estimation of functional regression model via functional dimension reduction abstract the functional linear regression model corresponding to a scalar response and a functional predictor is becoming increasingly common since the predictor is infinite dimensional some form of dimension reduction is essential there are two popular dimension reduction methods such as expanding the functional predictor or regression parameter function on the functional principal component basis or on a fixed bases such as b spline wavelet in the present paper we estimate the functional linear model by using the functional sufficient dimension reduction fsdr basis compared to the existing methods the proposed method is appealing because the fsdr basis is related to both the functional predictor and the response variable whereas the functional principal component basis is only related to the functional predictor and the fixed basis b spline wavelet is independent from both the functional predictor and the response variable our techniques involve methods for giving a new expansion for the predictor for giving a specific expression for the regression parameter function for estimating the fsdr space by a new method and some asymptotical properties about the regression parameter function and the prediction for the test samples numerical studies including both simulation studies and applications on real life data are presented to demonstrate the accuracy of the proposed method

dynamical solution of the on line minority game we solve the dynamics of the on line minority game mg with general types of decision noise using generating functional techniques a la de dominicis and the temporal regularization procedure of bedeaux et al the result is a macroscopic dynamical theory in the form of closed equations for correlation and response functions defined via an effective continuous time single trader process which are exact in both the ergodic and in the non ergodic regime of the mg our solution also explains why although one cannot formally truncate the kramers moyal expansion of the process after the fokker planck term upon doing so one still finds the correct solution that the previously proposed diffusion matrices for the fokker planck term are incomplete and how previously proposed approximations of the market volatility can be traced back to ergodicity assumptions

identification of continuous time models the paper considers the problem of estimation of the transfer function of a continuous time dynamic system in the presence of colored noise the author introduces an operator transformation that allows for keeping a continuous time parametrization whereas the parameter estimation can be made by means of a discrete time maximum likelihood algorithm a comparison is made between the performance of the new method in comparison with a standard identification of an armax model the method is useful in cases where it is important to estimate the coefficients of a continuous time transfer function and where it is important to maintain a physical interpretation of the transfer function results

stochastic decision making in the minority game abstract a discussion is presented of the effects of stochasticity in the decision making of agents in the minority game both simulational and analytic results are reported and discussed for both additive and multiplicative noise as a function of the ratio d of information dimension to number of agents a phase transition separates a low d non ergodic phase from a high d ergodic phase for additive noise the critical dc is temperature independent but for multiplicative noise dc t decreases with t additive noise does not affect the asymptotic behaviour for d dc but is relevant below dc multiplicative noise has consequence for all d

supervised learning with restricted training sets a generating functional analysis we study the dynamics of supervised on line learning of realizable tasks in feed forward neural networks we focus on the regime where the number of examples used for training is proportional to the number of input channels n using generating functional techniques from spin glass theory we are able to average over the composition of the training set and transform the problem for n to an effective single pattern system described completely by the student autocovariance the student teacher overlap and the student response function with exact closed equations our method applies to arbitrary learning rules i e not necessarily of a gradient descent type the resulting exact macroscopic dynamical equations can be integrated without finite size effects up to any degree of accuracy but their main value is in providing an exact and simple starting point for analytical approximation schemes finally we show how in the region of absent anomalous response and using the hypothesis that as in detailed balance systems the short time part of the various operators can be transformed away one can describe the stationary state of the network succesfully by a set of coupled equations involving only four scalar order parameters

broken ergodicity and memory in the minority game we study the dynamics of the batch minority game with market impact correction using generating functional techniques to carry out the quenched disorder average we find that the assumption of weak long term memory which one usually makes in order to calculate ergodic stationary states breaks down when the persistent autocorrelation becomes larger than cc0 772 we show that this condition remarkably coincides with the at line found in an earlier static calculation this result suggests a new scenario for ergodicity breaking in disordered systems

mixed population minority game with generalized strategies we present a quantitative theory based on crowd effects for the market volatility in a minority game played by a mixed population below a critical concentration of generalized strategy players we find that the volatility in the crowded regime remains above the random coin toss value regardless of the temperature controlling strategy use our theory yields good agreement with numerical simulations

