Approximations and boundary conditions for continuous-time threshold autoregressive processes

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.

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.

On continuous-time threshold ARMA processes

Large sample inference based on multiple observations from nonlinear autoregressive processes

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

We consider the problem of sampling continuous-time auto-regressive processes on a uniform grid. We investigate whether a given sampled process originates from a single continuous-time model, and address this uniqueness problem by introducing an alternative description of poles in the complex plane. We then utilize Kronecker's approximation theorem and prove that the set of non-unique continuous-time AR(2) models has Lebesgue measure zero in this plane. This is a key aspect in current estimation algorithms that use sampled data, as it allows one to remove the sampling rate constraint that is imposed currently.

Autoregression and irregular sampling: Filtering

It is well known that neurons in the sensorimotor cortex of primates tend to synchronously activate and generate a pulsating motor command, and such rhythmic cortical drive induces voluntary contraction of muscles. Using frequency analysis, numerous experimental data have revealed that even a single neuron in the motor system transmits a rhythmic signal with a frequency much higher than its own spike rate, although this seems odd when we consider the sampling theorem. To prove such physiological findings, we investigated the transmission mechanism of a neural signal using both mathematical and simulation approaches. For the mathematical consideration, we sampled an analytical signal as both an evenly spaced and unevenly spaced time series in order to simulate regularly and irregularly spiking neurons. The results of the discrete Fourier transform suggest that only data sampled at irregular intervals contained the exact spectrum of the input signal under the condition where the sampling rate was even lower than the input frequency. By observing the signal structure, we concluded that harmonics-cancellation was achieved due to irregular sampling intervals. We also simulated both regularly and irregularly spiking neurons using Hodgkin. Huxley's neural membrane model. When we applied sinusoidal current inflow of 135.0 Hz to the neuron, the neuron spiked with constant intervals (this case is comparable to the regularly sampled data). In this case, the spectrum did not reflect the input signal. However, in the case of sinusoidal input with frequency of 132.0 Hz, the spectrum showed the input frequency content. The neural spike rate (that is, the output frequency), was much lower than that of the input frequency. From these results, we concluded that fluctuation of spiking in neural output might play an important role for transmitting information in the input signal.

Nowadays , nonlinear time series and artificial neural networks (ANN) models are used for forecasting in the field of business, agriculture and soon. Recent studies have shown, ANN have been successfully used for forecasting of financial and agriculture data series The classical methods used for time series prediction like Box-Jenkins or ARIMA assumes that there is a linear relationship between inputs and outputs. ANN have more advantages that can approximate to model both linear and nonlinear structures in time series, they are not able to handling both structures equally well. The autoregressive integrated moving average (ARIMA) model and two ANN models namely, Radial basis function neural networks (RBFNN), and Elman recurrent neural networks (ERNN) methods were applied to Hyderabad airport traffic data. The data obtained for 15 years from 2002–2003 to 2016–2017 about domestic and international passenger of International Airport of Hyderabad, India. In this research paper, we compared the performances of ARIMA, RBFNN and ERNN were based on three measures: mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE). The results showed that RBFNN obtained the smallest MAE, MAPE and RMSE in both the modeling and forecasting processes. The performances of the three models ranked in ascending order were: ARIMA, ERNN and the RBFNN model. Keywords: T ime series, forecasting, artificial neural networks, ARIMA models, radial basis function neural networks, and Elman recurrent neural networks Cite this Article R. Ramakrishna, Berhe Aregay, Tewodros Gebregergs. The Comparison in Time Series Forecasting of Air Traffic Data by Autoregressive Integrated Moving Average Model, Radial Basis Function and Elman Recurrent Neural Networks. Research & Reviews: Journal of Statistics . 2018; 7(3): 75–90p.

This paper investigates nonlinear time series modelling using the general state-dependent autoregressive model. To achieve the estimate of the model, an attempt is made to approximate the state-dependent parameter by employing the Gaussian radial basis function for its universal approximation capability. As a result, a radial basis function-based autoregressive (RBF-AR) model is derived which has a form similar to a generalized exponential autoregressive model. To reach the applicability of the RBF-AR model, the evolutionary programming algorithm is employed to select a suitable set of radial basis function centres. By applying the resulting model to some complex data, it is shown that the RBF-AR model can not only reconstruct the dynamics of given nonlinear time series effectively, but also give much better fitting to complextime series than the approach of directly RBF neural network modelling. Therefore, as a paradigm combining a statistical model and neural network, the RBF-AR model has better performance than the RBF neural network especially for its problem reduction of the curse of dimensionality which is usually regarded as one of the potential main difficulties of RBF neural network modelling.

