hurst entropy a method to determine predictability in a binary series based on a fractal related process shannon s concept of information is related to predictability in a binary series the value of information relies on the frequency of 0 s and 1 s or how it is expected to occur however information entropy does not consider the bias in randomness related to autocorrelation in fact it is possible for a binary temporal series to carry both short and long term memories related to the sequential distribution of 0 s and 1 s although the hurst exponent measures the range of autocorrelation there is a lack of mathematical connection between information entropy and autocorrelation present in the series to fill this important gap we combined numerical simulations and an analytical approach to determine how information entropy changes according to the frequency of 0 s and 1 s and the hurst exponent indeed we were able to determine how predictability depends on both parameters our findings are certainly useful to several fields when binary times series are applied such as neuroscience to econophysics

development of a formal likelihood function for improved bayesian inference of ephemeral catchments 1 the application of formal bayesian inferential approaches in hydrologic modeling is often criticized for requiring explicit assumptions to be made about the distribution of the errors via the likelihood function these assumptions can be adequate in some situations but often little attention is paid to the selection of an appropriate likelihood function this paper investigates the application of bayesian methods in modeling ephemeral catchments we consider two modeling case studies including a synthetically generated data set and a real data set of an arid australian catchment the case studies investigate some typical forms of the likelihood function which have been applied widely by previous researchers as well as introducing new implementations aimed at better addressing likelihood function assumptions in dry catchments the results of the case studies indicate the importance of explicitly accounting for model residuals that are highly positively skewed due to the presence of many zeros zero inflation arising from the dry spells experienced by the catchment specifically the form of the likelihood function was found to significantly impact the calibrated parameter posterior distributions which in turn have the potential to greatly affect the uncertainty estimates in each application the likelihood function that explicitly accounted for the nonconstant variance of the errors and the zero inflation of the errors resulted in 1 equivalent or better fits to the observed discharge in both timing and volume 2 superior estimation of the uncertainty as measured by the reliability and sharpness metrics and 3 more linear quantile quantile plots indicating the errors were more closely matched to the assumptions of this form of the likelihood function

log normal distribution from a process that is not multiplicative but is additive the central limit theorem ensures that a sum of random variables tends to a gaussian distribution as their total number tends to infinity however for a class of positive random variables we find that the sum tends faster to a log normal distribution although the sum tends eventually to a gaussian distribution the distribution of the sum is always close to a log normal distribution rather than to any gaussian distribution if the summands are numerous enough this is in contrast to the current consensus that any log normal distribution is due to a product of random variables i e a multiplicative process or equivalently to nonlinearity of the system in fact the log normal distribution is also observable for a sum i e an additive process that is typical of linear systems we show conditions for such a sum an analytical example and an application to random scalar fields such as those of turbulence

persistent random motion with maximally correlated fluctuations how often should a random walker change its direction of motion in order to maximize correlation in velocity fluctuations over a finite time interval we address this optimal diffusion problem in the context of the one dimensional persistent random walk where we evaluate the correlation and mutual information in velocity trajectories as a function of the persistence level and the observation time we find the optimal persistence level corresponds to the average number of direction reversals asymptotically scaling as the square root of the observation time this square root scaling law makes the relative growth between the average number of direction reversals and the persistence length invariant with respect to changes in the overall time duration of the random walk

on a family of weierstrass type root finding methods with accelerated convergence kyurkchiev and andreev 1985 constructed an infinite sequence of weierstrass type iterative methods for approximating all zeros of a polynomial simultaneously the first member of this sequence of iterative methods is the famous method of weierstrass 1891 and the second one is the method of nourein 1977 for a given integer n 1 the nth method of this family has the order of convergence n 1 currently in the literature there are only local convergence results for these methods the main purpose of this paper is to present semilocal convergence results for the weierstrass type methods under computationally verifiable initial conditions and with computationally verifiable a posteriori error estimates

