Higher-order Moments Research Articles

Bitcoin halving and the integration of cryptocurrency and forex markets: An analysis of the higher-order moment spillovers

This paper introduces a flexible semi-nonparametric model, the so-called SNP-VSK model, to analyze the higher-order moment interactions among cryptocurrencies and the forex markets, i.e., accounting for conditional volatility, skewness, and kurtosis spillovers. These transmissions among cryptos are identified considering Bitcoin halving as a factor of price uncertainty. To study the degree of integration of cryptocurrency markets in the economy, spillover effects with the forex market are considered, showing that Bitcoin and Ethereum are the prime receivers of positive volatility transmission, a sign of integration with the currency markets, but also potential sources of financial instability.

On the Averaging and Closure of Fiber Orientation Tensors in Virtual Process Chains

Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space.

Evaluation of stochastic particle Bhatnagar–Gross–Krook methods with a focus on velocity distribution function

For precise application of Bhatnagar–Gross–Krook (BGK) methods, assessing its accuracy in non-equilibrium flows is necessary. Generally, this assessment relies on macroscopic parameters, which are moments of the velocity distribution function (VDF). However, in non-equilibrium flows, the significance of each moment diminishes as the VDF deviates from the Maxwellian VDF. This study investigates the VDF in non-equilibrium flows. Two Prandtl-corrected BGK methods, the ellipsoidal statistical BGK and Shakhov BGK (SBGK), are compared with the direct simulation Monte Carlo method. To observe the VDF while excluding the effects of convection, the homogeneous relaxation of the initial non-equilibrium state is analyzed. The VDF in Couette flow and normal shock waves, where collision and convection coexist, is then examined. When comparing the accuracy of the BGK methods using higher-order moments, inconsistencies are observed. However, when comparing the VDFs, the SBGK method reproduces the non-equilibrium VDF more accurately. The results demonstrate the importance of the VDF in the evaluation of non-equilibrium flows.

Multiresonant all-dielectric metasurfaces based on high-order multipole coupling in the visible.

In many cases, optical metasurfaces are studied in the single-resonant regime. However, a multiresonant behavior can enable multiband devices with reduced footprint, and is desired for applications such as display pixels, multispectral imaging and sensing. Multiresonances are typically achieved by engineering the array lattice (e.g., to obtain several surface lattice resonances), or by adopting a unit cell hosting one (or more than one) nanostructure with some optimized geometry to support multiple resonances. Here, we present a study on how to achieve multiresonant metasurfaces in the visible spectral range by exploiting high-order multipoles in dielectric (e.g., diamond or titanium dioxide) nanostructures. We show that in a simple metasurface (for a fixed particle and lattice geometry) one can achieve triple resonance occurring nearly at RGB (red, green, and blue) wavelengths. Based on analytical and numerical analysis, we demonstrate that the physical mechanism enabling the multiresonance behavior is the lattice induced coupling (energy exchange) between high-order Mie-type multipoles moments of the metasurface's particles. We discuss the influence on the resonances of the metasurface's finite size, surrounding material, polarization, and lattice shape, and suggest control strategies to enable the optical tunability of these resonances.

The skewness of the distance-redshift relation in ΛCDM

Starting from a recently proposed framework for the evaluation of the cosmological averages, we evaluate the higher-order moments for the distribution of a given observable. Then, we explicitly discuss the case of the Hubble-Lemaître diagram and evaluate its skewness at the leading order in the cosmological perturbative expansion of the gravitational potential. In particular, we focus on perturbations of the luminosity distance due to gravitational lensing. Finally, we discuss our findings in view of recent numerical relativistic simulations, confirming that the skewness in the Hubble-Lemaître diagram primarily originates from the late-time matter bispectrum, with other line-of-sight projection effects being sub-dominant.

Condition-based opportunistic maintenance strategy for multi-component wind turbines by using stochastic differential equations

The components of wind turbines are complex in structure and the working environment is harsh, which makes wind turbines face problems such as high failure rates and high maintenance costs. In this paper, the stochastic differential equation model has been established for the harsh operating environment of wind turbines, and used Brownian motion to simulate random disturbances; aiming at the problem of high failure rate of wind turbines, based on Weibull distribution, a new model has been established by combining operating time and equipment state to calculate the failure rate; in the analysis of monitoring data, the Higher-Order Moment method and Bayesian method were used to solve the parameters. The opportunity maintenance threshold curve and preventive maintenance threshold curve were obtained by analyzing Time-Based Maintenance and Condition-Based Maintenance. Therefore, the Condition-Based Opportunistic Maintenance strategy was obtained. The effectiveness of the proposed method was finally verified by arithmetic examples.

