Orthonormal Decomposition Research Articles

A new kernel regression approach for robustified L 2 boosting

We investigate L 2 boosting in the context of kernel regression. Kernel smoothers, in general, lack appealing traits like symmetry and positive definiteness, which are critical not only for understanding theoretical aspects but also for achieving good practical performance. We consider a projection-based smoother (Huang and Chen 2008) that is symmetric, positive-definite, and shrinking. Theoretical results based on the orthonormal decomposition of the smoother reveal additional insights into the boosting algorithm. In our asymptotic framework, we may replace the full-rank smoother with a low-rank approximation. We demonstrate that the smoother’s low-rank (d n ) is bounded above by O(h −1), where h is the bandwidth. Our numerical findings show that, in terms of prediction accuracy, low-rank smoothers may outperform full-rank smoothers. Furthermore, we show that the boosting estimator with a low-rank smoother achieves the optimal convergence rate. Finally, to improve the performance of the boosting algorithm in the presence of outliers, we propose a novel robustified boosting algorithm that can be used with any smoother discussed in the study. We investigate the numerical performance of the proposed approaches using simulations and a real application.

Deformation decomposition method of discrete element for irregular structures

In order to satisfy the demand of quantitative analysis of deformation properties of arbitrary irregular structures, a deformation decomposition method for arbitrary triangular element and arbitrary tetrahedral element is proposed based on mechanical equilibrium and mathematical orthogonality. The projection coefficients of various basic deformations of the structure can be obtained by applying the deformation decomposition method to the model divided by triangular or tetrahedral element. The quantitative and visual analysis of the structural deformation performance are realized. The work can provide theoretical basis for the orthonormal decomposition of arbitrary irregular elements and greatly improve the orthonormal deformation decomposition method.

Indagando aspectos evolutivos con filogenias: reloj molecular y otras técnicas útiles en biología comparada

The first step for evolutionary studies is usually to establish a hypothesis of relationships between members of the study group. After this, there is a wide range of possibilities depending on the researcher's interest. This contribution presents the generalities of some of the methodologies most commonly used in macroevolutionary studies: the molecular clock and the reconstruction of ancestral characters. Information on other useful techniques for comparative studies that use a phylogenetic framework is also presented, such as phylogenetic signal estimation, independent contrasts, orthonormal decomposition, and phylogenetic principal component analysis.

Phylogenetic signal analysis in the basicranium of Ursidae (Carnivora, Mammalia).

Ursidae is a monophyletic group comprised of three subfamilies: Tremarctinae, Ursinae and Ailuropodinae, all of which have a rich geographical distribution. The phylogenetic relationships within the Ursidae group have been underexamined, especially regarding morphological traits such as the basicranium. Importantly, the basicranium is a highly complex region that covers a small portion of the skull, combining both structural and functional aspects that determine its morphology. Phylogenetic hypotheses of the Ursidae (including Tremarctinae) have been made based on morphological characters that considers skull, mandible and teeth features, while specific characters of the auditory region and basicranium have not been taken into account. To do this, we analyse the shape and size macroevolution of the basicranium of Ursidae, testing its morphological disparity in a phylogenetic context, which is quantified by means of the phylogenetic signal. We investigated phylogenetical autocorrelation by shape (depicted by Principal Components Analysis scores from previous published analyses) and basicranium size (depicted by centroid size, CS) using an orthonormal decomposition analysis and Abouheif C mean. The main advantages of these methods are that they rely exclusively on cladogram topology and do not require branch-length estimates. Also, an optimisation of the ancestral nodes was performed using TNT 1.5 software. In relation to the phylogenetic signal, both methods showed similar results: the presence of autocorrelation was detected in PC1 and PC2, while in PC3, PC4 and PC5 and in the size of the basicranium (CS), the absence of autocorrelation occurred. The most significant nodes (where there is autocorrelation) are the basal nodes ‘Ursidae’ and ‘Ursinae-Tremarctinae’. Within this last group, distinctive basicranium morphology is observed, being more conservative in Tremarctinae than in Ursinae. The differences between these subfamilies could be related to historical events involving varying food and environmental preferences. The high phylogenetic signal in the node Tremarctinae probably indicates that the basicranium configuration of these bears was obtained early in their evolutionary history. Finally, our results of the basicranium and skull length ratios indicate that in Tremarctinae, the basicranium size was not determined by phylogeny but instead by other factors, such as adaptive responses to climatic changes and competition with other carnivores.

