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A number of spatial statistic measurements such as Moran’s I and Geary’s C can be used for spatial autocorrelation analysis. Spatial autocorrelation modeling proceeded from the 1-dimension autocorrelation of time series analysis, with time lag replaced by spatial weights so that the autocorrelation functions degenerated to autocorrelation coefficients. This paper develops 2-dimensional spatial autocorrelation functions based on the Moran index using the relative staircase function as a weight function to yield a spatial weight matrix with a displacement parameter. The displacement bears analogy with the time lag in time series analysis. Based on the spatial displacement parameter, two types of spatial autocorrelation functions are constructed for 2-dimensional spatial analysis. Then the partial spatial autocorrelation functions are derived by using the Yule-Walker recursive equation. The spatial autocorrelation functions are generalized to the autocorrelation functions based on Geary’s coefficient and Getis’ index. As an example, the new analytical framework was applied to the spatial autocorrelation modeling of Chinese cities. A conclusion can be reached that it is an effective method to build an autocorrelation function based on the relative step function. The spatial autocorrelation functions can be employed to reveal deep geographical information and perform spatial dynamic analysis, and lay the foundation for the scaling analysis of spatial correlation.

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.

This paper introduces a novel approach for the offline estimation of stationary moving average processes, further extending it to efficient online estimation of non-stationary processes. The novelty lies in a unique technique to solve the autocorrelation function matching problem leveraging that the autocorrelation function of a colored noise is equal to the autocorrelation function of the coefficients of the moving average process. This enables the derivation of a system of nonlinear equations to be solved for estimating the model parameters. Unlike conventional methods, this approach uses the Newton-Raphson and Levenberg–Marquardt algorithms to efficiently find the solution. A key finding is the demonstration of multiple symmetrical solutions and the provision of necessary conditions for solution feasibility. In the non-stationary case, the estimation complexity is notably reduced, resulting in a triangular system of linear equations solvable by backward substitution. For online parameter estimation of non-stationary processes, a new recursive formula is introduced to update the sample autocorrelation function, integrating exponential forgetting of older samples to enable parameter adaptation. Numerical experiments confirm the method’s effectiveness and evaluate its performance compared to existing techniques.

Drought is a complex natural phenomenon, so, precise prediction of drought is an effective mitigation tool for measuring the negative consequences on agriculture, ecosystems, hydrology, and water resources. The purpose of this research was to explore the potential capability of support vector regression (SVR) integrated with two meta-heuristic algorithms i.e., Grey Wolf Optimizer (GWO), and Spotted Hyena Optimizer (SHO), for meteorological drought (MD) prediction by utilizing EDI (effective drought index). For this objective, the two-hybrid SVR–GWO, and SVR–SHO models were constructed at Kumaon and Garhwal regions of Uttarakhand State (India). The EDI was computed in both study regions by using monthly rainfall data series to calibrate and validate the advanced hybrid SVR models. The autocorrelation function (ACF) and partial-ACF (PACF) were utilized to determine the optimal inputs (antecedent EDI) for EDI prediction. The results produced by the hybrid SVR models were compared with the calculated (observed) values by employing the statistical indicators and through graphical inspection. A comparison of results demonstrates that the hybrid SVR–GWO model outperformed to the SVR–SHO models for all study stations located in Kumaon and Garhwal regions. Also, the results highlighted the better suitability, supremacy, and convergence behavior of meta-heuristic algorithms (i.e., GWO and SHO) for meteorological drought prediction in the study regions.

A theory is developed in a stochastic climate model for understanding the general features of the seasonal predictability barrier (PB), which is characterized by a band of maximum decline in autocorrelation function phase-locked to a particular season. Our theory determines the forcing threshold, timing, and intensity of the seasonal PB as a function of the damping rate and seasonal forcing. A seasonal PB is found to be an intrinsic feature of a stochastic climate system forced by either seasonal growth rate or seasonal noise forcing. A PB is generated when the seasonal forcing, relative to the damping rate, exceeds a modest threshold. Once generated, all the PBs occur in the same calendar month, forming a seasonal PB. The PB season is determined by the decline of the seasonal forcing as well as the delayed response associated with damping. As such, for a realistic weak damping, the PB season is locked close to the minimum SST variance under the seasonal growth-rate forcing, but after the minimum SST variance under the seasonal noise forcing. The intensity of the PB is determined mainly by the amplitude of the seasonal forcing. The theory is able to explain the general features of the seasonal PB of the observed SST variability over the world. In the tropics, a seasonal PB is generated mainly by a strong seasonal growth rate, whereas in the extratropics a seasonal PB is generated mainly by a strong seasonal noise forcing. Our theory provides a general framework for the understanding of the seasonal PB of climate variability.