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

smoothing splines estimators for functional linear regression the paper considers functional linear regression where scalar responses y1 yn are modeled in dependence of random functions x1 xn we propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty theoretical analysis concentrates on the error in an out ofsample prediction of the response for a new random function xn 1 it is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors we then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves for the case of models with errors in variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves the methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

frequency weighted model reduction via quasi kalman decomposition a frequency weighted model reduction method based on quasi kalman decomposition is proposed for discrete systems the method is essentially an extension of the technique presented by lu and lee 1985 to the frequency weighted case and can handle both single sided as well as double sided weights two illustrative examples show the effectiveness of the method and results are compared with the method developed by enns 1984

classification of functional fragments by regularized linear classifiers with domain selection we consider classification of functional data into two groups by linear classifiers based on one dimensional projections of functions we reformulate the task of finding the best classifier as an optimization problem and solve it by the conjugate gradient method with early stopping the principal component method and the ridge method we study the empirical version with finite training samples consisting of incomplete functions observed on different subsets of the domain and show that the optimal possibly zero misclassification probability can be achieved in the limit along a possibly nonconvergent empirical regularization path we propose a domain extension and selection procedure that finds the best domain beyond the common observation domain of all curves in a simulation study we compare the different regularization methods and investigate the performance of domain selection our method is illustrated on a medical dataset where we observe a substantial improvement of classification accuracy due to domain extension

continuous time dynamics of the thermal minority game we study the continuous time dynamics of the thermal minority game we find that the dynamical equations of the model reduce to a set of stochastic differential equations for an interacting disordered system with nontrivial random diffusion this is the simplest microscopic description which accounts for all the features of the system within this framework we study the phase structure of the model and find that its macroscopic properties strongly depend on the initial conditions

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

minority game with sk interactions a batch minority game with fake random history and additional sk like quenched interaction is introduced and analysed a mixing parameter quantifies the admixture and dictates the relative dominance of the two contributions if 0 agent decisions are based on their strategies and point scores alone as in the pure minority game whereas for 0 the agents also communicate with each other directly and update their points accordingly keeping the minority game dynamics in which the agents points are updated in parallel at each time step the aim is to understand what happens if instead of simply using the normal strategy based decisions the agents also take account of an effective field generated by the other agents it is shown that the sk interaction introduces a noise term which is broader than that in the normal minority game and which furthermore kills the normal phase transition it is also shown that the same effect would occur if instead of an sk interaction gaussian distributed quenched random fields are added by calculating order parameters in the time translational invariant phase we show that the system is persistent in a ergodic phase both simulational and analytical results are presented

most predictive design points for functional data predictors we suggest a way of reducing the very high dimension of a functional predictor x to a low number of dimensions chosen so as to give the best predictive performance specifically if x is observed on a fine grid of design points t 1 t r we propose a method for choosing a small subset of these say t i1 t ik to optimize the prediction of a response variable y the values t ij are referred to as the most predictive design points or covariates for a given value of k and are computed using information contained in a set of independent observations x i y i of x y the algorithm is based on local linear regression and calculations can be accelerated using linear regression to preselect the design points boosting can be employed to further improve the predictive performance we illustrate the usefulness of our ideas through simulations and examples drawn from chemometrics and we develop theoretical arguments showing that the methodology can be applied successfully in a range of settings

closed form grammians and model reduction for flexible space structures the author derives analytical expressions for the grammians of a model given in modal coordinates for the dynamics of a flexible space structure fss these expressions are then used as the basis for a new algorithm to compute a dominant reduced order model for such a system in an efficient manner the resulting reduced order model is obtained directly rather than by truncating a balanced model of the entire system thus giving rise to significant efficiency gains for the typical fss case where the reduced order model is of very much lower order than the original system

on the use of reproducing kernel hilbert spaces in functional classification the hajek feldman dichotomy establishes that two gaussian measures are either mutually absolutely continuous with respect to each other and hence there is a radon nikodym density for each measure

irrelevance of memory in the minority game by means of extensive numerical simulations we show that all the distinctive features of the minority game introduced by challet and zhang physica a 256 514 1998 are completely independent of the memory of the agents the only crucial requirement is that all the individuals must possess the same information irrespective of whether this information is true or false

