Orthonormal Expansion Research Articles

Orthonormal expansions for translation-invariant kernels

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of ℒ2(R). This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

Unbiased particle conformation extraction from scattering spectra using orthonormal basis expansions

A strategy is outlined for quantitatively evaluating the particle density profiles from small-angle scattering spectra of dilute solutions. The approach employs an orthonormal basis function expansion method, enabling the determination of characteristic mass distributions in self-assembled structures without the need for a specific structural model. Through computational benchmarking, the efficacy of this approach is validated by effectively reconstructing the density profile of soft-ball systems with varying fuzziness from their scattering signatures. The feasibility of the method is demonstrated by fitting small-angle neutron scattering data obtained from Pluronic L64 micelles at different temperatures. This proposed approach is both simple and analytical, eliminating the requirement for a presumptive structural model in scattering analysis. The new method could therefore facilitate quantitative descriptions of complex nanoscopic structures inherent to numerous soft-matter systems using small-angle scattering techniques.

Orthonormal eigenfunction expansions for sixth-order boundary value problems

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. To solve such problems, we first derive a complete set of odd and even orthonormal eigenfunctions — resembling trigonometric sines and cosines, as well as the so-called “beam” functions. These functions intrinsically satisfy boundary conditions (BCs) of relevance to thin-film flows, since they are the solutions of a self-adjoint sixth-order Sturm–Liouville BVP with the same BCs. Next, we propose a Galerkin spectral approach for sixth-order problems; namely the sought function as well as all its derivatives and terms appearing in the differential equation are expanded into an infinite series with respect to the derived complete orthonormal (CON) set of eigenfunctions. The unknown coefficients in the series expansion are determined by solving the algebraic system derived by taking successive inner products with each member of the CON set of eigenfunctions. The proposed method and its convergence are demonstrated by solving two model sixth-order BVPs.

Equiconvergence for perturbed Jacobi polynomial expansions

We show asymptotic expansions of the eigenfunctions of certain perturbations of the Jacobi operator in a bounded interval, deducing equiconvergence results between expansions with respect to the associated orthonormal basis and expansions with respect to the cosine basis. Several results for pointwise convergence then follow.

Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative

<abstract><p>In this paper, a new orthonormal wavelets based on the orthonormal Euler polynomials (OEPs) is constructed to approximate the numerical solution of time-fractional Cattaneo equation with Caputo-Fabrizio derivative. By applying the Gram-Schmidt orthonormalization process on sets of Euler polynomials of various degrees, an explicit representation of OEPs is obtained. The convergence analysis and error estimate of the orthonormal Euler wavelets expansion are studied. The exact formula of Caputo-Fabrizio fractional integral of orthonormal Euler wavelets are derived using Laplace transform. The applicability and validity of the proposed method are verified by some numerical examples.</p></abstract>

Trend filtering for functional data

Despite increasing accessibility to function data, effective methods for flexibly estimating underlying functional trend are still scarce. We thereby develop a functional version of trend filtering for estimating trend of functional data indexed by time or on general graph by extending the conventional trend filtering, a powerful nonparametric trend estimation technique for scalar data. We formulate the new trend filtering by introducing penalty terms based on ‐norm of the differences of adjacent trend functions. We develop an efficient iteration algorithm for optimizing the objective function obtained by orthonormal basis expansion. Furthermore, we introduce additional penalty terms to eliminate redundant basis functions, which leads to automatic adaptation of the number of basis functions. The tuning parameter in the proposed method is selected via cross validation. We demonstrate the proposed method is locally adaptive and can identify change points through simulation studies and applications to real‐world datasets.

