Terms Of Error Research Articles

Prime and Möbius correlations for very short intervals in $\fq[x

abstract: We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):=\{f(x) + a : a \in\fp\}$ for $f(x)\in\fp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\"{o}bius $\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M\"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\"{o}bius sums is {\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\"{o}bius $\mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)\in\fq[x]$ and intervals of the form $f(x)+a$ for $a\in\fq$, where $p$ is fixed and $q=p^{l}$ grows.

Rough solutions of the relativistic Euler equations

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory.

Asymptotic Completeness for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in Yu (Modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition, 2021). Starting from a global solution u u to the quasilinear wave equation, we rigorously show that well chosen asymptotic variables solve the same reduced system with small error terms. This allows us to recover the scattering data for our system, as well as to construct a matching exact solution to the reduced system.

Novel hybrid data-driven modeling integrating variational modal decomposition and dual-stage self-attention model: Applied to industrial petrochemical process

Data-driven modeling methods are extensively employed in the vast landscape of petrochemical industries. Due to their accuracy and rapidity, these models play a guiding role in the simulation and control of industrial petrochemical processes, such as the fluid catalytic cracking (FCC) process. However, in industrial process, there is a high degree of nonlinearity between influencing variables and predictive variables, making it challenging for models to accurately capture the relationship. Additionally, there are temporal and spatial characteristics, which conventional models struggle to learn. To tackle this challenge, a novel hybrid data-driven modeling is proposed, which combines variational mode decomposition (VMD) and dual-stage attention long short-term memory (DA-LSTM), incorporating error compensation (EC). VMD decomposes the predictive variables into multiple components to alleviate the nonlinear between the influencing and predictive variables. Each component is predicted by the DA-LSTM model, a dual-stage self-attention model that incorporates feature and time attention mechanisms. The EC method predicts the error term, primarily originating from residuals in the VMD decomposition and inaccuracies in the predictions made by DA-LSTM. The superiority of the proposed model is verified by the prediction of industrial data in FCC, Tennessee Eastman and debutanizer column processes.

Exploiting residual errors in nonlinear online prediction

We introduce a novel online (or sequential) nonlinear prediction approach that incorporates the residuals, i.e., prediction errors in the past observations, as additional features for the current data. Including the past error terms in an online prediction algorithm naturally improves prediction performance significantly since this information is essential for an algorithm to adjust itself based on its past errors. These terms are well exploited in many linear statistical models such as ARMA, SES, and Holts-Winters models. However, the past error terms are rarely or in a certain sense not optimally exploited in nonlinear prediction models since training them requires complex nonlinear state-space modeling. To this end, for the first time in the literature, we introduce a nonlinear prediction framework that utilizes not only the current features but also the past error terms as additional features, thereby exploiting the residual state information in the error terms, i.e., the model’s performance on the past samples. Since the new feature vectors contain error terms that change with every update, our algorithm jointly optimizes the model parameters and the feature vectors simultaneously. We achieve this by introducing new update equations that handle the effects resulting from the changes in the feature vectors in an online manner. We use soft decision trees and neural networks as the nonlinear prediction algorithms since these are the most widely used methods in highly publicized competitions. However, as we show, our methods are generic and any algorithm supporting gradient calculations can be straightforwardly used. We show through our experiments on the well-known real-life competition datasets that our method significantly outperforms the state-of-the-art. We also provide the implementation of our approach including the source code to facilitate reproducibility (https://github.com/ahmetberkerkoc/SDT-ARMA).

Bond behavior of deformed rebar in concrete containing recycled brick aggregate and waste rubber

The reuse of aggregates and crumb rubber (CR) in concrete is gaining interest globally for reducing the adverse effects on the environment and maintaining construction sustainability. The bond strength of reinforcing rebars in such concrete is one of the key issues to be addressed before their practical application, particularly in structural elements. Pull-out tests were performed on 80 specimens to assess the bond performance of deformed rebars in rubberized recycled brick aggregate concrete (RRBAC). After investigating the strength properties of RRBAC, pull-out tests were performed on two sizes of cylindrical specimens. RBA%, CR%, rebar diameter (d), embedment length (le), and concrete cover-to-rebar diameter ratio (c/d) were among the test variables in this bond behavior investigation. Bond stress-slip relationships were obtained, examined, and average bond strengths were compared with the available codes. Concrete split failure was observed in 79% of the tested specimens, whereas rebar fracture was not seen in any of the tests. The bond strength was found to be decreased with the inclusion of RBAs and CRs though relative improvement was observed for higher percentage RBAs. Additionally, regression analysis was performed on the experimental data to produce a closed-form equation for forecasting the bond strength for RRBAC. With a strong coefficient of determination (R2 = 0.97) and low values of error terms, the predicted bond strengths were seen to be a good agreement with the test results.

