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Streamflow prediction is an important matter for the water resources management and the design of hydraulic structures that can be built on rivers. Recently, it has become a widely studied research field where data obtained from stream gauge stations can be utilized for creating estimating models by resorting to different methods such as machine and deep learning techniques. In this study, we performed monthly streamflow predictions by using the following data-driven methods of machine learning: linear regression, support vector regression, random forest and deep learning (DL) models to compare the performances of ML's and DL's techniques. A general workflow that can be applied to similar regions is presented. An estimating model containing six-input combinations and time-lagged streamflow data is improved by means of the autocorrelation function (ACF) and partial autocorrelation function (PACF). Furthermore, moving average is used as a smoothing technique to make the dataset more stable and reduce the effects of noise data. A comparative evaluation has been conducted to determine the performances of the above-mentioned methods. In this study, we proposed four different DL models and compared them with existing techniques. For the comparison of the results, we used evaluation criteria such as Nash–Sutcliffe efficiency (NSE), mean square error (MSE) and percent bias (PBIAS). The experimental results indicate that our bidirectional gated recurrent units (BiGRU) model outperforms both ML algorithms and existing solutions with 0.971 NSE, 0.001 MSE and − 1.536 PBIAS scores.

Accurate forecasting of carbon price is important and fundamental for anticipating the changing trends of the energy market, and, thus, to provide a valid reference for establishing power industry policy. However, carbon price forecasting is complicated owing to the nonlinear and non-stationary characteristics of carbon prices. In this paper, a combined forecasting model based on variational mode decomposition (VMD) and spiking neural networks (SNNs) is proposed. An original carbon price series is firstly decomposed into a series of relatively stable components through VMD to simplify the interference and coupling across characteristic information of different scales in the data. Then, a SNN forecasting model is built for each component, and the partial autocorrelation function (PACF) is used to determine the input variables for each SNN model. The final forecasting result for the original carbon price can be obtained by aggregating the forecasting results of all the components. Actual InterContinental Exchange (ICE) carbon price data is used for simulation, and comprehensive evaluation criteria are proposed for quantitative error evaluation. Simulation results and analysis suggest that the proposed VMD-SNN forecasting model outperforms conventional models in terms of forecasting accuracy and reliability.

In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, i.e., which also fulfil the Yule–Walker equations and thus provide the same PACF characterization as AR models, increased a lot. This motivates the use of the PACF test also for such count processes. By computing the sample PACF based on the raw data or the Pearson residuals, respectively, findings are usually evaluated based on well-known asymptotic results. However, the conditions for these asymptotics are generally not fulfilled for AR-type count processes, which deteriorates the performance of the PACF test in such cases. Thus, we present different implementations of the PACF test for AR-type count processes, which rely on several bootstrap schemes for count times series. We compare them in simulations with the asymptotic results, and we illustrate them with an application to a real-world data example.

Daily maximum temperature of four different regions in Kerala, India, from 01/01/2019 to 31/12/2020, is recorded and is used for modelling and forecasting. The forecasting methods used are Autoregressive integrated moving average (ARIMA), Seasonal Autoregressive integrated moving average (SARIMA) and Autoregressive fractional integrated moving average (ARFIMA). The comparison of forecasting performance was based on percentage accuracy, mean squared error (MSE) and mean absolute error (MAE). The models used can capture the variations of time series data. All the models exhibit reasonably good performance in predicting the daily maximum temperature. ARFIMA model gives the least forecast errors compared to other models.

Accurate monitoring and forecasting of drought are crucial. They play a vital role in the optimal functioning of irrigation systems, risk management, drought readiness, and alleviation. In this work, Artificial Intelligence (AI) models, comprising Multi-layer Perceptron Neural Network (MLPNN) and Co-Active Neuro-Fuzzy Inference System (CANFIS), and regression, model including Multiple Linear Regression (MLR), were investigated for multi-scalar Standardized Precipitation Index (SPI) prediction in the Garhwal region of Uttarakhand State, India. The SPI was computed on six different scales, i.e., 1-, 3-, 6-, 9-, 12-, and 24-month, by deploying monthly rainfall information of available years. The significant lags as inputs for the MLPNN, CANFIS, and MLR models were obtained by utilizing Partial Autocorrelation Function (PACF) with a significant level equal to 5% for SPI-1, SPI-3, SPI-6, SPI-9, SPI-12, and SPI-24. The predicted multi-scalar SPI values utilizing the MLPNN, CANFIS, and MLR models were compared with calculated SPI of multi-time scales through different performance evaluation indicators and visual interpretation. The appraisals of results indicated that CANFIS performance was more reliable for drought prediction at Dehradun (3-, 6-, 9-, and 12-month scales), Chamoli and Tehri Garhwal (1-, 3-, 6-, 9-, and 12-month scales), Haridwar and Pauri Garhwal (1-, 3-, 6-, and 9-month scales), Rudraprayag (1-, 3-, and 6-month scales), and Uttarkashi (3-month scale) stations. The MLPNN model was best at Dehradun (1- and 24- month scales), Tehri Garhwal and Chamoli (24-month scale), Haridwar (12- and 24-month scales), Pauri Garhwal (12-month scale), Rudraprayag (9-, 12-, and 24-month), and Uttarkashi (1- and 6-month scales) stations, while the MLR model was found to be optimal at Pauri Garhwal (24-month scale) and Uttarkashi (9-, 12-, and 24-month scales) stations. Furthermore, the modeling approach can foster a straightforward and trustworthy expert intelligent mechanism for projecting multi-scalar SPI and decision making for remedial arrangements to tackle meteorological drought at the stations under study.

