autoregressive age period cohort models summary age period cohort analysis of vital data has received much attention recently and it is already well known that the exact linear relation of the three time factors creates a non identifiability problem previous studies have shown that the curvature terms of these factors are estimable but the linear trends are not however little attention has been paid to the possibility that the effects due to cohort and or period might change through time stochastically rather than deterministically and hence display a stochastic trend in this paper we model the cohort effects as an ar 1 process and use lung cancer mortality data from 1966 to 1990 for males in taiwan as an example the parameters are identifiable in the proposed model and the estimates are found to be stable however the assumption made in the model should be carefully considered before using our methodology

quasi periodically forced nonlinear helmholtz oscillators in this paper we study a model that describes the dynamics of the tidal elevation within an almost enclosed short basin that is connected to a tidal sea by a narrow strait this model has the form of a forced nonlinear helmholtz oscillator the forcing is prescribed by the tide at sea and has a quasi periodic character of a special nature since the difference between the frequencies of the lunar and solar components of the forcing tide are very small we focus on the interactions between the nonlinearity in the oscillator caused by the geometry of the basin and the external forcing tide the behavior of small amplitude solutions of the weakly and quasi periodically forced helmholtz oscillator is studied by a combination of the averaging method and the melnikov method we construct three different melnikov functions these melnikov functions enable us to distinguish between six structurally different types of chaotic behavior

the estimation of age period and cohort effects for vital rates in models for vital rates which include effects due to age period and cohort there is aliasing due to a linear dependence among these three factors this dependence arises both when age and period intervals are equal and when they are not one solution to the dependence is to set an arbitrary constraint on the parameters estimable functions of the parameters are invariant to the particular constraint applied for evenly spaced intervals deviations from linearity are estimable but only a linear function of the three slopes is estimable when age and period intervals have different widths further aliasing occurs it is assumed that the number of deaths in the numerator of the rate equation has a poisson distribution the calculations are illustrated with data on mortality from prostate cancer among nonwhites in the u s

chaotic motions near homoclinic manifolds and resonant tori in quasiperiodic perturbations of planar hamiltonian systems abstract we study chaotic dynamics of nonlinear oscillators with the form of a two frequency quasiperiodic perturbation of a planar hamiltonian system possessing a homoclinic orbit whose interior contains a one parameter family of periodic orbits in the extended phase space the unperturbed system has a three dimensional homoclinic manifold and a one parameter family of invariant 3 tori using melnikov s technique and the second order averaging method we show that chaotic motions may exist near the unperturbed homoclinic manifold and the unperturbed resonant tori these chaotic motions result from transverse intersection between the stable and unstable manifolds of normally hyperbolic invariant tori and are characterized by a generalization of the bernoulli shift we also give an example for the quasiperiodically forced duffing oscillator and demonstrate the existence chaotic motions by numerical simulation

tests of hypotheses in overdispersed poisson regression and other quasi likelihood models abstract test statistics for evaluating the significance of added variables in a regression equation are developed for mixed poisson models where the structural parameter that determines the mean variance relationship var 2 is estimated by the method of moments and the regression coefficients are estimated by quasi likelihood the formulas presented for test statistics and related estimating equations are applicable generally to quasi likelihood models specified by an arbitrary mean value function x together with a variance function v that contains one or more unknown parameters two versions of the wald and score tests are investigated one calculated from the usual model based covariance matrix whose validity depends on correct specification of the variance function and another using an empirical covariance matrix that has a more general asymptotic justification monte carlo simulations demonstrate that the quasi likelihood method of moments ql m procedures yield ap

correlated poisson models for age period cohort analysis age period cohort apc models are widely used to analyze population level rates particularly cancer incidence and mortality these models are used for descriptive epidemiology comparative risk analysis and extrapolating future disease burden traditional apc models have 2 major limitations 1 they lack parsimony because they require estimation of deviations from linear trends for each level of age period and cohort and 2 rates observed at similar ages periods and cohorts are treated as independent ignoring any correlations between them that may lead to biased parameter estimates and inefficient standard errors we propose a novel approach to estimation of apc models using a spatially correlated poisson model that accounts for over dispersion and correlations in age period and cohort simultaneously we treat the outcome of interest as event rates occurring over a grid defined by values of age period and cohort rates defined in this manner lend themselves to well established approaches from spatial statistics in which correlation among proximate observations may be modeled using a spatial random effect through simulations we show that in the presence of spatial dependence and over dispersion 1 the correlated poisson model attains lower aic 2 the traditional apc model produces biased trend parameter estimates and 3 the correlated poisson model corrects most of this bias we illustrate our approach using brain and breast cancer incidence rates from the surveillance epidemiology and end results program of the united states our approach can be easily extended to accommodate comparative risk analyses and interpolation of cells in the lexis with sparse data

