transition phenomena in cellular automata rule space abstract we define several qualitative classes of cellular automata ca behavior based on various statistical measures and describe how the space of all cellular automata is organized as a cellular automaton is changed by varying entries in its rule table abrupt changes in qualitative behavior may occur these abrupt changes have the character of bifurcations in smooth dynamical systems or of phase transitions in statistical mechanical systems the most complex ca rules exhibit many of the characteristics of second order transitions suggesting an association between computation complexity and critical phenomena

subgroup identification based on differential effect search a recursive partitioning method for establishing response to treatment in patient subpopulations we propose a novel recursive partitioning method for identifying subgroups of subjects with enhanced treatment effects based on a differential effect search algorithm the idea is to build a collection of subgroups by recursively partitioning a database into two subgroups at each parent group such that the treatment effect within one of the two subgroups is maximized compared with the other subgroup the process of data splitting continues until a predefined stopping condition has been satisfied the method is similar to interaction tree approaches that allow incorporation of a treatment by split interaction in the splitting criterion however unlike other tree based methods this method searches only within specific regions of the covariate space and generates multiple subgroups of potential interest we develop this method and provide guidance on key topics of interest that include generating multiple promising subgroups using different splitting criteria choosing optimal values of complexity parameters via cross validation and addressing type i error rate inflation inherent in data mining applications using a resampling based method we evaluate the operating characteristics of the procedure using a simulation study and illustrate the method with a clinical trial example copyright 2011 john wiley sons ltd

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

symmetric and arbitrarily high order birkhoff hermite time integrators and their long time behaviour for solving nonlinear klein gordon equations abstract the klein gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering its computation represents a major challenge the main theme of this paper is the construction of symmetric and arbitrarily high order time integrators for the nonlinear klein gordon equation by integrating birkhoff hermite interpolation polynomials to this end under the assumption of periodic boundary conditions we begin with the formulation of the nonlinear klein gordon equation as an abstract second order ordinary differential equation ode and its operator variation of constants formula we then derive a symmetric and arbitrarily high order birkhoff hermite time integration formula for the nonlinear abstract ode accordingly the stability convergence and long time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix subject to suitable temporal and spatial smoothness a remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for pdes in the literature numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches

criticality in cellular automata abstract using recent results obtained for the transition to turbulence via spatiotemporal intermittency in extended dynamical systems critical cellular automata rules are built thanks to a systematic procedure the continuous phase transition observed in a coupled map lattice is translated into a sequence of cellular automata rules for which the critical properties of the original system are reproduced as precisely as desired it is shown that criticality as understood from the point of view of statistical mechanics is intimately related to the various characteristics of wolfram s class iv rules this suggests in turn the picture of a critical surface in the space of rules and provides the basis for a discussion of the problem of classification schemes we review recent cellular automata studies where questions linked to criticality arise and argue that they are unified in light of our results leaving the relationship between computational and statistical characterizations of critical rules as the central problem for future studies on this subject

chaos in networks of two dimensional homogeneous cellular automata abstract we carefully selected 16 rules from all 65 536 possible rules on a two dimensional network to create the corresponding 16 homogeneous cellular automata varying the initial state in the computer simulation we investigate systematically the properties of the final states of these cellular automata

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

inference for subgroup analysis with a structured logistic normal mixture model in this article we propose a statistical model for the purpose of identifying a subgroup that has an enhanced treatment effect as well as the variables that are predictive of the subgroup membership the need for such subgroup identification arises in clinical trials and in market segmentation analysis by using a structured logistic normal mixture model our proposed framework enables us to perform a confirmatory statistical test for the existence of subgroups and at the same time to construct predictive scores for the subgroup membership the inferential procedure proposed in the article is built on the recent literature on hypothesis testing for gaussian mixtures but the structured logistic normal mixture model enjoys some distinctive properties that are unavailable to the simpler gaussian mixture models with the bootstrap approximations the proposed tests are shown to be powerful and equally importantly insensitive to the choice of tuning parameters as an illustration we analyze a dataset fr

remarks on the rule classification of symmetric two dimensional homogeneous cellular automata abstract results generated in time by the dynamic of some homogeneous cellular automata from random initial states are presented geometrical connections between a rule and a final pattern are analyzed a necessary condition for a stabilization of two macroscopic functions contents and activity is formulated also

