Nonlinear Autoregressive Time Series Research Articles

Nonlinear semiparametric autoregressive models based on a class of two-piece distributions

In this study, we examined the well-known nonlinear autoregressive time series model in which innovations follow the flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family. The mentioned class of distributions is a rich class of distributions family that covers the robust symmetric/asymmetric light/heavy-tailed distributions. The nonlinear part of the autoregressive time series model is estimated via the semiparametric and nonparametric curve estimation based on the conditional least square method and nonparametric kernel approach. The maximum likelihood (ML) estimates of the model parameters, using a suitable hierarchical representation of the TP-SMN family on the model are obtained via an expectation-maximization type algorithm. Performances and usability of the proposed model and estimates are shown through simulation studies and a real dataset.

A robust class of nonlinear autoregressive models with regression function and dependent innovations using semiparametric kernel estimation

In this study, researchers examine a nonlinear autoregressive (NLAR) time-series model with regression function and dependent innovations in which the errors of the model follow the two-piece scale mixtures of normal (TP-SMN) distributions. Robustness and atypical forms of the proposed class of two-piece distributions along with the flexibility of the nonlinear autoregressive model develop desirable properties which can be applied to several types of datasets. The nonlinear regression function part of the autoregressive time-series model is estimated via the semiparametric and nonparametric curve estimation based on the conditional least square method and nonparametric kernel approach. The maximum likelihood (ML) estimates of the model parameters, using a suitable hierarchical representation of the TP-SMN family on the model are obtained via an expectation maximization (EM)-type algorithm. Performances and usefulness of the proposed model and estimates are shown via simulation studies and a real dataset.

Gradient-based Parameter Estimation for a Nonlinear Exponential Autoregressive Time-series Model by Using the Multi-innovation

Gradient-based Parameter Estimation for a Nonlinear Exponential Autoregressive Time-series Model by Using the Multi-innovation

Shock trace prediction by reduced models for a viscous stochastic Burgers equation.

Viscous shocks are a particular type of extreme event in nonlinear multiscale systems, and their representation requires small scales. Model reduction can thus play an essential role in reducing the computational cost for the prediction of shocks. Yet, reduced models typically aim to approximate large-scale dominating dynamics, which do not resolve the small scales by design. To resolve this representation barrier, we introduce a new qualitative characterization of the space-time locations of shocks, named the "shock trace," via a space-time indicator function based on an empirical resolution-adaptive threshold. Unlike exact shocks, the shock traces can be captured within the representation capacity of the large scales, thus facilitating the forecast of the timing and locations of the shocks utilizing reduced models. Within the context of a viscous stochastic Burgers equation, we show that a data-driven reduced model, in the form of nonlinear autoregression (NAR) time series models, can accurately predict the random shock traces, with relatively low rates of false predictions. Furthermore, the NAR model, which includes nonlinear closure terms to approximate the feedback from the small scales, significantly outperforms the corresponding Galerkin truncated model in the scenario of either noiseless or noisy observations. The results illustrate the importance of the data-driven closure terms in the NAR model, which account for the effects of the unresolved dynamics brought by nonlinear interactions.

Joint Parameter and Time-Delay Estimation for a Class of Nonlinear Time-Series Models

Nonlinear time-series modeling is fundamental to a wide variety of control and prediction problems. This letter focuses on the joint parameter and time-delay estimation for an extended version of the nonlinear exponential autoregressive (ExpAR) time-series model. To address the difficulties posed by the unknown time-delay and improve the estimation accuracy, we first employ the redundant rule to transform the ExpAR model into an augmented identification model. Then we invoke the multi-innovation theory to enhance data utilization and propose a new algorithm that combines stochastic gradient descent with discrete search for estimating the unknown model parameters and time-delay. The simulation results show that by properly adjusting the innovation length, the estimation accuracy of the proposed multi-innovation algorithm can significantly exceed that of the single-innovation algorithm.

Penalized estimation of threshold auto-regressive models with many components and thresholds.

