Figure 1

The first operation of MSsE is coarse graining. We divide the time series {x₁, x₂ … x_M} into non-overlapping boxes of size N (scale "N"). The median of each local box is taken, thus producing a new time series $\left\{{\mathrm{y}}_1^{\mathrm{N}},{\mathrm{y}}_2^{\mathrm{N}}\dots {\mathrm{y}}_{\mathrm{k}}^{\mathrm{N}}\right\}$. The second stage in forming the sign time series is to transform the coarse-grained series into yet another new series by taking the directions in its change. We measure the change against a threshold value and acquire the sign series $\left\{{\mathrm{b}}_1^{\mathrm{N}},{\mathrm{b}}_2^{\mathrm{N}}\dots {\mathrm{b}}_{\mathrm{k}}^{\mathrm{N}}\right\}$, where ${\mathrm{b}}_{\mathrm{i}}^{\mathrm{N}}$ is either +1 or -1, depending on whether the corresponding ${\mathrm{y}}_{\mathrm{i}}^{\mathrm{N}}$ is increasing or decreasing. To quantify the complexity of the sign sequence, we sort all sequences into categories of sub-sets consist of L consecutive binary bits (L-bit; L = 8 in this study). The probability distribution of all patterns of sub-sets is recorded. To avoid over-counting similar patterns, the data sequence of total length L should be divided into multiple m-dimensional vectors; each consists of m consecutive bits {(b₁, b₂, … b_m); (b₂, b₃, … b_{m +1}); …}. The conditional probability is determined numerically by the ratio of number of each paired vectors which are of exactly same binary codes for dimension "m+1" to the number for the identical vectors of dimension "m". By identifying the patterns of the same conditional probability, it allows us to rank the m-bit patterns according to the information they imply (large rank number means lower conditional probability). The expectation value of the rank conceptually indicates the degree of uncertainty.