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

predictive model assessment for count data we discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for count data our proposals include a nonrandomized version of the probability integral transform marginal calibration diagrams and proper scoring rules such as the predictive deviance in case studies we critique count regression models for patent data and assess the predictive performance of bayesian age period cohort models for larynx cancer counts in germany the toolbox applies in bayesian or classical and parametric or nonparametric settings and to any type of ordered discrete outcomes

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

emergence of quasiunits in the one dimensional zhang model we study the zhang model of sandpile on a one dimensional chain of length l where a random amount of energy is added at a randomly chosen site at each time step we show that in spite of this randomness in the input energy the probability distribution function of energy at a site in the steady state is sharply peaked and the width of the peak decreases as l 1 2 for large l we discuss how the energy added at one time is distributed among different sites by topplings with time we relate this distribution to the time dependent probability distribution of the position of a marked grain in the one dimensional abelian model with discrete heights we argue that in the large l limit the variance of energy at site x has a scaling form l 1 g x l where g xi varies as ln 1 xi for small xi which agrees very well with the results from numerical simulations

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

height correlations in the abelian sandpile model the authors study the distribution of heights in the self organized critical state of the abelian sandpile model on a d dimensional hypercubic lattice they calculate analytically the concentration of sites having minimum allowed value in the critical state they also calculate in the critical state the probability that the heights at two sites separated by a distance r would both have minimum values and show that the lowest order r dependent term in it varies as r 2d for large r

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

age period cohort models using smoothing splines a generalized additive model approach age period cohort apc models are used to analyze temporal trends in disease or mortality rates dealing with linear dependency among associated effects of age period and cohort however the nature of sparseness in such data has severely limited the use of apc models to deal with these practical limitations and issues we advocate cubic smoothing splines we show that the methods of estimable functions proposed in the framework of generalized linear models can still be considered to solve the non identifiability problem when the model fitting is within the framework of generalized additive models with cubic smoothing splines through simulation studies we evaluate the performance of the cubic smoothing splines in terms of the mean squared errors of estimable functions our results support the use of cubic smoothing splines for apc modeling with sparse but unaggregated data from a lexis diagram copyright 2013 john wiley sons ltd

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

the natural variability of vital rates and associated statistics the first concern of this work is the development of approximations to the distributions of crude mortality rates age specific mortality rates age standardized rates standardized mortality ratios and the like for the case of a closed population or period study it is found that assuming poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate poisson distribution it is suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic poisson distribution for the number of deaths with mean proportional to the population size and an exponential function of a linear combination of the explanatories such a model is fit to mortality data for canadian females classified by age and year a dynamic variant of the model is further fit to the time series of total female deaths alone by year the models with extra poisson variation are found to lead to substantially improved fits excerpt

toppling distributions in one dimensional abelian sandpiles we consider toppling distributions for the abelian sandpile model in one dimension we study the avalanche mass and duration distributions in the thermodynamic limit we also investigate their dependence on the seeding distribution which describes how sand is dropped none of the models shows criticality

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

monte carlo confidence intervals des intervalles de confiance non parametriques obtenus par la methode de monte carlo sont utiles quand les intervalles obtenus par les methodes analytiques sont indisponibles ou ne correspondent pas au niveau nominal on fournit des formules permettant de generer un intervalle de niveau donne et de quantifier la variation due a la methode de monte carlo

fundamental study from sandpiles to sand automata we introduce a new model for simulating natural phenomena we address several issues topology basic set properties like injectivity and surjectivity reversibility and decidability questions about a special kind of conservation law called grain conservation and ultimate periodicity

regression by local fitting methods properties and computational algorithms abstract local regression is a procedure for estimating regression surfaces by the local fitting of linear or quadratic functions of the independent variables in a moving fashion that is analogous to how a moving average is computed for a time series the advantage of the methodology over the global fitting of parametric functions of the independent variables by least squares the current paradigm in regression studies is that a much wider class of regression functions can be estimated without distortion in this paper we discuss the methods their statistical properties and computational algorithms

