theory and lattice structure of complex paraunitary filterbanks with filters of hermitian symmetry antisymmetry properties the theory of the real coefficient linear phase filterbank lpfb is extended to the complex case in two ways leading to two generalized classes of m channel filterbanks one is the symmetric antisymmetric filterbank safb where all filters are symmetric or antisymmetric the other is the complex linear phase filterbank clpfb where all filters are hermitian symmetric or hermitian antisymmetric and hence have the linear phase property necessary conditions on the filter symmetry polarity and lengths for the existence of permissible solutions are investigated complete and minimal lattice structures are developed for the paraunitary safb and paraunitary clpfb where the channel number m is arbitrary even or odd and the subband filters could have different lengths with the elementary unitary matrices in the structure of the paraunitary safb constrained to be real and orthogonal the structure covers the most general real coefficient paraunitary lpfbs compared with the existing results the number of parameters is reduced significantly

a high order time accurate loosely coupled solution algorithm for unsteady conjugate heat transfer problems abstract thermal interaction of fluids and solids or conjugate heat transfer cht is encountered in many engineering applications noting that time accurate computations of transient cht problems can be computationally expensive we consider the use of high order implicit time integration schemes which have the potential to be computationally more efficient relative to the commonly used second order implicit schemes for thermally weak couplings we present a loosely coupled solution algorithm where high order implicit explicit imex runge kutta schemes are employed for time integration the imex schemes consist of the explicit first stage singly diagonally implicit runge kutta esdirk schemes for advancing the solution in time within each separate fluid and solid subdomain and the explicit runge kutta erk schemes for explicit integration of part of the coupling terms by considering a numerical example an unsteady conjugate natural convection in an enclosure temporal order preservation of the coupling algorithm without subiterating is demonstrated in addition the stability of the loosely coupled algorithm is investigated numerically for the cht test case when the ratio of the thermal effusivities of the fluid and solid subdomains is much smaller than unity using large fourier numbers of the subdomains is possible indicating that time step size is restricted by accuracy rather than stability furthermore the computational work temporal precision character of several time integration schemes in solving the cht test case is compared over a range of accuracy requirements for time accurate solutions the fourth and fifth order imex schemes are 1 5 times more efficient than crank nicolson and 2 7 times more efficient than bdf2 the computational gain is higher for smaller tolerances

numerical comparisons of high order nonlinear solvers for the transient navier stokes equations based on homotopy and perturbation techniques efficient solvers for the unsteady navier stokes equations are presented a classic time stepping scheme is combined with high order nonlinear solvers coupling homotopy and a perturbation technique polynomial and rational representations are used to approximate the unknowns of the problem a pseudo residual criterion is proposed to improve the efficiency of the solvers the numerical example considered in this paper is the time periodic two dimensional flow around a circular cylinder comparisons with the classical first order newton raphson solver are performed numerical results reveal that a lower number of matrix factorization is needed with the proposed methods decreasing the computational effort

nonparametric bootstrap confidence intervals for discrete regression functions abstract this is a study of a binary response bootstrap for constructing confidence intervals in a nonparametric setup some first order asymptotic results are proved for this method which hold true when generalizing to the multinomial case this method is illustrated by means of an application to real medical data

boundary filter optimization for segmentation based subband coding this paper presents boundary optimization techniques for the nonexpansive decomposition of arbitrary length signals with multirate filterbanks both biorthogonal and paraunitary filterbanks are considered the paper shows how matching moments and orthonormality can be imposed as additional conditions during the boundary filter optimization process it provides direct solutions to the problem of finding good boundary filters for the following cases a biorthogonal boundary filters with exactly matching moments and b orthonormal boundary filters with almost matching moments with the proposed methods numerical optimization is only needed if orthonormality and exactly matching moments are demanded the proposed direct solutions are applicable to systems with a large number of subbands and or very long filter impulse responses design examples show that the methods allow the design of boundary filters with good frequency selectivity

perfect reconstruction fir filter banks some properties and factorizations perfect reconstruction finite impulse response fir filter banks are analyzed both in the z transform and time domains showing the alternatives between designs in the two domains various classes of perfect reconstruction schemes are indicated and relations between previously known systems are given windowed modulated filter banks with low computational complexity and perfect reconstruction are shown new factorizations of polyphase filter matrices leading in particular to linear phase filters are given the computational complexity and the architecture of the new structures are indicated

