an empirical non parametric likelihood family of data based benford like distributions abstract a mathematical expression known as benford s law provides an example of an unexpected relationship among randomly selected sequences of first significant digits fsds newcomb note on the frequency of use of the different digits in natural numbers am j math 4 1881 39 40 and later benford the law of anomalous numbers proc am philos soc 78 4 1938 551 572 conjectured that fsds would exhibit a weakly monotonic decreasing distribution and proposed a frequency proportional to the logarithmic rule unfortunately the benford fsd function does not hold for a wide range of scale invariant multiplicative data to confront this problem we use information theoretic methods to develop a data based family of alternative benford like exponential distributions that provide null hypotheses for testing purposes two data sets are used to illustrate the performance of generalized benford like distributions

decoding least effort and scaling in signal frequency distributions abstract here assuming a general communication model where objects map to signals a power function for the distribution of signal frequencies is derived the model relies on the satisfaction of the receiver hearer communicative needs when the entropy of the number of objects per signal is maximized evidence of power distributions in a linguistic context some of them with exponents clearly different from the typical 2 of zipf s law is reviewed and expanded we support the view that zipf s law reflects some sort of optimization but following a novel realistic approach where signals e g words are used according to the objects e g meanings they are linked to our results strongly suggest that many systems in nature use non trivial strategies for easing the interpretation of a signal interestingly constraining just the number of interpretations of signals does not lead to scaling

an approach to time series analysis and arma spectral estimation this paper presents an approach to time series analysis and arma spectral estimation from only the output data corrupted by noise it is shown that the generalized not well known modified yule walker myw equations hold when the residual is some correlated noise to solve such equations a new version of the generalized least squares gls method is proposed yielding ar parameter estimates with higher accuracy this gls method can also be used to enhance the estimation accuracy of ar parameters for short and noisy data in case of the myw equation holding theoretically furthermore a simple procedure for improving ma parameter estimates is studied our approach as cadzow s singular value decomposition svd method has provided significantly higher performance spectral estimates for a low order arma model than those obtained via usual techniques which are based upon direct solution of the myw equations and which use a high order model without considering an additive noise

structural properties of dendrite groups let g be the homeomorphism group of a dendrite we study the normal subgroups of g for instance there are uncountably many non isomorphic such groups g that are simple groups moreover these groups can be chosen so that any isometric g action on any metric space has a bounded orbit in particular they have the fixed point property fh

stochastic model simplification mathematical models defined by a structure and a set of parameter variation or uncertainty may be simplified both by structure reduction and parameter set reduction first the approximation of high order and time varying linear gaussian models by low order and time invariant ones is considered the proposed approach is based on maximizing the probabilistic ambiguity between the actual model and the approximate one and is applicable to general stochastic linear models reducing a model set defined on a set of parameter variation or uncertainty to a single fixed parameter model or a finite model group is then considered the set reduction criteria give rise to a min max optimization problem and a min max min problem which is converted to a constrained min max problem the algorithmic solution of the optimization problems is considered in detail along with several approximation and discretization schemes the application and the validity of the proposed approach are examined in view of traditional design considerations by solving numerical examples for several structure and set reduction problems

extreme amenability of abelian l0 groups we show that for any abelian topological group g and arbitrary diffused submeasure every continuous action of l0 g on a compact space has a fixed point this generalizes earlier results of herer and christensen glasner furstenberg and weiss and farah and solecki this also answers a question posed by farah and solecki in particular it implies that if h is of the form l0 r then h is extremely amenable if and only if h has no nontrivial characters which gives an evidence for an affirmative answer to a question of pestov the proof is based on estimates of chromatic numbers of certain graphs on z n it uses tools from algebraic topology and builds on the work of farah and solecki

stochastic aspects of one dimensional discrete dynamical systems benford s law benford s law owes its discovery to the grubby pages hypothesis a 19th century observation made by simon newcomb that the beginning pages of logarithm books were grubbier than the last few pages implying that scientists referenced the values toward the front of the books more frequently if a data set satisfies benford s law then it s significant digits will have a logarithmic distribution which favors smaller significant digits in this article we demonstrate two ways of creating discrete one dimensional dynamical systems that satisfy benford s law we also develop a numerical simulation methodology that we use to study dynamical systems when analytical results are not readily available

