Order Statistics and Benford's Law

Fix a base B and let zeta have the standard exponential distribution; the distribution of digits of zeta base B is known to be very close to Benford's Law. If there exists a C such that the distribution of digits of C times the elements of some set is the same as that of zeta, we say that set exhibits shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1, >..., X_N be independent identically distributed random variables. If the X_i's are drawn from the uniform distribution on [0,L], then as N\to\infty the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior (with a shift of \log_B L/N \bmod 1). By differentiating the cumulative distribution function of the logarithms modulo 1, applying Poisson Summation and then integrating the resulting expression, we derive rapidly converging explicit formulas measuring the deviations from Benford's Law. Fix a delta in (0,1) and choose N independent random variables from any compactly supported distribution with uniformly bounded first and second derivatives and a second order Taylor series expansion at each point. The distribution of digits of any N^\delta consecutive differences \emph{and} all N-1 normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the un-normalized differences converges to Benford's Law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.Comment: 14 pages, 2 figures, version 4: Version 3: most of the numerical simulations on shifted exponential behavior have been suppressed (though are available from the authors upon request). Version 4: a referee pointed out that we need epsilon > 1/3 - delta/2 in the proof of Theorem 1.5; this has now been adde

Kenosis and identities: pneumatological pointers

In the thesis a methodology of understanding and explicating Christian faith consistent with the mystery of the simultaneous close connection and radical difference of God, human beings and the physical-organic cosmos environment is been mapped out. The theanthropocosmic principle as an expression of the mystery functions as the heuristic key in opening up the notion of kenosis (and incarnation) of Jesus Christ and the Holy Spirit within the scope of the enduring interaction of . The Spirit in the kenotic sense of the word connects and differentiates the overall processes of being and becoming, here and there, now and then of the mystery of the `presences' of God, human beings and the natural cosmic world in being there (Dasein), being thus and thus (Sosein) and being dynamically actual (Aktsein). God acts in terms of the Spirit's operational kenotic presence within the margins of the creatureliness of people and the natural cosmic world as the kenotic clothing of God. A dynamic interpretation of the integral and differential character of being and becoming suggests that making sense of the dynamics of the formation of identities and identification is an ever ongoing endeavour. It implies a continuous process of negotiation whilst experiencing various continuums, remaining open-ended in an ever-increasing sense of wonder and mystery of "exitus a Deo-reditus in Deum".Systematic Theology and theological EthicsD. Th. (Systematic Theology

Measurement and assessment of transient torques in a three-phase squirrel cage induction motor

ThesisComputer simulation models have been developed to predict the magnitude of transient torques when starting and reswitching induction motors (De Sarkar & Berg, 1970:1031, Ghani, 1988:106 and Krause & Thomas, 1965:1038). In these models the starting or running parameters are used so that the squirrel cage induction motor has been represented by its single cage model and the effects of deep rotor bars have not been considered. During transient conditions, in an induction machine with deep bars, the deep bar effect can significantly influence the rotor time constant of the machine, which is an important parameter during reswitching operations. A model was developed by McCulloch, landy, levy & Macleod that accommodates the change in rotor resistance and leakage inductance as the rotor speed changes and it takes into account the effect of saturating the stator and rotor leakage reactances. During the implementation of the research project the motor dynamic equations have been simulated using the CASED (Computer Analysis and Simulation of Electric Drives ) software package developed by McCulloch et al for the simulation of the dynamic performance of motors with the addition of variable speed drive systems. It allows one to predict behavior with or without the improved bar models. The equations are formulated in the direct - quadrature axis form. Similar models have been developed ( Klingshirn & Jordan, 1970 : 1038, Lipo & Consoli, 1984 : 180, Smith, 1990 : 48 and Siemon, 1992 : 412) but more experimental measurements where required to validate the models. This would mean developing and uSing methods to measure the transient characteristics of the motor. Therefore, the measuring system forms a vital part of this work. Successful measurements were achieved of starting and restarting transients on a 75 kW induction motor. The very acceptable correlation obtained between measured and predicted results, using the deep bar model, was very encouraging and without any doubt shows the advantage of using these improved models when predicting performance. This research project has met its objectives by showing how important it is to use the improved model. It is shown in chapter 3 that if the fixed parameter model is used the correlation between measured and predicted results is very poor. When starting a loaded induction motor direct on line, its rotor windings are subjected to mechanical and thermal stresses due the sudden inrush of cu rrent. The performance of the rotor windings, under these conditions, is affected if the motor has loose or broken rotor bars. Techniques have been developed to detect rotor winding malfunctions i.e. broken rotor bars ( Tavner & Penman, 1987:259). During completion of the project there was an opportunity to assess whether a broken bar in the cage of the motor affects the transient behaviour. If a deviation in the transient behaviour of the motor indicates the presence of broken rotor bar(s), the torque measurement system can be economically used as a diagnostic tool to detect broken rotor bars. For th is reason, measurements on an identical machine having a broken cage where taken. These results are also shown in this thesis and it is interesting to note that it was found that the position of the broken bar affected the transient torque developed by the motor during run-up. Although no reason for this is offered in th is work, the author believes that this finding can form the basis for further intensive research. In the project the background information on the equations used for the simulation of the dynamic performance of a squirrel cage induction machine and the CASED computer simulation package are discussed. A method used to measure the torque-speed curve of the motors is also described. The simulated and measured results of the transient starting- and reswitching torque on the healthy motor are discussed. The reswitching transient torques on the healthy motor and the motor with the broken rotor bar are compared . Finally, the measured results of the starting torque on the motor with the broken rotor bar are shown and evaluated

