M1 - 6 - Random Walk with Notes

"A man starts from a point O and walks L yards in a straight line; he then turns through any angle whatever and walks another L yards in a second straight line. He repeats this process N times."

- K. Pearson, Nature 1905.

A "random walk" is defined as a mathematical formalisation of a trajectory that consists of taking successive random steps (wiki).  


A similar concept to the "random walk" is a "stochastic process", defined as: a collection of random variables often used to represent the evolution of some random value, or system, over time (wiki).   

Physicist Leonard Mlodinow's book Drunkards Walk is about the role of chance in human endeavours.  It is an interesting and accessible read, though one which is generally resigned about things. The subtitle, "How Randomness Rules Our Lives" is illustrative of this as well as the following larger-bit: 

In the scientific study of random processes the drunkard’s walk is the archetype. In our lives it also provides an apt model, for like the granules of pollen floating in the Brownian fluid, we’re continually nudged in this direction and then that one by random events. As a result, although statistical regularities can be found in social data, the future of particular individuals is impossible to predict, and for our particular achievements, our jobs, our friends, our finances, we all owe more to chance than many people realize. On the following pages, I shall argue, furthermore, that in all except the simplest real-life endeavors unforeseeable or unpredictable forces cannot be avoided, and moreover those random forces and our reactions to them account for much of what constitutes our particular path in life. I will begin my argument by exploring an apparent contradiction to that idea: if the future is really chaotic and unpredictable, why, after events have occurred, does it often seem as if we should have been able to foresee them? (p.63)

To test out the idea I decided to actually go for a Random Walk.  


I download an app for my smartphone which claims to be random and use it to decide which direction to walk every time I come to an intersection.   

Above is the GPS track I had while conducting the random walk in the city of Rome, March 22, 2012 from 2100-2200 hours. 

Some notes taken on the random walk:

A: 

I walk four times by the door of my own house

B: 

I walk 3 times past the entrance to a restaurant in which some VIP is eating, and in front of which sits a half dozen black gleaming Mercedes with security cards milling about in front in preparation for the imminent departure. I am a security threat. 

E: 

I try with all my strength to predict which direction the stochastic process will take me: sometimes I guess right, most often times wrong, and I frustratingly walk back upon the street I have walked on 3 times before.

F: 

I walk up half the Janiculum, then am instructed to turn mid-slope and head back, walking once again on streets I had just walked: the stochastic process almost brought me to the top of a mountain.  

G:

If I wish to walk half-way up a hill and past the door of my own home 3 or 4 times moving only a few hundred meters in an hour of  continuous walking, then I should use a stochastic process to inform my daily decisions.   

The conclusion above, in Excerpt "G" is very similar to that of Karl Pearson, inventor of the "Random Walk" as formulated in a 1905 edition of Nature: "The lesson... is that in open country the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!" 

Sources:

K. Pearson. Nature 72, 294; 318; 342 (1905) link

Mlodinow, Leonard. The Drunkard's Walk: How Randomness Rules Our Lives. New York: Pantheon, 2008. Print.

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