a discrete schrodinger equation via optimal transport on graphs abstract in 1966 edward nelson presented an interesting derivation of the schrodinger equation using brownian motion recently this derivation is linked to the theory of optimal transport which shows that the schrodinger equation is a hamiltonian system on the probability density manifold equipped with the wasserstein metric in this paper we consider similar matters on a finite graph by using discrete optimal transport and its corresponding nelson s approach we derive a discrete schrodinger equation on a finite graph the proposed system is quite different from the commonly referred discretized schrodinger equations it is a system of nonlinear ordinary differential equations odes with many desirable properties several numerical examples are presented to illustrate the properties

applying the method of surrogate data to cyclic time series the surrogate data methodology is used to test a given time series for membership of specific classes of dynamical systems currently there are three algorithms that are widely applied in the literature the most general of these tests the hypothesis of nonlinearly scaled linearly filtered noise however these tests and the many extensions of them that have been suggested are inappropriate for data exhibiting strong cyclic components for such data it is more natural to ask if there exist any long term of period longer than the data cycle length determinism in this paper we discuss existing techniques that attempt to address this hypothesis and introduce a new approach this new approach generates surrogates that are constrained i e they look like the data and for cyclic time series tests the null hypothesis of a periodic orbit with uncorrelated noise we examine various alternative implementations of this algorithm applying it to a variety of known test systems and experimental time series with unknown dynamics 2002 elsevier science b v all rights reserved

gradual multifractal reconstruction of time series formulation of the method and an application to the coupling between stock market indices and their holder exponents abstract a technique termed gradual multifractal reconstruction gmr is formulated a continuum is defined from a signal that preserves the pointwise holder exponent multifractal structure of a signal but randomises the locations of the original data values with respect to this 0 to the original signal itself 1 we demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual tree complex wavelet transform that is the 0 end of the continuum is realised using the recently proposed iterated amplitude adjusted wavelet transform algorithm keylock 2017 that fully randomises the wavelet phases this is extended to the gmr formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion an econophysics application of the technique is presented the relation between the normalised log returns and their holder exponents for the daily returns of eight financial indices are compared one particularly noticeable result is the change for the two american indices nasdaq 100 and s p 500 from a non significant to a strongly significant as determined using gmr cross correlation between the returns and their holder exponents from before the 2008 crash to afterwards this is also reflected in the skewness of the phase difference distributions which exhibit a geographical structure with asian markets not exhibiting significant skewness in contrast to those from elsewhere globally

estimating dimension from small samples abstract details of a new algorithm to estimate the dimension of a measure from point samples are described the new algorithm emphasizes that dimension need not be the same at all scales that is there need not be a scaling region the algorithm is verified with gaussian measures and is found to be reliable and have less demanding data requirements than conventional estimators using certain convergence properties of gaussian measures it is shown that good estimates can be made from very small samples

the density manifold and configuration space quantization on developpe la structure geometrique differentielle d une variete de frechet de densites ceci fournit un cadre geometrique a la quantification reliee a la mecanique stochastique de nelson on etudie les structures riemanniennes et symplectiques de la variete densite et on deduit l equation de schrodinger a partir d un principe variationnel

test your surrogate data before you test for nonlinearity the schemes for the generation of surrogate data in order to test the null hypothesis of linear stochastic process undergoing nonlinear static transform are investigated as to their consistency in representing the null hypothesis in particular we pinpoint some important caveats of the prominent algorithm of amplitude adjusted fourier transform surrogates aaft and compare it to the iterated aaft which is more consistent in representing the null hypothesis it turns out that in many applications with real data the inferences of nonlinearity after marginal rejection of the null hypothesis were premature and have to be reinvestigated taking into account the inaccuracies in the aaft algorithm mainly concerning the mismatching of the linear correlations in order to deal with such inaccuracies we propose the use of linear together with nonlinear polynomials as discriminating statistics the application of this setup to some well known real data sets cautions against the use of the aaft algorithm