Vector-Valued Nonuniform Wavelet Packets

ABSTRACTWe have introduced vector-valued nonuniform multiresolution analysis(VNUMRA) in [12] where the associated subspace V0 of has, an orthonormal basis, a collection of translates of a vector valued function ϕ of the form {ϕ(x−λ)}λ∈Λ where Λ = {0,r∕N}+2ℤ, N≥1 is an integer and r is an odd integer with 1≤r≤2N−1 such that r and N are relatively prime and ℤ is the set of all integers and the corresponding dilation factor is 2N. The notion of vector-valued nonuniform wavelet packets is introduced in this paper and their various properties are investigated. The orthonormal basis of is constructed from these wavelet packets and the orthonormal decomposition relation is also given.

Evolution of body size in anteaters and sloths (Xenarthra, Pilosa): phylogeny, metabolism, diet and substrate preferences

ABSTRACTPilosa include anteaters (Vermilingua) and sloths (Folivora). Modern tree sloths are represented by two genera,BradypusandCholoepus(both around 4–6 kg), whereas the fossil record is very diverse, with approximately 90 genera ranging in age from the Oligocene to the early Holocene. Fossil sloths include four main clades, Megalonychidae, Megatheriidae, Nothrotheriidae, and Mylodontidae, ranging in size from tens of kilograms to several tons. Modern Vermilingua are represented by three genera,Cyclopes,TamanduaandMyrmecophaga, with a size range from 0.25 kg to about 30 kg, and their fossil record is scarce and fragmentary. The dependence of the body size on phylogenetic pattern of Pilosa is analysed here, according to current cladistic hypotheses. Orthonormal decomposition analysis and Abouheif C-mean were performed. Statistics were significantly different from the null-hypothesis, supporting the hypothesis that body size variation correlates with the phylogenetic pattern. Most of the correlation is concentrated within Vermilingua, and less within Mylodontidae, Megatheriidae, Nothrotheriidae and Megalonychidae. Influence of basal metabolic rate (BMR), dietary habits and substrate preference is discussed. In anteaters, specialised insectivory is proposed as the primary constraint on body size evolution. In the case of sloths, mylodontids, megatheriids and nothrotheriids show increasing body size through time; whereas megalonychids retain a wider diversity of sizes. Interplay between BMR and dietary habits appears to be the main factor in shaping evolution of sloth body size.

Ortho-diffusion decompositions of graph-based representation of images

This paper proposes a new feature representation methodology for graph-based data. Initially, random walks on matrices of pairwise data similarities are considered. A diffusion process is embedded into orthonormal decompositions of such matrices at various scales while enabling a data reduction mechanism as well. At each scale, the QR orthonormal decomposition algorithm, alternating with diffusions and data reduction stages is applied recursively on the given graph-based data representations. The proposed methodology is used in extracting complex feature representations from images, which are then used for image matching and in face recognition. In the face recognition application, both global appearance models and semantic representations of biometric features are considered. Both the correlation and the covariance of images of human faces are considered for the training stage when using global appearance models. The proposed data representation is shown to be robust in face recognition applications, when face images are represented in low resolution and when they are corrupted by noise.

Predictor relative importance and matching regression parameters

Predictor importance in applied regression modeling gives the main operational tools for managers and decision-makers. The paper considers estimation of predictors' importance in regression using measures introduced in works by Gibson and R. Johnson (GJ), then modified by Green, Carroll, and DeSarbo, and developed further by J. Johnson (JJ). These indices of importance are based on the orthonormal decomposition of the data matrix, and the work shows how to improve this approximation. Using predictor importance, the regression coefficients can also be adjusted to reach the best data fit and to be meaningful and interpretable. The results are compared with the robust to multicollinearity, but computationally difficult, Shapley value regression (SVR). They show that the JJ index is good for importance estimation, but the GJ index outperforms it if both predictor importance and coefficients of regression are needed; hence, this index (GJ) can be used in place of the more computationally intensive estimation by SVR. The results can be easily estimated by the considered approach that is very useful in practical regression modeling and analysis, especially for big data.

