the universal relation between scaling exponents in first passage percolation it has been conjectured in numerous physics papers that in ordinary rst passage percolation on integer lattices the uctuation exponent and the wandering exponent are related through the universal relation 2 1 irrespective of the dimension this is sometimes called the kpz relation between the two exponents this article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense

robust mse equalizer design for mimo communication systems in the presence of model uncertainties this paper considers a robust mean square error mse equalizer design problem for multiple input multiple output mimo communication systems with imperfect channel and noise information at the receiver when the channel state information csi and the noise covariance are known exactly at the receiver a minimum mean square error mmse equalizer can be employed to estimate the transmitted signal however in actual systems it is necessary to take into account channel and noise estimation errors we consider here a worst case equalizer design problem where the goal is to find the equalizer minimizing the equalization mse for the least favorable channel model within a neighborhood of the estimated model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and estimated channel models lagrangian optimization is used to convert this min max problem into a convex min min problem over a convex domain which is solved by interchanging the minimization order the robust mse equalizer and associated least favorable channel model can then be obtained by solving numerically a scalar convex minimization problem simulation results are presented to demonstrate the mse and bit error rate ber performance of robust equalizers when applied to the least favorable channel model

the divergence of fluctuations for shape in first passage percolation we consider the first passage percolation model on z d for d 2 in this model we assign independently to each edge the value zero with probability p and the value one with probability 1 p we denote by t 0 the passage time from the origin to for r d and

estimation for the linear model with uncertain covariance matrices we derive a maximum a posteriori estimator for the linear observation model where the signal and noise covariance matrices are both uncertain the uncertainties are treated probabilistically by modeling the covariance matrices with prior inverse wishart distributions the nonconvex problem of jointly estimating the signal of interest and the covariance matrices is tackled by a computationally efficient fixed point iteration as well as an approximate variational bayes solution the statistical performance of estimators is compared numerically to state of the art estimators from the literature and shown to perform favorably

local tail bounds for functions of independent random variables it is shown that functions defined on 0 1 r 1 n satisfying certain conditions of bounded differences that guarantee sub gaussian tail behavior also satisfy a much stronger local sub gaussian property for self bounding and configuration functions we derive analogous locally subexponential behavior the key tool is talagrand s ann probab 22 1994 1576 1587 variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on 0 1 r 1 n for r 2

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

cooperation in spatial prisoner s dilemma with two types of players for increasing number of neighbors we study a spatial two strategy cooperation and defection prisoner s dilemma game with two types a and b of players located on the sites of a square lattice the evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor the strategy adoption probability is reduced by a prefactor w 1 from the players of type b we consider the competition between two opposite effects when increasing the number of neighbors k 4 8 and 24 within a range of the portion of influential players type a the inhomogeneous activity in strategy transfer yields a relevant increase dependent on w in the density of cooperators the noise dependence of this phenomenon is also discussed by evaluating phase diagrams

minimax mse ratio estimation with signal covariance uncertainties in continuation to an earlier work we further consider the problem of robust estimation of a random vector or signal with an uncertain covariance matrix that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix while in the earlier work we developed and proposed a competitive minimax approach of minimizing the worst case mean squared error mse difference regret criterion here we study in the same spirit the minimum worst case mse ratio regret criterion namely the worst case ratio rather than difference between the mse attainable using a linear estimator ignorant of the exact signal covariance and the minimum mse mmse attainable by optimum linear estimation with a known signal covariance we present the optimal linear estimator under this criterion in two ways the first is as a solution to a certain semidefinite programming sdp problem and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure we then show that the linear minimax ratio regret estimator can also be interpreted as the mmse estimator that minimizes the mse for a certain choice of signal covariance that depends on the uncertainty region we demonstrate that in applications the proposed minimax mse ratio regret approach may outperform the well known minimax mse approach the minimax mse difference regret approach and the plug in approach where in the latter one uses the mmse estimator with an estimated covariance matrix replacing the true unknown covariance

stochastic dynamics of invasion and fixation we study evolutionary game dynamics in finite populations we analyze an evolutionary process which we call pairwise comparison for which we adopt the ubiquitous fermi distribution function from statistical mechanics the inverse temperature in this process controls the intensity of selection leading to a unified framework for evolutionary dynamics at all intensities of selection from random drift to imitation dynamics we derive a simple closed formula that determines the feasibility of cooperation in finite populations whenever cooperation is modeled in terms of any symmetric two person game in contrast with previous results the present formula is valid at all intensities of selection and for any initial condition we investigate the evolutionary dynamics of cooperators in finite populations and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations as a function of a given initial number of cooperators showing how this interplay strongly affects the approach to fixation of a given trait in finite populations leading to counterintuitive results at different intensities of selection

minimax mse estimation of deterministic parameters with noise covariance uncertainties in this paper a minimax mean squared error mse estimator is developed for estimating an unknown deterministic parameter vector in a linear model subject to noise covariance uncertainties the estimator is designed to minimize the worst case mse across all norm bounded parameter vectors and all noise covariance matrices in a given region of uncertainty the minimax estimator is shown to have the same form as the estimator that minimizes the worst case mse over all norm bounded vectors for a least favorable choice of the noise covariance matrix an example demonstrating the performance advantage of the minimax mse approach over the least squares and weighted least squares methods is presented

