lower bounds of the topological entropy of maps of y abstract we give the best lower bound of the topological entropy of a continuous map of the space y z c z 3 0 1 into itself with 0 0 as a function of its set of periods

scrambled sets of continuous maps of 1 dimensional polyhedra let k be a 1 dimensional simplicial complex in r3 without isolated vertexes x iki be the polyhedron of k with the metric dk induced by k and f x x be a continuous map in this paper we prove that if k is finite then the interior of every scrambled set of f in x is empty we also show that if k is an infinite complex then there exist continuous maps from x to itself having scrambled sets with nonempty interiors and if x r or r then there exist c maps of x with the whole space x being a scrambled set

curvature measures for multiresponse regression models a multiresponse model presented in this study is two fold first a nonuniformity measure is developed to measure the validity of the approximation of the determinant contours by generalised least squares contours secondly curvature measures similar to those of bates watts are calculated for the generalised least squares model to evaluate the validity of the linear approximation of the model functions in a neighbourhood of the parameter estimates

parameter transformations for improved approximate confidence regions in nonlinear least squares in a previous paper it was shown that parameter effects nonlinearities of a nonlinear regression model experimental design parameterization combination can be quantified by means of a parameter effects curvature array a based on second derivatives of the model function in this paper the individual terms of a are interpreted and local compensation methods are suggested a method of computing the parameter effects array corresponding to a transformed set of parameters is given and we discuss how this result could be used to determine reparameterizations which have zero local parameter effects nonlinearity

a note on parameter effects curvature abstract the parameter effects curvature measure proposed by bates and watts 1980 is examined for a growth model and the fieller creasy problem exact confidence regions are constructed and compared to linear approximation regions for the growth model the agreement between the regions is good despite high curvature in the fieller creasy problem it is shown that the agreement can be quite poor despite low curvature

curvature adjusted optimal design of sampling times for the inference of pharmacokinetic compartment models in pharmacokinetics compartment models are often used to describe the time course of blood concentration after the administration of a drug in this article we propose an optimal design criterion for precise estimation of parameters included in the compartment model and illustrate the non sequential design of sampling times of blood drug concentration data in individual pharmacokinetics the proposed optimal design criterion minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated by the curvature array therefore the proposed criterion considers the intrinsic and parameter effects nonlinearity underlying the compartment model and so is applicable in a pharmacokinetic experiment where the sample size of the blood drug concentration data is quite small

ridge analysis with noise variables ridge analysis is a graphical and inferential method for exploring optimum factor levels of a response surface at fixed distances from the center of the experimental region this article proposes an approach to doing a ridge analysis for optimizing a response surface in the presence of noise variables we extend the ridge analysis method of peterson to include some of the factors as noise variables this approach allows an investigator to explore factor combinations that lower the mean squared error about a target value while at the same time keeping track of how much the mean response differs from the target value it also allows an investigator to compute a simultaneous confidence band about the root mean squared error about a target value this provides a guidance band to aid in determining optimal levels of operation a variety of factor constraints can be imposed including those found in mixture experiments in addition we propose a modification of our approach that can be used for larger is bett

power transformations and reparameterizations in nonlinear regression models a two stage procedure is proposed to achieve normality and homoscedasticity of the classical error assumptions and to remove nonlinearity of the regression function this systematic approach involves power transformations and reparameterization test statistics are obtained to assess the necessity of power transformations and the validity of homoscedasticity of the errors in addition nonlinearity measures are provided to diagnose the accuracy of linearization based approximate confidence regions for parameters numerical illustrations of the two stage procedure arc presented

coexistence of uniquely ergodic subsystems of interval mappings we prove that if i f has a uniquely ergodic subsystem x f x then for every s f i f has a uniquely ergodic subsystem y f y such that f s where i 0 1 f c i i

parameter plots in nonlinear regression summary we propose graphical methods for displaying relevant information on a selected parameter from a normal nonlinear regression model it is shown that the usual extension of added variable plots from linear to nonlinear regression can fail to reveal important diagnostic information and that this information can be recovered by using a parameter plot that depends on selected elements of the parameter effects curvature array