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

a conditional error function approach for subgroup selection in adaptive clinical trials growing interest in personalised medicine and targeted therapies is leading to an increase in the importance of subgroup analyses if it is planned to view treatment comparisons in both a predefined subgroup and the full population as co primary analyses it is important that the statistical analysis controls the familywise type i error rate spiessens and debois cont clin trials 2010 31 647 656 recently proposed an approach specific for this setting which incorporates an assumption about the correlation based on the known sizes of the different groups and showed that this is more powerful than generic multiple comparisons procedures such as the bonferroni correction if recruitment is slow relative to the length of time taken to observe the outcome it may be efficient to conduct an interim analysis in this paper we propose a new method for an adaptive clinical trial with co primary analyses in a predefined subgroup and the full population based on the conditional error function principle the methodology is generic in that we assume test statistics can be taken to be normally distributed rather than making any specific distributional assumptions about individual patient data in a simulation study we demonstrate that the new method is more powerful than previously suggested analysis strategies furthermore we show how the method can be extended to situations when the selection is not based on the final but on an early outcome we use a case study in a targeted therapy in oncology to illustrate the use of the proposed methodology with non normal outcomes copyright 2012 john wiley sons ltd

a spline collocation approach for the numerical solution of a generalized nonlinear klein gordon equation in this paper a finite element collocation approach using cubic b splines is employed for the numerical solution of a generalized form of the nonlinear klein gordon equation the efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine gordon equation the numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels

fourier collocation method for solving nonlinear klein gordon equation a fourier collocation scheme is proposed for solving the periodic problem of nonlinear klein gordon equation it keeps the energy discretely and is shown to be stable or generalized stable its convergence is investigated numerical results are also presented

stability and convergence of trigonometric integrator pseudospectral discretization for n coupled nonlinear klein gordon equations abstract this work serves as an improvement on a recent paper dong 2013 9 in which the n coupled nonlinear klein gordon equations were solved numerically by a fully explicit trigonometric integrator fourier pseudospectral tifp method this tifp method is second order accurate in time and spectral order accurate in space however in the previous work there was an absence of rigorous stability and convergence analysis moreover numerical studies in this work suggest that this tifp method suffers from a stability condition o h and h refer to time step and space mesh size to relax such a restriction while keeping the convergence properties and explicitness we propose two modified tifp methods motivated by the mollified impulse and gautschi type integrators for oscillatory odes for the modifications considered here linear stability and rigorous error estimates in the energy space are carried out which are the main achievements gained in this work meanwhile numerical results are also presented ideas of this work also suggest a general framework for proposing and analyzing efficient numerical methods for coupled wave type equations

classification of three dimensional cellular automata abstract cellular automata on a simple cubic lattice are classified according to their fixed points and periods the computer simulations show drastic differences with respect to two dimensions

towards the classification of all boolean cellular automata abstract a classification scheme for all boolean cellular automata models involving an arbitrary number of inputs is proposed the scheme is based upon counting the number of stable points for a given rule i e points whose time development is stable with respect to small perturbations when applied to all cellular automata with 2 3 4 or 5 boolean inputs the latter involving more than 4000 million cases it yields in each case a nontrivial classification i e there is no single class in which nearly all the rules lie in addition it is shown how rules satisfying a given boolean differential equation can easily be obtained

boolean derivatives on cellular automata abstract boolean derivatives are applied to cellular automata they extract the linear part of arbitrary rules and provide an intrinsic classification derivatives are applied to tracing complexity growth and to lattice gases a complete table of derivatives is given for one dimensional elementary cellular automata

