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.

Prediction models that capture and use the structure of state-space dynamics can be very effective. In practice, however, one rarely has access to full information about that structure, and accurate reconstruction of the dynamics from scalar time-series data-e.g., via delaycoordinate embedding-can be a real challenge. In this paper, we show that forecast models that employ incomplete embeddings of the dynamics can produce surprisingly accurate predictions of the state of a dynamical system. In particular, we demonstrate the effectiveness of a simple near-neighbor forecast technique that works with a two-dimensional embedding. Even though correctness of the topology is not guaranteed for incomplete reconstructions like this, the dynamical structure that they capture allows for accurate predictions-in many cases, even more accurate than predictions generated using a full embedding. This could be very useful in the context of real-time forecasting, where the human effort required to produce a correct delay-coordinate embedding is prohibitive. LEAD PARAGRAPHPrediction models constructed from state-space dynamics have a long and rich history, dating back to roulette and beyond. A major stumbling block in the application of these models in real-world situations is the need to reconstruct the dynamics from scalar time-series data-e.g., via delay-coordinate embedding. This procedure, which is the topic of a large and active body of literature, involves estimation of two free parameters: the dimension m of the reconstruction space and the delay, τ , between the observations that make up the coordinates in that space. Estimating good values for these parameters is not trivial; it requires the proper mathematics, attention to the data requirements, computational effort, and expert interpretation of the results of the calculations. This is a major challenge if one is interested in real-time forecasting, especially when the systems involved operate a) Electronic mail: joshua.garland@colorado.edu b) Electronic mail: lizb@colorado.edu on fast time scales. In this paper, we show that the full effort of delay-coordinate embedding is not always necessary when one is building forecast models, and can indeed be overkill. Using synthetic time-series data generated from the Lorenz-96 atmospheric model and real data from a computer performance experiment, we demonstrate that a two-dimensional embedding of scalar timeseries data from a dynamical system gives simple forecast methods enough traction to generate accurate predictions of the future course of those dynamics-sometimes even more accurate than predictions created using the full embedding. Since incomplete embeddings do not preserve the topology of the full dynamics, this is interesting from a mathematical standpoint. It is also potentially useful in practice. This reduced-order forecasting strategy involves only one free parameter (τ ), good values for which, we believe, can be estimated 'on the fly' using information-theoretic and/or machine-learning algorithms. As suc...

Delay-coordinate reconstruction is a proven modeling strategy for building effective forecasts of nonlinear time series. The first step in this process is the estimation of good values for two parameters, the time delay and the embedding dimension. Many heuristics and strategies have been proposed in the literature for estimating these values. Few, if any, of these methods were developed with forecasting in mind, however, and their results are not optimal for that purpose. Even so, these heuristics-intended for other applications-are routinely used when building delay coordinate reconstruction-based forecast models. In this paper, we propose an alternate strategy for choosing optimal parameter values for forecast methods that are based on delay-coordinate reconstructions. The basic calculation involves maximizing the shared information between each delay vector and the future state of the system. We illustrate the effectiveness of this method on several synthetic and experimental systems, showing that this metric can be calculated quickly and reliably from a relatively short time series, and that it provides a direct indication of how well a near-neighbor based forecasting method will work on a given delay reconstruction of that time series. This allows a practitioner to choose reconstruction parameters that avoid any pathologies, regardless of the underlying mechanism, and maximize the predictive information contained in the reconstruction.

Computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a number of free parameters and the computation of homology for a large number of simplices can be expensive. This paper is a study of how to avoid a full (diffeomorphic) reconstruction and how to decrease the computational burden. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a parsimonious simplicial complex-the "witness complex"-to compute its homology. Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds specified in the embedding theorems for assuring topological conjugacy between the true and reconstructed dynamics. We conjecture that this is because the requirements for reconstructing homology, are less stringent than those in the embedding theorems. In particular, to faithfully reconstruct the homology, a homeomorphism is sufficient-as opposed to a diffeomorphism, as is necessary for the full dynamics. We provide preliminary evidence that a homeomorphism, in the form of a delay-coordinate reconstruction map, may manifest at a lower dimension than that required to achieve an embedding.