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# Long Term Storage Capacity of Reservoirs

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... In Econophyiscs, the 'Hurst exponent' is often used to analyze financial market phenomena (Hurst, 1951). The Hurst exponent measure is a powerful tool for analyzing complex financial time series and is used to uncover various hidden patterns and events in financial markets. ...
... It is the period immediately following the recovery from the 2008 recession and includes several economic and political events and natural disasters such as Demonetization, the change of the ruling party as the Central government, and the Covid-19 pandemic. This model produces the Hurst (1951) values, making it possible to gauge the degree of herding behavior in the index. This study examines the behavior of BSE 100 investors. ...
... The following is how variance is determined: Step 5: By looking at the log-log plot of F q (s) versus s for each value of q; one may determine the scaling behavior of the fluctuation functions. F q (s) rises for a considerable value of s as a power law if the time series x(t) are long-range power-law correlated, where h(q) is the Hurst exponent used in the financial literature (Hurst, 1951). ...
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Herding has a history of igniting large, irrational market ups and downs, usually based on a lack of fundamental support. Intuitively, most herds start with an external shock. This empirical study seeks to detect shock-induced herding and the creation of nascent bubbles in the Indian stock market. Initially, the multifractal form of the detrended fluctuation analysis was applied. Then the Reformulated Hurst exponent for the Bombay stock exchange (BSE) was determined using Kantelhardt’s calibration. The investigation found evidence of high-level herding and a bubble in 2012, with a high value of Hurst Exponent (0.7349). The other years of the research period (2011, 2013, 2016, 2018, 2020–2021) observed mild to significant herding with comparatively lower Hurst values. The results confirm that herding behavior occurs during a crisis and harsh situations emitting shocks. The study concludes that shock-based herding is prevalent in all six shocks: the economic meltdown, commodities and currency devaluation, geo-political problems, the Central Bank’s decision on liquidity management, and the Pandemic. Additionally, the years following the Financial Crisis and the years of the Pandemic are when herding and bubble are prominent. AcknowledgmentsWe thank Dr. Bikramaditya Ghosh (Associate Professor, Symbiosis International University, Bangalore, India) for motivating us in this research. We also thank Dr. Natchimuthu N (Assistant Professor, Commerce, CHRIST (Deemed to be University), Bangalore, India) and Dr. Mahesh E. (Assistant Professor, Economics, CHRIST (Deemed to be University), Bangalore, India) for their support throughout this study.
... The Hurst exponent measures the long-term memory of time series, characterizing its persistent nature (Harold, 1951). Indeed, if  ∈ [0, 0.5) then the time series values switch between high and low values in adjacent pairs, if  ∈ (0.5, 1] then the time series has a long-term positive autocorrelation, while if  = 0.5 then the time series has a completely uncorrelated behavior (Harold, 1951;Mandelbrot, 1982). ...
... The Hurst exponent measures the long-term memory of time series, characterizing its persistent nature (Harold, 1951). Indeed, if  ∈ [0, 0.5) then the time series values switch between high and low values in adjacent pairs, if  ∈ (0.5, 1] then the time series has a long-term positive autocorrelation, while if  = 0.5 then the time series has a completely uncorrelated behavior (Harold, 1951;Mandelbrot, 1982). The singularity measures, e.g., the singularity width and the singularity spectrum width , are used to characterize the range of singularities and the range of Hausdorff dimensions that are present in a time series, quantifying the multifractal nature of a time series (Hausdorff, 1918;Mandelbrot, 1982;Ott, 2002). ...
... The evaluation of the stationarity of the time series is a crucial point for correctly evaluating complexity measures as for example the Hurst exponent () and the singularity features of time series ( , ( ), , ). Indeed, to evaluate these complexity measures is necessary that time series are stationary or non-stationary but characterized by stationary increments (Harold, 1951;Mandelbrot, 1982). Thus, the evaluation of the degree of stationarity permits us to select the range of timescales over which the time series are statistically stationary. ...
Article
The SYM-H and AE geomagnetic indices can be considered as proxies of the response of the Earth’s magnetosphere and ionosphere to solar magnetic activity. They indirectly monitor some electric current systems which flow in the ionosphere and magnetosphere whose dynamics are directly or indirectly related to the Sun-Earth interaction. Consequently, their temporal changes reflect processes occurring in the near-Earth space, which contribute differently to the overall magnetosphere-ionosphere dynamics. The aim of this work is to characterize the nature of these two geomagnetic indices by following a complex system approach and applying a novel formalism, e.g., the EMD-based dominant amplitude multifractal formalism (EMD-DAMF). A set of complexity measures, i.e., the Hurst exponent (H), the singularity width (Δα) and the spectrum width (Δf), is evaluated for both geomagnetic indices analyzing data recorded during the last two solar cycles. One of the most significant findings of this study is the absence of relevant differences between the two solar cycles in terms of complexity measures for both geomagnetic indices, suggesting that only the occurrence and the frequency of geomagnetic storms and substorms affect the Hurst exponent and the singularity widths of SYM-H and AE indices. Moreover, while the AE index complexity measures do not show a significant dependence on geomagnetic activity, the SYM-H index shows a reduction in its complexity features during the geomagnetic storms, manifesting a more persistent behavior and moving from a (mono)fractal-like to a multifractal-like behavior when passing from quiet to disturbed periods. Finally, our findings are consistent with previous works on the forecast horizon of the geomagnetic activity as well as on the relation between the high-latitude ionosphere and the low-latitude magnetosphere, thus confirming the importance of providing higher resolution measures for correctly dealing with several Space Weather phenomena.
... Step 1. Having observed the samples x 1 , x 2 , · · · , x n , we start estimatingd. Specific estimation method can be referred to Hurst (1951) [25]; ...
... Step 1. Having observed the samples x 1 , x 2 , · · · , x n , we start estimatingd. Specific estimation method can be referred to Hurst (1951) [25]; ...
Article
In this paper, the sieve bootstrap test for multiple change points in the mean of long memory sequence is studied. Firstly, the ANOVA test statistics for change points detection is obtained. Secondly, sieve bootstrap statistics is constructed and the consistency under the Mallows measure is proved. Finally, the effectiveness of the method was illustrated by simulation and example analysis. Simulation results show that our method can not only control the empirical size well but also have reasonable good power.
