Large sample inference based on multiple observations from nonlinear autoregressive processes

The electrocardiogram (ECG) constitutes one of the most useful diagnostic tools for evaluating the overall health of the heart. ECG interpretation involves the assessment of various heartbeat interval features, as well as ECG wave morphological patterns and variations. However, electrocardiographic alterations are associated with a multitude of factors, which sometimes are difficult to differentiate. In this paper, we propose a system identification based methodology that quantifies the dynamic effects of heart rate variability (HRV) and age on the ECG morphology. Specifically, samples of the ECG waveform of healthy young and elderly subjects were modeled as linear and nonlinear autoregressive processes (AR). The effects of HRV were investigated by considering the HRV time-series as an exogenous input to the system. Age-related ECG wave alterations were also examined by statistically comparing the differences in the predictive performance of the AR models in the two groups.

This paper examines the existence of limit cycles in Japanese macroeconomic variables using a threshold autoregressive (TAR) model. Recent business cycle theories are grouped into two main categories: (1) real business cycle and (2) endogenous business cycle. Real business cycle theory, which is modelled by an autoregressive (AR) model, has a linear dynamic system with consecutive exogenous shocks that cause cyclical deviations from a growth trend. On the other hand, endogenous business cycle theory, which includes limit cycles characterized by a TAR model, is based on a nonlinear dynamic system that has a mechanism that induces complicated economic fluctuations endogenously. Which theory is appropriate has significant implications for policy makers. Accordingly, it is necessary to investigate whether economic fluctuations depend on an endogenous mechanism or on exogenous shocks. To investigate this, we test for linearity in Japanese macroeconomic variables. The linearity test in the paper distinguishes the TAR model from the AR model. We find that most fluctuations have TAR processes. The results indicate that most Japanese macroeconomic data have limit cycles, and imply that the government could control the business cycle in Japan, even though the cyclical behaviour is extremely complicated.

Semiparametric estimation for partially linear models with [formula omitted]-weak dependent errors

Nowadays aircrafts are expected to perform varied and complex tasks which have presented unprecedented control challenges to the aero dynamicists and control engineers. This implies that linear characterization of aircrafts is not well enough to describe the systems characteristics for control purposes and nonlinear modelling techniques are required. Neural network based nonlinear characterization look promising in this regard. This paper investigates into the development of nonlinear modelling paradigms for modern air vehicles with application to a twin rotor multi-input-multi-output system (TRMS). The system is modelled using a nonlinear autoregressive process with external input (NARX) paradigm with a feedforward neural network. Four different types of conjugate gradient algorithms (CGAs) are used in this investigation for supervised learning of the network and their performances are compared in terms of input-output mapping and speed of convergence.

In this work, we implement the so-called matching-time estimators for estimating the entropy rate as well as the entropy production rate for symbolic sequences. These estimators are based on recurrence properties of the system, which have been shown to be appropriate for testing irreversibility, especially when the sequences have large correlations or memory. Based on limit theorems for matching times, we derive a maximum likelihood estimator for the entropy rate by assuming that we have a set of moderately short symbolic time series of finite random duration. We show that the proposed estimator has several properties that make it adequate for estimating the entropy rate and entropy production rate (or for testing the irreversibility) when the sample sequences have different lengths, such as the coding sequences of DNA. We test our approach with controlled examples of Markov chains, non-linear chaotic maps, and linear and non-linear autoregressive processes. We also implement our estimators for genomic sequences to show that the degree of irreversibility of coding sequences in human DNA is significantly larger than that for the corresponding non-coding sequences.

<p>The first chapter consists of an overview of the theory of empirical processes, covering an introduction to empirical processes in R, uniform empirical processes and function parametric empirical processes in Section 1.1. Section 1.2 contains an overview of the theory related to the law of the iterated logarithm for Brownian motion and the modulus of continuity for Brownian motion. Section 1.3 contains the theory of the limiting processes for the empirical process, most importantly Brownian motion, Brownian bridge and the connections and relationships between them, with distributions of selected statistics of Brownian motion and Brownian bridge derived from reflection principles. Section 1.4 contains an overview of the theory required to prove central limit results for the empirical processes, covering the theory of the space C and Donsker’s theorem. The second chapter covers research topics, starting with Fourier analysis of mixture distributions and associated theory in Section 2.1. Section 2.2 covers findings in a research problem about non-linear autoregressive processes. Section 2.3 introduces a martingale approach to testing a regression model. Section 2.4 links the theory of ranks and sequential ranks to the theory of empirical processes.</p>

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of $V$-geometrically ergodic Markov kernels indexed by a real parameter $u$, we give conditions under which the invariant probability distribution is differentiable with respect to $u$, in the sense of signed measures. Our results also complete the existing literature for the perturbation analysis of Markov chains, in particular when exponential moments are not finite. Our conditions are checked on several original examples of locally stationary processes such as integer-valued autoregressive processes, categorical time series or threshold autoregressive processes.

