Successfully predicting the future states of systems that are complex, stochastic, and potentially chaotic is a major challenge. Model forecasting error (FE) is the usual measure of success; however model predictions provide no insights into the potential for improvement. In short, the realized predictability of a specific model is uninformative about whether the system is inherently predictable or whether the chosen model is a poor match for the system and our observations thereof. Ideally, model proficiency would be judged with respect to the systems’ intrinsic predictability, the highest achievable predictability given the degree to which system dynamics are the result of deterministic vs. stochastic processes. Intrinsic predictability may be quantified with permutation entropy (PE), a model‐free, information‐theoretic measure of the complexity of a time series. By means of simulations, we show that a correlation exists between estimated PE and FE and show how stochasticity, process error, and chaotic dynamics affect the relationship. This relationship is verified for a data set of 461 empirical ecological time series. We show how deviations from the expected PE–FE relationship are related to covariates of data quality and the nonlinearity of ecological dynamics. These results demonstrate a theoretically grounded basis for a model‐free evaluation of a system's intrinsic predictability. Identifying the gap between the intrinsic and realized predictability of time series will enable researchers to understand whether forecasting proficiency is limited by the quality and quantity of their data or the ability of the chosen forecasting model to explain the data. Intrinsic predictability also provides a model‐free baseline of forecasting proficiency against which modeling efforts can be evaluated.

Follow this and additional works at: http://digitalcommons.unl.edu/geosciencefacpub Part of the Earth Sciences CommonsThis Article is brought to you for free and open access by the Earth and Atmospheric Sciences, Department of at DigitalCommons@University of Nebraska -Lincoln. It has been accepted for inclusion in Papers in the Earth and Atmospheric Sciences by an authorized administrator of DigitalCommons@University of Nebraska -Lincoln.Baker, P. A.; Fritz, Sherilyn C.; Garland, J.; and Ekdahl, E., "Holocene hydrologic variation at Lake Titicaca, Bolivia/Peru, and its relationship to North Atlantic climate variation" (2005). Papers in the Earth and Atmospheric Sciences. 37. http://digitalcommons.unl.edu/geosciencefacpub/37 IntroductionRecent studies from many sites in the Northern Hemisphere show centennial-to millennial-scale climate variation that has been correlated directly with changes in atmospheric radiocarbon production or with changes in North Atlantic oceanic circulation that also may be linked to atmospheric radiocarbon production and presumed solar variability (e.g. Bond et al., 1997Bond et al., , 2001Hodell et al., 2001;Neff et al., 2001;Fleitmann et al., 2003;Hu et al., 2003;Rohling and Pälike, 2005;Wang et al., 2005). However, despite a growing body of palaeoclimatic data from the Southern Hemisphere tropics of South America, patterns of hydrologic variation at these timescales are not as well known, because of the lack of records at suitably high resolution that span the entire Holocene. Thus, it is unclear how centennial-to-millennial variability in the climate system is manifest in the region, and how (or whether) these changes are correlated with climatic events in the Northern Hemisphere. In this paper we will address two questions: is there evidence in the Holocene record of Lake Titicaca for climatic changes that correlate with "Bond events" (Bond et al., 1997) of the North Atlantic and, if so, what is the phasing of these Altiplano events with respect to records from the North Atlantic and the Indian/Asian monsoon regions?Site description Lake Titicaca (16° S, 69° W) lies at 3810 m on the Altiplano of Boliva and Peru (Fig. 1), a high-elevation internally drained plateau. The lake consists of a large (7131 km 2 ) deep (maximum depth 284 m; mean depth 125 m) main basin and a smaller (1428 km 2 ) shallow basin (maximum depth 42 m; mean depth 9 m), connected at the Straits of Tiquina (25 m depth). Hydrologic inputs to the contemporary lake are balanced between direct rainfall (47%) and inflow from six major rivers (53%). Modern water export is primarily via evaporation (91%), with <9% loss via the sole surface outlet, the Rio Desaguadero at 3804 m elevation (Roche et al., 1992 AbstractA growing number of sites in the Northern Hemisphere show centennial-to millennial-scale climate variation that has been correlated with change in solar variability or with change in North Atlantic circulation. However, it is unclear how (or whether) these oscillations in the climate system are manifest in the ...

This paper provides insight into when, why, and how forecast strategies fail when they are applied to complicated time series. We conjecture that the inherent complexity of real-world time-series data-which results from the dimension, nonlinearity, and non-stationarity of the generating process, as well as from measurement issues like noise, aggregation, and finite data length-is both empirically quantifiable and directly correlated with predictability. In particular, we argue that redundancy is an effective way to measure complexity and predictive structure in an experimental time series and that weighted permutation entropy is an effective way to estimate that redundancy. To validate these conjectures, we study 120 different time-series data sets. For each time series, we construct predictions using a wide variety of forecast models, then compare the accuracy of the predictions with the permutation entropy of that time series. We use the results to develop a model-free heuristic that can help practitioners recognize when a particular prediction method is not well matched to the task at hand: that is, when the time series has more predictive structure than that method can capture and exploit.

Explaining how and why some species evolved to have more complex social structures than others has been a long-term goal for many researchers in animal behavior because it would provide important insight into the links between evolution and ecology, sociality, and cognition. However, despite long-standing interest, the evolution of social complexity is still poorly understood. This may be due in part to researchers focusing on the feasibility of quantifying aspects of sociality, rather than what features are characteristic of animal social complexity in the first place. Any given approach to studying complexity can tell us some things about animal sociality, but may miss others, so it is critical to decide first how to conceptualize complexity before jumping in to quantifying it. Here, we briefly summarize five existing approaches to measuring social complexity. Then, we highlight three fundamental concepts that are commonly used in the field of complex systems: (1) scales of organization, (2) compression, and (3) emergence. All of these concepts are applicable to the study of animal social systems, but are not often explicitly addressed in existing social complexity measures. We discuss how these concepts can provide a rigorous foundation for conceptualizing social complexity, the potential benefits of incorporating them, and how existing measures do (or do not) include them. Ultimately, researchers need to critically evaluate any measure of animal social complexity in order to balance the biological relevance of the aspect of sociality they are quantifying with the feasibility of obtaining enough data.

Animal social groups are complex systems that are likely to exhibit tipping points—which are defined as drastic shifts in the dynamics of systems that arise from small changes in environmental conditions—yet this concept has not been carefully applied to these systems. Here, we summarize the concepts behind tipping points and describe instances in which they are likely to occur in animal societies. We also offer ways in which the study of social tipping points can open up new lines of inquiry in behavioural ecology and generate novel questions, methods, and approaches in animal behaviour and other fields, including community and ecosystem ecology. While some behaviours of living systems are hard to predict, we argue that probing tipping points across animal societies and across tiers of biological organization—populations, communities, ecosystems—may help to reveal principles that transcend traditional disciplinary boundaries.