brief papers linear identification of arma processes a new method for estimating the parameters of an arma process is presented the method consists of three linear least squares estimations in the first an autoregressive model is fitted to the observation sequence yielding an estimate of the values of the driving white noise sequence linear least squares is then used to fit an arma model to the observation and estimated white noise sequences this model is used to filter the observation and estimated white noise sequences finally an arma model is fitted to the filtered sequences it is shown that the resultant estimator is p consistent the asymptotic bias tends to zero as the degree p of the autoregressive model tends to infinity and is p efficient the asymptotic efficiency approaches the theoretical maximum as p tends to infinity

understanding scaling through history dependent processes with collapsing sample space history dependent processes are ubiquitous in natural and social systems many such stochastic processes especially those that are associated with complex systems become more constrained as they unfold meaning that their sample space or their set of possible outcomes reduces as they age we demonstrate that these sample space reducing ssr processes necessarily lead to zipf s law in the rank distributions of their outcomes we show that by adding noise to ssr processes the corresponding rank distributions remain exact power laws p x x where the exponent directly corresponds to the mixing ratio of the ssr process and noise this allows us to give a precise meaning to the scaling exponent in terms of the degree to which a given process reduces its sample space as it unfolds noisy ssr processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from 2 to we discuss several applications showing how ssr processes can be used to understand zipf s law in word frequencies and how they are related to diffusion processes in directed networks or aging processes such as in fragmentation processes ssr processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative preferential or self organized critical processes

brief estimation of an n l n hammerstein wiener model estimation of a single input single output block oriented model is studied the model consists of a linear block embedded between two static nonlinear gains hence it is called n l n hammerstein wiener model first the model structure is motivated and the disturbance model is discussed the paper then concentrates on parameter estimation a relaxation iteration scheme is proposed by making use of a model structure in which the error is bilinear in parameters this leads to a simple algorithm which minimizes the original loss function the convergence and consistency of the algorithm are studied in order to reduce the variance error the obtained linear model is further reduced using frequency weighted model reduction simulation study will be used to illustrate the method

extreme amenability of l0 a ramsey theorem and levy groups we show that l0 h is extremely amenable for any diffused submeasure and any solvable compact group h this extends results of herer christensen and of glasner and furstenberg weiss proofs of these earlier results used spectral theory or concentration of measure our argument is based on a new ramsey theorem proved using ideas coming from combinatorial applications of algebraic topological methods using this work we give an example of a group which is extremely amenable and contains an increasing sequence of compact subgroups with dense union but which does not contain a levy sequence of compact subgroups with dense union this answers a question of pestov we also show that many levy groups have non levy sequences answering another question of pestov

arx modeling of unstable linear systems high order arx models can be used to approximate a quite general class of linear systems in a parametric model structure and well established methods can then be used to retrieve the true plant and noise models from the arx polynomials however this commonly used approach is only valid when the plant is stable or if the unstable poles are shared with the true noise model in this contribution we generalize this approach to allow the unstable poles not to be shared by introducing modifications to correctly retrieve the noise model and noise variance

on identification of stable systems and optimal approximation abstract approximative modelling of stable continuous time possibly infinite dimensional systems is studied based on an optimal approximation approach both approximation of analytical system representations system approximation as well as approximation of input output data based system estimates system identification are considered while special emphasis is given to approximative modelling in the h and hankel norms the l1 and l2 norm cases are also discussed the model sets considered here are finite dimensional systems and time shifted systems simple delay systems the theory of approximation numbers is shown to provide a convenient tool to study problems of identification of stable continuous time systems in a deterministic framework with close connections to complexity considerations laguerre fourier series methods and hankel operator techniques can be utilized to develop fully practical identification methods for continuous time possibly infinite dimensional systems

explaining the uneven distribution of numbers in nature the laws of benford and zipf abstract the distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of benford s law we analyze in detail this property for different data sets and give a general explanation for the origin of the benford s law in terms of multiplicative processes we show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity finally we derive a relation between the generalized benford s law and the popular zipf s law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

