Raw Series Research Articles

Refined offshore wind speed prediction: Leveraging a two-layer decomposition technique, gated recurrent unit, and kernel density estimation for precise point and interval forecasts

Precise forecasting of offshore wind speeds is paramount for various applications, including offshore wind energy production, disaster prevention and mitigation, and maritime navigation. This study introduces a novel model for point and interval prediction of offshore wind speed, incorporating an innovative two-layer decomposition technique, gated recurrent unit (GRU), and kernel density estimation (KDE). Initially, the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) is utilized to decompose the raw series into multiple subsequences, thus alleviating the inherent data nonstationarity. Subsequently, we employ the refined composite multiscale fuzzy entropy (RCMFE) to reconstruct the subsequences into multiple components, thereby reducing the computational complexity. Variational mode decomposition (VMD) is utilized to enhance the modeling accuracy to decompose the high-frequency component, thereby generating a set of submodels. The GRU, a specialized recurrent neural network architecture, is leveraged to forecast the remaining components and submodels, producing a series of subresults. The subresults are linearly summed to obtain the final point prediction result. Finally, based on point prediction, employing KDE to gauge the probability density distribution of prediction errors culminates in the derivation of the offshore wind speed prediction interval. The experimental results substantiate the superior predictive performance of the proposed prediction model.

Study of the River Discharge Alteration

This article aims to analyze the alteration in water discharge due to the building of one of the largest dams in Romania. Modifications in the hydrological patterns of the studied river were emphasized by a complex technique that includes decomposition models of the series into trends, seasonal indices, and random components, as well as into Intrinsic Mode Functions (IMFs). The Mann–Kendall trend test indicates the existence of different positive slopes for the subseries S1 and S2 (before and after the inception of the Siriu dam, respectively) built from the raw series, S. The stationarity hypothesis was rejected for all series. The multifractal analysis shows two different patterns of the data series. After decomposing the subseries S1 and S2, it resulted that the seasonality indices are not the same. Moreover, the seasonal variations decreased after building the dam. Empirical Mode Decomposition (EMD) unveils different short- and long-term patterns of the series before and after building the dam, concluding that there is a significant alteration in the river discharge after the dam’s inception.

Estimation of the respiratory mortality attributable to comprehensive effects of air pollutants in China using an enhanced seasonal and self-adaptive multivariate grey convolution model

Exposure to air pollutants has become a major global health threat and is a dominant contributor to increasing respiratory mortality. Accurate projections of respiratory mortality are crucial to mitigating the adverse influences of air pollution on public health and guiding long-term policy implementation. However, the forecasting models' accuracy is difficult to guarantee because seasonal factors perturb the raw series of air pollutants and respiratory mortality. This paper proposes an enhanced seasonal and self-adaptive multivariate grey convolution model to solve this dilemma. The innovations of the proposed model reside in the following areas: firstly, a seasonal operator vector of multidimensional data is initially built to digitize the seasonality of air pollutants and related mortality. Secondly, a multivariate grey convolution model with an adaptive time-lag cumulative optimization item is put forward. The weight coefficients of each driving variable at different periods are intended to adapt dynamically. Thirdly, the self-adaptive time-delay coefficient is determined by the Particle Swarm Algorithm, enhancing the accuracy of predictions. Theoretically, the generalizability and adaptability of the newly presented model are well-validated. Moreover, its practical applicability and higher forecasting performance are further verified in estimating the respiratory mortality attributable to the comprehensive effects of air pollutants in China.

Modelling the dynamics of cryptocurrency prices for risk hedging: The case of Bitcoin, Ethereum, and Litecoin

