Error Function Research Articles

Spatiotemporal Analysis of Sonar Detection Range in Luzon Strait

Sonar serves as a critical submarine detection apparatus for naval vessels, with its detection range forming the foundation of its overall performance in underwater surveillance. The Luzon Strait, in the eastern part of the South China Sea, presents a complex hydrographic setting that profoundly influences sonar performance, necessitating mastery of the detection range variation for enhanced anti-submarine operational efficiency. This study employs the Bellhop acoustic propagation model to estimate the transmission loss. Subsequently, a detection probability integration approach is applied to determine the sonar detection range in the Luzon Strait from 2019 to 2023, which is then subjected to statistical analysis. The findings indicate the following. (1) During the summer and autumn, the shallow mixed layer fails to generate a surface duct, resulting in shorter detection ranges that are primarily dependent on the water depth. In the Shallow Water Zone (<150 m), frequent interactions between sound waves and the sea boundaries lead to considerable acoustic energy attenuation, maintaining a short detection range. In the Intermediate Depth Zone (150–2500 m), sound rays retain adequate energy post-seabed reflection, extending the sonar detection to 5–8 km. Beyond 2500 m, the diminishing reflective energy restricts the range to 2–5 km. (2) Conversely, in the winter and spring, the formation of a surface duct becomes the predominant determinant of the detection range, capable of exceeding 10 km, overshadowing the influence of the water depth.

Adaptive fixed-time tracking control for nonlinear systems subject to asymmetric input saturation

For nonlinear systems with asymmetric input saturation, an adaptive fixed-time control strategy via multi-dimensional Taylor network (MTN) is proposed to effectively solve the tracking control problem. Firstly, the Gaussian error function and the differential mean value theorem are used to simulate the asymmetric input saturation model, that is, the nonlinear model is transformed into a linear model with a bounded perturbation. Secondly, MTNs are applied to control the design process, which approximate the nonlinear structures with linear combination of polynomials. Thirdly, in addition to the tracking error converging to an arbitrarily small neighborhood around the origin within a fixed time, the proposed control scheme not only guarantees that any signal in the controlled system remains bounded, but also avoids the problem of finite-time control relying on the initial state. Finally, three examples are used to validate the feasibility and superiority of the scheme.

Two types of anti-noise integral enhanced recurrent neural networks for solving time-varying complex quadratic programming

The time-varying complex quadratic programming problem plays an important role in scientific research and engineering applications, and has received extensive attention. In this paper, by solving the problem in different number domains, two types of novel anti-noise integral enhanced recurrent neural networks (IERNNs) are proposed and analyzed. Specifically, the IERNN-A model solves the problems in the complex number domain, while the IERNN-B model solves the problems by converting the problems from the complex number domain into the real number domain. Both the IERNN-A and IERNN-B have the same design paradigm, i.e., an integral-type error function is first defined to obtain good anti-noise capability, and then substitute it into a differential-driven design formula to ensure convergence. Theory analysis and simulation results verify the global convergence and robustness of the proposed IERNNs. Furthermore, if a linear activation is used, the exponential convergence rate is obtained. Meanwhile, when time-varying bounded noise exists, the IERNNs are input-to-state stable. In addition, compared with other art-of-the-state methods, the proposed methods have faster convergence speed, stronger noise-suppression ability, and no overshoot, which are verified by the simulation and application of manipulator trajectory tracking.

Upper bound for the second and third Hankel determinants of analytic functions associated with the error function and q-convolution combination

Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class Rα,ϒλ,q(δ,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{R}_{\\alpha ,\\Upsilon}^{\\lambda ,q}(\\delta ,\\eta )$\\end{document} of analytic functions was studied: Rα,ϒλ,q(δ,η):={F:ℜ((1−δ+2η)Hϒλ,qF(ζ)ζ+(δ−2η)(Hϒλ,qF(ζ))′+ηζ(Hϒλ,qF(ζ))″)}>α(0≦α<1).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{R}_{\\alpha ,\\Upsilon }^{\\lambda ,q}(\\delta ,\\eta ) \\\\ &\\quad := \\biggl\\{ \\mathcal{ F}: {\\Re} \\biggl( (1-\\delta +2\\eta ) \\frac{\\mathcal{H}_{\\Upsilon }^{\\lambda ,q}\\mathcal{F}(\\zeta )}{\\zeta}+(\\delta -2\\eta ) \\bigl(\\mathcal{H} _{\\Upsilon}^{\\lambda ,q}\\mathcal{F}(\\zeta ) \\bigr) ^{{ \\prime}}+\\eta \\zeta \\bigl( \\mathcal{H}_{\\Upsilon}^{\\lambda ,q} \\mathcal{F}( \\zeta ) \\bigr) ^{{{\\prime \\prime}}} \\biggr) \\biggr\\} \\\\ &\\quad >\\alpha \\quad (0\\leqq \\alpha < 1). \\end{aligned}$$ \\end{document}For these general analytic functions F∈Rβ,ϒλ,q(δ,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F}\\in \\mathcal{R}_{\\beta ,\\Upsilon}^{\\lambda ,q}(\\delta , \\eta )$\\end{document}, we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.

Heuristic energy-based cyclic peptide design.

Rational computational design is crucial to the pursuit of novel drugs and therapeutic agents. Meso-scale cyclic peptides, which consist of 7-40 amino acid residues, are of particular interest due to their conformational rigidity, binding specificity, degradation resistance, and potential cell permeability. Because there are few natural cyclic peptides, de novo design involving non-canonical amino acids is a potentially useful goal. Here, we develop an efficient pipeline (CyclicChamp) for cyclic peptide design. After converting the cyclic constraint into an error function, we employ a variant of simulated annealing to search for low-energy peptide backbones while maintaining peptide closure. Compared to the previous random sampling approach, which was capable of sampling conformations of cyclic peptides of up to 14 residues, our method both greatly accelerates the computation speed for sampling conformations of small macrocycles (ca. 7 residues), and addresses the high-dimensionality challenge that large macrocycle designs often encounter. As a result, CyclicChamp makes conformational sampling tractable for 15- to 24-residue cyclic peptides, thus permitting the design of macrocycles in this size range. Microsecond-length molecular dynamics simulations on the resulting 15, 20, and 24 amino acid cyclic designs identify trajectories with kinetic stability. To test their thermodynamic stability, we perform additional replica exchange molecular dynamics simulations and generate free energy surfaces. Two 15-residue designs and one 20-residue design emerge as promising candidates, along with one viable 24-residue candidate.

Efficient Aperture Fill Time Correction for Wideband Sparse Array Using Improved Variable Fractional Delay Filters.

To solve the problem of aperture fill time (AFT) for wideband sparse arrays, variable fractional delay (VFD) FIR filters are applied to eliminate linear coupling between spatial and time domains. However, the large dimensions of the filter coefficient matrix result in high system complexity. To alleviate the computational burden of solving VFD filter coefficients, a novel multi-regultion minimax (MRMM) model utilizing the sparse representation technique has been presented. The error function is constrained by the introduction of L2-norm and L1-norm regularizations within the minimax criterion. The L2-norm effectively resolves the problems of overfitting and non-unique solutions that arise in the sparse optimization of traditional minimax (MM) models. Meanwhile, the use of multiple L1-norms enables the optimal design of the smallest sub-filter number and order of the VFD filter. To solve the established nonconvex model, an improved sequential-alternating direction method of multipliers (S-ADMM) algorithm for filter coefficients is proposed, which utilizes sequential alternation to iteratively update multiple soft-thresholding problems. The experimental results show that the optimized VFD filter reduces system complexity significantly and corrects AFT effectively in a wideband sparse array.

