Trace Criterion Research Articles

Space-time convolutional codes over GF(p) for two transmit antennas

We present new space-time trellis codes for two transmit antennas and p-PSK modulations, where p=3. 5. 7.11. 13.17, satisfying the rank and the determinant or the trace criteria. The system utilizes a rate 1/2 convolutional encoder over GF(p), p a prime. Some encoder properties are presented that simplify the code search.

Asymptotic expansions of the null distributions for the Dempster trace criterion

In this paper, we study the Dempster trace criterion. When the number of variables and the dimension of the null hypothesis are large relative to sample size, we derive the asymptotic distribution and Cornish-Fisher expansion of the Dempster trace criterion in the cases such that the covariance matrix is known and that the covariance matrix is unknown. Finally, we study the accuracy of the asymptotic expansion by the numerical simulation

Minimax designs for optimum mixtures

In a mixture experiment the measured response is assumed to depend only on the relative proportion of ingredients or components present in the mixture. Scheffe [1958. Experiments with mixtures. Journal of Royal Statistical Society B 20, pp. 344–360; 1963. Simplex-centroid design for experiments with mixtures. Journal of Royal Statistical Society B 25, 235–263] first systematically considered this problem and introduced different models and designs suitable in such situations. Optimum designs for the estimation of parameters of different mixture models are available in the literature. The problem of estimating the optimum proportion of mixture components is of great practical importance. Pal and Mandal [2006. Optimum designs for optimum mixtures. Statistics and Probability Letters 76, 1369–1379] first attempted to find a solution to this problem using the trace criterion, assuming prior knowledge about the optimum mixing proportions. In this paper the minimax criterion has been employed to find a solution to the above problem.

On optimality of a second order rotatable design in three demensions

In this paper, we use the existing second order rotatable design to for a basis for selecting optimum design based on the known classical optimality criteria. We estimate the particular values of the moments used for the existing design and determine the information matrix. The exact values of the particular cases in the family of matrix means, Φ p , with p &#8712 (-∞1) are obtained which are integral in discriminating design optimality. Keywords :Optimality criteria, Rotatability, Information matrix, Determinant criterion, Average- variance criterion, Eigen-value criterion, and Trace criterion > East African Journal of Statistics Vol. 1 (2) 2006: pp. 123-129

Professor Siotani’s Contributions to Statistics

Professor Minoru Siotani has contributed to statistics through the study of various procedures for multivariate analysis, in which the distributional problem is one of the most important topics. He derived asymptotic expansions for distributions of statistics and their percentile points in multivariate analysis. His work is published in several refereed journals listed below and is reviewed by Dudewicz [D] and Fujikoshi [F], which are published in the special issue of the American Journal of Mathematical and Management Sciences (AJMMS) in honor of Professor Siotani on his 70th birthday. Most of his contributions are detailed in Siotani, Hayakawa, and Fujikoshi [S]. This book has been read by a number of statisticians and graduate students and is cited in many articles published in international journals. One of the great contributions by Siotani is the derivation of the asymptotic expansions for the distribution of generalized Hotelling’s T 2 0 statistic for testing linear hypotheses in a multivariate linear model, say, for instance, to test a hypothesis about multinormal means. He derived asymptotic expansions not only for null distributions but also nonnull distributions ([7], [10], [23]). As a result of this work, it became possible to compare the power of three criteria, the T 2 0 criterion, the likelihood ratio criterion, and the Bartlett-Nanda-Pillai trace criterion. He also derived asymptotic expansions for the distribution of T 2 max statistic, which is used in simultaneous statistical inferences such as

High Rate Recursive Space-Time Trellis Code Designed for Serially Concatenated Space-Time Codes

Using several high rate recursive convolutional codes as the basic element and the trace criteria as the designing principle, a new kind of recursive space-time trellis code with more flexible and higher data rate is presented for the serially concatenated space-time code. When 2 b -ary modulation and N transmit antennas are used, the data rate of the new code can be arranged from b bps/Hz to Nb-1bps/Hz by modifying the number of recursive convolutional codes and the data rate of each code.

Geometrical aspects of first-order optical systems

We reconsider the basic properties of ray-transfer matrices for first-order optical systems from a geometrical viewpoint. In the paraxial regime of scalar wave optics, there is a wide family of beams for which the action of a ray-transfer matrix can be fully represented as a bilinear transformation on the upper complex half-plane, which is the hyperbolic plane. Alternatively, this action can be also viewed in the unit disc. In both cases, we use a simple trace criterion that arranges all first-order systems in three classes with a clear geometrical meaning: they represent rotations, translations, or parallel displacements. We analyze in detail the relevant example of an optical resonator.