Cryptocurrencies have, over the years, gained an unprecedented prominence in financial discourse, with the market fielding over 5,300 digital currencies and reaching over $2 trillion in market capitalisation in 2022. The surge in market values of digital currencies and their popularity in the world of e-commerce have remained unabated and equally received special attention from researchers focusing on identifying the underlying factors that drive changes in their market values. Thus, this study models the dynamics of the prices of cryptocurrencies alongside their interconnectedness, focusing on Bitcoin, Ethereum, and Litecoin along the time and frequency dimensions of monthly data from 1 March 2016 to 05/31/2022. Based on the ARDL model, results show that the volume of transactions of Bitcoin, Ethereum, and Litecoin, oil prices, and gold prices exert a more significant positive influence on their prices in the longrun than in the shortrun. However, the publicity of the selected cryptocurrencies (google search rates) does not significantly influence their prices. Interestingly, results from the Wavelet Granger causality tests show no causality between the raw series of Bitcoin, Ethereum, and Litecoin prices. However, a bi-directional causality exists between Bitcoin and Ethereum prices during the longrun in their low frequencies, a unidirectional causality running from Bitcoin to Litecoin prices during the longrun in their low frequencies, and a unidirectional causality running from Litecoin to Ethereum prices during the shortrun, medium run and longrun in their high, medium, and low frequencies. These findings have profound implications for the global financial market and investor decisions.

Can Twitter-based economic uncertainty predict safe-haven assets under all market conditions and investment horizons?

This paper examines the Granger causality from Twitter-based economic uncertainty (TEU) to three safe-haven assets – Bitcoin, gold, and US10 year Treasury notes. Using daily data (June 1, 2011–August 30, 2021) and causality-in-quantiles and wavelets methods, the results indicate variability in the causality between the mean and variance, as well as the market conditions. TEU Granger-causes the returns and volatility of Treasuries, the volatility but not returns of Bitcoin, and neither the volatility nor the returns of gold for the raw series, and the causality is mostly significant at low and middle quantiles for Bitcoin and Treasuries. We include other risk factors and confirm the variability in the causality. Considering the possibility of a hidden causality over various frequency domains due to investors' heterogeneous expectations and perceptions of risk, the wavelet transforms-based causality tests reveal an increase in the predictability of risk indicators under specific investment horizons and market conditions. During the pandemic, TEU strongly predicts future volatility of Treasury and Bitcoin returns, reflecting the importance of social-media posts for safe-haven pricing. These findings highlight the benefits of applying the causality-in-quantiles test to decomposed series to determine the contribution of each scale to the causality over various market conditions.

Forecasting equity risk premium: A new method based on wavelet de‐noising

AbstractForecasting equity risk premium is notoriously difficult due to a mass of data noise in the raw series and the absence of clear tendency. Using the monthly S&P 500 excess returns from 1927:01 to 2018:12, we first de‐noise the in‐sample original returns series via wavelet method to capture the basic trend of equity risk premium, and then propose forecasting models to obtain one‐step forward out‐of‐sample predicted values based on the de‐noised returns. Our new models can provide substantially superior out‐of‐sample performance compared with other competing models and the historical average. Sizeable economic gains can be realized by a mean–variance investor if they allocate their asset through the new approach. Our findings are robust under different settings from both statistical and economic perspectives.

HomogWS-se: a century-long homogenized dataset of near-surface wind speed observations since 1925 rescued in Sweden

Abstract. Creating a century-long homogenized near-surface wind speed observation dataset is essential to improve our current knowledge about the uncertainty and causes of wind speed stilling and recovery. Here, we rescued paper-based records of wind speed measurements dating back to the 1920s at 13 stations in Sweden and established a four-step homogenization procedure to generate the first 10-member centennial homogenized wind speed dataset (HomogWS-se) for community use. Results show that about 38 % of the detected change points were confirmed by the known metadata events, and the average segment length split by the change points is ∼11.3 years. Compared with the raw wind speed series, the homogenized series is more continuous and lacks significant non-climatic jumps. The homogenized series presents an initial wind speed stilling and subsequent recovery until the 1990s, whereas the raw series fluctuates with no clear trend before the 1970s. The homogenized series shows a 25 % reduction in the wind speed stilling during 1990–2005 than the raw series, and this reduction is significant when considering the homogenization uncertainty. The homogenized wind speed series exhibits a significantly stronger correlation with the North Atlantic oscillation index than that of the raw series (0.54 vs. 0.29). These results highlight the importance of the century-long homogenized series in increasing our ability to detect and attribute multidecadal variability and changes in wind speed. The proposed homogenization procedure enables other countries or regions to rescue their early climate data and jointly build global long-term high-quality datasets. HomogWS-se is publicly available from the Zenodo repository at https://doi.org/10.5281/zenodo.5850264 (Zhou et al., 2022).

