Linear Invariance Research Articles

A two-parameter model of dispersion aversion

The idea of representing choice under uncertainty as a trade-off between mean returns and some measure of risk or uncertainty is fundamental to the analysis of investment decisions. In this paper, we show that preferences can be characterized in this way, even in the absence of objective probabilities. We develop a model of uncertainty averse preferences that is based on a mean and a measure of the dispersion of the state-wise utility of an act. The dispersion measure exhibits positive linear homogeneity, sub-additivity, translation invariance and complementary symmetry. Since preferences are only weakly separable in terms of these two summary statistics, the uncertainty premium need not be constant. We generalize the concept of decreasing absolute risk aversion. Further we derive two-fund separation and asset pricing results analogous to those that hold for the standard CAPM.

Translation Invariance in the Polynomial Kernel Space and Its Applications in kNN Classification

In this paper, a new technique is presented to measure dissimilarity in kernel space providing scaling and translation invariance. The motivation comes from signal/image processing, where classifiers are often required to ensure invariance against linear transforms, since in many cases linear transforms do not affect the content of a signal/image for a human observer. We examine the theoretical background of linear invariance in the polynomial kernel space, introduce the centered correlation and centered Euclidean dissimilarity in kernel space, deduce formulas to compute it efficiently and test the proposed dissimilarity measures with the kNN classifier. The experimental results show that the presented techniques are highly competitive in similarity or dissimilarity based classification methods.

Evaluation of critical storm duration rainfall estimates used in flood hydrology in South Africa

Design rainfall comprises of a depth and duration associated with a given probability of exceedance or return period. The purpose of the study was to evaluate and compare the methods used in flood hydrology to estimate depth-duration-frequency (DDF) relationships of design rainfall in South Africa based on the critical storm duration or time of concentration (TC) of a catchment. The influence of the type of rainfall, areal and temporal distribution of rainfall were also investigated to establish if a relationship exists between the catchment area, TC and areal reduction factors (ARFs). The DDF relationships based on the least-square regression analyses of Log-Extreme Value Type 1 distributions, the modified Hershfield equation, the regionalised South African Weather Service (SAWS) n-day design rainfall data and the Regional Linear Moment Algorithm and Scale Invariance (RLMA&SI) approach were compared in 3 distinctive TC-ranges. The results showed that the RMLA&SI approach can be considered as the preferred DDF relationship in future design flood estimations. The results also showed that a direct relationship exists between the catchment area and TC, thus ARFs can be explicitly expressed in terms of only the catchment area. Keywords: Rainfall, depth-duration-frequency, time of concentration, areal reduction factors, design flood

Compensating for noise and perturbances in linear object control

We consider the problem of constructing a robust control system for a linear stationary object under unmeasurable perturbances acting on the object. The perturbances are unbounded on the input of the system and bounded on the output. Based on the system inclusion technique, we consider necessary and sufficient conditions for the linear stationary systems' invariance to perturbances. With the auxiliary contour method, we compensate for the undesirable effects of perturbances on controlled variables. We give an illustrative example and computer modeling results.

Two-stage updating pheromone for invariant ant colony optimization algorithm

Ant colony optimization (ACO) is a metaheuristic approach for combinatorial optimization problems. With the introduction of hypercube framework, invariance property of ACO algorithms draws more attention. In this paper, we propose a novel two-stage updating pheromone for invariant ant colony optimization (TSIACO) algorithm. Compared with standard ACO algorithms, TSIACO algorithm uses solution order other than solution itself as independent variable for quality function. In addition, the pheromone trail is updated with two stages: in one stage, the first r iterative optimal solutions are employed to enhance search capability, and in another stage, only optimal solution is used to accelerate the speed of convergence. And besides, the pheromone value is limited to an interval. We prove that TSIACO not only has the property of linear transformational invariance but also has translational invariance. We also prove that the pheromone trail can limit to the interval (0, 1]. Computational results on the traveling salesman problem show the effectiveness of TSIACO algorithm.

Linear invariance of locally biholomorphic mappings in the unit ball of a [formula omitted]-triple

We study the notion of linear invariance on the unit ball of a J B ⁎ -triple X, and we obtain some connection between the norm-order of a linear invariant family and the starlikeness of order 1/2. Also, we give some result concerning the radius of univalence of some linear invariant families. Finally, if the dimension of X is finite and if the norm-order of a linear invariant family is finite, then we prove the normality of the linear invariant family and we also obtain upper bounds on the distortion and the growth of mappings in a linear invariant family with specified norm-order. In particular, our results are valid for the classical Cartan domains and the unit polydisc.

Inequality averse multi-utilitarian bargaining solutions

This paper introduces and analyzes the class of inequality averse multi-utilitarian solutions for cooperative bargaining problems. We show that generalized Gini solutions and inequality averse Choquet solutions are particular cases of this new multi-valued solution concept and provide a complete characterization in which an invariance property, consisting of a weakening of both the linear invariance axiom in Blackorby et al. (Econometrica 62:1161–1178, 1994) and the restricted invariance axiom in Ok and Zhou (Games Econ Behav 33:249–264, 2000), plays an important role. Moreover, by relaxing the assumptions involved in the characterization, the class is extended to include inequality loving multi-utilitarian solutions which are also studied in the paper.