... doi: bioRxiv preprint describing microscopic dynamics is the observed diffusion constant (ODC), which provides meaningful information regarding the quantity of displacement, and which might remain constant in an isotropic medium or vary in space and time in complex environments. Specifically, different motion regimens can be classified by computing ODC and by combining it with a measure of freedom of motion, such as the variation of Mean Square Displacement (MSD) over time 5 or the Hurst exponent (henceforth referred to as the Slope of the Moment Scaling Spectrum or SMSS) 47,48 , thereby representing individual trajectories as points in phase space 40,49 . In addition to representing a massive data reduction, this approach facilitates the classification of the mobility characteristics of multiple particles all at once without arbitrary selection. ...
... Work presented here addresses the question of how accurately rate of displacement, as measured by the observed diffusion constant (ODC) 5,57 , and freedom of motion, as measured by the Hurst exponent, also known as the slope of the moment scaling spectrum or SMSS 40,47,49,59 , can be estimated from the analysis of finite length single-particle trajectories whose individual points are significantly affected by localization errors. ...
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Quantitative analysis of microscopy images is ideally suited for understanding the functional biological correlates of individual molecular species identified by one of the several available “omics” techniques. Due to advances in fluorescent labeling, microscopy engineering and image processing, it is now possible to routinely observe and quantitatively analyze at high temporal and spatial resolution the real-time behavior of thousands of individual cellular structures as they perform their functional task inside living systems. Despite the central role of microscopic imaging in modern biology, unbiased inference, valid interpretation, scientific reproducibility and results dissemination are hampered by the still prevalent need for subjective interpretation of image data and by the limited attention given to the quantitative assessment and reporting of the error associated with each measurement or calculation, and on its effect on downstream analysis steps (i.e., error propagation). One of the mainstays of bioimage analysis is represented by single-particle tracking (SPT)1–5, which coupled with the mathematical analysis of trajectories and with the interpretative modelling of motion modalities, is of key importance for the quantitative understanding of the heterogeneous intracellular dynamic behavior of fluorescently-labeled individual cellular structures, vesicles, virions and single-molecules. Despite substantial advances, the evaluation of analytical error propagation through SPT and motion analysis pipelines is absent from most available tools 6. This severely hinders the critical evaluation, comparison, reproducibility and integration of results emerging from different laboratories, at different times, under different experimental conditions and using different model systems. Here we describe a novel, algorithmic-centric, Monte Carlo method to assess the effect of experimental parameters such as signal to noise ratio (SNR), particle detection error, trajectory length, and the diffusivity characteristics of the moving particle on the uncertainty associated with motion type classification The method is easily extensible to a wide variety of SPT algorithms, is made widely available via its implementation in our Open Microscopy Environment inteGrated Analysis (OMEGA) software tool for the management and analysis of tracking data 7, and forms an integral part of our Minimum Information About Particle Tracking Experiments (MIAPTE) data model 8.
... The Hurst exponent (Hurst, 1951), proposed by a hydrologist, is a valid method for quantifying the long-term dependence of time series. Recently, it has been extensively applied to measure the stability of the NDVI dynamics. ...
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Elucidating the response mechanism of variation in vegetation trend to determinant is of great value to environmental resource management, particularly significant in the ecologically fragile area. The Liaohe River Basin (LRB) is a key part of eco-security in China, which has experienced apparent climatic variations and intensified human activities in recent decades. Yet, it still remains not clear about drivers in shaping the spatio-temporal patterns of vegetation growth. Here, the normalized difference vegetation index (NDVI) was utilized to investigate the spatio-temporal variation of vegetation coverage from 2000 to 2019. Then, we incorporated partial derivatives analysis to conduct attribution analyses of vegetation greening in light of the meteorological data. The prime findings are as follows: (1) The vegetation coverage in the LRB presented a growing state in the recent 20 years at a rate of 0.0031/a, with significant spatial and temporal heterogeneity due to its slope; (2) The attribution results showed that the average contribution of precipitation, temperature, and solar radiation to the NDVI changes in the LRB was 0.00205/a, 0.00008/a, and − 0.00028/a, respectively. (3) The climatic change played the most dominant role in influencing vegetation activities as a result of the relative contributions of 59.68% of NDVI changes (40.32% contributed by anthropogenic activities); (4) LULC dynamics were characterized by an increase in forest land and large-scale ecological afforestation projects, which increase vegetation coverage. Conversely, urbanization adversely affected vegetation variations. Understanding the findings of this study is expected to offer further scientific support and practical implications for monitoring the local vegetation status.
... Fractal theory attempts to explain complex processes by determining simple underlying processes. The study of fractal theory began with Hurst's discovery of the long-range correlation in runoff records (Hurst 1951). Hurst rst used rescaled range (R/S) analysis to study the long-range correlation in time-series records of natural phenomena, and with the use of the calculated Hurst exponent, R/S analysis could provide a process for the quanti cation of the memory of time series (Mandelbrot and Wallis 1969 Weron 2002). ...
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Groundwater resources are important natural resources that must be appropriately managed. Because groundwater level fluctuation typically exhibits non-stationarity, revealing its complex characteristics is of scientific and practical significance for understanding the response mechanism of the groundwater level to natural or human factors. Therefore, employing multifractal analysis to detect groundwater level variation irregularities is necessary. In this study, multifractal detrended fluctuation analysis (MF-DFA) was applied to study the multifractal characteristics of the groundwater level in the Baotu Spring Basin and further detect the complexity of groundwater level variation. The main results indicate that groundwater level variation in the Baotu Spring Basin exhibited multifractal characteristics, and multifractality originated from broad probability density function (PDF) and the long-range correlation of the hydrological series. The groundwater level fluctuations in wells 358 and 361 exhibited a high complexity, those in wells 287 and 268 were moderately complex, and the groundwater level fluctuations in wells 257 and 305 were characterized by a low complexity. The spatial variability of hydrogeological conditions resulted in spatial heterogeneity in the groundwater level complexity. This study could provide important reference value for the analysis of the nonlinear response mechanism of groundwater to its influencing factors and the development of hydrological models.