multivariate copula analysis toolbox mvcat describing dependence and underlying uncertainty using a bayesian framework we present a newly developed multivariate copula analysis toolbox mvcat which includes a wide range of copula families with different levels of complexity mvcat employs a bayesian framework with a residual based gaussian likelihood function for inferring copula parameters and estimating the underlying uncertainties the contribution of this paper is threefold a providing a bayesian framework to approximate the predictive uncertainties of fitted copulas b introducing a hybrid evolution markov chain monte carlo mcmc approach designed for numerical estimation of the posterior distribution of copula parameters and c enabling the community to explore a wide range of copulas and evaluate them relative to the fitting uncertainties we show that the commonly used local optimization methods for copula parameter estimation often get trapped in local minima the proposed method however addresses this limitation and improves describing the dependence structure mvcat also enables evaluation of uncertainties relative to the length of record which is fundamental to a wide range of applications such as multivariate frequency analysis

information transmission of mean and variance coding in integrate and fire neurons neurons process information by translating continuous signals into patterns of discrete spike times an open question is how much information these spike times contain about signals which modulate either the mean or the variance of the somatic currents in neurons as is observed experimentally here we calculate the exact information contained in discrete spike times about a continuous signal in both encoding strategies we show that the information content about mean modulating signals is generally substantially larger than about variance modulating signals for biological parameters our analysis further reveals that higher information transmission is associated with a larger proportion of nonlinear signal encoding our study measures the complete information content of mean and variance coding and provides a method to determine what fraction of the total information is linearly decodable

integrated approach to the assessment of long range correlation in time series data to assess whether a given time series can be modeled by a stochastic process possessing long range correlation one usually applies one of two types of analysis methods the spectral method and the random walk analysis the first objective of this work is to show that each one of these methods used alone can be susceptible to producing false results we thus advocate an integrated approach which requires the use of both methods in a consistent fashion we provide the theoretical foundation of this approach and illustrate the main ideas using examples the second objective relates to the observation of long range anticorrelation hurst exponent hl1 2 in real world time series data the very peculiar nature of such processes is emphasized in light of the stringent condition under which such processes can occur using examples we discuss the possible factors that could contribute to the false claim of long range anticorrelations and demonstrate the particular importance of the integrated approach in this case

information theoretical analysis of resting state eeg microstate sequences non markovianity non stationarity and periodicities we present an information theoretical analysis of temporal dependencies in eeg microstate sequences during wakeful rest we interpret microstate sequences as discrete stochastic processes where each state corresponds to a representative scalp potential topography testing low order markovianity of these discrete sequences directly we find that none of the recordings fulfils the markov property of order 0 1 or 2 further analyses show that the microstate transition matrix is non stationary over time in 80 window size 10 s 60 window size 20 s and 44 window size 40 s of the subjects and that transition matrices are asymmetric in 14 20 70 subjects to assess temporal dependencies globally the time lagged mutual information function autoinformation function of each sequence is compared to the first order markov model defined by the classical transition matrix approach the autoinformation function for the markovian case is derived analytically and numerically for experimental data we find non markovian behaviour in the range of the main eeg frequency bands where distinct periodicities related to the subject s eeg frequency spectrum appear in particular the microstate clustering algorithm induces frequency doubling with respect to the eeg power spectral density while the tail of the autoinformation function asymptotically reaches the first order markov confidence interval for time lags above 1000 ms in summary our results show that resting state microstate sequences are non markovian processes which inherit periodicities from the underlying eeg dynamics our results interpolate between two diverging models of microstate dynamics memoryless markov models on one side and long range correlated models on the other microstate sequences display more complex temporal dependencies than captured by the transition matrix approach in the range of the main eeg frequency bands but show finite memory content in the long run

generalized likelihood uncertainty estimation glue and approximate bayesian computation what s the connection 1 there has been a recent debate in the hydrological community about the relative merits of the informal generalized likelihood uncertainty estimation glue approach to uncertainty assessment in hydrological modeling versus formal probabilistic approaches some recent literature has suggested that the methods can give similar results in practice when properly applied in this note we show that the connection between formal bayes and glue is not merely operational but goes deeper with glue corresponding to a certain approximate bayesian procedure even when the generalized likelihood is not a true likelihood the connection we describe relates to recent approximate bayesian computation abc methods originating in genetics abc algorithms involve the use of a kernel function and the generalized likelihood in glue can be thought of as relating to this kernel function rather than to the model likelihood two interpretations of glue emerge one as a computational approximation to a bayes procedure for a certain error free model and the second as an exact bayes procedure for a perturbation of that model in which the truncation of the generalized likelihood in glue plays a role the intent of this study is to encourage cross fertilization of ideas regarding glue and abc in hydrologic applications the connection we outline suggests the possibility of combining a formal likelihood with a kernel based on a generalized likelihood within the abc framework and also allows advanced abc computational methods to be used in glue applications the model based interpretation of glue may also be helpful in partially illuminating the implicit assumptions in different choices of generalized likelihood