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

constrained realization monte carlo method for hypothesis testing abstract we compare two theoretically distinct approaches to generate artificial or surrogate data for testing hypotheses about a given data set the first and more straightforward approach is to fit a single best model to the original data and then to generate surrogate data sets that are typical realizations of that model the second approach concentrates not on the model but directly on the original data it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics examples of these two approaches are provided for two simple cases a test for deviations from a gaussian distribution and a test for serial dependence in a time series additionally we consider test for nonlinearity in time series based on a fourier transform ft method and on more conventional autoregressive moving average arma fits to the data the comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal a statistic is pivotal if its distribution is the same for all processes consistent with the null hypothesis the typical realization method requires that the discriminating statistic satisfy this property the constrained realization approach on the other hand does not share this requirement and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic

a wavelet based method for surrogate data generation hypothesis testing based on surrogate data has emerged as a popular way to test the null hypothesis that a signal is a realisation of a linear gaussian stochastic process if these surrogates are constrained to the values and power spectrum of the original data there is no need to formulate a pivotal test statistic in this paper a method is presented for generating constrained surrogates using a wavelet transform introducing a threshold above which wavelet detail coefficients are pinned to their original values such surrogates avoid problems of nonstationarity for pseudo periodic data and appear to be more robust than conventional approaches for situations where period modulation is corrupting a gaussian stochastic process when used for generating ensemble realisations of a process the approach used here avoids some of the difficulties of methods based on simple randomisation of wavelet coefficients

testing stationarity with surrogates a time frequency approach an operational framework is developed for testing stationarity relatively to an observation scale in both stochastic and deterministic contexts the proposed method is based on a comparison between global and local time frequency features the originality is to make use of a family of stationary surrogates for defining the null hypothesis of stationarity and to base on them two different statistical tests the first one makes use of suitably chosen distances between local and global spectra whereas the second one is implemented as a one class classifier the time frequency features extracted from the surrogates being interpreted as a learning set for stationarity the principle of the method and of its two variations is presented and some results are shown on typical models of signals that can be thought of as stationary or nonstationary depending on the observation scale used

constrained surrogate time series with preservation of the mean and variance structure a method is presented for generating surrogates that are constrained realizations of a time series but which preserve the local mean and variance of the original signal the method is based on the popular iterated amplitude adjusted fourier transform method but makes use of a wavelet transform to constrain behavior in the time domain using this method it is possible to test for local changes in the nonlinear properties of the signal we present an example for a change in hurst exponent in a time series produced by fractional brownian motion

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

some geometric calculations on wasserstein space we compute the riemannian connection and curvature for the wasserstein space of a smooth compact riemannian manifold

adaptive linear step up procedures that control the false discovery rate we provide a new two stage procedure in which the linear step up procedure is used in stage one to estimate mo providing a new level q which is used in the linear step up procedure in the second stage we prove that a general form of the two stage procedure controls the false discovery rate at the desired level q this framework enables us to study analytically the properties of other procedures that exist in the literature a simulation study is presented that shows that two stage adaptive procedures improve in power over the original procedure mainly because they provide tighter control of the false discovery rate we further study the performance of the current suggestions some variations of the procedures and previous suggestions in the case where the test statistics are positively dependent a case for which the original procedure controls the false discovery rate in the setting studied here the newly proposed two stage procedure is the only one that controls the false discovery rate the procedures are illustrated with two examples of biological importance

improvements to surrogate data methods for nonstationary time series the method of surrogate data has been extensively applied to hypothesis testing of system linearity when only one realization of the system a time series is known normally surrogate data should preserve the linear stochastic structure and the amplitude distribution of the original series classical surrogate data methods such as random permutation amplitude adjusted fourier transform or iterative amplitude adjusted fourier transform are successful at preserving one or both of these features in stationary cases however they always produce stationary surrogates hence existing nonstationarity could be interpreted as dynamic nonlinearity certain modifications have been proposed that additionally preserve some nonstationarity at the expense of reproducing a great deal of nonlinearity however even those methods generally fail to preserve the trend i e global nonstationarity in the mean of the original series this is the case of time series with unit roots in their autoregressive structure additionally those methods based on fourier transform either need first and last values in the original series to match or they need to select a piece of the original series with matching ends these conditions are often inapplicable and the resulting surrogates are adversely affected by the well known artefact problem in this study we propose a simple technique that applied within existing fourier transform based methods generates surrogate data that jointly preserve the aforementioned characteristics of the original series including even strong trends moreover our technique avoids the negative effects of end mismatch several artificial and real stationary and nonstationary linear and nonlinear time series are examined in order to demonstrate the advantages of the methods corresponding surrogate data are produced with the classical and with the proposed methods and the results are compared