... The Hurst's rescaled range (R/S) analysis proposed by Hurst (1951), is used to detect the persistence effect of the changing trend of the precipitation variability, and quantify it with Hurst exponent (H). As a non-parametric analysis method, Hurst test based on R/S analysis has been widely used in the fields of hydrology, climatology, economics, geology, and geochemistry due to of its robustness Tatli, 2015;Sánche et al., 2008). ...
... We also verify whether the distribution of dengue incidence cases follows a normal distribution using the Anderson-Darling test [47]. Additionally, we check the long-term dependency or self-similarity using the Hurst exponent [48]. Furthermore, we employ the non-parametric Granger causality (GC) test [49] to verify the causal relationship between rainfall and dengue outbreak. ...
... Several methodologies can be used to assess long memory in a series (SOUZA et al., 2006), among them stand out the classic R/S analysis, by Hurst (1951) and Mandelbrot (1972); the modified R/S analysis, by Lo (1991) (see TABAK & CAJUEIRO, 2007); the method for estimation of the fractional integration parameter proposed by Geweke and Porter-Hudak (1983); the semiparametric estimator by log-periodogram, by Robinson et al. (1995a); the Gaussian semiparametric estimator, by Robinson et al. (1995b); and the V/S analysis developed by Giraitis et al. (2003) and Cajueiro and Tabak (2005). The present work is similar to the studies carried out by Lean and Smyth (2009), Gil-Alana et al. (2010), Tsoumas (2011, 2012) and Barros et al. (2011Barros et al. ( , 2012Barros et al. ( , 2016, because it employs the fractional integration methodology to identify the degree of persistence of the series, especially that of Barros et al. (2012), which uses the method proposed by Robinson et al. (1995a) to estimate the fractional parameter. ...
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The present work aims to analyze the behavior of orange juice exports, in volume and monetary value, considering its main destinations, aiming to understand how these series behaved in the face of market shocks between 1997 and 2019. For that, the methodology used was based on the method proposed by Robinson et al. (1995a), which tests the presence of long memory across time series data, and on the structural break test proposed by Andrews and Ploberger (1994). According to the results, all analyzed series show no long memory. This result indicates that shocks that occurred in the period had a temporary effect on the flow of Brazilian orange juice exports, with no long-term effect on the trajectory of the series. These results allow us to conclude that the destinations analyzed here are solid markets for Brazilian orange juice export, and that short-term variations in export flows indicate a search for bargain prices among agents who purchase the Brazilian drink.
... We also verify whether the distribution of dengue incidence cases follows a normal distribution using the Anderson-Darling test [53]. Additionally, we check the long-term dependency or self-similarity using the Hurst exponent [40]. Furthermore, we employ the non-parametric Granger causality (GC) test [35] to verify the causal relationship between rainfall and dengue outbreak. ...
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Dengue fever is a virulent disease spreading over 100 tropical and subtropical countries in Africa, the Americas, and Asia. This arboviral disease affects around 400 million people globally, severely distressing the healthcare systems. The unavailability of a specific drug and ready-to-use vaccine makes the situation worse. Hence, policymakers must rely on early warning systems to control intervention-related decisions. Forecasts routinely provide critical information for dangerous epidemic events. However, the available forecasting models (e.g., weather-driven mechanistic, statistical time series, and machine learning models) lack a clear understanding of different components to improve prediction accuracy and often provide unstable and unreliable forecasts. This study proposes an ensemble wavelet neural network with exogenous factor(s) (XEWNet) model that can produce reliable estimates for dengue outbreak prediction for three geographical regions, namely San Juan, Iquitos, and Ahmedabad. The proposed XEWNet model is flexible and can easily incorporate exogenous climate variable(s) confirmed by statistical causality tests in its scalable framework. The proposed model is an integrated approach that uses wavelet transformation into an ensemble neural network framework that helps in generating more reliable long-term forecasts. The proposed XEWNet allows complex non-linear relationships between the dengue incidence cases and rainfall; however, mathematically interpretable, fast in execution, and easily comprehensible. The proposal's competitiveness is measured using computational experiments based on various statistical metrics and several statistical comparison tests. In comparison with statistical, machine learning, and deep learning methods, our proposed XEWNet performs better in 75% of the cases for short-term and long-term forecasting of dengue incidence.
... Here, the [anomalous-diffusion MSD] exponent β = 2H is twice the Hurst exponent [131] and K 2H is the generalized diffusion coefficient. The exponent H = 1/2 demarcates the situations of persistent (positive) and antipersistent (negative) correlations of particle displacements realized for FBM in the case 1 > H > 1/2 and 0 < H < 1/2, correspondingly, see Refs. ...
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How does a systematic time-dependence of the diffusion coefficient D(t) affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the ensemble-and time-averaged mean-squared displacements (MSD x 2 (∆) and TAMSD δ 2 (∆)) of FBM featuring x 2 (∆) = δ 2 (∆) ∝ ∆ 2H (where H is the Hurst exponent) changes in the presence of a power-law deterministically varying diffusivity D(t) ∝ t α−1 , germane to the process of scaled Brownian motion (SBM), determining the strength of fractional Gaussian noise. The resulting compound "scaled-fractional" Brownian motion or FBM-SBM is found to be nonergodic, with x 2 (∆) ∝ ∆ α+2H−1 and δ 2 (∆) ∝ ∆ 2H. We also detect a stalling/constant behavior of the MSDs for very subdiffusive SBM and FBM, when α + 2H − 1 < 0. The distribution of particle displacements for FBM-SBM remains Gaussian, as for the parent processes of FBM and SBM, in the entire region of scaling exponents (0 < α < 2 and 0 < H < 1). The FBM-SBM process is aging in a manner similar to SBM. The velocity autocorrelation function (ACF) of particle increments of FBM-SBM exhibits a dip when the parent FBM process is subdiffusive. Both for sub-and superdiffusive FBM contributions to the FBM-SBM process, the SBM exponent affects the long-time decay exponent of the ACF. Applications of the FBM-SBM-amalgamated process to the analysis of single-particle-tracking (SPT) data are discussed. A comparative tabulated overview of recent experimental (mainly SPT) and computational datasets amenable for interpretation in terms of FBM-, SBM-, and FBM-SBM-like models of diffusion culminates the presentation. The statistical aspects of the dynamics of a wide range of biological systems is compared, from nanosized beads in living cells, to chromosomal loci, to water diffusion in the brain, and, finally, to patterns of animal movements.
