On risk concentration for convex combinations of linear estimators

We consider the estimation problem for an unknown vector β ∈ Rp in a linear model Y = Xβ + σξ, where ξ ∈ Rⁿ is a standard discrete white Gaussian noise and X is a known n × p matrix with n ≥ p. It is assumed that p is large and X is an ill-conditioned matrix. To estimate β in this situation, we use a family of spectral regularizations of the maximum likelihood method β^α(Y) = H ^α(X ^T X) β ^?(Y), α ∈ R⁺, where β ^?(Y) is the maximum likelihood estimate for β and {H ^α(·): R⁺ → [0, 1], α ∈ R⁺} is a given ordered family of functions indexed by a regularization parameter α. The final estimate for β is constructed as a convex combination (in α) of the estimates β^α(Y) with weights chosen based on the observations Y. We present inequalities for large deviations of the norm of the prediction error of this method.     

Estimation of body and tail distribution under extreme events for reliability analysis

In the past decades, many reliability analyses have been developed and applied to engineering fields considering uncertainties of input and output random variables as normal distributions. However, when input uncertainty is taken into the system as extreme events such as weather, temperature, environmental conditions etc., output distribution cannot be described by normal distribution. On the other hand, one of distributions to analyze reliability of a system under extreme events is generalized Pareto distribution. Generalized Pareto distribution has been developed and applied for modelling extreme events. However, conventional methods estimate only the shape and scale parameters by assuming that the location parameter is chosen by experiences focused only on the tail distribution. However, since the tail distribution affected by the body distribution and vice versa, both the body and tail distributions should be considered when the parameters of distribution are estimated. In this study, therefore, a new parameter estimation method is proposed to determine shape, scale and location parameters simultaneously by combining likelihood functions of body and tail distributions using Akaike information criterion and generalized Pareto distribution, respectively. Finally, the parameters of body and tail distributions are estimated by maximum likelihood estimation. The proposed method is verified by using mathematical examples with and without inclusion of extreme events. Results show that the proposed method can estimate parameters and distributions for body and tail distributions as well as the more accurate reliability of system under extreme events.     

Asymptotic assessments of CRC error probabilities in some telecommunication protocols

We study the value distributions for the control cyclic redundancy check (CRC) of length k, drawn at the data section of volume n. The behavior of CRC value distribution is examined at large n and fixed values of k (k = const, n → ∞). With the application of the character theory, we find the conditions of asymptomatic uniformity of the CRC distribution. The asymptomatic results can be applied during the assessment of errors of a series of protocols such as USB, X.25, HDLC, Bluetooth, Ethernet, etc.     

Noise Level Estimation in High-Dimensional Linear Models

We consider the problem of estimating the noise level σ² in a Gaussian linear model Y = Xβ+σξ, where ξ ∈ ?ⁿ is a standard discrete white Gaussian noise and β ∈ ?^p an unknown nuisance vector. It is assumed that X is a known ill-conditioned n × p matrix with n ≥ p and with large dimension p. In this situation the vector β is estimated with the help of spectral regularization of the maximum likelihood estimate, and the noise level estimate is computed with the help of adaptive (i.e., data-driven) normalization of the quadratic prediction error. For this estimate, we compute its concentration rate around the pseudo-estimate ||Y ? Xβ||²/n.     

On Cosmology in Nonlinear Multidimensional Gravity with Multiple Factor Spaces

Within the scope of multidimensional Kaluza–Klein gravity with nonlinear curvature terms and two spherical extra spaces of dimensions m and n, we study the properties of an effective action for the scale factors of extra dimensions. Dimensional reduction leads to an effective 4D multiscalar-tensor theory. Based on qualitative estimates of the Casimir energy contribution at a physically reasonable length scale, we demonstrate the existence of such sets of the initial parameters of the theory in the case m = n that provide a minimum of the effective potential at this scale which yields a fine-tuned value of the effective 4D cosmological constant The corresponding size of extra dimensions depends of which conformal frame is interpreted as the observational one: it is about three orders of magnitude larger than the standard Planck length if we adhere to the Einstein frame, but it is n-dependent in the Jordan frame, and its invisibility requirement restricts the total dimension to values D = 4 + 2n ≤ 20.     

