Bivariate Example Research Articles

Stochastic preemptive goal programming to balance goal achievements under uncertainty

AbstractStructured decision‐making in the presence of conflicting goals is difficult, and even more so when accounting for uncertainties in the goals or constraints. In this article, we propose a new approach to multi‐criteria decision‐making that extends the deterministic preemptive goal programming approach to account for such uncertainties. The uncertainties may be characterized in various ways including a Bayesian network or extensive Monte Carlo multi‐variate output. We contend the proposed stochastic preemptive goal programming approach is particularly applicable when better goal achievement increases uncertainty or induces risk to other goals. Resulting solutions tend to be a balance of the goals' achievements and robust to minor changes to the goals or constraints. We demonstrate the approach using three illustrative examples: a univariate example, a bivariate example, and a stock portfolio optimization example along with an application to determine military requests for absence. Our preliminary results suggest that a stochastic preemptive goal programming approach represents an effective means of analysing multi‐criteria decision‐making problems under uncertainty.

A Novel TRUST-TECH-Enabled Trajectory-Unified Methodology for Computing Multiple Optimal Solutions of Constrained Nonlinear Optimization: Theory and Computation

A novel trajectory-unified (TJU) methodology, enabled by transformation under stability-retraining equilibria characterization (TRUST-TECH, TT) method, is proposed to systematically and deterministically compute a set of multiple local optimal solutions (LOSs), if not all of them, for constrained nonlinear optimization problems. The proposed TT-TJU methodology is composed of two key phases. During Phase I, it robustly computes a LOS; while during Phase II, it systematically computes a set of neighboring LOSs. A theoretical foundation is developed for the proposed TT-TJU methodology. The existence of an optimal solution is shown to be equivalent to the existence of a stable equilibrium point of a class of nonlinear dynamical system derived from the constrained optimization problem under study. From a computational viewpoint, Phase I is numerically implemented in a 3-stage procedure for fast and robust computation of an optimal solution. The second phase makes use of the TRUST-TECH methodology to lead the search away from a computed optimal solution and continue to deterministically compute neighboring LOSs. The proposed methodology has been evaluated on an illustrative bivariate example and a benchmark problem with 110 variables and 55 constraints with promising results.

On Rereading D. Basu’s Jointly Sufficient Statistic Example Made Up of Two Ancillaries and Miscellany

D. Basu gave a striking bivariate normal example, N2(0, 0, 1, 1, ρ) with an unknown correlation coefficient ρ, − 1 < ρ < 1, where the jointly sufficient statistic (X1, X2) consists of two ancillary statistics X1, X2. We exhibit examples of ancillary statistics involving both X1, X2 followed by other variations. A situation is highlighted where a jointly minimal sufficient statistic (X1, X2) is made up of ancillary statistics X1, X2. Then, we illustrate a concrete multivariable full model which questions the customary practice of contrasting non-constant reduced models via deletion of one covariate at-a-time.

Performance of translation approach for modeling correlated non-normal variables

It is common to construct a consistent multivariate distribution from non-normal marginals and Pearson product–moment correlations using the well known translation approach. A practical variant of this approach is to match the Spearman rank correlations of the measured data, rather than the Pearson correlations. In this paper, the performance of these translation methods is evaluated based on their abilities to match the following exact solutions from one benchmark bivariate example where the joint distribution is known: (1) high order joint moments, (2) joint probability density functions (PDFs), and (3) probabilities of failure. It is not surprising to find significant errors in the joint moments and PDFs. However, it is interesting to observe that the Pearson and Spearman methods produce very similar results and neither method is consistently more accurate or more conservative than the other in terms of probabilities of failure. In addition, the maximum error in the probability of failure may not be associated with a large correlation. It can happen at an intermediate correlation.

Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants

Resultants are defined in the sparse (or toric) context in order to exploit the structure of the polynomials as expressed by their Newton polytopes. Since determinantal formulae are not always possible, the most efficient general method for computing resultants is rational formulae. This is made possible by Macaulay’s famous determinantal formula in the dense homogeneous case, extended by D’Andrea to the sparse case. However, the latter requires a lifting of the Newton polytopes, defined recursively on the dimension. Our main contribution is a single-lifting function of the Newton polytopes, which avoids recursion, and yields a simpler method for computing Macaulay-type formulae of sparse resultants. We focus on the case of generalized unmixed systems, where all Newton polytopes are scaled copies of each other, and sketch how our approach may extend to mixed systems of up to four polynomials, as well as those whose Newton polytopes have a sufficiently different face structure. In the mixed subdivision used to construct the matrices, our algorithm defines significantly fewer cells than D’Andrea’s, though the matrix formulae are same. We discuss asymptotic complexity bounds and illustrate our results by fully studying a bivariate example.

Variance Shifts Identification Model of Bivariate Process Based on LS-SVM Pattern Recognizer

Multivariate Statistical Process Control (MSPC) techniques are effective tools for detecting the abnormalities of multivariate process variation. MSPC techniques are based on overall statistics; this has caused the difficulties in interpretation of the alarm signal, that is, MSPC charts do not provide the necessary information about which process variables (or subset of them) are responsible for the signal, and this task is left up to the quality engineers in production field. This article proposes a model based on LS-SVM pattern recognizer to diagnose the bivariate process abnormality in covariance matrix. The main property of this model is a supplement of MSPC |S| chart to identify the variable(s) which is (are) responsible for the process abnormality when |S| chart issue a warning signal. Through simulation experiment, the performance of the model is evaluated by accuracy rate of pattern recognition. The results indicate that the proposed model is an effective method to interpret the root causes of the process abnormality. A bivariate example is presented to illustrate the application of the proposed model.

Source(s) Identification of Variance Shifts in Bivariate Process Using LS-SVM Based Pattern Recognition Model

MSPC techniques are effective tools for detecting the abnormalities of process variation. But MSPC charts do not provide the necessary information about which process variables (or subset of them) are responsible for the signal. In order to identify the process abnormality in covariance matrix of bivariate process, this article proposes a model based on LS-SVM pattern recognizer and |S| chart method, the main property of this model is to identify the assignable causes through LS-SVM pattern recognizer technique when |S| chart issue a warning signal. The simulation results indicate that the proposed model is feasible and effective. A bivariate example is presented to illustrate the application of the proposed model.

Multivariate generalized Pareto distributions

Statistical inference for extremes has been a subject of intensive research over the past couple of decades. One approach is based on modelling exceedances of a random variable over a high threshold with the generalized Pareto (GP) distribution. This has proved to be an important way to apply extreme value theory in practice and is widely used. We introduce a multivariate analogue of the GP distribution and show that it is characterized by each of following two properties: first, exceedances asymptotically have a multivariate GP distribution if and only if maxima asymptotically are extreme value distributed; and second, the multivariate GP distribution is the only one which is preserved under change of exceedance levels. We also discuss a bivariate example and lower-dimensional marginal distributions.

Multivariate Population Balances via Moment and Monte Carlo Simulation Methods: An Important Sol Reaction Engineering Bivariate Example and “Mixed” Moments for the Estimation of Deposition, Scavenging, and Optical Properties for Populations of Nonspherical Suspended Particles

Reactors or crystallizers synthesizing valuable particles can be formally described by combining the laws of continuum transport theory with a population balance equation governing evolution of the “dispersed” (suspended) particle population. Early examples necessarily focused on highly idealized device configurations and populations described locally using only one particle state variable, i.e., “size” (length or volume). However, in almost every application of current/future importance, a multivariate description is required, for which the existing literature offers little guidance. We describe here our recent research on an instructive bivariate prototype of physical interest (coagulating, sintering nanoparticles of prescribed composition, etc.) that will, hopefully, motivate a broader attack on important multivariate population balance problems, including those describing continuous molecular mixtures. We illustrate the use of both physically based bivariate “mixed” moment and Monte Carlo simulation methods amenable to future generalization. Multivariate extensions, improved and fully coupled rate laws, and consideration of interacting (coexisting) populations will be required to deal with future sol reaction engineering problems using similar strategies/techniques.

Measurement error in epidemiology: the design of validation studies II: bivariate situation.