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

optimal robust classifiers qualitatively a filter is said to be robust if its performance degradation is acceptable for distributions close to the one for which it is optimal that is the one for which it has been designed this paper adapts the signal processing theory of optimal robust filters to classifiers the distribution class conditional distributions to which the classifier is to be applied is parameterized by a state vector and the principle issue is to choose a design state that is optimal in comparison to all other states relative to some measure of robustness a minimax robust classifier is one whose worst performance over all states is better than the worst performances of the other classifiers defined at the other states a bayesian robust classifier is one whose expected performance is better than the expected performances of the other classifiers the state corresponding to the bayesian robust classifier is called the maximally robust state minimax robust classifiers tend to give too much weight to states for which classification is very difficult and therefore our effort is focused on bayesian robust classifiers whereas the signal processing theory of robust filtering concentrates on design with full distributional knowledge and a fixed number of observation variables features design via training from sample data and feature selection are so important for classification that robustness optimality must be considered from these perspectives in particular for small samples in this context for a given sample size we will be concerned with the maximally robust state feature pair all definitions are independent of the classification rule however applications are only considered for linear and quadratic discriminant analysis for which there are parametric forms for the optimal discriminants

linear minimax regret estimation of deterministic parameters with bounded data uncertainties we develop a new linear estimator for estimating an unknown parameter vector x in a linear model in the presence of bounded data uncertainties the estimator is designed to minimize the worst case regret over all bounded data vectors namely the worst case difference between the mean squared error mse attainable using a linear estimator that does not know the true parameters x and the optimal mse attained using a linear estimator that knows x we demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least squares estimator as well as several other least squares alternatives

sublinear variance in first passage percolation for general distributions we prove that the variance of the passage time from the origin to a point x in first passage percolation on mathbb z d is sublinear in the distance to x when d ge 2 obeying the bound c vert x vert log vert x vert under minimal assumptions on the edge weight distribution the proof applies equally to absolutely continuous discrete and singular continuous distributions and mixtures thereof and requires only 2 log moments the main result extends work of benjamini kalai schramm ann prob 31 2003 and benaim rossignol ann inst henri poincare prob stat 3 2008

maximum entropy and robust prediction on a simplex the related problems of finite length robust prediction and maximizing spectral entropy over a simplex of covariance matrices are considered general properties of iterative solutions of these problems are developed and monotone convergence proofs are presented for two algorithms that provide such solutions the analogous problems for simplexes of spectral densities are also considered

a competitive mean squared error approach to beamforming we treat the problem of beamforming for signal estimation where the goal is to estimate a signal amplitude from a set of array observations conventional beamforming methods typically aim at maximizing the signal to interference plus noise ratio sinr however this does not guarantee a small mean squared error mse so that on average the resulting signal estimate can be far from the true signal here we consider strategies that attempt to minimize the mse between the estimated and unknown signal waveforms the methods we suggest all maximize the sinr but at the same time are designed to have good mse performance since the mse depends on the signal power which is unknown we develop competitive beamforming approaches that minimize a robust mse measure two design strategies are proposed minimax mse and minimax regret we demonstrate through numerical examples that the suggested minimax beamformers can outperform several existing standard and robust methods over a wide range of signal to noise ratio snr values finally we apply our techniques to subband beamforming and illustrate their advantage in estimating a wideband signal

cooperation for volunteering and partially random partnerships competition among cooperative defective and loner strategies is studied by considering an evolutionary prisoner s dilemma game for different partnerships in this game each player can adopt one of its coplayer s strategy with a probability depending on the difference of payoffs coming from games with the corresponding coplayers our attention is focused on the effects of annealed and quenched randomness in the partnership for fixed number of coplayers it is shown that only the loners survive if the four coplayers are chosen randomly mean field limit on the contrary on the square lattice all the three strategies are maintained by the cyclic invasions resulting in a self organizing spatial pattern if the fixed partnership is described by a regular small world structure then a homogeneous oscillation occurs in the population dynamics when the measure of quenched randomness exceeds a threshold value similar behavior with higher sensitivity to the randomness is found if temporary partners are substituted for the standard ones with some probability at each step of iteration

oded schramm s contributions to noise sensitivity we survey in this paper the main contributions of oded schramm related to noise sensitivity we will describe in particular his various works which focused on the spectral analysis of critical percolation and more generally of boolean functions his work on the shape fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

