General Error Analysis Research Articles

Quantitative trait loci identification for brain endophenotypes via new additive model with random networks.

The identification of quantitative trait loci (QTL) is critical to the study of causal relationships between genetic variations and disease abnormalities. We focus on identifying the QTLs associated to the brain endophenotypes in imaging genomics study for Alzheimer's Disease (AD). Existing research works mainly depict the association between single nucleotide polymorphisms (SNPs) and the brain endophenotypes via the linear methods, which may introduce high bias due to the simplicity of the models. Since the influence of QTLs on brain endophenotypes is quite complex, it is desired to design the appropriate non-linear models to investigate the associations of genotypes and endophenotypes. In this paper, we propose a new additive model to learn the non-linear associations between SNPs and brain endophenotypes in Alzheimer's disease. Our model can be flexibly employed to explain the non-linear influence of QTLs, thus is more adaptive for the complex distribution of the high-throughput biological data. Meanwhile, as an important computational learning theory contribution, we provide the generalization error analysis for the proposed approach. Unlike most previous theoretical analysis under independent and identically distributed samples assumption, our error bound is based on m-dependent observations, which is more appropriate for the high-throughput and noisy biological data. Experiments on the data from Alzheimer's Disease Neuroimaging Initiative (ADNI) cohort demonstrate the promising performance of our approach for identifying biological meaningful SNPs. An executable is available at https://github.com/littleq1991/additive_FNNRW.

Generalized error analysis of analytical coarse alignment formulations for stationary SINS

This communication presents additional results on a recently published work of the author [1], which addressed the stationary coarse alignment (CA) stage of strapdown inertial navigation systems (SINS). As main contribution of this communication, the error analysis proposed in [1] is extended, and novel general expressions for the SINS CA errors are derived, which are valid regardless of the inertial measurement unit (IMU) orientation, and present hence, greater practical applicability. The general error expressions prove to be similar to the simplified equations derived in [1], but with body frame coordinates replaced by navigation frame coordinates. Simulation results validates the adequacy of the outlined verifications, and are in agreement with experimental results found in the literature.

Spectral iterative method and convergence analysis for solving nonlinear fractional differential equation

In this study a new spectral iterative method (SIM) based on fractional interpolation is presented for solving nonlinear fractional differential equations (FDEs) involving Caputo derivative. This method is equipped with a pre-algorithm to find the singularity index of solution of the problem. This pre-algorithm gives us a real parameter as the index of the fractional interpolation basis, for which the SIM achieves the highest order of convergence. In comparison with some recent results about the error estimates for fractional approximations, a more accurate convergence rate has been attained. We have also proposed the order of convergence for fractional interpolation error under the L2-norm. Finally, general error analysis of SIM has been considered. The numerical results clearly demonstrate the capability of the proposed method.

Error bound of magnitude-preserving ranking with Fredholm kernel

Data-dependent Fredholm kernel has attracted much attention in machine learning literatures for its flexibility to utilize the empirical information. However, the previous theoretical results are limited to the classification or density ratio estimation problems. In this paper, we extend the framework of learning with Fredholm kernel to the ranking setting. A new magnitude-preserving ranking with Fredholm kernel is proposed, and its generalization error analysis is established by using the concentrate estimate techniques. The derived result implies that the proposed method can achieve the satisfactory learning rate with polynomial decay.

“比” construction generalized error analysis and teaching method

“比” construction generalized error analysis and teaching method

Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First, we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray–Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators.

Dynamic Error Analysis Method for Vibration Shape Reconstruction of Smart FBG Plate Structure

Shape reconstruction of aerospace plate structure is an important issue for safe operation of aerospace vehicles. One way to achieve such reconstruction is by constructing smart fiber Bragg grating (FBG) plate structure with discrete distributed FBG sensor arrays using reconstruction algorithms in which error analysis of reconstruction algorithm is a key link. Considering that traditional error analysis methods can only deal with static data, a new dynamic data error analysis method are proposed based on LMS algorithm for shape reconstruction of smart FBG plate structure. Firstly, smart FBG structure and orthogonal curved network based reconstruction method is introduced. Then, a dynamic error analysis model is proposed for dynamic reconstruction error analysis. Thirdly, the parameter identification is done for the proposed dynamic error analysis model based on least mean square (LMS) algorithm. Finally, an experimental verification platform is constructed and experimental dynamic reconstruction analysis is done. Experimental results show that the dynamic characteristics of the reconstruction performance for plate structure can be obtained accurately based on the proposed dynamic error analysis method. The proposed method can also be used for other data acquisition systems and data processing systems as a general error analysis method.