The bias in relative risk estimates caused by errors in measurement of the relevant exposure is being increasingly recognized in epidemiology. Estimation of the necessary correction factor to remove this bias for univariate exposure has been considered in an earlier paper. We consider here the multivariate situation in which non-differential errors in measurement can lead to incorrect identification of the variable most closely associated with disease. Estimation of the necessary correction factor when the true exposure is unobservable necessarily requires assumptions. We explore the robustness of the estimation to departures from a range of assumptions. The value of good biomarkers is demonstrated. We present a bivariate example in which failure to take account of measurement error leads to the incorrect exposure being identified as the important determinant of disease risk.

The Wavelet Element Method: Part I. Construction and Analysis

The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some Rn. The domain of interest is split into subdomains which are mapped to a simple reference domain, heren-dimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the reference domain such that mapping them to each subdomain and matching along the interfaces leads to a wavelet system on the domain. In this paper we use adapted biorthogonal wavelet systems on the interval in such a way that tensor products of these functions can be used for the construction of wavelet bases on the reference domain. We describe the matching procedure in any dimensionnin order to impose continuity and prove that it leads to a construction of a biorthogonal wavelet system on the domain. These wavelet systems characterize Sobolev spaces measuring both piecewise and global regularity. The construction is detailed for a bivariate example and an application to the numerical solution of second order partial differential equations is given.

A Method for Defining and Selecting Multivariate “Percentile” Exemplars: H-Contours

Designing to fit the “95th percentile man” can proceed once the “95th percentile man” is given some conceptual and statistical meaning. Because of the multivariate nature of the problem, there are a number of ways to define and construct such a “95th percentile man”. At present, a percentile figure is generally one which has the given percentile for height but has the rest of the body segments proportioned according to some formulae relating the segments to the height. It is well known that the such resultants are not satisfactory representatives of the population. As a result, one often tries to piece manikins together by attaching (e.g.) “5th percentile” arms onto a “95th percentile” tall figure, or otherwise combining different percentile segments in order to try to represent the considerable percentile variation in body segments existing within individuals in the population of a “95th percentile (height) man”. Methods for constructing more representative exemplars (or manikins) of a population are needed for multivariate design and analysis. This paper proposes a method based on “h-contours” which has the advantages of conceptually extending the univariate percentile concept as well as being suitable for design by constructing realistic multivariate exemplars. A multivariate Gaussian distribution is used for computational purposes. The proposed method is briefly contrasted with the “ellipse” method as proposed by others. A bivariate example is used to illustrate the concepts and computations.

Correlation of spatial climate/weather maps and the advantages of using the Mahalanobis metric in predictions

he skill in predicting spatially varying weather/climate maps depends on the definition of the measure of similarity between the maps. Under the justifiable approximation that the anomaly maps are distributed multinormally, it is shown analytically that the choice of weighting metric, used in defining the anomaly correlation between spatial maps, can change the resulting probability distribution of the correlation coefficient. The estimate of the numbers of degrees of freedom based on the variance of the correlation distribution can vary from unity up to the number of grid points depending on the choice of weighting metric. The (pseudo-) inverse of the sample covariance matrix acts as a special choice for the metric in that it gives a correlation distribution which has minimal kurtosis and maximum dimension. Minimal kurtosis suggests that the average predictive skill might be improved due to the rarer occurrence of troublesome outlier patterns far from the mean state. Maximum dimension has a disadvantage for analogue prediction schemes in that it gives the minimum number of analogue states. This metric also has an advantage in that it allows one to powerfully test the null hypothesis of multinormality by examining the second and third moments of the correlation coefficient which were introduced by Mardia as invariant measures of multivariate kurtosis and skewness. For these reasons, it is suggested that this metric could be usefully employed in the prediction of weather/climate and in fingerprinting anthropogenic climate change. The ideas are illustrated using the bivariate example of the observed monthly mean sea-level pressures at Darwin and Tahitifrom 1866—1995.

A bivariate normal example where the locally most powerful and distantly most powerful test statistics are beaten everywhere from a median p-value standpoint

In the bivariate normal, n=2 case, when testing H0:μx=μy=0,σ2 x=σ2 y=1, ρ=0 vs. H1:μx=μy=0,σ2 x=σ2 y=1, 0<ρ<1, it is shown that the median p-values given by the locally most powerful test and the distantly most powerful test are both beaten everywhere by the median of a third test.

