Nonlinear Autoregressive Neural Network Model Research Articles

Platinum and palladium price forecasting through neural networks

To many commodity market participants, forecasts of price series represent a critical task. In this work, nonlinear autoregressive neural network models’ potential is explored for forecasting daily prices series of platinum and palladium over about a fifty-year period. For this purpose, one hundred and twenty model settings are examined, including different training algorithms, numbers of hidden neurons and delays, and ratios used to segment the data. With the analysis, two models leading to stable and accurate forecast results are constructed for the prices of the two commodities. In particular, the models’ performance in terms of the relative root mean square error is 1.86% and 3.61% for platinum and palladium, respectively, for the overall sample. Results in this work could help technical forecasts and policy analysis. The forecast framework might be extended to other different commodities.

Price forecasting through neural networks for crude oil, heating oil, and natural gas

Building price projections of various energy commodities has long been an important endeavor for a wide range of participants in the energy market. We study the forecast problem in this paper by concentrating on four significant energy commodities. Using nonlinear autoregressive neural network models, we investigate the daily prices of WTI and Brent crude oil as well as the monthly prices of Henry Hub natural gas and New York Harbor No. 2 heating oil. We investigate prediction performance resulting from various model configurations, including training techniques, hidden neurons, delays, and data segmentation. Based on the investigation, relatively straightforward models are built that yield quite accurate and reliable performance. Specifically, performance in terms of relative root mean square errors is 1.96%/1.81%/9.75%/21.76%, 1.96%/1.80%/8.76%/14.41%, and 1.87%/1.78%/9.10%/16.97% for model training, validation, and testing, respectively, and the overall relative root mean square error is 1.95%/1.80%/9.51%/20.35% for the whole sample for WTI crude oil/Brent crude oil/New York Harbor No. 2 heating oil/Henry Hub natural gas. The outcomes of this projection might be used in technical analysis or integrated with other fundamental forecasts for policy analysis.

Automated Pricing and Replenishment Decision Modeling for Supermarket Vegetables

This paper analyzes the historical sales data of vegetable products in a fresh food supermarket, and explores the automatic pricing and replenishment decision-making scheme of vegetable products in this supermarket. Firstly, we integrate and clean the attached data, and then we solve the characteristics and interrelationships of individual products and categories by establishing SARIMA model and Spearman's correlation coefficient model, and then we use nonlinear regression model, nonlinear autoregressive neural network model and optimization model to realize automatic pricing and replenishment decision-making for vegetable products.

Time Series Modeling of Global Average Absolute Sea Level Change

This study aimed to demonstrate the effectiveness of time series models in modeling long-term records of global average absolute sea level changes from 1880 to 2014. Following the Box–Jenkins methodology, the ARIMA(0,1,2) model with drift was identified as the best-fit model for the time series due to its lowest AIC value. Using the LM algorithm, the results revealed that the NARNN model with 7 neurons in the hidden layer and 7 time delays exhibited the best performance among the nonlinear autoregressive neural network models, as indicated by its lower MSE. While ARIMA models excel in modeling linear problems within time series data, NARNN models are better suited for nonlinear patterns. However, a HYBRID model was explored, which combines the strengths of both ARIMA and NARNN models, offering the capability to address both linear and nonlinear aspects of time series data. The comparative analysis of this study demonstrated that the HYBRID model, with 6 neurons in the hidden layer and 7 time delays, outperformed the NARNN model with 7 neurons in the hidden layer and 7 time delays, as well as the ARIMA(0,1,2) model, with the lowest MSE in this study. These findings represent a significant step in time series forecasting by leveraging the strengths of both statistical and machine learning methods.

Nonlinear Autoregressive Neural Network for Forecasting COVID-19 Confirmed Cases in Malaysia

A nonlinear autoregressive neural network (NARNN) model is a feedforward neural network for handling complex nonlinear time series problem. In this study, the tangent sigmoid (tansig) activation function with different number of past values and different number of hidden neurons for NARNN model is determined. The COVID-19 daily confirmed cases in Malaysia are collected with different amount of sample used which are 100, 500 and 900. Therefore, data of 100, 500 and 900 days prior to 21 September 2022 are extracted for the NARNN model training, validation and testing procedure. The lowest average mean squared error (MSE) are considered as the best combination. Result shown that the past value 1:10 and number of neurons of 10 when the sample size is 100. At sample size 500, past values of 1:10 and number of neurons of 8 enables the model to perform at its best. Whereas for sample size 900, network setting of 1:5 past value and 5 hidden neurons gives the least MSE. Multi-step ahead time series forecasting is conducted to forecast the number of COVID-19 confirmed cases in 7 days which is from 22 to 28 September 2022. The result shown for 7-days-ahead confirmed cases forecasting Malaysia datasets, the best forecasting outcome occurs when 900 samples are inputted.

