degrees of freedom made of weakly nearest neighbour coupled oscillators the maximum lyapunov exponent lambda 1 exhibits a power law behaviour as a function of the coupling constant epsilon lambda 1 approximately epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements these power laws do not depend on n and moreover increasing n lambda 1 rapidly tends to an asymptotic value lambda 1 which only depends on epsilon and on the kind of probability distribution chosen for building up the matrices they also compute the spectrum of the lyapunov exponents and show that is has a thermodynamic limit of large n this suggests the existence of a kolmogorov entropy per degree of freedom proportional to lambda 1

bayesian quantile regression for censored data ummary in this paper we propose a semiparametric quantile regression model for censored survival data quantile regression permits covariates to affect survival differently at different stages in the follow up period thus providing a comprehensive study of the survival distribution we take a semiparametric approach representing the quantile process as a linear combination of basis functions the basis functions are chosen so that the prior for the quantile process is centered on a simple location scale model but flexible enough to accommodate a wide range of quantile processes we show in a simulation study that this approach is competitive with existing methods the method is illustrated using data from a drug treatment study where we find that the bayesian model often gives smaller measures of uncertainty than its competitors and thus identifies more significant effects

quantile smoothing splines summary although nonparametric regression has traditionally focused on the estimation of conditional mean functions nonparametric estimation of conditional quantile functions is often of substantial practical interest we explore a class of quantile smoothing splines defined as solutions to

liapunov exponents in high dimensional symplectic dynamics the existence of a thermodynamic limit of the distribution of liapunov exponents is numerically verified in a large class of symplectic models ranging from hamiltonian flows to maps and products of random matrices in the highly chaotic regime this distribution is approximately model independent near an integrable limit only a few exponents give a relevant contribution to the kolmogorov sinai entropy

a mean field approximation for the maximum lyapunov exponent of bernoulli trials of matrices the author considers the infinite products of d d random matrices xi which are taken according to a bernoulli distribution i e xi a with probability 1 p and xi b with probability the maximum lyapunov exponent is estimated by selecting a suitable periodic sequence one need only then compute the eigenvalues of one particular matrix which is obtained as a finite product of matrices a and b he also discusses the cases in which the method fails because of fluctuation effects

improving point and interval estimators of monotone functions by rearrangement suppose that a target function is monotonic and an available original estimate of this target function is not monotonic rearrangements univariate and multivariate transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function furthermore suppose an original confidence interval which covers the target function with probability at least 1 is defined by an upper and lower endpoint functions that are not monotonic then the rearranged confidence interval defined by the rearranged upper and lower endpoint functions is monotonic shorter in length in common norms than the original interval and covers the target function with probability at least 1 we illustrate the results with a growth chart example

characterization of the spatial complex behavior and transition to chaos in flow systems abstract we introduce a spatial lyapunov exponent to characterize the complex behavior of non chaotic but convectively unstable flow sytems this complexity is of spatial type and is due to sensitivity to the boundary conditions we show that there exists a relation between the spatial complexity index we define and the comoving lyapunov exponents in such systems the transition to chaos i e the occurrence of a positive lyapunov exponent can manifest itself in two different ways in the first case from neither chaotic nor spatially complex behavior to chaos one observes the typical scenario i e as the system size grows up the spectrum of the lyapunov exponents gives rise to a density in the second case when the chaos develops from a convectively unstable situation one observes only a finite number of positive lyapunov exponents

time dependent mode structure for lyapunov vectors as a collective movement in quasi one dimensional systems the time dependent mode structure of the lyapunov vectors associated with the stepwise structure of the lyapunov spectra and its relation to the momentum autocorrelation function are discussed in quasi one dimensional many hard disk systems we obtain the complete mode structures lyapunov modes for all components of the lyapunov vectors including the longitudinal and transverse components of both the spatial and momentum parts and their phase relations these mode structures are suggested by the form of the lyapunov vectors for the zero lyapunov exponents the spatial node structures of these modes are explained by the reflection properties of the hard walls used in the models our main result is that the largest time oscillating period of the lyapunov modes is twice as long as the time oscillating period of the longitudinal momentum autocorrelation function this relation is satisfied irrespective of the number of particles and the boundary conditions a simple explanation for this relation is given based on the form of the time dependent lyapunov mode

