Basis Expansion Coefficients Research Articles

Distribution amplitude and decay constant of 1S and 2S state light mesons in the light-front quark model

Studying meson structures is essential for gaining insights into the nonperturbative nature of Quantum Chromodynamics (QCD). This study will focus on calculating the decay constant and distribution amplitudes (DAs) of unflavored light mesons (π and ρ) in the 1S and 2S states using the Light Front Quark Model. This study utilizes the QCD-motivated Hamiltonian, taking into account both contact and smeared spin-spin interactions. The two lowest harmonic oscillator bases are employed in this work to achieve improved results for the 2S states. The study found the optimal mixing parameter in basis expansion coefficients to be θ = 10°. Light meson properties, including the mass spectrum, decay constant, and twist-2 DAs, are then predicted using model parameters fixed through the variational principle. While the contact spin-spin interaction yields poor accuracy, the results from the smeared interaction generally agree well with experimental data and other theoretical models with $ f_{\pi}^{1S}=130 $ MeV and $ f_{\rho}^{1S}=210 $ MeV in the mixed state. Unlike the case for the 1S state, it should be noted that the properties of the 2S state are sensitive to the mixing parameter θ. In addition, we observe that the decay constant for ρ(2S) is $ f_{\rho}^{2S}=116 $ MeV. While for π(2S) the decay constant is extremely small with the value of $ f_{\pi}^{2S}=0.9 $ MeV, which is mainly due to the dynamical chiral symmetry breaking.

An improved LSTM stock price prediction model based on multiple basis function expansion and two-layer Bagging algorithm

Using neural network technology to predict the time series data of stock prices is a hot issue in the field of computational science. For example, Long Short-Term memory network technology (LSTM) has been widely used in stock price prediction. However, if the original stock opening price series information is not enough to predict the future trend, the effect of LSTM model may not be ideal. In addition, improper selection of the step size will also make the prediction effect of the LSTM model poor. This paper proposes an improved LSTM model based on multi-view basis function expansion and Bagging algorithm. On the one hand, under a fixed step size, the intra-day price information of the stock is extracted by using a variety of basis function expansion forms as prior information, and then the Bagging algorithm is used to establish the relationship between the basis expansion coefficients and the prediction residual of the original LSTM model to predict the residual, thereby increasing the prediction accuracy. On the other hand, the problem of LSTM step size selection is adaptively solved by using the idea of model average instead of the model selection method, and the specific method is to combine the weighted average of LSTM models with different time lengths. The actual data analysis shows that the proposed method improves the prediction accuracy of the original LSTM model, and the t-test shows that the proposed method has significant robustness.

Learning Block Structured Graphs in Gaussian Graphical Models

A prior distribution for the underlying graph is introduced in the framework of Gaussian graphical models. Such a prior distribution induces a block structure in the graph’s adjacency matrix, allowing learning relationships between fixed groups of variables. A novel sampling strategy named Double Reversible Jumps Markov chain Monte Carlo is developed for learning block structured graphs under the conjugate G-Wishart prior. The algorithm proposes moves that add or remove not just a single edge of the graph but an entire group of edges. The method is then applied to smooth functional data. The classical smoothing procedure is improved by placing a graphical model on the basis expansion coefficients, providing an estimate of their conditional dependence structure. Since the elements of a B-Spline basis have compact support, the conditional dependence structure is reflected on well-defined portions of the domain. A known partition of the functional domain is exploited to investigate relationships among portions of the domain and improve the interpretability of the results. Supplementary materials for this article are available online.

Projection-based model reduction of dynamical systems using space–time subspace and machine learning

This paper considers the creation of parametric surrogate models for applications in science and engineering where the goal is to predict high-dimensional spatiotemporal output quantities of interest, such as pressure, temperature and displacement fields. The proposed methodology develops a low-dimensional parametrization of these quantities of interest using space–time bases combined with machine learning methods. In particular, the space–time solutions are sought in a low-dimensional space–time linear trial subspace that can be obtained by computing tensor decompositions of usual state-snapshots data. The mapping between the input parameters and the basis expansion coefficients (or generalized coordinates) is approximated using four different machine learning techniques: multivariate polynomial regression, k-nearest-neighbors, random forests and neural networks. The relative costs and effectiveness of the four machine learning techniques are explored through three engineering problems: steady heat conduction, unsteady heat conduction and unsteady advective–diffusive–reactive system. Numerical results demonstrate that the proposed method performs well in terms of both accuracy and computational cost, and highlights the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models.

Function-on-Function Partial Quantile Regression

A function-on-function linear quantile regression model, where both the response and predictors consist of random curves, is proposed by extending the classical quantile regression setting into the functional data to characterize the entire conditional distribution of functional response. In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. Although the functional random variables belong to an infinite-dimensional space, they are observed in a finite set of discrete-time points in practice. Thus, in our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. The finite-sample performance of the proposed method is compared with the functional partial least squares method. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.

A partial least squares approach for function-on-function interaction regression

A partial least squares regression is proposed for estimating the function-on-function regression model where a functional response and multiple functional predictors consist of random curves with quadratic and interaction effects. The direct estimation of a function-on-function regression model is usually an ill-posed problem. To overcome this difficulty, in practice, the functional data that belong to the infinite-dimensional space are generally projected into a finite-dimensional space of basis functions. The function-on-function regression model is converted to a multivariate regression model of the basis expansion coefficients. In the estimation phase of the proposed method, the functional variables are approximated by a finite-dimensional basis function expansion method. We show that the partial least squares regression constructed via a functional response, multiple functional predictors, and quadratic/interaction terms of the functional predictors is equivalent to the partial least squares regression constructed using basis expansions of functional variables. From the partial least squares regression of the basis expansions of functional variables, we provide an explicit formula for the partial least squares estimate of the coefficient function of the function-on-function regression model. Because the true forms of the models are generally unspecified, we propose a forward procedure for model selection. The finite sample performance of the proposed method is examined using several Monte Carlo experiments and two empirical data analyses, and the results were found to compare favorably with an existing method.

