Random Walks

 
I first encountered the idea of the Random Walk in my law and economics class; specifically as that concept relates to stock prices. The thrust of the theory that stock prices proceed along a random path, and particular trends or outcomes cannot be identified or predicted based on past performance. However, the price distribution of a given stock over time is identical for all other stocks (see the above example for a random distribution plot). The upshot of the theory is that it’s impossible to outperform the market merely through superior research or intelligent analytics. You can only outperform the market by accepting additional risk (that is more risk than what is demanded by the market). Essentially, the stock market is like gambling, and the outcome over any particular time is a crapshoot.

The simplest model for a random walk is the coin flip game. Turns out, flipping a coin over and over again has a similarly random distribution effect over a single dimension  (e.g., –5 to +5 for five flips of a coin).

What about over a 2-dimensional plane? Consider this very simple explanation:

Things in nature often move in complicated ways. You have probably watched the way a butterfly moves. The molecules of the air that you are breathing move in a similar way. This type of motion we call a random walk. You can also take a random walk.

Go out to a large open field and mark a spot on the ground. Take with you a coin like a nickel or a quarter. Stand on the spot and flip the coin. If the coin comes up heads, turn to the right and take a large step. If the coin comes up tails, turn to the left and take a large step. Keep doing this many times and see where you end up.

If you flip the coin 25 times you will probably be about five steps away from where you started. This is because five times five equals 25. How far would you expect to be if you flipped the coin 100 times? A random walk is not a very fast way to get anywhere!

When you try this, you will notice that sometimes you go much farther than you expect and sometimes you end up very close to where you started. But if you repeat it many times or get several of your friends to do it with you with coins of their own, the average distance should come out as expected. In science we can often predict what will happen on the average even when the process is random.

It’s this distributional effect that I think is so fascinating. One might cover a lot of ground and end up very close to where one started. Indeed, on any one or two dimensional plane with an infinite number of steps there is a 100% chance you will revisit the point where you started (with 3 or more dimensions the probability decreases). Over a finite grid, a random walker (or drunk walker) would touch every point, including his “home space”—perhaps many times.

What the hell am I talking about?

Music.

Specifically the random feature on my mp3 player. I’m a big fan of the random button. But it works differently on different devices. This is so because computers cannot generate generate random numbers on their own. They need a algorithm, either a model borrowed from nature (for a truly random number) or a pattern generated according to a formula (a pseudo-random number). MP3 players, which can’t really sample air molecule patterns, use the latter. The type of random listening experience you have therefore depends on your chosen device.

I used to have an I-River mp3 player that had a terrible random function. No matter what song I started with, it would consistently end up giving me the same songs in the same order. Despite its many features and considerable hard drive for the time, its terrible random generator made it a pretty bad choice for an mp3 player.

The iPod’s shuffle feature is more sophisticated. For starters, in a given shuffle session, it won’t return to songs you’ve already played. This is basically random-plus, because on a one-dimensional plane (i.e., our list of songs), we know that returning to the same song sooner or later is a certainty. In addition to purposefully avoiding songs you’ve played, iPods may also be biased towards songs you’ve rated more highly or songs you’ve played more often in previous sessions. The “session” idea is key to understanding the iPod’s random shortcomings: for any playlist of songs, the shuffle feature is reset each time the iPod is reset or synced. What this means is that despite not returning to songs within a session, iTunes’s bias towards your favorite songs might still prevent you from fully exploring a given playlist over several sessions. In other words, the distribution is more cramped than a truly random walk. You will miss stuff. There could be a significant number of songs you never hear. This limitation can be overcome, but it takes additional work.

This is one reason why, even though I have an iPod touch, I tend to use my Sansa e250 when I’m walking the dog and just want to hear some music. The Sansa random feature is truly random, meaning that (as with a random walk) I often often hear songs that have previously played, even though I haven’t reset the player. Sometimes I even hear a song that has just finished playing moments ago. So while, on a given walk, I might hear the same song repeated, over time I’ll experience a more natural, and less cramped distribution among songs. In fact, over a 2 to 3 week stretch between syncs, I come awfully close to hearing all 300-400 songs on a 2-GB player. Then, it’s time to charge the Sansa again and I’ll start over with a new list randomly drawn from 27,000 songs on my hard drive.

Why does having a more even distribution matter so much to me? Well, that’s really the essence of what this is about. I’ve been buying less new music and trying to really listen to the music I already have: those 27,000 songs stuck on my hard drive, so rarely visited or experienced. My random walks have been allowing me to consider these songs, in many cases for the first time. Obviously, going through them 2 GB at a time over each 3 or 4 week period isn’t the fastest way to do it, but I’ve been discovering new favorites and uncovering songs I’d long forgotten. Do I mind the repeats? Nope. Sometimes, hearing some of the same songs cycle through again is a pretty welcome adventure. It reminds me of the times before mp3s, when I’d spend whole afternoons or car trips repeating a single song or record.

I feel like I’m enjoying music again. Stumbling through the backroads, smelling the flowers, etc.

I’ve also been blogging about it on my personal blog. For each random walk I take with my dog Hank, I select a new Dog Walk Jam™, which is typically my favorite surprise from that particular stroll around the neighborhood. I’m up to #27. The best part? There’s a better distribution amongst genres, genders, sounds than there would be if I were consciously trusting my memory to tell you what I like. It feels more honest. It feels more…random.