minimax estimation of deterministic parameters in linear models with a random model matrix we consider the problem of estimating an unknown deterministic parameter vector in a linear model with a random model matrix with known second order statistics we first seek the linear estimator that minimizes the worst case mean squared error mse across all parameter vectors whose possibly weighted norm is bounded above we show that the minimax mse estimator can be found by solving a semidefinite programming problem and develop necessary and sufficient optimality conditions on the minimax mse estimator using these conditions we derive closed form expressions for the minimax mse estimator in some special cases we then demonstrate through examples that the minimax mse estimator can improve the performance over both a baysian approach and a least squares method we then consider the case in which the norm of the parameter vector is also bounded below since the minimax mse approach cannot account for a nonzero lower bound we consider in this case a minimax regret method in which we seek the estimator that minimizes the worst case difference between the mse attainable using a linear estimator that does not know the parameter vector and the optimal mse attained using a linear estimator that knows the parameter vector for analytical tractability we restrict our attention to the scalar case and develop a closed form expression for the minimax regret estimator

submean variance bound for effective resistance of random electric networks we study a model of random electric networks with bernoulli resistances in the case of the lattice mathbb z 2 we show that the point to point effective resistance between 0 and a vertex v has a variance of order at most rm log v frac 2 3 whereas its expected value is of order log v when v goes to infinity when d 2 expectation and variance are of the same order similar results are obtained in the context of p resistance the proofs rely on a modified poincare inequality due to falik and samorodnitsky 7

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

a competitive minimax approach to robust estimation of random parameters we consider the problem of estimating in the presence of model uncertainties a random vector x that is observed through a linear transformation h and corrupted by additive noise we first assume that both the covariance matrix of x and the transformation h are not completely specified and develop the linear estimator that minimizes the worst case mean squared error mse across all possible covariance matrices and transformations h in the region of uncertainty although the minimax approach has enjoyed widespread use in the design of robust methods we show that its performance is often unsatisfactory to improve the performance over the minimax mse estimator we develop a competitive minimax approach for the case where h is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst case regret namely the worst case difference between the mse attainable using a linear estimator ignorant of the signal covariance and the optimal mse attained using a linear estimator that knows the signal covariance the linear minimax regret estimator is shown to be equal to a minimum mse mmse estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region we demonstrate through examples that the minimax regret approach can improve the performance over both the minimax mse approach and a plug in approach in which the estimator is chosen to be equal to the mmse estimator with an estimated covariance matrix replacing the true unknown covariance we then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly gaussian is nonlinear we often do not lose much by restricting attention to linear estimators

learning mutation and long run equilibria in games an evolutionary model with a finite number of players and with stochastic mutations is analyzed the expansion and contraction of strategies is linked to their current relative success but mutuation perturbing the system from its deterministic evolution are present as well the focus is on the long run implications of ongoing mutations which drastically reduce the set of equilibria for 2 by 2 symmetric games with two symmetric strict nash equilibria the risk dominant equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected even though there is another strict nash equilibrium copyright 1993 by the econometric society

robust competitive estimation with signal and noise covariance uncertainties robust estimation of a random vector in a linear model in the presence of model uncertainties has been studied in several recent works while previous methods considered the case in which the uncertainty is in the signal covariance and possibly the model matrix but the noise covariance is assumed to be completely specified here we extend the results to the case where the noise statistics may also be subjected to uncertainties we propose several different approaches to robust estimation which differ in their assumptions on the given statistics in the first method we assume that the model matrix and both the signal and the noise covariance matrices are uncertain and develop a minimax mean squared error mse estimator that minimizes the worst case mse in the region of uncertainty the second strategy assumes that the model matrix is given and tries to uniformly approach the performance of the linear minimum mse estimator that knows the signal and noise covariances by minimizing a worst case regret measure the regret is defined as the difference or ratio between the mse attainable using a linear estimator ignorant of the signal and noise covariances and the minimum mse possible when the statistics are known as we show earlier solutions follow directly from our more general results however the approach taken here in developing the robust estimators is considerably simpler than previous methods

the rate distortion function on classes of sources determined by spectral capacities the quantity sup a in cal r r a d is considered where cal r is a class of homogeneous n parameter sources and r a d denotes the single letter mean square error mse rate distortion function for the individual source a in particular the case in which the class cal r is specified in terms of spectral information is treated for general classes of spectral measures whose upper measures are capacities in the sense of choquet alternating of order two this type of class includes many common models for spectral uncertainty such as mixture models spectral band models and neighborhoods generated by variation and prohorov metrics it is shown that each such class contains a worst case source whose rate distortion function achieves the supremum over the class for each value of distortion this source is characterized as having a spectral density that is a derivative in the sense of huber and strassen of the upper spectral measure with respect to lebesgue measure on pi pi n moreover it is shown that the spectral measure of the worst case source is closest in a sense defined by directed divergence to lebesgue measure which corresponds to a memoryless source numerical results are presented for the particular case in which the source spectral measure is a mixture of a gauss markov spectrum and an unknown contaminating component