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

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

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

crowd anticrowd theory of the minority game abstract the minority game is a simple yet highly non trivial agent based model for a complex adaptive system here we provide an explanation of the game s fluctuations which is both intuitive and quantitative and which applies over the entire parameter range of interest the physical idea behind our theory is to describe the interplay between crowds of like minded agents and their anticorrelated partners anticrowds

componentwise classification and clustering of functional data the infinite dimension of functional data can challenge conventional methods for classification and clustering a variety of techniques have been introduced to address this problem particularly in the case of prediction but the structural models that they involve can be too inaccurate or too abstract or too difficult to interpret for practitioners in this paper we develop approaches to adaptively choose components enabling classification and clustering to be reduced to finite dimensional problems we explore and discuss properties of these methodologies our techniques involve methods for estimating classifier error rate and cluster tightness and for choosing both the number of components and their locations to optimize these quantities a major attraction of this approach is that it allows identification of parts of the function domain that convey important information for classification and clustering it also permits us to determine regions that are relevant to one of these analyses but not the other copyright 2012 oxford university press

generalized strategies in the minority game we show analytically how the fluctuations i e standard deviation ensuremath sigma in the minority game can decrease below the random coin toss limit if the agents use more general stochastic strategies this suppression of ensuremath sigma results from a cancellation between the actions of a crowd in which agents act collectively and make the same decision and those of an anticrowd in which agents act collectively by making the opposite decision to the crowd

a statistical view of some chemometrics regression tools chemometrics is a field of chemistry that studies the application of statistical methods to chemical data analysis in addition to borrowing many techniques from the statistics and engineering literatures chemometrics itself has given rise to several new data analytical methods this article examines two methods commonly used in chemometrics for predictive modeling partial least squares and principal components regression from a statistical perspective the goal is to try to understand their apparent successes and in what situations they can be expected to work well and to compare them with other statistical methods intended for those situations these methods include ordinary least squares variable subset selection and ridge regression

restricted likelihood ratio tests for functional effects in the functional linear model the goal of our article is to provide a transparent robust and computationally feasible statistical approach for testing in the context of scalar on function linear regression models assuming linearity between response and predictors we are interested in testing for the necessity of functional effects our methods are motivated by and applied to a large longitudinal study involving diffusion tensor imaging of intracranial white matter tracts in a susceptible cohort in the context of this study we conduct hypothesis tests that are motivated by anatomical knowledge and support recent findings regarding the relationship between cognitive impairment and white matter demyelination r code and data are in the examples of refund rlrt pfr supplementary materials for this article are available online

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

generating functional analysis of the dynamics of the batch minority game with random external information we study the dynamics of the batch minority game with random external information using generating functional techniques introduced by de dominicis the relevant control parameter in this model is the ratio a5 p n of the number p of possible values for the external information over the number n of trading agents in the limit n we calculate the location a c of the phase transition signaling the onset of anomalous response and solve the statics for a a c exactly the temporal correlations in global market fluctuations turn out not to decay to zero for infinitely widely separated times for a a c the stationary state is shown to be nonunique for a 0 we analyze our equations in leading order in a and find asymptotic solutions with diverging volatility s5o a 21 2 as regularly observed in simulations but also asymptotic solutions with vanishing volatility s5o a 1 2 the former however are shown to emerge only if the agents initial strategy valuations are below a specific critical value

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

extent to which least squares cross validation minimises integrated square error in nonparametric density estimation let ho o and c be the windows which minimise mean integrated square error integrated square error and the least squares cross validatory criterion respectively for kernel density estimates it is argued that o not ho should be the benchmark for comparing different data driven approaches to the determination of window size asymptotic properties of ho o and c o and of differences between integrated square errors evaluated at these windows are derived it is shown that in comparison to the benchmark o the observable window c performs as well as the so called optimal but unattainable window ho to both first and second order

optimal designs for longitudinal and functional data summary we propose novel optimal designs for longitudinal data for the common situation where the resources for longitudinal data collection are limited by determining the optimal locations in time where measurements should be taken as for all optimal designs some prior information is needed to implement the optimal designs proposed we demonstrate that this prior information may come from a pilot longitudinal study that has irregularly measured and noisy measurements where for each subject one has available a small random number of repeated measurements that are randomly located on the domain a second possibility of interest is that a pilot study consists of densely measured functional data and one intends to take only a few measurements at strategically placed locations in the domain for the future collection of similar data we construct optimal designs by targeting two criteria optimal designs to recover the unknown underlying smooth random trajectory for each subject from a few optimally placed measurements such that squared prediction errors are minimized optimal designs that minimize prediction errors for functional linear regression with functional or longitudinal predictors and scalar responses again from a few optimally placed measurements the optimal designs proposed address the need for sparse data collection when planning longitudinal studies by taking advantage of the close connections between longitudinal and functional data analysis we demonstrate in simulations that the designs perform considerably better than randomly chosen design points and include a motivating data example from the baltimore longitudinal study of aging the designs are shown to have an asymptotic optimality property

