have a stochastic Christmas

The decorating of the Christmas tree, for me, is always a time to reflect on family, love, charity, and of course probability theory.

Each year, we choose a different selection of ornament “styles” from the huge collection of ornaments that we have in storage. Last year, it was doves and metal and glass ornaments. This year, it is sparrows, ducks, apples, and straw ornaments.  Except for the Queen Dove, she always sits at the very top of the tree. She’s the only one who can keep the other birds in line.

How do the ornaments get arranged?  “Randomly” is the way most people would describe it. This is what most people say they do:  “I just arrange them randomly on the tree.”

Christmas Tree DecorationsWhat do people mean when they say they arrange ornaments “randomly”? The fact is, there are different kinds of “random” that have different degrees of randomness. The easiest way to see this is to consider two examples of randomness in nature:

  • The drops of rain hitting a pavement: the pattern is “random” in the sense that the position of one raindrop has nothing to do with the position of another. You can know everything about where the last 99 raindrops fell on a particular segment on the sidewalk, and it will not help you to predict where the next raindrop will fall.
  • Birds perching on tree branches: the pattern is “random” in the sense that they don’t perch in order or follow a clear pattern, but there still seem to be some rules. For example, no two birds will perch right next to each other when there is room for them to spread out.

So I’m going to call these “inanimate randomness” (for the former) and “living randomness” (for the later). This is an interesting distinction because when mathematicians and statisticians (and other extremely uptight people) talk about randomness, they usually mean inanimate randomness. This is what they call “white noise” and it means that there is absolutely no relationship between any of the items being considered.

But the most common kind of randomness that human beings deal with every day is living randomness.  The position of trees in a forest is random, but two trees will not grow right on top of each other. They can’t, because they would choke each other of sunlight and nutrients.  For the same reason, huts in a tribal village and flowers in a field and birds flying in a flock will all exhibit this “slightly less than random” form of living randomness.

And when you arrange ornaments on a tree, you arrange them in a way that follows living randomness, because that is what feels right and looks right to us: two apples should not be butt-up-against each other; neither should two straw ornaments.

When you look at human psychology from an evolutionary standpoint, this actually explains a great deal. Human beings are absolutely terrible at identifying or generating true randomness.  When people are asked to say a “random” sequence of numbers, they will almost never produce two digits in a row (e.g. “3 3 8 4 2 6 6 4”), even though with real randomness (inanimate randomness) repetitions happen fairly often.  But repetition doesn’t “feel” random to people. And that is likely because it isn’t in the kind of randomness that we have evolved to deal with the most: living randomness.

Now, for the people who really want to geek out…

The statisticians’ term for “completely random” (where each point has no relation to any other) is “white noise”, and this is a term that is pretty common. It describes the dots on a television screen tuned to a dead channel and the frequencies coming from a radio that isn’t tuned to a station at all. You have probably heard the term “white noise” before. But statisticians have terms for other kinds of noise, too.

For example, imagine someone sitting at a piano and hitting a note. Then, the person flips a coin, and moves up one key if it is heads and down one key if it is tails. The person keeps this up to produce a sequence of notes.  In this case, there is a kind of randomness involved, but it’s a very low-level randomness. Each note has a very strong relationship with the previous note: it’s only one step away! This is called a “random walk” or “Brownian motion,” named after the first scientist to study it thoroughly.  Since totally random is called “white noise”, statisticians call this “low-level” randomness “Brown noise.” (Yes, that is the kind of sense of humor you can expect from a statistician.)

White noise is boring because it sounds like nothing at all: complete absence of pattern. Brown noise sounds boring because it is too predictable: random walks up and down. But Statisticians have also studied the intermediate level of randomness, that is right in between the two. For reasons best known to them, they have dubbed this “pink noise”. It is more predictable than “white noise” (which has no pattern at all) and less predictable than “brown noise” (which is boring). As Goldilocks would say: pink noise is “just right.”

You want to know what kinds of things qualify as “pink noise”?  Fractals. Faces. Birds perched on a tree. Plants growing in the ground. And (when applied to frequencies over time) music. In other words: all of the things that our minds automatically are drawn to as being interesting and somehow associated with life or creativity. The stuff that our minds are programmed to pay attention to and think are cool all follow a pink-noise distribution.

Isn’t that cool?

So happy holidays! And make sure that you listen to your pink noise holiday songs while you distribute your pink noise ornaments on your tree.

 



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