novel techniques in parameter estimation for fractional dynamical models arising from biological systems in recent years both parameter estimation and fractional calculus have attracted a considerable interest parameter estimation of the fractional dynamical models is a new topic in this paper we consider novel techniques for parameter estimation of fractional nonlinear dynamical models in systems biology first a computationally effective fractional predictor corrector method is proposed for simulating fractional complex dynamical models second we convert the parameter estimation of fractional complex dynamical models into a minimization problem of the unknown parameters third a modified hybrid simplex search mhss and a particle swarm optimization pso is proposed finally these techniques are applied to a dynamical model of competence induction in a cell with measurement error and noisy data some numerical results are given that demonstrate the effectiveness of the theoretical analysis

scaling laws and persistence in human brain activity abstract we study the temporal variability of human brain activity in timeseries of functional magnetic resonance imaging fmri data we find that these timeseries show scaling behavior which is quantified by computing various scaling exponents we demonstrate that mentally active zones are one to one related to large exponents or equivalently to highly temporally correlated processes while mentally inactive zones are well described by a simple random walk model activation maps of scaling exponents are presented and compared to standard results in fmri in contrast to standard model based activation analyses no prior knowledge of the experimental stimulation paradigm has to be assumed for extracting activation patterns from fmri timeseries we mention consequences of persistence in fmri timeseries for standard statistical analysis

large scale lognormality in turbulence modeled by the ornstein uhlenbeck process lognormality was found experimentally for coarse grained squared turbulence velocity and velocity increment when the coarsening scale is comparable to the correlation scale of the velocity mouri et al phys fluids 21 065107 2009 we investigate this large scale lognormality by using a simple stochastic process with correlation the ornstein uhlenbeck ou process it is shown that the ou process has a similar large scale lognormality which is studied numerically and analytically

a scaling law to predict the finite length performance of spatially coupled ldpc codes spatially coupled low density parity check sc ldpc codes are known to have excellent asymptotic properties much less is known regarding their finite length performance we propose a scaling law to predict the error probability of finite length spatially coupled code ensembles when transmission takes place over the binary erasure channel we discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters the ultimate goal of this line of research is to develop analytic tools for the design of sc ldpc codes under practical constraints

analytical and empirical fluctuation functions of the eeg microstate random walk short range vs long range correlations we analyze temporal autocorrelations and the scaling behaviour of eeg microstate sequences during wakeful rest we use the recently introduced random walk approach and compute its fluctuation function analytically under the null hypothesis of a short range correlated first order markov process the empirical fluctuation function and the hurst parameter h as a surrogate parameter of long range correlations are computed from 32 resting state eeg recordings and for a set of first order markov surrogate data sets with equilibrium distribution and transition matrices identical to the empirical data in order to distinguish short range correlations h 0 5 from previously reported long range correlations h 0 5 statistically confidence intervals for h and the fluctuation functions are constructed under the null hypothesis comparing three different estimation methods for h we find that only one data set consistently shows h 0 5 compatible with long range correlations whereas the majority of experimental data sets cannot be consistently distinguished from markovian scaling behaviour our analysis suggests that the scaling behaviour of resting state eeg microstate sequences though markedly different from uncorrelated zero order markov processes can often not be distinguished from a short range correlated first order markov process our results do not prove the microstate process to be markovian but challenge the approach to parametrize resting state eeg by single parameter models

