GeD spline estimation of multivariate Archimedean copulas

A new multivariate Archimedean copula estimation method is proposed in a non-parametric setting. The method uses the so-called Geometrically Designed splines (GeD splines) to represent the cdf of a random variable W_θ, obtained through the probability integral transform of an Archimedean copula with parameter θ. Sufficient conditions for the GeD spline estimator to possess the properties of the underlying theoretical cdf, K(θ,t), of W_θ, are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure, finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem are separated. Thus, the resulting spline estimator is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is truly multivariate, it brings about numerical efficiency and as a result can be applied with large volumes of data and for dimensions d≥2, as illustrated by the numerical examples presented.     

Discrete J-spectral factorization

This paper deals with the J-spectral factorization for general discrete rational matrices. A simple approach based on the Kalman filtering in Krein space is proposed. The main idea is to construct a stochastic state space filtering model in Krein space such that the spectral matrix of the output is equal to the rational matrix to be factorized. The spectral factor is then easily derived by using the generalized Kalman filtering in Krein space, which is similar to the H₂ spectral factorization. Our approach unifies the treatment of the H₂ spectral factorization and the J-spectral factorization. The applications of the derived results in H_∞ and risk-sensitive estimation for both nonsingular and singular systems are demonstrated.     

Spectral unmixing model to assess land cover fractions in Mongolian steppe regions

The land cover fractions (LCFs) and spectral reflectance of photosynthetic vegetation (PV), nonphotosynthetic vegetation (NPV), and bare soil were measured at 58 sites in semi-arid and arid regions of Mongolia in the summers of 2005 and 2006. These data sets allowed a detailed assessment of the impact of measurement geometry as represented by the solar zenith angle θ_s, sensor view zenith angle θ_v and azimuth view angle ? in the estimation of LCF values by means of the spectral unmixing model (SUM). The bidirectional distribution function (BRDF) was fitted to the reflectance data and then used to produce reflectance at various measurement geometries. LCFs from these reflectance data for a given combination of θ_s, θ_v, and ? were compared with visually determined LCFs. It was found that θ_s in the range of 30-45° produced a better agreement of LCFs. For θ_v, the agreement is not very sensitive to the choice of angle for the range 30-70°, although θ_v = 50° showed a slightly better performance. The azimuth view angle does not have strong influences to the LCF estimation, except for the case of ? = 180° (view toward the sun), which does not allow precise fitting of BRDF function over a tall vegetation site. Overall, this study verified the results of earlier studies obtained mostly for the American continents that SUM is capable of producing LCF estimates accurately and also found that its accuracy was, in general, much better than that by the more traditional approach of the supervised classification method (SCM) applied to images of a digital camera.     

Locally optimal tests for exponential distributions with type-I censoring

This article studies a locally optimal test φ^∗ for testing H₀:θ≥θ₀ versus H₁:θ<θ₀ for the lifetime parameter θ in an exponential distribution based on type-I censored data. Certain properties associated with φ^∗ are addressed. We compare the performance of φ^∗ with that of Spurrier and Wei’s [Spurrier, J.D., Wei, L.J., 1980. A test of the parameter of the exponential distribution in the type-I censoring case. J. Amer. Statist. Assoc. 75, 405-409] test φ_SW, which is based on the MLE of θ. The exact powers and asymptotic powers of φ^∗ and φ_SW are computed. The numerical results indicate that the power of φ^∗ is better than that of φ_SW when θ(0<θ<θ₀) is close to θ₀.     

Subsampling techniques and the Jackknife methodology in the estimation of the extremal index

For a sequence of independent, identically distributed random variables any limiting point process for the time normalized exceedances of high levels is a Poisson process. However, for stationary dependent sequences, under general local and asymptotic dependence restrictions, any limiting point process for the time normalized exceedances of high levels is a compound Poisson process, i.e., there is a clustering of high exceedances, where the underlying Poisson points represent cluster positions, and the multiplicities correspond to the cluster sizes. For such classes of stationary sequences there exists the extremal indexθ, 0?θ?1, directly related to the clustering of exceedances of high values. The extremal index θ is equal to one for independent, identically distributed sequences, i.e., high exceedances appear individually, and θ>0 for “almost all” cases of interest. The estimation of the extremal index through the use of the Generalized Jackknife methodology, possibly together with the use of subsampling techniques, is performed. Case studies in the fields of environment and finance will illustrate the performance of the new extremal index estimator comparatively to the classical one.     