nonparametric quantile frontier estimation under shape restriction abstract this paper proposes a shape restricted nonparametric quantile regression to estimate the frontier which acts as a benchmark for whether a decision making unit achieves top efficiency this method adopts a two step strategy first identifying fitted values that minimize an asymmetric absolute loss under the nondecreasing and concave shape restriction second constructing a nondecreasing and concave estimator that links these fitted values this method makes no assumption on the error distribution and the functional form experimental results on some artificial data sets clearly demonstrate its superiority over the classical linear quantile regression we also discuss how to enforce constraints to avoid quantile crossings between multiple estimated frontiers with different values of finally this paper shows that this method can be applied to estimate the production function when one has some prior knowledge about the error term

multifractal amplitude fluctuations in the transfer matrix approach to the statistics of disordered lattice models it is shown that the amplitude fluctuations of vector iterates in the transfer matrix approach to the statistics of the one dimensional random ising chain correspond to an exact strange set characterised by a multifractal spectrum the temperature and disorder dependence of the spectrum are also investigated

equivalence of kinetic theory and random matrix approaches to lyapunov spectra of hard sphere systems abstract in the study of chaotic behaviour of systems of many hard spheres lyapunov exponents of small absolute values exhibit interesting characteristics leading to speculations about connections to non equilibrium statistical mechanics analytical approaches to these exponents so far can be divided into two groups macroscopically oriented approaches using kinetic theory or hydrodynamics and more microscopically oriented random matrix approaches in quasi one dimensional systems in this paper i present an approach using random matrices and weak disorder expansion in an arbitrary number of dimensions correlations between subsequent collisions of a particle are taken into account it is shown that the results are identical to those of a previous approach based on an extended enskog equation i conclude that each approach has its merits and provides different insights into the approximations made which include the stoszahlansatz the continuum limit and the long wavelength approximation the comparison also gives insight into possible connections between lyapunov exponents and fluctuations

scaling law for the maximum lyapunov characteristic exponent of infinite product of random matrices the authors present a simple derivation for the scaling behaviour of the maximum lyapunov characteristic exponent lambda of infinite product of symplectic random matrices the considered random matrices depend on a parameter epsilon and lambda 0 for epsilon 0 they obtain lambda varies as epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements the results are in agreement with a previous numerical simulation

multilevel quantile function modeling with application to birth outcomes summary infants born preterm or small for gestational age have elevated rates of morbidity and mortality using birth certicate records in texas from 2002 2004 and environmental protection agency air pollution estimates we relate the quantile functions of birth weight and gestational age to ozone exposure and multiple predictors including parental age race and education level we introduce a semi parametric bayesian quantile approach that models the full quantile function rather than just a few quantile levels our multilevel quantile function model establishes relationships between birth weight and the predictors separately for each week of gestational age and between gestational age and the predictors separately across texas public health regions we permit these relationships to vary nonlinearly across gestational age spatial domain and quantile level and we unite them in a hierarchical model via a basis expansion on the regression coefficients that preserves interpretability very low birth weight is a primary concern so we leverage extreme value theory to supplement our model in the tail of the distribution gestational ages are rounded into weekly values so we present methodology for modeling quantile functions of discrete response data in a simulation study we show that pooling information across gestational age and quantile level substantially reduces mse of predictor effects relative to standard frequentist quantile regression wend that ozone is negatively associated with the lower tail of gestational age in south texas and across the distribution of birth weight for high gestational ages our methods are available in the r package bsquare