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

humans are able to self paced constant running accelerations until exhaustion abstract although it has been experimentally reported that speed variations is the optimal way of optimizing his pace for achieving a given distance in a minimal time we still do not know what the optimal speed variations i e accelerations are at first we have to check the hypothesis that human is able to accurately self pacing its acceleration and this even in a state of fatigue during exhaustive self pacing ramp runs for that purpose 3 males and 2 females middle aged recreational runners ran in random order three exhaustive acceleration trials we instructed the five runners to perform three self paced acceleration trials based on three acceleration intensity levels soft medium and hard we chose a descriptive modelling approach to analyse the behaviour of the runners once we knew that the runners were able to perceive three acceleration intensity levels we proposed a mean reverting process ornstein uhlenbeck to describe those accelerations d a t a t a d t d w t where a is the mean acceleration a t is the measured acceleration at each time interval t the ability of the runner to correct the variations around a mean acceleration and the human induced variations the goodness of fit of the ornstein uhlenbeck process highlights the fact that humans are able to maintain a constant acceleration and are able to precisely regulate their acceleration regardless of its intensity in a run leading to exhaustion in the range from 1 min 36 s to 20 min

reply to comment by chu et al on high dimensional posterior exploration of hydrologic models using multiple try dream zs and high performance computing publications water resources research commentary 10 1002 2013wr014425 this article is a reply to chu et al 2014 doi 10 1002 2012wr013341 reply to comment by chu et al on high dimensional posterior exploration of hydrologic models using multiple try dream zs and high performance computing jasper a vrugt 1 2 3 and eric laloy 4 correspondence to j a vrugt jasper uci edu citation vrugt j a and e laloy 2014 reply to comment chu et al on high dimensional posterior exploration of hydrologic models using multiple try dream zs and high performance computing water resour res 50 2781 2786 doi 10 1002 2013wr014425 received 15 july 2013 accepted 8 feb 2014 accepted article online 15 feb 2014 published online 21 mar 2014 department of civil and environmental engineering university of california irvine california usa 2 department of earth system science university of california irvine california usa 3 institute for biodiversity and ecosystems dynamics university of amsterdam amsterdam netherlands 4 institute for environment health and safety belgian nuclear research centre mol belgium 1 introduction the comment of chu et al 2014 hereafter referred to as cyg raises questions about some of the results presented in our paper laloy and vrugt 2012 hereinafter lv which is not to be confused with las vegas although appropriate concerning the subject of this work on monte carlo simulation as a preamble we would like to remark that the work presented in chu et al 2010 and lv 2012 concern two different elds of study cyg view the model calibration as an optimization problem and use common concepts to ef ciently nd a single realization of the parameter values that minimizes or maximizes if appropriate some user de ned objective function our work on the contrary uses bayesian principles coupled with mcmc simulation to derive a distribution of parameter values that honor the observed data this distribution summarizes parameter and model predictive simulation uncertainty a requirement for probabilistic analysis operational forecasting disentangling error sources and decision making the maxi mum a posteriori density map parameter values derived with mcmc simulation should reside in close vicinity of the best solution found with an optimization algorithm if the exact same data set prior distribu tion and likelihood objective function are used in this reply we assume that cyg used a correct imple mentation of the mt dream zs algorithm and similar data set prior and likelihood function as lv otherwise the comparative analysis is meaningless we emphasize this for three reasons first the results presented herein contradict cyg and are similar to those reported in lv but now with more trials plotted second contributions in physics horowitz et al 2012 toyli et al 2012 yale et al 2013 and geophysics linde and vrugt 2013 laloy et al 2012 rosas carbajal et al 2014 t lochbuehler et al summary statistics from training images as model constraints in probabilistic inversion geophysical journal international in review 2014 demonstrate proper convergence behavior of mt dream zs on complex and high dimensional targets involving hundreds of parameters third to justify their sp uci algorithm the original paper of chu et al 2010 portrays misleading results of the predecessor of dream called scem ua section 4 of this reply will address this latter issue in more detail we now respond to the comments of cyg we use different sections with numbering corresponding to cyg 2 computational time unit cyg nd the computational time unit ctu diagnostic to be a poor indicator of the performance of mt dream zs they suggest using the number of function evaluations or clock time instead the cpu time s scales linearly with ctu or cpu 5 actu where a s denotes the average time it takes to complete a sin gle function model evaluation as a is dependent on the processor speed hardware lv purposely reported the ctu values note that we neglect the actual run time of mt dream zs in the determination of a which is appropriate given the intended application of this algorithm to cpu intensive forward models we purposely do not use the number of function evaluations as performance diagnostic this metric does not properly convey the cpu time ctu of parallel algorithms such as dream zs or mt dream zs these vrugt and laloy c 2014 american geophysical union all rights reserved v