use of the method of manufactured solutions for the verification of conjugate heat transfer solvers this paper demonstrates the use of the method of manufactured solutions to verify the implementation of tightly coupled conjugate heat transfer for fluid solid solvers the interface conditions in the prescribed manufactured solutions were implemented to mimic real effects such as no slip and temperature heat flux match between the solid and fluid domains the newly developed solid heat transfer solver was verified in standalone mode using this prescribed manufactured solution and was found to have no apparent coding errors our pre existing in house compressible fluid solver eilmer was used to demonstrate the conjugate heat transfer implementation both the fluid and solid solvers showed an expected spatial order of convergence of 2 0 in the standalone mode the coupled conjugate heat transfer mode also showed no coding errors and demonstrated that the spatial order of convergence was again 2 0 the one sided spatial discretisation utilised to enforce the tight coupling for the interface conditions were effectively equivalent to a central difference hence the overall spatial order of the error convergence for the entire domain including the interface was 2 0 the method prescribed in this work can be extended for verification of other conjugate heat transfer solvers in particular for compressible flow scenarios where analytical solutions may not be readily available

analysis synthesis techniques for subband image coding analysis synthesis systems designed for low bit rate image coding their impact on overall system quality and their computational complexity are discussed the investigation focuses on the design of analysis synthesis systems for image coding and the perceptual impact of these systems at low bit rates two objectives are emphasized in developing these systems confining the total size of the subband images to be equal to the original image size and designing the filters so that perceptual distortion is not introduced by the analysis synthesis system methods based on circular convolution and symmetric extensions are developed and discussed in detail the theory design and implementation of both recursive and nonrecursive filtering systems are discussed methods are introduced which display advantages over conventional quadrature mirror filter based approaches

on m channel linear phase fir filter banks and application in image compression this paper investigates the theory and structure of a large subclass of m channel linear phase perfect reconstruction fir filter banks systems with analysis and synthesis filters of length l sub i k sub i m spl beta where spl beta is an arbitrary integer 0 spl les spl beta m and k sub i is any positive integer for this subclass of systems we first investigate the necessary conditions for the existence of linear phase perfect reconstruction filter banks lpprfbs next we develop a complete and minimal factorization for all even channel linear phase paraunitary systems the most general lapped orthogonal transforms to date finally several design examples as well as comparisons with previous generalized lapped orthogonal transforms genlots in image compression are presented to confirm the validity of the theory

filters for distortion free two band multirate filter banks in this paper conditions are given for a two band multi rate filter bank to be alias free and to have a unity frequency response it is shown that the class of quadrature mirror filters qmf s that satisfies these conditions is quite limited a class of filters which does satisfy these conditions is given and a simple procedure for designing filters from this class is presented with an example

numerical simulation of coupled fluid solid problems abstract a multi grid finite volume method is presented for the prediction of coupled fluid solid problems in complex geometries whereby a unified finite volume approach based on the concept of block structured grids is employed for the handling of the complexity of the geometry and the fluid solid coupling a high numerical efficiency is obtained with a non linear multi grid method using a pressure correction based scheme as a smoother a number of numerical experiments were used to verify the accuracy and numerical efficiency of the method additionally to illustrate the capabilities of the approach results for two practical applications are presented

adaptive time step control for the incompressible navier stokes equations abstract adaptive time stepping is an important tool in computational fluid dynamics for controlling the accuracy of simulations and for enhancing their efficiency this paper presents a systematic study of three classes of implicit and linearly implicit time stepping schemes with adaptive time step control applied to a 2d laminar flow around a cylinder schemes diagonal implicit runge kutta dirk methods and rosenbrock wanner row methods the time step is controlled using embedded methods it is shown that several row methods clearly outperform the more standard schemes and the dirk methods the results depend on a prescribed tolerance in the time step control algorithm whose appropriate choice varies from scheme to scheme

mixture of regression models with varying mixing proportions a semiparametric approach in this article we study a class of semiparametric mixtures of regression models in which the regression functions are linear functions of the predictors but the mixing proportions are smoothing functions of a covariate we propose a one step backfitting estimation procedure to achieve the optimal convergence rates for both regression parameters and the nonparametric functions of mixing proportions we derive the asymptotic bias and variance of the one step estimate and further establish its asymptotic normality a modified expectation maximization type em type estimation procedure is investigated we show that the modified em algorithms preserve the asymptotic ascent property numerical simulations are conducted to examine the finite sample performance of the estimation procedures the proposed methodology is further illustrated via an analysis of a real dataset

