Euclidean Mean Research Articles

On the measure of the cut locus of a Fréchet mean

One of the fundamental differences between the Central Limit Theorem for empirical Frechet means obtained in [Kendall and Le, ‘Limit theorems for empirical Frechet means of independent and non‐identically distributed manifold‐valued random variables’, Braz. J. Probab. Stat. 25 (2011) 323–352] and that for empirical Euclidean means lies on the assumption that the probability measure of the cut locus of the true Frechet mean is zero. In [Hotz and Huckemann, ‘Intrinsic means on the circle: uniqueness, locus and asymptotics’, Preprint, 2011, arXiv:1108.2141v1], the authors show that, in the case of a circle, this assumption holds automatically. This paper shows that this holds for any complete and connected Riemannian manifold, assuming that there are at least two minimal geodesics between the Frechet mean and any point in its cut locus and that, in fact, it can also be generalized to local minima of the

Fréchet means of curves for signal averaging and application to ECG data analysis

Signal averaging is the process that consists in computing a mean shape from a set of noisy signals. In the presence of geometric variability in time in the data, the usual Euclidean mean of the raw data yields a mean pattern that does not reflect the typical shape of the observed signals. In this setting, it is necessary to use alignment techniques for a precise synchronization of the signals, and then to average the aligned data to obtain a consistent mean shape. In this paper, we study the numerical performances of Frechet means of curves which are extensions of the usual Euclidean mean to spaces endowed with non-Euclidean metrics. This yields a new algorithm for signal averaging and for the estimation of the time variability of a set of signals. We apply this approach to the analysis of heartbeats from ECG records.

Measuring Distances in Space

Determining distances in space is a technical phenomenon; astronomers are trying to come up with a correct way to do it. It is not easy for most of us to imagine the truly immense scale of the universe. “Scale,” in this case, refers to the size of an object compared with its surroundings or another object (McGaugh). Distances in the space are vast from one celestial body to another; for instance, it takes a light signal 10.5 years to travel to the nearest star that has planets (Bonnet, 1992). The fact that light travels 30000 times faster than any fastest rockets renders human beings unable to reach to some planets even if they travelled their entire lifetime. To determine space distance, several methods with different variations are used or have been proposed. These methods, unfortunately, have faults. In this research, I clearly explain several of these methods and their faults and errors. We will also observe the universe as being three-dimensional and flat as explained in Euclidean: Euclidean means that all the geometry and lines (that are taught in mathematics and physics) properties applies. Measuring distance from earth to celestial bodies like star, sun and moon help astronauts to determine the size of the universe; also it helps to estimate the age of universe. Therefore it is important to use correct methods in estimating space distance.

A New Proof of the Pythagorean Theorem and Its Application to Element Decompositions in Topological Algebras

We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” (addressing unital algebraic structure rather than simply vector space structure). One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of different algebras.

Kubo Formulas for Second-Order Hydrodynamic Coefficients

At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity η and on five additional "second-order" hydrodynamical coefficients τ(Π), κ, λ₁, λ₂, and λ₃. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient λ₃ can be evaluated directly by Euclidean means and does not in general vanish.

A multi-dimensional scaling approach to shape analysis

SUMMARY We propose an alternative to Kendall's shape space for reflection shapes of configurations in Rm with k labelled vertices, where reflection shape consists of all the geometric information that is. invariant under compositions of similarity and reflection transformations. The proposed approach embeds the space of such shapes into the space P(k - 1) of (k - 1) x (k - 1) real symmetric positive semidefinite matrices, which is the closure of an open subset of a Euclidean space, and defines mean shape as the natural projection of Euclidean means in P(k - 1) on to the embedded copy of the shape space. This approach has strong connections with multi-dimensional scaling, and the mean shape so defined gives good approximations to other commonly used definitions of mean shape. We also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions.