Upper and lower estimates for the number of some <Emphasis Type="Italic">k</Emphasis>-dimensional subspaces of a given weight over a finite field

A fast simulation method is proposed that makes it possible to construct upper and lower estimates for the number of k-measurable subspaces (of arbitrary weight ω) of an n-measurable vector space over a Galois field containing q elements. Numerical examples demonstrate the accuracy of the estimates.     

Self-testing of multiprocessor systems with regular diagnostic connections

We propose an approach to organizing self-testing in a multiprocessor system under certain constraints; in particular, we use special diagnostic graphs, and the number of failures does not exceed a certain value T. We show that a test experiment where each of n processors is tested by two others lets one determine the state of all except possibly two processors for T = 4 and except one for T = 3. The total number of tests does not exceed 2n + 2.     

On Cosets of Weight 4 of Binary BCH Codes with Minimum Distance 8 and Exponential Sums

We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length n = 2 ^m (m odd) with minimum distance 8. In the previous paper [1], we described coset weight distributions of such codes for cosets of weight j = 1, 2, 3, 5, 6. Here we obtain exact expressions for the number of codewords of weight 4 in terms of exponential sums of three types, in particular, cubic sums and Kloosterman sums. This allows us to find the coset distribution of binary primitive (narrow-sense) BCH codes with minimum distance 8 and also to obtain some new results on Kloosterman sums over finite fields of characteristic 2.     

Camera Spectral Sensitivity and White Balance Estimation from Sky Images

Photometric camera calibration is often required in physics-based computer vision. There have been a number of studies to estimate camera response functions (gamma function), and vignetting effect from images. However less attention has been paid to camera spectral sensitivities and white balance settings. This is unfortunate, since those two properties significantly affect image colors. Motivated by this, a method to estimate camera spectral sensitivities and white balance setting jointly from images with sky regions is introduced. The basic idea is to use the sky regions to infer the sky spectra. Given sky images as the input and assuming the sun direction with respect to the camera viewing direction can be extracted, the proposed method estimates the turbidity of the sky by fitting the image intensities to a sky model. Subsequently, it calculates the sky spectra from the estimated turbidity. Having the sky \(RGB\) values and their corresponding spectra, the method estimates the camera spectral sensitivities together with the white balance setting. Precomputed basis functions of camera spectral sensitivities are used in the method for robust estimation. The whole method is novel and practical since, unlike existing methods, it uses sky images without additional hardware, assuming the geolocation of the captured sky is known. Experimental results using various real images show the effectiveness of the method.     

Constructing unbiased prediction limits on future outcomes under parametric uncertainty of underlying models via pivotal quantity averaging approach

This paper presents a new simple, efficient and useful technique for constructing lower and upper unbiased prediction limits on outcomes in future samples under parametric uncertainty of underlying models. For instance, consider a situation where such limits are required. A customer has placed an order for a product which has an underlying time-to-failure distribution. The terms of his purchase call for k monthly shipments. From each shipment the customer will select a random sample of q units and accept the shipment only if the smallest time to failure for this sample exceeds a specified lower limit. The manufacturer wishes to use the results of an experimental sample of n units to calculate this limit so that the probability is γ that all k shipments will be accepted. It is assumed that the n experimental units and the kq future units are random samples from the same population. In this paper, attention is restricted to invariant families of distributions. The pivotal quantity averaging approach used here emphasizes pivotal quantities relevant for obtaining ancillary statistics and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed pivotal quantity averaging approach is conceptually simple and easy to use. For illustration, a left-truncated Weibull, two-parameter exponential, and Pareto distribution are considered. A practical numerical example is given.     

