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## Measures of nonlinearity

A few programs in the package directly issue scalar quantities that can be used in nonlinearity testing. These are the zeroth order nonlinear predictors (predict and zeroth) which implement Eq.(5) and the time reversibility statistic (3). For a couple of other quantities, we have deliberately omitted a black box algorithm to turn the raw results into a single number. A typical example are the programs for dimension estimation (d2, c2naive, and c1) which compute correlation sums for ranges of length scales and embedding dimensions m. For dimension estimation, these curves have to be interpreted with due care to establish scaling behaviour and convergence with increasing m. Single numbers issued by black box routines have lead to too many spurious results in the literature. Researchers often forget that such numbers are not interpretable as fractal dimensions at all but only useful for comparison and classification. Without genuine scaling at small length scales, a data set that gives by some ad hoc method to estimate cannot be said to have more degrees of freedom, or be more ``complex'' than one that yields .

This said, users are welcome to write their own code to turn correlation integrals, local slopes (c2d), Takens' estimator (c2t), or Gaussian Kernel correlation integrals (c2g) into nonlinearity measures. The same situation is found for Lyapunov exponents (lyap_k, lyap_r), entropies (boxcount) and other quantities. Since all of these have already been described in Ref. [9], we refer the reader there for further details.

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