robust wiener filtering with non parametric spectral uncertainty in this article robust least squares filtering problems are considered for non parametric multivariate spectral uncertainty defined by the so called spectral band and generalised moment constraints its major aim is to provide a basis for computing approximate solutions to worst case wiener filtering optimisation problems involving causal filters and multivariate signals it hinges upon associating upper and lower bounds on the minimum worst case performance achievable with causal filters with linear cost linear matrix inequality lc lmi constraint optimisation problems on the basis of a lagrangean duality formulation for the worst case least squares performance of a given filter upper bounds on it are obtained as the optimal values of lc lmi problems then for linearly parameterised classes of filter transfer functions a causal filter which optimises such an upper bound on worst case performance can also be obtained from an lc lmi optimisation problem to estimate the amount of conservatism incur

optimal control of uncertain stochastic systems subject to total variation distance uncertainty this paper is concerned with optimization of uncertain stochastic systems in which uncertainty is described by a total variation distance constraint between the measures induced by the uncertain systems and the measure induced by the nominal system while the payoff is a linear functional of the uncertain measure robustness at the abstract setting is formulated as a minimax game in which the control seeks to minimize the payoff over the admissible controls while the uncertainty aims at maximizing it over the total variation distance constraint it is shown that the maximizing measure in the total variation distance constraint exists while the resulting payoff is a linear combination of l 1 and l infty norms further the maximizing measure is characterized by a linear combination of a tilted measure and the nominal measure giving rise to a payoff which is a nonlinear functional on the space of measures to be minimized over the admissible controls the abstract formulation and results are subsequ

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

nonlinear estimation for a class of systems this paper considers nonlinear estimation problems for classes of models and employs relative entropy to describe the uncertainty classes two optimization problems are formulated on general banach spaces and their solutions are sought 1 when the transition probability between the signal to be estimated x and the measurement y or stochastic kernel is unknown and 2 when the joint probability induced by the random variables rvs x y is unknown for both problems the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution the results include existence of the optimal measures using weak convergence techniques and properties associated with the estimate of the true distribution classical examples are chosen to illustrate the applicability of the results

robust deconvolution of deterministic and random signals we consider the problem of designing an estimation filter to recover a signal x n convolved with a linear time invariant lti filter h n and corrupted by additive noise our development treats the case in which the signal x n is deterministic and the case in which it is a stationary random process both formulations take advantage of some a priori knowledge on the class of underlying signals in the deterministic setting the signal is assumed to have bounded weighted energy in the stochastic setting the power spectra of the signal and noise are bounded at each frequency the difficulty encountered in these estimation problems is that the mean squared error mse at the output of the estimation filter depends on the problem unknowns and therefore cannot be minimized beginning with the deterministic setting we develop a minimax mse estimation filter that minimizes the worst case point wise mse between the true signal x n and the estimated signal over the class of bounded norm inputs we then establish that the mse at the output of the minimax mse filter is smaller than the mse at the output of the conventional inverse filter for all admissible signals next we treat the stochastic scenario for which we propose a minimax regret estimation filter to deal with the power spectrum uncertainties this filter is designed to minimize the worst case difference between the mse in the presence of power spectrum uncertainties and the mse of the wiener filter that knows the correct power spectra the minimax regret filter takes the entire uncertainty interval into account and as demonstrated through an example can often lead to improved performance over traditional minimax mse approaches for this problem

growth and roughness of the interface for ballistic deposition in ballistic deposition bd d 1 dimensional particles fall sequentially at random towards an initially flat large but bounded d dimensional surface and each particle sticks to the first point of contact for both lattice and continuum bd a law of large numbers in the thermodynamic limit establishes convergence of the mean height and surface width sample standard deviation of the height of the interface to constants h t and w t respectively depending on time t we show that h t is asymptotically linear in t while w t 2 grows at least logarithmically in t when d 1 we use duality results showing that w t can be interpreted as the standard deviation of the height for deposition onto a surface growing from a single point

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

error estimates and convergence rates for the stochastic homogenization of hamilton jacobi equations we present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level set convex hamilton jacobi equations in i i d random environments the first quantitative homogenization results for these equations in the stochastic setting by taking advantage of a connection between the metric approach to homogenization and the theory of first passage percolation we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime away from flat spot of the effective hamiltonian in the sub ballistic regime on the flat spot we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed without further assumptions at an arbitrarily slow rate we identify a necessary and sufficient condition on the law of the hamiltonian for an algebraic rate of convergence to hold in the sub ballistic regime and show under this hypothesis that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization our methods are novel and quite different from the techniques employed in the periodic setting although we benefit from previous works in both first passage percolation and homogenization the link between the rate of homogenization and the flat spot of the effective hamiltonian which is related to the nonexistence of correctors is a purely random phenomenon observed here for the first time