the first digit frequencies in data of turbulent flows abstract considering the first significant digits noted d in data sets of dissipation for turbulent flows the probability to find a given number d 1 or 2 or 9 would be 1 9 for a uniform distribution instead the probability closely follows newcomb benford s law namely p d log 1 1 d the discrepancies between newcomb benford s law and first digits frequencies in turbulent data are analysed through shannon s entropy the data sets are obtained with direct numerical simulations for two types of fluid flow an isotropic case initialized with a taylor green vortex and a channel flow results are in agreement with newcomb benford s law in nearly homogeneous cases and the discrepancies are related to intermittent events thus the scale invariance for the first significant digits which supports newcomb benford s law seems to be related to an equilibrium turbulent state namely with a significant inertial range a matlab octave program provided in appendix is such that part of the presented results can easily be replicated

benford s law a poisson perspective abstract benford s law is a counterintuitive statistical law asserting that the distribution of leading digits taken from a large ensemble of positive numerical values that range over many orders of scale is logarithmic rather than uniform as intuition suggests in this paper we explore benford s law from a poisson perspective considering ensembles of positive numerical values governed by poisson process statistics we show that this poisson setting naturally accommodates benford s law and i establish a poisson characterization and a poisson multidigit extension of benford s law ii study a system invariant leading digit distribution which generalizes benford s law and establish a poisson characterization and a poisson multidigit extension of this distribution iii explore the universal emergence of the system invariant leading digit distribution couple this universal emergence to the universal emergence of the weibull and frechet extreme value distributions and distinguish the special role of benford s law in this universal emergence iv study the continued fractions counterpart of the system invariant leading digit distribution and establish a poisson characterization of this distribution and v unveil the elemental connection between the system invariant leading digit distribution and its continued fractions counterpart this paper presents a panoramic poisson approach to benford s law to its system invariant generalization and to its continued fractions counterpart

benford s law and first letter of words a universal first letter law fll is derived and described it predicts the percentages of first letters for words in novels the fll is akin to benford s law bl of first digits which predicts

order reduction of siso nonminimum phase stochastic systems we consider the order reduction problem for single input single output nonminimum phase linear stochastic systems the problem is formulated as that of approximating the original system output process by another stochastic process generated via a reduced order model it is shown that in order to obtain a correct phase approximation it is necessary to consider some higher order statistics of the original process in addition to the usual second order statistics a computationally feasible approximation criterion is proposed and analyzed an example is also presented to illustrate the proposed method

an optimal instrumental variable method for arma spectral estimation a multistep iterative and fast recursive algorithm for autoregressive moving average arma spectral estimation is presented the ar parameters of an arma process are estimated using the extended instrumental variable eiv method the optimal choice of instruments prefilter and weighting matrix is investigated a bootstrapping procedure that has computational convenience is proposed for the algorithm the statistical analysis and experiments show that the optimal iv estimate is unbiased consistent efficient asymptotically normal and equivalent to the maximum likelihood ml estimate and the prediction error pe estimate and the proposed algorithm has the advantages of sharper resolution less frequency bias and better efficiency of convergence

on free actions minimal flows and a problem by ellis we exhibit natural classes of polish topological groups g such that every continuous action of g on a compact space has a fixed point and observe that every group with this property provides a solution in the negative to a 1969 problem by robert ellis as the ellis semigroup e u of the universal minimal g flow u being trivial is not isomorphic with the greatest g ambit further refining ouir construction we obtain a polish topological group g acting freely on the universal minimal flow u yet such that s g and e u are not isomorphic we also display polish topological groups acting effectively but not freely orn their universal minimal flows in fact we can produce examples of groups of all three types having any prescribed infinite weight our examples lead to dynamical conclusions for some groups of importance in analysis for instance both the full group of permutations s x of an infinite set equipped with the pointwise topology and the unitary group u x of an infinite dimensional hilbert space with the strong operator topology admit no free action on a compact space and the circle 1 forms the universal minimal flow for the topological group homeo 1 of orientation preservinig haomeomorphisms it also follows that a closed subgroup of an amenable topological group need not be amenable

the significant digit law in statistical physics abstract the occurrence of the nonzero leftmost digit i e 1 2 9 of numbers from many real world sources is not uniformly distributed as one might naively expect but instead the nature favors smaller ones according to a logarithmic distribution named benford s law we investigate three kinds of widely used physical statistics i e the boltzmann gibbs bg distribution the fermi dirac fd distribution and the bose einstein be distribution and find that the bg and fd distributions both fluctuate slightly in a periodic manner around benford s distribution with respect to the temperature of the system while the be distribution conforms to it exactly whatever the temperature is thus benford s law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature furthermore various elegant properties of benford s law especially the mantissa distribution of data sets are discussed

