consistent estimation of a general nonparametric regression function in time series abstract we propose an estimator of the conditional distribution of x t x t 1 x t 2 and the corresponding regression function e x t x t 1 x t 2 where the conditioning set is of infinite order we establish consistency of our estimator under stationarity and ergodicity conditions plus a mild smoothness condition

multiple probability assessments by dependent experts abstract when two or more information sources experts provide a decision maker with information on two or more random variables the decision maker using bayes s rule has an opportunity to a update a prior about the random variables and b calibrate the experts calibration is the process of adjusting the decision maker s likelihood about the experts assessments this article presents a model for this two way process and specializes to the case in which the experts assessment errors have a multivariate normal density in general we find that variables which the decision maker and the experts regard as independent a priori will be dependent a posteriori because of dependence in the assessment errors formulas for posterior densities are given for the normal model in this model the posterior density of the random variables depends on only a weighted average of the expert s means with weights that depend on the experts assessments of previously known quantities i also present a special case o

conglomerability of probability measures on boolean algebras for the case where b is a boolean algebra of events and p is a probability finitely additive definetti 1972 considered the question of conglomerability of p and found that in many circumstances this natural notion was equivalent to countable additivity of p schervish seidenfeld and kadane 1984 pursued these investigations on the connection between countable additivity and conglomerability in greater detail for the case where b is a algebra hill and lane 1985 and zame 1988 give alternative proofs this article is an extension for the most part of schervish seidenfeld and kadane s work to the case where b is an arbitrary boolean algebra the more restrictive notion of positive conglomerability for a class of algebras including the countable algebras complete algebras and inifinite product algebras is treated completely this class is described by the requirement that a 0 1 valued measure be countably additive if every countable family of negligible sets is contained within a negligible set i e corresponds to a p point of of the stone space in general positive conglomerability fails to be equivalent to countable additivity though the degree of failure is minor building on techniques of hill lane and zame we obtain partial results on conglomerability for non complete algebras

first order variance of travel time in nonstationary formations 1 evaluating uncertainty in solute transport under nonstationary flow conditions is a computationally demanding task this is particularly true for cases with a two point covariance function of log conductivity depending on the actual positions of the points rather than their distance vector these may occur when the geological formation exhibits a trend nonstationarity can also be the result of uncertainty in the trend parameters of the mean log conductivity value or it may originate from conditioning of the log conductivity field to measurements of for example head or conductivity we present an efficient numerical method for evaluating the variance of travel time in such formations we cover cases in which the nonstationary covariance functions are constructed from stationary counterparts either by scaling functions or by summation with nonstationary functions resulting from marginalization or conditioning we apply a matrix based first order second moment fosm method for uncertainty propagation using the continuous adjoint state method for coupled systems to evaluate the sensitivity matrix the resulting matrix matrix multiplications are accelerated by fast fourier transformation fft techniques after periodic embedding of the covariance matrices referring to the stationary counterparts the combination of these methods makes it possible to compute the travel time uncertainty in domains discretized by several hundred thousand log conductivity values on standard personal computers within a reasonable time frame for demonstration we apply the method to a binary medium and a medium exhibiting a continuous trend in the covariance function in the latter application we also demonstrate the effects of marginalization and conditioning

on testing the validity of sequential probability forecasts events are observed sequentially and at each stage a probability for the next event is assessed we consider the relationship between the sequence of probability forecasts and the sequence of outcomes we argue that the forecasts may be considered empirically valid when both these sequences are consistent with a common joint distribution for the events to aid in assessing validity we introduce various test statistics that measure in a natural way the empirical performance of the probability forecasts in the light of the outcomes obtained it is shown that under the null hypothesis of forecast validity such statistics will after suitable normalization have a standard normal or chi squared distribution virtually irrespective of the common joint distribution supposed to underlie both sequences

sensitivity analysis for steady state groundwater flow using adjoint operators adjoint sensitivity theory is currently being considered as a potential method for calculating the sensitivity of nuclear waste repository performance measures to the parameters of the system for groundwater flow systems performance measures of interest include piezometric heads in the vicinity of a waste site velocities or travel time in aquifers and mass discharge to biosphere points the parameters include recharge discharge rates prescribed boundary heads or fluxes formation thicknesses and hydraulic conductivities the derivative of a performance measure with respect to the system parameters is usually taken as a measure of sensitivity to calculate sensitivities adjoint sensitivity equations are formulated from the equations describing the primary problem the solution of the primary problem and the adjoint sensitivity problem enables the determination of all of the required derivatives and hence related sensitivity coefficients in this study adjoint sensitivity theory is developed for equations of two dimensional steady state flow in a confined aquifer both the primary flow equation and the adjoint sensitivity equation are solved using the galerkin finite element method the developed computer code is used to investigate the regional flow parameters of the leadville formation of the paradox basin in utah the results illustrate the sensitivity of calculated local heads to the boundary conditions alternatively local velocity related performance measures are more sensitive to hydraulic conductivities

strategic manipulation of empirical tests theories can be produced by experts seeking a reputation for having knowledge hence a tester could anticipate that theories may have been strategically produced by uninformed experts who want to pass an empirical test we show that with no restriction on the domain of permissible theories strategic experts cannot be discredited for an arbitrary but given number of periods no matter which test is used provided that the test does not reject the actual data generating process natural ways around this impossibility result include 1 assuming that unbounded data sets are available and 2 restricting the domain of permissible theories opening the possibility that the actual data generating process is rejected out of hand in both cases it is possible to dismiss strategic experts but only to a limited extent these results show significant limits on what data can accomplish when experts produce theories strategically

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

efficient geostatistical inverse methods for structured and unstructured grids 1 the geostatistical inverse approach of kitanidis 1995 requires the covariance matrix of a discretized spatial random variable and matrix multiplications of the covariance matrix the explicit computation of the full covariance matrix is prohibitive for large scale problems discrete spectral methods which help to reduce computational costs dramatically are restricted to regular grids to make the geostatistical inverse approach applicable to unstructured grids which may be of great interest for practical applications we use the functional parameterization of the spatial random field by the karhunen loeve kl expansion in which the spatial variable is parameterized by weighted base functions that are derived from the covariance function in the inverse approach the estimation of the spatial variable becomes estimating the weights of base functions the prior covariance matrix of the weights is the identity matrix the base functions are continuous in space and can be discretized in any fashion for the derivation of the base functions we embed the domain into a larger unit cell of a periodic domain then the base functions are sinusoidal and can be derived by spectral methods which are independent of the discretization used for forward and inverse modeling for illustration we present case studies using synthetic data to estimate hydraulic conductivities on regular and unstructured grids the results show that we can efficiently obtain a good estimate using the inverse method in conjunction with the kl expansion we also demonstrate how to estimate structural parameters and to generate conditional realizations in the framework of our method

computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow finite difference and finite element methods are frequently used to study aquifer flow however additional analysis is required when model parameters and hence predicted heads are uncertain computational algorithms are presented for steady and transient models in which aquifer storage coefficients transmissivities distributed inputs and boundary values may all be simultaneously uncertain innovative aspects of these algorithms include a new form of generalized boundary condition a concise discrete derivation of the adjoint problem for transient models with variable time steps an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation and a new efficient first order second moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values the techniques are presented in matrix form and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations details of matrix structures are provided for a two dimensional linear triangular finite element model

adjoint derived location and travel time probabilities for a multidimensional groundwater system backward location and travel time probabilities can be used to determine the former location of contamination in an aquifer for a contaminant parcel that was detected in an aquifer the backward location probability describes its position at some time prior to sampling and the backward travel time probability describes the amount of time required for it to travel to the sampling location from some upgradient position these probabilities which can provide information about the source of contamination are related to adjoint states of resident concentration the governing equations of the backward probabilities are adjoints of the forward governing equation e g the advection dispersion equation we derive these backward governing equations and their boundary and final conditions for both location and travel time probabilities in a multidimensional system each governing equation contains the adjoint of the advection dispersion operator and a load term that defines the particular adjoint state probability the load term depends on both the type of probability location or travel time and the sampling device pumping well or monitoring well with which the contamination was detected the adjoint equation can also be used to efficiently determine forward location and travel time probabilities describing the future location of groundwater contamination a feature most useful for delineating pumping well captures zones we illustrate the use of the backward model for obtaining location and travel time probabilities in a hypothetical two dimensional domain

impact of sampling volume on the probability density function of steady state concentration 1 in recent years statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second order stationary velocity fields the merit of accurately estimating the mean and variance of concentration however remains unclear without knowing the shape of the probability density function pdf in a setup where a conservative solute is continuously injected into a domain the concentration is bounded between zero and the concentration value in the injected solution at small travel distances close to the fringe of the plume an observation point may fall into the plume or outside so that the statistical concentration distribution clusters at the two limiting values obviously this results in non gaussian pdf s of concentration with increasing travel distance the lateral plume boundaries are smoothed resulting in increased probability of intermediate concentrations likewise averaging the concentration in a larger sampling volume as typically done in field measurements leads to higher probabilities of intermediate concentrations we present semianalytical results of concentration pdf s for measurements with point like or larger support volumes based on stochastic theory applied to stationary media to this end we employ a reversed auxiliary transport problem in which we use analytical expressions for first and second central spatial lateral moments with an assumed gaussian pdf for the uncertainty of the first lateral moment and gauss like shapes in individual cross sections the resulting concentration pdf can be reasonably fitted by beta distributions the results are compared to monte carlo simulations of flow and steady state transport in 3 d heterogeneous domains in both methods the shape of the concentration pdf changes with distance to the contaminant source near the source the distribution is multimodal whereas it becomes a unimodal beta distribution far away from the contaminant source the semianalytical and empirical pdf s differ slightly which we contribute to the numerical artifacts in the monte carlo simulations but also to hard assumptions made in the semianalytical approach our results imply that geostatistical techniques for interpolation and other statistical inferences based on gaussian distributions such as kriging and cokriging may be feasible only far away from the contaminant source for calculations near the source the beta like distribution of concentration should be accounted for

sensitivity methods for time continuous spatially discrete groundwater contaminant transport models existing sensitivity methods for spatially discrete groundwater contaminant transport models have been developed for time stepping numerical algorithms and cannot be readily used with time continuous approaches to transport simulation such as the laplace transform galerkin technique we develop direct and adjoint sensitivity methods in which sensitivity coefficients are computed in the laplace domain and inverted numerically to the time domain the methods are computationally efficient when used in conjunction with time continuous transport equations the relative efficiency of the two methods depends on the number of model parameters number of performance measures and number of spatial discretization nodes the adjoint method is favored when the number of performance measures is much smaller than the number of model parameters the adjoint method is limited in that performance measures are restricted to being linear functions of state variables a two dimensional transport example is developed in detail and sensitivities with respect to nodal hydraulic conductivities are computed in the problems analyzed the direct and adjoint methods are 9 to 156 times faster than the perturbation method with the computational savings increasing as the size of the problem is increased

strong singularity disjointness and strong finite additivity of finitely additive measures abstract the most striking difference between finitely additive measures and countably additive measures is that the hahn decomposition theorem may be only approximate if countable additivity fails a corollary is that whereas two singular countably additive measures are disjoint in giving full measure to disjoint sets this is usually only approximately true without countable additivity one natural question is which finitely additive measures are disjoint from all singular measures in the stone space setting the question becomes which radon measures have support disjoint from all singular radon measures if so the support is a p1 set of atalla which is more general than a p set construction of such measures is considered at length existence of nonmolecular nonatomic measures based on p1 points with support a p1 set is equivalent to the existence of a p1 homeomorph of n in the stone space if x is a positive localizable measure space then ba has l x as a band and and l1 x as a subband it is shown that the band complementary to l x consists of measures disjoint to so supp is a p1 set in the stone space of a notion of strong singularity to intermediate to disjointness and singularity is shown to be equivalent to strong finite additivity admitting countable partitions by null sets for elements of l x the band of l x orthogonal to l1 x consists of strongly finitely additive measures precisely when the measure algebra satisfies the chain condition for a nonreal valued measurable cardinal or when the maximal ideal space of l x does not have real valued measurable cellularity a in ba has a countably additive ideal of negligible sets iff it is not strongly finitely additive on any set of positive measure on the stone space this corresponds to radon measures with p sets as supports even for purely finitely additive measures of this type much of countable additive measure theory holds for null functions are precisely those with null support it is not known for 2x whether purely finitely additive measures of this type exist their existence may be independent of zfc and or other axioms such as martin s real valued measurable cardinal or measurable cardinal a locally compact space y is sham compact so that c x cb x cc x precisely when its remainder in any compactification is a p set in this case m x mb x mc x the latter strong of identities is the definition of sham sham compact spaces these are characterized by having a p1 set as remainder in any compactification new examples of sham compact open subsets of the maximal ideal space z of a positive localizable measure space x are given based on maharam s homogeneity character however no open subset of z can be sham sham compact without being sham compact this is true when z is replaced by any compact space which locally the support of a radon measure

a statistical methodology for estimating transport parameters theory and applications to one dimensional advectivec dispersive systems a simulation nonlinear multiple regression methodology for estimating parameters that characterize the transport of contaminants is developed and demonstrated finite difference contaminant transport simulation is combined with a nonlinear weighted least squares multiple regression procedure the technique provides optimal parameter estimates and gives statistics for assessing the reliability of these estimates under certain general assumptions about the distributions of the random measurement errors monte carlo analysis is used to estimate parameter reliability for a hypothetical homogeneous soil column for which concentration data contain large random measurement errors the value of data collected spatially versus data collected temporally was investigated for estimation of velocity dispersion coefficient effective porosity first order decay rate and zero order production the use of spatial data gave estimates that were 2 3 times more reliable than estimates based on temporal data for all parameters except velocity comparison of estimated linear and nonlinear confidence intervals based upon monte carlo analysis showed that the linear approximation is poor for dispersion coefficient and zero order production coefficient when data are collected over time in addition examples demonstrate transport parameter estimation for two real one dimensional systems first the longitudinal dispersivity and effective porosity of an unsaturated soil are estimated using laboratory column data we compare the reliability of estimates based upon data from individual laboratory experiments versus estimates based upon pooled data from several experiments second the simulation nonlinear regression procedure is extended to include an additional governing equation that describes delayed storage during contaminant transport the model is applied to analyze the trends variability and interrelationship of parameters in a mourtain stream in northern california