toward diagnostic model calibration and evaluation approximate bayesian computation the ever increasing pace of computational power along with continued advances in measurement technologies and improvements in process understanding has stimulated the development of increasingly complex hydrologic models that simulate soil moisture flow groundwater recharge surface runoff root water uptake and river discharge at different spatial and temporal scales reconciling these high order system models with perpetually larger volumes of field data is becoming more and more difficult particularly because classical likelihood based fitting methods lack the power to detect and pinpoint deficiencies in the model structure gupta et al 2008 has recently proposed steps amongst others toward the development of a more robust and powerful method of model evaluation their diagnostic approach uses signature behaviors and patterns observed in the input output data to illuminate to what degree a representation of the real world has been adequately achieved and how the model should be improved for the purpose of learning and scientific discovery in this paper we introduce approximate bayesian computation abc as a vehicle for diagnostic model evaluation this statistical methodology relaxes the need for an explicit likelihood function in favor of one or multiple different summary statistics rooted in hydrologic theory that together have a clearer and more compelling diagnostic power than some average measure of the size of the error residuals two illustrative case studies are used to demonstrate that abc is relatively easy to implement and readily employs signature based indices to analyze and pinpoint which part of the model is malfunctioning and in need of further improvement 2013 american geophysical union all rights reserved

mutual information identifies spurious hurst phenomena in resting state eeg and fmri data long range memory in time series is often quantified by the hurst exponent h a measure of the signal s variance across several time scales we analyze neurophysiological time series from electroencephalography eeg and functional magnetic resonance imaging fmri resting state experiments with two standard hurst exponent estimators and with the time lagged mutual information function applied to discretized versions of the signals a confidence interval for the mutual information function is obtained from surrogate markov processes with equilibrium distribution and transition matrix identical to the underlying signal for eeg signals we construct an additional mutual information confidence interval from a short range correlated tenth order autoregressive model we reproduce the previously described hurst phenomenon h 0 5 in the analytical amplitude of alpha frequency band oscillations in eeg microstate sequences and in fmri signals but we show that the hurst phenomenon occurs without long range memory in the information theoretical sense we find that the mutual information function of neurophysiological data behaves differently from fractional gaussian noise fgn for which the hurst phenomenon is a sufficient condition to prove long range memory two other well characterized short range correlated stochastic processes ornstein uhlenbeck cox ingersoll ross also yield h 0 5 whereas their mutual information functions lie within the markovian confidence intervals similar to neural signals in these processes which do not have long range memory by construction a spurious hurst phenomenon occurs due to slow relaxation times and heteroscedasticity time varying conditional variance in summary we find that mutual information correctly distinguishes long range from short range dependence in the theoretical and experimental cases discussed our results also suggest that the stationary fgn process is not sufficient to describe neural data which seem to belong to a more general class of stochastic processes in which multiscale variance effects produce hurst phenomena without long range dependence in our experimental data the hurst phenomenon and long range memory appear as different system properties that should be estimated and interpreted independently

functional linear regression with points of impact the paper considers functional linear regression where scalar responses y 1 ldots y n are modeled in dependence of i i d random functions x 1 ldots x n we study a generalization of the classical functional linear regression model it is assumed that there exists an unknown number of points of impact that is discrete observation times where the corresponding functional values possess significant influences on the response variable in addition to estimating a functional slope parameter the problem then is to determine the number and locations of points of impact as well as corresponding regression coefficients identifiability of the generalized model is considered in detail it is shown that points of impact are identifiable if the underlying process generating x 1 ldots x n possesses specific local variation examples are well known processes like the brownian motion fractional brownian motion or the ornstein uhlenbeck process the paper then proposes an easily implementable method for estimating the number and locations of points of impact it is shown that this number can be estimated consistently furthermore rates of convergence for location estimates regression coefficients and the slope parameter are derived finally some simulation results as well as a real data application are presented

stochastic filtering for diffusion processes with level crossings we provide a general framework for computing the state density of a noisy system given the sequence of hitting times of predefined thresholds our method relies on eigenfunction expansion corresponding to the fokker planck operator of the diffusion process for illustration we present a particular example in which the state and the noise are one dimensional gaussian processes and observations are generated when the magnitude of the observed signal is a multiple of some threshold value we present numerical simulations confirming the convergence and the accuracy of the recovered density estimator applications of the filtering methodology will be illustrated