... to be in satisfactory agreement with observations [152], and in tectonics, where major criticalities are a matter of fact ( [81], [86], [92]). In particular, the elastic rebound theory does not contemplate strong time clustering ( [77], [85]) and it completely ignores differences in tectonic settings [46], for instance regarding packaging of faulting (e.g. [140] for normal faults), length of faults, variability in stress drop, frequency-size scaling [149], aftershock productivity [39] and duration of seismic sequences [192]. ...
Thesis
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The dissipation of energy stored in Earth’s crust due to the action of exogenous and endogenous forces reveals the heterogeneity of a complex and self-organized physical system. While in the lower layers a visco-elastic behavior dominates, in the upper ones the brittleness of the rocks is the main reason of highly unpredictable phenomena, which alternate long quiescent periods with sudden activity, producing dramatic impact on the people’s lives. Catastrophes are sometimes forewarned by precursors, other times they come unexpectedly. Therefore, understanding the physics driving the evolution of fault systems is a crucial task to fine tune seismic prediction methods and for the mitigation of seismic risk. Statistical mechanics of critical and complex phenomena is an essential tool for the development of effective models and for the execution of reliable analysis. In this thesis, a simple model is introduced to explain the variability of the parameters of the two main scale laws describing seismic processes: the Gutenberg-Richter law and the Omori-Utsu law. Seismological observations are related to the statistical properties of seismicity and to local and regional tectonics. A possible mechanism to account for the different duration of the seismic sequences as a function of the geodinamic environment and odds in the spatial distribution of seismic events is proposed. The instantaneous Omori p-parameter is also suggested as a prognostic candidate for the occurrence of large aftershocks: it lingers stationary or decreases throughout the sequence only if the system is evolving towards stability, on the contrary, p increases if it is becoming more unstable. An innovative idea is then inspected to establish the proximity to the critical breaking point. It is based on the mechanical response of rocks to tidal perturbations, which is proven to be modified before the mainshock. Several relations between tidal stress and seismic parameters such as focal mechanism, depth, magnitude and geodynamic setting are investigated. This technique allows to identify different patterns related to the seismic cycle, marking the fingerprints of progressive crustal weakening. Destabilization seems to arise from two possible mechanisms compatible with the so called “preslip” and “cascade” models. The first is featured by a decreasing susceptibility to stress perturbation, anomalous geodetic deformation and seismic activity, while, on the other hand, the second shows seismic quiescence and increasing responsiveness. It also provides new insights about the magnitudes of impending earthquakes, which are proposed to come out from a large-scale nucleation phase whose duration depends on the magnitude of the incoming event according to the power-law relation T ≈ M^(1/3) for Mw ≤ 6.5. Theoretical results are then applied to considerable recent seismicity occurred in Italy, California, Mexico, Cascadia, Iceland, Greece and New Zealand, also taking into account the influence of ocean tides, whenever necessary. Moreover, it is suggested that spatial changes in seismic response to tides also allow to improve our knowledge of poorly understood local crustal dynamics, in particular where seismic catalogs are lacking and structural observations are ambiguous; while temporal swings reveal a complex mechanism for tidal modulation of energy dissipation, above all in the upper brittle crust.
... The Hurst exponent was employed to predict the future trend in the SPEI. The Hurst exponent was proposed by Hurst when analysing the hydrological data of the Nile River (Hurst, 1951). It is deemed an effective method to quantitatively describe the long-term dependence of timeseries information based on a large number of empirical studies. ...
Article
A deep understanding of the response characteristics of vegetation to drought is important for ecosystem and water resource management. In this study, we investigate spatiotemporal patterns of vegetation responses to droughts and their causes in the Qinling Mountains (QMs). The study showed that (1) there was a significant positive correlation (p < 0.05) between the Normalized Difference Degetation Index (NDVI) and Standardized Precipitation Evapotranspiration Index (SPEI) in 80.98% of the QMs. The most sensitive period of vegetation growth in response to drought was May-July. The average timescale of the vegetation response to drought changes was 11.88 months, where the vegetation response timescale on the northern slope (10.67 months) was shorter than that on the southern slope (12.02 months). (2) A clear relationship was identified between the vegetation response to drought and vegetation type and elevation, indicating that the overall vegetation sensitivity in the QMs is relatively high, with grasses being the most sensitive to drought conditions. Vegetation drought management should be focused on extremely sensitive and severely sensitive areas in aridified regions in the future.
... The Hurst parameter was originally defined by Hurst [13] as a measure of the autocorrelation of a time series. But it is well known that it can also determine the degree of 'roughness' of a trajectory, e.g., in terms of the fractal dimension of its graph or its p th variation. ...
Preprint
We say that a continuous real-valued function $x$ admits the Hurst roughness exponent $H$ if the $p^{\text{th}}$ variation of $x$ converges to zero if $p>1/H$ and to infinity if $p<1/H$. For the sample paths of many stochastic processes, such as fractional Brownian motion, the Hurst roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber--Schauder coefficients of $x$ under which the Hurst roughness exponent exists and is given as the limit of the classical Gladyshev estimates $\wh H_n(x)$. This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because no assumption whatsoever is made on the possible dynamics of the function $x$. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber--Schauder expansion of $x$. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence $(\wh H_n)_{n\in\bN}$. We also discuss how a dynamic change in the Hurst roughness parameter of a time series can be detected. Our results are illustrated by means of high-frequency financial times series.
... Changes in the relationship between precipitation and streamflow/runoff can be detected by statistical methods such as the Pettitt method (Pettitt, 1979) and the Cumulative Sum of Departures of Modulus Coefficient (CSDMC) (Hurst, 1951). It can also be analyzed by the Budyko method/model that quantifies the regional water balance from the perspectives of water and energy balance in different climate conditions (e.g., Fu et al., 2007;Donohue et al., 2010;Yang et al., 2011;Xu et al., 2013;Zhang et al., 2013c;Wang et al., 2016;Wang et al., 2018a). ...