GA-based optimal selection of PZMI features for face recognition

One of the key problems in automated face recognition system is that of handling the face image variation in terms of scale, rotation (in plane) and translation. One approach is fixing mentioned problems in recognition processes by extracting one linear transformation invariant feature. This paper presents a novel method for face recognition. Pseudo Zernike moment invariant (PZMI) which has linear transformation invariance properties and is robust in the presence of noise utilized to produce feature vectors. For decreasing computational complexity of feature extraction step, we use genetic algorithm (GA) to select the optimal feature set which contains optimal PZMI orders and corresponding repetitions. In addition, we have investigated the effect of PZMI orders on recognition rate in noisy images. Proposed scheme has been tested on the FERET database. Experimental results prove the advantages of the proposed method when compared with other PZMI-based face recognition systems.

The effect of alignment precision in neutron penumbral imaging

Based on Monte Carlo simulation of neutron penumbral imaging，the linear space invariance is researched. In the field of view，the points broadening linearly depends on the axial distance on the object plane，but the integrated intensity keeps linear space invariance. Using this result，the alignment precision is analyzed. The axial error of the targets center on the object plane should be controlled within 50μm. The point source should be less than 300μm away from the object plane. The aperture rotation should be within 0.1mrad.

Time-warp invariant pattern detection with bursting neurons

Sound patterns are defined by the temporal relations of their constituents, individual acoustic cues. Auditory systems need to extract these temporal relations to detect or classify sounds. In various cases, ranging from human speech to communication signals of grasshoppers, this pattern detection has been found to display invariance to temporal stretching or compression of the sound signal (‘linear time-warp invariance’). In this work, a four-neuron network model is introduced, designed to solve such a detection task for the example of grasshopper courtship songs. As an essential ingredient, the network contains neurons with intrinsic bursting dynamics, which allow them to encode durations between acoustic events in short, rapid sequences of spikes. As shown by analytical calculations and computer simulations, these neuronal dynamics result in a powerful mechanism for temporal integration. Finally, the network reads out the encoded temporal information by detecting equal activity of two such bursting neurons. This leads to the recognition of rhythmic patterns independent of temporal stretching or compression.

On the Convergence of Nonlinear Modes of a Finite Element Model

Convergence of finite element models is generally realized via observation of mesh independence. In linear systems invariance of linear modes to further mesh refinement is often used to assess mesh independence. These linear models are, however, often coupled with nonlinear elements such as CFD models, nonlinear control systems, or joint dynamics. The introduction of a single nonlinear element can significantly alter the degree of mesh refinement necessary for sufficient model accuracy. Application of nonlinear modal analysis [1,2] illustrates that using linear modal convergence as a measure of mesh quality in the presence of nonlinearities is inadequate. The convergence of the nonlinear normal modes of a simply supported beam modeled using finite elements is examined. A comparison is made to the solution of Boivin, Pierre, and Shaw [3]. Both methods suffer from the need for convergence in power series approximations. However, the finite element modeling method introduces the additional concern of mesh independence, even when the meshing the linear part of the model unless p‐type elements are used [4]. The importance of moving to a finite element approach for nonlinear modal analysis is the ability to solve problems of a more complex geometry for which no closed form solution exists. This case study demonstrates that a finite element model solution converges nearly as well as a continuous solution, and presents rough guidelines for the number of expansion terms and elements needed for various levels of solution accuracy. It also demonstrates that modal convergence occurs significantly more slowly in the nonlinear model than in the corresponding linear model. This illustrates that convergence of linear modes may be an inadequate measure of mesh independence when even a small part of a model is nonlinear.

Scale-Sets Image Analysis

This paper introduces a multi-scale theory of piecewise image modelling, called the scale-sets theory, and which can be regarded as a region-oriented scale-space theory. The first part of the paper studies the general structure of a geometrically unbiased region-oriented multi-scale image description and introduces the scale-sets representation, a representation which allows to handle such a description exactly. The second part of the paper deals with the way scale-sets image analyses can be built according to an energy minimization principle. We consider a rather general formulation of the partitioning problem which involves minimizing a two-term-based energy, of the form � C + D, where D is a goodness-of-fit term and C is a regularization term. We describe the way such energies arise from basic principles of approximate modelling and we relate them to operational rate/distorsion problems involved in lossy compression problems. We then show that an important subset of these energies constitutes a class of multi-scale energies in that the minimal cut of a hierarchy gets coarser and coarser as parameter � increases. This allows us to devise a fast dynamic-programming procedure to find the complete scale-sets representation of this family of minimal cuts. Considering then the construction of the hierarchy from which the minimal cuts are extracted, we end up with an exact and parameter-free algorithm to build scale-sets image descriptions whose sections constitute a monotone sequence of upward global minima of a multi-scale energy, which is called the algorithm. This algorithm can be viewed as a continuation method along the scale dimension or as a minimum pursuit along the operational rate/distorsion curve. Furthermore, the solution verifies a linear scale invariance property which allows to completely postpone the tuning of the scale parameter to a subsequent stage. For computational reasons, the scale climbing algorithm is approximated by a pair-wise region merging scheme: however the principal properties of the solutions are kept. Some results obtained with Mumford-Shah's piece-wise constant model and a variant are provided and different applications of the proposed multi-scale analyses are finally sketched.

