Univariate Problem Research Articles

BIVARIATE FREE KNOT SPLINES

We consider the least squares approximation of gridded 2D data by tensor product splines with free knots. The smoothing functional to be minimized—a generalization of the univariate Schoenberg functional—is chosen in such a way that the solution of the bivariate problem separates into the solution of a sequence of univariate problems in case of fixed knots. The resulting optimization problem is a constrained separable least squares problem with tensor product structure. Based on some ideas developed by the authors for the univariate case, an efficient method for solving the specially structured 2D problem is proposed, analyzed and tested on hand of some examples from the literature.

Finite Sample Performance of tests for Symmetry of the Errors in a Linear Model

The finite sample performance of a number of tests for symmetry of the distribution of the errors of a linear model is considered. The first family of tests is based on the discrepancy between two regression fits. The first fit is appropriate under symmetric errors while the second is appropriate for skewed as well as symmetric error distributions. The second family of procedures consists of tests for the univariate symmetry problem. Thus, in the linear model setting these tests are based on residuals. An extensive empirical study of the finite sample, null behavior of the tests is presented. The results of a power comparison among the tests is also discussed.

Higher order C( α) tests with applications to mixture models

A class of second-order C( α) tests is proposed for testing composite hypotheses when no optimal test among the (first-order) C( α) tests, as defined by Neyman (Probability and Statistics, The Harald Cramer Volume, Almqvist and Wiksell, Uppsala, Sweden, 1959, p. 213), exists. The form of an optimal test of the new subclass is also presented. The procedure is seen to be easily extendable to C( α) tests of third or higher orders. Two examples of univariate normal mixtures are considered where the procedure is successfully applied. For the corresponding multivariate models, union–intersection tests of Roy (Ann. Math. Statist. 24 (1953) 220) are derived by combining the above optimal tests of the univariate problems.

Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints

In this paper, Lipschitz univariate constrained global optimization problems where both the objective function and constraints can be multiextremal are considered. The constrained problem is reduced to a discontinuous unconstrained problem by the index scheme without introducing additional parameters or variables. A Branch-and-Bound method that does not use derivatives for solving the reduced problem is proposed. The method either determines the infeasibility of the original problem or finds lower and upper bounds for the global solution. Not all the constraints are evaluated during every iteration of the algorithm, providing a significant acceleration of the search. Convergence conditions of the new method are established. Extensive numerical experiments are presented.

Observation and background adjoint sensitivity in the adaptive observation‐targeting problem

AbstractRecent observation‐targeting field experiments, such as the Fronts and Atlantic Storm‐Track Experiment (FASTEX) and the NORth Pacific Experiment (NORPEX), have demonstrated that by using objective adjoint techniques it is possible, in advance, to identify regions of the atmosphere where forecast‐error growth in numerical forecast models is maximally sensitive to the error in the initial conditions. 'Qpically, such techniques produce a field of the sensitivity of some aspect of the forecast to the analysis field. This analysis sensitivity field is then used to identify promising targets for the deployment of additional observations during the flight‐planning phase of field experiments. While FASTEX and, particularly. NORPEX had a number of successful ‘hits’, where the addition of dropsondes reduced the forecast error, there were also failures.None of the objective techniques have involved any consideration of the characteristics of the data‐assimilation systems used in the analysis of the targeted observations. In particular, the interaction with the background field, interactions with other observations, and the background‐ and observationerror characteristics have been ignored. This can lead to potential mis‐sampling, conflict with other observations, and inefficient use of aircraft and expendables.In this study, the adjoint of a simplified data‐assimilation system is used to determine directly the sensitivity of the forecast aspect to the observations and the background field. The procedure is illustrated by using simplified linear contexts such as the one‐ and two‐dimensional horizontal univariate problem and the one‐dimensional direct radiance assimilation problem. Adaptive targeting tools, such as a single‐observation sensitivity map and a marginal observation sensitivity vector, are devised and tested. The possibility of determining when the forecast would be sensitive to the background field and/or the observations is demonstrated. Such dependencies are shown to be a function of the specified observation‐ and background‐error variances, the characteristic scales of analysis sensitivity vector and background‐error correlations, and the properties of forward (observation) operators.Although the present experiments concerned simplified assimilation systems and observation networks, extension of the technique to real situations is quite feasible. Obtaining the adjoint of a full three‐dimensional variational assimilation system is straightforward; moreover, the target areas are small and contain relatively few observations so the computational requirements are modest.Finally, the data‐assimilation adjoint theory can be used for a posteriori assessment of those sources of forecast error which are due to errors in the initial analysis.