Analytical Explanations of the Concept of the Fixed Term and the Random Error Term in Econometric Models

The error term or disturbance term plays an important role in the econometric model, and there are several reasons for including it in the econometric model, including issues related to measurement problems, the omission of variables from the econometric model, or for contingency reasons or related to abnormal human behavior from the general rule, or for inaccurate description of the models or inaccurate preparation of the data. The fixed term or intercept also has different meanings in the econometric model, such as the mathematical meaning, the econometric meaning, the fixed term without meaning, or the fixed term with a negative sign, and others.

Zeros of higher derivatives of Riemann zeta function

In this article, we extend the result of Conrey [4, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of ζ(k). Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's ηk-function cluster near the critical line.

Type I Tobit Bayesian Additive Regression Trees for censored outcome regression

Censoring occurs when an outcome is unobserved beyond some threshold value. Methods that do not account for censoring produce biased predictions of the unobserved outcome. This paper introduces Type I Tobit Bayesian Additive Regression Tree (TOBART-1) models for censored outcomes. Simulation results and real data applications demonstrate that TOBART-1 produces accurate predictions of censored outcomes. TOBART-1 provides posterior intervals for the conditional expectation and other quantities of interest. The error term distribution can have a large impact on the expectation of the censored outcome. Therefore, the error is flexibly modeled as a Dirichlet process mixture of normal distributions. An R package is available at https://github.com/EoghanONeill/TobitBART.

Bayesian Estimation of Simultaneous Regression Quantiles Using Hamiltonian Monte Carlo

The simultaneous estimation of multiple quantiles is a crucial statistical task that enables a thorough understanding of data distribution for robust analysis and decision-making. In this study, we adopt a Bayesian approach to tackle this critical task, employing the asymmetric Laplace distribution (ALD) as a flexible framework for quantile modeling. Our methodology implementation involves the Hamiltonian Monte Carlo (HMC) algorithm, building on the foundation laid in priorwork, where the error term is assumed to follow an ALD. Capitalizing on the interplay between two distinct quantiles of this distribution, we endorse a straightforward and fully Bayesian method that adheres to the non-crossing property of quantiles. Illustrated through simulated scenarios, we showcase the effectiveness of our approach in quantile estimation, enhancing precision via the HMC algorithm. The proposed method proves versatile, finding application in finance, environmental science, healthcare, and manufacturing, and contributing to sustainable development goals by fostering innovation and enhancing decision-making in diverse fields.

The effect of policy uncertainty on the volatility of bitcoin

PurposeThis paper aims to investigate the effect of the economic policy uncertainty (EPU), geopolitical risk (GPR) and climate policy uncertainty (CPU) of USA on Bitcoin volatility from August 2010 to August 2022.Design/methodology/approachIn this paper, the authors have adopted the empirical strategy of Yen and Cheng (2021), who modified volatility model of Wang and Yen (2019), and the authors use an OLS regression with Newey-West error term.FindingsThe results using OLS regression with Newey–West error term suggest that the cryptocurrency market could have hedge or safe-haven properties against EPU and geopolitical uncertainty. While the authors find that the CPU has a negative impact on the volatility of the bitcoin market. Hence, the authors expect climate and environmental changes, as well as indiscriminate energy consumption, to play a more important role in increasing Bitcoin price volatility, in the future.Originality/valueThis study has two implications. First, to the best of the authors’ knowledge, the study is the first to extend the discussion on the effect of dimensions of uncertainty on the volatility of Bitcoin. Second, in contrast to previous studies, this study can be considered as the first to examine the role of climate change in predicting the volatility of bitcoin. This paper contributes to the literature on volatility forecasting of cryptocurrency in two ways. First, the authors discuss volatility forecasting of Bitcoin using the effects of three dimensions of uncertainty of USA (EPU, GPR and CPU). Second, based on the empirical results, the authors show that cryptocurrency can be a good hedging tool against EPU and GPR risk. But the cryptocurrency cannot be a hedging tool against CPU risk, especially with the high risks and climatic changes that threaten the environment.