discrete time norms of flexible structures abstract the paper presents and analyses the h 2 h and hankel norms of flexible structures the analysis is conducted for the discrete time models of structures and compared with the continuous time results the structural state space models are presented in modal co ordinates closed form expressions for norms of structural modes are obtained and norms of a structure are determined from the modal norms the relationships between the hankel h and h 2 modal norms are derived in addition the paper shows that the discrete time hankel and h norms converge to the continuous time counterparts when the sampling time approaches zero however the h 2 norm does not

boosting for high dimensional linear models we prove that boosting with the squared error loss l 2 boosting is consistent for very high dimensional linear models where the number of predictor variables is allowed to grow essentially as fast as o exp sample size assuming that the true underlying regression function is sparse in terms of the 11 norm of the regression coefficients in the language of signal processing this means consistency for de noising using a strongly overcomplete dictionary if the underlying signal is sparse in terms of the l 1 norm we also propose here an aic based method for tuning namely for choosing the number of boosting iterations this makes l 2 boosting computationally attractive since it is not required to run the algorithm multiple times for cross validation as commonly used so far we demonstrate l 2 boosting for simulated data in particular where the predictor dimension is large in comparison to sample size and for a difficult tumor classification problem with gene expression microarray data

generating functional analysis of minority games with real market histories it is shown how the generating functional method of de dominicis can be used to solve the dynamics of the original version of the minority game mg in which agents observe real as opposed to fake market histories here one again finds exact closed equations for correlation and response functions but now these are defined in terms of two connected effective non markovian stochastic processes a single effective agent equation similar to that of the fake history models and a second effective equation for the overall market bid itself the latter is absent in fake history models the result is an exact theory from which one can calculate from first principles both the persistent observables in the mg and the distribution of history frequencies

incorporating inertia into multiagent systems we consider a model that demonstrates the crucial role of inertia and stickiness in multiagent systems based on the minority game the inertia of an agent is introduced into the game model by allowing agents to apply hypothesis testing when choosing their best strategies thereby reducing their reactivity toward changes in the environment we find by extensive numerical simulations that our game shows a remarkable improvement of global cooperation throughout the whole phase space in other words the maladaptation behavior due to over reaction of agents is removed these agents are also shown to be advantageous over the standard ones which are sometimes too sensitive to attain a fair success rate we also calculate analytically the minimum amount of inertia needed to achieve the above improvement our calculation is consistent with the numerical simulation results finally we review some related works in the field that show similar behaviors and compare them to our work

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

another balanced controller reduction algorithm the problem of controller reduction has been addressed by a number of authors 1 4 using the concept of balanced coordinates originally proposed by moore 5 this note establishes a more direct link between the riccati balancing methods in 2 4 and moore s work and proposes yet another candidate method as a logical extension of 2 4

principal component analysis of flexible systems open loop case a generic class of flexible systems characterized by finitely many lightly damped harmonic oscillators is analyzed by means of the open loop principal component analysis that is singular value analysis and gramian balancing as the main result it is shown that as the damping ratio goes to zero the balanced state coordinates are decoupled and coincide with the modal coordinates further simple formulas expressing the asymptotic singular values as functions of the modal parameters are derived

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

on balanced realizations of bounded real and positive real functions based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions the ober balanced canonical forms for the classes of bounded real and positive real functions are derived this uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials

statistical mechanics of the mixed majority minority game with random external information we study the asymptotic macroscopic properties of the mixed majority minority game modelling a population in which two types of heterogeneous adaptive agents namely fundamentalists driven by differentiation and trend followers driven by imitation interact the presence of a fraction f of trend followers is shown to induce a a significant loss of informational efficiency with respect to a pure minority game in particular an efficient unpredictable phase exists only for f 1 2 we solve the model by means of an approximate static replica theory and by a direct dynamical generating functional technique the two approaches coincide and match numerical results convincingly

variable selection for the prediction of c 0 1 valued autoregressive processes using reproducing kernel hilbert spaces a model for the prediction of functional time series is introduced where observations are assumed to be continuous random functions we model the dependence of the data with a nonstandard autoregr

brief paper balanced truncation model reduction for systems with inhomogeneous initial conditions we present a rigorous approach to extend balanced truncation model reduction btmr to systems with inhomogeneous initial conditions we provide an estimate for the error between the input output maps of the original and of the reduced initial value system and we illustrate numerically the superiority of our approach over the naive application of btmr when btmr is applied to linear time invariant systems with inhomogeneous initial conditions it is crucial that the initial data are well represented by the subspaces generated by btmr this requirement is often ignored or it is avoided by making the restrictive assumption that the initial data are zero to ensure that the initial data are well represented by the btmr subspaces we add auxiliary inputs determined by the initial data

