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This is the home site for "Fractal Analysis of Various Market Segments in the North American Electronics Industry," which is intended for use by those involved in business intelligence, strategic marketing, optimization of business operations, and business process modeling, (BPM,) i.e., managing the P&L. The document was released in December of 1995, and the draft notes are available as fractal.ps.gz, (2MB gzip compressed Postscript,) or, fractal.pdf, (6 MB PDF,) and fractal.tar.gz which is the Latex sources for the 700+ page document, including the C Sources to the 60 some programs used in the analysis which are available separately on the Utilities page. The software methodology is described on the Architecture page, and the analytical technique described on the Usage page. The paradigm of the non-linear high entropy dynamical system model is validated on the quality assurance, (QA) page. There is a Mathematical & Numerical Methods development, for those so inclined.

The document itself contains some 600 diagrams and graphs along with some 300 tables that present and compare some 90 fractal variables for each market segment analyzed. The data for the market segments was obtained from the US Department of Commerce and Federal Reserve, and include every sector in the North American electronics industry for at least 20 years prior to 1994-including general economic data such as US GDP, M2, employment, indicators, and T-Bill rates that affect the growth and investments in the industry. Additionally, the variables were used by simulation programs to fabricate time series that contained certain characteristics found in the the market segments for the purpose of verifying the methodology used. The methodology can be found on the Architecture page.

There is additional information regarding the concepts of econophysics contained in the FAQs page, and a Links page to related sites for a general survey of the subject. Collateral programs can be found on the Utilities page, which can be used to optimize business operations-specifically, minimizing decisional risk, while at the same time maximizing profitability, growth, or market share.

The NtropiX "Software For The Algorithmic Trading Of Equities," uses the NdustriX software code base for non-linear high entropy dynamical system analysis of financial markets. The NdeX "Program for Binary Searching a Constant Flat File Database," (i.e., time series,) can be used to implement "data blades" via partial key searches of financial time series, and the NformatiX "Software For Full Text Information Retrieval," can be used to devise complex searches for financial content information in the 10-Q filings from the US SEC's EDGAR database, BusinessWire, etc.

Historical Perspective

The theoretical foundations of the entropic methods used in modern finance were formalized by the mathematicians Jakob Bernoulli, (Ars Conjectandi, 1713,) and Abraham de Moivre a few years later, (The Doctrine of Chances.) The dynamics of simple fractal systems, (for example, Brownian motion and the fixed increment fractals,) appear in the formalization of repeated trial convergence, (sample average,) in the Law of Large Numbers, and the formal development of the normal, (or Gaussian,) distribution/bell curve. Specific applications to fractal dynamics were further developed by A. A. Markov (1903-1922,) and Norbert Wiener, (1874-1964.)

The concept of entropic analysis of equity prices is not new. It was first proposed by Louis Bachelier (1870-1946) and published in his 1900 doctoral thesis. His "theory of speculation," (Theorie de la Speculation,) was discounted by none other than Henri Poincare, observing that "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." Nevertheless, the thesis anticipated many of the mathematical discoveries made later by Wiener and Markov, and outlined the importance of such ideas in today's financial markets, stating that "it is evident that the present theory solves the majority of problems in the study of speculation by the calculus of probability."

J. L. Kelly, Jr. established the isomorphism between the information-theoretic concept of information rate in a binary symmetric channel and speculation under uncertainty. The concept was presented in the Bell System Technical Journal, (A New Interpretation of Information Rate,) in 1956. The importance of Kelly's contribution can not be underestimated since it made the large mathematical infrastructure of information theory, developed by Claude Shannon in the mid 1940's, (The Mathematical Theory of Communication,) applicable to the analysis and optimization of speculative endeavors.

In 1963, Benoit Mandelbrot, (The Variation of Certain Speculative Prices,) citing Wesley Claire Mitchell's work of 1915, (Wieser's Theory of Social Economics,) formulated a theory of price fluctuations in speculative markets based on the probability distributions discovered by the French mathematician Paul Levy. As pointed out by Mandelbrot, the so-called log-normal distribution is of interest in finance since wealth in a multi-agent system evolves into a log-normal distribution. The Gaussian/Normal distribution is a special case of the more general Levy distributions, and is often used as an approximation to log-normal distributions for mathematical expediency.

In 1965, Paul Samuelson formally proved Bachelier's empirical theories on the characteristics of speculative markets, (Proof that Properly Anticipated Prices Fluctuate Randomly.) In the early 1970's, Fisher Black, Myron S. Scholes, and Robert C. Merton, extended the theory into a methodology for virtually zero risk option and derivative pricing, and established the isomorphism between the standard deviation of the fluctuations in price of a financial instrument, and investment risk.

By the mid 1990's, largely through the efforts of W. Brian Arthur of the Santa Fe Institute, bounded rationality and self-referential indeterminism were generally accepted as the cause of the stochastic characteristics of speculative markets. Today, stochastic analysis is a firmly entrenched branch of information theory, and economic applications of entropic principles are found in such disciplines as finance, strategic marketing, and the optimization of business operations and process-see the NdustriX site for particulars.

For a more detailed historical perspective, "Choice under Risk and Uncertainty" is highly recommended as a general survey and tautology.

Mathematical Analysis & Numerical Methods

For a brief mathematical development and description of the associated numerical methods:

All are from the NtropiX Mailing List, and/or, NdustriX Mailing List distributions.

Historical Economics

For applications of quantitative analysis of non-Linear high entropy economic systems (e.g., econophysics,) to historical data, see the Historical Economics page on the author's site. The analysis includes the world GDP since the late stone age, (about -5000 CE,) and the US since 1792.


The programs and documents are freely available for Download and distributed as source code, at no charge, and provided under License.

Mailing list

There is a mailing list available for users of the NdustriX software suite and general discussion of applications of fractal and entropic methodologies. To join see the Mailing List page.


The NdustriX archive is at http://www.johncon.com/ndustrix/archive/.

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