The Deterministic Plus Stochastic Model of the Residual Signal and Its Applications

The modeling of speech production often relies on a source-filter approach. Although methods parameterizing the filter have nowadays reached a certain maturity, there is still a lot to be gained for several speech processing applications in finding an appropriate excitation model. This manuscript presents a Deterministic plus Stochastic Model (DSM) of the residual signal. The DSM consists of two contributions acting in two distinct spectral bands delimited by a maximum voiced frequency. Both components are extracted from an analysis performed on a speaker-dependent dataset of pitch-synchronous residual frames. The deterministic part models the low-frequency contents and arises from an orthonormal decomposition of these frames. As for the stochastic component, it is a high-frequency noise modulated both in time and frequency. Some interesting phonetic and computational properties of the DSM are also highlighted. The applicability of the DSM in two fields of speech processing is then studied. First, it is shown that incorporating the DSM vocoder in HMM-based speech synthesis enhances the delivered quality. The proposed approach turns out to significantly outperform the traditional pulse excitation and provides a quality equivalent to STRAIGHT. In a second application, the potential of glottal signatures derived from the proposed DSM is investigated for speaker identification purpose. Interestingly, these signatures are shown to lead to better recognition rates than other glottal-based methods.

The k-Yamabe problem

1.1. The k-Yamabe problem. The k-Yamabe problem is a higher order extension of the celebrated Yamabe problem for scalar curvature. It was initially proposed by Viaclovsky [72] and also arose in the study of Q-curvatures in [11]. Viaclovsky found that in a conformal metric, the resultant k-curvature equation can be expressed as an equation similar to the k-Hessian equation, which has been studied by many authors [77]. The k-Yamabe problem can be formulated as follows. Let (Mn, g) be a closed Riemannian manifold of dimension n ≥ 3. It is well-known that there is an orthonormal decomposition of the Riemannian curvature tensor

Coherent Swing Instability of Power Grids

We interpret and explain a phenomenon in short-term swing dynamics of multi-machine power grids that we term the Coherent Swing Instability (CSI). This is an undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines in a sub-grid coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance. We develop a minimal mathematical model of CSI for synchronous machines that are strongly coupled in a loop transmission network and weakly connected to the infinite bus. This model provides a dynamical origin of CSI: it is related to the escape from a potential well, or, more precisely, to exit across a separatrix in the dynamical system for the amplitude of the weak nonlinear mode that governs the collective motion of the machines. The linear oscillations between strongly coupled machines then act as perturbations on the nonlinear mode. Thus we reveal how the three different mode oscillations—local plant, inter-machine, and inter-area modes—interact to destabilize a power grid. Furthermore, we present a phenomenon of short-term swing dynamics in the New England (NE) 39-bus test system, which is a well-known benchmark model for power grid stability studies. Using a partial linearization of the nonlinear swing equations and the proper orthonormal decomposition, we show that CSI occurs in the NE test system, because it is a dynamical system with a nonlinear mode that is weak relative to the linear oscillatory modes.

A tensor decomposition approach to data compression and approximation of ND systems

The method of Proper Orthogonal Decompositions (POD) is a data-based method that is suitable for the reduction of large-scale distributed systems. In this paper we propose a generalization of the POD method so as to take the ND nature of a distributed model into account. This results in a novel procedure for model reduction of systems with multiple independent variables. Data in multiple independent variables is associated with the mathematical structure of a tensor. We show how orthonormal decompositions of this tensor can be used to derive suitable projection spaces. These projection spaces prove useful for determining reduced order models by performing Galerkin projections on equation residuals. We demonstrate how prior knowledge about the structure of the model reduction problem can be used to improve the quality of approximations. The tensor decomposition techniques are demonstrated on an application in data compression. The proposed model reduction procedure is illustrated on a heat diffusion problem.