Influence of Anomalies on the Models for Nitrogen Oxides and Ozone Series

Nowadays, observing, recording, and modeling the dynamics of atmospheric pollutants represent actual study areas given the effects of pollution on the population and ecosystems. The existence of aberrant values may influence reports on air quality when they are based on average values over a period. This may also influence the quality of models, which are further used in forecasting. Therefore, correct data collection and analysis is necessary before modeling. This study aimed to detect aberrant values in a nitrogen oxide concentration series recorded in the interval 1 January–8 June 2016 in Timisoara, Romania, and retrieved from the official reports of the National Network for Monitoring the Air Quality, Romania. Four methods were utilized, including the interquartile range (IQR), isolation forest, local outlier factor (LOF) methods, and the generalized extreme studentized deviate (GESD) test. Autoregressive integrated moving average (ARIMA), Generalized Regression Neural Networks (GRNN), and hybrid ARIMA-GRNN models were built for the series before and after the removal of aberrant values. The results show that the first approach provided a good model (from a statistical viewpoint) for the series after the anomalies removal. The best model was obtained by the hybrid ARIMA-GRNN. For example, for the raw NO2 series, the ARIMA model was not statistically validated, whereas, for the series without outliers, the ARIMA(1,1,1) was validated. The GRNN model for the raw series was able to learn the data well: R2 = 76.135%, the correlation between the actual and predicted values (rap) was 0.8778, the mean standard errors (MSE) = 0.177, the mean absolute error MAE = 0.2839, and the mean absolute percentage error MAPE = 9.9786. Still, on the test set, the results were worse: MSE = 1.5101, MAE = 0.8175, rap = 0.4482. For the series without outliers, the model was able to learn the data in the training set better than for the raw series (R2 = 0.996), whereas, on the test set, the results were not very good (R2 = 0.473). The performances of the hybrid ARIMA–GRNN on the initial series were not satisfactory on the test (the pattern of the computed values was almost linear) but were very good on the series without outliers (the correlation between the predicted values on the test set was very close to 1). The same was true for the models built for O3.

Homogenising meteorological variables: Impact on trends and associated climate indices

Daily precipitation, maximum and minimum air temperature series are homogenised over the Lake Chad Basin between 1979 and 2020 using two conceptually different homogenisation methods; the Adapted Caussinus-Mestre Algorithm for homogenising Networks of Temperature series (ACMANT) and the iterated standard normal homogeneity (CLIMATOL). Results show the existence of unnatural breakpoints for most of the station series. However, the two methods show a general improvement in the quality of climate series. The trend estimation of the homogenised series based on the modified Mann Kendall methodology shows different modifications of the trend’s magnitude for different periods. Overall, CLIMATOL and ACMANT exhibit nearly similar trend patterns, suggesting the credibility of both homogenisation methods. Relative to the base period of analysis (1981–2010), the anomaly classification for the entire basin between 1979 and 2020 into dry-warm, wet-warm, dry-cold and wet-cold are misrepresented by the raw series compared to the homogenised series. Such erroneous representations resulting from inhomogeneities in raw climate series could misinform decisions akin to climate change assessment and water resources management strategies, thereby reducing the adaptive-capability of the basin’s inhabitants to climate change effects. Our study demonstrates the importance of robust homogenisation of climate series to mitigate inhomogeneity errors and improve the quality of information when observations are used in climate and hydrological studies.

PYME: an integrated platform for high-throughput nanoscopy

Single-molecule localization microscopy (SMLM) offers a ∼10-fold improvement in resolution over conventional, diffraction-limited fluorescence microscopy, which comes at the expense of acquisition time, data volume, and analysis overhead. For (d)STORM/(F)PALM, raw series typically consist of 10,000 to 100,000 frames acquired at 50 frames per second (FPS), meaning a day of manual imaging yields only on the order of ten fields of view. While efforts have been made to ease the tedium of acquisition by automating SMLM microscopes, the achieved throughputs are not dissimilar from manual imaging.

Improved CEEMDAN, GA, and SVR Model for Oil Price Forecasting.

Accurate prediction of crude oil prices (COPs) is a challenge for academia and industry. Therefore, the present research developed a new CEEMDAN-GA-SVR hybrid model to predict COPs, incorporating complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), a genetic algorithm (GA), and support vector regression machine (SVR). First, our team utilized CEEMDAN to realize the decomposition of a raw series of COPs into a group of comparatively simpler subseries. Second, SVR was utilized to predict values for every decomposed subseries separately. Owing to the intricate parametric settings of SVR, GA was employed to achieve the parametric optimisation of SVR during forecast. Then, our team assembled the forecasted values of the entire subseries as the forecasted values of the CEEMDAN-GA-SVR model. After a series of experiments and comparison of the results, we discovered that the CEEMDAN-GA-SVR model remarkably outperformed single and ensemble benchmark models, as displayed by a case study finished based on a time series of weekly Brent COPs.

Random Subspace Ensembles of Fully Convolutional Network for Time Series Classification

Time series classification (TSC) task is one of the most significant topics in data mining. Among all methods for this issue, the deep-learning-based shows superior performance for its good adaption to raw series data and automatic extraction of features. However, rare eyes are kept on composing ensembles of these superior individual classifiers to achieve further breakthroughs. The existing deep learning ensembles NNE did a heavy work of combining 60 individuals but did not maximize the deserving improvement, since it merely pays attention to the diversity of individuals but ignores their accuracy. In this paper, we propose to construct an ensemble of Full Convolutional Neural Networks (FCN) by Random Subspace Method (RSM), named RSM-FCN. FCN is a simple but outstanding individual classifier and RSM is suitable for high dimensional data such as time series, but there are few instances. Thus, the combination of these strengths, RSM-FCN provides a highly cost-effective approach to yield promising results. Experiments on the UCR dataset demonstrate the effectiveness and reasonability of the proposed method.

Detecting heavy rainfall using anomaly-based percentile thresholds of predictors derived from GNSS-PWV

Nowadays, tropospheric products obtained from the Global Navigation Satellite Systems (GNSS) observations, e.g., precipitable water vapor (PWV), have heralded a new era of GNSS meteorological applications, especially for the detection of heavy rainfall. In meteorological studies, the anomaly temporal series of an atmospheric variable is widely used to investigate the deviations of its raw series from a certain “normal” cycle, which is defined based on a specific purpose, e.g., its responses to a specific weather event. In this study, a new model for detecting heavy rainfall using anomaly-based percentile thresholds of seven predictors derived from PWV was established. The seven predictors, which can effectively reflect the complete picture of the variations in the PWV series prior to heavy rainfall events, include hourly PWV value and its six types of derivatives. The diurnal mean values and anomaly-based percentile thresholds for these predictors were obtained based on their raw time series over the 8-year period 2010–2017 at the co-located HKSC-KP stations. Then these values were applied to the sample data over the period 2018–2019 for determining their anomalies and series of abnormality. Finally, the detection results were compared with the hourly rainfall records for evaluation. Results showed that 97.6% of heavy rainfalls were correctly detected with an average lead time of 4.13 h. The seasonal false alarm rate of 13.4% from the new model was reduced in comparison to existing models. By conducting the verification experiments of the new model at another two pair of stations in the Hong Kong region, similar results were also obtained. These results all indicate that the anomaly-based percentile thresholds of predictors derived from PWV can be effectively applied to the detection of heavy rainfall events.

Estimation of fractional cycle bias for GPS/BDS-2/Galileo based on international GNSS monitoring and assessment system observations using the uncombined PPP model

The Fractional Cycle Bias (FCB) product is crucial for the Ambiguity Resolution (AR) in Precise Point Positioning (PPP). Different from the traditional method using the ionospheric-free ambiguity which is formed by the Wide Lane (WL) and Narrow Lane (NL) combinations, the uncombined PPP model is flexible and effective to generate the FCB products. This study presents the FCB estimation method based on the multi-Global Navigation Satellite System (GNSS) precise satellite orbit and clock corrections from the international GNSS Monitoring and Assessment System (iGMAS) observations using the uncombined PPP model. The dual-frequency raw ambiguities are combined by the integer coefficients (4,− 3) and (1,− 1) to directly estimate the FCBs. The details of FCB estimation are described with the Global Positioning System (GPS), BeiDou-2 Navigation Satellite System (BDS-2) and Galileo Navigation Satellite System (Galileo). For the estimated FCBs, the Root Mean Squares (RMSs) of the posterior residuals are smaller than 0.1 cycles, which indicates a high consistency for the float ambiguities. The stability of the WL FCBs series is better than 0.02 cycles for the three GNSS systems, while the STandard Deviation (STD) of the NL FCBs for BDS-2 is larger than 0.139 cycles. The combined FCBs have better stability than the raw series. With the multi-GNSS FCB products, the PPP AR for GPS/BDS-2/Galileo is demonstrated using the raw observations. For hourly static positioning results, the performance of the PPP AR with the three-system observations is improved by 42.6%, but only 13.1% for kinematic positioning results. The results indicate that precise and reliable positioning can be achieved with the PPP AR of GPS/BDS-2/Galileo, supported by multi-GNSS satellite orbit, clock, and FCB products based on iGMAS.