Some studies on nonpolynomial interpolation and error analysis

In this paper, we propose nonpolynomial and Hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation. The generalized error analysis results for nonpolynomial and Hermite nonpolynomial interpolations are derived. We establish theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series, and propose solution to Runge’s phenomenon through nonpolynomial interpolation. The Hermite nonpolynomial cubic spline, nonpolynomial cubic spline interpolation methods and their error analysis are presented. Numerical simulations are carried out for the analysis of error in cubic spline interpolations. Proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations, and to Empirical Mode Decomposition (EMD) to illustrate practical usefulness of the results.

Error Analysis for Constrained First-Order System Least-Squares Finite-Element Methods

In this paper, a general error analysis is provided for finite-element discretizations of partial differential equations in a saddle-point form with divergence constraint. In particular, this extends upon the work of [J. H. Adler and P. S. Vassilevski, Springer Proc. Math. Statist. 45, Springer, New York, 2013, pp. 1--19], giving a general error estimate for finite-element problems augmented with a divergence constraint and showing that these estimates are obtained for problems such as diffusion and Stokes' using the first-order system least-squares (FOSLS) finite-element method. The main result is that by enforcing the constraint on a ${\mathbf H}^1$-equivalent FOSLS formulation one maintains optimal convergence of the FOSLS functional (i.e., the energy norm of the error) while guaranteeing the conservation of the divergence constraint (i.e., mass conservation in some examples). The error estimates and results depend on using finite elements for the constraint space that are inf-sup stable when paired with the spaces used for the original unknowns. This includes using discontinuous spaces on coarse meshes and pairing with standard bilinear or biquadratic elements in order to confirm the results.

Analytical Evaluation of Amplification Factors, Stability, and Error Analysis of Square Finite Element Solution for the Kinematic Wave Shallow Water Equations

AbstractA generalized error analysis is discussed in this paper to investigate solution accuracy for watershed simulation results using overland flow equations. The total or compounded error for watershed simulations may increase with increasing spatial resolution, scale, and number of iterations to convergence if inaccurate time-step is chosen for the numerical simulations. A relationship is derived between watershed scale error, overland flow plane error, finite element, and node errors. The scale versus compounded error relationship for watershed scale simulations is a function of element spatial discretization level, dynamic time-step criteria, and the relationship between element and nodal errors. The Courant number and wave number space solution is simulation optimal and would require separate solutions of Fourier-transform (and its inversion) of finite element nodal equations to obtain solution accuracy within 5%. Dynamic FEM square grid discretization criteria is derived from the results of eigenv...

Calibration for Realization Errors of Two-Channel Hybrid Filter Bank Analog-to-Digital Converters Based on Frequency-Dependent Model

Hybrid filter bank (HFB) analog-to-digital converters (ADCs) are used to achieve higher sampling rates for medium-to-high resolution applications in many communication systems. However, the analog filter realization errors and channel mismatches severely degrade the reconstruction performance of HFB ADCs. A general error analysis, which takes analog imperfections and channel mismatches into consideration based on a single-channel filtering (SCF) HFB architecture is presented in this paper. A frequency-dependent model (FDM) for sub-ADCs output is then derived to integrate all the realization errors to an equivalent channel-transfer function (ECTF). Based on the ECTF, the perfect reconstruction condition is derived as the optimization object for the synthesis filter banks. Finally, a two-channel SCF HFB ADC prototype with 12-bit resolution and 200 MHz sampling rate is implemented. After mismatch correction with 64-tap finite impulse response (FIR) filters, 75.75 dB of spurious-free dynamic range (SFDR) is achieved over the whole bandwidth. The experimental results show that the average SFDR is enhanced by about 10–25 dB compared with the traditional calibration method.

An efficient pricing method for rainbow options based on two-dimensional modified sine–sine series expansions

An efficient option pricing method based on the Fourier cosine series expansion was proposed by Fang and Oosterlee [A novel option pricing method based on Fourier-cosine series expansions, SIAM J. Sci. Comput. 31 (2008), pp. 826–848], and recently it has been used in two-dimensional cases by Ruijter and Oosterlee [Two-dimensional Fourier cosine series expansion method for pricing financial options, SIAM J. Sci. Comput. 34 (2012), pp. B642–B671]. In this paper, we consider a modification of the Fourier sine–sine expansion, whereby in [−1, 1] the sin(π nx) functions are replaced by sin(π(n−)x), n≥1, for pricing rainbow options. Feasible approaches for obtaining the pricing formulae are presented. A numerical comparison and a flexible and general error analysis are considered to prove the effectiveness of the method. Practical applications to basket options and correlation options under different models are also given. They indicate satisfactory convergence and efficiency as expected.

Error Analysis of Coefficient-Based Regularized Algorithm for Density-Level Detection

In this letter, we consider a density-level detection (DLD) problem by a coefficient-based classification framework with [Formula: see text]-regularizer and data-dependent hypothesis spaces. Although the data-dependent characteristic of the algorithm provides flexibility and adaptivity for DLD, it leads to difficulty in generalization error analysis. To overcome this difficulty, an error decomposition is introduced from an established classification framework. On the basis of this decomposition, the estimate of the learning rate is obtained by using Rademacher average and stepping-stone techniques. In particular, the estimate is independent of the capacity assumption used in the previous literature.