Salience theory value spillovers between China’s systemically important banks: evidence from quantile connectedness

Analyzing the interdependencies among financial institutions is critical for designing systemic risk monitoring mechanisms; however, most existing research focuses on the first moment of the return distribution, which falls into the conventional models of choice under risk. Previous literature has observed the scarcity of investors’ attention and processing power, which makes the traditional theory of choice under risk more vulnerable and brings the salience theory that accommodates investors’ cognitive limitations to our attention. Motivated by evidence of salience theory value (STV) containing unique information not captured by traditional higher-order moments, we employ a quantile connectedness approach to examine the STV interconnectedness of China’s systemically important banks (C-SIBs). The quantile approach allows us to uncover the dynamic STV interconnectedness of C-SIBs under normal, bearish, and bullish market conditions and is well-suited to extreme risk problems. Our results show that the C-SIBs system is asymmetrically interconnected across quantiles and at higher levels under bullish than bearish market conditions. Principally, a bank’s performance in the C-SIBs system depends on its systemic importance and market conditions. Furthermore, the comparative analysis indicates that STV could provide more information than higher-order moments in capturing the dynamic change in the C-SIBs system and detecting some market events more precisely. These results have important implications for policymakers and market participants to formulate regulatory policy and design risk management strategies.

Correction to: Analytical and numerical techniques for damage localization of rectangular plates using higher-order moments of inertia

Correction to: Analytical and numerical techniques for damage localization of rectangular plates using higher-order moments of inertia

Higher-order moment connectedness between stock and commodity markets and portfolio management

This study examines the spillover in high-order moments for major stock markets in Europe, Japan, the UK, and the US (STOXX50, FTSE100, SP500, and NIKKEI225), and two representative commodities (Brent crude oil and gold futures) using 5-min data from January 1, 2020, to May 31, 2022. The results show that spillovers vary across order moments, which are larger for realized volatility and jumps than for realized skewness and kurtosis. Moreover, gold is a net receiver of spillovers for all order moments, whereas oil switches from a net receiver of spillovers under both realized volatility and jumps to a net transmitter of spillovers in realized skewness and kurtosis spillovers. The US stock market is a net transmitter of spillovers in all realized moments, whereas other stock markets shift from net receivers to net contributors based on the moments. Furthermore, spillovers in high-order moments vary over time, and their trends behave differently over time. The spillovers in high-order moments increase during different phases of the COVID-19 and Ukraine-Russia wars. These findings have significant implications for fund allocations and financial risk management.

Stability and mean growth rate of stochastic Solow model driven by jump–diffusion process

This work focuses on the stability and mean growth rate of stochastic Solow growth model. The uncertainty in our model is generated by the technology part, which includes fluctuations on the technological accumulation and jump growth from new inventions and ideas. We introduce the criteria of stability, including stochastic stability and exponential stability, and the sufficient condition for the existence of balanced growth path. Furthermore, we precisely calculate the higher-order moment of effective capital–labor ratio and the mean growth rate of major economic quantities on the balanced growth path. Our analysis sheds light on the impact of various stochastic effects on economic quantities and provides insight into the stability of the stochastic Solow model with technological uncertainty.

Predicting low-occurrence wind speed at the pedestrian levels of simplified arrays: Artificial neural network models versus statistical methods

Low-occurrence wind speed in urban areas has a profound impact on the daily activities of people. Although several statistical methods were developed for estimating the low-occurrence wind speeds, their accuracy and flexibility still need improvements. Based on the inference from the favorable outcomes of statistical models in which low-occurrence wind speeds can be estimated by the statistics including high-order moments, in this study, the artificial neural network (ANN) models were designed to estimate the low-occurrence wind speeds based on the moments. After the sensitivity tests for the dimensions, hidden layers, epochs and training data ratios of the ANN models, the optimal setting was applied to predict the low-occurrence wind speeds of two building array cases including square and staggered layouts. The applicability of the ANN models was also evaluated for the extrapolation method between two building array cases and compared with the traditional statistical methods. It was found that the ANN models with the higher-order statistic input showed better performance than those with the second-order statistic input. The results obtained from the interpolation method are more accurate than those from the extrapolation method. Compared with traditional statistical methods, the ANN models show greater superiority in the prediction accuracy. According to the comprehensive evaluation of the ANN models, although they require ample training datasets connecting the statistics and low-occurrence wind speeds, it is proved to be a promising approach for estimating the low-occurrence wind speeds in urban wind environment.