calibration sufficiency and domination considerations for bayesian probability assessors abstract degroot and fienberg 1982a recently considered various aspects of the problem of evaluating the performance of probability appraisers after briefly reviewing their notions of calibration and sufficiency we introduce related ideas of semicalibration and domination and consider their relationship to the earlier concepts we then discuss some simple bayesian mechanisms for making probability assessments and study their calibration semicalibration sufficiency and domination properties finally several results concerning the comparison of finite dichotomous experiments relevant to the present work are collected in an appendix

a stochastic inverse solution for transient groundwater flow parameter identification and reliability analysis in this paper a stochastic approach is developed for solving the inverse problem of parameter identification for transient groundwater flow adjoint state equations are derived for the stochastic partial differential equation spde relating transient head and log hydraulic conductivity perturbations the derived equations can be used to calculate the covariance matrix of head observations and the cross covariance matrix between head observations and log hydraulic conductivity measurements for each observation time as well as the covariance and cross covariance matrices between different observation times the number of simulation runs required for obtaining these matrices is equal to the number of head observations the reliability of model prediction is evaluated through the variance estimate method using adjoint sensitivity analysis and the cokriging estimate the reliability evaluation of model prediction can be used as an aid in the optimization of experimental design for sampling strategies a numerical example is given to illustrate the stochastic inverse procedure for a general transient flow problem the example shows that using head observations of all observation times simultaneously produces much better results than using them sequentially commonly known as the quasi steady approach the example also shows that it is possible to obtain a reliable stochastic inverse solution for meeting a given prediction objective based on a limited number of observations

a proposed upstream weight numerical method for simulating pollutant transport in groundwater an alternative upstream weighting numerical method is proposed to simulate the transport process in groundwater under a variety of conditions the discretized system of the solute transport equations is derived from the integral form that expresses the balance of mass in each element an upstream weight that depends on the local peclet number is directly added to the basis functions when convective transport dominates the dispersive transport this technique can eliminate the oscillation of solutions efficiently with only a nominal increase of computation effort over that of the general linear triangular galerkin finite element method the accuracy of the proposed numerical model is tested against two classical problems for which analytical solutions exist the agreement is nearly exact an additional numerical example is used to illustrate the applicability of this model

identification of parameter structure in groundwater inverse problem this paper develops an optimization method that can be used to identify parameter structure as well as its values in connection with the inverse problem in groundwater modeling the complexity of the parameter structure is determined by the quantity and quality of observation data output least square error is used as the criterion of identification sensitivity coefficients at finite element nodes are evaluated by the variational method the problem of identifying the parameter structure is formulated as a problem of combinatorial optimization a new parameterization technique is presented for describing the parameter structure it can represent various distributions of parameters from piecewise constant to continuous variability based on this representation the combinatorial optimization problem for determining the parameter structure is then solved by a simple one at a time searching procedure numerical experimentations are used to demonstrate the efficiency and practicality of the proposed method

incorporation of prior information on parameters into nonlinear regression groundwater flow models 1 theory prior information on the parameters of a groundwater flow model can be used to improve parameter estimates obtained from nonlinear regression solution of a modeling problem two scales of prior information can be available 1 prior information having known reliability that is bias and random error structure and 2 prior information consisting of best available estimates of unknown reliability a regression method that incorporates the second scale of prior information assumes the prior information to be fixed for any particular analysis to produce improved although biased parameter estimates approximate optimization of two auxiliary parameters of the formulation is used to help minimize the bias which is almost always much smaller than that resulting from standard ridge regression it is shown that if both scales of prior information are available then a combined regression analysis may be made

a logic of probability with application to the foundations of statistics a simple example shows that the classical theory of probability implies more than one can deduce via kolmogorov s calculus of probability developing dawid s ideas i propose a new calculus of probability which is free from this drawback this calculus naturally leads to a new interpretation of probability i argue that attempts to create a general empirical theory of probability should be abandoned and we should content ourselves with the logic of probability establishing relations between probabilistic theories and observations my approach to the logic of probability is based on a variant of ville s principle of the excluded gambling strategy

a proposed stepwise regression method for model structure identification this paper proposes a new methodology for constructing groundwater models the proposed methodology which determines simultaneously both model structure and model parameters is based on the following ideas 1 when solving the inverse problem different model structures always produce different model parameters 2 since the number of possible model structures of an aquifer is infinite the number of possible representative parameters is also infinite 3 to obtain a set of appropriate representative model parameters we must have an appropriate model structure and 4 an appropriate model structure should be determined not only by observation data and prior information but also by the accuracy requirements of model applications in this proposed methodology we start with a homogeneous model structure and step by step gradually increase the complexity of the model structure at each level of complexity we calculate not only the fitting residual of parameter identification but also the error of model structure to determine if a more complex model structure is needed the model structure error of using one model structure to replace another model structure is defined by a maximum minimum max min problem that is based on the distance between the two models and is measured in parameter observation and prediction or decision spaces this proposed methodology is used to solve a hypothetical remediation design problem in which the true hydraulic conductivity is a random field with a certain trend we have found that for the example problem virtually identical pumping policy is obtained when a five zone model with an optimized zonation pattern is used to represent the nonstationary random field we have also found that observation errors have minimum impact on management solution in comparison with structure errors to calculate the model structure error for this example the inverse solution is coupled with a management problem we have also developed an effective iteration method to handle nonlinear water quality constraints

a score regression approach to assess calibration of continuous probabilistic predictions calibration the statistical consistency of forecast distributions and the observations is a central requirement for probabilistic predictions calibration of continuous forecasts is typically assessed using the probability integral transform histogram in this article we propose significance tests based on scoring rules to assess calibration of continuous predictive distributions for an ideal normal forecast we derive the first two moments of two commonly used scoring rules the logarithmic and the continuous ranked probability score this naturally leads to the construction of two unconditional tests for normal predictions more generally we propose a novel score regression approach where the individual scores are regressed on suitable functions of the predictive variance this conditional approach is applicable even for certain nonnormal predictions based on the dawid sebastiani score two case studies illustrate that the score regression approach has typically more power in detecting miscalibrated forecasts than the other approaches considered including a recently proposed technique based on conditional exceedance probability curves

on optimal sequential prediction for general processes this paper considers several aspects of the sequential prediction problem for unbounded nonstationary processes under pth power loss spl lscr sub p u v u v sup p 1 p spl infin in the first part it is shown that bayes prediction schemes are cesaro optimal under general conditions that cesaro optimal prediction schemes are unique in a natural sense and that cesaro optimality is equivalent to a form of weak calibration connections between calibration and stronger forms of optimality are considered extensions of the existence and uniqueness results to generalized prediction and prediction from observations with additive noise are established for binary processes it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the hamming loss in the second part of the paper it is shown how to construct from a countable family of prediction schemes a single composite scheme whose asymptotic performance on any suitable process dominates the performance of each member of the family the construction is based on aggregating methods for individual binary sequences using the construction some results of algoet 1994 1995 on the existence of cesaro optimal schemes for families of ergodic processes are rederived in a direct way and extended to unbounded processes