Short-Term Wind Speed Forecasting Using Nonlinear Autoregressive Neural Network: A Case Study in Kocaeli-Türkiye

Recently, wind energy has been utilized globally as a renewable, sustainable, and eco-friendly energy source. However, wind energy’s unpredictable and stochastic nature influences its entry into the national electrical grid. An effective wind speed prediction is required to meet these challenges. In this article, the Nonlinear Autoregressive Neural Network (NARNN) model is used and investigated for short-term wind speed forecasting by taking a dataset from the Kandıra wind farm in Kocaeli- Türkiye. The crux of the paper is to improve the actual application of the existing NARNN model with factual data using a different number of neurons of the hidden layer, delays, and training functions in the learning phase called the model’s hyperparameters. The mean squared error (MSE) and determination coefficient (R2) are used as performance measures. As a result, the hyperparameter optimization for wind speed prediction using the NARNN increased the forecasting performance. Suggested NARNN model is compared with its exogenous version (NARXNN) using three extra inputs. It is observed that NARNN is not falling behind NARXNN because they provide close results, and NARNN has been shorter to run. Likewise, the learning algorithms were also compared, and it turned out that Bayesian Regularization (BR) is the best learning algorithm. Still, Levenberg Marquardt (LM) algorithm is much faster to execute and provides close results to BR.

China mainland new energy index price forecasting with the neural network

For policymakers and investors, forecasting prices of energy indices has always been an important task. The present work focuses on the Chinese market and explores the daily price forecasting problem for the new energy index in the mainland during the period spanning January 4, 2016 – December 31, 2020. Our analysis is facilitated through the nonlinear autoregressive neural network model and one hundred and twenty model settings are tested in the fields of the algorithm for training the model, the number of hidden neurons utilized, the number of delays utilized, and the ratio utilized for segmenting the price series into different phases. Analysis here leads to the construction of a rather simple model based upon four delays and two hidden neurons with the Levenberg-Marquardt algorithm for training, which generates accurate and stable forecast results, with relative root mean square errors below 1.80% and mean absolute percentage errors below 1.30% across training, validation, and testing phases and for the overall sample. This constructed model also leads to statistically significantly better forecasting performance than a linear autoregressive model at the 1% significance level based upon the modified Diebold-Mariano test. The model built could be used as part of policy analysis for policymakers and decision making for investors. The forecasting results might also benefit design of similar energy indices by offering reference information in terms of price paths projected through the model.

Integrating singular spectrum analysis and nonlinear autoregressive neural network for stock price forecasting

<span>The main objective of stock market investors is to maximize their gains. As a result, stock price forecasting has not lost interest in recent decades. Nevertheless, stock prices are influenced by news, rumor, and various economic factors. Moreover, the characteristics of specific stock markets can differ significantly between countries and regions, based on size, liquidity, and regulations. Accordingly, it is difficult to predict stock prices that are volatile and noisy. This paper presents a hybrid model combining singular spectrum analysis (SSA) and nonlinear autoregressive neural network (NARNN) to forecast close prices of stocks. The model starts by applying the SSA to decompose the price series into various components. Each component is then used to train a NARNN for future price forecasting. In comparison to the autoregressive integrated moving average (ARIMA) and NARNN models, the SSA-NARNN model performs better, demonstrating the effectiveness of SSA in extracting hidden information and reducing the noise of price series.</span>

Prognostics for remaining useful life estimation in proton exchange membrane fuel cell by dynamic recurrent neural networks

This paper proposes a nonlinear dynamic recurrent neural network (DRNN) prognostics method for predicting the performance degradation trend and estimating the remaining useful life (RUL) of a proton exchange membrane fuel cell (PEMFC). The conducted DRNN prognostic methods are based on a nonlinear autoregressive neural network (NARNN) model and nonlinear autoregressive with exogenous inputs neural network (NARXNN) model, with the goal of taking actions to extend the life of the PEMFC. To effectively remove data noise or spikes and provide smooth reconstruction, a locally weighted regression (LWR) method is used for pre-processing. The DRNN model parameters for the time lags are processed by an autocorrelation function (ACF). For comparing the predictions within the different DRNN models, the data sets are divided into three different prognostic configurations for predicting and estimating the remaining life of the PEMFC. The results show that the NARXNN model has the best predictions for the degradation trend and accurate remaining useful life, i.e. better than the NARNN model. Finally, the prognostic capability of the proposed DRNN method is compared with those in the literature.