finite time lyapunov exponents for products of random transformations i show how continuous products of random transformations constrained by a generic group structure can be studied by using iwasawa s decomposition into angular diagonal and shear degrees of freedom in the case of a gaussian process a set of variables adapted to the iwasawa decomposition and still having a gaussian distribution is introduced and used to compute the statistics of the finite time lyapunov spectrum of the process the variables also allow to show the exponential freezing of the shear degrees of freedom which contain information about the lyapunov eigenvectors

large volume limit of the distribution of characteristic exponents in turbulence for spatially extended conservative or dissipative physical systems it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity in the case of a turbulent viscous fluid however this simple idea is complicated by the phenomenon of intermittency in the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation these bounds have a reasonable large volume behavior for two dimensional fluids a particularly striking result is obtained the total information creation is bounded above by a fixed multiple of the total energy dissipation at fixed viscosity the distribution of characteristic exponents is estimated in an intermittent model of turbulence see 7 and it is found that a change of behavior occurs at the valued 2 6 of the self similarity dimension

the fixed scale transformation approach to spatiotemporal intermittency in coupled map lattices abstract a one dimensional array of nonlinear maps with laplacian couplings is studied at the phase transition between the asymptotically turbulent and laminar phases along the critical boundary where spatiotemporal intermittency is observed turbulent sites form a fractal set the fixed scale transformation approach allows us to simultaneously determine the fractal dimension of the turbulent sites and the invariant asymptotic measure defining scale invariant conditional probabilities and taking into account the nearest neighbour correlations exactly we determine an asymptotic distribution function for the variables pertaining to sites on or neighbouring the turbulent clusters at any given level of coarse graining and obtain the fractal dimension analytically

on the construction of fuzzy measures a relation defined on a s algebra called relative likelihood is introduced it is shown how to construct fuzzy measures which are compatible with a relative likelihood

modeling regression quantile process using monotone b splines abstractquantile regression as an alternative to conditional mean regression i e least square regression is widely used in many areas it can be used to study the covariate effects on the entire response distribution by fitting quantile regression models at multiple different quantiles or even fitting the entire regression quantile process however estimating the regression quantile process is inherently difficult because the induced conditional quantile function needs to be monotone at all covariate values in this article we proposed a regression quantile process estimation method based on monotone b splines the proposed method can easily ensure the validity of the regression quantile process and offers a concise framework for variable selection and adaptive complexity control we thoroughly investigated the properties of the proposed procedure both theoretically and numerically we also used a case study on wind power generation to demonstrate its use and effectiveness in real problems supplem

an analytic estimate of the maximum lyapunov exponent in products of tridiagonal random matrices in this work we present a theoretical and numerical study of the behaviour of the maximum lyapunov exponent in products of random tridiagonal matrices in the limit of small coupling and small fluctuations such a problem is directly motivated by the investigation of coupled map lattices in a regime where the chaotic properties are quite robust and yet a complete understanding has still not been reached we derive some approximate analytic expressions by introducing a suitable continuous time formulation of the evolution equation as a first result we show that the perturbation of the lyapunov exponent due to the coupling depends only on a single scaling parameter which in the case of strictly positive multipliers is the ratio of the coupling strength with the variance of local multipliers an explicit expression for the lyapunov exponent is obtained by mapping the original problem onto a chain of nonlinear langevin equations which are eventually reduced to a single stochastic equation the probability distribution of this dynamical equation provides an excellent description for the behaviour of the lyapunov exponent finally multipliers with random signs are also considered finding that the lyapunov exponent again depends on a single scaling parameter which however has a different expression

quantile correlations and quantile autoregressive modeling in this article we propose two important measures quantile correlation qcor and quantile partial correlation qpcor we then apply them to quantile autoregressive qar models and introduce two valuable quantities the quantile autocorrelation function qacf and the quantile partial autocorrelation function qpacf this allows us to extend the box jenkins three stage procedure model identification model parameter estimation and model diagnostic checking from classical autoregressive models to quantile autoregressive models specifically the qpacf of an observed time series can be employed to identify the autoregressive order while the qacf of residuals obtained from the fitted model can be used to assess the model adequacy we not only demonstrate the asymptotic properties of qcor and qpcor but also show the large sample results of qacf qpacf and the quantile version of the box pierce test moreover we obtain the bootstrap approximations to the distributions of parameter estimators and p