Gait asymmetry is often observed in populations with varying degrees of neuromuscular control. While changes in vertical ground reaction force (vGRF) peak magnitude are associated with altered limb loading that can be observed during asymmetric gait, the challenge is identifying techniques with the sensitivity to detect these altered movement patterns. Autoregressive (AR) modeling has successfully delineated between healthy and pathological gait during running; but has been little explored in walking. Thus, AR modeling was implemented to assess differences in vGRF pattern dynamics during symmetric and asymmetric walking. We hypothesized that the AR model coefficients would better detect differences amongst the symmetric and asymmetric walking conditions than the vGRF peak magnitude mean. Seventeen healthy individuals performed a protocol that involved walking on a split-belt instrumented treadmill at different symmetric (0.75m/s, 1.0 m/s, 1.5 m/s) and asymmetric (Side 1: 0.75m/s-Side 2:1.0 m/s; Side 1:1.0m/s-Side 2:1.5 m/s) gait conditions. Vertical ground reaction force peaks extracted during the weight-acceptance and propulsive phase of each step were used to construct a vGRF peak time series. Then, a second order AR model was fit to the vGRF peak waveform data to determine the AR model coefficients. The resulting AR coefficients were plotted on a stationarity triangle and their distance from the triangle centroid was computed. Significant differences in vGRF patterns were detected amongst the symmetric and asymmetric conditions using the AR modeling coefficients (p = 0.01); however, no differences were found when comparing vGRF peak magnitude means. These findings suggest that AR modeling has the sensitivity to identify differences in gait asymmetry that could aid in monitoring rehabilitation progression.

Autoregressive (AR) modeling is a very popular method for time series analysis. Being linear by nature, it obviously fails to adequately describe nonlinear systems. In this paper, we propose a kernel-based AR modeling, by combining two main concepts in kernel machines. One the one hand, we map samples to some nonlinear feature space, where an AR model is considered. We show that the model parameters can be determined without the need to exhibit the nonlinear map, by computing inner products thanks to the kernel trick. On the other hand, we propose a prediction scheme, where the prediction in the feature space is mapped back into the input space, the original samples space. For this purpose, a pre-image technique is derived to predict the future back in the input space. The efficiency of the proposed method is illustrated on real-life time-series, by comparing it to other linear and nonlinear time series prediction techniques.

Accurate prediction of time series is complex due to nonlinear characteristics but can play a significant role in practical problem. In this paper, a novel varying‐coefficient hybrid model is proposed to accurately predict the nonlinear time series. A set of fuzzy radial basis function (FRBF) neural networks is used to approximate the varying functional coefficients of the state‐dependent autoregressive model with exogenous variables (SD‐ARX). The obtained model is called the fuzzy radial basis function network‐based autoregressive model with exogenous variables (FRBF‐ARX), which combines the advantages of the FRBF in function approximation and the SD‐ARX model in nonlinear dynamics description. Then, a structured nonlinear parameter optimization method (SNPOM) and the modified multifold cross‐validation criterion are used to estimate the parameters of the proposed varying‐coefficient FRBF‐ARX model. The performances of the FRBF‐ARX model are used to predict the PM2.5 concentration and simulated SISO nonlinear process, respectively, and the performances of the proposed model are also compared and discussed. The experimental results show that the FRBF‐ARX model has better performances of accuracy on nonlinear time series forecasting than that of other models.

On guaranteed parameter estimation of a multiparameter linear regression process

An increasing number of researchers in psychology are collecting intensive longitudinal data in order to study psychological processes on an intraindividual level. An increasingly popular way to analyze these data is autoregressive time series modeling; either by modeling the repeated measures for a single individual using classic n = 1 autoregressive models, or by using multilevel extensions of these models, with the dynamics for each individual modeled at Level 1 and interindividual differences in these dynamics modeled at Level 2. However, while it is widely accepted in psychology that psychological measurements usually contain a certain amount of measurement error, the issue of measurement error is largely neglected in applied psychological (autoregressive) time series modeling: The regular autoregressive model incorporates innovations, or "dynamic errors," but not measurement error. In this article we discuss the concepts of reliability and measurement error in the context of dynamic (VAR(1)) models, and the consequences of disregarding measurement error variance in the data. For this purpose, we present a preliminary model that accounts for measurement error for constructs that are measured with a single indicator. We further discuss how this model could be used to investigate the between-person reliability of the measurements, as well as the (person-specific) within-person reliabilities and any individual differences in these reliabilities. We illustrate the consequences of assuming perfect reliability, the preliminary model, and reliabilities, using an empirical application in which we relate women's general positive affect to their positive affect concerning their romantic relationship. (PsycINFO Database Record (c) 2019 APA, all rights reserved).