Algorithm 1.

Estimating template dimension from measurements y (idealized version)

1: state estimates of dimension derived from values of dimension 2: phase estimates derived from 3: estimate of the cycle derived by taking as a Fourier series in 4: choose phase for use as a Poincaré section 5: interpolate to obtain the values , e.g. using linear interpolation 6: offsets from cycle 7: input–output pairs of the return map, each of dimension 8: for as many samples as wanted do 9: bootstrap to obtain , a set of input–output pairs taken with replacement from 10: estimate a return map matrix by least-squares regression of 11: end for 12: the (empirical) distribution of matrices at the Poincaré section . 13: find maximal dimension d such that the smallest d eigenvalues of M∼[ϕ] follow the circular law, i.e. the distribution of eigenvalues of random matrices for some scale s. 14: return the codimension of d as the statistically significant template dimension