Update of Prior Probabilities by Minimal Divergence

The present paper investigates the update of an empirical probability distribution with the results of a new set of observations. The update reproduces the new observations and interpolates using prior information. The optimal update is obtained by minimizing either the Hellinger distance or the quadratic Bregman divergence. The results obtained by the two methods differ. Updates with information about conditional probabilities are considered as well.

Empirical probability distribution validity based on accumulating statistics of observations by controlling the moving average and root-mean-square deviation

Knowing probability distributions for calculating expected values is always required in the engineering practice and other fields. Commonly, probability distributions are not always available. Moreover, the distribution type may not be reliably determined. In this case, an empirical distribution should be built directly from the observations. Therefore, the goal is to develop a methodology of accumulating and processing observation data so that the respective empirical distribution would be close enough to the unknown real distribution. For this, criteria regarding sufficiency of observations and the distribution validity are to be substantiated. As a result, a methodology is presente О.М. Мелкозьорова1, С.Г. Рассомахінd that considers the empirical probability distribution validity with respect to the parameter’s expected value. Values of the parameter are registered during a period of observations or measurements of the parameter. On this basis, empirical probabilities are calculated, where every next period the previous registration data are used as well. Every period gives an approximation to the parameter’s expected value using those empirical probabilities. The methodology using the moving averages and root-mean-square deviations asserts that the respective empirical distribution is valid (i.e., it is sufficiently close to the unknown real distribution) if the parameter’s expected value approximations become scattered very little for at least the three window multiple-of-2 widths by three successive windows. This criterion also implies the sufficiency of observation periods, although the sufficiency of observations per period is not claimed. The validity strongly depends on the volume of observations per period.

Probabilistic short term wind power forecasts using deep neural networks with discrete target classes

Usually, neural networks trained on historical feed-in time series of wind turbines deterministically predict power output over the next hours to days. Here, the training goal is to minimise a scalar cost function, often the root mean square error (RMSE) between network output and target values. Yet similar to the analog ensemble (AnEn) method, the training algorithm can also be adapted to analyse the uncertainty of the power output from the spread of possible targets found in the historical data for a certain meteorological situation. In this study, the uncertainty estimate is achieved by discretising the continuous time series of power targets into several bins (classes). For each forecast horizon, a neural network then predicts the probability of power output falling into each of the bins, resulting in an empirical probability distribution. Similiar to the AnEn method, the proposed method avoids the use of costly numerical weather prediction (NWP) ensemble runs, although a selection of several deterministic NWP forecasts as input is helpful. Using state-of-the-art deep learning technology, we applied our method to a large region and a single wind farm. MAE scores of the 50-percentile were on par with or better than comparable deterministic forecasts. The corresponding Continuous Ranked Probability Score (CRPS) was even lower. Future work will investigate the overdispersiveness sometimes observed, and extend the method to solar power forecasts.

Inverse Gaussian distribution of wave set-up heights along a shoreline with complicated geometry

The phenomenon of wave set-up may substantially contribute to the formation of devastating coastal flooding in certain coastal sections. We study empirical probability distribution of the occurrence of different set-up heights in section of coastline near Tallinn in the Gulf of Finland, the eastern Baltic Sea. The shoreline in the study area is often attacked by high waves from various directions and also has a complex geometry. Shown is that this distribution substantially deviates from the Rayleigh or the Weibull distribution that usually reflect the distribution of different wave heights. The distribution of wave set-up heights matches a Wald (inverse Gaussian) distribution along the entire study area. Even though different sections of the study area are open to different directions and host substantially different wave regimes, the leading term of the exponent in the associated inverse Gaussian distribution varies insignificantly along the study area and generally is close to −1. This appearance signals that extreme set-up events are substantially more probable that it could be expected from the probability of occurrence of severe seas. This feature is invariant with respect to the orientation of the coastline and with respect to the properties of local wave climate.

Evaluation of soil thermal diffusivity algorithms at two equatorial sites in West Africa

<p>This study presents comparisons between six algorithms used in the calculation of apparent thermal diffusivity (K_h) of the topsoil during measurement campaigns conducted at two equatorial sites. It further investigates the effects of transient and seasonal variations in soil moisture content (theta) on the estimation of K_h. The data used comprise soil temperatures (T) measured at depths of 0.05 m and 0.10 m, and theta within the period of transition from the dry season to the wet season at Ile Ife (7.55˚ N, 4.55˚ E), and for the peak of the wet season at Ibadan (7.44˚ N, 3.90˚ E). The thermal diffusivity, K_h, was calculated from six algorithms, of: harmonic, arctangent, logarithmic, amplitude, phase, and conduction-convection. The reliability of these algorithms was tested using their values to model T at a depth of 0.10 m, where direct measurements were available. The algorithms were further evaluated with statistical indices, including the empirical probability distribution function of the differences between the measured and modeled temperatures ([delta capitalized]T). The maximum absolute values of [delta capitalized]T for the six algorithms investigated were: 0.5˚C, 0.5˚C, 0.5˚C, 1˚C, 1˚C and 1˚C, respectively. K_h showed an increasing trend as theta increased from the dry season to the peak of the wet season, with R² = 0.70 for the harmonic algorithm. The accuracy of all of the algorithms in modeling T reduced with transient variations of theta. The harmonic, arctangent and logarithmic algorithms were the most appropriate for calculating K_h for the region of study. The empirical relation between theta and K_h and the values of K_h obtained in this study can be used to improve the accuracy of meteorological and hydrological models.</p>

Limits and characteristics of the multifractal behaviour of a high-resolution rainfall time series

The multifractal properties of a 2-year time series of 8-min rainfall intensity observations are investigated. The empirical probability distribution function suggests a hyperbolic intermittency with divergence of moment of order greater than 2. The power spectrum E(f) of the series obeys a power law form E(f)=f -0.66 in the range of scales 8 min to approximately 3 days. The variation of the average statistical moments with scale shows that the series is characterized by a multifractal behaviour between 8 min and approximately 11 days. The multifractal parameters associated with universality were estimated to be α=0.63 and C1=0,44 by using the Double Trace Moment, DTM, technique. The moment scaling functions obtained from the empirical values and the universal expression are in good agreement in the approximate range 1≤q≤3. Outside of this range, however, differences exist which may be related to either limitations of the data or an inexact estimation of the parameters by DTM. The evident multifractal nature of rainfall time series is encouraging since it may lead to new and improved ways of processing rainfall data used in hydrological calculations.