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specific process one can apply it to a piece of raw data, and then see how the results compare with those obtained from all possible sequences.

If the process is sufficiently simple then by using traditional mathematics one can sometimes work out fairly completely what will happen with all possible sequences. But in the vast majority of cases this cannot be done, and so in practice one has no choice but just to compare with results obtained by sampling some fairly limited collection of possible sequences.

Under these circumstances therefore it becomes quite unrealistic to notice subtle deviations from average behavior. And indeed the only reliable strategy is usually just to look for cases in which there are huge differences between results for particular pieces of data and for typical sequences. For any such differences provide clear evidence that the data cannot in fact be considered random.

As an example of what can happen when simple processes are applied to data, the pictures on the facing page show the results of evolution according to various cellular automaton rules, with initial conditions given by the sequences from page 594. On each row the first picture illustrates the typical behavior of each cellular automaton. And the point is that if the sequences used as initial conditions for the other pictures are to be considered random then the behavior they yield should be similar.

But what we see is that in many cases the behavior actually obtained is dramatically different. And what this means is that in such cases statistical analysis based on simple cellular automata succeeds in recognizing that the sequences are not in fact random.

But what about sequences like (g) and (h)? With these sequences none of the simple cellular automaton rules shown here yield behavior that can readily be distinguished from what is typical. And indeed this is what I have found for all simple cellular automata that I have searched.

So from this we must conclude that—just as with all the other methods of perception and analysis discussed in this chapter—statistical analysis, even with some generalization, cannot readily recognize that sequences like (g) and (h) are anything but completely random—even though at an underlying level these sequences were generated by quite simple rules.

From Stephen Wolfram: A New Kind of Science [citation]