Digital analysis and the reduction of auditor litigation risk

https://egrove.olemiss.edu/dl_proceedings/1113/thumbnail.jp

Short term load forecasting using neural networks

Published ArticleSeveral forecasting models are available for research in predicting the shape of electric load curves. The development of Artificial Intelligence (AI), especially Artificial Neural Networks (ANN), can be applied to model short term load forecasting. Because of their input-output mapping ability, ANN's are well-suited for load forecasting applications. ANN's have been used extensively as time series predictors; these can include feed-forward networks that make use of a sliding window over the input data sequence. Using a combination of a time series and a neural network prediction method, the past events of the load data can be explored and used to train a neural network to predict the next load point. In this study, an investigation into the use of ANN's for short term load forecasting for Bloemfontein, Free State has been conducted with the MATLAB Neural Network Toolbox where ANN capabilities in load forecasting, with the use of only load history as input values, are demonstrated

Developing a neural network model to predict the electrical load demand in the Mangaung municipal area

Thesis (D. Tech. (Engineering: Electric)) -- Central University of technology, 2012Because power generation relies heavily on electricity demand, consumers are required to wisely manage their loads to consolidate the power utility‟s optimal power generation efforts. Consequently, accurate and reliable electric load forecasting systems are required. Prior to the present situation, there were various forecasting models developed primarily for electric load forecasting. Modelling short term load forecasting using artificial neural networks has recently been proposed by researchers. This project developed a model for short term load forecasting using a neural network. The concept was tested by evaluating the forecasting potential of the basic feedforward and the cascade forward neural network models. The test results showed that the cascade forward model is more efficient for this forecasting investigation. The final model is intended to be a basis for a real forecasting application. The neural model was tested using actual load data of the Bloemfontein reticulation network to predict its load for half an hour in advance. The cascade forward network demonstrates a mean absolute percentage error of less than 5% when tested using four years of utility data. In addition to reporting the summary statistics of the mean absolute percentage error, an alternate method using correlation coefficients for presenting load forecasting performance results are shown. This research proposes that a 6:1:1 cascade forward neural network can be trained with data from a month of a year and forecast the load for the same month of the following year. This research presents a new time series modeling for short term load forecasting, which can model the forecast of the half-hourly loads of weekdays, as well as of weekends and public holidays. Obtained results from extensive testing on the Bloemfontein power system network confirm the validity of the developed forecasting approach. This model can be implemented for on-line testing application to adopt a final view of its usefulness

First Digit Distribution of Hadron Full Width

A phenomenological law, called Benford's law, states that the occurrence of the first digit, i.e., $1,2,...,9$, of numbers from many real world sources is not uniformly distributed, but instead favors smaller ones according to a logarithmic distribution. We investigate, for the first time, the first digit distribution of the full widths of mesons and baryons in the well defined science domain of particle physics systematically, and find that they agree excellently with the Benford distribution. We also discuss several general properties of Benford's law, i.e., the law is scale-invariant, base-invariant, and power-invariant. This means that the lifetimes of hadrons follow also Benford's law.Comment: 8 latex pages, 4 figures, final version in journal publicatio

Tracking the Equator Into the Paleogene (abstract of paper presented at AGU Fall Meeting, San Francisco, 8-12 Dec 2003)

Stratigraphy has been compiled for 63 tropical Pacific drill sites that sample lower Neogene and Paleogene sediments. These Sites derive from drilling on DSDP Leg 5 through ODP Leg 199. All Sites have been put on the biostratigraphic and paleomagnetic timescale refined by Leg 199 scientists. Sediment accumulation rates have been calculated for ten intervals ranging in age from 10 Ma to 56 Ma. A simple fixed hotspot model was used for Pacific lithospheric plate rotation in reconstructing the position of the selected sites for each of these ten intervals. The reconstruction of all intervals show the development of a tongue of relatively high accumulation rates associated with the oceanographic divergence at the geographic equator. The estimated position of the geographic equator based on these reconstructions lies consistently south of the position of the equator based on the rotation model used. However, the southward displacement is generally 2 degrees of latitude or less. We believe that this relatively small disagreement between the two estimates of equatorial position back to 56 Ma indicates: 1) Whatever hotspot movement that may have occurred in the interval between 40 and 56 Ma did not affect the motion of the Pacific plate; its motion after 40 Ma appears to have been approximately the same as before 40 Ma. 2) The estimated rate of true polar wander during the interval of 40 - 56 Ma must be very small (~0.125$\deg$/m.y.) and is probably not significant (i.e., well within the error of these reconstructions)

Theory of Analogous Force on Number Sets

A general statistical thermodynamic theory that considers given sequences of x-integers to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions p_{x} for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering p_{x} to describe complex phenomena, the theory leads to derive a distinct analogous force f_{x} on number sets proportional to $(\frac{\partial p_{x}}{\partial x} )_{T}$ at an analogous system temperature T. In particular, this yields to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed.Comment: RevTeX, PostScript Fig, To Appear Phys.