Posts

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 $1^\infty != 1$

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 area of a circle using elementary methods

Risk Resonance

Resonance is a compelling component of the study of mechanical systems, as it helps use identify "special situations" where a monotonic perspective on parameter dependence fails, perhaps un-expectedly.

Despite it's clear importance in mechanics, there is little discussion of the importance of resonance in probabilistic systems, as far as I can tell. This seems something worth correcting.

First, we won't be able to find a perfect parallel -- mechanical resonances depend on forcing, momentum, rotational effects, and superposition, which we won't find in simple probabilistic models. But if we widen our views a little, we can still make some progress. Classical resonance appears because of two competing effects in a system -- a natural resonance frequency and a forcing frequency. When these two frequencies are out of alignment, oscillation amplitudes respond monotonely; a faster forcing frequency leads to a smaller amplitude response. But in ranges where the frequencies coincide, we see a peak in the amplitude of oscillations, which we call resonance.

One analog for this in a random system could be how the probability of an outcome responds when it depends on two different competing paths. The simplest such example I can think of is a 4-state continuous-time Markov chain with two possible outcomes. Here's a diagram.

1 3 2 4 b c a d

We start at state 1, and ask, "What's the probability of ending up in state 4, rather than state 3?" A simple calculation shows the probability of ending up in 4 is \[\frac{cd}{(a+c)(b+d)}.\] This probability responds monotonely to each of the parameters. Increasing rate $c$ or $d$ increases the chance of reaching state $4$. Increasing rate $a$ or $b$ decreases the chance of reaching state $4$. This behavior is actually very general -- with a simple application of Cramer's rule for solving linear systems, we can show that the probability of any outcome will respond monotonely to perturbations of any individual transition rate.

However, if the rates are related to each other, the relationship may become more complicated. For instance, suppose that $a$ and $d$ are always equal, so that the chance of reaching state 4 is actually \[\frac{ca}{(a+c)(b+a)}.\] Now, if we start with $a=0$, there is no chance of getting to state 4. Increasing $a$ a little increases the probability of reaching state $4$. But as $a$ get's really big, the chance of getting to state 4 again goes back to zero. Turns out, we are most likely to get to state 4 if $a = \sqrt{bc}$! For example, if $b=c=1$, ...

Produced by GNUPLOT 4.4 patchlevel 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 2 4 6 8 10 12 14 Probability of reaching state 4 a

This is risk resonance. It's a situation where some factor in a system couples pathways in a way that can make an outcome respond non-monotonely. Another example is the case where two 1-directional random walks collide.

a a a a a a b b b b b b

The probability of reaching the red state when we start at the upper left corner is \[ \frac{6 a^2 b^2}{(a+b)^4} \] which is maximized when $a=b$.

Produced by GNUPLOT 4.4 patchlevel 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.01 0.1 1 10 100 Hitting Probability Relative transition rate (a/b)

The take-home? When 2 or more transitions in a Markov process are dependent on a common factor, outcomes may not respond monotonely to factor perturbations -- changes that reduce risk in one setting may increase risk in a separate setting. Identifying situations of potential risk resonance will help us prevent manage it.