Multivariate Polynomial Function Research Articles

Multivariate polynomial regression by an explainable sigma-pi neural network

<p>Over the years, data-driven regression on univariate functions has been extensively studied. However, fast, effective, and stable algorithms for multivariate function fitting are still lacking. Recently, Kolmogorov-Arnold networks have garnered significant attention among scholars due to their superior accuracy and interpretability compared to multi-layer perceptrons. In this paper, we have demonstrated that the sigma-pi neural network, a form of Kolmogorov-Arnold networks, can efficiently fit multivariate polynomial functions, including fractional-order multivariate polynomials. Three examples were employed to illustrate the regression performance of the designed neural networks. The explainable sigma-pi neural network will lay the groundwork for further development of general tools for multivariate nonlinear function regression problems.</p>

Protecting Function Privacy and Input Privacy in the Publicly Verifiable Outsourcing Computation of Polynomial Functions

With the prevalence of cloud computing, the outsourcing of computation has gained significant attention. Clients with limited computing power often outsource complex computing tasks to the cloud to save on computing resources and costs. In outsourcing the computation of functions, a function owner delegates a cloud server to perform the function’s computation on the input received from the user. There are three primary security concerns associated with this process: protecting function privacy for the function owner, protecting input privacy for the user and guaranteeing that the cloud server performs the computation correctly. Existing works have only addressed privately verifiable outsourcing computation with privacy or publicly verifiable outsourcing computation without input privacy or function privacy. By using the technologies of homomorphic encryption, proxy re-encryption and verifiable computation, we propose the first publicly verifiable outsourcing computation scheme that achieves both input privacy and function privacy for matrix functions, which can be extended to arbitrary multivariate polynomial functions. We additionally provide a faster privately verifiable method. Moreover, the function owner retains control over the function.

Interbirth interval and maternal anaemia in 21 sub-Saharan African countries: A fractional-polynomial analysis.

BackgroundMaternal anaemia is a global public health problem contributing to adverse maternal and perinatal outcomes. In addition to other risk factors, interbirth interval has been identified as a potentially modifiable risk factor of maternal anaemia. However, the current evidence for the association between interbirth interval and maternal anaemia remains inconclusive. Hence, this study examined the association between the interbirth interval and maternal anaemia in sub-Saharan Africa.MethodsWe conducted a multinational cross-sectional study of interbirth interval (time between two singleton live births) and maternal anaemia (haemoglobin levels < 12 g/dl for non-pregnant women, < 11 g/dl for pregnant women) for 21 sub-Saharan African countries using the most recent nationally representative Demographic and Health Surveys, 2010–2017. A weighted multivariable fractional polynomial function was used to estimate the non-linear relationship between interbirth interval and maternal anaemia, considering interbirth interval as a continuous variable and adjusting for potential confounders. Analyses were stratified by reproductive classification (non-pregnant and pregnant women).ResultsThere were 81,693 women included in the study (89.2% non-pregnant, 10.8% pregnant). Of all women, 32.2% were in their postpartum period. Overall, 36.9% of women had anaemia (36.0% of non-pregnant and 44.3% of pregnant women). Of the participants, 15% had a short interbirth interval (<24 months), and 16% had a long interbirth interval (≥ 60 months). We found that both short and longer interbirth intervals were associated with an increased risk of maternal anaemia in a dose-response fashion. Relatively a lower risk of maternal anaemia was observed between 24 and 40 months of interbirth intervals.ConclusionsOur findings suggest that both short and longer interbirth intervals were associated with an increased risk of maternal anaemia in sub-Saharan Africa.

Algebraic methods to study the dimension of supersmooth spline spaces

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera–Schenck–Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.

Method for identifying the impact load condition of thin-walled structure damage based on PSO-BP neural network.

