Nonlinear, Non-stationary and Seasonal Time Series Forecasting Using Different Methods Coupled with Data Preprocessing

Normalized Difference Vegetation Index (NDVI) time series is one of the most important instruments in precision agriculture. Forecasting of this index in precision agriculture allows us to define problems related to growth rates of agricultural crops in time. This Doctoral Thesis is devoted to the analysis and forecasting of nonlinear and nonstationary NDVI index time series with the use of data pre-processing, signal processing, linear algebra and machine learning methods. The aim of Doctoral Thesis is to develop a forecasting system of normalized difference vegetation index time series based on signal decomposition and sub-signal approximation approach, specialized data processing methods and machine learning. In the Thesis, NDVI time series forecasting system (NDVI FS) is proposed that makes short-term time series value forecasting for the next period using historical observations. In the developed system framework, an innovative approach to using variational mode decomposition method in NDVI time series forecasting task is realized. There is also offered an innovative approach to trained forecasting model transfer, which allows using this model to forecast other NDVI time series obtained from neighbourhood pixels.

Accurate clustering of time series is a challenging problem for data arising from areas such as financial markets, biomedical studies, and environmental sciences, especially when some, or all, of the series exhibit nonlinearity and nonstationarity. When a subset of the series exhibits nonlinear characteristics, frequency domain clustering methods based on higher-order spectral properties, such as the bispectra or trispectra are useful. While these methods address nonlinearity, they rely on the assumption of series stationarity. We propose the Bispectral Smooth Localized Complex EXponential (BSLEX) approach for clustering nonlinear and nonstationary time series. BSLEX is an extension of the SLEX approach for linear, nonstationary series, and overcomes the challenges of both nonlinearity and nonstationarity through smooth partitions of the nonstationary time series into stationary subsets in a dyadic fashion. The performance of the BSLEX approach is illustrated via simulation where several nonstationary or nonlinear time series are clustered, as well as via accurate clustering of the records of 16 seismic events, eight of which are earthquakes and eight are explosions. We illustrate the utility of the approach by clustering S&P 100 financial returns.

MODerate-resolution Imaging Spectroradiometer (MODIS) vegetation index (VI) time-series data are among the most widely utilized remote sensing datasets. To improve the quality of MODIS VI time-series data, most prior methods have focused on correcting negatively biased VI noise by approaching the upper envelope of the VI time series. Such treatment, however, may cause overcorrections on some true local low VI values, resulting in inaccurate simulations of vegetation phenological characteristics. In addition, another challenge in reconstructing MODIS VI time series is to fill temporally continuous gaps. The earlier spatiotemporal Savitzky-Golay (STSG) method tackled this problem by utilizing multiyear VI data, but its performance heavily relies on the consistency of data across different years. In this study, we proposed an improved STSG (iSTSG) method. The new method accounts for the autocorrelation within the VI time series and fills missing values in the VI time series by leveraging spatiotemporal VI data from the current year alone. Furthermore, iSTSG incorporates an indicator to quantify potential overcorrections in the VI time series, aiming to more accurately simulate phenological characteristics. The experiments to reconstruct MODIS normalized difference VI (NDVI) time-series product (MOD13A2) at four typical sites (a million square kilometers for each site) suggest two clear advantages in iSTSG over the iterative SG (called Chen-SG) and STSG methods. First, iSTSG more accurately reconstructs the annual NDVI time series, exhibiting the smallest mean absolute differences (MADs) between the smoothed and the simulated reference NDVI time series (0.012, 0.018, and 0.020 for iSTSG, STSG, and Chen-SG, respectively). Second, iSTSG more effectively simulates phenological characteristics in the NDVI time series, including the onset dates for vegetation greenup and dormancy, as well as the crop harvest period. The advantages of iSTSG were also demonstrated when applied to the successor of MODIS, Visible Infrared Imaging Radiometer Suite (VIIRS) VI time-series product (VNP13A1). iSTSG can be implemented on the Google Earth Engine (GEE), offering significant benefits for various applications, particularly in crop mapping and vegetation/crop phenology studies.

