Parametric Polynomial Research Articles

Evaluating accuracy of diagnostic tests without conditional independence assumption.

Evaluating the accuracy (ie, estimating the sensitivity and specificity) of new diagnostic tests without the presence of a gold standard is of practical meaning and has been the subject of intensive study for several decades. Existing methods use 2 or more diagnostic tests under several basic assumptions and then estimate the accuracy parameters via the maximum likelihood estimation. One of the basic assumptions is the conditional independence of the tests given the disease status. This assumption is impractical in many real applications in veterinary research. Several methods have been proposed with various dependence models to relax this assumption. However, these methods impose subjective dependence structures, which may not be practical and may introduce additional nuisance parameters. In this article, we propose a simple method for addressing this problem without the conditional independence assumption, using an empirical conditioning approach. The proposed method reduces to the popular Hui-Walter model in the case of conditional independence. Also, our likelihood function is of order-2 polynomial in parameters, while that of Hui-Walter is of order-3. The reduced model complexity increases the stability in estimation. Simulation studies are conducted to evaluate the performance of the proposed method, which shows overall smaller biases in estimation and is more stable than the existing method, especially when tests are conditionally dependent. Two real data examples are used to illustrate the proposed method.

Computing zeta functions of generic projective hypersurfaces in larger characteristic

We give improvements of the deformation method for computing the zeta function of a generic projective hypersurface in characteristic~$p$ that either reduce the dependence on~$p$ of the time complexity to $\tilde{O}(p^{1/2})$ or that of the space complexity to $\tilde{O}(\log(p))$ while remaining polynomial in the other input parameters.

Multivariate Splines-Based Asymmetric Aerodynamic Modeling of Morphing Aircraft

The aerodynamic parameters of morphing aircraft can vary significantly in the morphing process; the aerodynamic data model is complex and high-order nonlinear. This paper presents a new method of aerodynamic modeling of morphing aircraft by using multivariate splines. Firstly, the aerodynamic coefficient model of morphing aircraft based on the multivariate splines model structure is presented. This model can describe symmetric and asymmetric aerodynamic forces. Then, the multivariate simple splines estimation process is introduced in detail and the concrete formulation is deduced. Finally, the final model structure is determined by error analysis, and the aerodynamic parameter polynomial model in the global coordinate system is obtained by coordinate transformation. The results show that this method can accurately describe the aerodynamic model of morphing aircraft during arbitrary symmetric and asymmetric morphing process without predicting the specific model structure of the aerodynamic parameters related to varying process.

Direct and inverse identification of constitutive parameters from the structure of soft tissues. Part 1: micro- and nanostructure of collagen fibers

Soft tissues are characterized by a nonlinear mechanical response, highly affected by the multiscale structure of collagen fibers. The effectiveness and the calibration of constitutive models play a major role on the reliability and the applicability of computational models in biomechanics. This paper presents a procedure for the identification of the relationship between collagen-related structural features in soft tissues with model parameters of classical polynomial- and exponential-based constitutive models. Histological features at microscale, as well as biochemical and biophysical properties at nanoscale, are addressed by employing a multiscale structural description of soft tissue mechanics as benchmark data set. Both the direct (from structure to parameters) and the inverse (from parameters to structure) problem are addressed. Suitable optimization problems are introduced for accurate numerical and approximated analytical direct relationships. The inverse identification has been addressed by providing also a measure of the reliability of the computed estimates. Results show the effectiveness of the proposed strategies and allow to discuss the fitting capabilities of classical constitutive approaches in terms of parameters identification.

A convex polynomial model for planar sliding mechanics: theory, application, and experimental validation

We propose a polynomial model for planar sliding mechanics. For the force–motion mapping, we treat the set of generalized friction loads as the 1-sublevel set of a polynomial whose gradient directions correspond to generalized velocities. The polynomial is confined to be convex even-degree homogeneous in order to obey the maximum work inequality, symmetry, shape invariance in scale, and fast invertibility. We present a simple and statistically efficient model identification procedure using a sum-of-squares convex relaxation. We then derive the kinematic contact model that resolves the contact modes and instantaneous object motion given a position controlled manipulator action. The inherently stochastic object-to-surface friction distributions are modeled by sampling polynomial parameters from distributions that preserve sum-of-squares convexity. Thanks to the model smoothness, the mechanics of patch contact is captured while being computationally efficient without mode selection at support points. Simulation and robotic experiments on pushing and grasping validate the accuracy and efficiency of our approach.