Thin-walled structures (TWS) were widely used in engineering equipment, and may be subjected to impact loads to produce different degrees of structural damage during application. However, it is a difficult problem to determine the impact load conditions for these structural damages. In this study, we developed a novel method of identifying the impact load condition of the thin-walled structure damage, which is based on particle swarm optimization-backpropagation (PSO-BP) neural network. First, the known impact position and velocity are applied to the finite element model (FEM) of the TWS to produce permanent plastic deformation, and to fit the characteristic shape of the deformation is needed by invoking the multivariate polynomial function. Then, the method is devoted to build a basic data set. With impact position and velocity as input and function coefficients as output, a model of extended PSO-BP neural network is established. Besides, the basic sample set is expanded to solve the lack of samples. Ultimately, utilizing the expanded total sample set as training data, function coefficients, impact position and velocity will be outputted. On the basis of the known functional coefficients of deformed surfaces, a model of predictive PSO-BP neural network is established and predicted. Furthermore, we predicted the collision position and velocity using a conventional BP neural network in the same way. Finally, the predicted impact position and velocity is compared with the analysis results of the FEM, which verifies that the PSO-BP neural network algorithm has high accuracy.

Constrained global optimization of multivariate polynomials using polynomial B-spline form and B-spline consistency prune approach

In this paper, we propose basic and improved algorithms based on polynomial B-spline form for constrained global optimization of multivariate polynomial functions. The proposed algorithms are based on a branch-and-bound framework. In improved algorithm we introduce several new ingredients, such as B-spline box consistency and B-spline hull consistency algorithm to prune the search regions and make the search more efficient. The performance of the basic and improved algorithm is tested and compared on set of test problems. The results of the tests show the superiority of the improved algorithm over the basic algorithm in terms of the chosen performance metrics for 7 out-off 11 test problems. We compare optimal value of global minimum obtained using the proposed algorithms with CENSO, GloptiPoly and several state-of-the-art NLP solvers, on set of 11 test problems. The results of the tests show the superiority of the proposed algorithm and CENSO solver (open source solver for global optimization of B-spline constrained problem) in that it always captures the global minimum to the user-specified accuracy.

Statistical uncertainties of the vn{2k} harmonics from Q cumulants

Analytic formulas to calculate statistical uncertainties of $v_{n}\{2k\}$ harmonics extracted from the Q-cumulants are presented. The Q-cumulants are multivariate polynomial functions of the weighted means of $2m$-particle azimuthal correlations, $\llangle 2m \rrangle$. Variances and covariances of the $\llangle 2m \rrangle$ are included in the analytic formulas of the uncertainties that can be calculated simultaneously with the calculations of the $v_{n}\{2k\}$ harmonics. The calculations are performed using a simple toy model which roughly simulates elliptic flow azimuthal anisotropy with magnitudes around 0.05. The results are compared with the results obtained by the many data sub-sets, and by the bootstrapping method. The first one is a common way of estimation of the statistical uncertainties of the $v_{n}\{2k\}$ harmonics in a real experiment. In order to increase precision in the measurement of the $v_{n}\{2k\}$ harmonics, a large number of 15000 sub-sets and the same number of the re-sampling in the bootstrap method is used. A very good agreement in the comparison with the proposed way of the analytic calculation of the $v_{n}\{2k\}$ statistical uncertainties is found. Additionally, a recurrence equation for Q-cumulants calculation is also presented.

Symbolic Variable Elimination for Discrete and Continuous Graphical Models

Probabilistic reasoning in the real-world often requires inference incontinuous variable graphical models, yet there are few methods for exact, closed-form inference when joint distributions are non-Gaussian. To address this inferential deficit, we introduce SVE -- a symbolic extension of the well-known variable elimination algorithm to perform exact inference in an expressive class of mixed discrete and continuous variable graphical models whose conditional probability functions can be well-approximated as piecewise combinations of polynomials with bounded support. Using this representation, we show that we can compute all of the SVE operations exactly and in closed-form, which crucially includes definite integration w.r.t. multivariate piecewise polynomial functions. To aid in the efficient computation and compact representation of this solution, we use an extended algebraic decision diagram (XADD) data structure that supports all SVE operations. We provide illustrative results for SVE on probabilistic inference queries inspired by robotics localization and tracking applications that mix various continuous distributions; this represents the first time a general closed-form exact solution has been proposed for this expressive class of discrete/continuous graphical models.