Linear relative canonical analysis of Euclidean random variables, asymptotic study and some applications

We introduce the Linear Relative Canonical Analysis (LRCA) of Euclidean random variables. Then similar properties than for usual linear Canonical Analysis are obtained. Furthermore, we develop an asymptotic study of LRCA and apply the obtained results to tests for lack of relative linear association, dimensionality and invariance.

Automatic radiometric normalization of multitemporal satellite imagery

The linear scale invariance of the multivariate alteration detection (MAD) transformation is used to obtain invariant pixels for automatic relative radiometric normalization of time series of multispectral data. Normalization by means of ordinary least squares regression method is compared with normalization using orthogonal regression. The procedure is applied to Landsat TM images over Nevada, Landsat ETM+ images over Morocco, and SPOT HRV images over Kenya. Results from this new automatic, combined MAD/orthogonal regression method, based on statistical analysis of test pixels not used in the actual normalization, compare favorably with results from normalization from manually obtained time-invariant features.

Automatic radiometric normalization of multitemporal satellite imagery

The linear scale invariance of the multivariate alteration detection (MAD) transformation is used to obtain invariant pixels for automatic relative radiometric normalization of time series of multispectral data. Normalization by means of ordinary least squares regression method is compared with normalization using orthogonal regression. The procedure is applied to Landsat TM images over Nevada, Landsat ETM+ images over Morocco, and SPOT HRV images over Kenya. Results from this new automatic, combined MAD/orthogonal regression method, based on statistical analysis of test pixels not used in the actual normalization, compare favorably with results from normalization from manually obtained time-invariant features.

Doubly close-to-convex functions

We introduce the class L( β, γ) of holomorphic, locally univalent functions in the unit disk D={z: |z|<1} , which we call the class of doubly close-to-convex functions. This notion unifies the earlier known extensions. The class L( β, γ) appears to be linear invariant. First of all we determine the region of variability {w: w= logf′(r), f∈L(β,γ)} for fixed z, | z|= r<1, which give us the exact rotation theorem. The rotation theorem and linear invariance allows us to find the sharp value for the radius of close-to-convexity and bound for the radius of univalence. Moreover, it was helpful as well in finding the sharp region for α∈ R , for which the integral ∫ 0 z(f′(t)) α dt , f∈ L( β, γ), is univalent. Because L( β, γ) reduces to β-close-to-convex functions ( γ=0) and to convex functions ( β=0 and γ=0), the obtained results generalize several corresponding ones for these classes. We improve as well the value of the radius of univalence for the class considered by Hengartner and Schober (Proc. Amer. Math. Soc. 28 (1971) 519–524) from 0.345 to 0.577.

Linear Invariance of Locally Biholomorphic Mappings in Hilbert Spaces

We study the notion of linear invariance on the unit ball of a complex Hilbert space X and we obtain some connections between the norm-order of a linear invariant family (L.I.F.) and the quasi-convexity. Also, we give some results concerning the radius of univalence of some linear invariant families. Finally we find the norm-order of normalized convex mappings in the unit ball of X and some lower bounds for the radius of convexity and the radius of starlikeness for a certain L.I.F. in X .

Invariance control for a class of cascade nonlinear systems

We consider the control of partially linear cascade systems using switching control of the states of the linear subsystem. We give sufficient conditions under which feedback of the linear states with switching gains guarantees both exponential stability of the linear subsystem and positive invariance of a prespecified region in state space. We refer to a control scheme incorporating these two objectives as invariance control. Semiglobal asymptotic stabilization follows under some additional conditions. The key idea of our design is to keep a given state space region positively invariant by switching on the boundary of the region. Thus, the transient response of the system can be kept within prescribed bounds which is important in many practical applications. Our approach can also be viewed as an alternative to high gain designs. The results in this paper can be extended to nonlinear cascades and even to noncascaded systems.

A small sample calibration method for the empirical likelihood ratio

We present a calibration method for improving the coverage accuracy of the empirical likelihood ratio confidence interval for the mean. The method is made possible by a linear transformation invariance property of its coverage level. Simulation results show that, for non-normal distributions, the coverage level of the normal distribution calibrated empirical likelihood ratio confidence interval is comparable to that of the estimated Bartlett-corrected interval. For normal distributions, its coverage level is exact and it is competitive to the t-interval in terms of the variance and expected value of the interval length.

On the linear shift invariance of photon diffusion imaging in highly scattering media

On the linear shift invariance of photon diffusion imaging in highly scattering media