But you’re not alone. My physics colleagues almost universally say random when they mean uncorrelated.

Yay! I’ve been saying this for ages but it rarely seems to get through. Another example: a data set is never Gaussian.

I’m not sure if you were trying to correct me, or just adding to my comment; but to clarify, I’m well aware that Technical Analysts do not consider stock price movements random (by ‘adding to the list’, I meant that TAs see patterns in random fluctuations). My reference to Mandelbrot was a nod to the fact that it has been shown that many (perhaps all) of the features that a TA would look for in stock price movements can be generated by, for example, multifractal models of the stock market. With regards to “Who gives a damn [how you make money]“, I think the key issue here is that if you have been making money by chance, based on a deeply flawed idea (be it technical analysis or some inappropriate stochastic model), then one day you may find that those flaws catch up to you, and relieve you of your capital.

Also, if our memory device is very limited, then we have no choice but to describe very complex states by means of a probability distribution with comparable entropy.

Finally, given an initial single state with low complexity, the system is highly likely to be found in a state of high complexity later on, again because there are exponentially many more states of high versus low complexity. This is the complexity version of the 2nd law of thermodynamics, valid even for perfectly closed systems with unitary time evolution.

So, as you know, given a probability distribution rho_i, where i labels the possible states, the entropy is defined by:

entropy = -(sum i) rho_i log rho_i.

It vanishes for a pure state–that is, when rho_i is equal to one for a single state and zero for all other states. The entropy is maximal if we have no information and must assign equal probability to all states. Given knowledge of the average energy of the system, we must maximize the entropy (i.e., our ignorance) subject to the constraint that has the given value, and we then obtain the familiar canonical ensemble of thermodynamics.

Entropy can be measured in bits, provided that we take our logarithm base 2. (Changing the base is equivalent to multiplying the formula by a uniform constant.) As an example, the entropy of an unknown binary string n bits long, and therefore having 2^n possible states each with equal probability 1/2^n, is just n bits, as expected.

Meanwhile, the algorithmic complexity of a single state can be defined in several ways. One simple definition is that it’s the number of bits in the shortest string needed to communicate the value of the state to a second individual. A set of 10 vertical arrows obviously requires a shorter description than a set of 10 randomly oriented arrows.

Another definition of the complexity of a state is that its the number of bits in a miinimal algorithm or computer program needed to generate the value of the state.

Now, you are correct that there remains some ambiguity in the definition of complexity. I could always change my labelings so that a state |1010111001> became |1111111111>. Obviously the definition of a complexity function depends on our labeling scheme for the states.

But the crucial fact is that once we’ve chosen a complexity scheme, whatever our choice, there will always be exponentially more states of high complexity than low complexity, because the complexity function defines a one-to-one correspondence between the states of the system and the set of minimal binary messages representing them. That’s the crucial feature of complexity. That fact ensures the various properties I described earlier.

You are right, I got the pictures from Stephen Jay Gould’s book and used them with appropriate credit in my book From Cosmos to Chaos .

You will also find the same pair of images in various places around the web.

Peter

03, 09, 27, 81, 43, 29, 87, 61, 83, 49, 47, 41, 23, 69, 07 (A)

looks as random as the sequence

37, 74, 11, 48, 85, 22, 59, 96, 33, 70, 07, 44, 81, 18, 55 (B)

But the degrees of their stochasticity can be more objectively measured by the Kolmogorov parameter. It can be proved that that the stochasticity probability is approximately 4,700 times higher for the sequence (A) than for the sequence (B).