Veg-W2TCN: A parallel hybrid forecasting framework for non-stationary time series using wavelet and temporal convolution network model

HJ-1A/B is the first small satellite constellation built by China for environmental and disaster monitoring and forecasting. The satellite group has a 2-day repetition cycle and 30 m spatial resolution (charge coupled device camera). Thus, HJ-1A/B can provide hyper-temporal normalized difference vegetation index (NDVI) time series with a medium–high spatial resolution. However, the quality of the HJ NDVI time series can be abnormally low due to a number of factors, such as cloud cover, continuous fog, and haze. In the rainy season or in areas with serious atmospheric pollution, low-quality series often appear in succession, which is referred to as an abnormal segment. Neither the composition method nor quality flags satisfactorily solve this problem; therefore, a large amount of noise and long periods of abnormally low values often remain in HJ NDVI time series. This article presents a method to reconstruct the abnormal segments in HJ NDVI time series with the assistance of Moderate Resolution Imaging Spectroradiometer (MODIS) NDVI time series. The cointegration test was adopted to decide whether MODIS can be used for the reconstruction of NDVI time series for the corresponding HJ image pixels. Statistical quality control methods were used for singling out the abnormal segments in the HJ NDVI time series and establishing an error correction model that combines MODIS and HJ NDVI time series to perform the reconstruction. The study area is located in Jiangsu Province, China. Four-year (2009–2012) HJ multispectral images that cover the study area were used. The results show that abnormal segments in the HJ NDVI time series can be corrected using the proposed method. In a particular year, this method can decrease the root mean square error between the HJ NDVI time series and the reference sequence by 52.5%.

Artificial Neural networks (ANN) are powerful and effective tools in time-series applications. The first aim of this paper is to diagnose better and more efficient ANN models (Back Propagation, Radial Basis Function Neural networks (RBF), and Recurrent neural networks) in solving the linear and nonlinear time-series behavior. The second aim is dealing with finding accurate estimators as the convergence sometimes is stack in the local minima. It is one of the problems that can bias the test of the robustness of the ANN in time series forecasting. To determine the best or the optimal ANN models, forecast Skill (SS) employed to measure the efficiency of the performance of ANN models. The mean square error and the absolute mean square error were also used to measure the accuracy of the estimation for methods used. The important result obtained in this paper is that the optimal neural network was the Backpropagation (BP) and Recurrent neural networks (RNN) to solve time series, whether linear, semilinear, or non-linear. Besides, the result proved that the inefficiency and inaccuracy (failure) of RBF in solving nonlinear time series. However, RBF shows good efficiency in the case of linear or semi-linear time series only. It overcomes the problem of local minimum. The results showed improvements in the modern methods for time series forecasting.

Time series forecasting research area generally aims at improving prediction accuracy. Discrete wavelet transform (DWT) has been applied to time series for decomposing it into approximation and detail. Nevertheless, typically, the property of the approximation and the detail are presumed as either linear or nonlinear. Actually, the purpose of the DWT is not decomposing the original time series into linear and nonlinear time series. Hence, this paper develops a new hybrid model of autoregressive integrated moving average (ARIMA), artificial neural network (ANN), and the DWT without prior assumption on linear and nonlinear property of the approximation and the detail. The different Khashei and Bijari’s hybrid models involving the ARIMA and the ANN are built for the approximation and the detail in order to extract their both linear and nonlinear components and fit the relationship between the components as the function instead of additive relationship. Finally, the forecasted approximation and detail are combined to obtain final forecasting. The prediction capability of the proposed model is examined with two well-known time series: the sunspot and the Canadian lynx time series. The results show that the proposed model has the best performance in all two data sets and all three measures (i.e. MSE, MAE and MAPE).