Attasi nD Systems and Polynomial System Solving

Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.

Development of a cable suspension and balance system and its novel calibration methods for effective wind tunnel tests

Based on the advantages of the cable-driven parallel robot (CDPR), we developed a cable suspension and balance system (CSBS) as a multi-purpose device for wind tunnel tests with motion control and load measurement capability. The CSBS was designed to serve as both a wind tunnel model mount and balance.This paper suggests two novel methodologies to improve the CSBS performance, one for more accurate motion control and the other for improved load measurement. The motion control correction was implemented by applying a multi-variable polynomial function that adjusts motion commands to generate the required response utilizing the measured motion data. For more accurate load measurement, a multi-layer perceptron (MLP) is proposed as a correction function. This MLP was trained to provide corrected three forces and three moments according to the reference balance data.Several motion tests, loading tests, and statistical analyses were carried out before and after correction to investigate the correction effectiveness. Differences between the motion command and response were dramatically reduced by the polynomial correction, and the MLP learning methodology effectively enhanced the load measurement accuracy of the CSBS.Using both of the suggested corrections, the CSBS was equipped with enhanced motion control and load measurement performance. Several wind tunnel tests were conducted to examine the CSBS reliability as a model mount and load measurement system, and test results showed reasonably good agreement with reference data for a cylinder model and a NACA0015 airfoil model.

Hyperchaotic Behavior in the Novel Memristor-Based Symmetric Circuit System

A novel five-dimensional (5D) memristor-based symmetric circuit, which consists of two symmetric capacitors, two symmetric inductors and only one memristor is presented in this article. The multivariable first order and multivariable second order polynomial functions are used for the internal state function of the memristor respectively. Theoretical and simulation analyses of the novel memristive circuit are investigated using equilibrium points, phase portraits, bifurcation diagrams and Lyapunov exponent spectra, etc. Complex chaotic behaviors are observed and analyzed through simulation results. The first order internal state function memristor-based symmetric circuit system can only exhibit chaotic behavior whereas the second order internal state function memristor-based symmetric circuit system can generate not only chaotic attractors, but also hyperchaotic attractors in proper parameters.

Removing limiting factors for Leeds Test Object TO.10 usage

The Leeds Test Object TO.10 is routinely used to provide a subjective estimate of Signal to Noise Ratio (SNR) as a measure of overall image quality. Currently, calibrated contrast values are provided for a limited set of discrete peak voltage (65, 70, 75, 80 kVp) and copper filtration thickness (1.0, 1.5, 2.0 mm Cu) combinations. However, it can be challenging to attain these exact settings on modern interventional imaging systems incorporating Automatic Dose Rate Controls (ADRC) and varying amounts of additional copper filtration. These limit the accuracy of results obtained thus representing significant limiting factors for the TO.10. We describe two methods of removing these limiting factors: a three-dimensional (3D) Matlab interpolation and extrapolation algorithm, and a multivariate-polynomial function, the coefficients of which can be stored in Excel. Both methods make use of the available contrast values to generate contrast curves for any kVp and mm Cu combination. Results obtained from both methods are presented as Threshold Index (H_T(A)) curves modelled by best fit log-polynomials. Their accuracies are evaluated through comparison with H_T(A) curves obtained under calibrated conditions. Both methods are found to produce more accurate estimates of detail contrasts for non-standard kVp and mm Cu combinations. Although an inherent error of approximately 15% is associated with this type of contrast detail analysis, the ability to analyse TO.10 data for non-standard kVp and mm Cu combinations offered by modern systems increases the accuracy of calculated H_T(A)’s. These methods further reduce the time required for image quality tests and room downtime.

