Mixing Bandt-Pompe and Lempel-Ziv approaches: another way to analyze the complexity of continuous-states sequences

In this paper, we propose to mix the approach underlying Bandt-Pompe permutation entropy with Lempel-Ziv complexity, to design what we call Lempel-Ziv permutation complexity. The principle consists of two steps: (i) transformation of a continuous-state series that is intrinsically multivariate or arises from embedding into a sequence of permutation vectors, where the components are the positions of the components of the initial vector when re-arranged; (ii) performing the Lempel-Ziv complexity for this series of `symbols', as part of a discrete finite-size alphabet. On the one hand, the permutation entropy of Bandt-Pompe aims at the study of the entropy of such a sequence; i.e., the entropy of patterns in a sequence (e.g., local increases or decreases). On the other hand, the Lempel-Ziv complexity of a discrete-state sequence aims at the study of the temporal organization of the symbols (i.e., the rate of compressibility of the sequence). Thus, the Lempel-Ziv permutation complexity aims to take advantage of both of these methods. The potential from such a combined approach - of a permutation procedure and a complexity analysis - is evaluated through the illustration of some simulated data and some real data. In both cases, we compare the individual approaches and the combined approach.Comment: 30 pages, 4 figure

Chapter 4: Aquatic Insect Respiration

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Plant-herbivore interactions in streams near Mt. St. Helens

1. In four separate field experiments near Mount St Helens (Washington, U.S.A.) during 1986, the grazing effects of two large benthic herbivores, tadpoles of the tailed frog Ascaphus truei and larvae of the caddisfly Dicosmoecus gilvipes, were investigated using streamside channels and in-stream manipulations. In the experimental channels, abundances of periphyton and small benthic invertebrates declined significantly with increasing density of these larger herbivores. 2. In eleven small, high-gradient streams affected to varying degrees by the May 1980 eruption, in-stream platforms were used to reduce grazing by A, truei tadpoles on tile substrates. Single platforms erected in each tributary and compared to grazed controls revealed only minor grazing effects, and no significant differences among streams varying in disturbance intensity (and, consequently, tadpole density). However, results probably were confounded by high variability among streams in factors other than tadpole abundance. 3. Grazing effects were further examined in two unshaded streams with different tadpole densities, using five platforms per stream. In the stream with five tadpoles m−2, grazing reduced periphyton biomass by 98% and chlorophyll a by 82%. In the stream lacking tadpoles, no significant grazing effects were revealed. Low algal abundance on both platforms and controls, and high invertebrate density in that stream (c. 30000m−2) suggests that grazing by small, vagile invertebrates was approximately equivalent to that of tadpoles. 4. The influence of large benthic herbivores on algal and invertebrate communities in streams of Mount St Helens can be important, but reponses vary spatially in relation to stream disturbance history, local environmental factors, and herbivore distributional patterns and abundanc