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

the game of normal numbers we introduce a two player game where at each period one player say player 2 chooses a distribution and the other player player 1 chooses a realization player 1 wins the game if the sequence of realized outcomes is normal with respect to the sequence of distributions we present a pure winning strategy of player 1 and thereby provide a universal algorithm that generates a normal sequence for any discrete stochastic process it turns out that to select thenth digit the algorithm conductso n2 calculations the proof uses approachability in infinite dimensional spaces lehrer 2002

non stochastic infinite and finite sequences abstract combining outcomes of coin tossing and transducer algorithms it is possible to generate with probability close to 1 very pathological sequences for which computable probabilistic forecasting is impossible these sequences are not random with respect to any reasonable probability distribution a natural consequence from the definition of such sequences is that each simple measure of the set of all such sequences is equal to 0 it was kolmogorov s and levin s idea to estimate the probability of generating of such sequences in the combinations of probabilistic and algorithmic processes 8 14 21 we collect several results in this direction for infinite sequences and asymptotic results for finite sequences including estimation of space and time of losing randomness for time bounded forecasting systems a correction to 22

coupled inverse problems in groundwater modeling 1 sensitivity analysis and parameter identification some basic concepts and methods for solving the coupled inverse problem in groundwater modeling are presented in a series of two papers because there are crossover effects between state variables and parmeters coupled inverse problems are different from single inverse problems in several aspects especially in the selection of objective functions identifiability and experimental design this paper discusses the general definition of coupled inverse problems posed in the framework of multiobjective optimization in order to obtain gradient vectors of objective functions as well as all sensitivity coefficients a general procedure for deriving the adjoint state equations and their associated conditions is described using the rules of adjoint operation presented in this paper the derivation of adjoint problems becomes very easy even for nonlinear and transient coupled problems as examples the adjoint problem of flow mass transport or flow heat transport the adjoint problem of saltwater intrusion and the adjoint problem of two phase flow are derived numerical experiments on a two dimensional flow mass transport problem are given to explain the procedure of sensitivity analysis and parameter identification for coupled problems the hydraulic conductivity can be identified by using observations of head and or solute concentration

some stochastic properties of memoryless individual sequences an individual sequence of real numbers is memoryless if no continuous markov prediction scheme of finite order can outperform the best constant predictor under the squared loss it is established that memoryless sequences satisfy an elementary law of large numbers and sliding block versions of hoeffding s inequality and the central limit theorem it is further established that memoryless binary sequences have convergent sample averages of every order and that their limiting distributions are bernoulli several examples and sources of memoryless sequences are given and it is shown how memoryless binary sequences may be constructed from aggregating methods for sequential prediction

two dimensional characterization of hydraulic heterogeneity by multiple pumping tests 1 the conventional analysis of pumping tests by type curve methods is based on the assumption of a homogeneous aquifer applying these techniques to pumping test data from real heterogeneous aquifers leads to estimates of the hydraulic parameters that depend on the choice of the pumping and observation well positions in this paper we test whether these values may be viewed as pseudo local values of transmissivity and storativity which can be interpolated by kriging we compare such estimates to those obtained by geostatistical inverse modeling where heterogeneity is assumed in all stages of estimation we use drawdown data from multiple pumping tests conducted at the test site in krauthausen germany the geometric mean values of transmissivity and storativity determined by type curve analysis are very close to those obtained by geostatistical inversion but the conventional approach failed to resolve the spatial variability of transmissivity in contrast the estimate from geostatistical inversion reveals more structure this indicates that the estimates of the type curve approaches can not be treated as pseudo local values concerning storativity both analysis methods show strong fluctuations because the variability of all terms making up the storativity is small we believe that the estimated variability of storativity is biased we examine the influence of measurement error on estimating structural parameters of covariance functions in the inversion we obtain larger correlation lengths and smaller prior variances if we trust the measured data less

bayesian calibration of a large scale geothermal reservoir model by a new adaptive delayed acceptance metropolis hastings algorithm 1 the aim of this research is to estimate the parameters of a large scale numerical model of a geothermal reservoir using markov chain monte carlo mcmc sampling within the framework of bayesian inference all feasible parameters that are consistent with the measured data are summarized by the posterior distribution and hence parameter estimation and uncertainty quantification are both given by calculating expected values of statistics of interest over the posterior distribution it appears to be computationally infeasible to use the standard metropolis hastings algorithm mh to sample the high dimensional computationally expensive posterior distribution to improve the sampling efficiency a new adaptive delayed acceptance mh algorithm adamh is implemented to adaptively build a stochastic model of the error introduced by the use of a reduced order model this use of adaptivity differs from existing adaptive mcmc algorithms that tune proposal distributions of the metropolis hastings algorithm mh though adamh also implements that technique for the 3 d geothermal reservoir model we present here adamh shows a great improvement in the computational efficiency of the mcmc sampling and promising results for parameter estimation and uncertainty quantification are obtained this algorithm could offer significant improvement in computational efficiency when implementing sample based inference in other large scale inverse problems

fuzzy set theory observer bias and probability theory abstract it is known that given certain seemingly implausible assumption probability functions act like fuzzy set membership functions this paper shows that these seemingly implausible assumptions are satisfied in probabilistic situations where the sole uncertainty arises from unknown but systematic observer bias now such situations could be considered as on the borderline between the domain of applicability of probability functions and the domain of fuzzy set membership functions hence the convergence between the rules for combining probabilistic functions and those for combining fuzzy set membership functions in these borderline instances is intuitively quite satisfying this convergence also suggests a number of other ways in which possibility theory and probability theory are interrelated