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

the fuzzy integral on product spaces for nsa measures abstract this paper deals with the study of fuzzy integrals defined as a generalization of kruse s fuzzy integral 4 for fuzzy nsa measures given by weber 17 as opposed to sugeno s measures several properties and convergence theorems such as the monotone convergence theorem the dominated convergence theorem and fatou s lemma are shown for this fuzzy integral a fuzzy nsa measure will be constructed on a product space based on product of these kinds of measures with the same additive generator and we extend the above integral on to a product space further a version of fubini s theorem will be proved

fuzzy integrals and conditional fuzzy measures we introduce a fuzzy integral for fuzzy events with respect to l additive fuzzy measures this integral is the canonical generalization of the lehesgue integral a radon nikodym like theorem is used to give the definition of the conditional fuzziness

a review of the hydrodynamic lyapunov modes of hard disk systems a review of the results obtained for hard disk fluids confined to a quasi one dimensional qod system is presented one of the main achievements in recent years has been determining the hydrodynamic lyapunov modes hlms as covariant stable and unstable manifolds defined by k vector analogues of the zero exponent subspace of conserved quantities the tangent space dynamics allows an interpretation of the hlms as reduced hydrodynamic fields over the perturbations and is expanded upon in this paper the time evolution of these modes is governed by the dynamics of conjugate pairs of stable and unstable directions each pair is seen to interact in a subspace almost completely separated from other vectors the modes undergo a rotation until they are collinear with the stable or unstable manifold the angle distributions between these covariant stable and unstable hlms are also determined and tangencies are not observed for a generic chaotic trajectory this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

noncrossing quantile regression curve estimation since quantile regression curves are estimated individually the quantile curves can cross leading to an invalid distribution for the response a simple constrained version of quantile regression is proposed to avoid the crossing problem for both linear and nonparametric quantile curves a simulation study and a reanalysis of tropical cyclone intensity data shows the usefulness of the procedure asymptotic properties of the estimator are equivalent to the typical approach under standard conditions and the proposed estimator reduces to the classical one if there is no crossing the performance of the constrained estimator has shown significant improvement by adding smoothing and stability across the quantile levels

generalized lyapunov exponents in high dimensional chaotic dynamics and products of large random matrices we study the behavior of the generalized lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsd for products of random matrices without any particular structure the generalized lyapunov exponents become equal in this limit and the value of one of the generalized lyapunov exponents is obtained by simple arguments on the contrary for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized lyapunov exponents remains different ford indicating that high dimensionality cannot always destroy intermittency

two families of fuzzy integrals abstract in this paper we present two definitions of fuzzy integrals arising from the t seminorm and the t semiconorm concepts we show that sugeno s fuzzy integral is a particular case of them and we study their principal properties finally we suggest some applications of these results

power law behavior of lyapunov exponents in some conservative dynamical systems abstract three billiards whose border depends on a parameter e are considered for e 0 they are integrable while for e 0 they are known to be k systems the maximal lyapunov exponent 1 is numerically computed and a power law behavior 1 e is found with 1 2 for all billiards a related abstract problem is then considered precisely the case of an infinite product of conservative 2 2 random matrices which are perturbations of commuting parabolic ones analogous computations give here two power laws with 1 2 or 2 3 depending on the probability law used to construct random matrices all these phenomena seem to have an universal character a heuristic mean field theory is also proposed based on the formal interpretation of the onset of stochasticity in our dynamical systems as a phase transition

distribution of characteristic exponents in the thermodynamic limit the existence of the thermodynamic limit for the spectrum of the lyapunov characteristic exponents is numerically investigated for the fermi pasta ulam p model we show that the shape of the spectrum for energy density well above the equipartition threshold e allows the kolmogorov sinai entropy to be expressed simply in terms of the maximum exponent i the presence of a power law behaviour e is investigated the analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some universal properties

