Creating and detecting specious randomness
Jonas Almlöf, Gemma Vall Llosera, Elisabet Arvidsson, Gunnar Björk
Abstract
Abstract We present a new test of non-randomness that tests both the lower and the upper critical limit of a $\chi ^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -statistic. While checking the upper critical value has been employed by other tests, we argue that also the lower critical value should be examined for non-randomness. To this end, we prepare a binary sequence where all possible bit strings of a certain length occurs the same number of times and demonstrate that such sequences pass a well-known suite of tests for non-randomness. We show that such sequences can be compressed, and therefore are somewhat predictable and thus not fully random. The presented test can detect such non-randomness, and its novelty rests on analysing fixed-length bit string frequencies that lie closer to the a priori probabilities than could be expected by chance alone.