A Note on Rounding Error in The Variables: Understanding Its Influence on Statistical Estimators

When estimating the variance, higher moments, and regression coefficients using rounded data, bias is commonly introduced into these estimates. Sheppard’s correction can be used to approximately adjust for this bias under certain conditions. This study investegated the conditions that allow for the valid application of Sheppard’s corrections. A significant observation from the research was that the rounding error-often displaying a uniform distribution-was not independent of the original, unrounded variable. The study provided a thorough investigation of instances where variables, distributed uniformly, undergo rounding. This investigation yielded crucial insights into the statistical behavior and potential ramifications of such variables within statistical analyses. An integral part of this study was the computation and discussion of the density function of the rounding error. Deriving this density function facilitated a nuanced analysis of the differences between the moments of rounded data and the original, unrounded data. Sheppard’s regularities, under which Sheppard’s correction is applied, were discussed. The letter also carefully considered the effects of rounding error on regression estimates. In this context, Sheppard’s correction was implemented to the biased regression parameters to evaluate its efficacy in providing more accurate estimations. Further, our research provides new insights into how rounding can affect statistical outcomes, depending on the underlying distribution of the data. We have discovered that the impact is minimal under uniform distribution assumptions but becomes significantly pronounced under non-uniform distributions. This highlights the importance of considering the distributional characteristics of data when applying rounding corrections to ensure the reliability of statistical inferences.

Robust hyperspectral unmixing using weighted group sparsity and minimum volume regularization

ABSTRACT Hyperspectral unmixing (HU) has been a vital technique in hyperspectral image processing. Over the past years, many algorithms have been developed based on the nonnegative matrix factorization (NMF) framework. However, the conventional methods based on the NMF need prior information of the number of endmembers, which is usually estimated by a separate procedure. Therefore, the precision of estimating the number of endmembers will impact the subsequent unmixing effect. Thus, collaborating between the two processes is desired to avoid the propagation of errors. This paper proposes a new NMF-based method, which can estimate the number of endmembers and extract the endmembers and the abundance maps. Firstly, we introduce the weighted group sparsity regularization to adaptively eliminate the abundances corresponding to the redundant endmembers, while the abundances of the true endmembers are preserved. Then the number of endmembers can be determined by the reconstruction error function and the l 2 -norm of the abundance vectors. Secondly, since the simplex determined by the endmembers has the minimum volume among all simplexes that circumscribe the data points, we impose the total variational regularization to force the endmembers to be close to each other to minimize the simplex volume. And because the total variational regularization avoids the use of noisy pixels, it can decrease the influence of noise and improve the robustness of the solution. For the proposed model, we design an effective algorithm to solve it. Also, the experimental results based on the synthetic data and real data demonstrate that the proposed method can identify the endmembers number well and achieve robust unmixing results.

Photodegradation, kinetics and non-linear error functions of methylene blue dye using SrZrO3 perovskite photocatalyst

The degradation of methylene blue dye-contaminated wastewater via photocatalysis is an efficient approach towards environmental remediation. The SrZrO3 perovskite photocatalyst was synthesized using the modified Pechini sol-gel method, and characterized using XRD, FESEM, FTIR, and UV–visible spectrophotometer. Crystallite size obtained by the Scherrer and Williamson-Hall methods were 45.56 and 44.50 nm, respectively. The sample exhibited an orthorhombic crystal structure. The optical bandgap was estimated to be 5.31 eV. Kinetic study for the degradation of methylene blue dye using SrZrO3 in the presence of H2O2 showed complete decontamination of the methylene blue dye in the time interval of 90–120 min under visible light irradiation. The degradation efficiency was above 80.1 % under illumination and less than 40.1 % in dark. Kinetic studies were performed by varying the dose of photocatalyst and initial concentration of methylene blue. It was observed that higher dose of the photocatalyst and lower concentration of the contaminant produced higher rate of degradation. The solution pH 3 and catalysts dosage of 200 mg yielded the highest rate of degradation. The experimental data fitted best into the Psuedosecond order kinetic model with the highest R2 value, indicating a strong linear relationship. The experimental data was subjected into nonlinear error functions, where the Psuedo-second order kinetic model demonstrated lower values of error functions. The results suggest that the prepared SrZrO3 photocatalyst coupled H2O2 is a promising photocatalyst for the decontamination of methylene blue dye under visible light irradiation.

