This tutorial introduces how to perform both univariate and
multivariate early warning signal (EWS) assessments using
EWSmethods
. It will give examples of rolling and expanding
window approaches for univariate data, introduce traitbased composite
EWSs and then conclude with an example of multivariate EWSs.
Greater detail on each function can be found at the Reference page.
EWSmethods
comes bundled with two data objects which
allow you to practice using the uniEWS()
and
multiEWS()
functions in both transitioning and
nontransitioning data before applying it to your own use case.
"simTransComms"
contains three replicate datasets of a
simulated five species community that has been driven to transition by
the introduction of an invasive species (following Dakos 2018). This
will be our multivariate dataset when using multiEWS()
although we may also use each time series in isolation in
uniEWS()
.
"CODrecovery"
contains three replicate datasets of a
simulated cod ( Gadus morhua ) population that transitions from
an overfished to a recovered state following the relaxation of fishing
pressure. This data was first published by Clements et al.
2019. While univariate, "CODrecovery"
provides extra
information on the body size of cod individuals which will improve
composite EWSs estimated by uniEWS()
.
We can visualise a community from each of these datasets using the code below:
matplot(simTransComms$community1[,3:7], type = "l", xlab = "Time", ylab = "Density", main = "Transitioning five species community")
plot(x = CODrecovery$scenario2$time, y = CODrecovery$scenario2$biomass, type = "l", xlab = "Year", ylab = "Abundance", main = "Recovering cod population")
These plots show that a transition takes place at
time ~= 180
in "simTransComms$community1"
and
year ~= 2050
in "CODrecovery$scenario2"
.
EWSmethods
helpfully provides this information in each
dataset under the inflection_pt
column.
simTransComms  CODrecovery 

191.5  2055 
However, EWS assessments are only meaningful if performed on data
prior to a transition. As EWsmethods
provides the time
point of transition for both datasets, we can truncate our time series
to pretransition data only.
pre_simTransComms < subset(simTransComms$community1,time < inflection_pt)
pre_CODrecovery < subset(CODrecovery$scenario2,time < inflection_pt)
In reality, EWSs will be assessed in realtime with the presence of past/present tipping points often unknown. If past transitions are known to have occurred, it may be prudent to follow the suggestions of O’Brien & Clements (2021) who show that the occurrence of a historic transition can mask an oncoming event and that only using data post the historic transition improves EWS reliability.
Now the data has been loaded and truncated, it can now be passed to
uniEWS()
and multiEWS()
to perform EWS
assessments.
EWSmethods
provides two computational approaches to
calculate univariate EWSs via the uniEWS()
function 
rolling vs expanding windows. The difference between the two is evident
in the figure below but simply, rolling windows estimate EWSs in subsets
of the overall time series before ‘rolling’ on one data point and
reassessing, Conversely, expanding windows add data points sequentially
in to an ‘expanding’ assessment and then standardises against the
running mean and standard deviation of the previous window.
Both computational approaches are able to calculate the same EWS
indicators. A brief outline of each can be found in the following table
as well as their reference code in uniEWS()
for the
metrics =
argument.
EWS indicator  Description  uniEWS() metric code 