Asymptotic Results of a High Dimensional MANOVA Test and Power Comparison When the Dimension is Large Compared to the Sample Size

This paper is concerned with Dempster trace criterion for multivariate linear hypothesis which was proposed for high dimensional situation. First we derive asymptotic null and nonnull distributions of Dempster trace criterion when both the sample size and the dimension tend to infinity. Our approximations are examined through some numerical experiments. Next we compare the power of Dempster trace criterion with the ones of three classical criteria; likelihood ratio criterion, Lawley-Hotelling trace criterion, and Bartlett-Nanda-Pillai trace criterion when the dimension is large compared to the sample size.

Hyperbolic reflections as fundamental building blocks for multilayer optics.

We re-elaborate on the basic properties of lossless multilayers by using bilinear transformations. We study some interesting properties of the multilayer transfer function in the unit disk, showing that hyperbolic geometry turns out to be an essential tool for understanding multilayer action. We use a simple trace criterion to separate multilayers into three classes that represent rotations, translations, or parallel displacements. Moreover, we show that these three actions can be decomposed as a product of two reflections in hyperbolic lines. Therefore, we conclude that hyperbolic reflections can be considered as the basic pieces for a deeper understanding of multilayer optics.

Asymptotic expansions for the distributions of multivariate basic statistics and one-way MANOVA tests under nonnormality

In this paper, we first derive asymptotic expansions for the distributions of some multivariate basic statistics, especially, the joint distribution of sample mean vector and sample covariance matrix when the sample size is large. Next, these results are applied to obtain asymptotic expansions of the null distributions of some one-way MANOVA test statistics under nonnormality. The test statistics include the likelihood ratio criterion, the Lawley–Hotelling trace criterion and the Bartlett–Nanda–Pillai trace criterion which have been proposed under normality. It is shown that the Bartlett corrections under normality do work also for a general nonnormal population. Some numerical results based on these asymptotic expansions are given for several nonnormal populations.

Geometrical setting for the classification of multilayers.

We elaborate on the consequences of the factorization of the transfer matrix of any lossless multilayer in terms of three basic matrices of simple interpretation. By considering the bilinear transformation that this transfer matrix induces in the complex plane, we introduce the concept of multilayer transfer function and study its properties in the unit disk. In this geometrical setting, our factorization translates into three actions that can be viewed as the basic components for understanding the multilayer behavior. Additionally, we introduce a simple trace criterion that allows us to classify multilayers into three types with properties closely related to one (and only one) of these three basic matrices. We apply this approach to analyze some practical examples that are typical of these types of matrices.

Asymptotic expansions of the null distributions of test statistics for multivariate linear hypothesis under nonnormality

This paper is concerned with the distributions of some test statistics for a multivariate linear hypothesis under nonnormality. The test statistics considered include the likelihood ratio statistic, the Lawley-Hotelling trace criterion and the BartlettNanda-Pillai trace criterion, under normality. We derive asymptotic expansions of the null distributions of these test statistics up to the order n , where n is the sample size, under nonnormality. It is shown that our general results can be e¤ectively obtained by deriving an asymptotic expansion of the distribution of a multivariate t-statistic. As special cases of our general results our asymptotic expansions are given for Hotelling’s T 2 statistic, one-way MANOVA test statistics, etc. Numerical accuracies of asymptotic expansion approximations are examined. The validity of the expansions is also discussed. Moreover, we will find conditions such that the Bartlett correction in the normal case implies an improved w-approximation, even under nonnormality.

Simple trace criterion for classification of multilayers

The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that sorts the multilayers into three classes, each of whose properties are closely related to one (and only one) of the three basic matrices.

Limiting Behaviour of Bounding Ellipsoids for State Estimation

Ellipsoidal technique is an advantageous and helpful tool in state estimation of dynamic systems with bounded disturbances; it gives a reliable outer approximations on reachable sets. In this paper the problem concerning asymptotic behaviour of ellipsoidal estimates is considered for linear discrete-time systems without measurements. At the core of that question, some benefits and useful properties were noted using the trace criterion in ellipsoidal calculus algorithms - specifically the boundedness and convergence of approximations. The minimal ellipsoid that contains the limiting reachable set is compared with the recursive algorithm of state estimation.