scale invariance of human electroencephalogram signals in sleep in this paper we investigate the dynamical properties of electroencephalogram eeg signals of humans in sleep by using a modified random walk method we demonstrate that scale invariance is embedded in eeg signals after a detrending procedure is applied furthermore we study the dynamical evolution of the probability density function pdf of the detrended eeg signals by nonextensive statistical modeling it displays a scale independent property which is markedly different from the usual scale dependent pdf evolution and cannot be described by the fokker planck equation

l2 model reduction and variance reduction in this contribution we examine certain variance properties of model reduction the focus is on l 2 model reduction but some general results are also presented these general results can be used to analyze various other model reduction schemes the models we study are finite impulse response fir and output error oe models we compare the variance of two estimated models the first one is estimated directly from data and the other one is computed by reducing a high order model by l 2 model reduction in the fir case we show that it is never better to estimate the model directly from data compared to estimating it via l 2 model reduction of a high order fir model for oe models we show that the reduced model has the same variance as the directly estimated one if the reduced model class used contains the true system

fixed point properties for semigroups of nonlinear mappings on unbounded sets abstract a well known result of w ray asserts that if c is an unbounded convex subset of a hilbert space then there is a nonexpansive mapping t c c that has no fixed point in this paper we establish some common fixed point properties for a semitopological semigroup s of nonexpansive mappings acting on a closed convex subset c of a hilbert space assuming that there is a point c c with a bounded orbit and assuming that certain subspace of c b s has a left invariant mean left invariant mean or amenability is an important notion in harmonic analysis of semigroups and groups introduced by von neumann in 1929 28 and formalized by day in 1957 5 in our investigation we use the notion of common attractive points introduced recently by s atsushiba and w takahashi

brief variance analysis of l2 model reduction when undermodeling the output error case in this contribution variance properties of l 2 model reduction are studied that is given an estimated model of high order we study the resulting variance of an l 2 reduced approximation the main result of the paper is showing that estimating a low order output error oe model via l 2 model reduction of a high order model gives a smaller variance compared to estimating a low order model directly from data in case of undermodeling this has previously been shown to hold for finite impulse response models but is in this paper extended to general linear oe models

benford s law and the fsd distribution of economic behavioral micro data abstract in this paper we focus on the first significant digit fsd distribution of european micro income data and use information theoretic entropy based methods to investigate the degree to which benford s fsd law is consistent with the nature of these economic behavioral systems we demonstrate that benford s law is not an empirical phenomenon that occurs only in important distributions in physical statistics but that it also arises in self organizing dynamic economic behavioral systems the empirical likelihood member of the minimum divergence entropy family is used to recover country based income fsd probability density functions and to demonstrate the implications of using a benford prior reference distribution in economic behavioral system information recovery

arma spectral estimation of narrow band processes via model reduction the problem of estimating autoregressive moving average arma models for narrowband processes is considered the following approach is proposed estimate a high order autoregressive ar approximation of the process by model reduction based on a truncated internally balanced realization or optimal hankel norm model reduction reduce the order of this high order ar estimate to find a lower order arma model this algorithm gives arma spectral estimates with excellent resolution properties without using iterative numerical minimization methods as for the maximum likelihood method how to take the narrowband assumption into account in the model reduction step is discussed in detail

incidence of q statistics in rank distributions we show that size rank distributions with power law decay often only over a limited extent observed in a vast number of instances in a widespread family of systems obey tsallis statistics the theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the lyapunov exponent is negligible or vanishes the relevant statistical mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints and the resulting duality of entropy indexes is seen to portray physically relevant information whereas the value of the index fixes the distribution s power law exponent that for the dual index 2 ensures the extensivity of the deformed entropy