In the last few decades, a broad strand of literature in finance has implemented artificial neural networks as a forecasting method. The major advantage of this approach is the possibility to approximate any linear and nonlinear behaviors without knowing the structure of the data generating process. This makes it suitable for forecasting time series which exhibit long-memory and nonlinear dependencies, like conditional volatility. In this article, the predictive performance of feed-forward and recurrent neural networks (RNNs) was compared, particularly focusing on the recently developed long short-term memory (LSTM) network and nonlinear autoregressive model process with eXogenous input (NARX) network, with traditional econometric approaches. The results show that RNNs are able to outperform all the traditional econometric methods. Additionally, capturing long-range dependence through LSTM and NARX models seems to improve the forecasting accuracy also in a highly volatile period.

As the global population grows, ensuring sustainable food production has become critical. Marine aquaculture provides a sustainable and scalable source of protein; however, its continued expansion requires the development of novel technologies that enable remote management and autonomous operations. Digital twin technology emerges as a transformative tool for realizing this goal, yet its adoption remains limited. Fish net cages—flexible, floating structures—are critical but vulnerable components of aquaculture systems. Exposed to harsh and dynamic marine conditions, they experience substantial hydrodynamic loads that can cause structural damage leading to fish escapes, environmental impacts, and financial losses. We propose a multifidelity surrogate modeling framework for integration into a digital twin that enables real-time monitoring of net cage structural dynamics under stochastic marine conditions. At the core of the framework lies the nonlinear autoregressive Gaussian process method, which captures complex, nonlinear cross-correlations between models of varying fidelity. It combines low-fidelity simulation data with a limited set of high-fidelity field sensor measurements, which, although accurate, are costly and spatially sparse. The framework was validated at the SINTEF ACE fish farm in Norway, where the digital twin assimilates online metocean data to accurately predict net cage displacements and mooring line loads, closely matching field measurements. This approach is especially valuable in data-scarce environments, offering rapid predictions and real-time structural representation. Beyond monitoring, the developed digital twin enables proactive assessment of structural integrity and supports remote operations with unmanned underwater vehicles. Finally, we compare Gaussian processes and graph convolutional networks for predicting net cage deformation, demonstrating the superior ability of the latter to capture in complex structural behaviors.

In this work, we propose a study of the mean annual rainfall time series in order to evaluate the climate changes pattern over time. If the analysis of this time series is carried out correctly, it can contribute to improve planning and policy development. That is why we consider the problem of mathematical modeling and analysis of the mean annual rainfall of Morocco between 1901 and 2020 using descriptive statistics, structure changes analysis, spectral analysis and a nonlinear Exponential Autoregressive (EXPAR) processes to reproduce the behavior of this time series. The results indicate three main breakpoints and show that the time series has a remarkable cycles about 60, 18 and 6 years with a global decrease tendency about 0.56 mm per year. Furthermore, we have justified the choice of using a non-linear EXPAR processes rather than a linear traditional one and provided a good fitted EXPAR model.

Estimating the parameters of the autoregressive (AR) random process is a problem that has been well-studied. In many applications, only noisy measurements of AR process are available. The effect of the additive noise is that the system can be modeled as an AR model with colored noise, even when the measurement noise is white, where the correlation matrix depends on the AR parameters. Because of the correlation, it is expedient to compute using multiple stacked observations. Performing a weighted least-squares estimation of the AR parameters using an inverse covariance weighting can provide significantly better parameter estimates, with improvement increasing with the stack depth. The estimation algorithm is essentially a vector RLS adaptive filter, with time-varying covariance matrix. Different ways of estimating the unknown covariance are presented, as well as a method to estimate the variances of the AR and observation noise. The notation is extended to vector autoregressive (VAR) processes. Simulation results demonstrate performance improvements in coefficient error and in spectrum estimation.

On continuous-time threshold ARMA processes

A complex nonlinear exponential autoregressive (CNEAR) process which models the boundary coordinate sequence for invariant feature extraction to recognize arbitrary shapes on a plane is presented. All the CNEAR coefficients can be synchronically calculated by using a neural network which is simple in structure and, therefore, easy in implementation. The coefficients are adopted to constitute the feature set which are proven to be invariant to the transformation of a boundary such as translation, rotation, scale and choice of the starting point in tracing the boundary. Afterwards, the feature set is used as the input to a complex multilayer perceptron (C-MLP) network for learning and classification. Experimental results show that complicated shapes can be recognized in high accuracy, even in the low-order model. It is also seen that the classification method has a good degree of fault tolerance when noise is present.

A note on the geometric ergodicity of a nonlinear AR-ARCH model