Set-Membership Estimation with the Trace Criterion made Simpler than with the Determinant Criterion

The problem considered is outer ellipsoidal bounding of the state of a MIMO linear discrete-time model with bounded process perturbations and measurement noise. In the literature, two main measures of the size of bounding ellipsoids have been considered, namely the determinant and trace criteria. The determinant criterion is directly related to the volume of the ellipsoid and has been employed most frequently. It may, however, lead to narrow elongated ellipsoids. The trace algorithm tends to avoid this drawback, but has earned a reputation for complexity. We show that it can be made even simpler globally than the determinant algorithm. A complete filter based on the trace criterion is presented.

Ellipsoidal Bounding Techniques for Parameter Tracking

This paper deals with outerbounding time-varying parameter vectors. The approach proposed is based on ellipsoidal techniques that offer an attractive alternative to least squares and Kalman filtering. Two measures of the size of an ellipsoid are considered, namely its volume and the sum of the squares of its semi-saxes. They correspond to the determinant and trace criteria respectively. The approach is oriented towards analytic solutions.

A Method for Improving the Large-Sample Chi-Squared Approximations to Some Multivariate Test Statistics

SYNOPTIC ABSTRACTIt is well known that Bartlett adjustment yields an improvement on the chi-squared approximation to the likelihood ratio statistic. However, there exist some multivariate tests for which such an improvement is not possible by a usual Bartlett adjustment. Typical examples are the Lawley-Hotelling trace criterion and the Bartlett-Nanda-Pillai trace criterion for testing the multivariate linear hypothesis under normality. The purpose of this paper is to improve the large-sample chi-squared approximation to a general statistic T including these test statistics by constructing a monotone function = Qn(T).

Trace Versus Determinant in Ellipsoidal Outer-Bounding, with Application to State Estimation

Computing minimal-size ellipsoids guaranteed to contain the sum or the intersection of possibly degenerate ellipsoids are the basic tools of a bounded-error counterpart to Kalman filtering. In this paper, two measures of the size of an ellipsoid are considered and compared, namely its volume and the sum of the squares of its semi-axes, which respectively correspond to the determinant and trace criteria. The approach is oriented towards analytic solution, sometimes at the cost of suboptimality. and much attention is paid to recursive algorithms, which are applied to state estimation.

Suboptimal Solutions with Output Feedback Form for Decentralized Control Problems.

In the optimal control problem, all results are based upon the assumption of centrality, that is, all the information available about the system and decision making based upon this information arises at a single input. The design leads to systems in which every sensor output affects every actuator input. In the modelling and design of multi player control problems, the objectives in the optimal choice may be met by formulating the control problem as a differential game. The purpose of this work is to determine the decentralized control strategies with output feedback form which guarantee certain values obtained from the performance indices. We determine the suboptimal control strategies such that obtained performance indices are bounded by certain values of suboptimality degree γi where the bounded values are defined as γiJ^-i* for the linear closed-loop Nash game. Thus, the suboptimal control strategies are not equilibrium solutions. The control strategies with output feedback form require neither state estimation nor reconstruction. In this work, the control strategies guarantee the trace criteria associated with each player which are interpreted as expected values of standard quadratic criteria. Sufficient conditions for suboptimal control strategies with output feedback form are given for the infinite problem.

The effect of realistic ground impedance on the accuracy of ray tracing

The heuristic model of sound propagation in the atmosphere has been developed by L’Espérance et al. [Appl. Acoust. 37, 111–139 (1992)] to provide rapid calculation of sound-pressure levels. This model uses an analytic ray trace method to calculate ray paths from the source to the receiver. The effects of finite ground impedance are treated by calculating the spherical wave reflection factor as if the atmosphere was homogeneous. The actual reflection angles and times of flight are used in this formulation. Since the speed of sound varies linearly with height, no focusing factors are calculated. The heuristic model has been compared to full wave calculations and is found to agree well at long ranges. In this paper the agreement between the heuristic model and the fast field program is investigated in detail. Criteria for the accuracy of ray tracing are discussed. It is demonstrated that the heuristic model produces realistic predictions even though certain criteria are violated. The finite ground impedance is shown to remove those rays for which the ray tracing criteria do not hold.