Three-dimensional line-of-sight-angle-constrained leader-following cooperative interception guidance law with prespecified impact time

To address the problem of multi-missile cooperative interception against maneuvering targets at a prespecified impact time and desired Line-of-Sight (LOS) angles in Three-Dimensional (3D) space, this paper proposes a 3D leader-following cooperative interception guidance law. First, in the LOS direction of the leader, an impact time-controlled guidance law is derived based on the fixed-time stability theory, which enables the leader to complete the interception task at a prespecified impact time. Next, in the LOS direction of the followers, by introducing a time consensus tracking error function, a fixed-time consensus tracking guidance law is investigated to guarantee the consensus tracking convergence of the time-to-go. Then, in the direction normal to the LOS, by combining the designed global integral sliding mode surface and the second-order Sliding Mode Control (SMC) theory, an innovative 3D LOS-angle-constrained interception guidance law is developed, which eliminates the reaching phase in the traditional sliding mode guidance laws and effectively saves energy consumption. Moreover, it effectively suppresses the chattering phenomenon while avoiding the singularity issue, and compensates for unknown interference caused by target maneuvering online, making it convenient for practical engineering applications. Finally, theoretical proof analysis and multiple sets of numerical simulation results verify the effectiveness, superiority, and robustness of the investigated guidance law.

Probabilistic Corrosion-free Service Life of the RCC Structures Made with OPC and Ultra-fine Slag-based Binders

The primary factor in the deterioration of reinforced concrete is chloride-induced corrosion. It has a direct impact on the service life of the structure and has been the subject of much research over the past 5 decades. One of the most cost-effective ways to lessen chloride-induced corrosion was identified to be the use of more supplementary cementitious material (SCM). Moreover, when SCMs are ground down to a smaller size, the resulting particles serve as microfillers, which have a significant impact on the improvement of mechanical properties and durability. This paper highlights the influence of grinding slag into an Ultra Fine Slag (UFS) on chloride ion penetration in the mortar specimen. A bulk diffusion test was performed on a mortar replaced with UFS which show a synergic reduction in chloride diffusivity in comparison with control mixes. In addition, an attempt is made on prediction of probabilistic service life estimation by Error function model.

Research on signal equalization technology based on visible light communication

Abstract Visible Light Communication (VLC) transmits signals through visible light and realizes the two functions of lighting and communication at the same time without the need for other hardware equipment. There are still some shortcomings in the visible light communication system. The optical signal propagated indoors is affected by inter-symbol interference (ISI) and external light noise at the receiving end. The signal will be severely distorted. These factors will significantly reduce the accuracy of the received signal decision, thereby seriously affecting the performance of the visible light communication system. Given the problems of intersymbol interference and optical noise in indoor visible light systems, this article studies adaptive equalization technology and modulation and demodulation technology suitable for visible light communication systems and improves the signal-receiving end equalization. Based on the existing equalizer structure, we combined the advantages of two different equalizers and optimized their combined structure parameters. On this basis, we added the MAP algorithm and proposed a hyperbolic tangent function to improve the traditional error function. The defects speed up the convergence speed without changing the mean square error. Experimental results show that the algorithm has improved in convergence performance. The algorithm’s convergence speed is 53.6% higher than the traditional algorithm, the steady-state error is reduced by about 6 dB, and the steady-state error of the algorithm is reduced by about 5 dB respectively.

Analysis of Tapered Fibre Optic Surface Plasmon Resonance Bio-Sensor Chip With Highly Perturbed Taper Profiles.

