Discrete Observations Research Articles

Asymptotic normality of estimators for all parameters in the Vasicek model by discrete observations

We continue to study the joint estimation of all three parameters of the Vasicek model, based on equidistant observations of a trajectory. We consider strongly consistent estimators previously constructed in Prykhodko and Ralchenko (2022) and establish their joint asymptotic normality. The theoretical results are illustrated by a numerical study that confirms our theoretical findings.

Application of Discrete Variable-Gain-Based Self-Immunity Control to Flywheel Energy Storage Systems

For the study of the trade-off between steady-state error and transient response in control systems for flywheel energy storage, a controller with a discrete variable gain is proposed. This controller aims to adapt to changes in the system state by dynamically adjusting the controller gain to optimize the system’s anti-disturbance performance. Theoretical analysis and mathematical derivation demonstrate that increasing the observer gain can significantly enhance the system’s anti-disturbance capability. However, this increase also results in overshooting, which highlights the limitations of traditional control methods in achieving both system stability and anti-disturbance performance. A discrete variable-gain extended state observer is designed. The gain of this observer can be adaptively adjusted according to a system state value, enabling the effective control of both steady-state error and transient response. Additionally, the stability of the proposed control method was analyzed and verified, ensuring its effectiveness and reliability for practical applications. Finally, the effectiveness of the proposed method in improving system performance is demonstrated by simulation results.

Prediction intervals and bands with improved coverage for functional data under noisy discrete observation

Prediction intervals and bands with improved coverage for functional data under noisy discrete observation

Simulated Exact Confidence Intervals: With Applications to Censored Exponential Reliability Data

ABSTRACTA method for constructing exact simulated confidence intervals is presented, valid for situations with both discrete and continuous observations. The idea of the method is to invert a data generating function, which needs not be monotone, and where special attention is taken when the data generating function contains jumps. The method is applied to obtain exact confidence intervals for certain types of censored data from exponential distributions. The censoring schemes under study are earlier treated in the literature, and a comparison to these approaches is considered. The connection to fiducial inference is discussed, and a difference in the paradigm of obtaining intervals for the parameter is studied.

Ozone Detector Based on Ultraviolet Observations on the Martian Surface

Ozone plays a key role in both atmospheric chemistry and UV absorption in planetary atmospheres. On Mars, upper-tropospheric ozone has been widely characterized by space-based instruments. However, surface ozone remains poorly characterized, hindered by the limited sensitivity of orbiters to the lowest scale height of the atmosphere and challenges in delivering payloads to the surface of Mars, which have prevented, to date, the measurement of ozone from the surface of Mars. Systematic measurements from the Martian surface could advance our knowledge of the atmospheric chemistry and habitability potential of this planet. NASA’s Mars 2020 mission includes the first ozone detector deployed on the Martian surface, which is based on discrete photometric observations in the ultraviolet band, a simple technology that could obtain the first insights into total ozone abundance in preparation for more sophisticated measurement techniques. This paper describes the Mars 2020 ozone detector and its retrieval algorithm, including its performance under different sources of uncertainty and the potential application of the retrieval algorithm on other missions, such as NASA’s Mars Science Laboratory. Pre-landing simulations using the UVISMART radiative transfer model suggest that the retrieval is robust and that it can deal with common issues affecting surface operations in Martian missions, although the expected low ozone abundance and instrument uncertainties could challenge its characterization in tropical latitudes of the planet. Other space missions will potentially include sensors of similar technology.

Parameter estimation for second-order SPDEs in multiple space dimensions

AbstractWe analyse a second-order SPDE model in multiple space dimensions and develop estimators for the parameters of this model based on discrete observations of a solution in time and space on a bounded domain. While parameter estimation for one and two spatial dimensions was established in recent literature, this is the first work which generalizes the theory to a general, multi-dimensional framework. Our approach builds upon realized volatilities, enabling the construction of an oracle estimator for volatility within the underlying model. Furthermore, we show that the realized volatilities have an asymptotic illustration as response of a log-linear model with spatial explanatory variable. This yields novel and efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. For proving central limit theorems, we use a high-frequency observation scheme. To showcase our results, we conduct a Monte Carlo simulation.