jacobi free and complex free method for finding simultaneously all zeros of polynomials having only real zeros abstract there are many methods for finding simultaneously all zeros of polynomials however a complex arithmetic or jacobian matrix is needed in these methods even if the polynomial is real and only has real zeros in this paper a new method is proposed for finding simultaneously all zeros of real polynomials having only real zeros that does not require any complex arithmetic and jacobian matrix within iteration it is based on vieta s theorem and broyden s method numerical experiences show that the new method is very effective for the test polynomials

exact numerical simulation of the ornstein uhlenbeck process and its integral a numerical simulation algorithm that is exact for any time step ensuremath delta tg0 is derived for the ornstein uhlenbeck process x t and its time integral y t the algorithm allows one to make efficient unapproximated simulations of for instance the velocity and position components of a particle undergoing brownian motion and the electric current and transported charge in a simple r l circuit provided appropriate values are assigned to the ornstein uhlenbeck relaxation time ensuremath tau and diffusion constant c a simple taylor expansion in ensuremath delta t of the exact simulation formulas shows how the first order simulation formulas which are implicit in the langevin equation for x t and the defining equation for y t are modified in second order the exact simulation algorithm is used here to illustrate the zero ensuremath tau limit theorem textcopyright 1996 the american physical society

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

dynamics of electroencephalogram entropy and pitfalls of scaling detection in recent studies a number of research groups have determined that human electroencephalograms eeg have scaling properties in particular a crossover between two regions with different scaling exponents has been reported herein we study the time evolution of diffusion entropy to elucidate the scaling of eeg time series for a cohort of 20 awake healthy volunteers with closed eyes we find that the diffusion entropy of eeg increments obtained from eeg waveforms by differencing exhibits three features short time growth an alpha wave related oscillation whose amplitude gradually decays in time and asymptotic saturation which is achieved after approximately 1 s this analysis suggests a linear stochastic ornstein uhlenbeck langevin equation with a quasiperiodic forcing whose frequency and or amplitude may vary in time as the model for the underlying dynamics this model captures the salient properties of eeg dynamics in particular both the experimental and simulated eeg time series exhibit short time scaling which is broken by a strong periodic component such as alpha waves the saturation of eeg diffusion entropy precludes the existence of asymptotic scaling we find that the crossover between two scaling regions seen in detrended fluctuation analysis dfa of eeg increments does not originate from the underlying dynamics but is merely an artifact of the algorithm this artifact is rooted in the failure of the trend plus signal paradigm of dfa

monte carlo methods of inference for implicit statistical models a prescribed statistical model is a parametric specification of the distribution of a random vector whilst an implicit statistical model is one defined at a more fundamental level in terms of a generating stochastic mechanism this paper develops methods of inference which can be used for implicit statistical models whose distribution theory is intractable the kernel method of probability density estimation is advocated for estimating a log likelihood from simulations of such a model the development and testing of an algorithm for maximizing this estimated log likelihood function is described an illustrative example involving a stochastic model for quantal response assays is given possible applications of the maximization algorithm to ad hoc methods of parameter estimation are noted briefly and illustrated by an example involving a model for the spatial pattern of displaced amacrine cells in the retina of a rabbit

markov processes hurst exponents and nonlinear diffusion equations with application to finance abstract we show by explicit closed form calculations that a hurst exponent h 1 2 does not necessarily imply long time correlations like those found in fractional brownian motion fbm we construct a large set of scaling solutions of fokker planck partial differential equations pdes where h 1 2 thus markov processes which by construction have no long time correlations can have h 1 2 if a markov process scales with hurst exponent h 1 2 then it simply means that the process has nonstationary increments for the scaling solutions we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient d x t is specified as an example we generate a class of student t like densities from the class of quadratic diffusion coefficients notably the tsallis density is one member of that large class the tsallis density is usually thought to result from a nonlinear diffusion equation but instead we explicitly show that it follows from a markov process generated by a linear fokker planck equation and therefore from a corresponding langevin equation having a tsallis density with h 1 2 therefore does not imply dynamics with correlated signals e g like those of fbm a short review of the requirements for fbm is given for clarity and we explain why the usual simple argument that h 1 2 implies correlations fails for markov processes with scaling solutions finally we discuss the question of scaling of the full green function g x t x t of the fokker planck pdes