Thanks to their simplicity and interpretable structure, autoregressive processes are widely used to model time series data. However, many real time series data sets exhibit non-linear patterns, requiring nonlinear modeling. The threshold Auto-Regressive (TAR) process provides a family of non-linear auto-regressive time series models in which the process dynamics are specific step functions of a thresholding variable. While estimation and inference for low-dimensional TAR models have been investigated, high-dimensional TAR models have received less attention. In this article, we develop a new framework for estimating high-dimensional TAR models, and propose two different sparsity-inducing penalties. The first penalty corresponds to a natural extension of classical TAR model to high-dimensional settings, where the same threshold is enforced for all model parameters. Our second penalty develops a more flexible TAR model, where different thresholds are allowed for different auto-regressive coefficients. We show that both penalized estimation strategies can be utilized in a three-step procedure that consistently learns both the thresholds and the corresponding auto-regressive coefficients. However, our theoretical and empirical investigations show that the direct extension of the TAR model is not appropriate for high-dimensional settings and is better suited for moderate dimensions. In contrast, the more flexible extension of the TAR model leads to consistent estimation and superior empirical performance in high dimensions.

Forecasting of Photovoltaic Power by Means of Non-Linear Auto-Regressive Exogenous Artificial Neural Network and Time Series Analysis

In this research paper, a nonlinear autoregressive with exogenous input (NARX) model of the nonlinear system based on neural network and time series analysis is proposed to deal with the one-month forecast of the produced power from photovoltaic modules (PVM). The PVM is a monocrystalline cell with a rated production of 175 watts that is placed at Heliopolis University, Bilbéis city, Egypt. The NARX model is considered powerful enough to emulate the nonlinear dynamic state-space model. It is extensively performed to resolve a variety of problems and is mainly important in complex process control. Moreover, the NARX method is selected because of its quick learning and completion times, as well as high appropriateness, and is distinguished by advantageous dynamics and interference resistance. The neural network (NN) is trained and optimized with three algorithms, the Levenberg–Marquardt Algorithm (NARX-LMA), the Bayesian Regularization Algorithm (NARX-BRA) and the Scaled Conjugate Gradient Algorithm (NARX-SCGA), to attain the best performance. The forecasted results using the NARX method based on the three algorithms are compared with experimentally measured data. The NARX-LMA, NARX-BRA and NARX-SCGA models are validated using statistical criteria. In general, weather conditions have a significant impact on the execution and quality of the results.

Robust estimation for Binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies

A topical problem of robust statistical estimation of parameters for binomial conditionally nonlinear autoregressive (BiCNAR) time series under innovation outliers is considered. This problem is solved by means of s-order Markov properties for observed time series and probabilistic properties of multivariate conditional frequencies of the future state under its s-prehistory. The new robust statistical estimator ζˆ called frequencies-based estimator (FBE) is constructed for the BiCNAR parameters under innovation outliers with arbitrary discrete probability distribution having some fixed known expectation. Under mild regularity conditions the constructed FBE is shown to have the robustness properties: consistency and asymptotic normality with obtained asymptotic covariance matrix. FBE also has computational advantages: an explicit form and a fast recursive re-estimation algorithm for extension of the model. Asymptotic risk functional and its minimum are evaluated using Fisher information matrix for the considered model. Sensitivity analysis of the statistical estimator ζ̃ for the BiCNAR parameters, that is constructed for the hypothetical model without outliers, is carried out for the situation with innovation outliers: ζ̃ is shown to be inconsistent in this situation, its bias and the instability coefficient are evaluated and analyzed. The robust estimator ζˆ has a free parameter — weight matrix H. The optimal weight matrix H∗ is found by minimization of the asymptotic risk w.r.t. H. Statistical estimator for H∗ based on the observed time series is constructed. Results of multiple computer experiments on simulated and real data illustrate the theory.