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

a comparison of several methods of solving nonlinear regression groundwater flow problems computational efficiency and computer memory requirements for four methods of minimizing functions were compared for four test nonlinear regression steady state groundwater flow problems the fastest methods were the marquardt and quasi linearization methods which required almost identical computer times and numbers of iterations the next fastest was the quasi newton method and last was the fletcher reeves method which did not converge in 100 iterations for two of the problems the fastest method per iteration was the fletcher reeves method and this was followed closely by the quasi newton method the marquardt and quasi linearization methods were slower for all four methods the speed per iteration was directly related to the number of parameters in the model however this effect was much more pronounced for the marquardt and quasi linearization methods than for the other two hence the quasi newton and perhaps fletcher reeves method might be more efficient than either the marquardt or quasi linearization methods if the number of parameters in a particular model were large although this remains to be proven the marquardt method required somewhat less central memory than the quasi linearization metilod for three of the four problems for all four problems the quasi newton method required roughly two thirds to three quarters of the memory required by the marquardt method and the fletcher reeves method required slightly less memory than the quasi newton method memory requirements were not excessive for any of the four methods

bayesian nonparametric calibration and combination of predictive distributions abstractwe introduce a bayesian approach to predictive density calibration and combination that accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights building on the work of ranjan and gneiting we use infinite beta mixtures for the calibration the proposed bayesian nonparametric approach takes advantage of the flexibility of dirichlet process mixtures to achieve any continuous deformation of linearly combined predictive distributions the inference procedure is based on combination gibbs and slice sampling we provide some conditions under which the proposed probabilistic calibration converges in terms of weak posterior consistency to the true underlying density for both cases of iid and markovian observations this calibration property improves upon the earlier calibration approaches we study the methodology in simulation examples with fat tails and multimodal densities and apply it to density forecasts of dail

a composite l1 parameter estimator for model fitting in groundwater flow and solute transport simulation this paper proposes a composite l1 parameter estimator to solve the inverse problems in groundwater flow and solute transport modeling the estimator is formulated using a weighted l1 norm as the error measure between the vector of the observed hydraulic heads and solute concentrations and the corresponding vector of the heads and concentrations computed from a finite element model for steady state groundwater flow and solute transport in a two dimensional areal confined aquifer system the parameters currently considered are the hydraulic conductivities the dispersivities the porosities of the aquifer system and the solute source concentration s the gradients of the state variables with respect to the parameters are computed analytically using a formulation stemming from the finite element model for the state variables the solutions to two hypothetical groundwater parameter estimation problems are presented to illustrate that the proposed estimator does have the property of being fairly accurate and more significantly being robust in handling observation data containing certain outliers

simultaneous inversion of hydrogeologic and thermal data 1 theory and application using hydraulic head data one goal in solving the inverse problem in groundwater hydrology is to obtain parameter estimates for use in a simulation model a philosophy about the inverse is adopted here that takes the form of constructing a variety of models under different norms large differences in parameter estimates obtained using different norms suggests the possibility of an incorrect interpretational model an optimization procedure is used that is based on the constrained simplex method this algorithm easily handles constraints does not require derivatives of the functional with respect to the parameters and can be employed to minimize complicated functionals the algorithm produces a cluster of search points in parameter space about the global minimum of the objective function each of these points generates data that matches observed values within a prescribed tolerance the centroid of the search points is taken as the best estimate of the parameters the covariances of the estimated parameters and estimated data can also be computed this inverse technique is applied to a sparse inaccurate hydraulic head data set observed at downie slide british columbia canada for this cross sectional problem with noisy data and a strong drift component interpolation of hydraulic head values using kriging techniques was not possible the domain is represented as a homogeneous isotropic medium four parameters are estimated the free surface position of the water table and three boundary conditions further simulations using a synthetic data set with assumed properties similar to that of downie slide show that with noise free data and an adequate spacing between measurement points it is possible to interpolate an unbiased estimate of hydraulic head data at all nodes in the equivalent discretized domain when the inverse technique is applied the hydraulic conductivity structure is correctly identified the implications for downie slide are that in order to resolve more than a simple hydraulic conductivity structure accurate field measurements of hydraulic head a better spacing between measurements and more detailed knowledge of the boundary conditions are required

parameter estimation for coupled unsaturated flow and transport this paper deals with the estimation of soil hydraulic and transport parameters from transient unsaturated flow and tracer experiments using a combined simulations optimization approach hydraulic properties are defined by a modified form of van genuchten s 1980 parametric model for two phase permeability saturation pressure relations and transport properties are defined by an empirical parametric dispersion model a nonlinear weighted least squares algorithm is used to estimate unknown model parameters by minimizing deviations between concentrations water contents and pressure heads obtained from hypothetical infiltration redistribution evaporation experiments and those predicted by solving a numerical model of coupled unsaturated flow and transport simultaneous estimation of hydraulic and transport properties is found to yield smaller estimation errors for model parameters than sequential inversion of hydraulic properties from water content and pressure head data followed by inversion for transport properties from concentration data effects of random noise in data measurements soil layering and choice of improper parametric model on the parameter estimation process are discussed

a new goodness of fit test for event forecasting and its application to credit defaults we develop a new goodness of fit test for validating the performance of probability forecasts our test statistic is particularly powerful under sparseness and dependence in the observed data to build our test statistic we start from a formal definition of calibrated forecasts which we operationalize by introducing two components the first component tests the level of the estimated probabilities the second validates the shape measuring the differentiation between high and low probability events after constructing test statistics for both level and shape we provide a global goodness of fit statistic which is asymptotically 2 distributed in a simulation exercise we find that our approach is correctly sized and more powerful than alternative statistics in particular our shape statistic is significantly more powerful than the kolmogorov smirnov test under independence our global test has significantly greater power than the popular hosmer lemeshow s 2 test moreover even under dependence our global test remains correctly sized and consistent as a timely and important empirical application of our method we study the validation of a forecasting model for credit default events this paper was accepted by wei xiong finance

impact of numerical artifact of the forward model in the inverse solution of density dependent flow problem we investigate the effect of computational error on the inversion of a density dependent flow and transport model using seawat and ucode 2005 in an inverse identification of hydraulic conductivity and dispersivity using head and concentration data from a 2 d laboratory experiment we investigated inversions using three different solution schemes including variation of number of particles and time step length in terms of the three aspects the shape and smoothness of the objective function surface the consequent impacts to the optimization and the resulting pareto analyses this study demonstrates that the inversion is very sensitive to the choice of the forward model solution scheme in particular standard finite difference methods provide the smoothest objective function surface however this is obtained at the cost of numerical artifacts that can lead to erroneous warping of the objective function surface total variation diminishing tvd schemes limit these impacts at the cost of more computation time while the hybrid method of characteristics hmoc approach with increased particle numbers and or reduced time step gives both smoothed and accurate objective function surface use of the most accurate methods tvd and hmoc did lead to successful inversion of the two parameters however with distinct results for pareto analyses these results illuminate the sensitivity of the inversion to a number of aspects of the forward solution of the density driven flow problem and reveal that parameter values may result that are erroneous but that counteract numerical errors in the solution

calibration based empirical probability probability forecasts for a sequence of uncertain events may be compared with the outcomes of those events by means of a natural criterion of empirical validity calibration it is shown that any two sequences of forecasts which both meet this criterion must be in asymptotic agreement these agreed values can then be considered as correct objective probability forecasts for the particular sequence of outcome results obtained however the objective forecasts vary with the extent of the information taken into account when they are formulated we thus obtain a general theory of empirical probability relative to an information base this theory does not require that such probabilities be interpreted in terms of repeated trials of the same event some implications of this theory are discussed

