Analytical tools and resources
The Ocean Tipping Points team has collected a wide variety of methods for identifying nonlinearities or regime shifts in your own system, or early warning indicators of change. In the tables below we distill strengths and weaknesses of the various approaches and point you to where you can learn more.
We also provide a set of recommended criteria, approaches, and methods to detect and characterize regime shifts and their driverresponse mechanisms, synthesized from the literature. This process can help you to definitively identify nonlinear regime shifts in your own ecosystem and characterize their relationships to potential drivers.
Methods for detecting regime shifts and quantifying nonlinear relationships
Generalized additive model (GAM) 

Output  Identifies shape and strength of nonlinear relationships between ecological condition and ecosystem driver(s) 
Strengths  Identifies key drivers 
Flexible in its ability to fit any shape relationship  
Weaknesses  Correlative 
No directionality of relationships  
Computationally complex  
May overestimate degree of nonlinearity if overfitting is not controlled  
More information & examples  Hastie and Tibshirani (1990), Guisan et al. (2002) 
Software & code examples  gam package for R 
Change point analysis (e.g., Sequential ttest on the mean, STARS) 

Output  Identifies point of inflection in relationship between ecological condition and ecosystem driver(s), i.e. the threshold 
Strengths  Identifies location of the threshold or regime shift and corresponding driver/pressure level 
Identifies leading and lagging indicators  
Weaknesses  Correlative 
No directionality of relationships  
Does not explicitly take autocorrelation into account  
More information & examples  Rodionov (2006), Cury et al. (2011), Matteson and James (2014), Karr et al. (2015) 
Software & code examples  VBA for Excel at www.BeringClimate.noaa.gov and strucchange, changepoint, cpm, bcp packages for R 
Redundancy analysis (RDA) 

Output  Identifies nonlinear relationships between ecological condition and ecosystem driver(s) 
Determines likelihood of regime shift  
Strengths  Accommodates multivariate datasets 
Identifies regime shifts  
Weaknesses  Correlative 
No directionality of relationships  
More information & examples  Makarenkov and Legendre (2002), Borcard et al. (2011) 
Software & code examples  rda function in vegan: community ecology R package 
Principal components analysis (PCA) 

Output  Identifies key periods of ecosystem change and associated driver(s) 
Strengths  Accommodates multivariate datasets 
Facilitates linking time of ecosystem change to driver number and level  
Does not require a priori hypothesis of regime shift year(s)  
Weaknesses  Correlative 
No directionality of relationships  
No statistical significance of relationships  
More information & examples  Hare and Mantua (2000), Möllmann et al. (2009), Tomczak et al. (2013) 
Software & code examples  princomp and prcomp in the R Stats package 
Boosted regression trees 

Output  Identifies potentially significant direct and indirect effects of drivers on ecosystem components 
Strengths  Identifies indirect effects 
Facilitates experimental and observational studies of ecosystem effects  
Weaknesses  Correlative 
No directionality of relationships  
More information & examples  De’Ath (2007), Elith et al. (2008) 
Software & code examples  gbm R package 
Methods for early warning indicator analyses
Multivariate autoregressive statespace model (MARSS) 

Output  Identifies how nonlinear changes are related to biotic processes and changes in outside drivers 
Quantifies interaction strength between driver(s) and response variable(s)  
Strengths  Accommodates multivariate datasets 
Identifies drivers and ecosystem responses that could serve as early warning indicators  
Quantifies interaction strengths among drivers  
Weaknesses  Correlational 
Requires significant data input  
More information & examples  Zuur et al. (2003), Hampton and Schindler (2006), Holmes et al. (2012), Hampton et al. (2013) 
Software & code examples  MAR1 and MARSS R packages; Matlab code (Ives et al., 2003) 
Structural equation modeling (SEM) 

Output  Predicts how an ecosystem is likely to respond to changes in direct and indirect drivers 
Strengths  Predicts directionality and strength of relationship between driver and ecosystem response 
Accommodates wide range of data types  
Allows for incorporation of feedback loops and twoway interactions  
Weaknesses  Requires significant data inputs 
Requires a priori understanding of ecosystem  
Does not incorporate nonlinearities in relationships  
More information & examples  Grace (2008), Grace et al. (2010), Thrush et al. (2012), Fox et al. (2015) 
Software & code examples  sem R package 
Regime shift indicators (e.g., variance; autocorrelation; critical slowing down and flickering) 

Output  Provides early warning of threshold dynamics and regime shifts in spatial and temporal data sets 
Strengths  Accommodates wide range of data, including spatial and temporal data 
Allows early identification of threshold dynamics and regime shifts  
Weaknesses  Requires significant data inputs 
Usually retrospective  
May not be transferable across systems  
More information & examples  Dakos et al (2010, 2012), (Veraart et al., 2012), Litzow et al (2013), 
Software & code examples  nlme R package 
For more information on code and methods that can be used to identify early warning indicators, visit The Early Warning Signals Toolbox website: http://www.earlywarningsignals.org
Analytical process for detecting and systematically characterizing regime shifts
Based on the tools identified above and criteria outlined by Bestelmeyer and colleagues (2011) and Collie and colleagues (2004) we have put together a suggested process for detecting and characterizing regime shifts and their driverresponse mechanisms. Below you will find a stepwise process meant to help guide you through the best available criteria, approaches, and methods to detect and characterize shifts in your own system. This process has been synthesized and adapted in large part from Scheffer and Carpenter 2003, Collie et al. 2004, and Bestelmeyer et al. 2011.
Regime Shifts Questions (from Collie et al. 2004)  Question Details (from Collie et al. 2004)  Approach (adapted from Bestelmeyer et al. 2011)  Approach Rationale (adapted from Bestelmeyer et al. 2011)  Methods 