Application of Nonlinear Autoregressive Neural Network Model to Forecast Local Mean Sea Level

The primary purpose of this study was to apply the nonlinear autoregressive neural network to model the long-term records of the monthly mean sea level from January 1978 to October 2020 at Grand Isle, Louisiana, as extracted from the National Oceanic and Atmospheric Administration Tides and Currents database. In this study, the empirical results revealed that the Bayesian Regularization algorithm was the best-suit training algorithm for its high regression R-value and low mean square error compared to the Levenberg-Marquardt and Scaled Conjugate Gradient algorithms for the nonlinear autoregressive neural network. Understanding past sea levels is important for the analysis of current and future sea level changes. In order to sustain these observations, research programs utilizing the existing data should be able to improve our understanding and significantly narrow our projections of ffuture sea-level changes.

Thermal coal price forecasting via the neural network

• We build a non-linear auto-regressive neural network model for forecasts of thermal coal prices. • Closing prices recorded on a daily basis of thermal coal traded in China Zhengzhou Commodity Exchange are used for analysis. • The model is simple and leads to performance of good accuracy and stabilities. • The model could serve as technical forecast tools and for policy analysis. Thermal coal price forecasts represent an essential issue to investors and policy makers, given its importance as a strategic energy source. The current work aims at exploring usefulness of non-linear auto-regressive neural networks for this forecast problem based upon a data-set of closing prices recorded on a daily basis of thermal coal traded in China Zhengzhou Commodity Exchange during January 4, 2016 – December 31, 2020, which is an important financial index not sufficiently explored in the literature in terms of its price forecasts. Through testing a variety of model settings over algorithms, delays, hidden neurons, and data splitting ratios, the model that produces performance of good accuracy and stabilities is reached. Particularly, the model has five delays and ten hidden neurons and is constructed with the Levenberg-Marquardt algorithm based on the ratio of 80%–10%–10% of the data for training–validation–testing. It leads to relative root mean square errors of 1.48%, 1.49%, and 1.47% for the training, validation, and testing phases, respectively. Usefulness of neural networks for the price forecast issue of thermal coal is demonstrated. Forecast results here could serve as standalone technical forecasts and be combined with other forecasts when conducting policy analysis that involves forming perspectives of trends in prices.

Time Series forecasting global price of bananas using Hybrid ARIMA-NARNN model

<abstract> <p>This study tried to demonstrate the role of Time Series models in a modeling and forecasting process using publicly available long-term records of monthly global price of bananas during the period of January 1990 to November 2021 reported in the International Monetary Fund. Following the Box–Jenkins methodology, an ARIMA (2,1,4) with a drift model was selected as the best-fit model for the Time Series, according to its lowest AIC value. Using the Levenberg-Marquardt algorithm, the results revealed that the NARNN model with 12 neurons in the hidden layer and 6 times delays provided the best performance in the nonlinear autoregressive neural network models at its smaller MSE value. The ARIMA and NARNN models are good at modelling linear and nonlinear problems for the Time Series, respectively. However, using the HYBRID model, a combination of the ARIMA and NARNN models that has both linear and nonlinear modeling capabilities can be a better choice for modeling the Time Series. The comparative results revealed that the HYBRID model with 11 neurons in the hidden layer and 3 times delays yielded higher accuracy than the NARNN model with 12 neurons in the hidden layer and 6 times delays, and the ARIMA (2,1,4) with a drift model, according to its lowest MSE in this study.</p> </abstract>

Predviđanje brzine vetra zasnovano na dugotrajnoj memoriji primenom nelinearne autoregresivne neuronske mreže

Globally, wind power is a technologically matured and commercially accepted technology. However, intermittent and fluctuating wind speed makes it difficult to connect it directly to the grid. It becomes less attractive from the quality and continuous power supply point of view. Nevertheless, the wind speed is affected by meteorological parameters like temperature, pressure, and relative humidity and may be predicted better using all of these parameters or some of the theses as inputs. Since the weather conditions of a particular month repeat approximately after ten years and sometimes even year to year depending on geographical location. This study investigates the errors associated with predicting the wind speed of a particular calendar month using the historical data of the same calendar month in the previous years. Authors propose a strategy for long-term prediction of wind speed based on two nonlinear autoregressive neural network models, (i) nonlinear autoregressive neural network and (ii) nonlinear autoregressive neural networks with exogenous inputs. The models are developed by training the networks with hourly mean wind speed values for seven years, from 2011 to 2017, for three sites in the Eastern Province of Saudi Arabia. These models are used to predict the wind speed for 2018, and the results are compared with the measured data. Both models' effectiveness is evaluated by considering the impact of the exogenous parameters (temperature and atmospheric pressure). The study found that the prediction accuracy of wind speed in long-term forecasting depends not only on the location but also on the repeatability of training samples across the years.

