Is Bach's brain a Markov chain? Recurrence quantification to assess Markov order for short, symbolic, musical...

It is rarely possible to precisely characterise the system underlying a series of observations. Hypothesis testing, which involves assessing simple assumptions about driving mechanisms, provides hope that we can at least rule out certain possibilities regarding the nature of the system. Unfortunately, the brevity, nonstationarity, and symbolic nature of certain time series of interest undermines traditional hypothesis tests. Fortunately, recurrence quantification analysis (RQA) has an established record of success in understanding short and nonstationary time series. We evaluate the suitability of measures of RQA as test statistics in surrogate data tests of the hypothesis that ten compositions by the Baroque composer J. S. Bach (1685-1750) arose from a Markov chain. More specifically, we estimate the size (the rate at which true hypotheses are incorrectly rejected) and power (the rate at which false hypotheses are correctly rejected) from empirical rejection rates across 1000 realisations, for each of the ten compositions, of the surrogate algorithm. We compare hypothesis tests based on RQA measures to tests based on the conditional entropy, an established test statistic for surrogate data tests of Markov order, and find that the RQA measure $$ provides more consistent rejection of the fairly implausible hypothesis that Bach's brain was a Markov chain.