Probabilities and Processes
Alison Ord
(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf

Introduction

This is me. I am a geologist.

I have a small company called Mineral Dynamics, and write and learn through the University of Western Australia

alison.ord@uwa.edu.au

Aim

I explore the application of nonlinear dynamics to understanding how rocks deform . I would really like to understand the underlying dynamics of mineralising systems.

I noted last year that study of these systems has now entered the “big data” era with more and more data collected at finer and finer scales 

& that in the present era where “big data” are generated routinely by mineral exploration companies 

it is essential to have rigorous procedures for extracting quantitative information from these large data sets for developing rigorous ways of prediction 

and of deriving knowledge and understanding from such expensive data .

I continue the search for the mathematics most supportive of prediction within geology. This year, it is probabilities.

I ask: may specific probability distributions be characteristic of specific mineralisation styles?

 what may this tell us about growth mechanisms ?

Approach

In any system not at equilibrium fluxes of energy and/or mass from outside the system (forcing processes) drive the system further from equilibrium whereas dissipative processes within the system drive the system towards equilibrium. Thus the growth of the system results from a balance between forcing and dissipative processes. 

Savageau analysed a number of competitive processes and showed that a variety of probability distributions for the products of these processes can arise from such competition. 

In systems where the growth processes constrain the mean and/or the variance of variants to be fixed or small, the Central Limit Theorem and the Law of Large Numbers mean that the resultant probability distributions will be the Normal or the Log Normal; other constraints on the mean or variance result in other distributions. 

However the processes involved in nonlinear, open systems commonly result in very large (or infinite) variances and imprecision in the quantification of the mean.

For systems where the mean and variance are large the Generalised Central Limit theorem applies and specifies a family of stable distributions known as the Generalised Extreme Value (GEV) distributions. 

This family contains just three members: the Gumbel, Fréchet and Weibull distributions. 

It turns out that these three distributions characterise the probability distributions for many elements (including the distributions of mineralisation and alteration assemblages) within orogenic gold deposits. 

Rocha and Aleixo present the case that these distributions arise from processes involving nucleation, competitive growth and extinction. We follow that line of argument here.

This contrasts with the common view that such distributions should be Gaussian, log-normal or Pareto (power-law or fractal). 

Thus the probability distributions can give valuable constraints on the nature of the attractor for the system as well as give information on the growth kinetics of the system.

Mineralising Systems

Sunrise Dam, Western Australia

Methodology

(Legended[, "Data"])

Instead of adopting a parametric approach where one attempts to fit the data to a pre-decided probability distribution we adopt a non-parametric, data-driven approach where, from a library of probability distributions in Mathematica®, we ask: which distribution fits the data best?

FindDistribution[data,5]

jdist=EstimatedDistribution[data,GammaDistribution[α,β,χ,δ]]

jd=ProbabilityPlot[data,jdist,PlotLegends->{"Gamma4"},PlotStyle->Blue,PlotLabel->"truncated_dolomite"];

The outputs are a probability plot that shows the sample data plotted against the best fit distribution; a perfect fit is represented as a diagonal line on this plot, 
and a cumulative distribution plot for the best fit estimate, along with a number of other distributions that have inferior fits, with the sample data superimposed.

Results

Probability plots

(Legended[, ({{(LineLegend["Gamma4"])}, {(LineLegend["GompertzM4"])}, {(LineLegend["Gumbel2"])}, {(LineLegend["Pareto type IV"])}})])

(Legended[, ({{(LineLegend["Pareto type II"])}, {(LineLegend["Pareto type I"])}, {(LineLegend["Frechet3"])}, {(LineLegend["Frechet2"])}})])

Cumulative distribution plots

(Legended[, ({{"Data"}, {(PointLegend["Gamma4"])}, {(PointLegend["GompertzM4"])}, {(PointLegend["Gumbel2"])}, {(PointLegend["Pareto type IV"])}})])

(Legended[, ({{"Data"}, {(PointLegend["Pareto type II"])}, {(PointLegend["Pareto type I"])}, {(PointLegend["Frechet3"])}, {(PointLegend["Frechet2"])}})])

Best fit distributions

(Legended[, ({{(LineLegend["Pareto type II"])}, {(LineLegend["Frechet3"])}, {(LineLegend["Frechet2"])}})])

(Legended[, ({{"Data"}, {(PointLegend["Pareto type II"])}, {(PointLegend["Frechet3"])}, {(PointLegend["Frechet2"])}})])

The probability distributions for mineral abundance are members of the Generalised Extreme Value family, reflecting the details of their growth kinetics.

Fragmentation

Physical foundations for distributions for fragmentation

Breccias and Cataclasites

The production of breccias and cataclasites is commonly proposed to result in power-law or log- normal probability distributions for fragment (grain) size .

Results

We find that in both natural and experimental examples, the common best fit probability distributions for the complete distributions are members of the Generalised Gamma (GG), Generalised Extreme Value (GEV) and Generalised Pareto (GP) families; 
power - law and log-normal distributions are commonly, but not always, poor fits to the data. 

An hierarchical sequence, GG → GEV → GP, emerges as the sample mean of the fragment size decreases.

Thank you

Bruce Hobbs

Wolfram

I have just come across a quote from Mind Matters NEWS (22 March 2021) - that Wolfram has not made computers creative but he certainly took a lot of the drudgery out of the profession. 

I agree. 

Without mathematica, I very much doubt that I would have got as far as I have in exploring the myriad distributions described in the literature. I have not so far read about a distribution which I have not been able to find within mathematica's library of built-in distributions. 

Thank you all.

References

References

Cheng, Z., Redner, S, 1990. Kinetics of fragmentation. J. Phys. A 23, 1233. 
Redner, S. 1990. Statistical Models for the Fracture of Disordered Media, Eds H. J. Herrmann and S. Roux. North-Holland. 
Rocha, J. L., Aleixo, S. M. 2013. An extension of Gompertzian growth dynamics; Weibull and Fréchet models. Mathematical Biosciences and Engineering. 10, 379 - 398. 
Savageau, M. A. 1979. Growth of complex systems can be related to the properties of their underlying determinants. Proc. Nat. Acad. Sci. USA. 76, 5413-5417.  
Savageau, M. A. 1980. Growth equations: A general equation and a survey of special cases. Mathematical Biosciences48, 267 -278.

Submitted manuscripts

Hobbs, B. E., Ord, A, Blenkinsop, T.G. The spatial distributions of mineralisation. Submitted to Journal of Structural Geology.  2021
Ord, A., Hobbs, B. E. Orogenic gold deposits as nonlinear systems. Nonlinear analysis of data. Submitted to Ore Geology Reviews.  2021
Ord, A., Blenkinsop, T. G., Hobbs, B. E. Fragment size distributions in brittle deformed rocks. Submitted to Journal of Structural Geology. 2021