A Unified Gas-Kinetic Particle Method for Radiation Transport in an Anisotropic Scattering Medium.

In this paper, a unified gas kinetic particle (UGKP) method is developed for radiative transfer in both absorbing and anisotropic scattering media. This numerical method is constructed based on our theoretical work on the model reduction for an anisotropic scattering system. The macroscopic solver of this method directly solves the macroscopic anisotropic diffusion equations, eliminating the need to solve higher-order moment equations. The reconstruction of macroscopic scattering source in the microscopic solver, based on the multiscale equivalent phase function we proposed in this work, has also been simplified as one single scattering process, significantly reducing the computational costs. The proposed method has also the property of asymptotic preserving. In the optically thick regime, the proposed method solves the diffusion limit equations for an anisotropic system. In the optically thin regime, the kinetic processes of photon transport are simulated. The consistency and efficiency of the proposed method have been validated by numerical tests in a wide range of flow regimes. The novel equivalent scattering source reconstruction can be used for various transport processes, and the proposed numerical scheme is widely applicable in high-energy density engineering applications.

Probability distributions arising from isoperimetric random triangles

We analyze the family of triangles whose sides come from a random subdivision of a given line segment into three segments. The usual geometric measurements on these random triangles (heights, bisectors, medians, angles, area, radii of the incircle, excircles, circumcircle) become random variables for which we determine the distribution function, the probability density, the expectation, the variance and higher order moments. This work can serve as a basis for activities at the college or university level. It is located at the crossroads between probability, geometry, integral calculus, special functions and computer algebra.

Piecewise SOS-convex moment optimization and applications via exact semi-definite programs

This paper presents exact Semi-Definite Program (SDP) reformulations for infinite-dimensional moment optimization problems involving a new class of piecewise Sum-of-Squares (SOS)-convex functions and projected spectrahedral support sets. These reformulations show that solving a single SDP finds the optimal value and an optimal probability measure of the original moment problem. This is done by establishing an SOS representation for the non-negativity of a piecewise SOS-convex function over a projected spectrahedron. Finally, as an application and a proof-of-concept illustration, the paper presents numerical results for the Newsvendor and revenue maximization problems with higher-order moments by solving their equivalent SDP reformulations. These reformulations promise a flexible and efficient approach to solving these models. The main novelty of the present work in relation to the recent research lies in finding the solution to moment problems, for the first time, with piecewise SOS-convex functions from their numerically tractable exact SDP reformulations.

Handy formulas for binomial moments

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications. This paper introduces novel formulas for binomial moments in the form of polynomials in the variance rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in Python’s symbolic algebra package. The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct asymptotically sharp estimates for central binomial moments are established, improving upon previously known partial results.

Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility

<abstract><p>In an endeavor to encapsulate the dual aspects of volatility progression and periodicity inherent in autocorrelation frameworks demonstrated by various nonlinear time series, a novel conceptualization emerges—the periodic threshold autoregressive stochastic volatility (PTAR-SV) model. This model served as a viable alternative to the conventional periodic threshold generalized autoregressive conditional heteroskedasticity (TGARCH) process. The inherent probabilistic framework of the PTAR-SV model incorporated certain essential features, including strict periodic stationarity, enhancing its analytical robustness. Additionally, this study established the conditions for higher-order moments to exist within the PTAR-SV model. The autocovariance structure pertaining to the powers of the PTAR-SV process has been studied. The process of parameter estimation was scrutinized via the quasi-maximum likelihood technique. This estimation approach involved assessing likelihood using prediction error decomposition and Kalman filtering. Moreover, we extended our analysis to include a Bayesian Markov chain Monte Carlo (MCMC) method based on Griddy-Gibbs sampling, particularly suitable when the distribution of model innovations follows a standard Gaussian. Through a simulation study, we evaluated the performances of both the quasi-maximum likelihood (QML) and Bayesian Griddy Gibbs estimates, providing valuable insights into their respective strengths and weaknesses. Finally, we applied our newly developed methodology to model the spot rates of the euro against the Algerian dinar, demonstrating its applicability and efficacy in real-world financial modeling scenarios.</p></abstract>

