The Lempel-Ziv complexity (LZ) is a popular tool to quantify the uncertainty contained in time series data. In particular, LZ measures how “diverse” are the patterns that are present in a particular signal. The LZ method was introduced back in 1976 by the electrical engineers Abraham Lempel and Jacob Ziv to study binary sequences [Lempel1976]. These core ideas were later extended [Ziv1977,Ziv1978a] to become the basis of the well-known zip compression algorithm, which is used every day by millions of people to compress files. In parallel, this algorithm has been used by scientists to study the diversity of patterns in signals of diverse nature. In particular, studies on EEG brain activity have been carried out using LZ for more than 20 years, with some early studies focusing on epilepsy [Radhakrishnan1998] and depth of anaesthesia [Zhang2001], and more modern ones on other altered states of consciousness [Schartner2017]. Outside neuroscience, LZ has also been used to study the complexity of DNA sequences [Gusev1999] and ventricular fibrillation [Zhang1999], and has become a well-established tool for biomedical data analysis [Aboy2006].

LZ is calculated in two steps. First, the value of a given signal $X$ of length $T$ is binarised. The standard way of doing this is calculating its mean value and turning each data point above it to 1s and each point below it to 0s; however, other possibilities have also been used (thresholding over the median, using the Hilbert transform, etc…). As a second step, the resulting binary sequence is scanned sequentially looking for distinct structures or patterns, building up a dictionary that summarises the sequences seen so far. Finally, the LZ index is determined by the length of this dictionary; i.e., is the number of distinct patterns found, denoted by $C(X)$. Note that regular signals can be characterized by a small number of patterns and hence have low LZ complexity, while irregular signals require long dictionaries and hence have a high LZ complexity.

Following the reasoning above, the LZ method identifies signal complexity with richness of content [Mitchell2009]. In this way, a signal is regarded as complex if it is not possible to provide a brief (i.e. compressed) representation of it.

A popular way of making sense of LZ is by considering a proxy for estimating the value of the Kolmogorov complexity, which is the length of the shortest computer program that reproduce a given pattern [Vitanyi1997]. However, we argue that this view is brittle in theory and of limited use in practice. As a matter of fact, the comparison between the Kolmogorov complexities of finite strings is highly problematic [Cover2006]. The Kolmogorov complexity of a finite string differs by an additive constant depending on the Turing machine used to compress it. The problem is that for finite of any length the magnitude of this constant is unbounded, making comparisons between Kolmogorov complexities estimated from finite data essentially meaningless.

A simpler and more straightforward interpretation of LZ is by to focus on the quantity

which is known to be an efficient estimator of the entropy rate of $X$ [Ziv1978b]. The entropy rate is a quantity from Information Theory, which measures how many bits of innovation are introduced by each new data sample [Cover2006]. Moreover, the entropy rate is a good measure of how hard it is to predict the next value of a sequence. In effect, one half of the entropy rate approximates the probability of making an error with the best informed guess about the next sample [Feder1994]. Note that while LZ can – strictly speaking – be applied to non-stationary data, the interpretation as entropy rate assumes that the data is stationary. The previous interpretation is valid to the so called LZ76, which was defined by [Lempel1976]. There are two other versions of the LZ algorithm, known as LZ77 and LZ78 according to the year in which they were published. The numerical value of LZ differs when computed by the different versions; however, by using appropriate scaling any of these can be made to converge to the entropy rate of the process. However, in our experience LZ76 converge faster than LZ77 and LZ78, and also it computational time is much faster.

Sample Python code to calculate LZ76 complexity can be found in this website.

### References

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