A generalized framework for reduced‐order modeling of a wind turbine wake

AbstractA reduced‐order model for a wind turbine wake is sought from large eddy simulation data. Fluctuating velocity fields are combined in the correlation tensor to form the kernel of the proper orthogonal decomposition (POD). Proper orthogonal decomposition modes resulting from the decomposition represent the spatially coherent turbulence structures in the wind turbine wake; eigenvalues delineate the relative amount of turbulent kinetic energy associated with each mode. Back‐projecting the POD modes onto the velocity snapshots produces dynamic coefficients that express the amplitude of each mode in time. A reduced‐order model of the wind turbine wake (wakeROM) is defined through a series of polynomial parameters that quantify mode interaction and the evolution of each POD mode coefficients. The resulting system of ordinary differential equations models the wind turbine wake composed only of the large‐scale turbulent dynamics identified by the POD. Tikhonov regularization is used to recalibrate the dynamical system by adding additional constraints to the minimization seeking polynomial parameters, reducing error in the modeled mode coefficients. The wakeROM is periodically reinitialized with new initial conditions found by relating the incoming turbulent velocity to the POD mode coefficients through a series of open‐loop transfer functions. The wakeROM reproduces mode coefficients to within 25.2%, quantified through the normalized root‐mean‐square error. A high‐level view of the modeling approach is provided as a platform to discuss promising research directions, alternate processes that could benefit stability and efficiency, and desired extensions of the wakeROM.

Circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials

The quality of the approximation of circular arcs by parametric polynomials is usually measured by the Hausdorff distance. It is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property. In this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs. A general approach to this problem is considered and corresponding (nonlinear) equations are derived. For the approximants possessing the maximal order of geometric smoothness, a scalar nonlinear equation is analyzed in detail for the parabolic, the cubic and the quartic case. The existence of the admissible solution is confirmed. Moreover, the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved. Theoretical results are confirmed by numerical examples.

PPolyNets: Achieving High Prediction Accuracy and Efficiency With Parametric Polynomial Activations

Recently, to implement cloud-based machine learning approaches while maintaining data privacy, a scheme named CryptoNets is proposed to perform prediction on encrypted data using neural networks. By applying the leveled homomorphic encryption scheme, CryptoNets enables the cloud server to securely run the computation process without participations of other parties. Since the encryption scheme only supports polynomial operations, the authors simply use square activation as a substitution of the conventional activations, obtaining a relatively low prediction accuracy on the MNIST dataset (98.95%). Later work try to improve the accuracy by using polynomial approximations of the ReLU activation with large neural networks, which introduce heavy computation cost, making the prediction process impractical. In this paper, to achieve better prediction performance, we propose new parametric polynomial (PPoly) activations, which can adaptively learn the parameters during the training phase. Using our PPoly activations, we achieve higher accuracy (99.33%, 99.64%, and 99.70%) with shallow and narrow networks, guaranteeing the efficiency of prediction process. We conduct extensive experiments to show the expressiveness of our PPoly activations and discuss the tradeoff between accuracy and efficiency for the prediction on encrypted data.

Exact and Closed-Form CRLBS for High-Order Kinematic Parameters Estimation Using LFM Coherent Pulse Train

In radar signal processing, the detection and parameter estimation of high-speed maneuvering targets, which often utilize a coherent pulse train signal with linear frequency modulation, have been receiving increasing attention. Fundamentally and significantly, the Cramer–Rao lower bound (CRLB), as a cornerstone for evaluating the estimation performance of high-order kinematic parameters, has been derived and investigated here. In this paper, a 2-D echo signal model expressed in the fast-frequency and slow-time domains is adopted. The scaled orthogonal Legendre polynomials are deliberately introduced to solve the inverse problem of the Fisher information matrix, and then, a linear mapping relationship between different polynomial parameters can be used to obtain the analytical CRLB expressions. The main contributions included are: 1) the CRLBs, which are exact and closed form, have been extended to arbitrary motion model orders and reference time instants; 2) the influences of the motion model order, the reference time instant, as well as the radar parameters on the CRLBs are exploited comprehensively; and 3) some specific cases, including four low-order motion models and two preferred reference times, are also presented to better demonstrate the CRLB performance relationships. It highlights the fact that the reference time instant corresponding to the middle of the pulse train is a reasonable and compromised choice for parameter estimation, although it is not necessarily optimal for the kinematic parameters of all models and orders. The above research results are illustrated with numerical simulations and further verified using the maximum likelihood estimation method combined with Monte Carlo experiments.