A simple approach to the problem of 3-D reconstruction

A relatively simple mathematical procedure for the reconstruction of the 3-dimensional (3D) image of the left ventricle (LV) of the heart is presented. The method is based on the assumption that every ray whoch emanates from the midpoint of the long axis of the 3D body crosses the surface boundary of the ventricle at one and only one point. The coordinates r_i, φ_i, θ_i of the data points on, say, the outer boundary, (i.e., the epicardium) are calculated in a spherical coordinate system having its origin in the midpoint of the long axis. The problem of defining the coordinates of a prescribed grid point on the boundary is treated as an interpolation problem for the function r = r(φ, θ), defined in the rectangle 0 ≤ φ ≤ 2π; 0 ≤ θ ≤ π with r_i given in the points (φ_i, θ_i).     

Load-sharing system model and its application to the real data set

This study deals with the classical and Bayesian estimation of the parameters of a k-components load-sharing parallel system model in which each component's lifetime follows Lindley distribution. Initially, the failure rate of each of the k components in the system is h(t,θ₁) until the first component failure. However, upon the first failure within the system, the failure rates of the remaining (k − 1) surviving components change to h(t,θ₂) and remain the same until next failure. After second failure, the failure rates of (k − 2) surviving components change to h(t,θ₃) and finally when the (k − 1)th component fails, the failure rate of the last surviving component becomes h(t,θ_k). In classical set up, the maximum likelihood estimates of the load share parameters, system reliability and hazard rate functions along with their standard errors are computed. 100 × (1 − γ)% confidence intervals and two bootstrap confidence intervals for the parameters have also been constructed. Further, by assuming Jeffrey's invariant and gamma priors of the unknown parameters, Bayes estimates along with their posterior standard errors and highest posterior density credible intervals of the parameters are obtained. Markov Chain Monte Carlo technique such as Metropolis–Hastings algorithm has been utilized to generate draws from the posterior densities of the parameters.     

Empirical Bayesian analysis of the Poisson intervention and incidence parameters

In some population the AIDS/HIV incidence rate θ∈(0,∞) is altered in the middle of a data collection period due to preventive treatments imposed by the health service agencies. The intervened Poisson (IP) model in [Comput. Programs Biomed. 17 (1983) 89; Biometrics 48 (1985) 559] was introduced which is appropriate to analyze data of this type. However, the classical approach leading to the maximum likelihood (ML), moment (M) or minimum variance unbiased (MVU) estimator of θ is mathematically formidable and practically inconvenient as far as sequentially updating the estimate when new data arrive. Previous subjective Bayesian work has been done to overcome these issues. Hence, there is a need to devise a more practical empirical Bayesian technique to estimate θ, and it is done in this article. The results are illustrated using a data on AIDS/HIV incidence in the state of Alabama. Advantages in the Bayesian intervened approach are cited.     

Automatic bandwidth selection for circular density estimation

Given angular data θ₁,…,θ_n∈[0,2π) a common objective is to estimate the density. In case that a kernel estimator is used, bandwidth selection is crucial to the performance. A “plug-in rule” for the bandwidth, which is based on the concentration of a reference density, namely, the von Mises distribution is obtained. It is seen that this is equivalent to the usual Euclidean plug-in rule in the case where the concentration becomes large. In case that the concentration parameter is unknown, alternative methods are explored which are intended to be robust to departures from the reference density. Simulations indicate that “wrapped estimators” can perform well in this context. The methods are applied to a real bivariate dataset concerning protein structure.     

On the degree of polynomial parameter-dependent Lyapunov functions for robust stability of single parameter-dependent LTI systems: A counter-example to Barmish's conjecture

In this paper, we consider the robust Hurwitz stability analysis problems of a single parameter-dependent matrix A(θ)?A₀+θA₁ over θ∈[-1,1], where A₀,A₁∈R^n×n with A₀ being Hurwitz stable. In particular, we are interested in the degree N of the polynomial parameter-dependent Lyapunov matrix (PPDLM) of the form that ensures the robust Hurwitz stability of A(θ) via . On the degree of PPDLMs, Barmish conjectured in early 90s that if there exists such P(θ), then there always exists a first-degree PPDLM P(θ)=P₀+θP₁ that meets the desired conditions, regardless of the size or rank of A₀ and A₁. The goal of this paper is to falsify this conjecture. More precisely, we will show a pair of the matrices A₀,A₁∈R^3×3 with A₀+θA₁ being Hurwitz stable for all θ∈[-1,1] and prove rigorously that the desired first-degree PPDLM does not exist for this particular pair. The proof is based on the recently developed techniques to deal with parametrized LMIs in an exact fashion and related duality arguments. From this counter-example, we can conclude that the conjecture posed by Barmish is not valid when n?3 in general.     

On the structure of generalized rough sets

In this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that

1.

Any reflexive binary relation determines a topology.

2.

