Lecture 12 More on Two-dimensional Random Walks and The Gaussian Probability Distribution Function

Transcription

1 Physical Principles in Biology Biology 355 Fall 217 Lecture 12 More on Two-dimensional Random Walks and The Gaussian Probability Distribution Function Monday, 18 September c David P. Goldenberg University of Utah goldenberg@biology.utah.edu

2 A Random Walk in Two Dimensions 1 Start at (, y) coordinates (,). 2 Choose a random direction, defined by the angle θ from the -ais. 3 Move distance δ in the chosen direction. 4 Repeat 2 and 3 another (n 1) times.

3 Major Results for a Two-Dimensional Random Walk For n steps of length δ: Displacement along the - and y-aes (or any other direction): Mean displacement: n = y n =. Mean-square displacement: n 2 = y 2 n = n δ 2,y = nδ 2 /2 RMS displacement: RMS( n ) = RMS(y n ) = (n/2)δ Distance from starting point, r: Mean-square displacement: r 2 n = 2n δ 2,y = nδ 2 RMS displacement: RMS(r n ) = nδ Just like the one-dimensional random walk! Mean displacement:?

4 An Implication Number of steps = n. Length of steps = δ Total distance: D t = nδ. If total distance is fied and δ is changed: n = D t /δ RMS(r) = nδ = (D t /δ) δ = D t δ Average distance from start to end increases with step length.

5 Clicker Question #1 For a 2-dimensional random walk of a total distance of 1 m and a step length of 5 m, which (if any) of the following are correct? Choose up to 2. 1 RMS(r) 5 m 2 r 2 5 m 2 3 r 2 25 m 2 4 RMS(r) 22 m 5 None of the above n = 1 m/5 m = 2 r 2 = nδ 2 = 2 (5 m 2 ) = 5 m 2

6 Final -Coordinate for 2-d Random Walks 5 Steps 1 Steps 2 2 y Percent

7 Final Distance from Origin for 2-d Random Walks 5 Steps 1 Steps 2 2 y Percent Why isn t the peak at r =? r 1 2 r 3

8 Why Isn t the Peak at r = rb r a p(r)dr = probability that the walk endpoint lies between r a and r b. 12 y r 3 p(r)dr = probability that the endpoint lies in the annulus (ring) dr thick.

9 Clicker Question #2 What is the area of the annulus? y πr 2 2 πdr 2 3 2πr 4 2πdr 5 2πrdr -2 2

10 Why isn t the Peak at r = 12 y r 3 The probability, p(r)dr, is proportional to the area of the annulus. The area increases with r: A = 2πrdr. The density of endpoints decreases with r. The two effects balance at the peak of the distribution.

11 Ants on a Walk for Food Brachymyrme depilis (25 s) Dorymyrme insanus (21 s) Do either look like a random walk? Pearce-Duvet, J. M. C., Elemens, C. P. H. & Feener, D. H. (211). Walking the line: search behavior and foraging success in ant species. Behavioral Ecology, 22,

12 Simple Variations on the Two-dimensional Random Walk Constrain change in direction. Introduce variation in step length.

13 A Plain Random Walk y Step length = 2 No. steps = 2

14 A Correlated Random Walk y Turn angle restricted to 9 to 9 Step length = 8 No. steps = 2

15 A Random Walk With a Distribution of Step Lengths y Turn angle restricted to 9 to 9 Half-Gaussian (bell curve) distribution of step lengths No. steps = 2

16 A Lévy Flight A random walk with a heavy-tailed distribution of step lengths y Turn angle restricted to 9 to 9 Pareto distribution of step lengths No. steps = 2

17 Clicker Question #3 What does the ant walk most resemble? 1 A plain random walk 2 A correlated random walk 3 A Lévy Flight Brachymyrme depilis (25 s)

18 Description of a Three-dimensional Random Walk z y Each step is defined by a tilt from the local z-ais (φ i ) and a rotation around the z-ais (θ i ). The end of each step lies on a sphere of radius δ. r 2 = nδ 2, and RMS(r) = nδ, just like in one and two dimensions.

19 Warning! Direction Change The Gaussian Probability Distribution Function

20 The Binomial Distribution Revisited The probability of k successes in n successive binary (yes/no) trials when the probability of success in each trial is p. p(k; n, p) = n! k!(n k)! pk (1 p) (n k)

21 Two Important Parameters for any Probability Distribution Function Mean (µ) For a discrete probability distribution (µ = k = E(k)): n µ = kp(k) k=1 For a continuous probability distribution (µ = = E()): µ = ma min p()d Variance (σ 2 ) For a discrete probability distribution: n σ 2 = p(k)(k µ) 2 k=1 For a continuous probability distribution: µ = ma min p()( µ) 2 d

22 Mean, Variance and Standard Deviation for The Binomial Distribution Mean µ = np Variance σ 2 = np(1 p) Standard deviation σ = σ 2 = np(1 p)

23 The Simplest Form of a Gaussian Function 1..8 f () = e

24 The Gaussian Probability Distribution Function p() = 1 2πσ 2 e ( µ) 2 2σ 2 Mean = µ Variance = σ 2 Standard deviation = σ

25 Mean and Variance for The Binomial Distribution Mean µ = np Variance σ 2 = np(1 p) Gaussian probability function to approimate the binomial distribution function: p(k) = 1 e (k µ) 2 2σ 2 = 2πσ 2 1 e (k np) 2πnp(1 p) 2 2np(1 p)

26 Approimation of Binomial Distributions by Gaussian Distributions n doesn t have to be very large for a pretty good approimation!

27 Approimation of Binomial Distributions by Gaussian Distributions It doesn t work so well if the binomial distribution is biased, with n.5. The Gaussian distribution is inherently symmetrical; the binomial distribution isn t. If n is large enough, the Gaussian distribution is a good approimation, even if n.5. A general rule of thumb: The Gaussian distribution is a good approimation if: n > 9 1 p p and n > 9 p 1 p