The error term of the sum of digital sum functions in arbitrary bases

Let $k$ be a non-negative integer and $q > 1$ be a positive integer. Let $s_q(k)$ be the sum of digits of $k$ written in base $q.$ In 1940, Bush proved that $A_q(x)=\sum_{k \leq x} s_q (k)$ is asymptotic to $\frac{q-1}{2}x \log_q x.$ In 1968, Trollope proved an explicit formula for the error term of $A_2(n-1),$ labeled by $-E_2(n),$ where $n$ is a positive integer. In 1975, Delange extended Trollope's result to an arbitrary base $q$ by another method and labeled the error term $nF_q(\log_q n).$ When $q=2,$ the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope's method to extend $-E_2(n)$ to $-E_q(n)$ with a general base $q,$ and we also proved directly that $nF_q(\log_q n)$ and $-E_q(n)$ are equal for any $q.$

Integrated deviance information criterion for spatial autoregressive models with heteroskedasticity

In this study, we introduce the integrated deviance information criterion (DIC) for nested and non-nested model selection problems in heteroskedastic spatial autoregressive models. In a Bayesian estimation setting, we assume that the idiosyncratic error terms of our spatial autoregressive model have a scale mixture of normal distributions, where the scale mixture variables are latent variables that induce heteroskedasticity. We first derive the integrated likelihood function by analytically integrating out the scale mixture variables from the complete-data likelihood function. We then use the integrated likelihood function to formulate the integrated DIC measure. We investigate the finite sample performance of the integrated DIC in selecting the true model in a simulation study. The simulation results show that the integrated DIC performs satisfactorily and can be useful for selecting the correct model in specification search exercises. Finally, in a spatially augmented economic growth model, we use the integrated DIC to choose the spatial weights matrix that leads to better predictive accuracy.

Accurate Range Modeling for High-Resolution Spaceborne Synthetic Aperture Radar.

Spaceborne synthetic aperture radar (SAR) is an advanced microwave imaging technology that provides all-weather and all-day target information. However, as spaceborne SAR resolution improves, traditional echo signal models based on airborne SAR design become inadequate due to the curved orbit, Earth rotation, and increased propagation distance. In this study, we propose an accurate range model for high-resolution spaceborne SAR by analyzing motion trajectory and Doppler parameters from the perspective of the space geometry of spaceborne SAR. We evaluate the accuracy of existing range models and propose an advanced equivalent squint range model (AESRM) that accurately fits the actual range history and compensates for high-order term errors by introducing third-order and fourth-order error terms while maintaining the simplicity of the traditional model. The proposed AESRM's concise two-dimensional frequency spectrum form facilitates the design of imaging algorithms. Point target simulations confirm the effectiveness of the proposed AESRM, demonstrating significant improvements in fitting accuracy for range histories characterized by nonlinear trajectories. The developed AESRM provides a robust foundation for designing imaging algorithms and enables higher resolution and more accurate radar imaging.

Quasi-Interpolation on Chebyshev Grids with Boundary Corrections

Quasi-interpolation is a powerful tool for approximating functions using radial basis functions (RBFs) such as Gaussian kernels. This avoids solving large systems of equations as in RBF interpolation. However, quasi-interpolation with Gaussian kernels on compact intervals can have significant errors near the boundaries. This paper proposes a quasi-interpolation method with Gaussian kernels using Chebyshev points and boundary corrections to improve the approximation near the boundaries. The boundary corrections use a linear approximation of the function beyond the interval to estimate the truncation error and add correction terms. Numerical studies on test functions show that the proposed method reduces errors significantly near boundaries compared to quasi-interpolation without corrections, for both equally spaced and Chebyshev points. The convergence and accuracy with the boundary corrections are generally better with Chebyshev points compared to equally spaced points. The proposed method provides an efficient way to perform quasi-interpolation on compact intervals while controlling the boundary errors. This study introduces a novel approach to quasi-interpolation modification, which significantly enhances convergence rates and minimizes errors at boundary points, thereby advancing the methods for boundary approximation.