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Climate warming on the cryosphere could change catchment precipitation‐runoff relation by additional water from glacier melting and more energy absorbed at the ground surface. These changes further alter the physical properties of the catchment. The combined effects of these changes on runoff are analyzed from upstream to downstream of the Yarlung‐Zangpo River (YR) basin in southeastern Tibetan Plateau using statistical methods and a modified Budyko equation with consideration of glacier melting. Major results show that in the study period of 1980–2015 there was a jump in the mean annual temperature and precipitation around 1997. Since 1997, accelerated glacier melt and the thaw of permafrost have contributed to a nearly 80.9% increase in surface runoff in the upstream region of the YR basin. Meanwhile, the increase of runoff in the same period in downstream areas of the basin with warm and wet climates is smaller and has been mostly from an increase in annual precipitation. Part of that increase is offset by changes in catchment properties following the warming, such as the increase in vegetation coverage. Results of runoff responses to climate change and catchment properties across different sub‐basins in the YR basin and between the two time periods separated in 1997 suggest that continued warming would reduce the buffering effect of glacial and permafrost on runoff in upstream areas of the basin, shortening the runoff response time to precipitation and increasing flood and drought vulnerability of the YR basin.
... Long-range dependent time-series were first observed in hydrology [Gra+17]. By looking at recordings of floods from the Nile river, Hurst [Hur51] observed very long correlations between past and current values. This phenomenon was then recognized in many different fields, be it finance or telecommunications [DOT03;Gra+17]. ...
Thesis
This dissertation studies unsupervised time-series modelling. We first focus on the problem of linearly predicting future values of a time-series under the assumption of long-range dependencies, which requires to take into account a large past. We introduce a family of causal and foveal wavelets which project past values on a subspace which is adapted to the problem, thereby reducing the variance of the associated estimators. We then investigate under which conditions non-linear predictors exhibit better performances than linear ones. Time-series which admit a sparse time-frequency representation, such as audio ones, satisfy those requirements, and we propose a prediction algorithm using such a representation. The last problem we tackle is audio time-series synthesis. We propose a new generation method relying on a deep convolutional neural network, with an encoder-decoder architecture, which allows to synthesize new realistic signals. Contrary to state-of-the-art methods, we explicitly use time-frequency properties of sounds to define an encoder with the scattering transform, while the decoder is trained to solve an inverse problem in an adapted metric.
... Mann-Kendall test with long-term persistence The MK2 and MK3 tests address the implications of short-term Environ Sci Pollut Res persistence in our time series. But long-term persistence (LTP) in hydroclimatic data, or the Hurst phenomenon (Hurst 1951), can also have a considerable influence on our success in detecting trends (Cohn and Lins 2005). LTP can manifest as groups of events with similar climatic characteristics, such as floods or droughts, occurring clustered in time on time scales of months to years or longer (Koutsoyiannis and Montanari 2007). ...
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Long-term streamflow trends are closely related to meteorological factors; understanding the relationships between them helps to improve water resources management in advance. In this study, we examined long-term annual and seasonal streamflow trends over 1961–2010 in 28 stations in the Songhua River Basin (SRB), China, using four kinds of trend detection methods and then determined the optimal meteorological predictors for SRB streamflow based on the multiple wavelet coherence. We found significant downward trends in annual streamflow in a large part of the study stations (varies from 10 to 18 for different methods), and fewer decreasing stations were detected when we consider the full autocorrelation and the long-term persistence in streamflow. In contrast to annual streamflow, fewer stations showed significant downward trends in summer and winter streamflow. Streamflow generally followed the pattern of precipitation (PRE); the largest streamflow changes occurred in summer and August monthly streamflow variation contributed the most to the annual streamflow variation. We found PRE and potential evapotranspiration (PET) combined was the optimal predictor for streamflow above Jiangqiao and on the Jiangqiao–Dalai section of the Songhua River; as for the Dalai–Harbin section and the Harbin–Jiamusi section, the optimal predictor combinations are PRE and number of rainy days (WET), and PRE and average monthly temperature (TMP) respectively.
... O expoente de escala (obtido como o coeficiente angular da regressão linear versus ) é chamado de expoente de Hurst generalizado e para séries temporais estacionárias, é idêntico ao conhecido expoente de Hurst (Hurst, 1951). Para series multifractais é a função decrescente de enquanto para séries temporais monofractais é independente de . ...
Article
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Em muitos países tropicais incluindo Brasil observou-se que as mudanças nos padrões de chuva causam inundações e secas com tendência a continuar se agravar durante século 21. Para diminuir as consequências na vida e saúde humana, atividades econômicas, ecossistemas e infraestrutura é necessário desenvolver modelos de previsão mais confiáveis. O primeiro passo nesta direção é uma análise detalhada da variabilidade climática na região estudada. Neste trabalho analisou-se propriedades multifractais das séries temporais do Índice de Precipitação Padronizado (SPI), desenvolvido para classificar condições secas/úmidas de acordo com severidade. Este índice foi calculado para diferentes escalas de tempo (1, 3, 6 e 12 meses) e analisado utilizando o método Multifractal detrended fluctuation analysis. Os parâmetros de complexidade do espectro multifractal (posição de máximo, largura e assimetria) junto com o expoente de Hurst, mostraram que as séries de SPI são geradas pelo processo multifractal com multifractalidade e persistência mais forte para maiores escalas de acumulação da chuva.
... It has been shown that time series data exhibit complex and dynamic behaviors whose autocorrelation (the cross-correlation of the signal with itself) can be characterized by power laws (Moret et al., 2003). Hurst was one of the first to identify a power law in a time series in the real world, specifically studying the Nile River and problems related to water storage using rescaled (R/S) statistics (Hurst, 1951). An alternative method, DFA, was proposed to detect long-range autocorrelations embedded in the mosaic structure of DNA because it avoids the spurious detection of apparent long-range autocorrelations (Peng et al., 1994). ...