lyapunov exponents and dimensions of chaotic neural networks dynamical properties of a network model composed of n continuous elements with randomly chosen asymmetrical couplings and reduced connectivity are studied the authors calculate numerically lyapunov exponents and dimensions for chaotic and non chaotic states of the network the dependence of the parameters can be described analytically in good agreement with the numerical data

non crossing non parametric estimates of quantile curves in this paper a new nonparametric estimate of conditional quantiles is proposed that avoids the problem of crossing quantile curves calculated for various p 2 0 1 the method uses an initial estimate of the conditional distribution function in a flrst step and solves the problem of inversion and monotonization with respect to p 2 0 1 simultaneously it is demonstrated that the new estimates are asymptotically normal distributed and asymptotically flrst order equivalent to quantile estimates obtained by local constant or local linear smoothing of the conditional distribution function the performance of the new procedure is illustrated by means of a simulation study and some comparisons with the currently available procedures which are similar in spirit with the proposed method are presented

methods for estimating a conditional distribution function abstract motivated by the problem of setting prediction intervals in time series analysis we suggest two new methods for conditional distribution estimation the first method is based on locally fitting a logistic model and is in the spirit of recent work on locally parametric techniques in density estimation it produces distribution estimators that may be of arbitrarily high order but nevertheless always lie between 0 and 1 the second method involves an adjusted form of the nadaraya watson estimator it preserves the bias and variance properties of a class of second order estimators introduced by yu and jones but has the added advantage of always being a distribution itself our methods also have application outside the time series setting for example to quantile estimation for independent data this problem motivated the work of yu and jones

joint estimation of quantile planes over arbitrary predictor spaces abstractin spite of the recent surge of interest in quantile regression joint estimation of linear quantile planes remains a great challenge in statistics and econometrics we propose a novel parameterization that characterizes any collection of noncrossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar vector and function valued parameters statistical models based on this parameterization inherit a fast computation of the likelihood function enabling penalized likelihood or bayesian approaches to model fitting we introduce a complete bayesian methodology by using gaussian process prior distributions on the function valued parameters and develop a robust and efficient markov chain monte carlo parameter estimation the resulting method is shown to offer posterior consistency under mild tail and regularity conditions we present several illustrative examples where the new method is compared against existing approaches and is found to o

on the effects of an uncertainty on the evolution law in dynamical systems abstract we show that a small perturbation of the evolution law of chaotic dynamical systems is practically equivalent to the finite precision on the knowledge of the initial condition in spite of the short predictability time t p on a single trajectory we found that the statistical properties are not sensitive to small changes in the evolution law or initial probability distribution this feature holds also for correlation functions at a delay larger than t p we discuss the roundoff effects on integrable systems

modeling persistent trends in distributions abstractwe present a nonparametric framework to model a short sequence of probability distributions that vary both due to underlying effects of sequential progression and confounding noise to distinguish between these two types of variation and estimate the sequential progression effects our approach leverages an assumption that these effects follow a persistent trend this work is motivated by the recent rise of single cell rna sequencing experiments over a brief time course which aim to identify genes relevant to the progression of a particular biological process across diverse cell populations while classical statistical tools focus on scalar response regression or order agnostic differences between distributions it is desirable in this setting to consider both the full distributions as well as the structure imposed by their ordering we introduce a new regression model for ordinal covariates where responses are univariate distributions and the underlying relationship reflects consistent changes i

intrinsic stochasticity with many degrees of freedom we study a simple model equation describing a system with an infinity of degrees of freedom which displays an intrinsically chaotic behavior some concepts of fully developed turbulence are discussed in relation to this model we also develop an approach based on lyapunov exponent measurements numerical results on the distribution of lyapunov numbers and the power spectrum of the associated lyapunov vectors are presented and briefly discussed