Fuel Quantity Estimation of Aircraft Supplementary Tank Using Markov Chain Monte Carlo Method

This paper presents an aircraft fuel quantity estimation method using the Markov Chain Monte Carlo (MCMC) method. Using the proposed method, fuel quantity uncertainty of an aircraft supplementary tank can be estimated when the roll and pitch attitudes of an aircraft change. Through reflecting uncertainties, the conservative bound of fuel quantity estimation results can be found, which is necessary for a reliable aircraft operation. The first step of the estimation process is a mathematical modeling of the fuel quantity in a supplementary tank. In the model, the fuel quantity is represented as a multivariate polynomial function of sensor output (i.e., frequency), aircraft roll and pitch angles. The parameter of the mathematical model is then estimated using the MCMC method. As an estimation result, the probability density function of the fuel quantity is provided, which accounts for the uncertainties caused from the developed mathematical model and measured data. The lower bound in the estimation result can be utilized as a conservative fuel quantity value for a reliable operation. To validate the proposed fuel quantity estimation approach, a test with known fuel quantity is performed.

Unified integrals involving product of multivariable polynomials and generalized Bessel functions

The aim of this paper is to evaluate two integral formulas involving a finite product of the generalized Bessel function of the first kind and multivariable polynomial functions which results are expressed in terms of the generalized Lauricella functions. The major results presented here are of general character and easily reducible to unique and well-known integral formulae.

On types of degenerate critical points of real polynomial functions

In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point c, which is degenerate, of a multivariate polynomial function f. To this end, we introduce the definition of faithful radius of c by means of the curve of tangency of f. We show that the type of c can be determined by the global extrema of f over the Euclidean ball centered at c with a faithful radius. We propose algorithms to compute faithful radius of c and determine its type.

Multivariate Polynomial Interpolation Based Indoor Fingerprinting Localization Using Bluetooth

Comparing with mature outdoor positioning technology, the demand for indoor positioning is also becoming increasingly extensive. Fingerprinting-based localization using Bluetooth is a promising method for indoor localization. The localization accuracy mainly depends on the quality of fingerprints database, and we have to collect a larger number of training sites in order to achieve finer-grained positioning. However, the site survey for constructing a fine-grained fingerprints database is very time consuming and labor intensive. In order to improve localization accuracy under the condition of without additional fingerprints, we use multivariate polynomial function to construct the fine-grained interpolated fingerprints database derived from the coarse-grained collected fingerprints. Extensive simulation results verify that the proposed multivariate polynomial function interpolation based fingerprinting localization method can improve localization accuracy effectively.

New Method to Optimize the Production Functions in the System of Safety in Operation Management

Abstract The Safety in Operation system represents the framework within which every technological process operates. Any technological parameter - production, quality, must be analyzed in terms of this system. Mathematical and/or empirical modeling, as the first stage in the determination of the technological processes optimum operation regimes, represents a “must” both in the phase of conception, but mainly in the operation analysis phase. From this standpoint, the introduction of the active optimization method based on scheduled experiment represents an efficient tool to relieve the extreme conditions and to get information for technological processes optimum management. The work is focused on the mathematical modeling of the production function for an assembly of 13 drawing frames from two Romanian units, in terms of maintainability and reliability parameters. After determining the polynomial that characterizes the model, the work investigates the optimization of non-linear multi-variable polynomial function, and explains the context of getting the extreme values within the multi-factorial space. The work defines in its structure the notions of the safety in operation system used in research, applies quantitative study on a system from textile spinning mill field (one notices reliable operation times, break-down times, number of failures and the production of the technical systems on a pre-established time horizon); it statistically validates and mathematically optimizes the results. The theoretical approach consists in an algorithm in a unitary software application that can be used as a tool in the decision problems appeared during the technological process.

