The previous parts of article discussed practical application of theoretical model BCI (Balanced Concentration Index) – TOMEM (Transitive-Optimal Market Equilibrium Model) based on physical indicators of energy mixes’ structure. **BCI-TOMEM: Nonlinear Model with Beta-distribution **

Prior to passing over to a more complicated aggregate variant of BCI-TOMEM, it would be appropriate to present brief theoretical and mathematic-statistical justification of the model, confirming its possible application to specifics in economy and other areas.

Different from majority of linear and exponential models, the TOMEM and new BCI are non-linear, but very close to the logistic and normal distribution.

The TOMEM and BCI models are built on the basis of an approach with the Beta statistical distribution.

- Designation: Beta(x|α, ß), B(x|α, ß) or simply B(a,b).

- Value range of argument x: 0≤x≤1.

- Parameters’ domain: a,b>0; a – shape parameter, b – scale parameter.

- Distribution density: expressed using gamma- or beta-function of parameters a and b.

- Mean: a/(a+b).

- Dispersion: ab/(a+b)2(a+b+1).

- Cumulative distribution: not expressed in elementary functions, still not particularly complex for determination.

**Beta Distribution – a Powerful and Flexible Statistical Method **

Regardless of the great popularity of standard distribution and the natural tendency to bring various models to it, in reality Beta distribution is most frequent in mathematical statistics and has a number of interesting features:

- For F-distribution and cumulative distribution of F-relation, when at n→∞, Beta distribution converges to Gamma distribution;

- Feller’s “arcsine law” in the probability calculus is a particular case of Beta distribution at a=b=0.5;

- Uniform distribution can be easily be expressed as Beta distribution at a=b=1: R(x|0.1)=B(x|1.1);

- Binomial distribution can be presented in an elegant way as Beta distribution: ß(m|n,p)=B(1-p|n-m, m+1);

- Symmetrical feature of the Beta distribution: B(x|a,b)=1-B(1-x|b,a).

Quite useful feature for Beta distribution is to take for different values of parameters (a,b) different curve profiles, which can be separated in sub-families suitable application in various economic activities, phase transfers in processes and chains and different analysis levels. It is widely used in studies and models about growth and marketing, risk analysis in finance and insurance, etc. Beta distribution is very suitable for variables representing a part of unity (or percentage) – for instance, market share, share in production, etc. Although standard Beta is “framed” within zero and one, with simple linear transformation it may be adjusted for any intervals. And, very important, Beta is used for process with natural low and/or top limits.

The basic function BCI (for more details see OGE #4, 2013, p. 34) is closest to beta-distribution and can be rather accurately determined by the following parameters: a>b and 0.5<a<1; b=a+0.4.

The TOMEM function is also closest to beta-distribution, but with other parameters at a<b and

2<a<2.5; b=a-0.62.

The function of actual distribution of diversification of energy mixes in countries and regions of the world in 2011 can be represented in beta- and gamma-distributions, which are rather close to the normal distribution, but are closest to gamma-distribution which is also convenient for application in further analytical studies. Values of the BCI basic regression can be also presented