Spectral Sets Research Articles

Divergence and Sufficiency for Convex Optimization

The notion of Bregman divergence and sufficiency will be defined on general convex state spaces. It is demonstrated that only spectral sets can have a Bregman divergence that satisfies a sufficiency condition. Positive elements with trace 1 in a Jordan algebra are examples of spectral sets, and the most important example is the set of density matrices with complex entries. It is conjectured that information theoretic considerations lead directly to the notion of Jordan algebra under some regularity conditions.

Double outlyingness analysis in quantitative spectral calibration: Implicit detection and intuitive categorization of outliers

In this study, outliers in the spectral calibration set were analyzed and categorized based on their error introducing patterns. A double outlyingness analysis (DOA) method for outlier detection and categorization was proposed as a tool to detail the error structures of outliers. Two outlyingness values were calculated based on a proposed procedure. The outlier diagnosis diagram based on the inlier model was drawn to distinguish four types of samples: type I (outlier), incorrect concentration(s) with contaminated spectral signals; type II (outlier), incorrect concentration(s) with uncontaminated spectral signals; type III (outlier), correct concentration(s) with contaminated spectral signals; type IV (inlier), correct concentration(s) with uncontaminated spectral signals. Four data sets for quantitative spectral calibrations were used to compare DOA and five existing methods. Results show that DOA is able to detect all types of outliers and provide a tool to analyze outlier structures.

Spectrality of digit sets and spectral self-affine measures

The present paper determines the spectrality of digit set $D$ relating to a spectral self-affine measure $\mu_{M,D}$. This is motivated by a conjecture of Dutkay, Han and Jorgensen. The conjecture states that $D$ is always a spectral set if $\mu_{M,D}$ is a spectral measure in the dimension $n=1$. For a self-affine measure $\mu_{M,D}$, we obtain several conditions for the digit set $D$ to be a spectral set. The result here provides some supportive evidence for the conjecture. It also generalizes the corresponding known result in a certain case.

Significance of Higher-Order Spectral Analysis in Infant Cry Classification

In this paper, higher-order spectral analysis is applied to infant cry signals for classification of normal infant cries from pathological infant cries. From the family of higher-order spectra, bispectrum is considered for the proposed task. Bispectrum is the Fourier transform of the third-order cumulant function. To extract features from the bispectrum, application of higher-order singular value decomposition theorem is proposed. Experimental results show the average classification accuracy of \({82.44} \pm {4.03}{ \%}\) and Matthew’s correlation coefficient (MCC) of 0.62 with proposed bispectrum features. In all of the experiments reported in this paper, support vector machine with radial basis function kernel is used as the pattern classifier. Performance of the proposed features is also compared with the state-of-the-art methods such as linear frequency cepstral coefficients, Mel frequency cepstral coefficients, perceptual linear prediction coefficients, linear prediction coefficients, linear prediction cepstral coefficients and perceptual linear prediction cepstral coefficients, and is found to be better than that given by these feature sets. The proposed bispectrum-based features are shown to be robust under signal degradation or noisy conditions at various SNR levels. Performance in the presence of noise is compared with the state-of-the-art spectral feature sets using MCC scores. In addition, effectiveness of cryunit segmentation in normal and pathological infant cry classification task is reported.

What Is Variable Bandwidth?

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where p > 0 is a strictly positive function. Denote by the orthogonal projection of Ap corresponding to the spectrum of Ap in ; the range of this projection is the space of functions of variable bandwidth with spectral set in Λ.We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems; second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum the main results say that, in a neighborhood of , a function of variable bandwidth behaves like a band‐limited function with local bandwidth .Although the formulation of the results is deceptively similar to the corresponding results for classical band‐limited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame‐theoretic methods; on the other hand, we need the precise spectral theory of Sturm‐Liouville operators and the scattering theory of one‐dimensional Schrödinger operators. © 2017 Wiley Periodicals, Inc.

Tests for complete K-spectral sets

Let Φ be a family of functions analytic in some neighborhood of a complex domain Ω, and let T be a Hilbert space operator whose spectrum is contained in Ω‾. Our typical result shows that under some extra conditions, if the closed unit disc is complete K′-spectral for φ(T) for every φ∈Φ, then Ω‾ is complete K-spectral for T for some constant K. In particular, we prove that under a geometric transversality condition, the intersection of finitely many K′-spectral sets for T is again K-spectral for some K≥K′. These theorems generalize and complement results by Mascioni, Stessin, Stampfli, Badea–Beckermann–Crouzeix and others. We also extend to non-convex domains a result by Putinar and Sandberg on the existence of a skew dilation of T to a normal operator with spectrum in ∂Ω. As a key tool, we use the results from our previous paper [11] on traces of analytic uniform algebras.