non normal limiting distribution for optimal alignment scores of strings in binary alphabets we consider two independent binary i i d random strings x and y of equal length n and the optimal alignments according to a symmetric scoring functions only we decompose the space of scoring functions into five components two of these components add a part to the optimal score which does not depend on the alignment and which is asymptotically normal we show that when we restrict the number of gaps sufficiently and add them only into one sequence then the alignment score can be decomposed into a part which is normal and has order o sqrt n and a part which is on a smaller order and tends to a tracy widom distribution adding gaps only into one sequence is equivalent to aligning a string with its descendants in case of mutations and deletes for testing relatedness of strings the normal part is irrelevant since it does not depend on the alignment hence it can be safely removed from the test statistic

inertia in strategy switching transforms the strategy evolution a recent experimental study traulsen et al proc natl acad sci 107 2962 2010 shows that human strategy updating involves both direct payoff comparison and the cost of switching strategy which is equivalent to inertia however it remains largely unclear how such a predisposed inertia affects 2 2 games in a well mixed population of finite size to address this issue the inertia bonus strategy switching cost is added to the learner payoff in the fermi process we find how inertia quantitatively shapes the stationary distribution and that stochastic stability under inertia exhibits three regimes with each covering seven regions in the plane spanned by two inertia parameters we also obtain the extended 1 3 rule with inertia and the speed criterion with inertia these two findings hold for a population above two we illustrate the above results in the framework of the prisoner s dilemma game as inertia varies two intriguing stationary distributions emerge the probability of coexistence state is maximized or those of two full states are simultaneously peaked our results may provide useful insights into how the inertia of changing status quo acts on the strategy evolution and in particular the evolution of cooperation

central limit theorem for first passage percolation time across thin cylinders we prove that first passage percolation times across thin cylinders of the form 0 n hn hn d 1 obey gaussian central limit theorems as long as hn grows slower than n1 d 1 it is an open question as to what is the fastest that hn can grow so that a gaussian clt still holds under the natural but unproven assumption about existence of fluctuation and transversal exponents and strict convexity of the limiting shape in the direction of 1 0 0 we prove that in dimensions 2 and 3 the clt holds all the way up to the height of the unrestricted geodesic we also provide some numerical evidence in support of the conjecture in dimension 2

lower bounds for parametric estimation with constraints a chapman robbins form of the barankin bound is used to derive a multiparameter cramer rao cr type lower bound on estimator error covariance when the parameter theta in r sup n is constrained to lie in a subset of the parameter space a simple form for the constrained cr bound is obtained when the constraint set theta sub c can be expressed as a smooth functional inequality constraint it is shown that the constrained cr bound is identical to the unconstrained cr bound at the regular points of theta sub c i e where no equality constraints are active on the other hand at those points theta in theta sub c where pure equality constraints are active the full rank fisher information matrix in the unconstrained cr bound must be replaced by a rank reduced fisher information matrix obtained as a projection of the full rank fisher matrix onto the tangent hyperplane of the full rank fisher matrix onto the tangent hyperplane of the constraint set at theta a necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable and examples for which the bound is achieved are presented in addition to providing a useful generalization of the cr bound the results permit analysis of the gain in achievable mse performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization

evolutionary prisoner s dilemma game on a square lattice a simplified prisoner s game is studied on a square lattice when the players interacting with their neighbors can follow two strategies to cooperate c or to defect d unconditionally the players updated in random sequence have a chance to adopt one of the neighboring strategies with a probability depending on the payoff difference using monte carlo simulations and dynamical cluster techniques we study the density c of cooperators in the stationary state this system exhibits a continuous transition between the two absorbing states when varying the value of temptation to defect in the limits stackrel ensuremath rightarrow c 0 and 1 we have observed critical transitions belonging to the universality class of directed percolation

coevolutionary dynamics from finite to infinite populations traditionally frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations however the relationship between deterministic and stochastic approaches remained unclear here we solve this problem by explicitly considering large populations in particular we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics moreover differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and depending on the population size can even invert the direction of the evolutionary process

intrinsically bayesian robust kalman filter an innovation process approach in many contemporary engineering problems model uncertainty is inherent because accurate system identification is virtually impossible owing to system complexity or lack of data on account of availability time or cost the situation can be treated by assuming that the true model belongs to an uncertainty class of models in this context an intrinsically bayesian robust ibr filter is one that is optimal relative to the cost function in the classical sense and the prior distribution over the uncertainty class in the bayesian sense ibr filters have previously been found for both wiener and granulometric morphological filtering in this paper we derive the ibr kalman filter that performs optimally relative to an uncertainty class of state space models introducing the notion of bayesian innovation process and the bayesian orthogonality principle we show how the problem of designing an ibr kalman filter can be reduced to a recursive system similar to the classical kalman recursive equations except with effective counterparts such as the effective kalman gain matrix after deriving the recursive ibr kalman equations for discrete time we use the limiting method to obtain the ibr kalman bucy equations for continuous time finally we demonstrate the utility of the proposed framework for two real world problems sensor networks and gene regulatory network inference