Quantum-memory-assisted entropic uncertainty in two-qubit Heisenberg <Emphasis Type="Italic">XYZ</Emphasis> chain with Dzyaloshinskii–Moriya interactions and effects of intrinsic decoherence

Quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in two-qubit Heisenberg XYZ spin chain model with Dzyaloshinskii–Moriya (DM) interaction has been investigated. The paper shows that the DM interactions and the spin interactions alone x, y, z directions can efficiently suppress the entropic uncertainty of Pauli observables (\(\sigma _{x}\) and \(\sigma _{z}\)), even make the entropic uncertainty close to zero. As well, it is pointed out that the entropic uncertainty reaches to zero at very low temperature, starts to increase with temperature after a threshold, and generally becomes constant at a fixed value. We also verified the Bob’s uncertainty about Alice’s measurement outcomes is anticorrelated with the sum of the accessible information of observer. Furthermore, the decoherence conditions including dephasing and noisy environments are considered. For the fixed initial state, the entropic uncertainty of the XYZ model with DM interaction in z-direction are independent of spin–spin coupling \(J_z\) and the anisotropy parameter \(\varDelta \). In the dephasing environment, the evolutions of entropic uncertainty and its lower bound \(U_{B}\) oscillate with the time and saturates at a finite value, and this value is varied with the purity parameter r of initial state. In the noisy environment, the entropic uncertainty and its lower bound monotonically increase with the time and will be stable at value 2 quickly. This is because the combined effects of the DM interaction and the decoherence force the various initial entanglement states to oscillate into an identical state, regardless of the value of \(D_{z}\) and the parameter r of initial state.     

Implementation of hybrid short-term load forecasting system with analysis of temperature sensitivities

Load forecasting is necessary for economic generation of power, economic allocation between plants (unit commitment scheduling), maintenance scheduling, and for system security such as peak load shaving by power interchange with interconnected utilities. A novel hybrid load forecasting algorithm, which combines the fuzzy support vector regression method and the linear extrapolation based on similar days method with the analysis of temperature sensitivities is presented in this paper. The fuzzy support vector regression method is used to consider the lower load-demands in weekends and Monday than on other weekdays. The normal load in weekdays is forecasted by the linear extrapolation based on similar days method. Moreover, the temperature sensitivities are used to improve the accuracy of the load forecasting in relation to the daily load and temperature. The result demonstrated the accuracy of the proposed load forecasting scheme.     

A meshless method for numerical solution of the coupled Schrödinger-KdV equations

This paper presents meshless method using RBF collocation scheme for the coupled Schrödinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are studied. L ₂ and L _∞ error norms between the results and exact solution is used as a performance measure. Moreover the results of numerical experiments are presented, and are compared with the findings of Finite Element method, finite difference Crank–Nicolson (CN) scheme and analytical solution to confirm the good accuracy of the presented scheme.     

On the Complexity of Fibonacci Coding

We show that converting an n-digit number from a binary to Fibonacci representation and backward can be realized by Boolean circuits of complexity O(M(n) log n), where M(n) is the complexity of integer multiplication. For a more general case of r-Fibonacci representations, the obtained complexity estimates are of the form \({2^O}{(\sqrt {\log n} )_n}\).     

On fractional faces of the metric polytope

The integrality recognition problem is considered on a sequence M _{n, k} of nested relaxations of a Boolean quadric polytope, including the rooted semimetric M _n and metric M _{n, 3} polytopes. The constraints of the metric polytope cut off all faces of the rooted semimetric polytope that contain only fractional vertices. This makes it possible to solve the integrality recognition problem on M _n in polynomial time. To solve the integrality recognition problem on the metric polytope, we consider the possibility of cutting off all fractional faces of M _{n, 3} by a certain relaxation M _{n, k} . The coordinates of points of the metric polytope are represented in homogeneous form as a three-dimensional block matrix. We show that in studying the question of cutting off the fractional faces of the metric polytope, it is sufficient to consider only constraints in the form of triangle inequalities.     