Evaluation of Feature Selection Methods for Object-Based Land Cover Mapping of Unmanned Aerial Vehicle Imagery Using Random Forest and Support Vector Machine Classifiers

The increased feature space available in object-based classification environments (e.g., extended spectral feature sets per object, shape properties, or textural features) has a high potential of improving classifications. However, the availability of a large number of derived features per segmented object can also lead to a time-consuming and subjective process of optimizing the feature subset. The objectives of this study are to evaluate the effect of the advanced feature selection methods of popular supervised classifiers (Support Vector Machines (SVM) and Random Forest (RF)) for the example of object-based mapping of an agricultural area using Unmanned Aerial Vehicle (UAV) imagery, in order to optimize their usage for object-based agriculture pattern recognition tasks. In this study, several advanced feature selection methods were divided into both types of classifiers (SVM and RF) to conduct further evaluations using five feature-importance-evaluation methods and three feature-subset-evaluation methods. A visualization method was used to measure the change pattern of mean classification accuracy with the increase of features used, and a two-tailed t-test was used to determine the difference between two population means for both repeated ten classification accuracies. This study mainly contribute to the uncertainty analysis of feature selection for object-based classification instead of the per-pixel method. The results highlight that the RF classifier is relatively insensitive to the number of input features, even for a small training set size, whereby a negative impact of feature set size on the classification accuracy of the SVM classifier was observed. Overall, the SVM Recursive Feature Elimination (SVM-RFE) seems to be an appropriate method for both groups of classifiers, while the Correlation-based Feature Selection (CFS) is the best feature-subset-evaluation method. Most importantly, this study verified that feature selection for both classifiers is crucial for the evolving field of Object-based Image Analysis (OBIA): It is highly advisable for feature selection to be performed before object-based classification, even though an adverse impact could sometimes be observed from the wrapper methods.

The Numerical Range is a $(1+\sqrt{2})$-Spectral Set

It is shown that the numerical range of a linear operator on a Hilbert space is a (complete) $(1{+}\sqrt{2})$-spectral set. The proof relies, among other things, on the behavior of the Cauchy transform of the conjugates of holomorphic functions.

Spectral sets for numerical range

We define and study a numerical-range analogue of the notion of spectral set. Among the results obtained are a positivity criterion and a dilation theorem, analogous to those already known for spectral sets. An important difference from the classical definition is the role played in the new definition by the base point. We present some examples to illustrate this aspect.

On decomposition of operators having Γ_3 as a spectral set

On decomposition of operators having Γ_3 as a spectral set

Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

The commutation principle of Ramírez, Seeger, and Sossa [SIAM J. Optim., 23 (2013), pp. 687--694] proved in the setting of Euclidean Jordan algebras says that when the sum of a Fréchet differentiable function $\Theta$ and a spectral function $F$ is minimized (maximized) over a spectral set $\Omega$, any local minimizer (respectively, maximizer) $a$ operator commutes with the Fréchet derivative $\Theta'(a)$. In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.

Spectral sets for numerical range

We define and study a numerical-range analogue of the notion of spectral set. Among the results obtained are a positivity criterion and a dilation theorem, analogous to those already known for spectral sets. An important difference from the classical definition is the role played in the new definition by the base point. We present some examples to illustrate this aspect.

On decomposition of operators having Γ_3 as a spectral set

On decomposition of operators having Γ_3 as a spectral set

Approximating the Real Structured Stability Radius with Frobenius-Norm Bounded Perturbations

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the $H_\infty$ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost is dominated by the computation of the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequenc...

Classification of visible and infrared hyperspectral images based on image segmentation and edge-preserving filtering

The classification of hyperspectral images with a few labeled samples is a major challenge which is difficult to meet unless some spatial characteristics can be exploited. In this study, we proposed a novel spectral-spatial hyperspectral image classification method that exploited spatial autocorrelation of hyperspectral images. First, image segmentation is performed on the hyperspectral image to assign each pixel to a homogeneous region. Second, the visible and infrared bands of hyperspectral image are partitioned into multiple subsets of adjacent bands, and each subset is merged into one band. Recursive edge-preserving filtering is performed on each merged band which utilizes the spectral information of neighborhood pixels. Third, the resulting spectral and spatial feature band set is classified using the SVM classifier. Finally, bilateral filtering is performed to remove “salt-and-pepper” noise in the classification result. To preserve the spatial structure of hyperspectral image, edge-preserving filtering is applied independently before and after the classification process. Experimental results on different hyperspectral images prove that the proposed spectral-spatial classification approach is robust and offers more classification accuracy than state-of-the-art methods when the number of labeled samples is small.