Multiple functional regression—II. Rotation followed by classical regression technique

A method and FORTRAN program is presented for calculating a multivariate functional regression, analogous to the reduced major axis regression for the bivariate situation. The method is based on rotating the coordinates of the experimental points with respect to the axes. The rotation is such that the direction of distances whose sums of squares are to be minimized now are parallel to the “ Y” axis. A classical regression, using matrix expressions, then is performed following which the best-fit line is rotated back to give its equation in the original space. The rotation is performed to give a regression in a direction proportional to the square roots of the total variances of the parameters in the regression (which is the reduced major axis regression for the bivariate situation). Standard errors of the regression coefficients are estimated by a Monte-Carlo method. The program also can perform a classical regression, simply be omitting the rotation phase. The two examples used in Part 1 are repeated. In both examples the program gives the same result for classical regression as other methods including the function minimization method of Part 1. The first example is a bivariate example and the program gives the same answer as analytical expressions for the reduced major axis regression. In the second, the multivariate example, the program gives the same result as the function minimization method of Part 1.

Fast Evaluation of the Distribution of the Durbin-Watson and Other Invariant Test Statistics in Time Series Regression

Abstract A method is given for evaluating p values in O(n) operations for a general class of invariant test statistics that can be expressed as the ratio of quadratic forms in time series regression residuals. The best known of these is the Durbin-Watson statistic, although several others have been discussed in the literature. The method is numerically exact in the sense that the user specifies the error tolerance at the outset. As with existing exact methods, the problem is reexpressed in terms of the distribution function of a single quadratic form in independent normals, which is evaluated by numerically inverting its characteristic function. Existing methods, however, calculate the characteristic function by reducing the matrix defining the quadratic form to either eigenvalue or tridiagonal form, each of which requires O(n 3) operations for sample size n, whereas the new method uses a modification of the Kalman filter to do it in O(n) operations. Moreover, the new method has minimal storage requirements, which are independent of n, whereas eigenvalue and tridiagonal methods have storage requirements increasing as n 2. The method extends to bivariate problems, where the Kalman filter algorithm has even greater computational advantages over the eigenvalue approach. The computational properties of the new method are illustrated for the Durbin-Watson statistic and a bivariate example.

Pre‐post Treatment Evaluation by Symmetry Testing in Square Contingency Table

AbstractStarting from an r × r contingency table derived from ordinal pre‐ and post treatment observations, the traditional Bowker symmetry test is suggested to be replaced by one of the following alternatives (1) a globals test of general improvement, (2) a set of simultaneous symmetry tests of differential improvement without assuming general improvement under H0 and (3) a set of simultaneous symmetry tests beyond assuming general improvement under H0. The tests are illustrated by a bivariate example of observing neuroticism and introversion binarily (+, ‐) before and after Rogerian psychotherapy in a sample of clients.

Estimation of location of discontinuity in a density

Here we propose a few estimators of θ, in addition to those studied in Goria (1978), the point of discontinuity of the probability density $$f(x,\theta ) = \frac{1}{{2\Gamma (\alpha )}}e^{ - |x - \theta |} |x - \theta |^{\alpha - 1} ,$$ for $$0< \alpha< 1, - \infty< x< \infty , - \infty< \theta< \infty .$$ We establish the consistency and the optimality of the Bayes and the maximum probability estimators. Despite their nice properties, these estimators are not easy to compute in this case and their effective computation depends on the knowledge of the exponent α. Hence, we propose another class of estimators, dependent upon the spacings of the observations, computable without actual knowledge of the value of α as long as it is known that α < α0 < 1: we show that these estimators converge at the best possible rate. We further demonstrate, using a modified version of the maximum probability estimator's technique, that the tails of the density do not substantially effect their efficiency. Finally a bivariate family of densities, having a ridge dependent on the parameter θ, is considered and it is shown that this family exhibits features similar to the univariate case, and thus, the necessary modifications of the arguments of the univariate case are utilized for the estimation of θ in this bivariate example.