Estimating non-linear persistence for impact assessment in European forests
Authors: Williams Tristan , Mahecha Miguel D. , Camps-Valls Gustau 
Published: February 2023

Persistence is an important characteristic of many complex systems in nature and of the Earth system in particular. Relating this statistical concept to physical properties of ecosystems is rather elusive, but reflects how long the system remains at a certain state before changing to a different one and is measured via the memory and dependence of values on past states [1]. Characterizing persistence in the terrestrial biosphere is very relevant to understand intrinsic properties of the system such as legacy effects of extreme climate events [2]. Such memory effects are often highly non-linear and therefore challenging to detect in observational records and poorly represented in Earth system models. This study estimates long and short-term non-linear persistence in eddy-covariance flux measurements and remote sensing products in European forests and the corresponding hydro-meteorological data. Characterizing persistence in the data allows us to make inferences on the interaction between Drought-Heat events, forest dynamics, and ecosystem resilience [3]. The comparison of in-situ and Earth Observation (EO) data allows us to infer how meaningful EO data are for monitoring complex dynamics in ecosystems.For short-term, spatio-temporal persistence, we use echo state networks using the technique suggested in [4] as an explainable AI (XAI) technique. In this context, the persistence of the system can be estimated by the model's response when the input fades abruptly. For the characterization of long-term persistence, we introduce a novel kernel extension of the well-established Detrended Fluctuation Analysis (DFA) [5], a method widely used in atmospheric sciences [1]. The DFA method is a scaling analysis that provides a simple quantitative parameter (the scaling exponent) to represent the correlation properties of a signal and a characteristic time of the event of interest. Unlike DFA, the proposed kernel DFA method can handle non-linear time-scales interactions.&#160;Estimating the non-linear persistence of forests and climate data allows us to relate characteristic times, crossover points between different scaling exponents, and short-term memory parameters with the duration and intensity of the events, as well as an indicator of change in the vegetation response to hydro-climatic conditions.&#160;[1] Salcedo-Sanz, S., et al. &#8220;Persistence in complex systems&#8221;. Physics Reports 957, 1-73, (2022).[2] Bastos, Ana, et al. &#8220;Direct and seasonal legacy effects of the 2018 heat wave and drought on European ecosystem productivity." Science advances 6.24 (2020)[3] Scheffer, M., Carpenter, S. R., Dakos, V. & van Nes, E. H. Generic indicators of ecological resilience: inferring the chance of a critical transition.&#160;Annu. Rev. Ecol. Evol. Syst.&#160;46, 145&#8211;167 (2015).[4] Barredo Arrieta, A., Gil-Lopez, S., La&#241;a, I. et al. On the post-hoc explainability of deep echo state networks for time series forecasting, image and video classification. Neural Comput & Applic 34, 10257&#8211;10277 (2022).[5] Peng, C&#8208;K., et al. "Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series." Chaos: an interdisciplinary journal of nonlinear science 5.1 (1995): 82-87.