on robust wiener signal estimation a minimax formulation is considered for the problem of designing robust linear causal estimators of continuous time signals in noise it is shown that the minimax estimation problem can generally be reduced to finding a least favorable pair of spectra in particular robust predictors filters and finite lag smoothers are shown to be just the optimal predictors filters and smoothers respectively for a least favorable pair this result can often greatly simplify the design of robust estimators

optimal classifiers with minimum expected error within a bayesian framework part ii properties and performance analysis in part i of this two part study we introduced a new optimal bayesian classification methodology that utilizes the same modeling framework proposed in bayesian minimum mean square error mmse error estimation optimal bayesian classification thus completes a bayesian theory of classification where both the classifier error and our estimate of the error may be simultaneously optimized and studied probabilistically within the assumed model having developed optimal bayesian classifiers in discrete and gaussian models in part i here we explore properties of optimal bayesian classifiers in particular invariance to invertible transformations convergence to the bayes classifier and a connection to bayesian robust classifiers we also explicitly derive optimal bayesian classifiers with non informative priors and explore relationships to linear and quadratic discriminant analysis lda and qda which may be viewed as plug in rules under gaussian modeling assumptions finally we present several simulations addressing the robustness of optimal bayesian classifiers to false modeling assumptions companion website http gsp tamu edu publications supplementary dalton12a

universal composite hypothesis testing a competitive minimax approach a novel approach is presented for the long standing problem of composite hypothesis testing in composite hypothesis testing unlike in simple hypothesis testing the probability function of the observed data given the hypothesis is uncertain as it depends on the unknown value of some parameter the proposed approach is to minimize the worst case ratio between the probability of error of a decision rule that is independent of the unknown parameters and the minimum probability of error attainable given the parameters the principal solution to this minimax problem is presented and the resulting decision rule is discussed since the exact solution is in general hard to find and a fortiori hard to implement an approximation method that yields an asymptotically minimax decision rule is proposed finally a variety of potential application areas are provided in signal processing and communications with special emphasis on universal decoding

robust matched filters for optical receivers the problem of designing optical receivers that are robust against uncertainty in the statistics of the observation process in photodetection is investigated in particular a modification in the design of the postdetection matched filter is proposed to account for possible uncertainty in the rate function of the incident light the rate of the dark current and in the statistics of the additive noise present at the input to the optical receiver this design is based on a game theoretic approach in which a filter is sought that has the maximum worst case output signal to noise ratio possible over the class of allowable statistics that is the design criterion is maximin signal to noise ratio a general characterization of maximin robust matched filters for observed poisson processes is presented in this context and specific solutions for several useful uncertainty models are obtained numerical results are presented for a specific example to illustrate the performance of the proposed technique

robust hypothesis testing for bounded classes of probability densities corresp least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds the results are shown to extend huber s solution for the epsilon contaminated classes which is contained as a special case a minimum distance property of the least favorable density pair is also made explicit

universality of weak selection weak selection which means a phenotype is slightly advantageous over another is an important limiting case in evolutionary biology recently it has been introduced into evolutionary game theory in evolutionary game dynamics the probability to be imitated or to reproduce depends on the performance in a game the influence of the game on the stochastic dynamics in finite populations is governed by the intensity of selection in many models of both unstructured and structured populations a key assumption allowing analytical calculations is weak selection which means that all individuals perform approximately equally well in the weak selection limit many different microscopic evolutionary models have the same or similar properties how universal is weak selection for those microscopic evolutionary processes we answer this question by investigating the fixation probability and the average fixation time not only up to linear but also up to higher orders in selection intensity we find universal higher order expansions which allow a rescaling of the selection intensity with this we can identify specific models which violate linear weak selection results such as the one third rule of coordination games in finite but large populations

on the p point uncertainty class corresp the p point uncertainty class also known as sakrison s class b is considered although the p point class is not a 2 alternating choquet capacity class it is shown how many of the robust signal processing and communication theory results which have been developed for capacity classes can be applied to the p point class by embedding it in a slightly larger extended p point class

minimax robust prediction of discrete time series we discuss a robust approach for predicting a weakly stationary discrete time series whose spectral density f is not exactly known we assume that we know that f mathfrak d where mathfrak d is a convex set of spectral densities fulfilling some not too stringent conditions we proof the existence of a most indeterministic density f0 in mathfrak d and we show that the classical optimal linear predictor for a time series with spectral density f0 is mini max robust in the sense that it minimizes the maximal possible prediction error

brief paper an analysis of the effects of spectral uncertainty on wiener filtering a representative selection of results from an extensive study of the performance of wiener filtering under spectral uncertainty is presented for a variety of spectral uncertainty models the wiener filter is shown to have undesirable sensitivity to even small deviations from those signal and noise spectral densities which were used to design the filter performance of a robust filter designed to have the best possible performance when the uncertainty is worst is also examined in most cases the robust filter s insensitivity to spectral uncertainty makes it preferable to the traditional wiener filter