A numerical model based on the Transfer matrix method (TMM) is proposed for the first time to study the gold coated tapered fibre optic surface plasmon resonance (SPR) with eight different types of taper profiles namely linear, exponential-linear, Gaussian, quadratic, sinusoidal, error function type and highly perturbed taper profile so-called chirp type of profile. The performance in terms of sensitivity, full width at half maximum (FWHM), detection accuracy (D.A.), amplitude dip, and half power points are estimated with respect to tapering ratio and choices of taper profile. It is found that sensitivity increased almost linearly with the taper ratio of each taper choice for the account of the reduction of detection accuracy. It has been found that sensitivity is highest for the case of chirp taper profile and lowest for the case of quadratic taper profile at low taper ratio. In this study, the aqueous solution is considered for sensor development which is adulterated by biomolecules species like DNA, blood samples, etc.

High-order allpass cascade-connected phase-system design utilizing fractional polynomial errors

ABSTRACT This paper proposes a nonlinear optimization method for designing a high-order recursive allpass digital phase system that comprises several cascade-connected second-order (SO) allpass sections. This method first splits a given phase design specification (approximation target) into a set of equal sub-specifications, and then the equal sub-specification acts as the ideal phase for each of the SO sections. Since an SO section can be designed via employing a linear programming to minimize a linear fractional error function, this splitting process generates an excellent starting point (good initial values for the coefficients of the cascaded SO sections) for further minimizing the overall approximation error. This paper first derives a wide variety of fractional polynomial error functions for designing cascade-connected allpass phase systems of different high orders, and then shows that a high-order cascade-connected allpass phase system can be designed by utilizing a nonlinear solver to minimize the peak deviation of such a fractional polynomial error. Using the derived fractional polynomial error functions in the system design enables the system designer to find an excellent starting point for further minimizing the approximation error and thus reach a convergent design solution. This paper first derives a set of fractional polynomial errors, and then adopts the eighth-order cascade-connected phase system design to exemplify the effectiveness of the fractional-polynomial-error-based design tactic. Furthermore, the stability of the cascade-connected high-order allpass system can be readily checked via confirming that all of the cascaded SO sections have their poles inside the unit circle in the complex plane.

Mean Absolute Directional Loss as a new loss function for machine learning problems in algorithmic investment strategies

This paper investigates the issue of an adequate loss function in the optimization of machine learning models used in the forecasting of financial time series for the purpose of algorithmic investment strategies (AIS) construction. We propose the Mean Absolute Directional Loss (MADL) function, solving important problems of classical forecast error functions in extracting information from forecasts to create efficient buy/sell signals in algorithmic investment strategies. MADL places appropriate emphasis not only on the quality of the point forecast but also on its impact on the rate of achievement by the investment system based on it. The introduction and detailed description of the theoretical properties of this new MADL loss function are our main contributions to the literature. In the empirical part of the study, based on the data from two different asset classes (cryptocurrencies: Bitcoin and commodities: Crude Oil), we show that our new loss function enables us to select better hyperparameters for the LSTM model and obtain more efficient investment strategies, with regard to risk-adjusted return metrics on the out-of-sample data.

Adaptive Weighted Data Fusion for Line Structured Light and Photometric Stereo Measurement System.

Line structured light (LSL) measurement systems can obtain high accuracy profiles, but the overall clarity relies greatly on the sampling interval of the scanning process. Photometric stereo (PS), on the other hand, is sensitive to tiny features but has poor geometrical accuracy. Cooperative measurement with these two methods is an effective way to ensure precision and clarity results. In this paper, an LSL-PS cooperative measurement system is brought out. The calibration methods used in the LSL and PS measurement system are given. Then, a data fusion algorithm with adaptive weights is proposed, where an error function that contains the 3D point cloud matching error and normal vector error is established. The weights, which are based on the angles of adjacent normal vectors, are also added to the error function. Afterward, the fusion results can be obtained by solving linear equations. From the experimental results, it can be seen that the proposed method has the advantages of both the LSL and PS methods. The 3D reconstruction results have the merits of high accuracy and high clarity.