Spectral sets and functions on Euclidean Jordan algebras

Spectral sets (functions) in Euclidean Jordan algebras are generalizations of permutation invariant sets (respectively, functions) in Rn. In this article, we study properties of such sets and functions and show how they are related to algebra automorphisms and majorization. We show that spectral sets/functions are indeed invariant under automorphisms, but the converse may not hold unless the algebra is Rn or simple. We study Schur-convex spectral functions and provide some applications. We also discuss the transfer principle and a related metaformula.

Classification of Tandem Mass Spectra for Identification of N- and O-linked Glycopeptides.

Analysis of intact glycopeptides by mass spectrometry is essential to determining the microheterogeneity of protein glycosylation. Higher-energy collisional dissociation (HCD) fragmentation of glycopeptides generates mono- or disaccharide ions called oxonium ions that carry information about the structure of the fragmented glycans. Here, we investigated the link between glycan structures and the intensity of oxonium ions in the spectra of glycopeptides and utilized this information to improve the identification of glycopeptides in biological samples. Tandem spectra of glycopeptides from fetuin, glycophorin A, ovalbumin and gp120 tryptic digests were used to build a spectral database of N- and O-linked glycopeptides. Logistic regression was applied to this database to develop model to distinguish between the spectra of N- and O-linked glycopeptides. Remarkably, the developed model was found to reliably distinguish between the N- and O-linked glycopeptides using the spectral features of the oxonium ions using verification spectral set. Finally, the performance of the developed predictive model was evaluated in HILIC enriched glycopeptides extracted from human serum. The results showed that pre-classification of tandem spectra based on their glycosylation type improved the identification of N-linked glycopeptides. The developed model facilitates interpretation of tandem mass spectrometry data for assignment of glycopeptides.

Unsupervised gene set testing based on random matrix theory.

BackgroundGene set testing, or pathway analysis, is a bioinformatics technique that performs statistical testing on biologically meaningful sets of genomic variables. Although originally developed for supervised analyses, i.e., to test the association between gene sets and an outcome variable, gene set testing also has important unsupervised applications, e.g., p-value weighting. For unsupervised testing, however, few effective gene set testing methods are available with support especially poor for several biologically relevant use cases.ResultsIn this paper, we describe two new unsupervised gene set testing methods based on random matrix theory, the Marc̆enko-Pastur Distribution Test (MPDT) and the Tracy-Widom Test (TWT), that support both self-contained and competitive null hypotheses. For the self-contained case, we contrast our proposed tests with the classic multivariate test based on a modified likelihood ratio criterion. For the competitive case, we compare the new tests against a competitive version of the classic test and our recently developed Spectral Gene Set Enrichment (SGSE) method. Evaluation of the TWT and MPDT methods is based on both simulation studies and a weighted p-value analysis of two real gene expression data sets using gene sets drawn from MSigDB collections.ConclusionsThe MPDT and TWT methods are novel and effective tools for unsupervised gene set analysis with superior statistical performance relative to existing techniques and the ability to generate biologically important results on real genomic data sets.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-1299-8) contains supplementary material, which is available to authorized users.

A Multidisciplinary Remote Sensing Ocean Color Sensor: Analysis of User Needs and Recommendations for Future Developments

Advancing remote sensing into the future to serve the expanding needs of the scientific community requires enhancing current capabilities of space-borne sensors and sustaining ocean color observations for diverse applications. In this study, an extensive set of experimental and space-borne measurements is analyzed for describing the operating characteristics of a global multidisciplinary ocean color sensor. Essential requirements for a successful mission include sensor characterization, high signal-to-noise ratio (SNR) and dynamic range, sensor stability, minimal polarization sensitivity, on-orbit calibration, vicarious calibration, atmospheric and in-water algorithms, product validation, and widely distributed regional and global products. To achieve high-quality products, an in-depth analysis of some key design aspects is provided which include spectral band sets, radiance levels, SNR, and dynamic range. These are essential characteristics of satellite sensors for measuring atmospheric signal in both ocean color and aerosol bands without saturation yet allowing high sensitivity measurements of water-leaving radiances and enabling ocean color measurements in aerosol and sun-glint contaminated regions of the ocean. Provisions to meet other requirements for improving retrievals of chlorophyll and phycocyanin fluorescence, partitioning algal and nonalgal color signals, and for implementing new approaches in atmospheric correction are further discussed. With these new capabilities and key design features, it will become feasible to resolve ocean color signals under most environmental circumstances and answer science questions related to changing conditions in the coastal and marine ecosystems and biogeochemical cycles due to climate change.

Tiling sets and spectral sets over finite fields

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6]). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Zp5 for all odd primes p and Zp4 for all odd primes p such that p≡3 mod 4. Although counterexamples in low dimensional groups over cyclic rings Zn were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis.