Stochastic Analysis of Hourly to Monthly Potential Evapotranspiration with a Focus on the Long-Range Dependence and Application with Reanalysis and Ground-Station Data

The stochastic structures of potential evaporation and evapotranspiration (PEV and PET or ETo) are analyzed using the ERA5 hourly reanalysis data and the Penman–Monteith model applied to the well-known CIMIS network. The latter includes high-quality ground meteorological samples with long lengths and simultaneous measurements of monthly incoming shortwave radiation, temperature, relative humidity, and wind speed. It is found that both the PEV and PET processes exhibit a moderate long-range dependence structure with a Hurst parameter of 0.64 and 0.69, respectively. Additionally, it is noted that their marginal structures are found to be light-tailed when estimated through the Pareto–Burr–Feller distribution function. Both results are consistent with the global-scale hydrological-cycle path, determined by all the above variables and rainfall, in terms of the marginal and dependence structures. Finally, it is discussed how the existence of, even moderate, long-range dependence can increase the variability and uncertainty of both processes and, thus, limit their predictability.

Application of Generalized Cauchy Process on Modeling the Long-Range Dependence and Self-Similarity of Sea Surface Chlorophyll Using 23 years of Remote Sensing Data

Understanding the temporal characteristics of sea surface chlorophyll (SSC) is helpful for marine environmental management. This study chose 10 time series of remote daily sea surface chlorophyll products from the European Space Agency during the period from July 29, 1998 to December 31, 2020. A generalized Cauchy model was employed to capture the local and global behaviors of sea surface chlorophyll from a fractal perspective; the fractal dimension D measures the local similarity while the Hurst parameter H measures the global long-range dependence. The generalized Cauchy model was fitted to the empirical autocorrelation function values of each SSC series. The results showed that the sea surface chlorophyll was multi-fractal in both space and time with the D values ranging from 1.0000 to 1.7964 and H values ranging from 0.6757 to 0.8431. Specifically, regarding the local behavior, 9 of the 10 series had low D values (<1.5), representing weak self-similarity; on the other hand, regarding the global behavior, high H values represent strong long-range dependence that may be a general phenomenon of daily sea surface chlorophyll.

Examination of Long Memory in Indian Stock Market: A Sectoral Juxtaposition

One of the situations encountered in time series analysis is long-range dependence, also known as Long memory. We investigated the presence of long memory in the Indian sectoral indices returns and investigated whether the long memory behaviour is affected by the data frequency. We applied the autoregressive fractionally integrated moving average (ARFIMA) models to 13 sectoral indices of the National Stock Exchange of India and examined the long memory in daily, monthly and quarterly return series. The results indicate the persistence in daily return series and anti-persistence in monthly and quarterly return series. Thus, we conclude that the frequency of data does have a significant effect on the behaviour of long memory patterns. The results will be helpful for present and potential investors, institutional investors, portfolio managers and policymakers to understand the dynamic nature of long memory in the Indian stock market.

Long range dependence of heavy-tailed random functions

We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

To draw or not to draw: understanding the temporal organization of drawing behaviour using fractal analyses

Studies on drawing often focused on spatial aspects of the finished products. Here, the drawing behaviour was studied by analysing its intermittent process, between drawing (i.e. marking a surface) and interruption (i.e. a pause in the marking gesture). To assess how this intermittence develops with age, we collected finger-drawings on a touchscreen by 185 individuals (children and adults). We measured the temporal structure of each drawing sequence to determine its complexity. To do this, we applied temporal fractal estimators to each drawing time series before combining them in a Principal Component Analysis procedure. The youngest children (3 years-old) drew in a more stereotypical way with long-range dependence detected in their alternations between states. Among older children and adults, the complexity of drawing sequences increased showing a less predictable behaviour as their drawings become more detailed and figurative. This study improves our understanding of the temporal aspects of drawing behaviour, and contributes to an objective understanding of its ontogeny.