Sparse Functional Principal Component Analysis in High Dimensions

Functional principal component analysis (FPCA) is a fundamental tool and has attracted increasing attention in recent decades, while existing methods are restricted to data with a single or finite number of random functions (much smaller than the sample size $n$). In this work, we focus on high-dimensional functional processes where the number of random functions $p$ is comparable to, or even much larger than $n$. Such data are ubiquitous in various fields such as neuroimaging analysis, and cannot be properly modeled by existing methods. We propose a new algorithm, called sparse FPCA, which is able to model principal eigenfunctions effectively under sensible sparsity regimes. While sparsity assumptions are standard in multivariate statistics, they have not been investigated in the complex context where not only is $p$ large, but also each variable itself is an intrinsically infinite-dimensional process. The sparsity structure motivates a thresholding rule that is easy to compute without nonparametric smoothing by exploiting the relationship between univariate orthonormal basis expansions and multivariate Kahunen-Lo\`eve (K-L) representations. We investigate the theoretical properties of the resulting estimators, and illustrate the performance with simulated and real data examples.

All-round improvement in DCT-based blind image watermarking with visual enhancement via denoising autoencoder

Discrete cosine transform (DCT)-based watermarking has received great success in the copyright protection of digital images. This study exploited the orthonormal expansion properties of DCT to achieve superior performance in computational efficiency and watermarking efficacy. In particular, we reformulated the relative modulation to allow the modification of paired DCT coefficients as if operable in the spatial domain. To account for human visual characteristics, we also adjusted the embedding strength adaptively according to image entropy. The use of a range regulation mechanism enabled perfect watermark retrieval. The pursuit of an ideal balance between robustness and imperceptibility could be benefited from the grey wolf optimizer. Using the above-discussed techniques, the proposed watermarking scheme demonstrates superior robustness in terms of bit error rate and normalized correlation coefficient with the peak signal-to-noise ratio tuned at around 38 dB. Compared to the regular DCT approach, the computational acceleration is on the scale of N2/(log2N)2, where N signifies the side length of the image block under processing. Furthermore, for the type of character-style watermark, we explored the feasibility of using a denoising autoencoder (DAE) to enhance the retrieved watermark comprehensibility. The proposed DAE was found to be capable of contributing a 60% reduction of the bit error rate, thus making the recovered watermark visually more recognizable.

Assessing Correlation between Heart Rate Variability Markers Based on Laguerre Expansion and Direct Measures of Sympathetic Activity during Incremental Head-up Tilt.

Traditional frequency domain analysis of heart period (HP) variability allows the estimation of the parasympathetic modulation directed to the heart but the sympathetic one remains largely unknown. Recently, sympathetic and parasympathetic activity indexes (SAI and PAI) have been proposed to address this issue. SAI and PAI were derived from HP variability via the application of an orthonormal Laguerre expansion allowing the separation of HP variations driven by sympathetic and parasympathetic outflows. In this study, SAI and PAI were validated against tonic and variability measures of muscle sympathetic nerve activity (MSNA) and more traditional markers derived from HP variability. Indexes were calculated in 12 healthy subjects (9 females, age from 20 to 36 years, median 22.5 years) undergoing incremental head-up tilt. Results showed that traditional HP and MSNA variability markers as well as SAI and PAI were modified in proportion to the magnitude of the postural challenge. However, SAI was not correlated with any MSNA markers and PAI was not linked to respiratory sinus arrhythmia. SAI and PAI can capture modifications of cardiac control induced by the orthostatic challenge but they might be weak surrogates of vagal and sympathetic activities and/or modulations.Clinical Relevance- SAI and PAI markers are useful to characterize cardiac control but poorly linked with autonomic nervous system state.

The Complex-Number Mortality Model (CNMM) based on orthonormal expansion of membership functions

Abstract The paper deals with a new fuzzy version of the Lee-Carter (LC) mortality model, in which mortality rates as well as parameters of the LC model are treated as triangular fuzzy numbers. As a starting point, the fuzzy Koissi-Shapiro (KS) approach is recalled. Based on this approach, a new fuzzy mortality model – CNMM – is formulated using orthonormal expansions of the inverse exponential membership functions of the model components. The paper includes numerical findings based on a case study with the use of the new mortality model compared to the results obtained with the standard LC model.