Two deep learning approaches to forecasting disaggregated freight flows: convolutional and encoder–decoder recurrent

Time series forecasting of disaggregated freight flow is a key issue in decision-making by port authorities. For this purpose and to test new deep learning techniques we have selected seven time series of imported goods from Morocco to Spain through the port of Algeciras, and we have tested two forecasting deep neural networks models: dilated causal convolutional and encoder–decoder recurrent. We have experimented with four different granularities for each series: quarterly, monthly, weekly and daily. The results show that our neural network models can manage these raw series without first removing seasonality or trend. We also highlight the ability of neural models to work with a fixed input size of one year, being able to make good predictions using the same input size for all granularities. The two deep learning models have globally improved the benchmarks of the M4 Competition of forecasting. Each neural network model obtains its best results under different circumstances: the recurrent one with daily granularity and intermittent series, and the convolutional one with weekly and monthly granularities.

Predictions and mitigation strategies of PM2.5 concentration in the Yangtze River Delta of China based on a novel nonlinear seasonal grey model.

High delicate particulate matter (PM2.5) concentration can seriously reduce air quality, destroy the environment, and even jeopardize human health. Accordingly, accurate prediction for PM2.5 plays a vital role in taking precautions against upcoming air ambient pollution incidents. However, due to the disturbance of seasonal and nonlinear characteristics in the raw series, pronounced forecasts are confronted with tremendous handicaps, even though for seasonal grey prediction models in the preceding researches. A novel seasonal nonlinear grey model is initially designed to address such issues by integrating the seasonal adjustment factor, the conventional Weibull Bernoulli grey model, and the cultural algorithm, simultaneously depicting the seasonality and nonlinearity of the original data. Experimental results from PM2.5 forecasting of four major cities (Shanghai, Nanjing, Hangzhou, and Hefei) in the YRD validate that the proposed model can obtain more accurate predictive results and stronger robustness, in comparison with grey prediction models (SNGBM(1,1) and SGM(1,1)), conventional econometric technology (SARIMA), and machine learning methods (LSSVM and BPNN) by employing accuracy levels. Finally, the future PM2.5 concentration is forecasted from 2020 to 2022 using the proposed model, which provides early warning information for policy-makers to develop PM2.5 alleviation strategies.

PYME: An Integrated Platform for High-Throughput Smart Nanoscopy

Single-Molecule Localization Microscopy (SMLM) offers a ∼10-fold improvement in resolution over conventional, diffraction-limited fluorescence microscopy, at the expense of acquisition time, data volume, and analysis overhead. For (d)STORM/(F)PALM, raw series typically require 10,000 to 100,000 frames acquired at 50 frames per second (FPS), meaning a day of diligent manual imaging yields only on the order of ten fields of view. Efforts have been made to ease the tedium of acquisition by automating SMLM microscopes.

A blended approach incorporating TVFEMD, PSR, NNCT-based multi-model fusion and hierarchy-based merged optimization algorithm for multi-step wind speed prediction

