Analytical tools and resources
The Ocean Tipping Points team has collected a wide variety of methods for identifying non-linearities 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 driver-response mechanisms, synthesized from the literature. This process can help you to definitively identify non-linear regime shifts in your own ecosystem and characterize their relationships to potential drivers.
Methods for detecting regime shifts and quantifying non-linear relationships
Generalized additive model (GAM) |
|
| Output | Identifies shape and strength of non-linear 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 t-test 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 non-linear 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 state-space model (MARSS) |
|
| Output | Identifies how non-linear 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 two-way interactions | |
| Weaknesses | Requires significant data inputs |
| Requires a priori understanding of ecosystem | |
| Does not incorporate non-linearities 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.early-warning-signals.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 driver-response mechanisms. Below you will find a step-wise 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 system-specific 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 driver-response 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 state-space 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 t-test 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 response-driver relationships before and after the breakpoint(s) with hysteresis one should expect different response-driver relationships before and after the breakpoint(s) |
- linear (lm) and non-linear (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 log-transformed 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, 243-251.
- 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.r-project.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, 89-100.
- 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, 2663-2669.
- Hare, S.R., and Mantua, N.J. (2000). Empirical evidence for North Pacific regime shifts in 1977 and 1989. Progress in Oceanography 47, 103-145.
- Hastie, T., and Tibshirani, R. (1990). Exploring the nature of covariate effects in the proportional hazards model. Biometrics 46, 1005-1016.
- Holmes, E.E., Ward, E.J., and Wills, K. (2012). Marss: Multivariate autoregressive state-space models for analyzing time-series data. The R Journal 4, 11-19.
- Ives, A., Dennis, B., Cottingham, K., and Carpenter, S. (2003). Estimating community stability and ecological interactions from time-series data. Ecological Monographs 73, 301-330.
- 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, 334-345.
- 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 soft-sediments highlight the potential for change in ecological resilience. Ecological Applications 22, 1213-1223.
- 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, 357-359.