confidence regions for parameter subsets in nonlinear regression on utilise la methode a deux niveaux proposee par hamilton watts et bates 1982 pour obtenir des approximations ameliorees des regions de confiance

confidence curves in nonlinear regression abstract standard wald confidence regions for parameters in a normal nonlinear regression model often fail to capture accurately the uncertainty of estimation as reflected by the corresponding profile log likelihood we present a graphical method along with a stable computational algorithm for inference on scalar parameters in a nonlinear regression model

nonlinear experiments optimal design and inference based on likelihood abstract nonlinear experiments involve response and regressors that are connected through a nonlinear regression type structure examples of nonlinear models include standard nonlinear regression logistic regression probit regression poisson regression gamma regression inverse gaussian regression and so on the fisher information associated with a nonlinear experiment is typically a complex nonlinear function of the unknown parameter of interest as a result we face an awkward situation designing an efficient experiment will require knowledge of the parameter but the purpose of the experiment is to generate data to yield parameter estimates our principal objective here is to investigate proper designing of nonlinear experiments that will let us construct efficient estimates of parameters we focus our attention on a very general nonlinear setup that includes many models commonly encountered in practice the experiments considered have two fundamental stages a static design in the initial stage

approximate confidence limits for a parameter function in nonlinear regression on propose une methode a deux etapes la premiere utilisant une transformation des parametres simples pour definir de nouveaux parametres la seconde necessitant l utilisation d une region de confiance ellipsoidale pour obtenir un intervalle de confiance

geometry of nonlinear least squares with applications to sloppy models and optimization parameter estimation by nonlinear least squares minimization is a common problem that has an elegant geometric interpretation the possible parameter values of a model induce a manifold within the space of data predictions the minimization problem is then to find the point on the manifold closest to the experimental data we show that the model manifolds of a large class of models known as sloppy models have many universal features they are characterized by a geometric series of widths extrinsic curvatures and parameter effect curvatures which we describe as a hyper ribbon a number of common difficulties in optimizing least squares problems are due to this common geometric structure first algorithms tend to run into the boundaries of the model manifold causing parameters to diverge or become unphysical before they have been optimized we introduce the model graph as an extension of the model manifold to remedy this problem we argue that appropriate priors can remove the boundaries and further improve the convergence rates we show that typical fits will have many evaporated parameters unless the data are very accurately known second bare model parameters are usually ill suited to describing model behavior cost contours in parameter space tend to form hierarchies of plateaus and long narrow canyons geometrically we understand this inconvenient parametrization as an extremely skewed coordinate basis and show that it induces a large parameter effect curvature on the manifold by constructing alternative coordinates based on geodesic motion we show that these long narrow canyons are transformed in many cases into a single quadratic isotropic basin we interpret the modified gauss newton and levenberg marquardt fitting algorithms as an euler approximation to geodesic motion in these natural coordinates on the model manifold and the model graph respectively by adding a geodesic acceleration adjustment to these algorithms we alleviate the difficulties from parameter effect curvature improving both efficiency and success rates at finding good fits

marginal curvatures and their usefulness in the analysis of nonlinear regression models abstract interval estimation for individual parameters in nonlinear regression is examined in relation to curvature of the solution locus criteria are developed to define severe curvature and distinguish cases in which linear methods may be used from those in which parameter transformations are necessary a formula for improving the linear approximation in cases of relatively mild curvature is proposed for example suppose that interest is focused on one specific parameter j from among 1 2 p in a nonlinear regression model a crude approach to setting 95 confidence limits to j might proceed as follows first calculate the standard error of the estimate as the square root of its asymptotic variance then call the confidence limits in this article higher order correction terms and are proposed that enable us to develop a power series expansion for the confidence limits as and respectively here denotes some estimate of 2 the variance of a single observation