probability distribution of intersymbol distances in random symbolic sequences applications to improving detection of keywords in texts and of amino acid clustering in proteins symbolic sequences have been extensively investigated in the past few years within the framework of statistical physics paradigmatic examples of such sequences are written texts and deoxyribonucleic acid dna and protein sequences in these examples the spatial distribution of a given symbol a word a dna motif an amino acid is a key property usually related to the symbol importance in the sequence the more uneven and far from random the symbol distribution the higher the relevance of the symbol to the sequence thus many techniques of analysis measure in some way the deviation of the symbol spatial distribution with respect to the random expectation the problem is then to know the spatial distribution corresponding to randomness which is typically considered to be either the geometric or the exponential distribution however these distributions are only valid for very large symbolic sequences and for many occurrences of the analyzed symbol here we obtain analytically the exact randomly expected spatial distribution valid for any sequence length and any symbol frequency and we study its main properties the knowledge of the distribution allows us to define a measure able to properly quantify the deviation from randomness of the symbol distribution especially for short sequences and low symbol frequency we apply the measure to the problem of keyword detection in written texts and to study amino acid clustering in protein sequences in texts we show how the results improve with respect to previous methods when short texts are analyzed in proteins which are typically short we show how the measure quantifies unambiguously the amino acid clustering and characterize its spatial distribution

benford s law first significant digit and distribution distances for testing the reliability of financial reports in developing countries abstract we discuss a common suspicion about reported financial data in 10 industrial sectors of the 6 so called main developing countries over the time interval 2000 2014 these data are examined through benford s law first significant digit and through distribution distances tests it is shown that several visually anomalous data have to be a priori removed thereafter the distributions much better follow the first digit significant law indicating the usefulness of a benford s law test from the research starting line the same holds true for distance tests a few outliers are pointed out

covid 19 flattening the curve and benford s law for many countries attempting to control the fast rising number of coronavirus cases and deaths the race is on to flatten the curve since the spread of coronavirus disease 2019 covid 19 has taken on pandemic proportions in the absence of significant control interventions the curve could be steep with the number of covid 19 cases growing exponentially in fact this level of proliferation may already be happening since the number of patients infected in italy closely follows an exponential trend thus we propose a test when the numbers are taken from an exponential distribution it has been demonstrated that they automatically follow benford s law bl as a result if the current control interventions are successful and we flatten the curve i e we slow the rate below an exponential growth rate then the number of infections or deaths will not obey bl for this reason bl may be useful for assessing the effects of the current control interventions and may be able to answer the question how flat is flat enough in this study we used an epidemic growth model in the presence of interventions to describe the potential for a flattened curve and then investigated whether the epidemic growth model followed bl for ten selected countries with a relatively high mortality rate among these countries south korea showed a particularly high degree of control intervention although all of the countries have aggressively fought the epidemic our analysis shows that all countries except for japan satisfied bl indicating the growth rates of covid 19 were close to an exponential trend based on the simulation table in this study bl test shows that the data from japan is incorrect

benford s law and complex atomic spectra we found that in transition arrays of complex atomic spectra the strengths of electric dipolar lines obey benford s law which means that their significant digits follow a logarithmic distribution favoring the smallest values this indicates that atomic processes result from the superposition of uncorrelated probability laws and that the occurrence of digits reflects the constraints induced by the selection rules furthermore benford law can be a useful test of theoretical spectroscopic models its applicability to the statistics of electric dipolar lines can be understood in the framework of random matrix theory and is consistent with the porter thomas law

reduced order models and controllers for continuous time stochastic systems an information theory approach reduced order models and controllers for continuous time stochastic systems are described the reduced order models are chosen to minimize the kullback leibler information distance klid between the outputs of the actual and reduced systems an lqg controller based on a reduced order system model is described a second reduced order controller is found to minimize the klid between the closed loop system outputs with the full and reduced order controllers

on amenability and groups of measurable maps abstract we show that if g is an amenable topological group then the topological group l 0 g of strongly measurable maps from 0 1 into g endowed with the topology of convergence in measure is whirly amenable hence extremely amenable conversely we prove that a topological group g is amenable if l 0 g is

an approximate predictor approach to reduced order models and controllers for distributed parameter systems two reduced order digital controllers for distributed parameter systems dps are described reduced order models approximate the optimal finite past predictor and error covariance for the full system to minimize an approximation to the kullback leibler information distance klid an lqg controller based on a reduced order system model is described a reduced order controller is found to minimize the klid between the closed loop system outputs with the full and reduced order controllers noncollocated control of a flexible beam is simulated