on prequential model assessment in life history analysis summary we extend the prequential approach introduced by a p dawid to continuous time marked point processes a prequential forecasting system is expressed by means of the stochastic intensities of the point process the exact independence and 1 exponentiality of the derived compensators are used to assess the predictive performance of the model and the inferential procedure

variational sensitivity analysis data requirements and parameter identification in a leaky aquifer system the adjoint state sensitivity method is developed to aid the analysis of a pumping test design for parameter identification in a leaky aquifer system the system considered consists of an unconfined aquifer overlaying an aquitard which overspreads a confined aquifer the adjoint equations are derived in the continuous form without discretizing the governing equations and are expressed in terms of convection diffusion equations with time dependent coefficients the multiple cell balance mcb method is used to solve the nonlinear governing equations as well as their adjoint equations algorithms are developed for calculating sensitivity coefficients of heads in each layer of an aquifer with respect to various parameters a concept called contribution of an observation is introduced which can be used to measure the usefulness of an observation in connection with data requirements for parameter identification it links experimental design with observation errors and the requirement of model application the sufficiency of an observation system for parameter identification can be evaluated at the design stage before extensive resources are committed to data collection a numerical example is given to verify the proposed method and to explain its applicability to the design of a pumping test

an upstream weight multiple cell balance finite element method for solving three dimensional convection dispersion equations an upstream weight multiple cell balance finite element method is developed to simulate the three dimensional contaminant transport in aquifers the method combines the concepts of local mass balance and finite element approximations when convective transport dominates the dispersive transport the proposed technique effectively eliminates the oscillation of numerical solutions with only a nominal increase of formulation effort over that of the general galerkin finite element method two numerical examples for which analytical solutions exist are used to verify the proposed numerical solutions an additional example is presented to demonstrate the practical applicability of the proposed method

parameter space methods in joint parameter estimation for groundwater flow models the joint estimation of model parameters using hydraulic head and tracer concentration data is examined using methods based on parameter space analysis response surfaces and confidence regions can indicate how multiple data sets interact to reduce parameter uncertainty each data set produces a unique response surface and confidence region if these surfaces are oriented differently for the two data sets parameter uncertainties are significantly reduced when the second data set is included in the inversion if these surfaces are similar for the two data sets the second data set will not reduce the parameter uncertainty significantly the axes of the linearized confidence ellipsoid are analyzed to determine the difference in orientation of the confidence regions for the two data sets the use of confidence regions can be extended to predict the value of a second data set in reducing the uncertainty of parameter estimates before the data are collected parameter space approaches are also introduced for selecting the relative weights for the individual data sets in joint parameter estimation the conventional method based on the analysis of data residuals is compared to three other methods with the weights selected to maximize parameter stability to minimize the volume of the confidence region or to minimize the longest axis of the confidence region each criterion can lead to substantially different weights applied to each of the data sets the application of these methods is demonstrated using hydraulic head measurements and 14c concentrations to calibrate a model of groundwater flow in the san juan basin new mexico

geostatistical inference of hydraulic conductivity and dispersivities from hydraulic heads and tracer data 1 in groundwater hydraulic heads and solute arrival times depend primarily on the hydraulic conductivity field and hydraulic boundary conditions the spread of breakthrough curves in contrast depends also on longitudinal and transverse dispersion coefficients the shape of point like measured breakthrough curves can be reproduced by simulations only when appropriate dispersivities are applied the values and spatial distributions of dispersivities depend on the resolution of the underlying hydraulic conductivity field we present a geostatistical method for the joint estimation of log conductivity and log dispersivities from measurements of hydraulic heads and temporal moments of local breakthrough curves the parameter fields are considered random space variables and they are conditioned on the measurements by bayesian inference the estimated longitudinal and transverse dispersivities are to be applied in conjunction with the estimated conductivity field we apply our technique to data of a technical scale tracer experiment in the particular application the amount and quality of measured data are sufficient to infer the conductivity distribution at a spatial resolution at which the spread of locally obtained breakthrough curves is dominated by pore scale transverse dispersion

any inspection is manipulable consider a sequential forecast that may vary with time and history a calibration test see dawid 1982 foster and vohra 1997 and kalai lehrer and smorodinsky 1999 compares the observed empirical frequency over a set of times with the average prediction over the same set of times each test checks different events at different times a forecasting scheme passes a calibration test at a sequence of state realizations if the gap between these two figures diminishes to zero in other words each test defines what is consistent with the observed facts differently a forecast is consistent with the observed facts according to a given test if the empirical frequency of the events checked coincides asymptotically with the average prediction of these events when the forecast and the true distribution according to which the states are selected are the same the forecast passes any calibration test with probability one a calibration test is therefore type i error free the calibration tests treated in the literature are pure after any history it is known what specific event for instance rain foggy is being considered here we introduce mixed calibration tests where after each history a randomly chosen event is checked a mixed test checks whether the average prediction of these randomly chosen events is asymptotically equal to the frequency of their realizations like a deterministic test a mixed calibration test is also type i error free the main result of this paper shows that there is no mixed calibration test that rejects any inaccurate forecast that is for any test of this kind there exists a forecasting scheme that is always consistent with the realized events a constructive method to produce this scheme is provided as well another interpretation of the main result involves a forecaster and an inspector the inspector randomly chooses a calibration test according to a known distribution it is shown that a forecaster without knowing anything regarding the way states are selected can pass almost every test chosen by the inspector independently of the sequence of realizations in other words the forecaster has a forecasting method that is consistent with the observed facts in the meaning induced by almost all tests instead of randomly choosing one test as discussed previously the inspector can employ a set of tests simultaneously the main result implies that if the inspector employs countably many calibration tests for instance all those computable by a turing machine a forecaster can likewise pass them all2 simultaneously it means that a consultant or a scientist e g a weather forecaster who knows the countable set of

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

combining probability forecasts summary linear pooling is by far the most popular method for combining probability forecasts however any non trivial weighted average of two or more distinct calibrated probability forecasts is necessarily uncalibrated and lacks sharpness in view of this linear pooling requires recalibration even in the ideal case in which the individual forecasts are calibrated towards this end we propose a beta transformed linear opinion pool for the aggregation of probability forecasts from distinct calibrated or uncalibrated sources the method fits an optimal non linearly recalibrated forecast combination by compositing a beta transform and the traditional linear opinion pool the technique is illustrated in a simulation example and in a case study on statistical and national weather service probability of precipitation forecasts

toward a reconciliation of the bayesian and frequentist approaches to point estimation abstract the bayesian and frequentist approaches to point estimation are reviewed the status of the debate regarding the use of one approach over the other is discussed and its inconclusive character is noted a criterion for comparing bayesian and frequentist estimators within a given experimental framework is proposed the competition between a bayesian and a frequentist is viewed as a contest with the following components a random observable a true prior distribution unknown to both statisticians an operational prior used by the bayesian a fixed frequentist rule used by the frequentist and a fixed loss criterion this competition is studied in the context of exponential families conjugate priors and squared error loss the class of operational priors that yield bayes estimators superior to the best frequentist estimator is characterized the implications of the existence of a threshold separating the space of operational priors into good and bad priors are explored and their relevance in ar

illustration and verification of adjoint sensitivity theory for steady state groundwater flow the application of adjoint sensitivity theory to steady state ground water flow is illustrated with three one dimensional flow problems adjoint states are analytically derived for four performance measures of these test problems hydraulic head at a point spatially average hydraulic head darcy velocity at a point and flux from a prescribed head boundary sensitivity coefficients are analytically calculated for average head the adjoint states are interpreted and their usefulness is discussed the implementation of a numerical adjoint sensitivity flow code to solve for these adjoint states is described and the computed adjoint states are used in the code to evaluate the sensitivities of model results to model input parameters the one dimensional flow problems provide a set of verification tests for the numerical code the numerical code successfully reproduces both the analytically derived adjoint states including those involving jump conditions and the sensitivity coefficients for model output values that are nonlinear with respect to model parameters