Integral pulse frequency modulation model driven by sympathovagal dynamics: Synthetic vs. real heart rate variability

Computational models that generate synthetic heart rate variability (HRV) series constitute important tools for the assessment of the effect of autonomic nervous system activity on cardiovascular control, and for the evaluation of novel algorithms using synthetic data. A widely used technique for synthetic HRV generation is the integral pulse frequency modulation (IPFM) model; however, IPFM relies on the HRV spectral paradigm, which cannot separate sympathetic and vagal oscillations that are overlapped in the low-frequency band (0.04–0.15 Hz). To overcome this limitation, a novel IPFM-inspired model driven by cardiac sympathetic and vagal dynamics estimated from HRV is proposed, where our recently developed sympathetic and parasympathetic activity indices that rely on orthonormal Laguerre expansions of the RR interval autoregressive kernels are exploited. The performance of the proposed model is evaluated by comparing the synthetic vs. real RR interval series in a simulation study involving postural changes, with real HRV data gathered from 10 healthy subjects. Moreover, the performance of the proposed model is compared with that of the standard IPFM to discern different autonomic control states associated with resting and postural changes. The results confirm that the proposed physiologically inspired model adequately predicts RR intervals during resting and postural changes. The proposed model clearly outperforms the standard IPFM method, considering both median error, and maximum error. The developed model provides valuable insights for a better understanding of the sympathovagal activity in the analysis of heartbeat dynamics.

Application of Walsh series to represent iterated Stratonovich stochastic integrals

The goal of the paper is to represent iterated Stratonovich stochastic integrals using orthonormal expansions of functions by the Walsh series, to construct an algorithm for modeling iterated Stratonovich stochastic integrals, and to test this algorithm by numerical experiments. Obtained results may be used in the constructing high-order numerical methods based on the Taylor –Stratonovich expansion and the Taylor– It^o expansion. Therefore, they can be applied in a lot of problems of physics, engineering, economics and finance, when it is necessary to solve numerically stochastic differential equations.

Kernel-based interpolation at approximate Fekete points

We construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion of the kernel. Uniform error estimates are proved for kernel interpolants at the resulting points. If the kernel is Gaussian, we show that the approximate Fekete points in one dimension are the solution to a convex optimisation problem and that the interpolants converge with a super-exponential rate. Numerical examples are provided for the Gaussian kernel.

Methods of decomposition of Boolean functions

The article considers on of the main types of Boolean functions conversion – their decomposition. The decomposition is being used in many applications for studying and designing complex objects and systems with logical methods. Firstly, the article observes decomposition of Boolean functions represented in the truth table. Among traditional decomposition approaches, the article describes formulas presented by Boole and Shannon and special methods of orthonormal expansions. Then, the paper considers contemporary decomposition techniques and their classification. Next, the article discusses representation of Boolean functions in normal forms and essential features of that representation, important for decomposition. After that, the article reviews decomposition techniques for boolean functions represented in conjunctive normal forms (CNF). The study mentions CNF partitioning as hypergraph partitioning. Then the paper discusses applications generating large CNFs. Minimization of large CNFs with traditional methods is impossible. Their minimization is possible only using decomposition techniques. The separate class of tasks is minimization of Horn formulas. Finally, at the conclusion, the study discusses decomposition for parallelization of SAT problem.

Orthonormal Polynomial Expansion of Different Types of Silver Nanoparticles Spectroscopic Data

Our Orthonormal Polynomial Expansion Method (OPEM) in one-dimensional version is applied to describe the original different silver nanoparticles spectroscopic data. We construct orthogonal (orthonormal) polynomials for presenting the curves. The corridors defined by errors of given data by the help of the weights construct the optimal behavior of sought for curve. We have received from experiments some curves in thousands points for analysis, but two of them are important—with ten and five laser pulses. In this way the paper is a continuation and a generalization of the investigation in the previous paper. We have chosen the subintervals in two curves. The most important subintervals of spectra data are investigated, where the minimum (Surface Plasmon Resonance Absorption) is looking for. This study describes the Ag nanoparticles produced by laser approach in a ZnO medium forming a AgNPs/ZnO nanocomposite heterostructure.

