Complexity analysis of riverflow time series

Abstract

I have used the Lempel–Ziv measure to assess the complexity in riverflow activity over England and Wales for the period 1867–2002. In particular, I have examined the reconstructed monthly riverflow time series from fifteen representative catchments in these regions and calculated the Lempel–Ziv Complexity (LZC) value for each time series. The results indicate that the LZC values in some catchments are close to each other while in others they differ significantly. In addition, I have divided the period 1867–2002 into four equal subintervals: (a) 1867–1900, (b) 1901–1934, (c) 1935–1968, (d) 1969–2002, and calculated the LZC values for the various time series in these subintervals. It is found that during the period 1969–2002, there is a decrease in complexity in most of the catchments in comparison to the subinterval 1935–1968. This complexity loss may be attributed to increased human intervention involving land and crop use, urbanization, commercial navigation and climatic changes due to human activity. Determining the complexity in the riverflow time series is important because an understanding of the extent of complexity may be useful in developing appropriate models of riverflow activity. The extent of complexity may also influence the predictability of the variability in riverflow dynamics.

Appendix

1.1 Calculation of Lempel–Ziv complexity

In their original work, Lempel and Ziv (1976) developed a measure of complexity to estimate the degree of disorder or irregularity in a sequence. The procedure consists of enumerating the repeating subsequences of all lengths using a complexity counter. For a time series, this is done by first creating an appropriate sequence by partitioning the time series. The Lempel–Ziv complexity analysis of a time series {x _i }, i = 1, 2, 3, …, N, can be carried out as follows. First, encode the time series by constructing a sequence S of the characters 0 and 1 written as: s(i), i = 1, 2, 3, …, n, according to the rule:

$$ s(i)\,\,\, = \,\,\,\left\{ \begin{aligned}{} & 0\quad {\rm if}\;x_{i} \,\, < \,\,x_{*} \\ & 1\quad {\rm if}\;x_{i} \,\, \ge \,\,\,x_{*} \, \\ \end{aligned} \right. $$

(1)

Here x _* is a chosen threshold. I shall choose the mean value of the time series to be the threshold. The mean value of the time series has often been used as the threshold (Zhang et al. 2001). Depending on the application, other encoding schemes are also used (Radhakrishnan et al. 2000; Small 2005).

To start the computation, the original sequence is scanned from left to right and a complexity counter c(n) is increased by one unit every time a new subsequence of consecutive characters is encountered. The complexity counter measures the number of distinct patterns contained in a given sequence. With this procedure the Lempel–Ziv complexity measure (LZC) can be computed using the following steps (see Gomez et al. 2006; Zhang et al. 2001).

• 1. Let P and Q denote two subsequences and PQ be the concatenation of P and Q. In addition, let PQπ be a sequence derived from PQ after its last character is deleted. Here the symbol π is used to denote the operation of deleting the last character in a sequence.

• 2. Let v(PQπ) denote the vocabulary of all different subsequences of PQπ.

• 3. At the beginning of the computation, the complexity counter is set to be one, i.e., c(n) = 1, with P = s(1), Q = s(2), PQ = s(1), s(2), and PQπ = s(1).

• 4. In general, P = s(1), s(2), …, s(r), and Q = s(r + 1). Then PQπ = s(1), s(2), s(3), …, s(r). If Q belongs to v(PQπ), then Q is a subsequence of PQπ, not a new sequence.

• 5. Renew Q to be s(r + 1), s(r + 2) and see if Q belongs to v(PQπ).

• 6. Repeat the previous steps until Q does not belong to v(PQπ). In this case, Q = s(r + 1), s(r + 2), …, s(r + i) is not a subsequence of PQπ = s(1), s(2), …, s(r + i − 1); so increase the counter by one.

• 7. Thereafter, P is renewed to be the sequence: P = s(1), s(2), …, s(r + i), with Q = s(r + i − 1).

• 8. Repeat the previous steps until Q is the last character. At this point, the complexity counter c(n) gives the number of different subsequences in P, which is equal to the Lempel–Ziv complexity (LZC).

Note that the above procedure uses only two operations: comparison and accumulation, and this makes the computation of c(n) easy to implement.

A Lempel–Ziv complexity measure which is independent of the length of the sequence can be obtained by normalizing c(n) with respect to the complexity of a random binary sequence: $b(n) = n/ \log _{2} n$. The normalized LZC measure is given by

$$ C(n) = \frac{1} {n}\,\,c(n)\,\log _{2} n $$

(2)

This normalized measure is used for estimating the LZC values in the present study. For a deterministic system that is composed of well defined cycles such as a periodic motion, C(n) has a value much smaller than 1, whereas for a stochastic system, the value of C(n) is close to 1.