A Modified Affine Arithmetic Method for Computational Error Analysis

Computational precision always has been a concern in digital systems design and optimization. This study presents a modified Affine Arithmetic method to calculate uncertainty range in the output of arithmetic units. We have considered five case studies, namely: quadratic equation, RGB to CrYCb system, fourth-order equation, multivariate polynomial functions and low pass filter, by which the proposed method is evaluated. Presented results indicate the modified affine arithmetic can outperform the traditional methods.

Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations

In this paper we study multivariate polynomial functions in complex variables and their corresponding symmetric tensor representations. The focus is to find conditions under which such complex polynomials always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, characterize the conditions for such complex polynomials to be real valued, and present their corresponding tensor representations. New notions of eigenvalues/eigenvectors for complex tensors are introduced, extending similar properties from the Hermitian matrices. Moreover, we study a property of the symmetric tensors, namely, the largest eigenvalue (in the absolute value sense) of a real symmetric tensor is equal to its largest singular value; the result is also known as Banach's theorem. We show that a similar result holds for the complex case as well. Finally, we discuss some applications of the new notion of eigenvalues/eigenvectors for the complex tensors.

Screening the geomechanical stability (thermal and mechanical) of shared multi-user CO2 storage assets: A simple effective tool applied to the Captain Sandstone Aquifer

Multi-user storage systems are anticipated in the near future to permanently store CO2 captured at industrial sources to meet emissions reductions targets. Multiple storage permit applications will be required to exploit the immense potential capacity within extensive CO2 storage assets. To retain 99% of the injected CO2 for 1000 years the geomechanical stability of the sealing strata above the pressurised storage reservoir is a key factor which needs to be included in the geo-engineering design of shared storage assets The potential for interaction of increased pressure at multiple injection sites needs to be predicted and assessed at a regional scale to assure the integrity at all existing sites before a storage permit is granted. Geomechanical models coupled with the expected fluid pressure response predict the stability of the storage asset during and after injection of CO2 at multiple injection sites, and can be used as a tool to ensure efficient utilisation of the storage capacity. The geomechanical analysis of the thermal stress as well as local and regional fluid pressure changes requires a detailed numerical evaluation, often at a resolution significantly higher than the data available. Coupling of regional-scale static geological models, dynamic multi-phase flow models and detailed geomechanical models requires extensive computational resources. Such models often produce seemingly detailed results, but are usually only one or two realisations of a system populated by a statistically generated parameter set. Limits on time and computational resources prevent more simulations within fixed time and financial budgets. To enable a more time and cost efficient methodology of assessing the geomechanical stability of potential storage sites we present a four-tier modelling approach with increasing complexity that allows an in-depth evaluation of the geomechanical stability at a regional scale of a multi-user storage asset taking into account the fluid pressure increase and the thermal stress impact on the stability of the strata sealing the CO2 store. The tiers include: (1) development of a geo-mechanical facies model of the storage system, (2) development of an analytical geomechanical model for the storage site static stress conditions, (3) fitting an empirical multivariable polynomial function to the analytical modal, and (4) conditioning the empirical function using coupled numerical THM modelling for dynamic stress conditions. The result is look up function which gives the maximum possible fluid pressure as a function of location. This approach significantly simplifies the computational requirements and time for the prediction of geomechanical behaviour. In addition to presenting this methodology, using the Captain Sandstone of the North Sea as an example, three key findings are further examined. Firstly, detailed analysis of the stress changes as a result of cold fluid injection suggests that the redistribution of thermal stress can, in some cases, be beneficial to the storage system depending on the stress bridging which occurs. Secondly, pressure plume migration over time in dipping strata, from deeper injection sites to shallower sites, needs to be taken into account. Thirdly, the nature of the strata underlying the storage formation is critical to the pressure increase in response to the fluid injection. The methodology developed in this paper enables a rapid and efficient screening of the dynamic geomechanical stability and facilitates an efficient coupling to diverse discrete multiphase fluid flow models using commonly available computational resources.