A test for bivariate two sample location problem

A consistent test for difference in locations between two bivariate populations is proposed, The test is similar as the Mann-Whitney test and depends on the exceedances of slopes of the two samples where slope for each sample observation is computed by taking the ratios of the observed values. In terms of the slopes, it reduces to a univariate problem, The power of the test has been compared with those of various existing tests by simulation. The proposed test statistic is compared with Mardia's(1967) test statistics, Peters-Randies(1991) test statistic, Wilcoxon's rank sum test. statistic and Hotelling' T2 test statistic using Monte Carlo technique. It performs better than other statistics compared for small differences in locations between two populations when underlying population is population 7(light tailed population) and sample size 15 and 18 respectively. When underlying population is population 6(heavy tailed population) and sample sizes are 15 and 18 it performas better than other statistic compared except Wilcoxon's rank sum test statistics for small differences in location between two populations. It performs better than Mardia's(1967) test statistic for large differences in location between two population when underlying population is bivariate normal mixture with probability p=0.5, population 6, Pearson type II population and Pearson type VII population for sample size 15 and 18 .Under bivariate normal population it performs as good as Mardia' (1967) test statistic for small differences in locations between two populations and sample sizes 15 and 18. For sample sizes 25 and 28 respectively it performs better than Mardia's (1967) test statistic when underlying population is population 6, Pearson type II population and Pearson type VII population

A new non-polynomial univariate interpolation formula of Hermite type

A new C∞ interpolant is presented for the univariate Hermite interpolation problem. It differs from the classical solution in that the interpolant is of non‐polynomial nature. Its basis functions are a set of simple, compact support, transcendental functions. The interpolant can be regarded as a truncated Multipoint Taylor series. It has essential singularities at the sample points, but is well behaved over the real axis and satisfies the given functional data. The interpolant converges to the underlying real‐analytic function when (i) the number of derivatives at each point tends to infinity and the number of sample points remains finite, and when (ii) the spacing between sample points tends to zero and the number of specified derivatives at each sample point remains finite.

Error propagation modelling in raster GIS: overlay operations

Performing data manipulations on maps that possess error as a result of the process of data collection leads to error propagation. The errors that are present in maps are modified by such operations in ways that may undermine the purposeofanalysisand lead to increased uncertainty in thevalidity ofthe conclusions that are drawn. This paper analyses how source map error propagates as a result of overlay operations. Geman and Geman's corruption model for individual source map error is used for the analysis which allows for attribute measurement error and location error that can then interact with the (true) source map geography. This paper reports theoretical results on the univariate overlay problem and then extends these results through simulation. Throughout a set of source maps and error processes are used with specified properties in order to examine in detail the interactions that can take place between the different elements of the source map structure and the error process. The paper uses ANOVA methods to quantify the contribution made by different components of the map and the measurement process to spatial (join count) and non-spatial (proportion of misclassified pixels and the Kappa index) descriptors of error. The paper concludes with a discussion of the usefulness of these results for managing error in spatial databases and for the development of automated uncertainty reporting.

Monte Carlo sampling in dual space for approximating the empirical halfspace distance

The Kolmogorov-Smirnov distance is an important tool for constructing confidence sets and tests in univariate problems. In multivariate settings, an analogous role is played by the halfspace distance, which has the merit of being invariant under linear transformations. However, the evaluation of the halfspace distance between two samples is a computationally very intensive combinatorial problem even in moderate dimensions, which severely restricts the use of the halfspace distance, especially in resampling procedures. To approximate this distance in a fast and data-dependent way, the notion of a dual measure is introduced. Based on geometric concepts, it will be shown how the above problem can be put as a density estimation problem using Monte Carlo sampling in a certain dual space. A central limit theorem for the empirical halfspace distance is derived and used as a gauge to compare the new procedure with a traditional random search.

On the minimization of correlated residuals

SUMMARY A variety of methods for solving overdetermined linear systems of equations are considered. The Gaussian method of least squares assumes a noise-free matrix of coefficients. The maximum-likelihood method assumes the same amount of noise in each column of the matrix and the resulting vector. Both methods represent special cases of the general problem of deriving unbiased solutions in the presence of noise. Here, the minimization of correlated residuals is introduced. A set of boundary conditions is required to adjust the statistical properties of the solution. This method minimizes the second term of the autocovariance function of the residuals and is independent of the noise. The operation of the method is outlined analytically using an univariate scatter problem and applied numerically to synthetic data of a linear regression problem. To demonstrate the applicability of the method in practical situations, geomagnetic response functions are derived from observational data using the Z : H method.

Regression Adjustment for Variables in Multivariate Quality Control

Multivariate process control problems are inherently more difficult than univariate problems. It is not always clear what type of multivariate statistic should be used, and the most statistically p...