The aim of this study was to determine the extent to which Sentinel-2 Normalised Difference Vegetation Index (NDVI) reflects soil moisture conditions, and whether this product offers an improvement over Landsat-8. Based on drought exposure, cloud-free imagery availability, and measured soil moisture, five sites in the Southwestern United States were selected. These sites, normally dry to arid, were in various states of drought. A secondary focus was therefore the performance of the NDVI under extreme conditions. Following supervised classification, the NDVI values for one-kilometre radius areas were calculated. Sentinel-2 NDVI variants using Spectral Bands 8 (10 m spatial resolution), 5, 6, 7, and 8A (20 m spatial resolution) were calculated. Landsat-8 NDVI was calculated at 30 m spatial resolution. Pearson correlation analysis was undertaken for NDVI against moisture at various depths. To assess the difference in correlation strength, a principal component analysis was performed on the combination of all bands and the combination of the new red-edge bands. Performance of the red-edge NDVI against the standard near infrared (NIR) was then evaluated using a Steiger comparison. No significant correlations between Landsat-8 NDVI and soil moisture were found. Significant correlations at depths of less than 30 cm were present between Sentinel-2 NDVI and soil moisture at three sites. The remaining two sites were characterised by low vegetation cover, suggesting a cover threshold of approximately 30–40% is required for a correlation to be present. At all sites of significant positive moisture to NDVI correlation, the linear combination of the red-edge bands produced stronger correlations than the poorer spectral but higher spatial resolution band. NDVI calculated using the higher spectral resolution bands may therefore be of greater use in this context than the higher spatial resolution option. Results suggest potential for the application of Sentinel-2 NDVI in soil moisture monitoring, even in extreme environments. To the best of our knowledge, this paper represents the first study of this kind using Sentinel-2.

Effects of Band Positioning and Bandwidth on NDVI Measurements of Tropical Savannas

High-precision maps of soil organic carbon (SOC) are beneficial for managing soil fertility and understanding the global carbon cycle. Digital soil mapping plays an important role in efficiently obtaining the spatial distribution of SOC, which contributes to precision agriculture. However, traditional soil-forming factors (i.e., terrain or climatic factors) have weak variability in low-relief areas, such as plains, and cannot reflect the spatial variation of soil attributes. Meanwhile, vegetation cover hinders the acquisition of the direct information of farmland soil. Thus, useful environmental variables should be utilized for SOC prediction and the digital mapping of such areas. SOC has an important effect on crop growth status, and remote sensing data can record the apparent spectral characteristics of crops. The normalized difference vegetation index (NDVI) is an important index reflecting crop growth and biomass. This study used NDVI time series data rather than traditional soil-forming factors to map SOC. Honghu City, located in the middle of the Jianghan Plain, was selected as the study region, and the NDVI time series data extracted from Landsat 8 were used as the auxiliary variables. SOC maps were estimated through stepwise linear regression (SLR), partial least squares regression (PLSR), support vector machine (SVM), and artificial neural network (ANN). Ordinary kriging (OK) was used as the reference model, while root mean square error of prediction (RMSEP) and coefficient of determination of prediction (R2P) were used to evaluate the model performance. Results showed that SOC had a significant positive correlation in July and August (0.17, 0.29) and a significant negative correlation in January, April, and December (−0.23, −0.27, and −0.23) with NDVI time series data. The best model for SOC prediction was generated by ANN, with the lowest RMSEP of 3.718 and highest R2P of 0.391, followed by SVM (RMSEP = 3.753, R2P = 0.361) and PLSR (RMSEP = 4.087, R2P = 0.283). The SLR model was the worst model, with the lowest R2P of 0.281 and highest RMSEP of 3.930. ANN and SVM were better than OK (RMSEP = 3.727, R2P = 0.372), whereas PLSR and SLR were worse than OK. Moreover, the prediction results using single-data NDVI or short time series NDVI showed low accuracy. The effect of the terrain factor on SOC prediction represented unsatisfactory results. All these results indicated that the NDVI time series data can be used for SOC mapping in plain areas and that the ANN model can maximally extract additional associated information between NDVI time series data and SOC. This study presented an effective method to overcome the selection of auxiliary variables for digital soil mapping in plain areas when the soil was covered with vegetation. This finding indicated that the time series characteristics of NDVI were conducive for predicting SOC in plains.