practical scheffe type credibility intervals for variables of a groundwater model simultaneous scheffe type credibility intervals the bayesian version of confidence intervals for variables of a groundwater flow model calibrated using a bayesian maximum a posteriori procedure were derived by cooley 1993b it was assumed that variances reflecting the expected differences between observed and model computed quantities used to calibrate the model are known whereas they would often be unknown for an actual model in this study the variances are regarded as unknown and variance variability from observation to observation is approximated by grouping the data so that each group is characterized by a uniform variance the credibility intervals are calculated from the posterior distribution which was developed by considering each group variance to be a random variable about which nothing is known a priori then eliminating it by integration numerical experiments using two test problems illustrate some characteristics of the credibility intervals nonlinearity of the statistical model greatly affected some of the credibility intervals indicating that credibility intervals computed using the standard linear model approximation may often be inadequate to characterize uncertainty for actual field problems the parameter characterizing the probability level for the credibility intervals was however accurately computed using a linear model approximation as compared with values calculated using second order and fully nonlinear formulations this allows the credibility intervals to be computed very efficiently

improved methodology for parameter inference in nonlinear hydrologic regression models a new method is developed for the construction of reliable marginal confidence intervals and joint confidence regions for the parameters of nonlinear hydrologic regression models a parameter power transformation is combined with measures of the asymptotic bias and asymptotic skewness of maximum likelihood estimators to determine the transformation constants which cause the bias or skewness to vanish these optimized constants are used to construct confidence intervals and regions for the transformed model parameters using linear regression theory the resulting confidence intervals and regions can be easily mapped into the original parameter space to give close approximations to likelihood method confidence intervals and regions for the model parameters unlike many other approaches to parameter transformation the procedure does not use a grid search to find the optimal transformation constants an example involving the fitting of the michaelis menten model to velocity discharge data from an australian gauging station is used to illustrate the usefulness of the methodology

accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions joint confidence and likelihood regions for the parameters in nonlinear regression models can be defined using the geometric concepts of sample space and solution locus using a quadratic approximation to the solution locus instead of the usual linear approximation it is shown that these inference regions correspond to ellipsoids on the tangent plane at the least squares point accurate approximate inference regions can be obtained by mapping these ellipsoids into the parameter space and measures of the effect of intrinsic nonlinearity on inference can be based on the difference between the tangent plane ellipsoids and the sphere which would be obtained using a linear approximation

fieller s theorem scheffe simultaneous confidence intervals and ratios of parameters of linear and nonlinear mixed effects models the issue of joint confidence region and simultaneous confidence interval estimation for ratios of the parameters of a nonlinear mixed effects model is addressed using a fieller s theorem approach the method presented is similarly applicable to linear mixed effects models in addition previous work on linear fixed effects models is demonstrated to be a special case of the present method the methodology is applied to the ratios of slopes in a study of gestational maturation of placental glucose transfer capacity in sheep

on catch effort methods of estimating animal abundance by making a clear distinction between catch and removal catch effort methods can be applied in broader circumstances than generally believed for instance animals can be added to a population rather than removed data on animal sightings can be treated as catch and more than one kind of data e g data on both sightings and kill can be used jointly to produce population estimates models and estimation procedures are presented which illustrate these possibilities also least squares estimation generally nonlinear can be used in conjunction with any likelihood model and we illustrate this procedure in a simulation study involving both natural and catch hazard functions properly formulated least squares estimates appear to be as accurate and precise as corresponding maximum likelihood estimates

identification via channels the authors main finding is that any object among doubly exponentially many objects can be identified in blocklength n with arbitrarily small error probability via a discrete memoryless channel dmc if randomization can be used for the encoding procedure a novel doubly exponential coding theorem is presented which determines the optimal r that is the identification capacity of the dmc as a function of its transmission probability matrix this identification capacity is a well known quantity namely shannon s transmission capacity for the dmc

normal approximation diagnostics for the cox model summary we discuss two diagnostic methods for assessing the accuracy of the normal approximated confidence region to the likelihood based confidence region for the cox proportional hazards model with censored data the proposed diagnostic methods are extensions of the contour measures of hodges 1987 journal of the american statistical association82 149 154 and cook and tsai 1990 journal of the american statistical association85 770 777 and the curvature measures of jennings 1986 journal of the american statistical association81 471 476 and cook and tsai 1990 these methods are also illustrated in a study of hepatocyte growth factor in patients with lung cancer and a mayo clinic randomized study of participants with primary biliary cirrhosis