Enhanced predictive optimization of methane dry reforming via ResponseSurface methodology and artificial neural network approaches: Insights using a novel nickel-strontium-zirconium-aluminum catalyst

This study investigates the molecular dynamics of methane dry reforming catalyzed by a novel nickel-strontium-zirconium-aluminum (5Ni+3Sr/10Zr+Al) catalyst, leveraging both Response Surface Methodology (RSM) and Radial Basis Function Neural Network (RBFNN) for predictive optimization. Focusing on the impact of operational parameters—hourly space velocity, reaction temperature, and CO2:CH4 mole ratio—on the conversion rates and formation of reaction components, we aim to predict optimal conditions and corresponding process variables. Through a comparison of a three-layer Feed Forward Neural Network, optimized at a 3:10:1 topology, with traditional RSM approaches, our findings highlight the superior predictive capabilities of ANN models. Notably, ANN demonstrated exceptional performance with Radj2and F_Ratio values significantly surpass those of RSM, alongside lower statistical error terms. This superiority is attributed to ANN's robust handling of nonlinear relationships between inputs and outputs, asserting its potential for enhancing predictive accuracy in chemical process optimization. At optimum predicted conditions like 1 CH4/CO2,750 °C reaction temperature, 12000 cm3g−1h−1 space velocity, NiSrZrAl outperformed with > 85 % CH4 and CO2 conversion with H2/CO ∼1 up to 20 h time on stream. Our research underscores the importance of integrating advanced modeling techniques for the efficient and accurate prediction of catalytic reactions, offering valuable insights for future applications in chemical engineering and catalysis.

A Factor‐Augmented New Keynesian Phillips Curve for the European Union Countries

AbstractIn this paper, a factor‐augmented version of the New Keynesian Phillips curve (NKPC) is assessed using a data set comprised of a large panel of European Union (EU) member countries. The factor‐augmentation is natural given that country‐level inflation rates are highly co‐moving. The presence of unattended common factors is important because it raises the issue of omitted variables bias, as the real marginal cost, which is a regressor of the NKPC, is likely to load on the same factors as inflation. One possibility here is to employ the regular instrumental variables approach. However, if the external instruments load on the same factors as the error term of the NKPC, the instruments would be invalid and the results would therefore likely be misleading. Motivated by this last observation, the present paper proposes a new estimator of the NKPC that allows for very general forms of factor dependencies and endogeneity. Our results provide evidence in support of the NKPC, but only after the presence of common factors has been appropriately accounted for.

Simulation Analysis of Phase Jitter in Differential Sampling of AC Waveforms Based on the Programmable Josephson Voltage Standard

The effect of phase jitter on differential sampling using the programmable Josephson voltage standard (PJVS) system is studied in this paper. A phase jitter model is established for the measured signal, and compensation coefficients for phase jitter removal are derived for three different post-processing methods based on the discrete Fourier transform algorithm (DFT). Based on our analysis, the phase jitter compensation coefficients are determined by the phase jitter angle distribution and harmonic order. Furthermore, after analyzing and simulating various common distributions, the phase jitter compensation coefficients have been verified. The simulation shows that when the standard deviation of the phase jitter angle is 20 ns, and the frequency of the measuring waveform is 3.46 kHz, the influence of the phase jitter is 1 × 10−7. The results of the simulation indicate that, in the differential sampling of AC waveforms using a PJVS system, phase jitter is one of the error terms for an uncertainty budget that cannot be neglected, particularly as the frequency of the measured waveforms increases.

Testing for homogeneous treatment effects in linear and nonparametric instrumental variable models

. The hypothesis of homogeneous treatment effects is central to the instrumental variables literature. This assumption signifies that treatment effects are constant across all subjects. It allows to interpret instrumental variable estimates as average treatment effects over the whole population of the study. When this assumption does not hold, the bias of instrumental variable estimators can be greater than that of naive estimators ignoring endogeneity. This article develops two tests for the assumption of homogeneous treatment effects when the treatment is endogenous and an instrumental variable is available. The tests leverage a covariable that is (jointly with the error terms) independent of a coordinate of the instrument. This covariate does not need to be exogenous. The first test assumes that the potential outcomes are linear in the regressors and is computationally simple. The second test is nonparametric and relies on Tikhonov regularization. The treatment can be either discrete or continuous. We show that the tests have asymptotically correct level and asymptotic power equal to one against a range of alternatives. Simulations demonstrate that the proposed tests attain excellent finite sample performances. The methodology is also applied to the evaluation of returns to schooling and demand estimation in a fish market.

Convergence analysis of a second order numerical scheme for the Flory–Huggins–Cahn–Hilliard–Navier–Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) error analysis for the phase variable and the ℓ∞(0,T;ℓ2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn–Hilliard–Navier–Stokes system with a singular energy potential.