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

prediction of cancer incidence in the nordic countries empirical comparison of different approaches prediction of the future number of cancer cases is of great interest to society the classical approach is to use the age period cohort model for making cancer incidence predictions we made an empirical comparison of different versions of this model using data from cancer registries in the nordic countries for the period 1958 1997 we have applied 15 different methods to 20 sites for each sex in denmark finland norway and sweden median absolute value of the relative difference between observed and predicted numbers of cases for these 160 combinations of site sex and country was calculated the medians varied between 10 4 per cent and 15 3 per cent in predictions 10 years ahead and between 15 1 per cent and 32 0 per cent for 20 year predictions we have four main conclusions i projecting current trends worked better than assuming that future rates are equal to present rates ii the method based on the multiplicative apc model often overestimated the number of cancer cases due to its exponential growth over time but using a power function to level off this growth improved the predictions iii projecting only half of the trend after the first 10 years also gave better long term predictions iv methods that emphasize trends in the last decade seem to perform better than those that include earlier time trends

bayesian extrapolation of space time trends in cancer registry data nous avons utilise un modele bayesien sur des donnees de mortalite par cancer de l estomac en allemagne de l ouest les donnees sont stratifiees par groupe d ge annee et district en utilisant sur cesdonnees un modele age periode cohorte avec une composante spatiale nous avons cherche a mettre en evidence des interactions spatic temporelles nous avons de plus etudie si des interactions periode composante spatiale ou cohorte composante spatiale etaient plus appropriees pour faire de la prediction sur les taux de mortalite pour chacun des elements du modele bayesien nous avons utilise des chaines de markov avec marche aleatoire gaussienne comme a priori les inferences statistiques ont ete obtenues a partir d un algorithme d echantillonnage par bloc recemment propose dans la litterature pour ce genre de modele

second order averaging and chaos in quasiperiodically forced weakly nonlinear oscillators abstract using the averaging method and melnikov s technique we study the chaotic dynamics of a class of quasiperiodically forced weakly nonlinear oscillators averaging is carried out to second order and the existence of transverse homoclinic tori resulting in chaotic dynamics for quasiperiodically forced oscillators is detected we also give an example for a duffing type weakly nonlinear oscillator subjected to combined parametric and external forcing in the case of double resonance in which primary external resonance and subharmonic or superharmonic parametric resonance exist simultaneously

modeling of time trends and interactions in vital rates using restricted regression splines summary for the analysis of time trends in incidence and mortality rates the age period cohort apc model has became a widely accepted method the considered data are arranged in a two way table by age group and calendar period which are mostly subdivided into 5 or 10 year intervals the disadvantage of this approach is the loss of information by data aggregation and the problems of estimating interactions in the two way layout without replications in this article we show how splines can be useful when yearly data i e 1 year age groups and 1 year periods are given the estimated spline curves are still smooth and represent yearly changes in the time trends further it is straightforward to include interaction terms by the tensor product of the spline functions if the data are given in a nonrectangular table e g 5 year age groups and 1 year periods the period and cohort variables can be parameterized by splines while the age variable is parameterized as fixed effect levels which leads to a semiparametric apc model an important methodological issue in developing the nonparametric and semiparametric models is stability of the estimated spline curve at the boundaries here cubic regression splines will be used which are constrained to be linear in the tails another point of importance is the nonidentifiability problem due to the linear dependency of the three time variables this will be handled by decomposing the basis of each spline by orthogonal projection into constant linear and nonlinear terms as suggested by holford 1983 biometrics 39 311 324 for the traditional apc model the advantage of using splines for yearly data compared to the traditional approach for aggregated data is the more accurate curve estimation for the nonlinear trend changes and the simple way of modeling interactions between the time variables the method will be demonstrated with hypothetical data as well as with cancer mortality data the analysis of time trends in vital rates such as incidence and mortality rates has had an increasing part in the epidemiological literature over the last years and is accompanied by a permanent discussion of methodological issues and developments three time variables that are often considered in such an investigation are life time age calendar time period and birth time cohort the age distribution of a population is the most important influence factor for cancer rates which should always be adjusted for in a time trend analysis period effects are changes in the rates which occur at a particular time consistently for each age group they are mostly due to new effective improvements in treatment or diagnosis or a change in disease classification at a certain date the third time variable is the cohort effect in most cases these can be interpreted as generation effects i e they reflect different lifestyle e g smoking behavior or different environmental exposure from generation to generation of course there are also other interpretations possible to explain the characteristics of period and cohort trends and an observed effect needs careful investigation probably the most comprehensive analysis of time trends in worldwide cancer rates is given in coleman et al 1993