On the relevance of realized quarticity for exchange rate volatility forecasts

<p>High-frequency tick data have proved helpful for forecasting volatility across asset classes. In the finite samples typically faced by practitioners, however, noise inherent in tick-level prices creates inaccuracies in model parameter estimates and resulting forecasts. A remedy proposed to alleviate these measurement errors is to include higher-order moments, more specifically the realized quarticity, in volatility prediction models. In this paper, we investigate the relevance of this approach in foreign exchange markets, as represented by EURUSD and USDJPY data from 2010 to 2022. Using well-established realized volatility models, we find that including realized quarticity leads to higher precision in daily, weekly, and monthly out-of-sample forecasts. These results are robust across estimation windows, evaluation metrics, and model specifications.</p>

On rotational invariance of higher-order moments of regular polyhedra

This paper covers aspects of establishing a relationship between the highestorder rotation-invariant moments of inertia and the order of symmetry of Platonic polyhedra. Moments of inertia about arbitrary, but centroidal axes are considered. After an introductory part which summarizes the possible applications of higher-order moments of area and inertia, the revision of the already solved two-dimensional version of this problem is presented, in other words the highest-order rotation-invariant moments of area about the origin of regular m-gons are studied. As a continuation, some aspects of the possible decomposition of spatial finite rotations into a sequence of rotations about given axes will be covered which are of great importance by the extension of the twodimensional problem into three dimensions. Finally, the solution of the 3D problem will be presented, emphasizing the differences between the behavior of the even- and oddorder moments which is present in the 2D case, too, and also between the behaviour of the invariant moments of the tetrahedral and the octahedral or icosahedral solids. As a last part, possible applications will be presented.

A study on automated improvement of securities trading strategies using machine learning optimization algorithms

Abstract Automation in securities trading offers advantages over human subjective trading, such as immunity to subjective emotional factors, high efficiency, and the ability to monitor multiple stocks simultaneously, making it a cutting-edge development path in the securities trading industry. In this paper, we first apply the concept of time-frequency decomposition, gradually moving from the first-order moments of securities prices to the higher-order moments. We then combine this with the EMD time-frequency decomposition method to analyze the securities price sequence and extract the characteristics of the securities price fluctuations. Finally, we use the differential long- and short-term memory network to construct an automatic optimization trading system. We compare the system’s performance with traditional technical analysis indexes, as well as the annualized returns of PPO and A2C models on various securities, to verify its performance under unilateral rising, oscillating rising, and plummeting quotes. Finally, we conducted a live test on 1000 GEM stocks. The system in this paper outperforms all traditional technical indicators, with an average annualized return of 71.85% at the lowest and 127.27% at the highest among 5 securities, demonstrating excellent performance. In the three quotes of Ningde Times, Aier Dental, and Goldfish that are rising one way, rising and falling over time, and rising again, the annualized returns of this paper’s system are 77.13%, 67.16%, and 12.66%, which are higher than those of the PPO and A2C models.

Markov-switching threshold stochastic volatility models with regime changes

<abstract><p>This paper introduces a comprehensive class of models known as Markov-Switching Threshold Stochastic Volatility (MS-TSV) models, specifically designed to address asymmetry and the leverage effect observed in the volatility of financial time series. Extending the classical threshold stochastic volatility model, our approach expresses the parameters governing log-volatility as a function of a homogeneous Markov chain with a finite state space. The primary goal of our proposed model is to capture the dynamic behavior of volatility driven by a Markov chain, enabling the accommodation of both gradual shifts due to economic forces and sudden changes caused by abnormal events. Following the model's definition, we derive several probabilistic properties of the MS-TSV models, including strict (or second-order) stationarity, causality, ergodicity, and the computation of higher-order moments. Additionally, we provide the expression for the covariance function of the squared (or powered) process. Furthermore, we establish the limit theory for the Quasi-Maximum Likelihood Estimator (QMLE) and demonstrate the strong consistency of this estimator. Finally, a simulation study is presented to assess the performance of the proposed estimation method.</p></abstract>