Subjective Breakdown Points of R-estimators Applied in Deformation Analysis

This paper presents practical aspect of the breakdown point theory in deformation analysis by applying R-estimators. The main aim of the paper is to determine impact of the probability of positive (or negative) gross errors and the number of such errors on the value of breakdown point of the estimates applied. Authors consider two types of networks: a levelling network and a horizontal one. Calculations are made for two cases, namely when observations are affected by gross errors in both measurement epochs or only in the second epoch. The main results are based on the Monte Carlo method, which is a very useful tool to solve such a geodetic problem. The simulations show that the breakdown point depends on the probability of positive gross errors but also on the number of epochs in which the gross errors occur. This is especially vivid in the case of levelling networks. Another interesting finding is that even if the number of gross errors exceeds the breakdown point, we can still get reasonable results; however, not always. Thus, the paper shows the probabilities that the method breaks down for several different cases. The paper includes some numerical tests, which provided practical information about the subjective breakdown points and their importance for R-estimates applied in deformation analysis.

Leading Edge Vortex Control on a Delta Wing With Dielectric Barrier Discharge Actuators

The interest in the active flow control based on dielectric barrier discharge (DBD) plasma actuators has increased rapidly in the past decade. Because of its features such as light weight, low power consumption, fast response and flexibility, the DBD plasma actuator is a promising technology in advancing the aerodynamic performance and maneuvering of unmanned aerial vehicles. In this study, DBD plasma actuators are employed on a full span delta wing with a 75 degree swept angle to control the leading edge vortices (LEV), which generate the vortex lift on the delta wing. The experiment is conducted in a low speed closed-loop wind tunnel and the Reynolds number based on the delta wing chord is 50,000. To fix the stagnation points, both leading edges are beveled on the windward sides at an angle of 35 degrees and actuators are insulated at the leading edges. These actuators are driven independently at a frequency of 20 kHz and a voltage of 12 kV in both continuous mode and periodic mode. The DBD actuators are calibrated using a pitot tube. Smoke flow visualization result indicates that the breakdown points of leading edge vortices can be significantly affected by DBD plasma actuators at the leading edge. In the asymmetric control case (only an actuator on one side is powered), the breakdown point of the LEV on the controlled side is greatly advanced while the one on the uncontrolled side is delayed; in the symmetric control case (actuators on both sides are powered), the control shifted the breakdown points of both LEVs further downstream. Particle image velocimetry (PIV) demonstrates clearly that the control caused by DBD actuators at the leading edge can influence the separation at the leading edges and also the shear layer vortices, which form the substructures around the primary vortices. As a result, breakdown points of LEVs are affected. Interestingly observed, the control leads to a contrary flow phenomenon: in the asymmetric case, the breakdown point of the LEV on the controlled side is advanced while, in the symmetric control case, the breakdown points of the LEV on both sides are delayed. The effects of reduced frequency and duty cycle on the control authority are also investigated experimentally. Control efficiencies of both continuous mode and periodic mode are discussed.

Conditional Covariance Prediction in Portfolio Analysis Using MCD and PCS Robust Multivariate Scatter Estimators

In this paper we compare two matrix estimators of multivariate scatter – the minimal covariance determinant estimator MCD with a new proposal an estimator minimizing an incongruence criterion PCS in a context of their applications in economics. We analyze the estimators using simulation studies and using empirical examples related to issues of portfolio building. In a decision process we often make use of multivariate scatter estimators. Incorrect value of these estimates may result in financial losses. In this paper we compare two robust multivariate scatter estimators – MCD (minimum covariance determinant) and recently proposed PCS (projection congruent subset), which are affine equivariant and have high breakdown points. In the empirical analysis we make use of them in the procedure of weights setting for minimum vari ance and equal risk contribution (ERC) portfolios.

PHASE TRANSITIONS IN ERROR CORRECTING AND COMPRESSED SENSING BY ℓ1 LINEAR PROGRAMMING

In correcting a real linear code y = Bx + w by ℓ1 linear programming, where the encoding matrix B ∈ ℝm × n has full rank with m ≥ n and the noise w ∈ ℝm is a sparse random vector, it is numerically observed that the breakdown points of 50% successes in recovering the input vector x ∈ ℝn from the corrupted oversampled measurement y lie on the Donoho–Tanner curves when reflected in their midpoint. The curves of 50% successes in solving underdetermined systems, z = Aw, by ℓ1 linear programming with uniformly distributed compressed sensing matrices A ∈ ℝd × m, where d < m and w is a sparse vector, have been numerically observed and recently shown to coincide with the Donoho–Tanner curves for normally-distributed compressed sensing matrices A derived from geometric combinatorics. When n ≤ m/2, correcting a linear code is faster if done directly by ℓ1 linear programming. However, when n > m/2, to save computing time, this problem can be transformed into an underdetermined compressed sensing problem, Aw = z := Ay, for the syndrome z by a full rank matrix A ∈ ℝd × m, d = m – n, such that AB = 0. For this purpose, to have equivalently high mean breakdown points by ℓ1 linear programming, one can use uniformly distributed random matrices A ∈ ℝ(m-n) × m and matrices B ∈ ℝm × n with orthonormal columns spanning the null space of A. Two exceptional cases have been found. Numerical results are collected in figures and tables.