Nonlinear Time Series Models Research Articles

Application of Density Plots and Time Series Modelling to the Analysis of Nitrogen Dioxides Measured by Low-Cost and Reference Sensors in Urban Areas

Temporal variability of NO2 concentrations measured by 28 Envirowatch E-MOTEs, 13 AQMesh pods, and eight reference sensors (five run by Sheffield City Council and three run by the Department for Environment, Food and Rural Affairs (DEFRA)) was analysed at different time scales (e.g., annual, weekly and diurnal cycles). Density plots and time variation plots were used to compare the distributions and temporal variability of NO2 concentrations. Long-term trends, both adjusted and non-adjusted, showed significant reductions in NO2 concentrations. At the Tinsley site, the non-adjusted trend was −0.94 (−1.12, −0.78) µgm−3/year, whereas the adjusted trend was −0.95 (−1.04, −0.86) µgm−3/year. At Devonshire Green, the non-adjusted trend was −1.21 (−1.91, −0.41) µgm−3/year and the adjusted trend was −1.26 (−1.57, −0.83) µgm−3/year. Furthermore, NO2 concentrations were analysed employing univariate linear and nonlinear time series models and their performance was compared with a more advanced time series model using two exogenous variables (NO and O3). For this purpose, time series data of NO, O3 and NO2 were obtained from a reference site in Sheffield, which were more accurate than the measurements from low-cost sensors and, therefore, more suitable for training and testing the model. In this article, the three main steps used for model development are discussed: (i) model specification for choosing appropriate values for p, d and q, (ii) model fitting (parameters estimation), and (iii) model diagnostic (testing the goodness of fit). The linear auto-regressive integrated moving average (ARIMA) performed better than the nonlinear counterpart; however, its performance in predicting NO2 concentration was inferior to ARIMA with exogenous variables (ARIMAX). Using cross-validation ARIMAX demonstrated strong association with the measured concentrations, with a correlation coefficient of 0.84 and RMSE of 9.90. ARIMAX can be used as an early warning tool for predicting potential pollution episodes in order to be proactive in adopting precautionary measures.

Stochastic analysis and invariant subspace method for handling option pricing with numerical simulation

‎ In this paper option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method.

Improved grey system models for predicting traffic parameters

In transportation applications such as real-time route guidance, ramp metering, congestion pricing and special events traffic management, accurate short-term traffic flow prediction is needed. For this purpose, this paper proposes several novel online Grey system models (GM): GM(1,1|cos(ωt)), GM(1,1|sin(ωt),cos(ωt)), and GM(1,1|e-at,sin(ωt),cos(ωt)). To evaluate the performance of the proposed models, they are compared against a set of benchmark models: GM(1,1) model, Grey Verhulst models with and without Fourier error corrections, linear time series model, and nonlinear time series model. The evaluation is performed using loop detector and probe vehicle data from California, Virginia, and Oregon. Among the benchmark models, the error corrected Grey Verhulst model with Fourier outperformed the GM(1,1) model, linear time series, and non-linear time series models. In turn, the three proposed models, GM(1,1|cos(ωt)), GM(1,1|sin(ωt),cos(ωt)), and GM(1,1|e-at,sin(ωt),cos(ωt)), outperformed the Grey Verhulst model in prediction by at least 65%, 16% and 11%, in terms of Root Mean Squared Error, and by 82%, 58% and 42%, in terms of Mean Absolute Percentage Error, respectively. It is observed that the proposed Grey system models are more adaptive to location (e.g., perform well for all roadway types) and traffic parameters (e.g., speed, travel time, occupancy, and volume), and they do not require as many data points for training (4 observations are found to be sufficient).

Parameter estimation for a class of radial basis function-based nonlinear time-series models with moving average noises

This paper focuses on the parameter estimation for radial basis function-based state-dependent autoregressive models with moving average noises (RBF-ARMA models). An extended projection algorithm is derived based on the negative gradient search. In order to reduce the sensitivity of the algorithm to noise and reduce the fluctuations of the parameter estimation errors, a modified extended stochastic gradient algorithm is proposed. By introducing a moving data window, a modified moving data window-based extended stochastic gradient algorithm is further developed to improve the parameter estimation accuracy. The simulation results show that the proposed algorithms can effectively estimate the parameters of the RBF-ARMA models.