Synchronization of Stochastic Hybrid Coupled Systems via Completely Aperiodically Intermittent Discrete Observation Control

Synchronization of Stochastic Hybrid Coupled Systems via Completely Aperiodically Intermittent Discrete Observation Control

Continuous-Time Multivariate Analysis

The starting point for much of multivariate analysis (MVA) is an n × p data matrix whose n rows represent observations and whose p columns represent variables. Some multivariate datasets, however, may be best conceptualized not as n discrete p-variate observations, but as p curves or functions defined on a common time interval. Here we introduce a framework for extending techniques of multivariate analysis to such settings. The proposed continuous-time multivariate analysis (CTMVA) framework rests on the assumption that the curves can be represented as linear combinations of basis functions such as B-splines, as in the Ramsay-Silverman representation of functional data; but whereas functional data analysis extends MVA to the case of observations that are curves rather than vectors—heuristically, n × p data with p infinite—we are instead concerned with what happens when n is infinite. We present continuous-time extensions of the classical MVA methods of covariance and correlation estimation, principal component analysis, Fisher’s linear discriminant analysis, and k-means clustering. We show that CTMVA can improve on the performance of classical MVA, in particular for correlation estimation and clustering, and can be applied in some settings where classical MVA cannot, including variables observed at disparate time points. CTMVA is illustrated with a novel perspective on a well-known Canadian weather dataset, and with applications to datasets involving international development, brain signals, and air quality. The proposed methods are implemented in the publicly available R package ctmva. Supplementary materials for this article are available online.

Analytically pricing volatility options and capped/floored volatility swaps with nonlinear payoffs in discrete observation case under the Merton jump-diffusion model driven by a nonhomogeneous Poisson process

In this paper, we introduce novel analytical solutions for valuating volatility derivatives, including volatility options and capped/floored volatility swaps, employing discrete sampling within the framework of the Merton jump-diffusion model, which is driven by a nonhomogeneous Poisson process. The absence of a comprehensive understanding of the probability distribution characterizing the realized variance has historically impeded the development of a robust analytical valuation approach for such instruments. Through the application of the cumulative distribution function of the realized variance conditional on Poisson jumps, we have derived explicit expectations for the derivative payoffs articulated as functions of the extremum values of the square root of the realized variance. We delineate precise pricing structures for an array of instruments, encompassing variance and volatility swaps, variance and volatility options, and their respective capped and floored variations, alongside establishing put-call parity and relationships for capped and floored positions. Complementing the theoretical advancements, we substantiate the practical efficacy and precision of our solutions via Monte Carlo simulations, articulated through multiple numerical examples. Conclusively, our analysis extends to the quantification of jump impacts on the fair strike prices of volatility derivatives with nonlinear payoffs, facilitated by our analytic pricing expressions.

Mean and covariance estimation for discretely observed high-dimensional functional data: Rates of convergence and division of observational regimes

Estimation of the mean and covariance parameters for functional data is a critical task, with local linear smoothing being a popular choice. In recent years, many scientific domains are producing multivariate functional data for which p, the number of curves per subject, is often much larger than the sample size n. In this setting of high-dimensional functional data, much of developed methodology relies on preliminary estimates of the unknown mean functions and the auto- and cross-covariance functions. This paper investigates the convergence rates of local linear estimators in terms of the maximal error across components and pairs of components for mean and covariance functions, respectively, in both L2 and uniform metrics. The local linear estimators utilize a generic weighting scheme that can adjust for differing numbers of discrete observations Nij across curves j and subjects i, where the Nij vary with n. Particular attention is given to the equal weight per observation (OBS) and equal weight per subject (SUBJ) weighting schemes. The theoretical results utilize novel applications of concentration inequalities for functional data and demonstrate that, similar to univariate functional data, the order of the Nij relative to p and n divides high-dimensional functional data into three regimes (sparse, dense, and ultra-dense), with the high-dimensional parametric convergence rate of log(p)/n1/2 being attainable in the latter two.

Spline-based methods for functional data on multivariate domains

Functional data analysis is typically performed in two steps: first, functionally representing discrete observations, and then applying functional methods to the so-represented data. The initial choice of a functional representation may have a significant impact on the second phase of the analysis, as shown in recent research, where data-driven spline bases outperformed the predefined rigid choice of functional representation. The method chooses an initial functional basis by an efficient placement of the knots using a simple machine-learning algorithm. The knot selection approach does not apply directly when the data are defined on domains of a higher dimension than one such as, for example, images. The reason is that in higher dimensions the convenient and numerically efficient spline spaces use tensor bases that require knots located on a lattice. This fundamentally limits flexible knot placement which is fundamental for the approach. The goal of this research is two-fold: first, to propose modified approaches that circumvent the issue by coding the irregular knot selection into the topology of the spaces of tensor-based splines; second, to apply the approach to a classification problem workflow for functional data that utilizes knot selection. The performance is preliminarily accessed on a benchmark dataset and shown to be comparable to or better than the previous methods.