Unified Array Manifold Decomposition Based on Spherical Harmonics and 2-D Fourier Basis

In this paper, we derive a unified framework for orthonormal decomposition of the array manifold on scalar fields. Such fields are encountered in cases where polarization is not considered, e.g., single polarized radio waves and acoustic pressure. The results generalize and unify different decompositions of the array manifold, found in recent literature, to arrays of arbitrary geometry including conformal arrays. The concept of equivalence matrix is introduced, establishing a link between the spherical harmonics and 2-D Fourier basis functions. Under some mild assumptions that typically hold in practice, a one-to-one relationship between spherical harmonic spectra and 2-D Fourier spectra may be established. Additionally, it is shown that the rows of the equivalence matrix and the 2-D Fourier spectra of the array manifold span the same subspace. With such results the spherical harmonic and 2-D Fourier decompositions of the array manifold vector, i.e., Wavefield Modeling and 2-D Effective Aperture Distribution Function (EADF) are shown to be equivalent. Results on the modeling capabilities of both orthonormal decompositions are obtained. Moreover, the equivalence matrix is shown to facilitate noise attenuation. A fast spherical harmonic transform with complexity <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> ) can be obtained by exploiting the equivalence matrix, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> represents the total number of points on the sphere. Finally, the equivalence matrix allows to gain more insight into the relation between rotating a function on the sphere and on the torus. These contributions facilitate high-resolution array processing both in elevation and azimuth irrespective of the array structure or imperfections.

Construction of Shape Features for the Representation of Full-Field Displacement/Strain Data

The achievement of high levels of confidence in finite element models involves their validation using measured responses such as static strains or vibration mode shapes. A huge amount of data with a high level of information redundancy is usually obtained in both the detailed finite element prediction and the full-field measurements so that achieving a meaningful validation becomes a challenging problem. In order to extract useful shape features from such data, image processing and pattern recognition techniques may be used. One of the most commonly adopted shape feature extraction procedures is the Fourier transform in which the original data may be expressed as a set of coefficients (coordinates) of the decomposition kernels (bases) in the feature space. Localised effects can be detected by the wavelet transform. The acquired shape features are succinct and therefore simplify the model validation, based on the full-field data, allowing it to be achieved in a more effective and efficient way. In this paper, full-field finite element strain patterns of a plate with a centred circular hole are considered. A special set of orthonormal shape decomposition kernels based on the circular Zernike polynomials are constructed by the Gram-Schmidt orthonormalization process. It is found that the strain patterns can suitably be represented by only a very small number of shape features from the derived kernels.

Numerical simulation of flows in tight-lattice fuel bundles

The flow in tight rod bundles is characterized by long-term, large-scale coherent patterns in the stream-wise direction. In the present work, the issue of simulating these structures through unsteady CFD simulations employing periodic boundary conditions in the stream-wise direction, will be addressed. The validity of the approach is assessed through the comparison of a large eddy simulation (LES) for similar flow conditions inside a simplified geometry and experimental data. A powerful statistical tool (proper orthonormal decomposition) is used to analyze the time varying solution. The flow field has been decomposed into a series of normal modes, identifying the structures responsible for the flow transfer between sub-channels. Additional insights on the physics of these coherent structures are obtained. An unsteady Reynolds averaged Navier–Stokes simulation (URANS) of the flow in a rod bundle has then been carried out. The comparison between numerical results and experimental results [Krauss, T., Meyer, L., 1998. Experimental investigation of turbulent transport of momentum and energy in a heated rod bundle. Nucl. Eng. Design 180, 185–206] proves that accuracy can be achieved for averaged statistics such as stream-wise velocity, turbulent intensity and wall shear stresses.

Wavelet-based statistics of energy transfer in high-resolution direct numerical simulation of three-dimensional homogeneous isotropic turbulence

Small-scale statistics of wavelet-based quantities of turbulence are examined, with particular attention to t α,β[ ι ], which is the wavelet-based transfer of the wavelet-based energy e α[ ι ] at location ι and scale r α to all scales smaller than r β (⩽r α). A divergence-free orthonormal wavelet decomposition method is applied to direct numerical simulation data of three-dimensional incompressible homogeneous isotropic turbulence with grid points and Taylor microscale Reynolds numbers up to 20483 and 732, respectively. We propose a new statistical measure for the degree of scale locality of wavelet-based energy flux to scales smaller than r β constructed from t α,β[ ι ]. At R λ = 732, the measure retains a constant value in the inertial subrange, a value about two-thirds as large as that in the dissipation range, though the flux is local in scale throughout the range, from the inertial subrange to the dissipation range. We confirm that the influence of energy transfer from the large-scale flow on the flux becomes smaller with decreasing scales and negligible in the dissipation range. Scale-dependent spatial correlation between t α,β[ ι ] and e α[ ι ] is discussed. We find that, when r β is in the inertial range, the conditional correlation between forward transfer, which is t α,β[ ι ] at the positions satisfying t α,β[ ι ]>0, and e α[ ι ] becomes stronger as scale r α increases from r β, while this behavior is reversed when r β is in the dissipation range.