Data-driven probabilistic small signal stability analysis for grid-connected PV systems

Probabilistic small signal stability analysis is useful to quantify the effect of uncertain parameters, such as the solar and wind generations, on system stability margin. In the literature, a semi-analytic method based on general polynomial chaos expansion (gPCE) has been proposed for probabilistic small signal stability analysis. The gPCE method can achieve high order accuracy with lower computation burden, but it needs a predefined probability distribution of random parameters, which may not be available in practical engineering applications. To reduce such dependency on the full information of probability distribution, this paper presents a data-driven polynomial chaos expansion (DD-PCE) method for assessing the impact of uncertain solar irradiance on the stability margin of a grid-connected photovoltaic (PV) system. In the proposed method, the system stability margin is characterized in terms of its critical mode damping ratio, and the functional relationship between the critical mode damping ratio and the uncertain solar irradiance is estimated by orthonormal polynomial expansion. The orthonormal polynomial basis is constructed using the moments of solar irradiance, and the coefficients of the orthonormal polynomial expansion are calculated by the collocation points of polynomial basis. The efficiency of the proposed method is validated by comparisons with gPCE method and Monte Carlo method.

A modified orthonormal polynomial series expansion tailored to thin beams undergoing slamming loads

In this paper, a novel version of orthonormal polynomial series expansion method is developed to analyze vibrations of bottom hull panels due to moving impacting fluid loads. The initial phase of slamming is considered in the presented mathematical model, wherein, an elastic linear Euler-Bernoulli beam of uniform thickness mimics the bottom panel structural behaviour. The framework developed, suppresses the Gibbs phenomenon, and captures the beam deflection, flexural moment, and shear force by direct integration. The proposed solution exhibits a higher convergence rate, compared to the conventional method, in terms of the number of involved shape functions.

Optimal transmission modes under atmosphere turbulence with transmitter/receiver aperture size constraint.

In this paper, an optimal mode set is proposed to maximize the receiving power for free space transmission under atmosphere turbulence with transmitter/receiver aperture size constraint. The optimal beam profiles are evaluated through eigenmode analysis of the Fredholm integral equation, which is mathematically equivalent to the eigen vector analysis of an infinitely large matrix. The matrix is formed by orthonormal basis expansion, and its element is the overlap integral of the orthonormal basis functions and the Fredholm kernel. If circular aperture is implemented, then it is rigorously proven in this work that the eigenmodes possess certain topological charges (i.e., they are the OAM modes). These OAM modes have specific radial beam profiles, which have been optimized to minimize the power loss and the inter-mode crosstalk. While the traditional OAM beams, such as the Laguerre-Gauss (LG) beams, suffer significant energy loss and inter-radial-mode crosstalk, the optimized beam profiles will remarkably reduce the penalties.

Continuous-Time DC Kernel—A Stable Generalized First-Order Spline Kernel

The stable spline (SS) kernel and the diagonal correlated (DC) kernel are two kernels that have been applied and studied extensively for kernel-based regularized LTI system identification. In this note, we show that similar to the derivation of the SS kernel, the continuous-time DC kernel can be derived by applying the same “stable” coordinate change to a “generalized” first-order spline kernel, and thus, can be interpreted as a stable generalized first-order spline kernel. This interpretation provides new facets to understand the properties of the DC kernel. In particular, we derive a new orthonormal basis expansion of the DC kernel and the explicit expression of the norm of the reproducing kernel Hilbert space associated with the DC kernel. Moreover, for the nonuniformly sampled DC kernel, we derive its maximum entropy property and show that its kernel matrix has tridiagonal inverse.

A new Monte Carlo method for estimation of time asymptotic parameters of polarized radiation

In this work we discuss the problem of estimation of time asymptotic parameters of polarized radiation flux in the scattering and absorbing media, illuminated by an external or an internal source. We construct a distinctive weighted Monte Carlo algorithm for evaluation of asymptotic parameters and time distribution density of polarized radiation flux. This algorithm is based on the randomized projective evaluation of the functionals via the orthonormal polynomial expansion of a special kind. We provide results of performed numerical simulation of the time distribution of the radiation intensity and degree of polarization, scattered by semi-infinite layer of the scattering and absorbing media.