analysis and application of high order implicit runge kutta schemes for unsteady conjugate heat transfer a strongly coupled approach abstract thermal interaction of fluids and solids or conjugate heat transfer cht is encountered in many engineering applications since time accurate computations of unsteady cht can be computationally demanding we consider the use of high order implicit time integration schemes which have the potential to be more efficient relative to the commonly used second order implicit schemes we present a strongly coupled solution algorithm where the high order l stable explicit first stage singly diagonally implicit runge kutta esdirk schemes are used to advance the solution in time within each separate fluid and solid subdomains furthermore the stability and rate of convergence of performing gauss seidel subiterations at each stage of the esdirk schemes are analyzed the results from solving a numerical example an unsteady conjugate natural convection in an enclosure show good agreement with the performed analytical stability analysis in addition the computational work temporal precision character of several schemes in solving a strongly coupled cht problem is compared over a range of accuracy requirements from the efficiency investigation it is observed that performing subiterations with the strongly coupled esdirk algorithm is more efficient than lowering time step size using a high order loosely coupled imex algorithm in addition by using the esdirk schemes gain in computational efficiency relative to crank nicolson is observed for time accurate solutions a factor of 1 4 using the fourth order esdirk the computational gain is higher for smaller tolerances

generalized unequal length lapped orthogonal transform for subband image coding generalized linear phase lapped orthogonal transforms with unequal length basis functions gullots are considered the length of each basis of the proposed gullot can be different from each other whereas all the bases of the conventional genlot are of equal length in general for image coding application the long basis for a low frequency band and the short basis for a high frequency one are desirable to reduce the blocking and the ringing artifact simultaneously therefore the gullot is suitable especially for a subband image coding in order to apply the gullot to a subband image coding we also investigate the size limited structure to process the finite length signal which is important in practice finally some design and image coding examples are shown to confirm the validity of the proposed gullot

linear phase paraunitary filter banks theory factorizations and designs m channel maximally decimated filter banks have been used in the past to decompose signals into subbands the theory of perfect reconstruction filter banks has also been studied extensively nonparaunitary systems with linear phase filters have also been designed the authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase specific instances of this problem have been addressed by other authors and linear phase paraunitary systems have been shown to exist this property is often desirable for several applications particularly in image processing they begin by answering several theoretical questions pertaining to linear phase paraunitary systems next they develop a minimal factorization for a large class of such systems this factorization will be proved to be complete for even m further they structurally impose the additional condition that the filters satisfy pairwise mirror image symmetry in the frequency domain this significantly reduces the number of parameters to be optimized in the design process they then demonstrate the use of these filter banks in the generation of m band orthonormal wavelets several design examples are also given to validate the theory

a development of symmetric extension method for subband image coding describes the development of a technique for subband image coding called the symmetric extension method which utilizes the nature of a symmetrically extended image to achieve high quality coding there are however some restrictions on the usable subband coding system the development is done based on the property of a symmetrically extended signal in the analysis process to remove the restriction the behavior of the sequence after two processes filtering and decimation is examined and formulated the development extends the application area of the technique

lattice structure for regular paraunitary linear phase filterbanks and m band orthogonal symmetric wavelets parauninary linear phase pulp m channel uniform filterbanks which are also known as the generalized lapped orthogonal transforms genlots can be designed and implemented using lattice structures this paper discusses how to impose regularity constraints onto the lattice structure of pulp filterbanks these conditions are expressed in term of the rotation angles of the lattice components by which the resulting filterbanks are guaranteed to have one or two degrees of regularity iterating these new regular filterbanks on the lowpass subband generates a large family of symmetric m band orthonormal wavelets design procedures with many design examples are presented smooth interpolation using regular pulp filterbanks is illustrated through image coding experiments where the novel m band wavelets consistently yield smoother reconstructed images and better perceptual quality