Steady flow toward a fully penetrating well takes place in a natural porous formation, where the erratic spatial variations, and the raising uncertainty, of the hydraulic conductivity K are modeled within a stochastic framework which regards the log‐conductivity, ln K, as a Gaussian, stationary, random field. The study provides second order moments of the flow variables by regarding the variance of the log‐conductivity as a perturbation parameter. Unlike similar studies on the topic, moments are expressed in a quite general (valid for any autocorrelation function of ln K) and very simple (from the computational stand point) form. It is shown that the (cross)variances, unlike the case of mean uniform flows, are not anymore stationary due to the dependence of the mean velocity upon the distance from the well. In particular, they vanish at the well because of the condition of given head along the well’s axis, whereas away from it they behave like those pertaining to a uniform flow. Then, theoretical results are applied to a couple (one serving for calibration and the other used for validation purposes) of pumping tests to illustrate how they can be used to determine the hydraulic properties of the aquifers. In particular, the concept of head‐factor is shown to be the key‐parameter to identify the statistical moments of the random field K. Plain Language Summary Flow toward a single well takes place in a porous formation, where the hydraulic conductivity is regarded as a random space function to account for its irregular spatial variability. A simple solution to this difficult problem is achieved by adopting some simplifying assumptions which apply to numerous real settings. Theoretical results are applied to a series of pumping tests in order to demonstrate their utility in the identification of aquifers' hydraulic properties. Key Points A simple, general formulation to compute second‐order moments is presented The head factor is introduced for a robust identification of aquifers' statistical parameters The application to pumping tests is illustrated and discussed

A new version of the fuzzy logic model, called the co-active neuro fuzzy inference system (CANFIS), is introduced for predicting standardized precipitation index (SPI). Multiple scales of drought information at six meteorological stations located in Uttarakhand State, India, are used. Different lead times of SPI were computed for prediction, including 1, 3, 6, 9, 12, and 24 months, with inputs abstracted by autocorrelation function (ACF) and partial-ACF (PACF) analysis at 5% significance level. The proposed CANFIS model was validated against two models: classical artificial intelligence model (e.g., multilayer perceptron neural network (MLPNN)) and regression model (e.g., multiple linear regression (MLR)). Several performance evaluation metrices (root mean square error, Nash-Sutcliffe efficiency, coefficient of correlation, and Willmott index), and graphical visualizations (scatter plot and Taylor diagram) were computed for the evaluation of model performance. Results indicated that the CANFIS model predicted the SPI better than the other models and prediction results were different for different meteorological stations. The proposed model can build a reliable expert intelligent system for predicting meteorological drought at multi-time scales and decision making for remedial schemes to cope with meteorological drought at the study stations and can help to maintain sustainable water resources management.

The Snow Microwave Radiative Transfer (SMRT) thermal emission and backscatter model was developed to determine uncertainties in forward modeling through intercomparison of different model ingredients. The model differs from established models by the high degree of flexibility in switching between different electromagnetic theories, representations of snow microstructure, and other modules involved in various calculation steps. SMRT v1.0 includes the dense media radiative transfer theory (DMRT), the improved Born approximation (IBA), and independent Rayleigh scatterers to compute the intrinsic electromagnetic properties of a snow layer. In the case of IBA, five different formulations of the autocorrelation function to describe the snow microstructure characteristics are available, including the sticky hard sphere model, for which close equivalence between the IBA and DMRT theories has been shown here. Validation is demonstrated against established theories and models. SMRT was used to identify that several former studies conducting simulations with in situ measured snow properties are now comparable and moreover appear to be quantitatively nearly equivalent. This study also proves that a third parameter is needed in addition to density and specific surface area to characterize the microstructure. The paper provides a comprehensive description of the mathematical basis of SMRT and its numerical implementation in Python. Modularity supports model extensions foreseen in future versions comprising other media (e.g., sea ice, frozen lakes), different scattering theories, rough surface models, or new microstructure models.

Gait movies recorded in daily clinical practice are usually not filmed with specific devices, which prevents neurologists benefitting from leveraging gait analysis technologies. Here we propose a novel unsupervised approach to quantifying gait features and to extract cadence from normal and parkinsonian gait movies recorded with a home video camera by applying OpenPose, a deep learning-based 2D-pose estimator that can obtain joint coordinates from pictures or videos recorded with a monocular camera. Our proposed method consisted of two distinct phases: obtaining sequential gait features from movies by extracting body joint coordinates with OpenPose; and estimating cadence of periodic gait steps from the sequential gait features using the short-time pitch detection approach. The cadence estimation of gait in its coronal plane (frontally viewed gait) as is frequently filmed in the daily clinical setting was successfully conducted in normal gait movies using the short-time autocorrelation function (ST-ACF). In cases of parkinsonian gait with prominent freezing of gait and involuntary oscillations, using ACF-based statistical distance metrics, we quantified the periodicity of each gait sequence; this metric clearly corresponded with the subjects' baseline disease statuses. The proposed method allows us to analyze gait movies that have been underutilized to date in a completely data-driven manner, and might broaden the range of movies for which gait analyses can be conducted.

Hand-foot-and-mouth disease_(HFMD) is one of the most typical diseases in children that is associated with high morbidity. Reliable forecasting is crucial for prevention and control. Recently, hybrid models have become popular, and wavelet analysis has been widely performed. Better prediction accuracy may be achieved using wavelet-based hybrid models. Thus, our aim is to forecast number of HFMD cases with wavelet-based hybrid models. We fitted a wavelet-based seasonal autoregressive integrated moving average (SARIMA)-neural network nonlinear autoregressive (NNAR) hybrid model with HFMD weekly cases from 2009 to 2016 in Zhengzhou, China. Additionally, a single SARIMA model, simplex NNAR model, and pure SARIMA-NNAR hybrid model were established for comparison and estimation. The wavelet-based SARIMA-NNAR hybrid model demonstrates excellent performance whether in fitting or forecasting compared with other models. Its fitted and forecasting time series are similar to the actual observed time series. The wavelet-based SARIMA-NNAR hybrid model fitted in this study is suitable for forecasting the number of HFMD cases. Hence, it will facilitate the prevention and control of HFMD.