Eigenvectors of Orthogonally Decomposable Functions

The eigendecomposition of quadratic forms (symmetric matrices) guaranteed by the spectral theorem is a foundational result in applied mathematics. Motivated by a shared structure found in inferential problems of recent interest---namely orthogonal tensor decompositions, independent component analysis (ICA), topic models, spectral clustering, and Gaussian mixture learning---we generalize the eigendecomposition from quadratic forms to a broad class of “orthogonally decomposable” functions. We identify a key role of convexity in our extension, and we generalize two traditional characterizations of eigenvectors: First, the eigenvectors of a quadratic form arise from the optima structure of the quadratic form on the sphere. Second, the eigenvectors are the fixed points of the power iteration. In our setting, we consider a simple first order generalization of the power method which we call gradient iteration. It leads to efficient and easily implementable methods for basis recovery. It includes influential machine learning methods such as cumulant-based FastICA and the tensor power iteration for orthogonally decomposable tensors as special cases. We provide a complete theoretical analysis of gradient iteration using the structure theory of discrete dynamical systems to show almost sure convergence and fast (superlinear) convergence rates. The analysis also extends to the case when the observed function is only approximately orthogonally decomposable, with bounds that are polynomial in dimension and other relevant parameters, such as perturbation size. Our perturbation results can be considered as a nonlinear version of the classical Davis--Kahan theorem for perturbations of eigenvectors of symmetric matrices.

Decoupled Parameter Estimation Methods for Hammerstein Systems by Using Filtering Technique

The implementation of parameter estimation of Hammerstein systems is much difficult due to the existing parameter products from the nonlinear block and the linear block. This paper directly decomposes the parameter coupling between the nonlinear part and the linear part in a Hammerstein system by using the estimated parameter polynomial of the coupled linear part to filter the Hammerstein system, transforms the Hammerstein system into two forms, and investigates two decoupled parameter estimation methods: the one-step decoupled least squares estimation method and the two-step decoupled least squares estimation method corresponding to the two forms. Furthermore, the computational complexity is compared between the proposed two estimation algorithms. The simulation results show the effectiveness of the proposed two estimation methods with a similar estimation accuracy.

Orbits of polynomial dynamical systems modulo primes

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over C \mathbb {C} . Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates.

In-Situ Systematic Error Correction for Digital Volume Correlation Using a Reference Sample

The self-heating effect of a laboratory X-ray computed tomography (CT) scanner causes slight change in its imaging geometry, which induces translation and dilatation (i.e., artificial displacement and strain) in reconstructed volume images recorded at different times. To realize high-accuracy internal full-field deformation measurements using digital volume correlation (DVC), these artificial displacements and strains associated with unstable CT imaging must be eliminated. In this work, an effective and easily implemented reference sample compensation (RSC) method is proposed for in-situ systematic error correction in DVC. The proposed method utilizes a stationary reference sample, which is placed beside the test sample to record the artificial displacement fields caused by the self-heating effect of CT scanners. The detected displacement fields are then fitted by a parametric polynomial model, which is used to remove the unwanted artificial deformations in the test sample. Rescan tests of a stationary sample and real uniaxial compression tests performed on copper foam specimens demonstrate the accuracy, efficacy, and practicality of the presented RSC method.

T1 mapping with the variable flip angle technique: A simple correction for insufficient spoiling of transverse magnetization.

The variable flip angle method derives T1 maps from radiofrequency-spoiled gradient-echo data sets, acquired with different flip angles α. Because the method assumes validity of the Ernst equation, insufficient spoiling of transverse magnetization yields errors in T1 estimation, depending on the chosen radiofrequency-spoiling phase increment (Δϕ). This paper presents a versatile correction method that uses modified flip angles α' to restore the validity of the Ernst equation. Spoiled gradient-echo signals were simulated for three commonly used phase increments Δϕ (50°/117°/150°), different values of α, repetition time (TR), T1 , and a T2 of 85 ms. For each parameter combination, α' (for which the Ernst equation yielded the same signal) and a correction factor CΔϕ (α, TR, T1 ) = α'/α were determined. CΔϕ was found to be independent of T1 and fitted as polynomial CΔϕ (α, TR), allowing to calculate α' for any protocol using this Δϕ. The accuracy of the correction method for T2 values deviating from 85 ms was also determined. The method was tested in vitro and in vivo for variable flip angle scans with different acquisition parameters. The technique considerably improved the accuracy of variable flip angle-based T1 maps in vitro and in vivo. The proposed method allows for a simple correction of insufficient spoiling in gradient-echo data. The required polynomial parameters are supplied for three common Δϕ. Magn Reson Med 79:3082-3092, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