The objective of this paper is to assess the distribution of the Partial Autocorrelation Function (PACF), both theoretically and empirically, emphasizing its crucial role in modeling and forecasting time series data. Additionally, it evaluates the deviation of the sum of sample PACF from normality: identifying the lag at which departure occurs. Our investigation reveals that the sum of the sample PACF, and consequently its components, diverges from the expected normal distribution beyond a certain lag. This observation challenges conventional assumptions in time series modeling and forecasting, indicating a necessity for reassessment of existing methodologies. Through our analysis, we illustrate the practical implications of our findings using real-world scenarios, highlighting their significance in unraveling complex data patterns. This study delves into 185 years of monthly Bank of England Rate data, utilizing this extensive dataset to conduct an empirical analysis. Furthermore, our research paves the way for future exploration, offering insights into the complexities and potential revisions in time series analysis, modeling, and forecasting.

The COVID-19 pandemic has profoundly impacted economic and social structures, directly affecting individuals’ lives. Deep learning models offer the potential to forecast future long-term trends and capture the temporal dependencies present in time series data. In this study, we propose leveraging the autocorrelation function (ACF) and the partial autocorrelation function (PACF) series as additional components to enhance the forecasting accuracy of our models. Our proposed method is applied to forecast COVID-19 time series data in twelve countries using the deep learning techniques of Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRU). When comparing the rankings average of mean absolute error and R-squared, the proposed models demonstrated superior performance in time series forecasting compared to the standard LSTM and GRU model. Specifically, the ACF-PACF-GRU model achieved the best median values for mean absolute percentage error (1.67 per cent for confirmed cases and 2.17 per cent for death cases) and root mean square error (1.92 for confirmed cases and 2.17 for death cases). Therefore, the proposed ACF-PACF-GRU model showed the highest performance in forecasting both confirmed and death cases. This research introduces a novel method for constructing effective time series models aimed at forecasting disease burdens, thereby aiding in epidemic control and the implementation of preventive measures.

PurposeMeteorological disasters pose a significant risk to people’s lives and safety, and accurate prediction of weather-related disaster losses is crucial for bolstering disaster prevention and mitigation capabilities and for addressing the challenges posed by climate change. Based on the uncertainty of meteorological disaster sequences, the damping accumulated autoregressive GM(1,1) model (DAARGM(1,1)) is proposed.Design/methodology/approachFirstly, the autoregressive terms of system characteristics are added to the damping-accumulated GM(1,1) model, and the partial autocorrelation function (PACF) is used to determine the order of the autoregressive terms. In addition, the optimal damping parameters are determined by the optimization algorithm.FindingsThe properties of the model were analyzed in terms of the stability of the model solution and the error of the restored value. By fitting and predicting the losses affected by meteorological disasters and comparing them with the results of four other grey models, the validity of the new model in fitting and prediction was verified.Originality/valueThe dynamic damping trend factor is introduced into the grey generation operator so that the grey model can flexibly adjust the accumulative order of the sequence. On the basis of the damping accumulated grey model, the autoregressive term of the system characteristics is introduced to take into account the influence of the previous data, which is more descriptive of the development trend of the time series itself and increases the effectiveness of the model.

With the development of the carbon market in China, research on the carbon price has received more and more attention in related fields. However, due to its nonlinearity and instability, the carbon price is undoubtedly difficult to predict using a single model. This paper proposes a new hybrid model for carbon price forecasting that combines fast ensemble empirical mode decomposition, sample entropy, phase space reconstruction, a partial autocorrelation function, and an extreme learning machine that has been improved by particle swarm optimization. The original carbon price series is decomposed using the fast ensemble empirical mode decomposition and sample entropy methods, which eliminate noise interference. Then, the phase space reconstruction and partial autocorrelation function methods are combined to determine the input and output variables in the forecasting models. An extreme learning machine optimized by particle swarm optimization was employed to forecast carbon prices. An empirical study based on carbon prices in three typical regional carbon markets in China found that this new hybrid model performed better than other comparable models.

Accurate wind speed forecasting is a fundamental element of wind power prediction. Thus, a new hybrid wind speed forecasting model, using variational mode decomposition (VMD), the partial autocorrelation function (PACF), and weighted regularized extreme learning machine (WRELM), is proposed to improve the accuracy of wind speed forecasting. First, the historic wind speed time series is decomposed into several intrinsic mode functions (IMFs). Second, the partial correlation of each IMF sequence is analyzed using PACF to select the optimal subfeature set for particular predictors of each IMF. Then, the predictors of each IMF are constructed in order to enhance its strength using WRELM. Finally, wind speed is obtained by adding up all the predictors. The experiment, using real wind speed data, verified the effectiveness and advancement of the new approach.