evaluation of prediction intervals for expressing uncertainties in groundwater flow model predictions we tested the accuracy of 95 individual prediction intervals for hydraulic heads streamflow gains and effective transmissivities computed by groundwater models of two danish aquifers to compute the intervals we assumed that each predicted value can be written as the sum of a computed dependent variable and a random error testing was accomplished by using a cross validation method and by using new field measurements of hydraulic heads and transmissivities that were not used to develop or calibrate the models the tested null hypotheses are that the coverage probability of the prediction intervals is not significantly smaller than the assumed probability 95 and that each tail probability is not significantly different from the assumed probability 2 5 in all cases tested these hypotheses were accepted at the 5 level of significance we therefore conclude that for the groundwater models of two real aquifers the individual prediction intervals appear to be accurate

stratification of continuous maps of an interval we define the notion of turbulence for a continuous map of an interval into the line and study its relation with periodic and homoclinic points we define also strongly simple orbits and show in particular that they represent periodic orbits with minimum entropy further results are obtained for unimodal maps with negative schwarzian which sharpen recent results of block and hart

standard errors for the eigenvalues in second order response surface models in the interpretation of the geometry of second order response surface models standard errors and confidence intervals for the eigenvalues of the second order coefficient matrix play an important role in this article we propose a new method for estimating the standard errors and hence approximate confidence intervals of these eigenvalues the method is simple in both concept and execution it involves the refitting of a full quadratic model after rotating the coordinate system to coincide with the canonical axes the estimated standard errors of the pure quadratic terms from this refitting are then used as approximate standard errors of the eigenvalues because this approach is based on the canonical form it is geometrically intuitive and easily taught our method is intended as a way for practitioners to get quick estimates of the standard errors of the eigenvalues in our justification of the approach we show that it is equivalent to using the delta method proposed by carter chinchilli and camp

a general approach to confidence regions for optimal factor levels of response surfaces summary for a response surface experiment an approximate hypothesis test and an associated confidence region is proposed for the minimizing or maximizing factor level configuration carter et al 1982 cancer research42 2963 2971 show that confidence regions for optimal conditions provide a way to make decisions about therapeutic synergism the response surface may be constrained to be within a specified bounded region these constraint regions can be quite general this allows for more realistic constraint modeling and a wide degree of applicability including constraints occurring in mixture experiments the usual assumption of a quadratic model is also generalized to include any regression model that is linear in the model parameters an intimate connection is established between this confidence region and the box hunter 1954 biometrika41 190 199 confidence region for a stationary point as a byproduct this methodology also provides a way to construct a confidence interval for the difference between the optimal mean response and the mean response at a specified factor level configuration the application of this confidence region is illustrated with two examples extensive simulations indicate that this confidence region has good coverage properties

primary cycles on the circle we consider cycles i e periodic orbits of continuous degree one maps of the circle one cycleaforces another if every such map that has a cycle which looks like the first also has a cycle which looks like the second we characterize primary cycles i e those which force no other cycle of the same period

converse coding theorems for identification via channels in identification via channels the sum of two types of error probabilities of identification goes to one as the block length of transmission tends to infinity at rates above capacity when channels satisfy some stochastic properties this is well known as a strong converse theorem for the identification via channels in this paper we prove that the sum of two error probabilities tends to one exponentially and derive an explicit lower bound of this exponent function

a profile based approach to parametric sensitivity analysis of nonlinear regression models predictions from a nonlinear regression model are subject to uncertainties propagated from the estimated parameters in the model parameters exerting the strongest influence on model predictions can be identified by a sensitivity analysis in this article a new parametric sensitivity measure is introduced based on the profiling algorithm developed by bates and watts for constructing likelihood intervals for the individual parameters in nonlinear regression models in contrast with traditional sensitivity coefficients this profile based sensitivity measure accounts for both correlation structure among the parameters and model nonlinearity it also provides sensitivity information over wide ranges of parameter uncertainties application of the proposed approach is illustrated with three examples