Precise wind speed prediction plays an essential role in wind farm planning. To enhance the wind speed forecasting accuracy, a blended approach incorporating time varying filtering-based empirical mode decomposition (TVFEMD), phase space reconstruction (PSR), no negative constraint theory (NNCT)-based multi-model fusion and hierarchy-based merged optimization algorithm is proposed in this paper. Firstly, raw wind speed series is decomposed through TVFEMD into a variety of intrinsic mode functions (IMFs) to decrease its non-stationarity, which are further reconstructed by PSR into input and output vectors of forecasting model. Then, NNCT-based multi-model fusion made up of diverse single forecasting models is established to predict wind speed series, of which the weight coefficients would influence the forecasting results significantly. Subsequently, a hierarchy-based merged optimization algorithm is proposed to optimize the weight coefficients which integrates global elite opposition-based learning strategy and hierarchy-based mechanism with whale optimization algorithm and differential evolution (GHWOADE). Furthermore, the final forecasting results of the raw series are obtained by accumulating the forecasting values of all decomposing subsequences. Eventually, the effectiveness of the proposed approach is verified by comprehensive experiments on wind speed series and benchmark datasets. The corresponding results are summarized as follows: (1) data pretreatment technology based on TVFEMD can effectively reduce the non-stationarity of raw wind speed series; (2) the NNCT-based multi-model fusion strategy can make up the shortcoming that single forecasting model cannot achieve well prediction results for all decomposed subsequences; (3) the proposed GHWOADE algorithm is employed to optimize weight coefficients of corresponding forecasting models, which contributes to promoting wind speed prediction precision.

Visibility Graph Complex Network Analysis on Seismic-Induced Variation in Groundwater Level of Nanxi Well, China

Groundwater level in wells, i.e., well water level (WWL) is an important index in hydrological monitoring during earthquakes. Due to the complex dynamics of groundwater, the WWL might change under seismic actions. This paper attempts to identify the long-term correlation between WWL and earthquakes, and disclose the topological features of groundwater dynamics. Taking Nanxi Well as an example, the authors conducted state space analysis on the raw series and trend of WWL to eliminate interferences like barometric pressure, rainfall, and solid tide, creating the trend time series. Then, the raw series and trend time series were converted into the raw visible graph (VG) network and trend VG network, respectively. Further, the global period was divided into five local time windows, and the two VG networks were compared by global aspect, local aspect, and topological properties of complex networks. The results show that the nodes of high degrees are closely related to the seismic response of the WWL in Nanxi Well; all VG networks are scale free and hierarchical; the seismic response of the WWL in the well is reflected by degree correlation; the community division of raw VG network was basically the same as that of trend VG network. The research findings provide insights to the seismic response of WWL and the dynamic fluctuation of groundwater level.

A compound framework for wind speed forecasting based on comprehensive feature selection, quantile regression incorporated into convolutional simplified long short-term memory network and residual error correction

A reliable wind speed forecasting framework can contribute to handling rational dispatching and safe operation for power system effectively. For this purpose, a novel compound framework coupling decomposition technique, subseries aggregation, synchronous optimization, improved deep network and residual error correction (REC) is investigated in this study. To begin with, time varying filter-based empirical mode decomposition (TVF-EMD) is employed to decompose the raw series into a set of subseries, which are further aggregated based on fuzzy entropy (FE) theory and approximation criterion. Then the synchronous optimization implemented by blended coding-based Harris hawks optimization (HHO) is adopted to optimize the parameters of phase space reconstruction (PSR) and applicable features for each aggregated subseries. Subsequently, quantile regression (QR) is incorporated into an improved deep network, namely convolutional simplified long short-term memory network (QRConvSLSTM), to deduce conditional quantiles for each aggregated subseries, in which the optimized arguments obtained above are applied to construct the optimal input matrixes. Later, the initial point forecasting results can be calculated on the basis of accumulating the conditional quantiles of all the aggregated subseries, while the corresponding error series is deduced therewith. Then the conditional quantiles of the error series are estimated by QRConvSLSTM in the light of REC strategy, after which the final conditional quantiles are calculated by summating the conditional quantiles of the raw series and the error series. Finally, kernel density estimation (KDE) is employed to estimate probabilistic density functions (PDF) of wind speed series in accordance to the final conditional quantiles. To validate the efficiency and effectiveness of the proposed compound framework, nine relevant models are performed on three datasets for comparative experiments, among which the results of point, interval and probability prediction are comprehensively demonstrated and analyzed. The experimental results illustrate that: (1) data preprocessing strategy integrating TVF-EMD and FE-based subseries aggregation contributes to balancing forecasting performance and timing computation properly; (2) the applicable deterministic and uncertainty forecasting results can be obtained by the improved deep network, namely QRConvSLSTM; (3) appropriate parameters of PSR and feature selection can be effectively optimized by the proposed synchronous optimization; (4) the application of REC possesses positive effects on further compensating the ultimate forecasting results.