State of Health Estimation Method for Lithium-Ion Batteries Based on Nonlinear Autoregressive Neural Network Model With Exogenous Input

Abstract The working state of lithium-ion batteries must be estimated accurately and efficiently in the battery management system. Building a model is the most prevalent way of predicting the battery's working state. Based on the variable order equivalent circuit model, this article examines the attenuation curve of battery capacity with the number of cycles. It identifies the order of the equivalent circuit model using Bayesian information criterion (BIC). Based on the correlation between capacity and resistance, this article concludes that there is a nonlinear correlation between model parameters and state of health (SOH). The nonlinear autoregressive neural network with exogenous input (NARX) is used to fit the nonlinear correlation for capacity regeneration. Then, the self-adaptive weight particle swarm optimization (SWPSO) method is suggested to train the neural network. Finally, single-battery and multibattery tests are planned to validate the accuracy of the SWPSO-NARX estimate of SOH. The experimental findings indicate that the SOH estimate effect is significant.

Automatic time series forecasting using nonlinear autoregressive neural network model with exogenous input

This study aims to determine an automatic forecasting method of univariate time series, using the nonlinear autoregressive neural network model with exogenous input (NARX). In this automatic setting, users only need to supply the input of time series. Then, an automatic forecasting algorithm sets up the appropriate features, estimate the parameters in the model, and calculate forecasts, without the users’ intervention. The algorithm method used include preprocessing, tests for trends, and the application of first differences. The time series were tested for seasonality, and seasonal differences were obtained from a successful analysis. These series were also linearly scaled to [−1, +1]. The autoregressive lags and hidden neurons were further selected through the stepwise and optimization algorithms, respectively. The 20 NARX models were fitted with different random starting weights, and the forecasts were combined using the ensemble operator, in order to obtain the final product. This proposed method was applied to real data, and its performance was compared with several available automatic models in the literature. The forecasting accuracy was also measured by mean squared error (MSE) and mean absolute percent error (MAPE), and the results showed that the proposed method outperformed the other automatic models.

A hybrid autoregressive fractionally integrated moving average and nonlinear autoregressive neural network model for short-term traffic flow prediction

Intelligent traffic control and guidance system is an effective way to solve urban traffic congestion, improve road capacity and guarantee drivers' travel safety, while short-term traffic flow prediction is the core of intelligent traffic control and guidance system. To investigate the long-term memory and the dynamic feature of short-time traffic flow time series, a hybrid model was proposed by integrating autoregressive fractionally integrated moving average (ARFIMA) model and nonlinear autoregressive (NAR) neural network model to predict short-time traffic flow, in which ARFIMA model can address the long-term memory of linear component and NAR neural network can accommodate the dynamic feature of nonlinear residual component. First, the ARFIMA model was employed to predict the linear component of traffic flow, and the results were compared with those of autoregressive integrated moving average (ARIMA) model. Next, the NAR neural network model was adopted to forecast the nonlinear residual components, and the weighted results were considered as the predicted flow of the hybrid model. The proposed hybrid model was validated by using the cross-sectional traffic flow data in California freeways obtained from the open-access PeMS database. The results showed that the ARFIMA model considering the long-term memory can effectively predict the short-term traffic flow, and the prediction accuracy of the hybrid model is better than that of the singular models. The findings provide an alternative for the short-term traffic flow prediction with lower error and higher accuracy.

Short-term prediction of COVID-19 spread using grey rolling model optimized by particle swarm optimization

The prediction of the spread of coronavirus disease 2019 (COVID-19) is vital in taking preventive and control measures to reduce human health damage. The Grey Modelling (1,1) is a popular approach used to construct a predictive model with a small-sized dataset.​ In this study, a hybrid model based on grey prediction and rolling mechanism optimized by particle swarm optimization algorithm (PSO) was applied to create short-term estimates of the total number of confirmed COVID-19 cases for three countries, Germany, Turkey, and the USA. A rolling mechanism that updates data in equal dimensions was applied to improve the forecasting accuracy of the models. The PSO algorithm was used to optimize the Grey Modelling parameters (1,1) to provide more robust and efficient solutions with minimum errors. To compare the accuracy of the predictive models, a nonlinear autoregressive neural network (NARNN) was also developed. According to the analysis results, Grey Rolling Modelling (1,1) optimized by PSO algorithm performs better than the classical Grey Modelling (1,1), Grey Rolling Modelling (1,1), and NARNN models for predicting the total number of confirmed COVID-19 cases. The present study can provide an important basis for countries to allocate health resources and formulate epidemic prevention policies effectively.