a composite grid solver for conjugate heat transfer in fluid structure systems we describe a numerical method for modeling temperature dependent fluid flow coupled to heat transfer in solids this approach to conjugate heat transfer can be used to compute transient and steady state solutions to a wide range of fluid solid systems in complex two and three dimensional geometry fluids are modeled with the temperature dependent incompressible navier stokes equations using the boussinesq approximation solids with heat transfer are modeled with the heat equation appropriate interface equations are applied to couple the solutions across different domains the computational region is divided into a number of sub domains corresponding to fluid domains and solid domains there may be multiple fluid domains and multiple solid domains each fluid or solid sub domain is discretized with an overlapping grid the entire region is associated with a composite grid which is the union of the overlapping grids for the sub domains a different physics solver fluid solver or solid solver is associated with each sub domain a higher level multi domain solver manages the entire solution process we propose and analyze some centered discrete approximations to the interface equations that have some desirable stability properties the coupled interface equations may be solved directly when using explicit time stepping methods in the sub domains resulting in a strongly coupled approach the stability of the interface treatment in this case is independent of the relative sizes of the material properties in the two domains with the time step only depending on the usual von neumann conditions for each sub domain for implicit time stepping methods we solve the interface equations in a weakly coupled fashion to avoid forming a coupled implicit system across all sub domains the convergence of this approach does depend on the relative sizes of the thermal conductivities and diffusivities we analyze different iteration strategies for solving these implicit equations including the use of mixed robin approximations at the interface numerical results are presented to illustrate the method the accuracy of the technique is verified using the method of analytic solutions and by computing the solution to some heat exchanger problems where the exact solution is known the technique is also applied to the modeling of an inertial confinement fusion hohlraum target and the flow of coolant past an hexagonal array of heated fuel rods the multi domain solver runs in parallel on distributed memory computers and some parallel results are provided

asymptotic confidence regions for kernel smoothing of a varying coefficient model with longitudinal data abstract we consider the estimation of the k 1 dimensional nonparametric component t of the varying coefficient model y t x t t t e t based on longitudinal observations yij x i tij tij i 1 n j 1 ni where tij is the jth observed design time point t of the ith subject and yij and x i tij are the real valued outcome and r k 1 valued covariate vectors of the ith subject at tij the subjects are independently selected but the repeated measurements within subject are possibly correlated asymptotic distributions are established for a kernel estimate of t that minimizes a local least squares criterion these asymptotic distributions are used to construct a class of approximate pointwise and simultaneous confidence regions for t applying these methods to an epidemiological study we show that our procedures are useful for predicting cd4 t helper lymphocytes cell changes among hiv human immunodeficiency virus infected persons the finite sample properties of o

generalized symmetric extension for size limited multirate filter banks a multirate mr filter bank is called size limited if the total number of output samples equals the number of input samples a method called symmetric extension improved performance in subband image compression systems compared to the earlier method of circular convolution however the symmetric extension method was developed only for two band uniform filter banks and required even length linear phase analysis filters the authors generalize the symmetric extension method to the m band possibly nonuniform filter banks where m spl ges 2 the length restriction on the analysis filters is relaxed

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

a posteriori error estimation and adaptive mesh refinement for a multiscale operator decomposition approach to fluid solid heat transfer we analyze a multiscale operator decomposition finite element method for a conjugate heat transfer problem consisting of a fluid and a solid coupled through a common boundary we derive accurate a posteriori error estimates that account for all sources of error and in particular the transfer of error between fluid and solid domains we use these estimates to guide adaptive mesh refinement in addition we provide compelling numerical evidence that the order of convergence of the operator decomposition method is limited by the accuracy of the transferred gradient information and adapt a so called boundary flux recovery method developed for elliptic problems in order to regain the optimal order of accuracy in an efficient manner in an appendix we provide an argument that explains the numerical results provided sufficient smoothness is assumed

boundary operation of 2 d nonseparable linear phase paraunitary filter banks this paper proposes a boundary operation technique of 2 d nonseparable linear phase paraunitary filter banks ns lppufbs for size limitation the proposed technique is based on a lattice structure consisting of the 2 d separable block discrete cosine transform and nonseparable support extension processes the bases are allowed to be anisotropic with the fixed critically subsampling overlapping orthogonal symmetric real valued and compact support properties first the blockwise implementation is developed so that the basis images can be locally controlled the local control of basis images is shown to maintain orthogonality this property leads a basis termination bt technique as a boundary operation the technique overcomes the drawback of ns lppufbs that the popular symmetric extension method is invalid through some experimental results of diagonal texture coding the significance of the bt is verified