model reductions of high order estimated models the asymptotic ml approach abstract the reduction of order of high order models obtained from an identification experiment is discussed from a statistical point of view the asymptotic maximum likelihood ml approach is defined to reduce the order of an estimated model this approach considers the maximum likelihood criterion given the asymptotic statistics both in the number of data and the order of the estimated model and corresponds to frequency weighted l 2 norm model reduction by using the insight from the asymptotic ml approach an identification algorithm is proposed based on a high order arx estimate and model reduction via a frequency weighted balanced realization the advantage of this algorithm is that iterative minimization methods are not required to find the estimate

parametric identification using weighted null space fitting in identification of dynamical systems the prediction error method with a quadratic cost function provides asymptotically efficient estimates under gaussian noise but in general it requires solving a nonconvex optimization problem which may imply convergence to nonglobal minima an alternative class of methods uses a nonparametric model as intermediate step to obtain the model of interest weighted null space fitting wnsf belongs to this class starting with the estimate of a nonparametric arx model with least squares then the reduction to a parametric model is a multistep procedure where each step consists of the solution of a quadratic optimization problem which can be obtained with weighted least squares the method is suitable for both open and closed loop data and can be applied to many common parametric model structures including output error armax and box jenkins the price to pay is the increase of dimensionality in the nonparametric model which needs to tend to infinity as function of the sample size for certain asymptotic statistical properties to hold in this paper we conduct a rigorous analysis of these properties namely consistency and asymptotic efficiency also we perform a simulation study illustrating the performance of wnsf and identify scenarios where it can be particularly advantageous compared with state of the art methods

design variables for bias distribution in transfer function estimation estimation of transfer functions of linear systems is one of the most common system identification problems several different design variables chosen by the user for the identification procedure affect the properties of the resulting estimate in this paper it is investigated how the choices of prefilters noise models sampling interval and prediction horizon i e the use of k step ahead prediction methods influence the estimate an important aspect is thai the true system is not assumed to be exactly represented within the chosen model set the estimate will thus be biased it is shown how the distribution of bias in the frequency domain is governed by a weighting function which emphasizes different frequency bands the weighting function in turn is a result of the previously listed design variables it is shown e g thai the common least squares method has a tendency to emphasize high frequencies and that this can be counteracted by prefiltering it is also shown that asymptotically it is only the prediction horizon itself and not how it is split up into sampling interval times number of predicted sampling instants that affects this weighting function

the law of the leading digits and the world religions abstract benford s law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger ones we investigate here numerical data on the country wise adherent distribution of seven major world religions i e christianity islam buddhism hinduism sikhism judaism and baha ism to see if the proportion of the leading digits occurring in the distribution conforms to benford s law we find that the adherent data of all the religions except christianity excellently does conform to benford s law furthermore unlike the adherent data on christianity the significant digit distribution of the three major christian denominations i e catholicism protestantism and orthodoxy obeys the law thus in spite of their complexity general laws can be established for the evolution of religious groups

technical communique linear estimation of arma processes the method of estimating arma parameters first discussed by mayne and firoozan is investigated it is shown that if at the first stage the order of the fitted autoregression is allowed to depend on the number of time points in a reasonable manner then it will still be true that the final estimate of the parameter vector will converge almost surely to the true value this is to be compared to the result in the original paper where the order is fixed and it is shown that as the sample size increases the estimate converges to a value which if the order of the autoregression is high enough will be arbitrarily near to the true value some comments are made on other extensions on the law of the iterated logarithm on the central limit theorem and on the choice of the order of the fitted autoregression the innovation sequence need not be gaussian and for the convergence result only a natural condition relating to prediction needs be imposed

on the validation of the newcomb benford law and the weibull distribution in neuromuscular transmission abstract the neuromuscular junction represents a relevant substrate for revealing important biophysical mechanisms of synaptic transmission in this context calcium ions are important in the synapse machinery providing the nerve impulse transmission to the muscle fiber in this work we carefully investigated whether intervals of spontaneous electrical activity recorded in seven different calcium concentrations conform to the newcomb benford law our analysis revealed that electrical discharge of neuromuscular junction yields the expected values for newcomb benford law for first and second digits for different calcium concentrations on the other hand first two digits archived conformity especially for concentrations much above the physiological level we next examined previous theoretical studies establishing a relation between the law and lognormal and weibull distributions we showed that the weibull distribution is more appropriate to fit the intervals as compared to the lognormal distribution altogether the present findings strongly suggest that the electrical discharge dynamics is characterized by base invariance and scale invariance being the time intervals also well described by weibull statistics