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In this study we investigate possible long-range trends in the cryptocurrency market. We employed the Hurst exponent in a sample covering the period from 1 January 2016 to 26 March 2021. We calculated the Hurst exponent in three non-overlapping consecutive windows and in the whole sample. Using these windows, we assessed the dynamic evolution in the structure and long-range trend behavior of the cryptocurrency market and evaluated possible changes in their behavior towards an efficient market. The innovation of this research is that we employ the Hurst exponent to identify the long-range properties, a tool that is seldomly used in analysis of this market. Furthermore, the use of both the R/S and the DFA analysis and the use of non-overlapping windows enhance our research’s novelty. Finally, we estimated the Hurst exponent for a wide sample of cryptocurrencies that covered more than 80% of the entire market for the last six years. The empirical results reveal that the returns follow a random walk making it difficult to accurately forecast them.
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This study develops the governing equations of unsteady multi-dimensional incompressible and compressible flow in fractional time and multi-fractional space. When their fractional powers in time and in multi-fractional space are specified to unit integer values, the developed fractional equations of continuity and momentum for incompressible and compressible fluid flow reduce to the classical Navier–Stokes equations. As such, these fractional governing equations for fluid flow may be interpreted as generalizations of the classical Navier–Stokes equations. The derived governing equations of fluid flow in fractional differentiation framework herein are nonlocal in time and space. Therefore, they can quantify the effects of initial and boundary conditions better than the classical Navier–Stokes equations. For the frictionless flow conditions, the corresponding fractional governing equations were also developed as a special case of the fractional governing equations of incompressible flow. When their derivative fractional powers are specified to unit integers, these equations are shown to reduce to the classical Euler equations. The numerical simulations are also performed to investigate the merits of the proposed fractional governing equations. It is shown that the developed equations are capable of simulating anomalous sub- and super-diffusion due to their nonlocal behavior in time and space.
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We demonstrate that when power scaling occurs for an individual tree and in a forest, there is great resulting simplicity notwithstanding the underlying complexity characterizing the system over many size scales. Our scaling framework unifies seemingly distinct trends in a forest and provides a simple yet promising approach to quantitatively understand a bewilderingly complex many-body system with imperfectly known interactions. We show that the effective dimension, Dtree, of a tree is close to 3 whereas a mature forest has Dforest approaching 1. We discuss the energy equivalence rule and show that the metabolic rate-mass relationship is a power law with an exponent D/(D + 1) in both cases leading to a Kleiber’s exponent of 3/4 for a tree and 1/2 for a forest. Our work has implications for understanding carbon sequestration and for climate science.
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In this paper, we empirically show the dynamics of daily wavelet-filtered (denoised) S&P 500 returns (2000–2020) to consist of an almost equally divided combination of stochastic and deterministic chaos, rendering the series unpredictable after expiration of the Lyapunov time, resulting in futile forecasting attempts. We observe a clear distinction of the nature of the underlying time series dynamics by applying a novel and combinatory chaos analysis framework by comparing the wavelet-filtered S&P 500 returns with surrogate datasets, Brownian motion returns, and a Lorenz system realisation. Furthermore, we are the first to observe the strange attractor of the daily S&P 500 return system graphically via Takens’ embedding and a spectral embedding in combination with Laplacian eigenmaps. Finally, we critically discuss the implications and future prospects of financial forecasting.
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The detrended fluctuation analysis (DFA) and its variants are popular methods to analyze the self-similarity of a signal. Two steps characterize them: firstly, the trend of the centered integrated signal is estimated and removed. Secondly, the properties of the so-called fluctuation function which is an approximation of the standard deviation of the resulting process is analyzed. However, it appears that the statistical mean was assumed to be equal to zero to obtain it. As there is no guarantee that this assumption is true a priori, this hypothesis is debatable. The purpose of this paper is to propose two alternative definitions of the fluctuation function. Then, we compare all of them based on a matrix formulation and the filter-based interpretation we recently proposed. This analysis will be useful to show that the approach proposed in the original paper remains a good compromise in terms of accuracy and computational cost.
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Due to climate change, extreme rainfall and drought events are becoming more and more frequent in several regions of the globe. We investigated the suitability of employing statistical and fractal (or scaling) methods to characterise extreme precipitation and drought events. The case of the island of Mauritius was considered, for which monthly mean rainfall data for the period January 1950 to December 2016 were analysed. The generalised extreme value distribution was used to extract the 10- and 20-year return levels and the Standardised Precipitation Index (SPI) was used to identify anomalous wet and dry events. A log-term correlation analysis was also performed to characterise the relationship between maximum rainfall and its duration. The results indicate that the 10-year return level is approximately between 500 mm and 850 mm and the 20-year return level is between 600 mm and 1000 mm. Results also show that the extreme maximum rainfall events occur mostly during austral summer (November to April) and could be related to the effects of tropical cyclones and La Niña events, while anomalous dry events were found to be significantly persistent with very long periods of drought. Moreover, there was a strong correlation between maximum rainfall and its duration. The methodology used in this work could be very useful in similar studies for other Small Island Developing States. Significance: • We show the usefulness of both statistical and fractal methods to understand occurrences of extreme precipitation events. • We identify anomalous wet and dry events in rainfall time-series data using the Standardised Precipitation Index.
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Catastrophic event in geotechnical engineering such as rock failure could cause serious casualties and property losses. Aiming at proposing an early-warning method for rock failure, precursors based on Hurst exponent and acoustic emission/microseismic activity monitoring were investigated. Firstly, Hurst exponent was introduced to the geotechnical field to reflect the long memory and fractal texture in time series. A modified R/S method was proposed with overlapping subseries, so fewer data were required for the calculation of Hurst exponent. Secondly, Hurst exponent was proposed varying with time, and the generation of Hurst exponent series was constituted. Finally, the failure of rock was studied with Hurst exponent, assisted with acoustic emission monitoring in laboratory testing and microseismic activity monitoring in field testing. Results show that the start of Hurst exponent parameter decrease and acoustic emission parameter increase could be viewed as the early-warning point of instability. Moreover, the drop in Hurst exponent combined with significant microseismic events or magnitude or sudden increase of cumulative apparent volume is defined as the early-warning point of instability. Hurst exponent with acoustic emission/microseismic activity monitoring for early-warning of instability is applicable and effective. The proposed method could be applied in disaster preventions with an artificial intellect.