Computational intelligence approaches and linear models in case studies of forecasting exchange rates

The forecasting and characterization of time series are very useful for experts to take appropriate decisions, to plan actions and to understand the time series patterns. However, there are a small number of methods that consider both objectives at the same time. In this paper, an algorithm for nonstationary and seasonal time series forecasting, with an evolving neuro-fuzzy Takagi-Sugeno (NF-TS) structure, is proposed. For this algorithm, the NF-TS inputs are unobservable patterns extracted from the time series by a decomposition technique. As experiment, a real seasonal time series was used to compare the forecasting performance of these proposed algorithm with an other similar NF-TS, whose inputs were formed by autoregressive data from the same time series. When there is available observations from time series, the NF-TS evolves its structure and adapt its parameters. If the data is not available, the proposed methodology needs to forecast the next value. In order to extract the unobservable components from time series, the Holt-Winters method optimized by Particle Swarm Optimization (PSO) approach was considered in this experiment.

Exponomial forecasts of nonstationary time series

Accurate mapping of the spatio-temporal dynamic information of conversion from -ropland into forest/grassland is necessary for studying hydro-ecological effects of implementation of Grain to Green Project (GTGP) on the Loess Plateau, China. Currently, the accuracy of change detection of farmland and forest/grassland at 30-m scale in this area is seriously affected by insufficient temporal information from observations and irregular fluctuations in vegetation greenness caused by precipitation and human activities. In this study, an innovative method for continuous change detection of cropland and forest/grassland using all available Landsat time-series data. The period with vegetation coverage is firstly identified using normalized difference vegetation index (NDVI) time series. The intra-annual NDVI time series is then developed at a 1-day resolution based on linear interpolation and S-G filtering using all available NDVI data during the period when vegetation types are stable. Vegetation type change is initially detected by comparing the NDVI of intra-annual composites and the newly observed NDVI. Finally, the time of change and classification for vegetation types are determined using decision tree rules developed using a combination of inter-annual and intra-annual NDVI temporal metrics. Validation results showed that the change detection was accurate, with an overall accuracy of 88.9% ± 1.0%, and a kappa coefficient of 0.86, and the time of change was successfully retrieved, with 85.2% of the change pixels attributed to within a 2-year deviation.

Quantification of root-zone soil moisture (RZSM) is crucial for agricultural applications and soil sciences. RZSM impacts processes such as vegetation transpiration and water percolation. Surface soil moisture (SSM) can be assessed through active and passive microwave remote sensing methods, but no current sensor enables direct RZSM retrieval. Spatial maps of RZSM can be retrieved via proxy observations (vegetation stress, water storage change, and surface soil moisture) or via land surface model predictions. In this study, we investigated the combination of surface soil moisture information with process-based inferred features involving artificial neural networks (ANNs). We considered the infiltration process through the soil water index (SWI) computed with a recursive exponential filter and the evaporation process through the evaporative efficiency computed based on a MODIS remote sensing dataset and simplified analytical model, while vegetation growth was expressed through normalized difference vegetation index (NDVI) time series. Several ANN models with different sets of features were developed. Training was conducted considering in situ stations distributed several areas worldwide characterized by different soil and climate patterns of the International Soil Moisture Network (ISMN), and testing was applied to stations of the same data hosting facility. The results indicate that the integration of process-based features into ANN models increased the overall performance over the reference model level in which only SSM features were considered. In arid and semi-arid areas, for instance, performance enhancement was observed when the evaporative efficiency was integrated into the ANN models. To assess the robustness of the approach, the trained models were applied on observation sites in Tunisia, Italy and South-India that are not part of ISMN. The results reveal that joint use of surface soil moisture, evaporative efficiency, NDVI and recursive exponential filter represented the best alternative for more accurate predictions in the case of Tunisia, where the mean correlation of the predicted RZSM based on SSM only sharply increased from 0.443 to 0.801 when process-based features were integrated into the ANN models in addition to SSM. However, process-based features have no to little added value in temperate to tropical conditions.