the first digit frequencies of prime numbers and riemann zeta zeros prime numbers seem to be distributed among the natural numbers with no law other than that of chance however their global distribution presents a quite remarkable smoothness such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes in this paper we show that a generalization of the well known first digit benford s law which addresses the rate of appearance of a given leading digit d in datasets describes with astonishing precision the statistical distribution of leading digits in the prime number sequence moreover a reciprocal version of this pattern also takes place in the sequence of the non trivial riemann zeta zeros we prove that the prime number theorem is in the final analysis responsible for these patterns

random texts exhibit zipf s law like word frequency distribution it is shown that the distribution of word frequencies for randomly generated texts is very similar to zipf s law observed in natural languages such as english the facts that the frequency of occurrence of a word is almost an inverse power law function of its rank and the exponent of this inverse power law is very close to 1 are largely due to the transformation from the word s length to its rank which stretches an exponential function to a power law function

the dependence of frequency distributions on multiple meanings of words codes and signs the dependence of the frequency distributions due to multiple meanings of words in a text is investigated by deleting letters by coding the words with fewer letters the number of meanings per code

scale free networks with exponent one a majority of studied models for scale free networks have degree distributions with exponents greater than two real networks however can demonstrate essentially more heavy tailed degree distributions we explore two models of scale free equilibrium networks that have the degree distribution exponent ensuremath gamma 1 p q ensuremath sim q ensuremath ensuremath gamma such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high our models exploit a rewiring mechanism they are local in the sense that no knowledge of the network structure apart from the immediate neighborhood of the vertices is required these models generate uncorrelated networks in the infinite size limit where they are solved explicitly we investigate finite size effects by the use of simulations we find that both models exhibit disassortative degree degree correlations for finite network sizes in addition we observe a markedly degree dependent clustering in the finite networks we indicate a real world network with a similar degree distribution

revisiting the benford law when the benford like distribution of leading digits in sets of numerical data is expectable abstract the benford law states that the frequencies of decimal digits at the first place of numbers corresponding to various kinds of statistical or experimental data are not equal changing from 0 3 for 1 to 0 04 for 9 the corresponding frequencies distribution is described by the logarithmic function as is shown in the present article the benford distribution is a particular case of a more general mathematical statement namely if a function describing the dependence between two measurable quantities has a positive second derivative then the mentioned above frequencies decrease for digits from 1 to 9 the exact benford distribution is valid for the exponential function only explicit expressions for frequencies of leading digits are obtained and specified for the power logarithmic and tangent functions as examples the kinematic experiment was performed to illustrate the above results also the tabulated data on thermal conductivities of liquids confirm the proposed formula for frequencies distribution

paper a new algorithm for recursive estimation of parameters in controlled arma processes a recursive algorithm for estimating the constant but unknown parameters of a controlled arma process is presented the algorithm is a recursive version of an off line algorithm using three stages of standard least squares in the first stage the parameters of a controlled ar model of degree p are estimated the residuals used in this stage are employed in the second stage to estimate the parameters of a controlled arma process the first two stages constitute a recursive version of durbin s algorithm the model obtained in the second stage is used to filter the input output and residuals and these filtered variables are used in the final stage to obtain improved estimates of the controlled arma process it is shown that the estimate is globally p consistent i e that the estimate converges a s as the number of data tends to infinity to a vector which in turn converges to the true parameter vector as the degree p of the ar model tends to infinity

nonextensivity and self affinity in the mammalian neuromuscular junction we study time series and the spontaneous miniature end plate potentials mepps of mammals recorded at neuromuscular junctions using two different approaches generalized thermostatistics and detrended fluctuation analysis dfa classical concepts establish that the magnitude of these potentials is characterized by gaussian statistics and that their intervals are randomly displayed first we show that mepp distributions adequately satisfy the q gaussian distributions that maximize the tsallis entropy indicating their nonextensive and nonequilibrium behavior we then examine the intervals between the miniature potentials via dfa where the profile of the intervals between events configures a deviation from the expected random behavior some possible physiological substrates for these findings are discussed