a spatiotemporal back calculation approach to estimate cancer incidence measures incidence rates are an important population level disease risk measure cancer incidence data in the united states which are collected by disease registries have been spatiotemporally sparse back calculation methods can yield incidence estimates for a spatial domain by solving a convolution equation that relates mortality to incidence through survival estimates we propose a novel back calculation approach that uses spatiotemporal age period cohort apc modeling to estimate incidence for spatial units within a region the method is applied to state specific lung cancer data in the united states for males ages 30 to 84 in years 1975 2004 seer data are used to model cancer progression from incidence to mortality along three timescales apc and across four regions using mortality data from national vital statistics system incidence is back calculated for unique birth cohorts in 49 states a conditionally autoregressive model with cubic splines for temporal effects is used to smooth back calculated estimates bayesian estimates of model parameters are obtained using integrated nested laplace approximation results show varying time trends in lung cancer risk across states which may quantify effects of state policies

identification of the age period cohort model and the extended chain ladder model we consider the identification problem that arises in the age period cohort models as well as in the extended chain ladder model we propose a canonical parameterization based on the accelerations of the trends in the three factors this parameterization is exactly identified and eases interpretation estimation and forecasting the canonical parameterization is applied to a class of index sets which have trapezoidal shapes including various lexis diagrams and the insurance reserving triangles

abelian sandpile model a systematic and simple method to find the correlation function of the abelian sandpile model up to any finite order is developed in addition an algorithm for evaluating the distribution function of the avalanche size p s exactly is also discovered along the same line this method is in general more efficient and accurate than the numerical simulation that is currently used in obtaining p s

abelian sandpile model on the bethe lattice the authors study bak tang and wiesenfeld s abelian sandpile model 1987 of self organised criticality on the bethe lattice exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained they also determine the probability distribution of the number of distinct sites that topple at least once the number of toplings at the origin and the total number of toplings in an avalanche the probability that an avalanche consists of more than n toplings varies as n 1 2 for large n the probability that its duration exceeds t decreases as 1 t for large t these exponents are the same as for the critical percolation clusters in mean field theory

approaches to fitting age period cohort models with unequal intervals age period cohort models have provided useful insights into the analysis of time trends for disease rates in spite of the well known identifiability problem unique parameter estimates that avoid arbitrary constraints are provided by estimable functions of the parameter estimates for data that are generated using equal interval widths for age and period the identifiability issue may be expressed in terms of the age period and cohort slopes however when the interval widths are not the same for age and period additional identifiability problems arise these may be represented in terms of macro trends which have the identical identifiability problem seen in the equal interval case and micro trends which are the source of the additional problems a framework for testing estimability is presented and a variety of potentially interesting functions of the parameters considered unlike the equal interval case drift is not estimable for unequal intervals but local drift may be in addition the available functions for forecasting are much more restrictive in the latter case this estimability problem induces cyclical patterns in the estimates of trend as is demonstrated using data on leukaemia in connecticut males but this can be avoided through the use of smoothing splines these methods of are illustrated for three year period and five year age intervals using data on lung cancer mortality in californian women

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

bayesian age period cohort models with versatile interactions and long term predictions mortality and population in finland 1878 2050 age period cohort apc models are widely used for studying time trends of disease incidence or mortality model identifiability has become less of a problem with bayesian apc models these models are usually based on random walk rw1 rw2 smoothing priors for long and complex time series and for long predicted periods these models as such may not be adequate we present two extensions for the apc models first we introduce flexible interactions between the age period and cohort effects based on a two dimensional conditional autoregressive smoothing prior on the age period plane our second extension uses autoregressive integrated ari models to provide reasonable long term predictions to illustrate the utility of our model framework we provide stochastic predictions for the finnish male and female population in 2010 2050 for that we first study and forecast all cause male and female mortality in finland 1878 2050 showing that using an interaction term is needed for fitting and interpreting the observed data we then provide population predictions using a cohort component model which also requires predictions for fertility and migration as our main conclusion ari models have better properties for predictions than the simple rw models do but mixing these prediction models with rw1 or rw2 smoothing priors for observed periods leads to a model that is not fully consistent further research with our model framework will concentrate on using a more consistent model for smoothing and prediction such as autoregressive integrated moving average models with state space methods or gaussian process priors