Modeling and Forecasting Long-Term Records of Mean Sea Level at Grand Isle, Louisiana: SARIMA, NARNN, and Mixed SARIMA-NARNN Models

This study tried to demonstrate the role of time series models in modeling and forecasting process using long-term records of monthly mean sea level from January 1978 to October 2020 at Grand Isle, Louisiana. Following the Box–Jenkins methodology, the ARIMA(1,1,1)(2,0,0)12 with drift model was selected to be the best fit model for the time series, according to its lowest AIC value. Using the LM algorithm, the results revealed that the NARNN model with 9 neurons in the hidden layer and 6 time delays provided the best performance in the nonlinear autoregressive neural network models at its smaller MSE value. The Mixed model, a combination of the SARIMA and NARNN models has both linear and nonlinear modelling capabilities can be a better choice for modelling the time series. The comparative results revealed that the Mixed-LM model with 9 neurons in the hidden layer and 3 time delays yielded higher accuracy than the NARNN-LM model with 9 neurons in the hidden layer and 6 time delays, and the ARIMA(1,1,1)(2,0,0)12 with drift model, according to its lowest MSE in this study. Thus, this study may provide an integrated modelling approach as a decision-making supportive method for formulating local mean sea level forecast in advance. Understanding past sea level is important for the analysis of current and future sea level changes. In order to sustain these observations, research programs utilizing the resulting data should be able to improve significantly our understanding and narrow projections of future sea level rise and variability.

Application of Nonlinear Autoregressive Neural Network to Model and Forecast Time Series Global Price of Bananas

The primary purpose of this study was to apply the nonlinear autoregressive neural network to model the long-term records of monthly global price of bananas from January 1990 to November 2020. The development of the optimal architecture for the nonlinear autoregressive neural network requires determination of time delays, the number of hidden neurons, and an efficient training algorithm. Through training of the nonlinear autoregressive neural network models, the prediction performance of the models was evaluated by its mean squared error value, the average squared difference between the observed and predicted values. In this study, the empirical results revealed that the NAR-BR model with 13 neurons in the hidden layer and 6 time delays provided the best performance at its smaller mean squared error value and yielded higher accuracy than the NAR-LM model with 12 neurons in the hidden layer and 4 time delays and NAR-SCG model with 12 neurons in the hidden layer and 6 time delays. Understanding past global price of bananas is important for the analyses of current and future global price of bananas changes. In order to sustain these observations, research programs utilizing the resulting data should be able to improve our understanding and narrow projections of future global price of bananas significantly.

Blood Glucose Level Prediction of Diabetic Type 1 Patients Using Nonlinear Autoregressive Neural Networks

Diabetes type 1 is a chronic disease which is increasing at an alarming rate throughout the world. Studies reveal that the complications associated with diabetes can be reduced by proper management of the disease by continuously monitoring and forecasting the blood glucose level of patients. Objective. The prior prediction of blood glucose level is necessary to overcome the lag time for insulin absorption in diabetic type 1 patients. Method. In this research, we use continuous glucose monitoring (CGM) data to predict future blood glucose level using the previous data points. We compare two neural network techniques. We apply the optimal feedforward neural network and then propose optimal nonlinear autoregressive neural networks for blood glucose prediction 15–30 minutes earlier for diabetic type 1 patients. We validate the proposed model with 2 virtual subjects using their 24-hour blood glucose level data. These two case studies have been compiled from AIDA, i.e., the freeware mathematical diabetes simulator. Results. In the prediction horizon (PH) of 15 and 30 minutes, improved results have been shown for minimal inputs for blood glucose level of a particular subject. Root mean square error (RMSE) is used for performance calculation. For the optimal feedforward neural network, the RMSE is 0.9984 and 3.78 ml/dl, and for the optimal nonlinear autoregressive neural network, it reduces the RMSE to 0.60 and 1.12 ml/dl for 15 min and 30 min prediction horizons, respectively, for subject 1. Similarly, for subject 2 for the optimal feedforward neural network, RMSE is 1.43 and 3.51 ml/dl which is improved using the optimal autoregressive neural network to 0.7911 and 1.6756 ml/dl for 15 min and 30 min prediction horizons, respectively. Validation. We further validate our proposed model using UCI machine learning datasets (Abalone and Servo), and it shows improved results on that as well. Conclusion and Future Work. The proposed optimal nonlinear autoregressive neural network model performs better than the feedforward neural network model for these time series data. In the future, we intend to investigate a greater collection of AIDA scenarios and data that are real and influence other factors of BGLs.