Panayiotis Dimitriadis

Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes

Three common stochastic tools, the climacogram i.e. variance of the time averaged process over averaging time scale, the autocovariance function and the power spectrum are compared to each other to assess each one’s advantages and disadvantages in stochastic modelling and statistical inference. Although in theory, all three are equivalent to each other (transformations one another expressing second order stochastic properties), in practical application their ability to characterize a geophysical process and their utility as statistical estimators may vary. In the analysis both Markovian and non Markovian stochastic processes, which have exponential and power-type autocovariances, respectively, are used. It is shown that, due to high bias in autocovariance estimation, as well as effects of process discretization and finite sample size, the power spectrum is also prone to bias and discretization errors as well as high uncertainty, which may misrepresent the process behaviour (e.g. Hurst phenomenon) if not taken into account. Moreover, it is shown that the classical climacogram estimator has small error as well as an expected value always positive, well-behaved and close to its mode (most probable value), all of which are important advantages in stochastic model building. In contrast, the power spectrum and the autocovariance do not have some of these properties. Therefore, when building a stochastic model, it seems beneficial to start from the climacogram, rather than the power spectrum or the autocovariance. The results are illustrated by a real world application based on the analysis of a long time series of high-frequency turbulent flow measurements.

Predictability in dice motion: how does it differ from hydro-meteorological processes?

ABSTRACT From ancient times dice have been used to denote randomness. A dice throw experiment is set up in order to examine the predictability of the die orientation through time using visualization techniques. We apply and compare a deterministic-chaotic model and a stochastic model and we show that both suggest predictability in die motion that deteriorates with time, just as in hydro-meteorological processes. Namely, a die’s trajectory can be predictable for short horizons and unpredictable for long ones. Furthermore, we show that the same models can be applied, with satisfactory results, to high temporal resolution time series of rainfall intensity and wind speed magnitude, occurring during mild and strong weather conditions. The difference among the experimental and two natural processes is in the time length of the high-predictability window, which is of the order of 0.1 s, 10 min and 1 h for dice, rainfall and wind processes, respectively.

Stochastic similarities between the microscale of turbulence and hydro-meteorological processes

ABSTRACT Turbulence is considered to generate and drive most geophysical processes. The simplest case is isotropic turbulence. In this paper, the most common three-dimensional power-spectrum-based models of isotropic turbulence are studied in terms of their stochastic properties. Such models often have a high order of complexity, lack stochastic interpretation and violate basic stochastic asymptotic properties, such as the theoretical limits of the Hurst coefficient, when Hurst-Kolmogorov behaviour is observed. A simpler and robust model (which incorporates self-similarity structures, e.g. fractal dimension and Hurst coefficient) is proposed using a climacogram-based stochastic framework and tested over high-resolution observational data of laboratory scale as well as hydro-meteorological observations of wind speed and precipitation intensities. Expressions of other stochastic tools such as the autocovariance and power spectrum are also produced from the model and show agreement with data. Finally, uncertainty, discretization and bias related errors are estimated for each stochastic tool, showing lower errors for the climacogram-based ones and larger for power spectrum ones.

Stochastic synthesis approximating any process dependence and distribution

An extension of the symmetric-moving-average (SMA) scheme is presented for stochastic synthesis of a stationary process for approximating any dependence structure and marginal distribution. The extended SMA model can exactly preserve an arbitrary second-order structure as well as the high order moments of a process, thus enabling a better approximation of any type of dependence (through the second-order statistics) and marginal distribution function (through statistical moments), respectively. Interestingly, by explicitly preserving the coefficient of kurtosis, it can also simulate certain aspects of intermittency, often characterizing the geophysical processes. Several applications with alternative hypothetical marginal distributions, as well as with real world processes, such as precipitation, wind speed and grid-turbulence, highlight the scheme’s wide range of applicability in stochastic generation and Monte-Carlo analysis. Particular emphasis is given on turbulence, in an attempt to simulate in a simple way several of its characteristics regarded as puzzles.

From Fractals to Stochastics: Seeking Theoretical Consistency in Analysis of Geophysical Data

Fractal-based techniques have opened new avenues in the analysis of geophysical data. On the other hand, there is often a lack of appreciation of both the statistical uncertainty in the results and the theoretical properties of the stochastic concepts associated with these techniques. Several examples are presented which illustrate suspect results of fractal techniques. It is proposed that concepts used in fractal analyses are stochastic concepts and the fractal techniques can readily be incorporated into the theory of stochastic processes. This would be beneficial in studying biases and uncertainties of results in a theoretically consistent framework, and in avoiding unfounded conclusions. In this respect, a general methodology for theoretically justified stochastic processes, which evolve in continuous time and stem from maximum entropy production considerations, is proposed. Some important modelling issues are discussed with focus on model identification and fitting often made using inappropriate methods. The theoretical framework is applied to several processes, including turbulent velocities measured every several microseconds, and wind and temperature measurements. The applications show that several peculiar behaviours observed in these processes are easily explained and reproduced by stochastic techniques.

The uncertainty of atmospheric processes in planning a hybrid renewable energy system for a non-connected island

Vasiliki Daniil, George Pouliasis, Eleni Zacharopoulou, Evangelos Demetriou, Georgia Manou, Maria Chalakatevaki, Iliana Parara, Xristina Georganta, Paraskevi Stamou, Sophia Karali, Evanthis Hadjimitsis, Giannis Koudouris, Evangelos Moschos, Dimitrios Roussis, Konstantinos Papoulakos, Aristotelis Koskinas, Giorgos Pollakis, Panagiota Gournari, Katerina Sakellari, Yiannis Moustakis, and the Stochastics in Energy Resources Management (NTUA)* Team

Stochastic investigation of precipitation process for climatic variability identification

The precipitation process is important not only to hydrometeorology but also to renewable energy resources management. We use a dataset consisting of daily and hourly records around the globe to identify statistical variability with emphasis on the last period. Specifically, we investigate the occurrence of mean, maximum and minimum values and we estimate statistical properties such as marginal probability distribution function and the type of decay of the climacogram (i.e. mean process variance vs. scale).