replica symmetry breaking in the minority game we extend and complete recent work concerning the analytic solution of the minority game nash equilibria ne of the game have been found to be related to the ground states of a disordered hamiltonian with replica symmetry breaking rsb signalling the presence of a large number of ne here we study the number of ne both analytically and numerically we then analyse the stability of the recently obtained replica symmetric solution and in the region where it becomes unstable derive the solution within one step rsb approximation we are finally able to draw a detailed phase diagram of the model

statistical mechanics of systems with heterogeneous agents minority games we study analytically a simple game theoretical model of heterogeneous interacting agents we show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz such a stationary state differs from the nash equilibrium where each agent maximizes her own utility the latter turns out to be characterized by a replica symmetry broken structure numerical results fully agree with our analytical findings

dynamics of a spherical minority game we present an exact dynamical solution of a spherical version of the batch minority game mg with random external information the control parameters in this model are the ratio of the number of possible values for the public information over the number of agents and the radius of the spherical constraint on the microscopic degrees of freedom we find a phase diagram with three phases two without anomalous response an oscillating versus a frozen state and a further frozen phase with divergent integrated response in contrast to standard mg versions we can also calculate the volatility exactly our study reveals similarities between the spherical and the conventional mg but also intriguing differences numerical simulations confirm our analytical results

effective history length of the minority game it is known that the memory is relevant in the symmetric phase of the minority game in our previous work we have successfully explained the quasi periodic behavior of the game in the symmetric phase with the help of the probability theory based on this explanation we are able to determine how the history length affects the variance of the system in this paper by using some particular types of fake history such as periodic type and random type we determine how efficient the history length has been used in the standard game furthermore the analysis on the effective history length strongly supports the result we proposed previously that there are three distinct phases in the minority game r 2007 elsevier b v all rights reserved

generating functional analysis of minority games with inner product strategy definitions we use generating functional methods to solve the so called inner product versions of the minority game mg with fake and or real market histories by generalizing the theory developed recently for look up table mgs with real histories the phase diagrams of the look up table and inner product mg versions are generally found to be identical with the exception of inner product mgs where histories are sampled linearly which are found to be structurally critical however we encounter interesting differences both in the theory where the role of the history frequency distribution in look up table mgs is taken over by the eigenvalue spectrum of a history covariance matrix in inner product mgs and in the static and dynamic phenomenology of the models our theoretical predictions are supported by numerical simulations

functional samples and bootstrap for predicting sulfur dioxide levels in this article we give enhancements of several functional techniques to forecast sulfur dioxide levels near a power plant the data are considered as a time series of curves assuming a lag one dependence the predictions are computed using the functional kernel with local bandwith and the linear autoregressive hilbertian model we carry out the estimation with a so called historical matrix which is a subsample that emphasizes uncommon shapes we introduce a bootstrap method to evaluate the range of the forecasts which uses fraiman and muniz s order for functional data finally we compare our functional techniques with neural networks and semiparametric methods and find that the former models are often more effective

generating functional analysis of batch minority games with arbitrary strategy numbers both the phenomenology and the theory of minority games mg with more than two strategies per agent are different from those of the conventional and extensively studied case s 2 mgs with s 2 exhibit nontrivial statistics of the frequencies with which the agents select from their available decision making strategies with far reaching implications in the few theoretical mg studies with s 2 published so far these statistics could not be calculated analytically this prevented solution even in ergodic stationary states equations for order parameters could only be closed approximately using simulation data here we carry out a generating functional analysis of fake history batch mgs with arbitrary values of s and give an analytical solution of the strategy frequency problem this leads to closed equations for order parameters in the ergodic regime exact expressions for strategy selection statistics and phase diagrams our results find perfect confirmation in numerical simulations

peter hall functional data analysis and random objects functional data analysis has become a major branch of nonparametric statistics and is a fast evolving field peter hall has made fundamental contributions to this area and its theoretical underpinnings he wrote more than 25 papers in functional data analysis between 1998 and 2016 and from 2005 on was a tenured faculty member with a 25 appointment in the department of statistics at the university of california davis this article describes aspects of his appointment and academic life in davis and also some of his major results in functional data analysis along with a brief history of this area it concludes with an outlook on new types of functional data and an emerging field of random objects that subsumes functional data analysis as it deals with more complex data structures