Abstract
A discrete stochastic process which closely follows the mantra repeated during Covid-19 about how epidemics grow is presented as a childs game of cards. When the results of multiple games are averaged the resulting difference equations match the differential equations of the standard SIR model and thus this minimal model is validated. Extending to an animal social network formed by voles allows comparison of results with those of the complete graph of the basic game. This allows questions to be asked about flattening the curve and seasonality. A discrete probability distribution for the ratio of growth to exponential growth is derived and seen to scale to the logit-normal distribution. Correlations for the moments and underlying mean and standard deviation are then used to create a stochastic differential equation model.A teachers guide to the modelling of epidemics
Affine returns on Bernoulli trials in finance
Abstract
Offset returns of stock price movements were used to model the Paris Bourse in 1900. This first Mathematical model of Brownian motion was superseded by Geometric Brownian motion in the 1960s. i.e. the normal distribution was replaced by the log-normal and offset returns by linear returns. The market crash of 1998 caused the latter model to be questioned. This paper extends the model to affine returns, matching the average behaviour seen on the S&P 500 to calculations from averages of the daily ups and downs along with their probabilities. i.e. expected behaviour to noise. Affine returns lead to a mixture distribution consisting of two components - the log-normal distribution and one other which looks like the logit-normal distribution scaled from [0, 1] to some other finite support. For the shrinking case it was shown that the distribution was not parameterisable.The affine growth of NBA players and other adults
Abstract
The heights of NBA players form a skewed distribution rather than follow the normal distribution as the textbooks suggest. There has long been disquiet around whether the normal is the appropriate distribution with some authors using the log-normal instead. For Bernoulli trials offset returns are approximated by the normal distribution, linear returns by the log-normal and it has been recently shown affine returns may be approximated by an appropriately scaled logit-normal distribution. A good fit for the heights of the NBA players is performed by this latter distribution whose parameters are derived using a mixture of Maximum Likelihood Estimation and a grid search. Fitting adult heights of the general population is also undertaken.The affine growth of river heights
Abstract
River heights form a skewed distribution which is variously fit by the log-normal, Gamma, Generalized Extreme Value, Weibull and Pareto distributions. For Bernoulli trials it has been recently shown affine returns may be approximated by an appropriately scaled logit-normal distribution. A good fit for some of the river heights is performed by this latter distribution whose parameters are derived using a mixture of Maximum Likelihood Estimation and a grid search.The affine growth of hurricane speeds
Abstract
Hurricane strengths form a skewed distribution which is variously fit by the Weibull or log-normal distributions. For Bernoulli trials it has been recently shown affine returns may be approximated by an appropriately scaled logit-normal distribution. A reasonable fit for some of the hurricane strengths is performed by this latter distribution whose parameters are derived using a mixture of Maximum Likelihood Estimation and a grid search.The affine growth of wave heights
Abstract
Wave heights in oceanography are modelled using the Rayleigh, Weibull, Generalized Gamma and log-normal distributions. For Bernoulli trials it has been recently shown affine returns may be approximated by an appropriately scaled logit-normal distribution. A good fit for some of the river heights is performed by this latter distribution whose parameters are derived using a mixture of Maximum Likelihood Estimation and a grid search.Abstract
Economists fit both the log-normal and Pareto distributions to wealth depending on whether they are on the right tail or not. For Bernoulli trials it has been recently shown affine returns may be approximated by an appropriately scaled logit-normal distribution. A good fit for some of the wealth statistics is performed by this latter distribution whose parameters are derived using a mixture of Maximum Likelihood Estimation and a grid search.The affine growth of insurance premiums
Abstract
Insurance premiums are modelled using the log-normal, Pareto, Gamma and Weibull distributions. For Bernoulli trials it has been recently shown affine returns may be approximated by an appropriately scaled logit-normal distribution. A good fit for some of the premiums is performed by this latter distribution whose parameters are derived using a mixture of Maximum Likelihood Estimation and a grid search.A discrete stochastic model of turbulence
Abstract
Direct Numerical Simulation of the decay of isotropic turbulence was performed using a discrete stochastic model adapted from Molecular Dynamics. Although resulting in the expected power law the resulting exponent was extreme.Rebalancing First Past The Post voting