stochastic approximation in monte carlo computation the wang landau wl algorithm is an adaptive markov chain monte carlo algorithm used to calculate the spectral density for a physical system a remarkable feature of the wl algorithm is that it is not trapped by local energy minima which is very important for systems with rugged energy landscapes this feature has led to many successful applications of the algorithm in statistical physics and biophysics however there does not exist rigorous theory to support its convergence and the estimates produced by the algorithm can reach only a limited statistical accuracy in this article we propose the stochastic approximation monte carlo samc algorithm which overcomes the shortcomings of the wl algorithm we establish a theorem concerning its convergence the estimates produced by samc can be improved continuously as the simulation proceeds samc also extends applications of the wl algorithm to continuum systems the potential uses of samc in statistics are discussed through two classes of applications i

construction of multivariate surrogate sets from nonlinear data using the wavelet transform abstract the use of surrogate data has become a crucial first step in the study of nonlinearity in time series data a widely used technique to construct surrogate data is to randomize the phases of the data in the fourier domain in this paper an alternative technique based on the resampling of wavelet coefficients is discussed this approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data it proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties removal of nonlinear structure and computational demands however the wavelet technique may have specific and distinct advantages in the application to complex data sets such as numerical analysis of turbulence and experimental brain imaging data where wavelets give a more parsimonious representation of spatio temporal patterns than fourier modes it is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

correlation dimension a pivotal statistic for non constrained realizations of composite hypotheses in surrogate data analysis abstract currently surrogate data analysis can be used to determine if data is consistent with various linear systems or something else a nonlinear system in this paper we propose an extension of these methods in an attempt to make more specific classifications within the class of nonlinear systems in the method of surrogate data one estimates the probability distribution of values of a test statistic for a set of experimental data under the assumption that the data is consistent with a given hypothesis if the probability distribution of the test statistic is different for different dynamical systems consistent with the hypothesis one must ensure that the surrogate generation technique generates surrogate data that are a good approximation to the data this is often achieved with a careful choice of surrogate generation method and for noise driven linear surrogates such methods are commonly used this paper argues that in many cases particularly for nonlinear hypotheses it is easier to select a test statistic for which the probability distribution of test statistic values is the same for all systems consistent with the hypothesis for most linear hypotheses one can use a reliable estimator of a dynamic invariant of the underlying class of processes for more complex nonlinear hypothesis it requires suitable restatement or cautious statement of the hypothesis using such statistics one can build nonlinear models of the data and apply the methods of surrogate data to determine if the data is consistent with a simulation from a broad class of models these ideas are illustrated with estimates of probability distribution functions for correlation dimension estimates of experimental and artificial data and linear and nonlinear hypotheses

bayesian subset modeling for high dimensional generalized linear models this article presents a new prior setting for high dimensional generalized linear models which leads to a bayesian subset regression bsr with the maximum a posteriori model approximately equivalent to the minimum extended bayesian information criterion model the consistency of the resulting posterior is established under mild conditions further a variable screening procedure is proposed based on the marginal inclusion probability which shares the same properties of sure screening and consistency with the existing sure independence screening sis and iterative sure independence screening isis procedures however since the proposed procedure makes use of joint information from all predictors it generally outperforms sis and isis in real applications this article also makes extensive comparisons of bsr with the popular penalized likelihood methods including lasso elastic net sis and isis the numerical results indicate that bsr can generally outperform the penalized likelihood methods the m

wasserstein hamiltonian flows abstract we establish kinetic hamiltonian flows in density space embedded with the l 2 wasserstein metric tensor we derive the euler lagrange equation in density space which introduces the associated hamiltonian flows we demonstrate that many classical equations such as vlasov equation schrodinger equation and schrodinger bridge problem can be rewritten as the formalism of hamiltonian flows in density space