minimax linear smoothing for capacities the case of correlated signals and noise results of an earlier paper giving minimax linear smoothers for the problem of estimating a homogeneous signal field in an additive orthogonal noise field when both have uncertain spectral properties are extended to the case in which the signal and noise fields are arbitrarily correlated as before spectral uncertainty is modeled by assuming that the spectral measures of the signal and noise fields lie in classes of measures generated by two alternating choquet capacities it is demonstrated that this problem admits a general solution in terms of the huber strassen derivative between the capacities that generate the uncertainty sets and that the least favorable spectra for smoothing in orthogonal noise are also the least favorable marginal spectra for smoothing in correlated noise the resulting filter is seen to be a zonal filter that also arises as the solution to an analogous problem in nonparametric minimax hypothesis testing these new results extend the applicability of minimax robust smoothing techniques to application involving signal dependent noise phenomena such as multipath and clutter which are usually difficult to model precisely

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

shape fluctuations are different in different directions we consider the first passage percolation model on z 2 in this model we assign independently to each edge e a passage time t e with a common distribution f let t u v be the passage time from u to v in this paper we show that whenever f 0 0 and f r p c it is known that there exists m such that for all n 2 t 0 0 n n m these results tell us that shape fluctuations not only depend on distribution f but also on direction when showing this result we find the following interesting geometrical property with the special distribution above any long piece with r edges in an optimal path from 0 0 to n 0 has to be very circuitous

intrinsically optimal bayesian robust filtering when designing optimal filters it is often unrealistic to assume that the statistical model is known perfectly the issue is then to design a robust filter that is optimal relative to an uncertainty class of processes robust filter design has been treated from minimax best worst case performance and bayesian best average performance perspectives heretofore the bayesian approach has involved finding a model specific optimal filter one that is optimal for some model in the uncertainty class lifting this constraint we optimize over the full class from which the original optimal filters were obtained for instance over all linear filters by extending the original characteristics determining the filter such as the power spectral density to effective characteristics that apply across the uncertainty class we demonstrate for both linear and morphological filtering that an intrinsically optimal bayesian robust filter can be represented in the same form as the standard solution to the optimal filter except via the effective characteristics solutions for intrinsic bayesian robust filters are more transparent and intuitive than solutions for model specific filters and also less tedious because effective characteristics push through the spectral theory into the bayesian setting whereas solutions in the model specific case depend on grinding out the optimization

evolutionary dynamics on graphs efficient method for weak selection investigating the evolutionary dynamics of game theoretical interactions in populations where individuals are arranged on a graph can be challenging in terms of computation time here we propose an efficient method to study any type of game on arbitrary graph structures for weak selection in this limit evolutionary game dynamics represents a first order correction to neutral evolution spatial correlations can be empirically determined under neutral evolution and provide the basis for formulating the game dynamics as a discrete markov process by incorporating a detailed description of the microscopic dynamics based on the neutral correlations this framework is then applied to one of the most intriguing questions in evolutionary biology the evolution of cooperation we demonstrate that degree heterogeneity of a graph impedes cooperation and that the success of tit for tat not only depends on the number of rounds but also on the degree of the graph moreover considering the mutation selection equilibrium shows that the symmetry of the stationary distribution of states under weak selection is skewed in favor of defectors for larger selection strengths in particular degree heterogeneity a prominent feature of scale free networks generally results in a more pronounced increase of the critical benefit to cost ratio required for evolution to favor cooperation as compared to regular graphs this conclusion is corroborated by an analysis of the effects of population structures on the fixation probabilities of strategies in general 2 2 games for different types of graphs computer simulations confirm the predictive power of our method and illustrate the improved accuracy as compared to previous studies

minimax robust deconvolution filters under stochastic parametric and noise uncertainties the author consider the design of robust deconvolution filters for linear discrete time systems with stochastic parameter and noise uncertainties it is assumed that some large but bounded uncertainties exist in the driving and measurement noise covariances as well as the second order statistics of stochastic parameters and initial conditions three kinds of minimax sensitivity criteria are used to develop the techniques to the synthesis of minimax deconvolution filters under uncertain linear stochastic systems their approach is based on saddle point theory and the sensitivity analysis of kalman filters the design algorithms give the recursive realization of the minimax deconvolution filters for the time varying uncertain systems under fairly general conditions for the time invariant uncertain case the existence and solutions of steady state deconvolution filters are further developed finally the utility of the minimax design approaches and the properties of the resulting minimax deconvolution filters are illustrated by a numerical example