subgroup detection and sample size calculation with proportional hazards regression for survival data in this paper we propose a testing procedure for detecting and estimating the subgroup with an enhanced treatment effect in survival data analysis here we consider a new proportional hazard model that includes a nonparametric component for the covariate effect in the control group and a subgroup treatment interaction effect defined by a change plane we develop a score type test for detecting the existence of the subgroup which is doubly robust against misspecification of the baseline effect model or the propensity score but not both under mild assumptions for censoring when the null hypothesis of no subgroup is rejected the change plane parameters that define the subgroup can be estimated on the basis of supremum of the normalized score statistic the asymptotic distributions of the proposed test statistic under the null and local alternative hypotheses are established on the basis of established asymptotic distributions we further propose a sample size calculation formula for detecting a given subgroup effect and derive a numerical algorithm for implementing the sample size calculation in clinical trial designs the performance of the proposed approach is evaluated by simulation studies an application to an aids clinical trial data is also given for illustration

evaluating markers for selecting a patient s treatment selecting the best treatment for a patient s disease may be facilitated by evaluating clinical characteristics or biomarker measurements at diagnosis we consider how to evaluate the potential impact of such measurements on treatment selection algorithms for example magnetic resonance neurographic imaging is potentially useful for deciding whether a patient should be treated surgically for carpal tunnel syndrome or should receive less invasive conservative therapy we propose a graphical display the selection impact si curve that shows the population response rate as a function of treatment selection criteria based on the marker the curve can be useful for choosing a treatment policy that incorporates information on the patient s marker value exceeding a threshold the si curve can be estimated using data from a comparative randomized trial conducted in the population as long as treatment assignment in the trial is independent of the predictive marker estimating the si curve is therefore part of a post hoc analysis to determine whether the marker identifies patients that are more likely to benefit from one treatment over another nonparametric and parametric estimates of the si curve are proposed in this article asymptotic distribution theory is used to evaluate the relative efficiencies of the estimators simulation studies show that inference is straightforward with realistic sample sizes we illustrate the si curve and statistical inference for it with data motivated by an ongoing trial of surgery versus conservative therapy for carpal tunnel syndrome

subgroup analysis via recursive partitioning subgroup analysis is an integral part of comparative analysis where assessing the treatment effect on a response is of central interest its goal is to determine the heterogeneity of the treatment effect across subpopulations in this paper we adapt the idea of recursive partitioning and introduce an interaction tree it procedure to conduct subgroup analysis the it procedure automatically facilitates a number of objectively defined subgroups in some of which the treatment effect is found prominent while in others the treatment has a negligible or even negative effect the standard cart breiman et al 1984 methodology is inherited to construct the tree structure also in order to extract factors that contribute to the heterogeneity of the treatment effect variable importance measure is made available via random forests of the interaction trees both simulated experiments and analysis of census wage data are presented for illustration

classification of triangular and honeycomb cellular automata abstract cellular automata on a two dimensional triangular and honeycomb lattice are classified according to their fixed points and periods the results are compared with other lattices in one of three dimensions we also compare the totalistic rules for all mentioned lattices

change plane analysis for subgroup detection and sample size calculation abstractwe propose a systematic method for testing and identifying a subgroup with an enhanced treatment effect we adopts a change plane technique to first test the existence of a subgroup and then identify the subgroup if the null hypothesis on nonexistence of such a subgroup is rejected a semiparametric model is considered for the response with an unspecified baseline function and an interaction between a subgroup indicator and treatment a doubly robust test statistic is constructed based on this model and asymptotic distributions of the test statistic under both null and local alternative hypotheses are derived moreover a sample size calculation method for subgroup detection is developed based on the proposed statistic the finite sample performance of the proposed test is evaluated via simulations finally the proposed methods for subgroup identification and sample size calculation are applied to a data from an aids study