Filter sequences


More thoughts on how power-law distributions come about: I haven't worked through the math on this yet, but it looks like multiplying several random variables together will produce a suitibly fat-tailed distribution. The more variables multiplied together, the fatter the tails. (i'm assuming these variables can get arbitrarily close to zero, or even go negative, so this does not just produce a log-normal distribution) The obvious thing to call the number of variables would be "dimension"... may or may not be appropriate.

What kind of process would have that kind of multiplying effect?

One possibility: a system in which the input passes through a sequence of filters to produce an output, each filter having a pseudo-random effect that may be characterized as multiplication by a (Gaussian) random variable.

Two examples:

So this looks kind of like it will scale beyond the level of individuals, which is nice.

A slight variation on this would be a feedback system, in which system state is dependant on previous system state, eg

current state = previous state * random variable + input * another random variable

The variance of the first random variable determines how self-absorbed the system is (literally how "autistic" it is, the word "autism" deriving from the root word "auto"). A measure of "dimension" is presumably derivable from this. If the variance exceeds one the "dimension" goes to infinity, and the system has an epileptic fit.

A feedback system could of course also contain multiple elements. For example, a kitten-kitten feedback system will display behaviours of power-law distributed random magnitude, somewhat time-correlated, which may be modulated to some small extent by external inputs.

I'm pretty happy with this. It seems to unify a lot of disparate ideas.

Addendum: How about memory? I'm guessing that memory is a major part of the pseudo-random function. Input of some state similar to a previous event will cause output of the state that previously followed. In the recursive model, this can lead to chains of recall. A chain of recall can be derailed or caused to switch tracks by external input. In an autistic person, the external input is weaker then the recursive input, so they are not as easily derailed.

Tying this in with that best-fit synthesis stuff... Consider a narrow column within the synthesis process. It has a recursive input from the previous line of synthesis, and external inputs from either side. What is needed to derail recall is a big disruption. The L1 metric does not produce extreme disruptions from the external inputs, very very roughly all disruptions are of much the same magnitude. The L2 metric does provide occasional big disruptions: it squares errors, and a bigish error will result in a really big derailment. Once derailed, there's less chance of locking back on to the memory too. ... ok, i don't think i've managed to explain this clearly, but it feels about right, trust me.

Another addendum: there's something of a hint here that the environment might have an effect on the level of autism. An environment with a steady stream of input might allow greater autism (in the sense of self-absorbtion) than one with occasional abrupt bursts. Greater autism is dangerous in the bursty environment because one of the bursts could push the person's brain into overload... too many neurons firing at once, metabolism overloading.

A trivial example of this would be sleep, where an absense of inputs requiring responses allows the brain to become almost entirely self-driven. To push the envelope of self-drive would require turning input way down (input possibly including some internal randomness generators as well as sensory input), resulting in near zero activity with occassional bursts. I would predict from this that non-REM sleep will show up on an EEG as having power-law behaviour with lower power than that of REM sleep, which in turn would have lower power than waking EEG patterns. (Sleep spindles?) This also points to deep non-REM sleep as being the most profound/brain-changing form of sleep.

Various forms of meditation and prayer would also achieve this effect.

Going the other way to try to cure the more extreme forms of autism is a dangerous game, if the person doesn't adapt they will simply overload. Would have to take the process pretty slowly. Could be worth the risk though. Would be something like techniques for reducing phobias.

Yet another addendum: Regarding distinction between recursive and multi-stage processing, i would note that in the brain even multi-stage processing would be endemically recursive. It's useful to feed high level interpretation back down to lower levels. In general we're looking at a collection of sub-systems coupled to each other with varying degrees strength and asymmetry.