2024-02-11: Symbolic algebra and typing

2023-08-01: Population waves

2023-05-18: Math of telephone billing mystery

2023-05-05: Franklin and DNA More information…

2023-04-25: On angle and dimension

2023-02-20: On Leonardo da Vinci and Gravity

2022-04-29: Fabricating Evidence to catch Carmen Sandiego

2022-03-04: Probabilistic law of the excluded middle

2020-05-04: Archimedes and the sphere

2019-05-16: Glow worms return

2019-04-11: Original memetic sin

2019-01-31: The theory of weight

2018-11-06: Origins of telephone network theory

2018-10-24: Modern thought

2018-09-10: Feeding a controversy

2018-06-11: Glow worm distribution

2018-04-23: Outlawing risk

2017-08-22: A rebuttal on the beauty in applying math

2017-04-22: Free googles book library

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-02-17: Dependent Non-Commuting Random Variable Systems

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: Mathematical Epidemiology citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-12: A Comment on How Biased Dispersal can Preclude Competitive Exclusion

2015-02-09: Hamilton's selfish-herd paradox

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-17: Ebola comments

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why \(1^{\infty} eq 1\)

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the circle area formula using elementary methods

The cult of the Levy flight

Don't let people using math-speak bully you into abandoning your common sense. If they cann't explain in simple terms, it's hogwash and they are probably selling snake oil.

While doing some research for a paper elaborating on the theory of ecological and genetic invasions across space, I stumbled on this rather disturbing video about Levy flights.

If you've never heard of a Levy flight, don't worry. Levy flights are like regular random walks but the step lengths are drawn from a power-law-like distribution with infinite variance, such as a Cauchy distribution. Levy flights are special because they not converge to the diffusion equation behavior revealed by Bachelier, Einstein, and Kolmogorov -- instead, they spread faster than a diffusion model predicts. Mandelbrot was initially the main proponent of Levy walks, which have close connections to geometric theory of fractals.

The Center for Homeland Defense and Security is affiliated with the Naval Postgraduate School (NPS) in Monterey, California (although the exact financial nature of this affiliation is unclear to me). What disturbs me about the Center for Homeland Defence and Security video is it's abuse of obtuse mathematical ideas with claims of providing simple answers to complex problems like where will the next terrorism event occur. You can develop better models based on common sense. True Levy flights require people to be able to travel infinitely fast, which of course can not happen. There is a classic fallacy in asymptotic analysis which seems to trip people up every generation.

Asymptotic analysis is a mathematical technique that attempts to extract relatively simple common components from a system and use these components to approximate the behavior of the original class of systems. Asymptotic analysis is an incredibly useful and common tool. Calculus is an example of asymptotic analysis where we the concept of infinitessimally small measures to obtain formulas for areas and volumes. The Gauss's normal distribution is an example of asymptotic analysis of sums of random variables. Maximum likelihood analysis is based on asymptotics. The renormalization theory used in physics to study phase transitions is another asymptotic analysis. And so on. Basically, asymptotic results give us a way of filtering the "wheat from the chaff", allowing us to through away complicated parts of an equation that only have a small effect on an answer and keeping a simple formula that gives us a mostly-correct answer about specific questions.

The common fallacy of asymptotic analysis to which I refer is one which confounds cause and effect. In asymptotic analysis, there are many different models that lead to the same asymptotic formula's. For instance, there are many different kinds of random walks -- some with fixed step sizes, some with variable step sizes, some that turn periodically, some that turn randomly and continuously, ... and as long as the step sizes are bounded, all of these random walks can be approximated by the same linear advection-diffusion partial-differential equations. But what this means for those of us interested in mechanism is that we can not tell what the underlying mechanism is, based on an observed match to an asymptotic law -- many different mechanisms would give the same result.

However, people, even professional researchers and scientists, often mistakenly assume that if the mechanism they have hypothesized gives the right asymptotic statistics, it must be correct. Not so. This would be akin to assuming correlation implied causation. Classical examples of this mistake would be the aristotelian conclusion that the sun revolves around the earth just because of the daily arc it follows across the sky. Nicole Oresme's 14th century criteque laid the logical fallacies of Aristotle's arguments bare, and the Copernican view eventually one the day. Similarly, natural philosophers could not help but conclude that since complicated things must be created, only powerful dieties can create very complicated things, and life itself is a complicate thing, then life must be the creation of a diety. Only when philosophers like Darwin and Wallace proposed another mechanism by which complicated organisms might evolve from simplier ones did the modern science of biology ignite.

The interprettation of power-law statistics as consequences of Levy flights is another (admittedly much smaller, but still problematic) error allong these lines.

This is an issue that really deserves a longer post, as there is a bit of pseudo-science culture around fractal-things that needs some pruning, but the day's work calls. Basic conclusion -- don't let people using math-speak bully you into abandoning your common sense. If they cann't explain it, is probably hogwash.