subordinated α stable ornstein uhlenbeck process as a tool for financial data description abstract the classical financial models are based on the standard brownian diffusion type processes however in the exhibition of some real market data like interest or exchange rates we observe characteristic periods of constant values moreover in the case of financial data the assumption of normality is often unsatisfied in such cases the popular vasicek model that is a mathematical system describing the evolution of interest rates based on the ornstein uhlenbeck process seems not to be applicable therefore we propose an alternative approach based on a combination of the popular ornstein uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior the probability density function of the proposed process can be described by a fokker planck type equation and therefore it can be examined as an extension of the basic ornstein uhlenbeck model in this paper we propose the parameters estimation method and calibrate the subordinated vasicek model to the interest rate data

the simultaneous approximation of polynomial roots abstract we give new proofs of some known results concerning an iterative method of newton type for the simultaneous calculation of all roots of a polynomial originally proposed by weierstrass 1 the previously known local convergence analysis of the method is simplified and sharpened we also propose a modified method with improved convergence properties global convergence of both methods is touched upon briefly

generalized ornstein uhlenbeck process by doob s theorem and the time evolution of financial prices abstract we generalize the ornstein uhlenbeck ou process using doob s theorem we relax the gaussian and stationary conditions assuming a linear and time homogeneous process the proposed generalization retains much of the simplicity of the original stochastic process while exhibiting a somewhat richer behavior analytical results are obtained using transition probability and the characteristic function formalism and compared with empirical stock market data which are notorious for the non gaussian behavior the analysis focus on the decay patterns and the convergence study of the first four cumulants considering the logarithmic returns of stock prices it is shown that the proposed model offers a good improvement over the classical ou model

assessment of long range correlation in time series how to avoid pitfalls due to the ubiquity of time series with long range correlation in many areas of science and engineering analysis and modeling of such data is an important problem while the field seems to be mature three major issues have not been satisfactorily resolved i many methods have been proposed to assess long range correlation in time series under what circumstances do they yield consistent results ii the mathematical theory of long range correlation concerns the behavior of the correlation of the time series for very large times a measured time series is finite however how can we relate the fractal scaling break at a specific time scale to important parameters of the data iii an important technique in assessing long range correlation in a time series is to construct a random walk process from the data under the assumption that the data are like a stationary noise process due to the difficulty in determining whether a time series is stationary or not however one cannot be 100 sure whether the data should be treated as a noise or a random walk process is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process and vice versa in this paper we seek to gain important insights into these issues by examining three model systems the autoregressive process of order 1 on off intermittency and l evy motions and considering an important engineering problem target detection within sea clutter radar returns we also provide a few rules of thumb to safeguard against misinterpretations of long range correlation in a time series and discuss relevance of this study to pattern recognition

quantifying encoding redundancy induced by rate correlations in poisson neurons temporal correlations in neuronal spike trains are known to introduce redundancy to stimulus encoding however exact methods to describe how these correlations impact neural information transmission quantitatively are lacking here we provide a general measure for the information carried by correlated rate modulations only neglecting other spike correlations and use it to investigate the effect of rate correlations on encoding redundancy we derive it analytically by calculating the mutual information between a time correlated rate modulating signal and the resulting spikes of poisson neurons whereas this information is determined by spike autocorrelations only the redundancy in information encoding due to rate correlations depends on both the distribution and the autocorrelation of the rate histogram we further demonstrate that at very small signal strengths the information carried by rate correlated spikes becomes identical to that of independent spikes in effect measuring the signal modulation depth in contrast a vanishing signal correlation time maximizes information but does not generally yield the information of independent spikes overall our study sheds light on the role of signal induced temporal correlations for neural coding by providing insight into how signal features shape redundancy and by establishing mathematical links between existing methods