A novel elastic net-based NGBMC(1,n) model with multi-objective optimization for nonlinear time series forecasting

Nonlinear grey Bernoulli multivariate model NGBMC (1, n) is known as a novel forecasting model for nonlinear time series with small samples. However, ill-posed problem would make it less efficient and even cause large errors. In order to improve its generality, a hybrid method combining Elastic Net and multi-objective optimization is introduced in this work. This method effectively solves the essential defect of the ill-posed problem of the NGBMC (1, n) model, making the NGBMC (1, n) model more stable, more reliable, and more interpretable. The parameter identification of the new model uses the alternating direction method of multipliers, and the nonlinear parameter of the model and the regularization parameter of Elastic Net regression are optimized by the multi-objective grey wolf optimizer (MOGWO). Eight numerical cases all show that the use of the Elastic Net regularization method and multi-objective optimization technology can significantly improve the prediction accuracy of the NGBMC (1, n) model for future data. In addition, the hybrid method combing the regularization and optimization strategies proposed in this paper is a general framework for the grey prediction models, which has a high potential in improving the other grey models.

Rejoinder of “On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures”

I am pleased that my review article has stimulated such broader and thoughtful discussions in probability theory, theoretical statistics, estimation methods, and applications. The discussants have ...

Discussion on paper ‘On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures’ by Zhengjun Zhang

I would like to take this opportunity to congratulate Zhengjun for his continuing contribution to extreme-value statistics in recent years. In this review paper, some fundamental theories on univar...

Discussion of ‘On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures’

We congratulate Prof. Zhang for this timely review on recent advances in extreme value theory for heterogeneous populations and on time series models for extreme observations. This is a substantial...

Discussion of ‘On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures’

I congratulate and thank the author for providing a systematic and thorough review of both classical approaches and modern developments on the modelling of extremal events and tail dependence in th...

Discussion on “on studying extreme values and systematic risks with nonlinear time series models and tail dependence measures”

Extreme value theory provides essential mathematical foundations for modelling tail risks and has wide applications. The emerging of big and heterogeneous data calls for the development of new extr...

MIMO Modeling by Learning Explicitly the Projection Space: The Maximum Correlation Ratio Cost Function

Maximal correlation measures the statistical dependence between two random variables and has been broadly used by statisticians. In this paper, we present a novel regression framework based on a special case of the maximal correlation using the correlation ratio to train a Multiple-input Multiple-output (MIMO) model called the Bank of Wiener Models (BWM). Nonlinear time series modeling under maximal correlation unifies the cost and the mapping function under the same mathematical optimization framework. This provides direct optimality (maximal statistical dependence) between BWM model outputs and the desired response, without the restriction of constructing an error signal as conventionally done in regression. Based on the idea of correlation ratio, we propose the Maximal Correlation Algorithm (MCA), which approximates directly the correlation ratio between BWM outputs and the desired response. As an important consequence, MCA modularizes the training of models with hidden layers and avoids the end-to-end training of backpropagation (BP), while improving the equivalent mapping capability. We further propose several possible multi layer arrangements to create deep networks trained with MCA. We demonstrate experimentally that MCA performs at the same level of output error or better when compared to a single and multiple-hidden-layer MLP trained with BP and the Mean Squared Error (MSE). We also show that the system identification capabilities of MCA are superior to MLPs trained with BP. Hence, MCA has great promise in both nonlinear system identification and machine learning.

On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures

ABSTRACT This review paper discusses advances of statistical inference in modeling extreme observations from multiple sources and heterogeneous populations. The paper starts briefly reviewing classical univariate/multivariate extreme value theory, tail equivalence, and tail (in)dependence. New extreme value theory for heterogeneous populations is then introduced. Time series models for maxima and extreme observations are the focus of the review. These models naturally form a new system with similar structures. They can be used as alternatives to the widely used ARMA models and GARCH models. Applications of these time series models can be in many fields. The paper discusses two important applications: systematic risks and extreme co-movements/large scale contagions.

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 .

Forecasting prices of dairy commodities – a comparison of linear and nonlinear models

Dairy commodity prices have become more volatile over the last 10–11 yr. The aim of this paper was to produce reliable price forecasts for the most frequently traded dairy commodities. Altogether five linear and nonlinear time series models were applied. The analysis reveals that prices of dairy commodities reached a structural breakpoint in 2006/2007. The results also show that a combination of linear and nonlinear models is useful in forecasting commodity prices. In this study, the price of cheese is the most difficult to forecast, but a simple autoregressive (AR) model performs reasonably well after 12 mo. Similarly, for butter the AR model performs the best, while for skimmed milk powder (Smp), whole milk powder (Wmp) and whey powder (Whp) the nonlinear methods are the most accurate. However, few of the differences between models are significant according to the Diebold–Mariano (DM) test. The findings could be of interest to the whole dairy industry.

SUBGEOMETRICALLY ERGODIC AUTOREGRESSIONS

In this paper, we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications, we also obtain stationarity and $\beta $ -mixing with subgeometrically decaying mixing coefficients.