an alternative computational algorithm for calculating the nonlinearity of regression models with two parameters this paper presents an alternative computational algorithm for finding the root mean square rms curvature as a nonlinearity measure of nonlinear regression model nlr model with two parameters specially the suggested algorithm is applied on a nlr model in which there is a conditionally linear parameter the algorithm is mainly based on the work presented in d m bates d g watts relative curvature measures of nonlinearity with discussion journal of the royal statistical society 42 1 1980 1 25 the algorithm is suited for implementation using computer algebra systems cas such as maple matlab macsyma and mathematica the researchers compare the results with the corresponding output of the s plus program for the same considering example and the results in some earlier works

a general approach to ridge analysis with confidence intervals in response surface methodology ridge analysis is a graphical technique for interpreting response surfaces particularly those of three or more dimensions standard ridge analysis techniques employ a second order polynomial regression model an approach to ridge analysis is presented here that generalizes its current scope this approach allows the use of a general regression model for the response surface within the class of linear models conservative or approximate simultaneous confidence intervals about the optimal mean responses for varying radii can be obtained these simultaneous confidence intervals form a guidance band to aid the experimenter in determining optimal levels of operation a variety of explanatory variable constraints including mixture constraints can also be imposed for the standard quadratic model these confidence intervals can be easily modified to obtain confidence bounds about the maximum or minimum eigenvalues of b the matrix of second order regression coefficients

multiresponse estimation with special application to linear systems of differential equations we present a multiresponse estimation procedure for parameters of systems described by as the procedure features a generalized gauss newton algorithm for optimizing the determinant criterion efficient evaluation of the expectation function and its derivatives directly from the reaction network or compartment diagram automatic determination of starting values and eficient computational procedures for handling linear constraints on the responses

general nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to the wiretap channel several nonasymptotic formulas are established in channel resolvability and identification capacity and they are applied to the wiretap channel by using these formulas the epsi capacities of the above three problems are considered in the most general setting where no structural assumptions such as the stationary memoryless property are made on a channel as a result we solve an open problem proposed by han and verduacute moreover we obtain lower bounds of the exponents of error probability and the wiretapper s information in the wiretap channel

inverse estimates from a multiresponse model summary in a multiresponse model a multivariate y is modelled as a nonlinear function of the independent variable x the inverse estimate of x for a given observation on y is the object of study in this paper a straightforward method is proposed for estimating x and setting confidence limits to the estimate an example of determining age from the lengths of horns of a rhinoceros is given animal based on horn lengths only we specifically deal with the situation in which the response functions are nonlinear in the unknown parameters the method proposed is to augment the original data set with the additional measure ments lag and vpg of the extra animal whose age is needed and a guessed age xi varying values of xg will generate varying values of the maximized likelihood conditional on xg and these define the profile likelihood kalbfleisch and sprott 1970 for xg this profile likelihood can be used and interpreted in a number of ways all basically intended to illustrate the reliability of the age estimate a cautionary word is required the step of associating the profile likelihood with classical confidence limits for the true value of xg is difficult and only approximate solutions are available to that extent this paper should not be regarded as proposing a definitive answer there are open questions raised in particular it would be valuable to quantify more precisely the sampling distribution of the profile likelihood function the problem posed here is a version of the multivariate calibration problem the literature although sometimes couched in more general terms invariably deals in detail only with response models that are linear in the unknown parameters 0 brown 1982 gives a broad review of the linear multivariate calibration problem and outlines both bayesian and non bayesian solutions hunter and lamboy 1981 review the linear problem from a bayesian viewpoint brown and sundberg 1987 1989 propose using the profile likelihood of x minder and whitney 1975 derived a marginal likelihood approach the classic paper on univariate linear calibration is that of fieller 1940 and many modern ideas can be related back to the fieller fiducial limits oman and wax 1984 give an interesting application of linear multivariate calibration in estimating fetal age given two

linear orderings and the sets of periods for star maps we show that baldwin s characterization of the set of periods of continuous self maps of the n star can be expressed in terms of a finite number of linear orderings