Robust Procedures for Testing Equality of Covariance Matrices

Normal-theory tests for common variance are very sensitive to departures from normality in both univariate and multivariate applications. For mild departures from normality and moderate sample size, nominal a = .01 produced Type I error rates in excess of .20 in simulation studies. Since a modification of Levene's test has been shown to perform well for the univariate problem, two multivariate generalizations of this test are proposed. Both procedures are robust to departures from normality, and are simple computationally. Comparisons of the power of these two procedures and a procedure proposed previously by Tiku and Balakrishnan (1985, Communications in StatisticsTheory and Methods 14, 3033-3051) indicate that the three methods should be viewed as complementary.

Exchangeability in Multivariate Markov Chain Models

Time-homogeneous Markov chain models with state space [0, 1]k are useful in analysis of binary follow-up data on k individuals that interact. The number of parameters increases exponentially with k so more restrictive models are imperative for statistical inference. The hypothesis that the matrix of transition probabilities is invariant under permutation of individuals is discussed. It is shown that if individuals are exchangeable, then the process counting the number of individuals occupying a given state is a Markov chain. This reduction of data is sufficient if either at most a single individual may change state between two consecutive time points or if a state is absorbing. Similar results are obtained for exchangeability within two subgroups. Inference in the multivariate process reduces to a univariate problem if individuals are independent given the group's previous response. It is shown how conditional independence could be tested assuming exchangeability. The different hypotheses re examined in an analysis of the occurrence of bacteria in milk samples of Danish dairy cattle.

Interpolation by bivariate quadratic splines on a four-directional mesh

We give sufficient conditions which guarantee that a set of points admits unique Lagrange interpolation by quadratic splines on a four-directional mesh. The poisedness of these sets will be proved by reducing one bivariate problem to a finite sequence of univariate problems.

GCDHEU: Heuristic polynomial GCD algorithm based on integer GCD computation

A heuristic algorithm, GCDHEU, is described for polynomial GCD computation over the integers. The algorithm is based on evaluation at a single large integer value (for each variable), integer GCD computation, and a single-point interpolation scheme. Timing comparisons show that this algorithm is very efficient for most univariate problems and it is also the algorithm of choice for many problems in up to four variables.

Software Reviews : STATLIB 1 & 2 Reviewed by Donald R. Ploch, University of Tennessee Publisher: PDS Inc., PO Box 3280, Wallingford, CT 06494 (telephone: 800-426-7279) Year of Publication: 1987 Price: $194.95 (includes the Personal Scientific Calculator Machine Specificity: IBM PC, XT, AT, or compatible ' System Requirements: CGA, math coprocessor, 192K RAM, one double- sided disk drive, IBM APL

sTATiIB it 2 is excellent for the analysis of small, experimental datasets by statistically inclined investigators who are familiar with ŒM'S version of APL for the Pc. The farther one is from that ideal type, the less satisfactory or more difficult to use the package will be. It contains a wide array of statistics for the analysis of univariate distributions, experimental designs, quality control, regression, and multivariate problems. Procedures are detailed and flexible.

On the identifiability of multivariate survival distribution functions

Let ( T 1, T 2) be a non-negative random vector which is subjected to censoring random intervals [ X 1, Y 1] and [ X 2, Y 2]. The censoring mechanism is such that the available informations on T 1 and T 2 are expressed by a pair of random vectors W=( W 1, W 2) and δ=( δ 1, δ 2), where W i =max(min( Y i , T i ), X i ) and δ i 1, ifX i⩽T i⩽Y i, i=1,2, 2, ifT i>Y i, i=1,2, 3, ifT i<Y i, i=1,2, In this paper we will show that under some mild conditions the joint survival function of T 1 and T 2 can be expressed uniquely as functional of observable joint survival functions. Our results extend recent works on the randomly right censored bivariate data case and on the univariate problem with double censoring to the bivariate data with double censoring.

Fixed width interval estimation for the multiple response callibration problem

A multiple response extension is proposed for the univariate calibration problem and a sequential solution is developed. The sequential sampling procedure is a generalization of existing univariate results of Perng and Tong (1974) and is based on the theory developed by Chow and Robbins (1965). It is shown that the proposed multiple response procedure is at least as good as and in many instances better than the univariate method

On the inadmissibility of preliminary-test estimators when the loss involves a complexity cost

Estimation-preceded-by-testing is studied in the context of estimating the mean vector of a multivariate normal distribution under squared error loss together with a complexity cost. It is shown that although the preliminary test estimator is admissible for the univariate problem (cf Meeden and Arnold (1979),J. Amer. Statist. Assoc.,74, 872–874), for dimensionp≧3, the estimator is inadmissible. A new preliminary test estimator is obtained, which depends on the cost for each component and dominates the usual preliminary test estimator.

On the characteristic function of multivariate Student t‐distribution

AbstractThe characteristic function of the multivariate Student t‐distribution is obtained, and it is shown that this characteristic function has the pedogogical virtue of reducing the multivariate problem to the analogous univariate problem. Applications of the characteristic function are discussed.