Соотношения для оценки ошибок установки углового положения в матричных имитаторах радиоэлектронной обстановки

Simplified formulas have been obtained to estimate the error in setting the angular position of the apparent center of radiation in a matrix simulator. The approximation of the error function is used for this purpose. It is shown that the well-known formula for calculating the error in simula¬ting radar reflections from point targets, when used to find the statistical characteristics of noise coordinates of distributed targets, requires simplifications. In this case, the error function can be divided into four sections, within which its approximation by polynomials of the first or second degree is allowed. At the same time, the error function has an oblique symmetry, which makes it possible to analyze only two sections, and to extend the results to four. The numerical enumeration method is used as a way to find an optimal division into sections. For the typical shapes of the main lobe of the directional pattern of the radar antenna, the coefficients of the approximating polynomials were obtained by the Nelder–Meade simplex method. It is shown that the use of approximation makes it possible to obtain analytical expressions for the moments of error in setting the angular position of the apparent radiation center. In particular, the relations for finding the mathematical expectation of an error are given.

Analysis of Fractional Advection Equation with Improved Homotopy Analysis Method

The enhanced homotopy analysis method was employed to solve the nonlinear time-fractional advection equation, resulting in the derivation of a series of solutions. The objective of this study is to minimize the absolute error by identifying the ideal value for a certain parameter, indicated as h, by the utilization of the residual error (RE) function associated with that parameter. The 3-dimensional graphs illustrating the absolute discrepancies between the solutions obtained from the exact approach and the modified homotopy analysis method have been made using the MAPLE software. Hence, the proposed approach demonstrates efficacy and appropriateness in tackling fractional partial differential equations.

Oscillation Times in Water Hammer Signatures: New Insights for the Evaluation of Diversion Effectiveness in Field Cases

Diversion is a crucial technique for effectively improving shale reservoir production by creating more complex fracture networks. Evaluating diversion effectiveness is necessary to optimize the parameters in hydraulic fracturing. Water hammer diagnostics, an emerging fracturing diagnosis technique, evaluate diversion effectiveness by analyzing water hammer signals. The water hammer attenuation, as indicated by the oscillation time, correlates with the complexity of fracture networks. However, it remains unclear whether the oscillation time is associated with diversion effectiveness. This paper elucidates the relationship between the water hammer oscillation time and diversion effectiveness by taking the probability of diversion and the treating pressure response as the evaluation criteria. Initially, a high-frequency pressure sensor was installed at the wellhead to sample the water hammer signals. Next, the oscillation times were determined using the feature extraction method. Simultaneously, the probability of diversion and the treating pressure response were calculated using the cepstrum error function and treating pressure curve, respectively. Then, the relationship between the oscillation time and diversion effectiveness was analyzed. Finally, a rapid judgment method for evaluating diversion effectiveness based on the water hammer oscillation time was proposed. The results indicated a negative correlation between the probability of diversion and the oscillation time, with higher probabilities resulting in lower oscillation times. The oscillation times exhibited a negative correlation with the treating pressure response, including the treating pressure increases and diversion pressure spikes, wherein a greater pressure differential led to lower oscillation times. Drawing from the statistics of a shale gas horizontal well in Sichuan, a better diversion effectiveness is associated with fewer oscillations, demonstrating a negative correlation between the diversion effectiveness and the oscillation time in water hammer signatures. Finally, a rapid judgment method for evaluating diversion effectiveness was proposed, utilizing the 95% confidence interval of the mean oscillation time. This paper offers useful insights into evaluating diversion performance in field cases.

Parameter tuning in the radial kernel-based partition of unity method by Bayesian optimization

In this paper, we employ Bayesian optimization to concurrently explore the optimal values for both the shape parameter and the radius in the partition of unity interpolation using radial basis functions. Bayesian optimization is a probabilistic, iterative approach that models the error function through a progressively self-updated Gaussian process. Meanwhile, the partition of unity approach harnesses a meshfree method, allowing us to significantly reduce computational expenses, particularly when considering a substantial number of scattered data points. This reduction in computational cost is achieved by decomposing the entire domain into several smaller subdomains, each of them with a variable radius. We provide an estimation of the complexity of our algorithm and carry out numerical experiments to illustrate the effectiveness of our approach, dealing with test and real-world datasets.