ESTIMATING EXTERIOR ORIENTATION PARAMETERS OF HYPERSPECTRAL BANDS BASED ON POLYNOMIAL MODELS

Abstract. Hyperspectral camera operating in sequential acquisition mode produces spectral bands that are not recorded at the same instant, thus having different exterior orientation parameters (EOPs) for each band. The study presents experiments on bundle adjustment with time-dependent polynomial models for band orientation of hyperspectral cubes sequentially collected. The technique was applied to a Rikola camera model. The purpose was to investigate the behaviour of the estimated polynomial parameters and the feasibility of using a minimum of bands to estimate EOPs. Simulated and real data were produced for the analysis of parameters and accuracy in ground points. The tests considered conventional bundle adjustment and the polynomial models. The results showed that both techniques were comparable, indicating that the time-dependent polynomial model can be used to estimate the EOPs of all spectral bands, without requiring a bundle adjustment of each band. The accuracy of the block adjustment was analysed based on the discrepancy obtained from checkpoints. The root mean square error (RMSE) indicated an accuracy of 1 GSD in planimetry and 1.5 GSD in altimetry, when using a minimum of four bands per cube.

CORRECTION OF OVERLAPPING MULTISPECTRAL LIDAR INTENSITY DATA: POLYNOMIAL APPROXIMATION OF RANGE AND ANGLE EFFECTS

Abstract. Recent development of radiometric calibration, correction and normalization approaches have facilitated the use of monochromatic LiDAR intensity and waveform data for land surface analysis and classification. Despite the recent successful attempts, the majority of existing approaches are mainly tailor made for monochromatic LiDAR toward specific land surface scenario. In view of the latest development of multispectral LiDAR sensor, such as the Optech Titan manufactured by Teledyne Optech, a more generic approach should be developed so that the radiometric correction model is able to handle and compensate the laser energy loss with respect to different wavelengths. In this study, we propose a semi-physical approach that aims to utilize high order polynomial functions to model the distortion effects due to the range and the angle. To estimate the parameters of the respect polynomial functions for the range and angle, our approach first locates a pair of closest points within the overlapping LiDAR data strips and subsequently uses a non-linear least squares adjustment to retrieve the polynomial parameters based on the Levenberg-Marquardt algorithm. The approach was tested on a multispectral airborne LiDAR dataset collected by the Optech Titan for the Petawawa Research Forest located in Ontario, Canada. The experimental results demonstrated that the coefficient of variation of the intensity of channel 1 (1550 nm), channel 2 (1064 nm) and channel 3 (532 nm) were reduced by 0.1 % to 39 %, 10 % to 45 % and 12 % to 54 %, respectively. The striping noises, no matter found within single strip and overlapping strips, were significantly reduced after implementing the proposed radiometric correction.

New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type $$_4F_{3}(1)$$ . With the aid of some standard reduction formulae such as Pfaff-Saalschutz’s and Watson’s identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.

A Linear Programming-based Iterative Approach to Stabilizing Polynomial Dynamics

In this paper, we consider the problem of synthesizing static output feedback controllers for stabilizing polynomial systems. We jointly synthesize a Lyapunov function and a static output feedback controller that stabilizes the system over a given subset of the state-space. Motivated by the numerical issues that are commonly faced using SOS (Sum of Squares)/SDP (Semi-Definite Programming) solvers, we examine a linear programming (LP) based alternative approach that can yield more precise results, in practice. Our approach uses Bernstein polynomials to relax parametric polynomial optimization problems into bilinear optimization problems (BP). Subsequently, we approach the bilinear inequalities using a modified alternating minimization approach that alternates between solving linear programs on complementary sets of variables. Finally, we provide a comparison between our approach and BMI (bilinear matrix inequalities) solvers that tackle the same problem. We conclude that LP/BP relaxation approach is promising and can be more efficient than SDP/BMI relaxations.

A compact optimization model for the tail assignment problem

This paper investigates a new model for the so-called Tail Assignment Problem, which consists in assigning a well-identified airplane to each flight leg of a given flight schedule, in order to minimize total cost (cost of operating the flights and possible maintenance costs) while complying with a number of operational constraints. The mathematical programming formulation proposed is compact (i.e., involves a number of 0−1 decision variables and constraints polynomial in the problem size parameters) and is shown to be of significantly reduced dimension as compared with previously known compact models. Computational experiments on series of realistic problem instances (obtained by random sampling from real-world data set) are reported. It is shown that with the proposed model, current state-of-the art MIP solvers can efficiently solve to exact optimality large instances representing 30-day flight schedules with typically up to 40 airplanes and 1500 flight legs connecting as many as 21 airports. The model also includes the main existing types of maintenance constraints, and extensive computational experiments are reported on problem instances of size typical of practical applications.

A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus

For each irreducible module of the symmetric group $\mathcal{S}_{N}$ there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the $N$-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages, arXiv:1511.06721] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The $N$-torus is divided into $(N-1)!$ connected components by the hyperplanes $x_{i}=x_{j}$, $i<j$, which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components.