If θ is a reflexive and symmetric relation on a set X, then O={A⊆X|θ_-(A)=A} is a topology such that A is open if and only if it is closed.

3.

Conversely, for every topological space (X,O) satisfying the condition that A is open if and only if it is closed, there exists a reflexive and symmetric relation R such that O={A⊆X|R_-(A)=A}.

4.

Let θ be an equivalence relation on X. For any pseudo ω-closed subset A of X, θ₋(A) is an ω-closed set if and only if ω(x, x, … , x) ∈ θ₋(A) for any x ∈ X.

Moreover we consider properties of generalized rough sets.     

Lagrangean relaxation and decomposition in an uncapacitated 2-hierarchal location-allocation problem

We consider an uncapacitated 2-hierarchal location-allocation problem where p₁ level 1 facilities and p₂ level 2 facilities are to be located among n(?p₁ + p₂) potential locations so as to minimize the total weighted travel distance to the facilities when θ, (0 ? θ ? 1) fraction of the demand from a level 1 facility is referred to a level 2 facility. At most one facility may be located at any location. In this model, a level 2 facility provides services in addition to services provided by a level 1 facility.The problem is formulated as a mathematical programming problem, relaxed and solved by a subgradient optimization procedure. The proposed procedure is illustrated with an example.     

Linear Differential Operators for Polynomial Equations

Given a squarefree polynomial P ∈ k₀[ x,y ], k₀a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P =  0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k₀, and calculate information concerning the Galois group of P over ___ k₀(x) as well as overk₀ (x).     

Motion of spin-half particles in the axially symmetric field of naked singularities of the static <Emphasis Type="Italic">q</Emphasis>-metric

Quantum-mechanical motion of a spin-half particle is examined in the axially symmetric fields of static naked singularities formed by a mass distribution with a quadrupole moment (q-metric). The analysis is performed by means of the method of effective potentials of the Dirac equation generalized to the case where radial and angular variables are not separated. If ?1 < q < qlim, |qlim| ? 1, where q is the quadrupolemoment in proper units, the naked singularities do not exclude the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For an oblate mass distribution, the naked singularities of the q-metric are separated from a Dirac particle by infinitely large repulsive barriers followed by a potential well which deepens while moving apart from the equator (from θ = θ _min or θ = π ? θ _min) toward the poles. The poles make an exception, and at 0 < q < q*, there are some points θ _i for particle states with j ≥ 3/2.     

A practical approximation algorithm for the LMS line estimator

The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Robust estimators are widely used because of their lack of sensitivity to outlying data points. The least median-of-squares (LMS) regression line estimator is among the best known robust estimators. Given a set of n points in the plane, it is defined to be the line that minimizes the median squared residual or, more generally, the line that minimizes the residual of any given quantile q, where 0<q?1. This problem is equivalent to finding the strip defined by two parallel lines of minimum vertical separation that encloses at least half of the points.The best known exact algorithm for this problem runs in O(n²) time. We consider two types of approximations, a residual approximation, which approximates the vertical height of the strip to within a given error bound ε_r?0, and a quantile approximation, which approximates the fraction of points that lie within the strip to within a given error bound ε_q?0. We present two randomized approximation algorithms for the LMS line estimator. The first is a conceptually simple quantile approximation algorithm, which given fixed q and ε_q>0 runs in O(nlogn) time. The second is a practical algorithm, which can solve both types of approximation problems or be used as an exact algorithm. We prove that when used as a quantile approximation, this algorithm's expected running time is . We present empirical evidence that the latter algorithm is quite efficient for a wide variety of input distributions, even when used as an exact algorithm.     

Analysis of an infinite multi-server queue with an optional service

This paper deals with an infinite-capacity multi-server queueing system with a second optional service (SOS) channel. The inter-arrival times of arriving customers, the service times of the first essential service (FES) and the SOS channel are all exponentially distributed. A customer may leave the system after the FES channel with a probability (1 − θ), or the completion of the FES may immediately require a SOS with a probability θ (0 ? θ ? 1). The formulae for computing the rate matrix and stationary probabilities are derived by means of a matrix analytical approach. A cost model is developed to simultaneously determine the optimal values of the number of servers and the two service rates at the minimal total expected cost per unit time. Quasi-Newton method and Particle Swarm Optimization (PSO) method are employed to deal with the optimization problem. Under optimal operating conditions, numerical results are provided from which several system performance measures are calculated based on the assumed numerical values of the system parameters.     