Stable convergence of conditional least squares estimators for supercritical continuous state and continuous time branching processes with immigration

We prove stable convergence of conditional least squares estimators of drift parameters for supercritical continuous state and continuous time branching processes with immigration based on discrete time observations.

The Fourier–Malliavin Volatility (FMVol) MATLAB® library

This paper presents the Fourier–Malliavin Volatility (FMVol) estimation library for MATLAB®. This library includes functions that implement Fourier–Malliavin estimators (see Malliavin and Mancino (2002, 2009)) of the volatility and co-volatility of continuous stochastic volatility processes and second-order quantities, like the quarticity (the squared volatility), the volatility of volatility and the leverage (the covariance between changes in the process and changes in its volatility). The Fourier–Malliavin method is fully non-parametric, does not require equally-spaced observations and is robust to measurement errors, or noise, without any preliminary bias correction or pre-treatment of the observations. Furthermore, in its multivariate version, it is intrinsically robust to irregular and asynchronous sampling. Although originally introduced for a specific application in financial econometrics, namely the estimation of asset volatilities, the Fourier–Malliavin method is a general method that can be applied whenever one is interested in reconstructing the latent volatility and second-order quantities of a continuous stochastic volatility process from discrete observations.

An approach for estimating microbial growth and chlorinated ethenes biotransformation in groundwater based on discrete observations

The bioremediation of chlorinated ethenes (CEs) contaminated groundwater is attracting increasingly attention in practical remediation projects. However, modelling of microbial metabolic processes under the constraints of substrate and environmental factors is inadequate. This study developed a new kinetic model, which incorporated the logistic model and Dual-Monod kinetic to represent the interaction between the controlled microbial growth and the bioavailable substrates in CE-contaminated groundwater. The proposed model was based on discrete observations to simulate microbial growth under the constraints of substrate and environmental conditions, reducing the amount of observational data required for the model. Meanwhile, the proposed model introduced two new kinetic parameters, the effective specific growth rate μeff and the real self-limiting coefficient of microbial growth keff,sl, to simplified the number of independent parameters. A parameter estimation method based on the quasi-Newton's algorithm for the proposed model was also developed. The model was validated based on the hypothetical data, experimental results, and a published dataset, demonstrated the successful simulation of microbial growth and the sequential biodegradation of PCE in groundwater systems (*E < 0.3). The monitoring duration and the sampling schedule have significant impacts on estimating the biological parameters, and large errors would be induced when the data from the periods of extremely low substrate concentration or microbial growth decline were involved in parameter estimation. The research suggested that the proposed kinetic model provided a new insight to express the limitation of microbial population growth due to the available substrates and environmental factors, and is hoping to be applied in actual CE-contaminated sites.

Efficient likelihood estimation of Heston model for novel climate-related financial contracts valuation

We propose novel Bitcoin-denominated derivatives contracts on carbon bonds. We consider a futures contract on carbon bonds where its price is expressed in terms of bitcoins. Then, we put forward options on a futures contract of the former type. Governments can use such contracts to hedge climate change and influence the prices of carbon bonds and cryptocurrencies. We show how these derivatives transfer volatility to the bitcoin market without a negative effect in the carbon bonds market. Since the aforementioned options are not yet traded in the market, we price them by assuming that the underlying is driven by the well-known Heston model, where the model parameters are estimated by a novel method based on Shannon wavelets. Heston model belongs to the class of stochastic volatility (SV) models. The discrete observations from the SV model can be seen as a state-space model, that is, a stochastic model in discrete-time which contains two sets of equations, the state equation and the observation equation. While the first describes the transition of a latent process in time, the second shows how an observer measures the latent process at each time period. We infer the properties of the latent variable by means of a filtering algorithm, and we estimate the parameters of the model via maximum likelihood. The evaluation of the likelihood function is a time-consuming task that involves updating and prediction steps of the state variable, leading to the computation of complicated integrals. We calculate these integrals by means of an integration method based on Shannon wavelets, and compare the root mean square error (RMSE) of the estimation with state-of-the-art methods. The results show that the RSME is dramatically reduced in a short CPU time with the use of wavelets.

