3 research outputs found
A Discrete Evolutionary Model for Chess Players' Ratings
The Elo system for rating chess players, also used in other games and sports,
was adopted by the World Chess Federation over four decades ago. Although not
without controversy, it is accepted as generally reliable and provides a method
for assessing players' strengths and ranking them in official tournaments.
It is generally accepted that the distribution of players' rating data is
approximately normal but, to date, no stochastic model of how the distribution
might have arisen has been proposed. We propose such an evolutionary stochastic
model, which models the arrival of players into the rating pool, the games they
play against each other, and how the results of these games affect their
ratings. Using a continuous approximation to the discrete model, we derive the
distribution for players' ratings at time as a normal distribution, where
the variance increases in time as a logarithmic function of . We validate
the model using published rating data from 2007 to 2010, showing that the
parameters obtained from the data can be recovered through simulations of the
stochastic model.
The distribution of players' ratings is only approximately normal and has
been shown to have a small negative skew. We show how to modify our
evolutionary stochastic model to take this skewness into account, and we
validate the modified model using the published official rating data.Comment: 17 pages, 4 figure
A multiplicative process for generating the rank-order distribution of UK election results
Human dynamics and sociophysics suggest statistical models that may explain and
provide us with a better understanding of social phenomena. Here we propose a generative
multiplicative decrease model that gives rise to a rank-order distribution and allows us
to analyse the results of the last three UK parliamentary elections. We provide empirical
evidence that the additive Weibull distribution, which can be generated from our model,
is a close fit to the electoral data, offering a novel interpretation of the recent election
results
Dynamics of hydrofracturing and microseismic signals in porous versus tight rocks
This work discusses the dynamic development of hydraulic fractures, their evolution and the resulting seismicity during fluid injection in a coupled numerical model. The model describes coupling between a solid that can fracture dynamically and a compressible fluid that can push back at the rock and open fractures. With a series of numerical simulations it is shown how the fracture pattern and seismicity change depending on changes in depth, injection rate, Young’s modulus and breaking strength. Simulations indicate that the Young’s modulus has the largest influence on the fracture dynamics and also the related seismicity. Simulations of rocks with a Young’s modulus smaller than 10 GPa show dominant mode I failure and a growth of fracture aperture with a decrease in Young’s modulus. Simulations of rocks with a Young’s modulus higher than 10 GPa show fractures with a constant aperture and fracture growth that is mainly governed by a growth in crack length and an increasing amount of mode II failure. This change in fracture geometry evolution has an effect on the observed seismicity. Rocks with a Young’s modulus of 10 GPa have the smallest moment magnitude while both decrease and increase of Young’s modulus value contribute to a growth of the seismic moment magnitude. The signal is further altered by non-linear change in dip and tensile angle depending on the Young’s modulus value.
It is proposed that two distinct failure regimes are observed in the simulations. Below 10 GPa a fracture propagates through growth in aperture, this causes the fracture tip to be under constant extension. For rocks above 10 GPa, the aperture is small and the fracture is under compression. In this case fracture growth is driven by stress intensification at the crack tip, which causes fracture opening to have greater proportion of mode II compared to mode I. To suppliment the observations made from numerical simulations, laboratory experiments with air injection into vertically orientated Hele-Shaw cell were carried out. Strain analysis of the recorded experiments showed stress regimes that are very similar to the ones observed during numerical simulations with soft rocks. In both cases negative strain fields could be observed in front of the fracture tip. This indicates that fracture propagation for soft materials is driven by tensile failure and walls being pushed apart. Further analysis on fracture propagation mechanisms and solid media response were carried out.
These results are applicable to the prediction of fracture dynamics and seismicity during fluid injection, especially since we see a transition from one failure regime to another at around 10 GPa, a Young’s modulus that lies in the middle of possible values for natural shale rocks