Approach to block entropy modeling and optimization

This paper introduces an approach to the block entropy modeling of stationary signals. The block entropies, constructed according to the Tsallis generalized formalism, are optimized with respect to the block length and the partition of the signal value domain, to appropriately measure the complexity of the signal. The optimal partition is known to be addressed by the Euclidean mean, expressing the signal optimized level based on the least squares fitting method. However, this is valid only for random signal values following a symmetric distribution. Alternatively, and within the framework of fitting methods based on non-Euclidean metrics, we implement the q th order means to consistently describe the optimal signal level, and to clearly present the difference between the optimal signal level and the optimal partition. The signal optimization is utilized for detecting the distribution modes, developing a technique, being resistant to the noise corruption, that can be useful for detecting the optimal partition. Moreover, the mechanisms that affect the optimal partition are identified and thoroughly investigated. We first consider random signals following an arbitrary distribution, where the block entropy modeling reveals that the optimal partition is located at the median. Thereafter, we consider persistent signals, specifying a degree of determinism, where the optimal partition is found to be driven far from the median and close to the persistent mode. The existence of persistent modes of small hitting time is the key point of this dissertation, highlighting their implications on the block entropy modeling. Finally, efforts towards block entropy modeling of non-stationary signals are discussed.

Geometric Means in a Novel Vector Space Structure on Symmetric Positive‐Definite Matrices

In this work we present a new generalization of the geometric mean of positive numbers on symmetric positive‐definite matrices, called Log‐Euclidean. The approach is based on two novel algebraic structures on symmetric positive‐definite matrices: first, a lie group structure which is compatible with the usual algebraic properties of this matrix space; second, a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. From bi‐invariant metrics on the Lie group structure, we define the Log‐Euclidean mean from a Riemannian point of view. This notion coincides with the usual Euclidean mean associated with the novel vector space structure. Furthermore, this means corresponds to an arithmetic mean in the domain of matrix logarithms. We detail the invariance properties of this novel geometric mean and compare it to the recently introduced affine‐invariant mean. The two means have the same determinant and are equal in a number of cases, yet they are not identical in general. Indeed, the Log‐Euclidean mean has a larger trace whenever they are not equal. Last but not least, the Log‐Euclidean mean is much easier to compute.

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

Análise da diversidade genética entre acessos de banco ativo de germoplasma de algodão

O conhecimento da diversidade genética entre um grupo de genitores é importante no melhoramento, sobretudo na identificação de combinações híbridas de maior heterozigose e maior efeito heterótico e, portanto, na recuperação de genótipos superiores nas gerações segregantes. Os métodos preditivos de diversidade têm sido muito utilizados porque dispensam a obtenção de combinações híbridas entre os genitores e normalmente usam uma medida de similaridade para avaliar a diversidade entre eles. Foram avaliados 221 acessos do banco ativo de germoplasma de algodão da Epamig, por meio da distância euclidiana média e posterior agrupamento dos indivíduos, de modo a permitir escolhas mais apropriadas de genitores para cruzamentos. Para isso foram consideradas 11 características morfológicas e de fibra. Foi detectada grande diversidade genética entre os 221 acessos reunidos em dez grupos distintos. A maior distância euclidiana (4,36) ocorreu entre os acessos S 8186 e T-295-1-1 e a menor (0,25) entre os acessos TX CACES 1-81 e TX CDPS 177.

Effect of a fault on in situ stresses studied by the distinct element method

In situ stress is an important parameter in rock mechanics, and reliable estimates of far-field stresses are indispensable for robust rock engineering analysis. Here, by using the combined finite-discrete element method, we simulate a series of stress fields of both synthetic and natural fracture networks to examine whether the Euclidean mean of local stresses can be used to estimate the far-field stress. The calculations show that given a large number of local stress measurements, their Euclidean mean gives a close approximation of the far-field stress state. Whereas when only a limited number of local stresses are available, the probability of obtaining a practically acceptable estimate of far-field stress increases when more stress measurements are involved. The required number of stress measurements for deriving an acceptable estimate varies with the geomechanical condition; in general, the larger the overall stress variability is, the more local stress measurements are needed. Our research findings suggest that given a limited number of stress measurements, which is often encountered in rock engineering projects, attention is needed when deriving the far-field stress state based on them, and simply using their mean as far-field stress for further rock structure design and numerical analysis may yield erroneous results.