Functional data clustering: a survey

Clustering techniques for functional data are reviewed. Four groups of clustering algorithms for functional data are proposed. The first group consists of methods working directly on the evaluation points of the curves. The second groups is defined by filtering methods which first approximate the curves into a finite basis of functions and second perform clustering using the basis expansion coefficients. The third groups is composed of methods which perform simultaneously dimensionality reduction of the curves and clustering, leading to functional representation of data depending on clusters. The last group consists of distance-based methods using clustering algorithms based on specific distances for functional data. A software review as well as an illustration of the application of these algorithms on real data are presented.

A Spatiotemporal Framework for Estimating Trial-to-Trial Amplitude Variation in Event-Related MEG/EEG

A spatiotemporal framework for estimating trial-to-trial variability in evoked response (ER) data is presented. Spatial and temporal bases capture the aspects of the response that are consistent across trials, while the basis expansion coefficients represent the variable components of the response. We focus on the simplest case of constant spatiotemporal response shape and varying amplitude across trials. Two different constraints on the amplitude evolution are employed to effectively integrate the individual responses and improve robustness at low SNR. The linear dynamical system response constraint estimates the current trial amplitude as an unknown constant scaling of the estimate in the previous trial plus zero-mean Gaussian noise with unknown variance. The independent response constraint estimates response amplitudes across trials as independent Gaussian random variables having unknown mean and variance. We develop a generalized expectation-maximization algorithm to obtain the maximum-likelihood (ML) estimates of the signal waveform, noise covariance matrix, and unknown constraint parameters. ML source localization is achieved by scanning the likelihood over different sets of spatial bases. We demonstrate the variability estimation and source localization effectiveness of the proposed algorithms using both real and simulated ER data.

Iterative joint time-variant channel estimation and multi-user detection for MC-CDMA

Joint time-variant channel estimation and multi-user detection are key building-blocks for wireless broadband communication for mobile users at vehicular speed. We propose an iterative receiver for a multi-carrier (MC) code division multiple access (CDMA) system in the uplink. Multi-user detection is implemented through iterative parallel interference cancellation and conditional linear minimum mean square error (MMSE) filtering. MC-CDMA is based on orthogonal frequency division multiplexing (OFDM), thus time-variant channel estimation can be performed for every subcarrier individually. The variation of a subcarrier over the duration of a data block is upper bounded by the maximum Doppler bandwidth which is determined by the maximum velocity of the users. We exploit results from the theory of time-concentrated and bandlimited sequences and apply a Slepian basis expansion for time-variant subcarrier estimation. This approach enables time-variant channel estimation without complete knowledge of the second-order statistics of the fading process. The square bias of the Slepian basis expansion is one order of magnitude smaller compared to the Fourier basis expansion. The square bias of the basis expansion is the determining factor for the performance of the iterative joint channel estimation and data detection. We present an iterative linear MMSE estimation algorithm for the basis expansion coefficients in a multi-user system. The consistent performance of the iterative receiver using the Slepian basis expansion is validated by simulations for a wide range of velocities

Auxiliary basis expansions for large-scale electronic structure calculations

One way to reduce the computational cost of electronic structure calculations is to use auxiliary basis expansions to approximate four-center integrals in terms of two- and three-center integrals, usually by using the variationally optimum Coulomb metric to determine the expansion coefficients. However, the long-range decay behavior of the auxiliary basis expansion coefficients has not been characterized. We find that this decay can be surprisingly slow. Numerical experiments on linear alkanes and a toy model both show that the decay can be as slow as 1/r in the distance between the auxiliary function and the fitted charge distribution. The Coulomb metric fitting equations also involve divergent matrix elements for extended systems treated with periodic boundary conditions. An attenuated Coulomb metric that is short-range can eliminate these oddities without substantially degrading calculated relative energies. The sparsity of the fit coefficients is assessed on simple hydrocarbon molecules and shows quite early onset of linear growth in the number of significant coefficients with system size using the attenuated Coulomb metric. Hence it is possible to design linear scaling auxiliary basis methods without additional approximations to treat large systems.

PARTICLE CREATION IN AN OSCILLATING SPHERICAL CAVITY

We study the creation of massless scalar particles from the quantum vacuum due to the dynamical Casimir effect by spherical shell with oscillating radius. In the case of a small amplitude of the oscillation, to solve the infinite set of coupled differential equations for the instantaneous basis expansion coefficients we use the method based on the time-dependent perturbation theory of the quantum mechanics. To the first order of the amplitude we derive the expressions for the number of the created particles for both parametric resonance and non-resonance cases.

PARTICLE CREATION BY MOVING SPHERICAL SHELL IN THE DYNAMICAL CASIMIR EFFECT

The creation of massless scalar particles from the quantum vacuum by spherical shell with time varying radius is studied. In the general case of motion the equations are derived for the instantaneous basis expansion coefficients. The examples are considered when the mean number of particles can be explicitly evaluated in the adiabatic approximation.