Secure state estimation of networked switched systems under denial-of-service attacks

This paper studies the problem of secure state estimation of networked switched systems in the presence of denial-of-service (DoS) attacks, as well as disturbances and measurement noise. Firstly, a state transformation rule is designed to partition the original system into two subsystems, facilitating the design of discrete and continuous state observers. Secondly, by modifying the traditional super-twisting sliding-mode method and taking into account the frequency and duration characteristics of DoS attacks, we employ dynamic differential properties between different modes to design a switching law identification strategy. We show that this strategy can accurately estimate the switching state without imposing any requirement on the switching times and sequences. Thirdly, based on the identified activated mode, a set of mode-dependent continuous state sliding-mode observers is designed, that achieves continuous state estimation in finite time. The practicality and applicability of the developed results are validated through numerical simulations.

Non-asymptotic statistical tests of the diffusion coefficient of stochastic differential equations

We develop several statistical tests of the determinant of the diffusion coefficient of a stochastic differential equation, based on discrete observations on a time interval [0,T] sampled with a time step Δ. Our main contribution is to control the test Type I and Type II errors in a non asymptotic setting, i.e. when the number of observations and the time step are fixed. The test statistics are calculated from the process increments. In dimension 1, the density of the test statistic is explicit. In dimension 2, the test statistic has no explicit density but upper and lower bounds are proved. We also propose a multiple testing procedure in dimension greater than 2. Every test is proved to be of a given non-asymptotic level and separability conditions to control their power are also provided. A numerical study illustrates the properties of the tests for stochastic processes with known or estimated drifts.

CALIPSO Observed Low Clouds Over the Subtropical Northeast Pacific: Summer Climatology and Interannual Variability

AbstractThis study examines the spatial distribution and temporal variations of low‐cloud top over the subtropical northeast Pacific (NEP), using observations from the Cloud‐Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO). A comparison between CALIPSO observations and in‐situ soundings reveals a distinct bimodal distribution of low‐cloud tops, characterized by peaks near the inversion base and transition layer, corresponding to shallow cumulus clouds below the stratocumulus deck and boundary‐layer decoupling phenomena. Climatologically, while traversing along a great circle from Los Angeles to Honolulu, the frequency of low cloud occurrence experiences a gradual reduction from 80% to 30%. Meanwhile, cloud top height increases from 0.6 to 1.6 km, and surface roughness varies from 50 to 300 m. This study marks the first presentation of long‐term characteristics of low clouds over a vast, remote area, expanding beyond discrete field campaign observations. Accurately characterizing the marine atmospheric boundary layer (MABL) including its depth and degree of decoupling, enables further investigation into the interannual variations of low clouds, using 14 years of CALIPSO observations. Significant interannual variability in areal fraction and vertical bimodality of low clouds emerges downstream of the anomalously cold water; increased cloud cover with weakened bimodality is associated with a shallow weakly decoupled MABL. This research emphasizes the need to consider the downstream effects in studying low cloud‐sea surface temperature feedback within climate models.

Universal and nonuniversal probability laws in Markovian open quantum dynamics subject to generalized reset processes.

We consider quantum-jump trajectories of Markovian open quantum systems subject to stochastic in time resets of their state to an initial configuration. The reset events provide a partitioning of quantum trajectories into consecutive time intervals, defining sequences of random variables from the values of a trajectory observable within each of the intervals. For observables related to functions of the quantum state, we show that the probability of certain orderings in the sequences obeys a universal law. This law does not depend on the chosen observable and, in the case of Poissonian reset processes, not even on the details of the dynamics. When considering (discrete) observables associated with the counting of quantum jumps, the probabilities in general lose their universal character. Universality is only recovered in cases when the probability of observing equal outcomes in the same sequence is vanishingly small, which we can achieve in a weak-reset-rate limit. Our results extend previous findings on classical stochastic processes [N. R. Smith et al., Europhys. Lett. 142, 51002 (2023)0295-507510.1209/0295-5075/acd79e] to the quantum domain and to state-dependent reset processes, shedding light on relevant aspects for the emergence of universal probability laws.

Resilient data‐driven asymmetric bipartite consensus for nonlinear multi‐agent systems against DoS attacks

AbstractIn this article, we study an unified resilient asymmetric bipartite consensus (URABC) problem for nonlinear multi‐agent systems with both cooperative and antagonistic interactions under denial‐of‐service (DoS) attacks. We first prove that the URABC problem is solved by stabilizing the neighborhood asymmetric bipartite consensus error. Then, we develop a distributed compact form dynamic linearization method to linearize the neighborhood asymmetric bipartite consensus error. By using an attack compensation mechanism to eliminate the adverse effects of DoS attacks and an extended discrete state observer to enhance the robustness against unknown dynamics, we finally propose a distributed resilient data‐driven adaptive control (DDAC) algorithm to solve the URABC problem. A numerical example validates the proposed results.