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

coupled estimation of flow and solute transport parameters a methodology for solving the coupled linear flow and transport inverse problem is presented it allows the estimation of aquifer parameters transmissivity storativity recharge boundary heads and flows leakage dispersivity molecular diffusion porosity retardation linear decay boundary concentrations using their prior estimates as well as head and concentration measurements as basic information transient flow and transport equations are solved using the finite element method in space and a weighted finite difference scheme in time flow and transport domains can be one two or quasi three dimensional applying linear one dimensional elements linear two dimensional triangles or bilinear two dimensional rectangles mixed at will time regimes can be transient or steady state in any equation flow or transport parameters are estimated on the basis of maximum likelihood theory which allows us to obtain information about parameters uncertainty minimization of the negative log likelihood is performed using marquardt s method the applicability of the method is shown by means of two real examples they illustrate that the proposed approach can successfully address model selection parameter uncertainty and nontrivial identifiability problems estimation of time dependent sources and the role of prior estimates are also addressed in the examples while the formulation of the problem leaves room for both stochastic and deterministic treatment of hydraulic parameters the latter has been adopted in both examples limitations of the approach are also apparent

a two step bayesian approach for propensity score analysis simulations and case study a two step bayesian propensity score approach is introduced that incorporates prior information in the propensity score equation and outcome equation without the problems associated with simultaneous bayesian propensity score approaches the corresponding variance estimators are also provided the two step bayesian propensity score is provided for three methods of implementation propensity score stratification weighting and optimal full matching three simulation studies and one case study are presented to elaborate the proposed two step bayesian propensity score approach results of the simulation studies reveal that greater precision in the propensity score equation yields better recovery of the frequentist based treatment effect a slight advantage is shown for the bayesian approach in small samples results also reveal that greater precision around the wrong treatment effect can lead to seriously distorted results however greater precision around the correct treatment effect parameter yields quite good results with slight improvement seen with greater precision in the propensity score equation a comparison of coverage rates for the conventional frequentist approach and proposed bayesian approach is also provided the case study reveals that credible intervals are wider than frequentist confidence intervals when priors are non informative

geostatistical inverse modeling of transient pumping tests using temporal moments of drawdown 1 pumping tests belong to the most common techniques of hydrogeological site assessment while the steady state drawdown is determined by the distribution of transmissivity alone the transient behavior is also influenced by the storativity field in geostatistical inverse modeling the spatial distributions of both transmissivity and storativity are inferred from the drawdown curves and prior information on the spatial correlation of the parameter fields so far however transient data have hardly been analyzed by geostatistical inverse methods because the computational effort is rather high in the present study we characterize the drawdown by its temporal moments we present moment generating equations and corresponding equations to compute the sensitivity of the temporal moments of drawdown with respect to the distributions of transmissivity and storativity we utilize these equations to infer the transmissivity and storativity fields from transient pumping tests using the quasi linear geostatistical approach of inverse modeling considering temporal moments rather than full drawdown curves drastically reduces the computational effort of the estimation procedure in test cases we show that the first two temporal moments are sufficient to characterize the drawdown curves we investigate how erroneous assumptions regarding the spatial variability of storativity affect the estimate of the transmissivity field and we analyze the effect of truncating the measured drawdown curves

cokriging estimation of the conductivity field under variably saturated flow conditions a linear estimator cokriging was applied to estimate hydraulic conductivity using pressure head solute concentration and solute arrival time measurements in a hypothetical heterogeneous vadose zone under steady state infiltrations at different degrees of saturation covariances and cross covariances required by the estimator were determined by a first order approximation in which sensitivity matrices were calculated using an adjoint state method the effectiveness of the pressure concentration and arrival time measurements for the estimator were then evaluated using two statistical criteria l1 and l2 norms i e the average absolute error and the mean square error of the estimated conductivity field results of our analysis showed that pressure head measurements at steady state flow provided the best estimation of hydraulic conductivity among the three types of measurements in addition head measurements of flow near saturation were found more useful for estimating conductivity than those at low saturations the arrival time measurements do not have any significant advantage over concentration factors such as variability linearity and ergodicity were discussed to explain advantage and limitation of each type of data set finally to take advantage of different types of data set e g head and concentration a computationally efficient estimation approach was developed to combine them sequentially to estimate the hydraulic conductivity field the conductivity field estimated by using this sequential approach proves to be better than all the previous estimates using one type of data set alone

estimation of historical groundwater contaminant distribution using the adjoint state method applied to geostatistical inverse modeling 1 as the incidence of groundwater contamination continues to grow a number of inverse modeling methods have been developed to address forensic groundwater problems in this work the geostatistical approach to inverse modeling is extended to allow for the recovery of the antecedent distribution of a contaminant at a given point back in time which is critical to the assessment of historical exposure to contamination such problems are typically strongly underdetermined with a large number of points at which the distribution is to be estimated to address this challenge the computational efficiency of the new method is increased through the application of the adjoint state method in addition the adjoint problem is presented in a format that allows for the reuse of existing groundwater flow and transport codes as modules in the inverse modeling algorithm as demonstrated in the presented applications the geostatistical approach combined with the adjoint state method allow for a historical multidimensional contaminant distribution to be recovered even in heterogeneous media where a numerical solution is required for the forward problem

on calibration error of randomized forecasting algorithms it has been recently shown that calibration with an error less than d 0 is almost surely guaranteed with a randomized forecasting algorithm where forecasts are obtained by random rounding the deterministic forecasts up to d we show that this error cannot be improved for a vast majority of sequences we prove that using a probabilistic algorithm we can effectively generate with probability close to one a sequence resistant to any randomized rounding forecasting with an error much smaller than d we also reformulate this result by means of a probabilistic game