Analysis of modified time series prediction techniques: A case study with bakery industry

In the current scenario, it is a very difficult task to predict the demand for a particular product in the market. It is very challenging work for the industries to manufacture the product without being acquainted with the actual sale. To meet the demand of customers and to make the supply chain process optimize various strategies are being adopted at each node of the supply chain. Manufacturing the product by the market consumption significantly affects the revenue generation of small and medium scale enterprises especially of the FMCG industry.In this paper, a case study of a medium-scale industry named “Gopal Ji Food Products.” Kosi Kalan (U.P.) has been performed, Highlights of this case study have been kept to know the demand for the number of boxes of biscuits which are to be distributed in the market. The primary work is done by collecting historical data of the requirement of boxes for the last 104 weeks (April 2017 to March 2018). With the help of this data, the demand pattern is known. After knowing the demand pattern, an attempt has been made to develop a prediction model using time series forecasting to acquaint with the number of boxes to be manufactured in advance. The result shows that the data obtained through Non-linear Autoregressive time series forecasting can be used for better planning and fulfilling market demand on time and making the whole supply chain more optimize, effective, and cost-efficient.

On Order and Regime Determination of SETAR Model in Modelling Nonlinear Stationary Time Series Data Structure: Application to Lafia Rainfall Data, Nasarawa State, Nigeria

The linear time series model refers to the class of models for which fixed correlation parameters can fully explain the dependency between two random variables, but many real-life circumstances, such as monthly unemployment results, supplies and demands, interest rate, exchange rate, share prices, rainfall, etc., violate the assumption of linearity. For fitting and forecasting of nonlinear time series data, the self-exiting threshold autoregressive (SETAR) model was suggested. Using R to generate random nonlinear autoregressive data, a Monte Carlo simulation was performed, the SETAR model was fitted to the simulated data and Lafia rainfall data, Nasarawa State, Nigeria to determine the best regime orders and/or scheme number to make future forecast. Using Mean Square Error (MSE) and Akaike Information Criteria (AIC), the relative performance of models was examined. At a specific autoregressive order, regime order, sample size and step ahead, the model with minimum criteria was considered as the best. The results show that the best autoregressive and regime orders to be chosen are 3rd and 2nd [SETAR (3, 2)] respectively for fitting and forecasting nonlinear autoregressive time series data with small and moderate sample sizes. As the sample size increases, the output of the four models increases. Finally, it is shown that when sample size and number of steps forward are increased, the efficiency and forecasting capacity of the four models improves.

On stability Conditions of Pareto Autoregressive model

This paper concerned with studding a stability conditions of the proposed non-linear autoregressive time series model Known as Pareto Autoregressive model, acronym is defined by Pareto . A dynamical method Known as local linearization approximation method was used to obtain the stability condition of a non-zero singular point of Pareto model. In addition, we obtain the orbital stability condition of a limit cycle in terms of model parameters when the Pareto possesses a limit cycle with period .

Data-Driven Model Reduction for Stochastic Burgers Equations.

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.

An autoregressive model based on the generalized hyperbolic distribution

AbstractWe define a nonlinear autoregressive time series model based on the generalized hyperbolic distribution in an attempt to model time series with non‐Gaussian features such as skewness and heavy tails. We show that the resulting process has a simple condition for stationarity and it is also ergodic. An empirical example with a forecasting experiment is presented to illustrate the features of the proposed model.

Is Japanese Retail Industrial Fuel Oil Price Stickier Downwards Than Upwards in Response to Oil Cost Shocks?

This paper examined the Japanese industrial fuel oil market for evidence of asymmetric price adjustment and rent-seeking following changes in crude oil prices. The study used the nonlinear autoregressive distributed lag (NARDL) modeling framework and monthly time series data for the period January 2005 to December 2015. The results indicate that Japanese industrial fuel oil market is fraught with sluggish speed of adjustment, which is typical of markets witnessing uncompetitive pricing and other irregular behaviours by retail firms. The results further indicate that Japanese industrial fuel oil market is bedeviled by the problem of short-run asymmetric price transmission from crude oil market, which is consistent with the rockets and fathers effect. However, the results did not show any evidence of rent-seeking since the observed short-run asymmetry is not obscured at pump. In view of the prevailing problem of rockets and feathers effect, the paper supports policies that will encourage continuous monitoring of the market in order to preserve competition and the overall social welfare. Keywords : Rockets and Feathers effect; Rent-seeking; Asymmetric Price Adjustment; Nonlinear ARDL model; Japan JEL Codes: Q43; D40; C22; N94. DOI : 10.7176/JESD/10-4-15