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Some Amazonia regions are vulnerable to natural disasters. Hurst analysis of hydrological events can provide more information than classical statistical methods. The objectives of the present work were (1) to search for nonlinear signatures in rainfall and temperature using nonlinear time series methods and (2) to identify different Hurst scaling regimes in the Standardized Precipitation-Evapotranspiration Index (SPEI hereafter) at multiple resolutions using R/S analysis. Data sets of monthly rainfall and monthly mean temperature correspond to Belem (Pará state) and Manaus (Amazonas state), Brazil. The study covered a temporal scale of fifty-nine years (1961–2019). We selected 1-month, 3-month, 6-month, 9-month, 12-month, 24-month and 48-month SPEI timescales. The main analyses were conducted on detrended time series. Previous to rescaled range analysis (R/S hereafter) application, we explored nonlinear properties of each raw and residual time series through linear and nonlinear stationarity tests. Based on local means, the undetrended temperature and rainfall time series showed evidence of heteroscedasticity. Second-order statistics was stationary in all cases. Each rainfall and temperature time series showed deterministic components ranging from 0.78 to 0.83. Drought/extremely wet events are probably related to deterministic processes. We found two long-term correlation zones with scaling regimes separated by a crossover starting in March 1964. Short-scale regimes in SPEI1 to SPEI48 could be due to self-organized criticality or finite sample effect. Positive SPEI values can persist in Belem in the future, while memory effect of negative SPEI values in Manaus suggests trend-reinforcing of different drought types in the future.
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Net primary productivity (NPP) has been widely used as the indicator of vegetation function and exhibits large spatial and temporal variations caused by numerous factors. Northwest China (NWC) is one of the driest regions in China, and water supply is the key determinant of NPP here. However, studies on the effects of water stress on NPP in NWC at the regional scale are still relatively lacking. Thus, in this study, based on a set of Moderate-Resolution Imaging Spectroradiometer (MODIS) NPP and evapotranspiration (ET) datasets, we quantified the response of NPP to water stress, which is indicated by crop water stress index (CWSI). Regional average of annual NPP in NWC showed an increasing trend during the study period, at a rate of 0.84 g C m−2 yr−1. At the province level, the NPP increase rates increased in the order of Ningxia (7.7%), Shaanxi (6.5%), Gansu (4.5%), Qinghai (3.8%), and Xinjiang (1.7%). NPP was negatively correlated with CWSI (p<0.05) in 73% of areas, indicating the key role of water stress in constraining NPP over this arid region. The effect of water stress on NPP changes with elevation. Water stress has the strongest negative impact on NPP in areas with elevations around 2000 m. In elevations above 5000 m, NPP is not limited by water stress, mostly positively correlated with CWSI. Our findings further clarify the importance of water stress in dryland ecosystems, while highlighting that elevation gradients can significantly affect the correlation between NPP and water stress.
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The Total Triangles Area (TTA) algorithm was introduced as a good alternative for the estimation of the Hurst exponent in Lotfalinezhad and Maleki (2020). However, a theoretical framework for the validity of the TTA algorithm was missing. The main goal of this paper is to provide such a theoretical framework. A slightly different approach to the TTA algorithm is also presented, which leads to the introduction of the Triangle Area (TA) algorithm. A comparison of the accuracy of these algorithms (also with respect to the GHE one) is also carried out with fractional Brownian motions, in order to provide some light about when it is better to apply each of the algorithms.
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In order for researchers to deliver robust evaluations of time series models, it often requires high volumes of data to ensure the appropriate level of rigor in testing. However, for many researchers, the lack of time series presents a barrier to a deeper evaluation. While researchers have developed and used synthetic datasets, the development of this data requires a methodological approach to testing the entire dataset against a set of metrics which capture the diversity of the dataset. Unless researchers are confident that their test datasets display a broad set of time series characteristics, it may favor one type of predictive model over another. This can have the effect of undermining the evaluation of new predictive methods. In this paper, we present a new approach to generating and evaluating a high number of time series data. The construction algorithm and validation framework are described in detail, together with an analysis of the level of diversity present in the synthetic dataset.
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Grassland, as the primary vegetation on the Qinghai-Tibet Plateau, has been increasingly influenced by water availability due to climate change in last decades. Therefore, identifying the evolution of drought becomes crucial to the efficient management of grassland. However, it is not yet well understood as to the quantitative relationship between vegetation variations and drought at different time scales. Taking Qinghai Province as a case, the effects of meteorological drought on vegetation were investigated. Multi-scale Standardized Precipitation Evapotranspiration Index (SPEI) considering evapotranspiration variables was used to indicate drought, and time series Normal Difference Vegetation Index (NDVI) to indicate the vegetation response. The results showed that SPEI values at different time scales reflected a complex dry and wet variation in this region. On a seasonal scale, more droughts occurred in summer and autumn. In general, the NDVI presented a rising trend in the east and southwest part and a decreasing trend in the northwest part of Qinghai Province from 1998 to 2012. Hurst indexes of NDVI revealed that 69.2% of the total vegetation was positively persistent (64.1% of persistent improvement and 5.1% of persistent degradation). Significant correlations were found for most of the SPEI values and the one year lagged NDVI, indicating vegetation made a time-lag response to drought. In addition, one month lagged NDVI made an obvious response to SPEI values at annual and biennial scales. Further analysis showed that all multiscale SPEI values have positive relationships with the NDVI trend and corresponding grassland degradation. The study highlighted the response of vegetation to meteorological drought at different time scales, which is available to predict vegetation change and further help to improve the utilization efficiency of water resources in the study region.