using discharge data to reduce structural deficits in a hydrological model with a bayesian inference approach and the implications for the prediction of critical source areas 1 a distributed hydrological model was used to simulate the distribution of fast runoff formation as a proxy for critical source areas for herbicide pollution in a small agricultural catchment in switzerland we tested to what degree predictions based on prior knowledge without local measurements could be improved upon relying on observed discharge this learning process consisted of five steps for the prior prediction step 1 knowledge of the model parameters was coarse and predictions were fairly uncertain in the second step discharge data were used to update the prior parameter distribution effects of uncertainty in input data and model structure were accounted for by an autoregressive error model this step decreased the width of the marginal distributions of parameters describing the lower boundary percolation rates but hardly affected soil hydraulic parameters residual analysis step 3 revealed model structure deficits we modified the model and in the subsequent bayesian updating step 4 the widths of the posterior marginal distributions were reduced for most parameters compared to those of the prior this incremental procedure led to a strong reduction in the uncertainty of the spatial prediction thus despite only using spatially integrated data discharge the spatially distributed effect of the improved model structure can be expected to improve the spatially distributed predictions also the fifth step consisted of a test with independent spatial data on herbicide losses and revealed ambiguous results the comparison depended critically on the ratio of event to preevent water that was discharged this ratio cannot be estimated from hydrological data only the results demonstrate that the value of local data is strongly dependent on a correct model structure an iterative procedure of bayesian updating model testing and model modification is suggested

hurst exponents markov processes and fractional brownian motion there is much confusion in the literature over hurst exponents recently we took a step in the direction of eliminating some of the confusion one purpose of this

a formal likelihood function for parameter and predictive inference of hydrologic models with correlated heteroscedastic and non gaussian errors estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions residual errors are often assumed to be independent and to be adequately described by a gaussian probability distribution with a mean of zero and a constant variance here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed a formal generalized likelihood function is presented which extends the applicability of previously used likelihood functions to situations where residual errors are correlated heteroscedastic and non gaussian with varying degrees of kurtosis and skewness the approach focuses on a correct statistical description of the data and the total model residuals without separating out various error sources application to bayesian uncertainty analysis of a conceptual rainfall runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors when applied to daily rainfall runoff data from a humid basin we find that 1 residual errors are much better described by a heteroscedastic first order auto correlated error model with a laplacian distribution function characterized by heavier tails than a gaussian distribution and 2 compared to a standard least squares approach proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference application to daily rainfall runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that 1 multiplicative bias factors can be used to compensate for some of the largest errors and 2 a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near zero flows we conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well

exact numerical simulation of power law noises many simulations of stochastic processes require colored noises i describe here an exact numerical method to simulate power law noises the method can be extended to more general colored noises and is exact for all time steps even when they are unevenly spaced as may often happen for astronomical data see e g n r lomb astrophys space sci 39 447 1976 the algorithm has a well behaved computational complexity it produces a nearly perfect gaussian noise and its computational efficiency depends on the required degree of noise gaussianity

evaluation of the path integral for flow through random porous media we present a path integral formulation of darcy s equation in one dimension with random permeability described by a correlated multivariate lognormal distribution this path integral is evaluated with the markov chain monte carlo method to obtain pressure distributions which are shown to agree with the solutions of the corresponding stochastic differential equation for dirichlet and neumann boundary conditions the extension of our approach to flow through random media in two and three dimensions is discussed

a fast algorithm for the first passage times of gauss markov processes with holder continuous boundaries even for simple diffusion processes treating first passage problems analytically proves intractable for generic barriers and existing numerical methods are inaccurate and computationally costly here we present a novel numerical method that is faster and has more tightly controlled accuracy our algorithm is a probabilistic variant of dichotomic search for the computation of first passage times through non negative homogeneously holder continuous boundaries by gauss markov processes these include the ornstein uhlenbeck process underlying the ubiquitous leaky integrate and fire model of neuronal excitation our method evaluates discrete points in a sample path exactly and refines this representation recursively only in regions where a passage is rigorously estimated to be probable e g when close to the boundary

markov analysis of stochastic resonance in a periodically driven integrate and fire neuron we model the dynamics of the leaky integrate and fire neuron under periodic stimulation as a markov process with respect to the stimulus phase this avoids the unrealistic assumption of a stimulus reset after each spike made in earlier papers and thus solves the long standing reset problem the neuron exhibits stochastic resonance both with respect to input noise intensity and stimulus frequency the latter resonance arises by matching the stimulus frequency to the refractory time of the neuron the markov approach can be generalized to other periodically driven stochastic processes containing a reset mechanism