Quantifying the thermal damping effect in underground vertical shafts using the nonlinear autoregressive with external input (NARX) algorithm

As air descends the intake shaft, its infrastructure, lining and the strata will emit heat during the night when the intake air is cool and, on the contrary, will absorb heat during the day when the temperature of the air becomes greater than that of the strata. This cyclic phenomenon, also known as the “thermal damping effect” will continue throughout the year reducing the effect of surface air temperature variation. The objective of this paper is to quantify the thermal damping effect in vertical underground airways. A nonlinear autoregressive time series with external input (NARX) algorithm was used as a novel method to predict the dry-bulb temperature (Td) at the bottom of intake shafts as a function of surface air temperature. Analyses demonstrated that the artificial neural network (ANN) model could accurately predict the temperature at the bottom of a shaft. Furthermore, an attempt was made to quantify typical “damping coefficient” for both production and ventilation shafts through simple linear regression models. Comparisons between the collected climatic data and the regression-based predictions show that a simple linear regression model provides an acceptable accuracy when predicting the Td at the bottom of intake shafts.

On the Dynamics of Complex Valued Nonlinear Autoregressive Time Series Models and Stochastic Limit Cycles

On the Dynamics of Complex Valued Nonlinear Autoregressive Time Series Models and Stochastic Limit Cycles

Strong consistency of the distribution estimator in the nonlinear autoregressive time series

This paper considers the uniform strong consistency of the error cumulative distribution function (CDF) estimator. Under appropriate assumptions, the classical Glivenko–Cantelli Theorem is obtained for the residual based empirical error CDF in the nonlinear autoregressive time series.

On the complex dynamics of functional-coefficients nonlinear autoregressive time series models

In this article, a new chaotic functional-coefficient nonlinear autoregressive time series model is formulated. Asymptotic stability of the equilibria of the skeleton is studied. Complex dynamics of the proposed model are investigated by means of the bifurcations, time series diagrams, and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation. Two chaotic indicators, namely, the fractal dimension and the Lyapunov exponent methods are investigated for the model.

Identification of Lags in Nonlinear Autoregressive Time Series Using a Flexible Fuzzy Model

This work proposes a method to find the set of the most influential lags and the rule structure of a Takagi---Sugeno---Kang (TSK) fuzzy model for time series applications. The proposed method resembles the techniques that prioritize lags, evaluating the proximity of nearby samples in the input space using the closeness of the corresponding target values. Clusters of samples are generated, and the consistency of the mapping between the predicted variable and the set of candidate past values is evaluated. A TSK model is established, and possible redundancies in the rule base are avoided. The proposed method is evaluated using simulated and real data. Several simulation experiments were conducted for five synthetic nonlinear autoregressive processes, two nonlinear vector autoregressive processes and eight benchmark time series. The results show a competitive performance in the mean square error and a promising ability to find a proper set of lags for a given autoregressive process.

On the use of estimating functions in monitoring time series for change points

A large class of estimators including maximum likelihood, least squares and M-estimators are based on estimating functions. In sequential change point detection related monitoring functions can be used to monitor new incoming observations based on an initial estimator, which is computationally efficient because possible numeric optimization is restricted to the initial estimation. In this work, we give general regularity conditions under which we derive the asymptotic null behavior of the corresponding tests in addition to their behavior under alternatives, where conditions become particularly simple for sufficiently smooth estimating and monitoring functions. These regularity conditions unify and extend a large amount of existing procedures in the literature, while they also allow us to derive monitoring schemes in time series that have not yet been considered in the literature including non-linear autoregressive time series and certain count time series such as binary or Poisson autoregressive models. We do not assume that the estimating and monitoring functions are equal or even of the same dimension, therefore, we allow for example to combine a non-robust but more precise initial estimator with a robust monitoring scheme. Some simulations and data examples illustrate the usefulness of the described procedures.