Complexity and Statistics: Tipping Points and CrashesOrganised by the Environmental Statistics Section of the Royal Statistical Society.Friday 22 October 2010 (10:30am - 5pm)RSS, 12 Errol Street, London, EC1Y 8LX |
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The meeting organisers are Chris Ferro and Nick Watkins. Please contact Chris Ferro for further details.
The meeting is kindly being sponsored by the
Industrial Mathematics Knowledge Transfer Network
andCambridge Centre for Risk Studies University of Cambridge, Judge Business School.
- Parallels between earthquake prediction, financial crash prediction and epileptic seizures predictions, available here.
- Climate tipping as a noisy bifurcation: a predictive technique available here and here
- Markovian, predictive, and conceivably causal representations of stochastic processes, available here.
- Applying degenerate fingerprinting in studying climate tipping points
- Change point detection using the informational approach with applications in atmospheric sciences
You need not be a member of the Royal Statistical Society to attend, but registration is required via the web form available here
Twenty free places for Retired/Student Fellows are available courtesy of the meeting sponsors and will be allocated on a first-come, first-served basis. Otherwise, registration fees, which cover lunch and refreshments, are as follows:
The meeting will take place at the Royal Statistical Society, 12 Errol Street, London, EC1Y 8LX.
Directions can be found here
TBA
In the first half of this contribution we review the bifurcations of dissipative dynamical systems. The co-dimension-one bifurcations, namely those which can be typically encountered under slowly evolving controls, can be classified as safe, explosive or dangerous. Focusing on the dangerous events, which could underlie climate tippings, we examine the precursors (in particular the slowing of transients) and the outcomes which can be indeterminate due to fractal basin boundaries.
It is often known, from modelling studies, that a certain mode of climate tipping is governed by an underlying bifurcation. For the case of a so-called fold, a commonly encountered bifurcation (of the oceanic thermohaline circulation, for example), we estimate how likely it is that the system escapes from its currently stable state due to noise before the tipping point is reached. Our analysis is based on simple normal forms, which makes it potentially useful whenever this type of tipping is identified (or suspected) in either climate models or measurements.
Drawing on this, we suggest a scheme of analysis that determines the best stochastic fit to the existing data. This provides the evolution rate of the effective control parameter, the (parabolic) variation of the stability coefficient, the path itself and its tipping point. By assessing the actual effective level of noise in the available time series, we are then able to make probability estimates of the time of tipping. In this vein, we examine, first, the output of a computer simulation for the end of greenhouse Earth about 34 million years ago when the climate tipped from a tropical state into an icehouse state with ice caps. Second, we use the algorithms to give probabilistic tipping estimates for the end of the most recent glaciation of the Earth using actual archaeological ice-core data.
Basically any stochastic process can be represented as a random function of a homogeneous, measured-valued Markov process. The Markovian states are minimal sufficient statistics for predicting the future of the original process. This has been independently discovered multiple times since the 1970s, by researchers in probability, dynamical systems, reinforcement learning and time series analysis, under names like "the measure-theoretic prediction process", "epsilon machines", "causal states", "observable operator models", and "predictive state representations". I will describe the mathematical construction and information-theoretic properties of these representations, touch briefly on their reconstruction from data, and finally consider under what conditions they may allow us to form causal models of dynamical processes.
Recently, Held and Kleinen employed lag-1 autocorrelation for detection of bifurcation in the modelled collapse of termohaline circulation in CLIMBER2 model [1]. Following their approach, we developed an alternative method based on Detrended Fluctuation Analysis (DFA) to monitor short-term correlations in fluctuations after removal of trends which may affect autocorrelations [2]. In addition, we consider lag-1 autocorrelation after removal of local linear trend. In these three techniques we calculate so-called “propagators”: ACF-propagator, ACF-propagator with trend removal and DFA-propagator. Noticeable trend of propagators towards critical value 1 indicates approaching of bifurcation or transition. Comparing ACF-propagators with and without trend removal allows us to detect the subsets of data affected by simple trends, and the DFA-propagator provides information about possible growing effects of short-term memory that may result in nonstationary behaviour of time series and lead to a bifurcation. Combined together, the techniques help distinguish between transitions and genuine bifurcations and allow us to study climate tipping points.
Ditlevsen and Johnsen [3] analysed autocorrelation and variance of GRIP ice-core data and concluded that Dansgaard-Oeschger (DO) events were noise-induced jumps between system double-well-potential. This kind of abrupt transitions may be unpredictable [4], providing no early warning signal of approaching a tipping point, but in the case of combined increase of autocorrelation and increase of memory effects, the early warning using degenerate fingerprinting is possible. Moreover, in systems with memory (autocorrelation exponent gamma less than 1), monitoring only ACF/DFA-propagators without monitoring variance is sufficient for bifurcation detection.
We apply propagator techniques to various artificial datasets and recorded climatic time series and discuss detected transitions and bifurcations in the context of the main tipping points described in [5].
- Held and Kleinen, GRL 2004
- Livina and Lenton, GRL 2007
- Ditlevsen and Johnsen, GRL 2010
- Livina, Ditlevsen and Lenton, submitted
- Lenton et al, PNAS 2008
A change point in a time series can be viewed as a time point at which the parameters of a statistical distribution or a statistical model change. Most change point detection techniques were developed to identify the most likely time for a shift and to test whether or not this shift occurred by comparing the model with a shift to a simpler model without a shift. Change points can be observed in a wide variety of fields (e.g. economy, social sciences and climate). Change point methods have been applied to climate time series to detect artificial or natural discontinuities and regime shifts.
Most change point approaches were designed to detect a specific type of shift: a change in the mean, in the variance, in both the mean and the variance or in the parameters of a regression moUse of uninitialized value in concatenation (.) or string at E:\listplex\SYSTEM\SCRIPTS\filearea.cgi line 455,
line 260. del, but not to discriminate between several types of changes. Furthermore, most change point methods make the hypothesis that the residuals are independent, but the presence of autocorrelation is a common feature of climate time series (especially at short time scales). If not taken into account in the analysis, the presence of positive autocorrelation can lead to the detection of false shifts. Several studies have started to take into account a first order autoregressive model in change point detection. The autocorrelation structure in climate time series can often be explained by the El Nino Southern Oscillation or by climate forcings such as volcanic eruptions and solar irradiance. These covariate effects can be integrated in the model to determine whether change points remain when taking them into account. In this work, the informational approach is used to discriminate between several types of changes (shifts in the mean, shifts in the variance, shift in the trend, shift in the relation with covariate effects or a combination of these different types of changes) by fitting a hierarchy of models. The autocorrelation structure is identified in each model (not restricted only to a first order autoregressive model) and integrated in the analysis. The usefulness of this approach to detect change points in atmospheric CO2 concentration, in the growth rate of atmospheric CO2 and in the sources and sinks of atmospheric CO2 is demonstrated through applications.