Position-invariant close-talking microphone array

Close-talking microphones are commonly used for speech pick-up in high-noise environments since they can significantly improve the signal-to-noise ratio (SNR) relative to conventional pressure microphones. Increased SNR is based on the assumption that unwanted noise sources are in the far field. However, close-talking microphones have to be carefully positioned or otherwise they can have reduced SNR improvement. A new close-talking microphone array is proposed that is orientation invariant with respect to the attenuation of far-field noise sources. It is shown that by appropriate processing of the close-talking microphone array signals, one can adaptively compensate for the distance and orientation of the microphone for a near-field source. The proposed close-talking array is based on a spatially orthonormal decomposition of the sound field for a near-field source. It consists of four or more pressure microphones that are mounted in the surface of a small rigid sphere. Finally, it is shown that decomposing the sound field into spatially orthonormal modes leads to a computationally efficient implementation. The orientation compensation is applicable not just to close-taking microphones but to directional microphones in general, e.g., podium microphones or hand-held vocal microphones.

Impulse response of a spherical microphone array (eigenmike)

Spherical microphone arrays are capable of complete three-dimensional sound field capture and robust noise cancellation [J. Meyer &amp; G.W. Elko, ‘‘A Highly Scalable Spherical Microphone Array Based on an Orthonormal Decomposition of the Soundfield,’’ in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, 2002, Vol. II, pp. 1781–1784]. The spherical array tested has 24 omni-directional pressure microphones arranged on a rigid sphere. The sound pressure from each sensor is decomposed into spherical harmonics (standard eigen analysis) which results in rotationally symmetric output along any axis (three dimensionally) through the center of the sphere. An accurate impulse response of this system is important to sound localization and tracking methods. Time history can also be used to enhance the prediction model for source movement. The impulse measurements on this array were made in a semi-anechoic chamber and are in good agreement with expected results. The three main characteristics of the impulse response that will be discussed are the delay times, amplitude differences, and frequency response.

Position independent close-talking microphone

Close-talking microphones are required for speech pick-up in high noise environments and are based on the assumption that unwanted noise sources are distant from a desired nearfield source. Common close-talking microphones in use today have to be carefully oriented relative to the desired nearfield source. To address this problem, a new close-talking microphone array is proposed that is orientation-invariant with respect to the attenuation of farfield noise sources. It is shown that by appropriate processing of the close-talking microphone array signals, one can adaptively compensate for the distance and orientation of the microphone for a nearfield source. The proposed close-talking array is based on an spatially orthonormal decomposition of the soundfield for a nearfield source. It consists of four or more pressure microphones that are mounted in the surface of a rigid sphere with a diameter on the order of a centimeter. Finally, it is shown that decomposing the soundfield into spatially orthonormal modes leads to a computationally efficient implementation.

Application of the Haar wavelet transform to detect microseismic signal arrivals

A discrete wavelet transform is applied for the time-scale representation of raw seismic data and subsequent identification of events of interest. The wavelet transform properties such as localization, which is essential for the analysis of transient signals, provide a filter to extract characteristics of interest such as energy and predominant time scales. This information is subsequently exploited for microseismic events detection. The sample sum of squares is partitioned on a scale-by-scale basis and analysed across the time scales to emphasise the signal phase arrival, retaining the scales at which the seismic events have larger energy. The orthonormal decomposition of the signal energy estimated by the wavelet variance into the retained scales provides a useful means of describing the change in the signal magnitude associated with the triggering events. This type of analysis discriminates between signal phase arrival and spurious signal triggering by the different magnitude of local relative energy, which is much smaller in the latter case. The relative energy across the scales also changes, with greater magnitudes at coarser resolutions in the pattern expected in a trace decomposition with only a random noise component.