Propagation of Uncertainties Using Improved Surrogate Models

We study the effect of various sources of error on the propagation of uncertain parameters and data through surrogate response surfaces approximating quantities of interest from stochastic differential equations. The main result centers on a novel approach for improving the pointwise accuracy of a surrogate with the use of an adjoint-based a posteriori estimate of its error. A general error analysis on propagated distribution functions for both forward and inverse problems is derived. To provide concrete examples focusing on the use of the improved surrogate, we consider standard polynomial spectral methods to approximate the surrogate. However, neither the definition of the improved surrogate nor the general error analysis for the computed distribution functions requires a specific surrogate formulation. Numerical results comparing pointwise errors in propagated distributions using a surrogate versus its improved counterpart demonstrate global decreases in both actual error and in error bounds for both f...

General Error Analysis in the Relationship between Free Thyroxine and Thyrotropin and Its Clinical Relevance

Background. This treatise investigates error sources in measurements applicable to the hypothalamus-pituitary-thyroid (HPT) system of analysis for homeostatic set point computation. The hypothalamus-pituitary transfer characteristic (HP curve) describes the relationship between plasma free thyroxine [FT4] and thyrotropin [TSH]. Objective. We define the origin, types, causes, and effects of errors that are commonly encountered in TFT measurements and examine how we can interpret these to construct a reliable HP function for set point establishment. Design and Methods. The error sources in the clinical measurement procedures are identified and analyzed in relation to the constructed HP model. Results. The main sources of measurement and interpretation uncertainties are (1) diurnal variations in [TSH], (2) TFT measurement variations influenced by timing of thyroid medications, (3) error sensitivity in ranges of [TSH] and [FT4] (laboratory assay dependent), (4) rounding/truncation of decimals in [FT4] which in turn amplify curve fitting errors in the [TSH] domain in the lower [FT4] range, (5) memory effects (rate-independent hysteresis effect). Conclusions. When the main uncertainties in thyroid function tests (TFT) are identified and analyzed, we can find the most acceptable model space with which we can construct the best HP function and the related set point area.

SIXTH ORDER EXPLICIT EXPONENTIAL ROSENBROCK-TYPE METHODS FOR SEMILINEAR PARABOLIC PROBLEMS

The paper is concerned with the numerical analysis of high-order exponential Rosenbrock-type integrators for large-scale systems of stiff differential equations. The analysis is performed in a semigroup framework of semilinear evolution equations in Banach space. By expanding the errors of the numerical methods in terms of the solution, we further derive new order conditions and thus allows us to construct higher-order methods. A new and more general stiff error analysis is presented to show the converge results for variable step sizes.

Consistency and error analysis of Prior-Knowledge-Based Kernel Regression

Incorporating prior knowledge (PK) into learning methods is an effective means to improve learning performance. The consistency and error theories of PK-based methods, which are of great theoretical importance, are still far from well established. Concentrating on the PK-based kernel regression, this paper proposes a methodology of analyzing the consistency and error. This methodology converts the specific methods firstly to a unified optimization problem and then to a unified solution expression, and a general consistency and error analysis tool is proposed and applied. A few examples are given to illustrate the analysis procedure.

Iterative methods for use with nonlinear discrete algebraic models

A family of multi-point iterative methods for solving nonlinear equations were described in Cordero and Torregrosa (2008) [4], and a general error analysis was given, always for a simple root. Here we study the order of convergence of such methods when we have multiple roots. We prove that the order of convergence goes down to 1 but, when the multiplicity n is known, it may be raised to 2 by using different types of correction. For n unknown, we show that some methods of this family converge faster than the classical Newton’s method. In addition, we provide various numerical tests which confirm or improve on theoretical results and allow us to compare some methods of the aforementioned family.

A Fourier Transform Method for Spread Option Pricing

Spread options are a fundamental class of derivative contracts written on multiple assets and are widely traded in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient, and flexible enough to apply in general asset models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the payoff function. Our detailed investigation, including a flexible and general error analysis, proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of spread option prices. It is found to be easy to implement, stable, efficient, and applicable in a wide variety of asset pricing models.

Asymptotic analysis of Bayesian generalization error with Newton diagram

Statistical learning machines that have singularities in the parameter space, such as hidden Markov models, Bayesian networks, and neural networks, are widely used in the field of information engineering. Singularities in the parameter space determine the accuracy of estimation in the Bayesian scenario. The Newton diagram in algebraic geometry is recognized as an effective method by which to investigate a singularity. The present paper proposes a new technique to plug the diagram in the Bayesian analysis. The proposed technique allows the generalization error to be clarified and provides a foundation for an efficient model selection. We apply the proposed technique to mixtures of binomial distributions.