Threshold quantum secret sharing based on single qubit

Based on unitary phase shift operation on single qubit in association with Shamir’s (t, n) secret sharing, a (t, n) threshold quantum secret sharing scheme (or (t, n)-QSS) is proposed to share both classical information and quantum states. The scheme uses decoy photons to prevent eavesdropping and employs the secret in Shamir’s scheme as the private value to guarantee the correctness of secret reconstruction. Analyses show it is resistant to typical intercept-and-resend attack, entangle-and-measure attack and participant attacks such as entanglement swapping attack. Moreover, it is easier to realize in physic and more practical in applications when compared with related ones. By the method in our scheme, new (t, n)-QSS schemes can be easily constructed using other classical (t, n) secret sharing.     

An extrapolation approach for the solution of singular optima

In this paper, an efficient extrapolation approach for the solutions of singular problems in structural topology optimization subjected to stress constraints is presented. On the basis of the ε-relaxed formulation recently proposed by the authors, the sensitivities of the active design variables and the Lagrange multipliers associated with the active constraints with respect to the relaxation parameter at the current optimum of the relaxed problem are derived. Through the use of these sensitivities, a singular optimum can be obtained by employing the extrapolation technique at a relatively large value of ε, thus great computation efforts associated with the continuation approach for the solution of singular optimum can be saved. Several numerical examples illustrate the effectiveness of the proposed approach. Received February 15, 1999     

Reliable region predictions for automated valuation models

Accurate property valuation is important for property purchasers, investors and for mortgage-providers to assess credit risk in the mortgage market. Automated valuation models (AVM) are being developed to provide cheap, objective valuations that allow dynamic updating of property values over the term of a mortgage. A useful feature of automated valuations is to provide a region of plausible price estimates for each individual property, rather than just a single point estimate. This would allow buyers and sellers to understand uncertainty on pricing individual properties and mortgage providers to include conservatism in their credit risk assessment. In this study, Conformal Predictors (CP) are used to provide such region predictions, whilst strictly controlling for predictive accuracy. We show how an AVM can be constructed using a CP, based on an underlying k-nearest neighbours approach. Time trend in property prices is dealt with by assuming a systematic effect over time and adjusting prices in the training data accordingly. The AVM is tested on a large data set of London property prices. Region predictions are shown to be reliable and the efficiency, ie region width, of property price predictions is investigated. In particular, a regression model is constructed to model the uncertainty in price prediction linked to property characteristics.     

A Level-Set Method for Computing the Eigenvalues of Elliptic Operators Defined on Compact Hypersurfaces

We demonstrate, through separation of variables and estimates from the semi-classical analysis of the Schrödinger operator, that the eigenvalues of an elliptic operator defined on a compact hypersurface in ? ⁿ can be found by solving an elliptic eigenvalue problem in a bounded domain Ω?? ⁿ . The latter problem is solved using standard finite element methods on the Cartesian grid. We also discuss the application of these ideas to solving evolution equations on surfaces, including a new proof of a result due to Greer (J. Sci. Comput. 29(3):321–351, 2006).     

Efficient Money Burning in General Domains

We study mechanism design where the objective is to maximize the residual surplus, i.e., the total value of the outcome minus the payments charged to the agents, by truthful mechanisms. The motivation comes from applications where the payments charged are not in the form of actual monetary transfers, but take the form of wasted resources. We consider a general mechanism design setting with m discrete outcomes and n multidimensional agents. We present two randomized truthful mechanisms that extract an O(logm) fraction of the maximum social surplus as residual surplus. The first mechanism achieves an O(logm)-approximation to the social surplus, which is improved to an O(1)-approximation by the second mechanism. An interesting feature of the second mechanism is that it optimizes over an appropriately restricted space of probability distributions, thus achieving an efficient tradeoff between social surplus and the total amount of payments charged to the agents.