Polarimetric SAR characterization of man-made structures in urban areas using normalized circular-pol correlation coefficients

Polarimetric Synthetic Aperture Radar (SAR) backscatter from man-made structures in urban areas is quite different than backscatter from predominantly natural areas. Backscatter from natural areas is often reflection symmetric; i.e., characterized by near zero values for covariance matrix off-diagonal terms of the form 〈S_HVS_HH^?〉, 〈S_HVS_VV^?〉 and their conjugates. A new approach is proposed to detect scattering from non-reflection symmetric structures using circular-pol, RR-LL, correlation coefficients, |ρ|. This method creates a normalization term, |ρ₀|, and then forms a ratio, |ρ|/|ρ₀|. The normalization term, |ρ₀|, contains the same diagonal terms of the covariance matrix. The 〈S_HVS_HH^?〉 and 〈S_HVS_VV^?〉 off-diagonal terms and their conjugates are purposely set to zero. The ratio, |ρ|/|ρ₀|, is rewritten as a product of separable helicity (τ) and orientation angle (θ) dependencies. The mathematical form of the τ dependence is a resonant singularity, or pole, term. This pole significantly enhances returns from man-made, high helicity, non-reflection symmetric structures. These structures have values of τ near the resonance value at τ = ± 1. Natural scatterers possess very strong RR / LL symmetry (τ ≈ 0) and the pole response for them is correspondingly weak. The dependence of |ρ|/|ρ₀| on the orientation angle (θ) is known from previous studies to be useful for measuring urban building alignments (relative to the azimuth direction) and measuring surface topography. The ratio |ρ|/|ρ₀| reduces much of the un-needed image detail of backscatter variations from natural areas of different surface roughness. This image simplification further facilitates detection of localized man-made targets.     

A sum-over-paths extension of edit distances accounting for all sequence alignments

This paper introduces a simple Sum-over-Paths (SoP) formulation of string edit distances accounting for all possible alignments between two sequences, and extends related previous work from bioinformatics to the case of graphs with cycles. Each alignment ℘, with a total cost C(℘), is assigned a probability of occurrence P(℘)=exp[−θC(℘)]/Z where Z is a normalization factor. Therefore, good alignments (having a low cost) are favored over bad alignments (having a high cost). The expected cost ∑_℘∈PC(℘)exp[−θC(℘)]/Z computed over all possible alignments ℘∈P defines the SoP edit distance. When θ→∞, only the best alignments matter and the measure reduces to the standard edit distance. The rationale behind this definition is the following: for some applications, two sequences sharing many good alignments should be considered as more similar than two sequences having only one single good, optimal, alignment in common. In other words, sub-optimal alignments could also be taken into account. Forward/backward recurrences allowing to efficiently compute the expected cost are developed. Virtually any Viterbi-like sequence comparison algorithm computed on a lattice can be generalized in the same way; for instance, a SoP longest common subsequence is also developed. Pattern classification tasks performed on five data sets show that the new measures usually outperform the standard ones and, in any case, never perform significantly worse, at the expense of tuning the parameter θ.     

Characterization of analytic phase signals

In many cases, a real-valued signal χ(t) may be associated with a complex-valued signal a(t)e^iθ(t), the analytic signal associated with χ(t) with the characteristic properties χ(t) = a(t) cosθ(t) and H(a(·)cosθ(·))(t) = a(t)sinθ(t). Using such obtained amplitude-frequency modulation the instantaneous frequency of χ(t) at the time t₀ may be defined to be θ′(t₀), provided θ′(t₀) ≥ 0. The purpose of this note is to characterize, in terms of analytic functions, the unimodular functions F(t) = C(t) + iS(t),C²(t) + S² (t) = 1, a.e., that satisfy HC(t) = S(t). This corresponds to the case a(t) ≡ 1 in the above formulation. We show that a unimodular function satisfies the required condition if and only if it is the boundary value of a so called inner function in the upper-half complex plane. We also give, through an explicit formula, a large class of functions of which the parametrization C(t) = cosθ(t) is available and the extra condition θ′(t) ≥ 0, a.e. is enjoyed. This class of functions contains Blaschke products in the upper-half complex plane as a proper subclass studied by Picinbono in [1].     

Redefined fuzzy implicative filters

Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (θ, ψ)-fuzzy implicative filters where θ, ψ are any two of {∈, q, ∈ ∨ q, ∈ ∧ q} with θ ≠ ∈ ∧ q is introduced, and related properties are discussed. Relations between (∈ ∨ q, ∈ ∨ q)-fuzzy implicative filters and (∈, ∈ ∨ q)-fuzzy implicative filters are investigated, and conditions for an (∈, ∈ ∨ q)-fuzzy implicative filter to be an (∈, ∈)-fuzzy implicative filter are provided. Characterizations of (∈, ∈ ∨ q)-fuzzy implicative filters are given, and conditions for a fuzzy set to be a (q, ∈ ∨ q)-fuzzy implicative filter are provided.