solving the estimation identification problem in two phase flow modeling in this paper a procedure is presented to solve the estimation identification problem in two phase flow modeling given discrete observations made on the system response an optimum parameter set is derived for an appropriate conceptual model by solving the inverse problem using standard optimization techniques subsequently a detailed error analysis is performed and nonlinearity effects are considered we discuss the iterative process of model identification and parameter estimation for a ventilation test performed at the grimsel rock laboratory switzerland a numerical model of the ventilation drift and the surrounding crystalline rock matrix is developed evaporation of moisture at the drift surface and the propagation of the unsaturated zone into the formation are simulated a sensitivity analysis is performed to identify the parameters to be estimated absolute permeability and two parameters of van genuchten s characteristic curves are subsequently determined based on measurements of negative water potentials evaporation rates and gas pressure data the performance of the minimization algorithm and the system behavior for the optimum parameter set are discussed the study shows that a field experiment conducted under two phase flow conditions can be successfully reproduced by taking into account a variety of physical processes and that it is possible to reliably determinemore the two phase hydraulic properties that are related to the given conceptual model 32 refs 8 figs 7 tabs less

incorporation of prior information on parameters into nonlinear regression groundwater flow models 2 applications this paper investigates factors influencing the degree of improvement in estimates of parameters of a nonlinear regression groundwater flow model by incorporating prior information of unknown reliability consideration of expected behavior of the regression solutions and results of a hypothetical modeling problem lead to several general conclusions first if the parameters are properly scaled linearized expressions for the mean square error mse in parameter estimates of a nonlinear model will often behave very nearly as if the model were linear second by using prior information the mse in properly scaled parameters can be reduced greatly over the mse of ordinary least squares estimates of parameters third plots of estimated mse and the estimated standard deviation of mse versus an auxiliary parameter the ridge parameter specifying the degree of influence of the prior information on regression results can help determine the potential for improvement of parameter estimates fourth proposed criteria can be used to make appropriate choices for the ridge parameter and another parameter expressing degree of overall bias in the prior information results of a case study of truckee meadows reno sparks area washoe county nevada conform closely to the results of the hypothetical problem in the truckee meadows case incorporation of prior information did not greatly change the parameter estimates from those obtained by ordinary least squares however the analysis showed that both sets of estimates are more reliable than suggested by the standard errors from ordinary least squares

jordan fields and improper integrals abstract an integral is defined for not necessarily bounded functions with respect to a finitely additive possibly infinite measure in particular this strictly includes the familiar case of absolutely convergent improper riemann integration in euclidean space however the measure need not have a countably additive extension and no topology is assumed on the underlying space the jordan field of sets with integrable characteristic functions is used to characterize the integrable functions as those that are nearly jordan measurable and their integrals are shown to equal the lebesgue integrals of the measures of the appropriate jordan sets

a wavelet approach to adjoint state sensitivity computation for steady state differential equations 1 the computation of the sensitivity matrix is the most time consuming part of any parameter estimation algorithm that requires sensitivity coefficients an efficient wavelet approach to adjoint sensitivity analysis is proposed to reduce the computational cost of obtaining sensitivity coefficients the method exploits a wavelet reduction of the data space to reduce the size of the linear system encountered in steady state adjoint equations in this regard wavelet transform is used as a data compression tool numerical examples applied to spatial data are used to verify and show the effectiveness of the method

notes on jordan fields based on an article by maharam abstract results of maharam on jordan fields associated with a finitely additive measure are extended to take into consideration the natural topology induced by an algebra of sets namely that induced by the stone space based on this the question of when the jordan field is a algebra and the question of the completeness of associated l 1 spaces are answered

internal regret with partial monitoring calibration based optimal algorithms we provide consistent random algorithms for sequential decision under partial monitoring when the decision maker does not observe the outcomes but receives instead random feedback signals those algorithms have no internal regret in the sense that on the set of stages where the decision maker chose his action according to a given law the average payoff could not have been improved in average by using any other fixed law they are based on a generalization of calibration no longer defined in terms of a voronoi diagram but instead of a laguerre diagram a more general concept this allows us to bound for the first time in this general framework the expected average internal as well as the usual external regret at stage n by o n 1 3 which is known to be optimal

probabilistic forecasts calibration and sharpness summary probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions we propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only a simple theoretical framework allows us to distinguish between probabilistic calibration exceedance calibration and marginal calibration we propose and study tools for checking calibration and sharpness among them the probability integral transform histogram marginal calibration plots the sharpness diagram and proper scoring rules the diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the stateline wind energy centre in the us pacific northwest in combination with cross validation or in the time series context our proposal provides very general nonparametric alternatives to the use of information criteria for model diagnostics and model selection

coupled inverse problems in groundwater modeling 2 identifiability and experimental design parameter identifiability and experimental design in the context of solving the coupled inverse problem in groundwater modeling are considered in part 2 of the two paper series the following three new definitions of extended identifiability for a distributed parameter system are presented interval identifiability ini prediction equivalence identifiability pei and management equivalence identifiability mei the uniqueness requirement of the inverse solution is relaxed by these definitions an identified parameter is said to be extendedly identifiable if it is able to satisfy an accuracy requirement in a specific application of the model when a generalized least squares norm is used the proposed identifiabilities are applicable to any coupled problems in which the state variables and parameters may have different order of magnitudes a salient feature of the new definitions is that the extended identifiabilities are closely related to experimental design consequently the sufficiency of an experimental design for assuring a given level of an extended identifiability can be estimated prior to field experiments the methodology developed can be used to determine the reliability of an experimental design even for nonlinear models a numerical example on the management of a wastewater disposal is given to explain how the concepts and methods presented in this paper can be used for solving practical problems

calibration with many checking rules each period an outcome out of finitely many possibilities is observed for simplicity assume two possible outcomes a and b each period a forecaster announces the probability of a occurring next period based on the past consider an arbitrary subsequence of periods e g odd periods even periods all periods in which b is observed etc given an integer n divide any such subsequence into associated sub subsequences in which the forecast for a is between i n i 1 n i 0 1 n we compare the forecasts and the outcomes realized next period separately in each of these subsubsequences given any countable partition of 0 1 and any countable collection of subsequences we construct a forecasting scheme such that for all infinite strings of data the long run average forecast for a matches the long run frequency of realized a s

further thoughts on testing for causality with econometric models abstract the ontological basis for causality testing must be some empirically interpretable system leading one to specify a logically valid a priori law that can be shown to exist such paradigms as conventional linear and non linear models be they structural or time series tend to embody contradictory assumptions that unfortunately make causal interpretability problematic the paper demonstrates how these difficulties can be avoided using a stochastic coefficient approach the final part of this article is devoted to a discussion of probabilistic logic as a valid tool for scientific analysis and interpretation of causal relationships

evaluating probability forecasts probability forecasts of events are routinely used in climate predictions in forecasting default probabilities on bank loans or in estimating the probability of a patient s positive response to treatment scoring rules have long been used to assess the efficacy of the forecast probabilities after observing the occurrence or nonoccurrence of the predicted events we develop herein a statistical theory for scoring rules and propose an alternative approach to the evaluation of probability forecasts this approach uses loss functions relating the predicted to the actual probabilities of the events and applies martingale theory to exploit the temporal structure between the forecast and the subsequent occurrence or nonoccurrence of the event