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

quantifying the objective cost of uncertainty in complex dynamical systems real world problems often involve complex systems that cannot be perfectly modeled or identified and many engineering applications aim to design operators that can perform reliably in the presence of such uncertainty in this paper we propose a novel bayesian framework for objective based uncertainty quantification uq which quantifies the uncertainty in a given system based on the expected increase of the operational cost that it induces this measure of uncertainty called mocu mean objective cost of uncertainty provides a practical way of quantifying the effect of various types of system uncertainties on the operation of interest furthermore the proposed uq framework provides a general mathematical basis for designing robust operators and it can be applied to diverse applications including robust filtering classification and control we demonstrate the utility and effectiveness of the proposed framework by applying it to the problem of robust structural intervention of gene regulatory networks an important application in translational genomics

minimax linear smoothing for capacities abstract minimax linear smoothers are considered for the problem of estimating a homogeneous signal field in an additive orthogonal noise field a minimax game with the quadratic mean estimation error as an objective function is used to formulate this problem uncertainty in signal and noise field spectra is modeled using general nonparametric classes of measures proposed by huber and strassen for the problem of minimax hypothesis testing these classes which are described in terms of choquet alternating capacities of order 2 include the conventional models for spectral uncertainty and admit a general solution to the minimax linear smoothing problem author

robust mean squared error estimation in the presence of model uncertainties we consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties where the vector x is known to satisfy a weighted norm constraint we first assume that the model is known exactly and seek the linear estimator that minimizes the worst case mean squared error mse across all possible values of x we show that for an arbitrary choice of weighting the optimal minimax mse estimator can be formulated as a solution to a semidefinite programming problem sdp which can be solved very efficiently we then develop a closed form expression for the minimax mse estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of mayer and willke with a specific choice of shrinkage factor that explicitly takes the prior information into account next we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst case mse across all possible values of x and all possible values of the model matrix as we show the robust minimax mse estimator can also be formulated as a solution to an sdp finally we demonstrate through several examples that the minimax mse estimator can significantly increase the performance over the conventional least squares estimator and when the model matrix is subject to uncertainties the robust minimax mse estimator can lead to a considerable improvement in performance over the minimax mse estimator

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

first passage percolation on mathbb z 2 a simulation study first passage percolation on mathbb z 2 is a model for describing the spread of an infection on the sites of the square lattice the infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges in this paper the speed of the growth and the shape of the infected set is studied by aid of large scale computer simulations with focus on continuous passage time distributions it is found that the most important quantity for determining the value of the time constant which indicates the inverse asymptotic speed of the growth is mathbf e min tau 1 ldots tau 4 where tau 1 ldots tau 4 are i i d passage time variables the relation is linear for a large class of passage time distributions furthermore the directional time constants are seen to be increasing when moving from the axis towards the diagonal so that the limiting shape is contained in a circle with radius defined by the speed along the axes the shape comes closer to the circle for distributions with larger variability

robust training sequence design for correlated mimo channel estimation we study how to design a worst case robust training sequence for multiple input multiple output mimo channel estimation we consider mean squared error of channel estimates as the figure of merit which is a function of second order statistics of the mimo channel i e channel covariance matrix in order to optimize training sequences under a total power constraint in practical applications the channel covariance matrix is not known perfectly thus the main aspect of our design is to improve robustness of the training sequences against possible uncertainties in the available channel covariance matrix using a deterministic uncertainty model we formulate a robust training sequence design as a minimax optimization problem where we take such imperfections into account we investigate the robust design problem assuming the general case of an arbitrarily correlated mimo channel and a non empty compact convex uncertainty set we prove that such a problem admits a globally optimal solution by exploiting the convex concave structure of the objective function and propose numerical algorithms to address the robust training design problem we proceed the analysis by considering multiple input single output miso channels and kronecker structured mimo channels along with unitarily invariant uncertainty sets for these scenarios we show that the problem is diagonalized by the eigenvectors of the nominal covariance matrices so that the robust design is significantly simplified from a complex matrix variable problem to a real vector variable power allocation problem for the miso channel we provide closed form solutions for the robust training sequences with the uncertainty sets defined by the spectral norm and nuclear norm

transversal fluctuations for increasing subsequences on the plane consider a realization of a poisson process in 2 with intensity 1 and take a maximal up right path from the origin to n n consisting of line segments between the points where maximal means that it contains as many points as possible the number of points in such a path has fluctuations of order n where 1 3 bdj here we show that typical deviations of a maximal path from the diagonal x y is of order n with 2 3 this is consistent with the scaling identity 2 1 which is believed to hold in many random growth models

a sublinear variance bound for solutions of a random hamilton jacobi equation we estimate the variance of the value function for a random optimal control problem the value function is the solution w of a hamilton jacobi equation with random hamiltonian h p x k p v x in dimension d 2 it is known that homogenization occurs as 0 but little is known about the statistical fluctuations of w our main result shows that the variance of the solution w is bounded by o log the proof relies on a modified poincare inequality of talagrand

an rkhs approach to robust l sup 2 estimation and signal detection the application of rkhs reproducing kernel hilbert space theory to the problems of robust signal detection and estimation is investigated it is shown that this approach provides a general and unified framework in which to analyze the problems of l sup 2 estimation matched filtering and quadratic detection in the presence of uncertainties regarding the second order structure of the random processes involved