Posts

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 $4$.

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 most likely to get to state 4 if $a = \sqrt{bc}$!

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.