1. Is there a discrete step function or intervention in the time series?  A significant step is a necessary condition for a regime shift. However, the type of regime shift cannot be inferred from time series alone.  Select driver and response variables based on systemspecific analysis, literature review, and expert understanding of ecosystem dynamics and analyze temporal patterns in these variables. Locate and statistically test one or more breakpoints in response variable time series data.  Researchers should hesitate to infer response mechanisms based solely on the presence of threshold patterns in biological response variables; analyses of driverresponse relationships provide stronger tests of such inferences. Detection of one or more breakpoints suggests that an abrupt transition may have occurred. 
 Locally weighted scatterplot smoother (LOESS): Cleveland and Devlin 1988, Bestelmeyer et al. 2011  Redundancy analysis (RDA)  Principal components analysis (PCA)  Boosted regression trees  Multivariate autoregressive statespace model (MARSS)  Structural equation modeling (SEM)  strucchange R package, cumulative sum (CUSUM) plot, residual sums of squares (RSS) and the Bayesian Information Criterion (BIC): Bestelmeyer et al. 2011, Zeileis et al. 2002  Aggregate standard deviates (ASD) compositing: Mantua 2004  Change point analysis (e.g., Sequential ttest on the mean, STARS)  Intervention analysis (IA) and autoregressive moving average modeling (ARMA): Noakes 1986, Jenkins 1976, Mantua 2004 
2. Does the response state variable(s) have a bimodal (or multimodal) distribution?  Answering this question in the affirmative indicates the occurrence of a regime shift but not the type.  Statistically test unimodality of frequency distributions of response variables  A linear tracking model should yield a unimodal distribution, whereas a threshold or hysteresis model should yield a bimodal distribution 
 Histograms and density smoothers. Test departures from unimodality using Hartigan’s dip test: Hartigan and Hartigan 1985, Bestelmeyer et al. 2011 
3. Is there a different functional relationship between driver and response in different regimes?  A positive answer to this question demonstrates a regime shift but not the type.  Assess relationship between response variables and drivers before and after breakpoints  With linear tracking one should expect similar responsedriver relationships before and after the breakpoint(s) with hysteresis one should expect different responsedriver relationships before and after the breakpoint(s) 
 linear (lm) and nonlinear (nls) regression in R stats library: Bestelmeyer et al. 2011  Generalized additive model (GAM) 
4. Does the system switch to an alternative state when perturbed?  Assuming that the forcing variable is known, the system should switch states when this variable changes. A positive answer to this question indicates a discontinuous regime shift.  
5. Does the system have a different trajectory when the forcing variable increases, compared to when it decreases?  If yes, this is evidence for hysteresis and the existence of a discontinuous regime shift.  
6. Does the second derivative of the time series have peaks?  This effect, if present, is likely to be subtle. It should be observed in mathematical models, but perhaps not in noisy data.  Calculate temporal variance (a leading indicator used to forecast state transitions) of response variables  Abrupt increases in variance can be used as a leading indicator of abrupt transitions in hysteresis models (Carpenter and Brock 2006) 
 rollapply function in the R zoo library: Bestelmeyer et al. 2011  Regime shift indicators (e.g., coefficient of variation; standard deviation of logtransformed data; skewness): Dakos et al. 2010, 2012, Veraart et al., 2012, Litzow et al 2013 
References
 De'ath, G. (2007). Boosted trees for ecological modeling and prediction. Ecology 88, 243251.
 Fox, J., Nie, Z., Byrnes, J., Culbertson, M., Debroy, S., Friendly, M., Jones, R.H., Kraner, A., and Monetter, G. (2015). SEM: Structural Equation Models [Online]. Available: http://cran.rproject.org/web/packages/sem/sem.pdf [Accessed].
 Guisan, A., Edwards, T.C., and Hastie, T. (2002). Generalized linear and generalized additive models in studies of species distributions: setting the scene. Ecological modelling 157, 89100.
 Hampton, S.E., Holmes, E.E., Scheef, L.P., Scheuerell, M.D., Katz, S.L., Pendleton, D.E., and Ward, E.J. (2013). Quantifying effects of abiotic and biotic drivers on community dynamics with multivariate autoregressive (MAR) models. Ecology 94, 26632669.
 Hare, S.R., and Mantua, N.J. (2000). Empirical evidence for North Pacific regime shifts in 1977 and 1989. Progress in Oceanography 47, 103145.
 Hastie, T., and Tibshirani, R. (1990). Exploring the nature of covariate effects in the proportional hazards model. Biometrics 46, 10051016.
 Holmes, E.E., Ward, E.J., and Wills, K. (2012). Marss: Multivariate autoregressive statespace models for analyzing timeseries data. The R Journal 4, 1119.
 Ives, A., Dennis, B., Cottingham, K., and Carpenter, S. (2003). Estimating community stability and ecological interactions from timeseries data. Ecological Monographs 73, 301330.
 Matteson, D.S., and James, N.A. (2014). A nonparametric approach for multiple change point analysis of multivariate data. Journal of the American Statistical Association 109, 334345.
 Rodionov, S.N. (2006). Use of prewhitening in climate regime shift detection. Geophysical Research Letters 33.
 Thrush, S., Hewitt, J., and Lohrer, A. (2012). Interaction networks in coastal softsediments highlight the potential for change in ecological resilience. Ecological Applications 22, 12131223.
 Veraart, A.J., Faassen, E.J., Dakos, V., Van Nes, E.H., Lürling, M., and Scheffer, M. (2012). Recovery rates reflect distance to a tipping point in a living system. Nature 481, 357359.