Developing hybrid time series and artificial intelligence models for estimating air temperatures

Air temperature is a vital meteorological variable required in many applications, such as agricultural and soil sciences, meteorological and climatological studies, etc. Given the importance of this variable, this study seeks to estimate minimum (Tmin), maximum (Tmax), and mean (T) air temperatures by applying a linear autoregressive (AR) time series model and then developing a hybrid model by means of coupling the AR and a non-linear time series model, namely autoregressive conditional heteroscedasticity (ARCH). Hence, the hybrid AR-ARCH model was tested. To that end, the Tmin, Tmax, and T data from 1986 to 2015 at two weather stations located in Northwestern Iran were used for both daily and monthly time scales. The results showed that the hybrid time series model (i.e., AR-ARCH) performed better than the single AR for estimating the air temperature parameters at the study sites. Multi-layer perceptron (MLP) was then employed to estimate the air temperatures using lagged temperature data as input predictors. Next, the single AR and hybrid AR-ARCH time series models were utilized to implement the hybrid MLP-AR and MLP-AR-ARCH models. It is worth noting that developing the hybrid MLP-AR and MLP-AR-ARCH models, as well as AR-ARCH one is the novelty of this study. Three statistical metrics including root mean square error (RMSE), mean absolute error (MAE), and normalized RMSE (NRMSE) were used to investigate the performance of whole the developed models. The hybrid MLP-AR and MLP-AR-ARCH models were found to perform better than the single MLP when estimating the daily and monthly Tmin, Tmax, and T; however, the MLP-AR models outperformed the MLP-AR-ARCH ones. At the end of this study, the performance of MLP was evaluated under an external condition (i.e., estimating the temperature components at any particular site using the temperature data of an adjacent location). The results indicated that the temperature data of a nearby station can be used for estimating the temperatures of a desired station. Most accurate results during the test stage were obtained under a local assessment through the hybrid MLP-AR(1) at the Tabriz station when estimating the monthly Tmax (RMSE = 0.199 °C, MAE = 0.159 °C, NRMSE = 1.012%) and hybrid MLP-AR(12) at the Urmia station when estimating the daily Tmax (RMSE = 0.364 °C, MAE = 0.277 °C, NRMSE = 1.911%).

Bitcoin option pricing with a SETAR-GARCH model

This paper aims to study the pricing of Bitcoin options with a view to incorporating both conditional heteroscedasticity and regime switching in Bitcoin returns. Specifically, a nonlinear time series model combining both the self-exciting threshold autoregressive (SETAR) model and the generalized autoregressive conditional heteroscedastic (GARCH) model is adopted for modeling Bitcoin return dynamics. Specifically, the SETAR model is used to model regime switching and the Heston-Nandi GARCH model is adopted to model conditional heteroscedasticity. Both the conditional Esscher transform and the variance-dependent pricing kernel are used to specify pricing kernels. Numerical studies on the Bitcoin option prices using real bitcoins data are presented.

Nonlinear time series models for the North Atlantic Oscillation

Abstract. The North Atlantic Oscillation (NAO) is the dominant mode of climate variability over the North Atlantic basin and has a significant impact on seasonal climate and surface weather conditions. This is the result of complex and nonlinear interactions between many spatio-temporal scales. Here, the authors study a number of linear and nonlinear models for a station-based time series of the daily winter NAO index. It is found that nonlinear autoregressive models, including both short and long lags, perform excellently in reproducing the characteristic statistical properties of the NAO, such as skewness and fat tails of the distribution, and the different timescales of the two phases. As a spin-off of the modelling procedure, we can deduce that the interannual dependence of the NAO mostly affects the positive phase, and that timescales of 1 to 3 weeks are more dominant for the negative phase. Furthermore, the statistical properties of the model make it useful for the generation of realistic climate noise.

Specification of initial Kalman recursions of symmetric nonlinear state-space model

A new class of nonlinear Time Series model referred to as Symmetric Nonlinear State-Space Model (SNSSM) was successfully developed using Kalman filter methodology. Some vital properties of the SNSSM such as optimal Kalman gain and optimal filter state covariance were derived. We finally initialized the filter which enabled us obtained the initial Kalman recursions under stationarity and nonstationarity assumptions. Under the former, the mean and variance were obtained unconditionally using Kronecker products and vec operator. But under the later, the mean and variance/covariance of the system were conditionally obtained using a well-known marginal and conditional property of multivariate normal distribution. It is expected that the former will be better than the later if the system is stationary, otherwise the later will be better.

Two‐stage recursive identification algorithms for a class of nonlinear time series models with colored noise

SummaryThis article concentrates on the recursive identification algorithms for the exponential autoregressive model with moving average noise. Using the decomposition technique, we transform the original identification model into a linear and nonlinear subidentification model and derive a two‐stage least squares (LS) extended stochastic gradient (ESG) algorithm. In order to improve the parameter estimation accuracy, we employ the multi‐innovation identification theory and develop a two‐stage LS multi‐innovation ESG algorithm. A simulation example is provided to test the effectiveness of the proposed algorithms.