recursive estimation of mixed autoregressive moving average order summary the order p q of an autoregressive moving average sequence y t may be estimated by minimizing a criterion log 2 p q log t t with respect to p and q where 2 is the maximum likelihood estimate of the variance of the innovations t it is suggested that instead a2 be estimated from a series of regressions of y t on y t 1 y t 2 y t p 5 t 1 t q where the t are obtained by fitting a long autoregression to the data it is shown how the sequence of regressions may for p q be economically recursively calculated by embedding them in a sequence of bivariate autoregressions asymptotic properties of the procedure are established under very general conditions

network properties of written human language we investigate the nature of written human language within the framework of complex network theory in particular we analyze the topology of orwell s 1984 focusing on the local properties of the network such as the properties of the nearest neighbors and the clustering coefficient we find a composite power law behavior for both the average nearest neighbor s degree and average clustering coefficient as a function of the vertex degree this implies the existence of different functional classes of vertices furthermore we find that the second order vertex correlations are an essential component of the network architecture to model our empirical results we extend a previously introduced model for language due to dorogovtsev and mendes we propose an accelerated growing network model that contains three growth mechanisms linear preferential attachment local preferential attachment and the random growth of a predetermined small finite subset of initial vertices we find that with these elementary stochastic rules we are able to produce a network showing syntacticlike structures

continuous time stochastic model simplification approximation of high order and time varying linear continuous time gaussian models by low order and time invariant models is considered the recent work of baram and be eri pertaining to the discrete time systems is extended to continuous time system models the model simplification is carried out by maximizing the probabilistic ambiguity between the actual system and the approximate model differences between the continuous time and the discrete time model simplification problems are examined the case when the observation noise covariance of the approximate model differs from that of the actual system presents technical difficulties not present in the discrete time version

randomness versus specifics for word frequency distributions the text length dependence of real word frequency distributions can be connected to the general properties of a random book it is pointed out that this finding has strong implications when decidi

size dependent word frequencies and translational invariance of books it is shown that a real novel shares many characteristic features with a null model in which the words are randomly distributed throughout the text such a common feature is a certain translational

on the nature of benford s law we study multiplicative and a ne sequences of real numbers de1ned by n j 1 j n j j where j and j are sequences of positive real numbers in the multiplicative case j 0 for all j we investigate the conditions under which the leading digits k of n j have the following probability distribution known as benford s law p k log10 k 1 k we present two main results first we show that contrary to the usual assumption in the literature j does not necessarily need to come from a chaotic or independent random process for benford s law to hold the multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing second we give conditions under which the distribution of the 1rst digits of an a ne process displays benford s law our proofs use techniques from ergodic theory c 2002 elsevier science b v all rights reserved

beyond the zipf mandelbrot law in quantitative linguistics abstract in this paper the zipf mandelbrot law is revisited in the context of linguistics despite its widespread popularity the zipf mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of words contained in some given corpus in particular we focus our attention on the important deviations that become statistically relevant as larger corpora are considered and that ultimately could be understood as salient features of the underlying complex process of language generation finally it is shown that all the different observed regimes can be accurately encompassed within a single mathematical framework recently introduced by c tsallis

first digit law in nonextensive statistics nonextensive statistics characterized by a nonextensive parameter q is a promising and practically useful generalization of the boltzmann statistics to describe power law behaviors from physical and social observations we here explore the unevenness of the first digit distribution of nonextensive statistics analytically and numerically we find that the first digit distribution follows benford s law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature the fluctuation decreases when q increases and the result converges to benford s law exactly as q approaches 2 the relevant regularities between nonextensive statistics and benford s law are also presented and discussed

a fast estimation method for arma processes an improvement of a batch algorithm for parameter estimation of an arma process is presented a fast and iterative algorithm is presented and provides estimates with the same asymptotic distribution as the maximum likelihood estimates this algorithm is a modified version of the method suggested by mayne and firoozan 1982 linear identification of arma processes automatica 18 461 466 and requires one less regression than the previous method if only one pass is executed

asymptotic properties of black box identification of transfer functions the problem of estimating the transfer function of a linear stochastic system is considered the transfer function is parametrized as a black box and no given order is chosen a priori this means that the model orders may increase to infinity when the number of observed data tends to infinity the consistency and convergence properties of the resulting transfer function estimates are investigated asymptotic expressions for the variances and distributions of these estimates are also derived for the case that the model orders increase it is shown that the variance of the transfer function estimate at a certain frequency is asymptotically given by the noise to signal ratio at that frequency mulliplied by the model order to number of data points ratio