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

bootstrapping neural networks knowledge about the distribution of a statistical estimator is important for various purposes such as the construction of confidence intervals for model parameters or the determination of critical values of tests a widely used method to estimate this distribution is the so called boot strap which is based on an imitation of the probabilistic structure of the data generating process on the basis of the information provided by a given set of random observations in this article we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space we establish consistency results for bootstrap estimates of the distribution of parameter estimates

constructing a bootstrap confidence interval for the unknown concentration in radioimmunoassay the statistical problem associated with radioimmunoassay is known as calibration or inverse regression in the current study we propose a bootstrap procedure aimed at constructing an inverse confidence interval for the univariate calibration problem the calibration curve is estimated either parametrically or by non parametric regression the methods are illustrated by an example

bootstrap and wild bootstrap for high dimensional linear models in this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of f test statistics in high dimensional linear models an asymptotic approach will be chosen where the dimension p of the model may increase for sample size n rightarrow infty the range of validity will be compared for the normal approximation and for the bootstrap procedures furthermore it will be argued that the rates of convergence are different for the bootstrap procedures in this asymptotic framework this is in contrast to the usual asymptotic approach where p is fixed

m channel nonlinear phase filter banks in image compression structure design and signal extension this paper describes a transform coding technique based on m channel perfect reconstruction filter banks with a nonlinear phase in contrast with linear phase filter banks the nonlinearity provides an extra degree of freedom that can be used to design a more efficient transform we present new lattice structures of paraunitary and perfect reconstruction biorthogonal filter banks which can be implemented with a lower computational complexity and or represented by a few free parameters through the decomposition of the lattice blocks and the displacement across the delay block we further discuss a smooth extension method for nonlinear phase filter banks to obtain a nonexpansive transform the promise of our proposed approaches is demonstrated through several design examples extended signals and compression results

symmetric extension methods for m channel linear phase perfect reconstruction filter banks the symmetric extension method has been shown to he an efficient way for subband processing of finite length sequences this paper presents an extension of this method to general linear phase perfect reconstruction filter banks we derive constraints on the length and symmetry polarity of the permissible filter banks and propose a new design algorithm in the algorithm different symmetric sequences are formulated in a unified form based on the circular symmetry framework the length constraints in symmetrically extending the input sequence and windowing the subband sequences are investigated the effect of shifting the input sequence is included when the algorithm is applied to equal length filter banks we explicitly show that symmetric extension methods can always be constructed to replace the circular convolution approach

structures for m channel perfect reconstruction fir qmf banks which yield linear phase analysis filters the authors develop structures for fir finite impulse response perfect reconstruction qmf quadrature mirror filter banks which cover a subclass of systems that yield linear phase analysis filters for arbitrary m the parameters of these structures can be optimized to design analysis filters with minimum stopband energy which at the same time have linear phase and satisfy the perfect reconstruction property if there are m subbands then depending on whether the coefficients h sub k n of each analysis filter are symmetric or antisymmetric several combinations of filter banks are possible some of these permit perfect reconstruction and some do not for a given m the authors develop a formula for the number of combinations for a subclass of linear phase perfect reconstruction structures as an example they elaborate on a perfect reconstruction linear phase lattice structure for three channels the lattice structure is such that regardless of the parameter values the qmf bank enjoys the perfect reconstruction property while at the same time the analysis filters have linear phase a design example is presented along with tables of impulse response coefficients

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

application of implicit explicit high order runge kutta methods to discontinuous galerkin schemes despite the popularity of high order explicit runge kutta erk methods for integrating semi discrete systems of equations erk methods suffer from severe stability based time step restrictions for very stiff problems we implement a discontinuous galerkin finite element method dgfem along with recently introduced high order implicit explicit runge kutta imex rk schemes to overcome geometry induced stiffness in fluid flow problems the imex algorithms solve the non stiff portions of the domain using explicit methods and isolate and solve the more expensive stiff portions using an l stable stiffly accurate explicit singly diagonally implicit runge kutta method esdirk furthermore we apply adaptive time step controllers based on the embedded temporal error predictors we demonstrate in a number of numerical test problems that imex methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid induced stiffness

confidence bands in nonparametric time series regression we consider nonparametric estimation of mean regression and conditional variance or volatility functions in nonlinear stochastic regression models simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct the imposed dependence structure allows applications in many linear and nonlinear auto regressive processes the results are applied to the s p 500 index data

simultaneous inference of linear models with time varying coefficients summary the paper considers construction of simultaneous confidence tubes for time varying regression coefficients in functional linear models using a gaussian approximation result for non stationary multiple time series we show that the constructed simultaneous confidence tubes have asymptotically correct nominal coverage probabilities our results are applied to the problem of testing whether the regression coefficients are of certain parametric forms which is a fundamental problem in the inference of functional linear models as an application we analyse an environmental data set and study the association between levels of pollutants and hospital admissions