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In order to improve the prediction performance in oil price forecasting, a novel memory-trait-driven decomposition-reconstruction-ensemble learning paradigm is proposed for oil price forecasting. The proposed methodology consists of four steps, i.e., data decomposition for original complex time series, component reconstruction for decomposed components, individual prediction for the reconstructed components, and ensemble output based on the individual component prediction results, which are all driven by memory traits. For verification purpose, the West Texas Intermediate (WTI) crude oil spot prices are used as the sample data. The experimental results demonstrated that the proposed methodology can produce the better and more robust results relative to the benchmarking models listed in this study. This indicates that the proposed memory-trait-driven decomposition-reconstruction-ensemble methodology can be used as a promising solution to oil price prediction with the traits of memory.
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This study examines the potential of Shanghai crude oil (SCO) futures as a benchmark in the Asian market. We investigate the market efficiency and long-term equilibrium of SCO futures in comparison with global benchmarks, such as West Texas Intermediate, Brent, and Dubai crude oil futures. Despite the weak market integration between SCO futures and other international benchmarks, we find strong evidence that their market efficiency and long-term equilibrium do not significantly differ. We explain how current market properties are achieved using the information flow from international crude oil to the SCO futures market. Our findings present implications for investors and policymakers based on the unilateral information flow at the level and rise–fall pattern in the price series: (1) investors could exploit this pattern’s predictability in their investment strategy, and (2) regulators could implement open trading policies that would enable SCO futures to integrate with global benchmarks.
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This paper reports measurements of radon in groundwater conducted in China between Janu-ary 2013 and December, 2013. During measurements, a great Ms=6.6 earthquake occurred in the Gansu province China. The paper analyses data derived from three nearby stations via monofractal and multifractal detrended fluctuations analysis and explores whether pre-earthquake fractal and long memory trends exist in the time series. Several critical epochs with characteristic pre-seismic long-lasting fractal patterns are identified in segmented parts of the series. In the whole data series, the fluctuation function, scaling exponent and generalised Hurst exponent depend on the fractal scales which indicates multifractality. The multifractal spectrum of the raw, shuffled and truncated series showed a significant increase degree of fluctuation that can be explained as a pre-earthquake indicators attributed the Gansu earthquake due to its very high magnitude.
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Timely and accurate monitoring of the spatiotemporal changes in drought is very important for the reduction in the social losses caused by drought. The Optimized Meteorological Drought Index (OMDI), originally established in southwestern China, showed great potential for drought monitoring over large regions on a large scale. However, the applicability of the index requires further evaluation, especially when used throughout China, which has a different agricultural divisions, variable climatic conditions, complex terrain and diverse land cover. In addition, the OMDI model relies on training data to construct local parameters for the model. On a large scale, it is of great significance to use multisource remote sensing data sets to construct OMDI model parameters. In this paper, the constrained optimization method was used to establish weights for the MODIS-derived Vegetation Conditional Index (VCI), TRMM-derived Precipitation Condition Index (PCI), and GLDAS-derived Soil Moisture Condition Index (SMCI) and calculate the OMDI based on the Standard Precipitation Index (SPI), Standardized Precipitation Evapotranspiration Index (SPEI) and weather stations. The accuracy of the OMDI model was evaluated by using the correlation coefficient. Moreover, the spatiotemporal changes in drought were also analyzed through trend analysis, Mann-Kendall (MK) statistics and the Hurst index on the monthly and annual scales. The results showed that (1) the highest positive correlation between the OMDI and the SPI was SPI-1, which was higher than that for any other month interval, such as 3 months, 6 months, 9 months and 12 months of the SPI. The results indicated that the OMDI was suitable to monitor meteorological drought. (2) In the nine agricultural subareas in China, the degree of drought in the Yangtze River (DYR) area had the most severe evolution and change frequency. This region was very sensitive to drought in the past two decades. (3) The area with OMDI variation coefficient less than 0.1 accounted for 94%, indicating that the degree of drought fluctuates little; The linear tendency rate is 0.0004, and the area greater than 0 reaches 66.44%, indicating that the drought is developing in a lightning trend. (4) The Hurst index value is mostly higher than 0.5 (the area ratio is 56.31%), and the area of "Positive-Consistent" and "Negative- Opposite" accounted for 54.02%, indicating that more than half of China's area drought changes will show a trend of mitigation in the future.
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In this paper, while examining the factorical effects of COVID-19 on different layers of capital market and economy, we examine market efficiency during Corona virus pandemic, as well as correlation and relationship between capital market indicators and market efficiency with COVID-19 data in Iran. Hurst exponent is able to identify fractal dimensions of time series data in various capital market indices and COVID-19 is analyzed by using newly modified index. In addition, calculated trend indicates inefficiency during virus outbreak. Analysis of real time series data deficit in Iran from 22 February to 23 September 2020, which reveal interesting and new facts. In this paper, we use modified Hurst exponent index that is one of contribution which is a new technique and confirms output of results.
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The Qinghai-Tibet Plateau (QTP) is an area sensitive to global climate change, and land use/land cover change (LUCC) plays a vital role in regulating climate system at different temporal and spatial scales. In this study, we analyzed the temporal and spatial trend of precipitation and the characteristics of LUCC on the QTP. Meanwhile, we also used the normalized difference vegetation index (NDVI) as an indicator of LUCC to discuss the relationship between LUCC and precipitation. The results show the following: (1) Annual precipitation showed a fluctuant upward trend at a rate of 11.5 mm/decade in this area from 1967 to 2016; three periods (i.e., 22 years, 12 years, and 2 years) of oscillations in annual precipitation were observed, in which expectant 22 years is the main oscillation period. It was predicted that QTP will still be in the stage of increasing precipitation. (2) The LUCC of the plateau changed apparently from 1980 to 2018. The area of grassland decreased by 9.47%, and the area of unused land increased by 7.25%. From the perspective of spatial distribution, the transfer of grassland to unused land occurred in the western part of the QTP, while the reverse transfer was mainly distributed in the northwestern part of the QTP. (3) NDVI in the northern and southwestern parts of the QTP is positively correlated with precipitation, while negative correlations are mainly distributed in the southeast of the QTP, including parts of Sichuan and Yunnan Province. Our results show that precipitation in the QTP has shown a fluctuating growth trend in recent years, and precipitation and NDVI are mainly positively correlated. Furthermore, we hope that this work can provide a theoretical basis for predicting regional hydrology, climate change, and LUCC research.
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