Standard deviation  Increasing variance/standard deviation is observed approaching a transition  "SD" 
Coefficient of variation  Equivalent to SD as is simply SD at time t divided by the mean SD of the time series  "cv" 
Autocorrelation at lag1  Autocorrelation (similarity between successive observations) increases approaching a transition. The value of this indicator can be estimated as either the autocorrelation coefficient estimated from a first order autoregressive model or the estimated autocorrelation function at lag1  "ar1"  autoregressive model,
"acf"  autocorrelation function 
Skewness  At a transition, the distribution of values in the time series can become asymmetric  "skew" 
Kurtosis  Kurtosis represents the system reaching more extreme values in the presence of a transition. Due to the increased presence of rare values in the time series, the tails of the observation distribution widen  "kurt" 
Return rate  The inverse of the firstorder term of a fitted autoregressive AR(1) model. Return rate is the primary quantity impacted by CSD – return rate decreases as a tipping point is approached  "rr" 
Density ratio  Spectral reddening (high variance at low frequencies) occurs near transition. The density ratio quantifies the degree of reddening as the ratio of the spectral density at low frequency to the spectral density at high frequency  "dr" 
The rolling window approach is the most commonly used form of EWS
computation due to the work of Dakos et al 2012 and the
earlywarnings
package. uniEWS()
accepts a
method =
and a winsize =
argument which calls
the expanding window method and creates a rolling window
winsize
% of the total time series length.
Lets use an example where we are interested in the autocorrelation,
variance and skewness of one of the five species in
pre_simTransComms
. First we supply a dataframe/matrix of n
x 2 dimensions (first column is an equally time sequence and the second
is the abundance/biomass time series) and the EWS indicator metrics. The
remaining arguments specify the form of computation, window size and
plotting characteristics.
rolling_ews_eg < uniEWS(data = pre_simTransComms[,c(2,5)],
metrics = c("ar1","SD","skew"),
method = "rolling",
winsize = 50)
Note how all EWS indicators begin to trend upwards at
time ~= 170
which results in the positive Kendall Tau
correlation coefficient indicative of an oncoming transition/tipping
point.
Let’s explore the alternative expanding window approach. All we need
to change in uniEWS()
is the method =
argument, and replace winsize =
with
burn_in =
. Instead of specifying the size of the rolling
window, burn_in =
dictates the number of datapoints
uniEWS()
is to use to ‘train’ the algorithm. This mitigates
the high number of falsepositive signals resulting from the short time
series length and high variability when few data points are supplied at
the beginning of assessment (O’Brien & Clements, 2021).
expanding_ews_eg < uniEWS(data = pre_simTransComms[,c(2,5)],
metrics = c("ar1","SD","skew"),
method = "expanding",
burn_in = 50,
threshold = 2)
Similar to the rolling window approach, EWS indicators calculated by
expanding windows have exceeded the 2σ threshold for more than two
consecutive time points and thus identified warnings from
time ~= 170
. However we are more confident in this
conclusion as the composite metrics also display this warning
(ar1 + SD + skew
). These composite metrics simple sum the
standardised individual indicator strengths together and are known to
provide a more reliable signal than lone indicators (Clements &
Ozgul, 2016).
The final contribution by uniEWS()
is the ability to
integrate multiple information sources in the assessment. For example,
including body size estimates improves assessment reliability by
reducing false positive rate whilst increasing the number of true
positives (Clements and Ozgul 2016, Baruah et al. 2020).
uniEWS()
consequently accepts a trait =
argument where an additional trait time series can be combined with the
other abundancebased EWSs as a composite metric. This capability is
only available if method = "expanding"
and
metrics =
contains "trait"
trait_ews_eg < uniEWS(data = pre_CODrecovery[,c(2,3)],
metrics = c("ar1","SD","trait"), #note "trait" is provided here
method = "expanding",
trait = pre_CODrecovery$mean.size, #and here
burn_in = 15, #small burn_in due to shorter time series
threshold = 2)
A more powerful and informative form of EWS are multivariate EWSs. These indicators combine multiple time series/measurements of the focal system to provide a community level assessment of transition risk.
There are two primary forms of multivariate EWS, those which are
averaged univariate EWS across all time series and those which are
assessments made on a dimension reduction of all representative time
series. A brief outline of each can be found in the following table as
well as their reference code in multEWS()
for the
metrics =
argument. See Weinans et al. (2021) for
a rigorous testing of these multivariate EWSs in a simulated system.
Multivariate EWS indicator  Description  multiEWS() metric code 
Average or dimension reduction technique 

Mean standard deviation  Average variance across all time series representing the system  "meanSD" 
Average 
Max standard deviation  The variance of the time series with the highest variance of all assessed time series  "maxSD" 
Average 
Mean autocorrelation at lag1  Average autocorrelation across all time series representing the system  "meanAR" 
Average 
Max autocorrelation at lag1  The autocorrelation of the time series with the highest autocorrelation of all assessed time series  "maxAR" 
Average 
Dominant MAF (maximum autocorrelation factor) eigenvalue  The minimum eigenvalue of the system following MAF dimension reduction  "eigenMAF" 
Dimension reduction 
MAF autocorrelation  The autocorrelation of the data projected on to the first MAF – i.e. the autocorrelation of the first MAF.  "mafAR" 
Dimension reduction 
MAF standard deviation  The variance of the data projected on to the first MAF – i.e. the variance of the first MAF  "mafSD" 
Dimension reduction 
First PC (principal component) autocorrelation  The autocorrelation of the data projected on to the first PC – i.e. the autocorrelation of the first PC  "pcaAR" 
Dimension reduction 
First PC standard deviation/ Explained variance  The variance of the data projected on to the first PC – i.e. the variance of the first PC  "pcaSD" 
Dimension reduction 
First PC standard deviation/ Explained variance  The variance of the data projected on to the first PC – i.e. the variance of the first PC  "pcaSD" 
Dimension reduction 
Dominant eigenvalue of the covariance matrix  The maximum eigenvalue of the covariance matrix between all representative time series  "eigenCOV" 
Neither 
Maximum covariance  The maximum value of the covariance matrix between all representative time series.  "maxCOV" 
Neither 
Mutual information  The ‘amount of information’ one time series contains on another.  "mutINFO" 
Neither 
Using multiEWS()
we can estimate each of these
multivariate indicators in the same way as uniEWS()

specifying the method =
, winsize =
and/or
burn_in =
 but must provide a n x m dataframe/matrix of
all representative time series. The first column must again be an
equally spaced time sequence.
A rolling window assessment would therefore be coded as such:
multi_ews_eg < multiEWS(data = pre_simTransComms[,2:7],
metrics = c("meanAR","maxAR","meanSD","maxSD","eigenMAF","mafAR","mafSD","pcaAR","pcaSD","eigenCOV","maxCOV","mutINFO"),
method = "rolling",
winsize = 50)
Many of these indicators are postively correlated with time and therefore are ‘warnings’.
We could also use expanding windows to achieve a similar result.
Note  no composite metric is computed in
multiEWS()
as it is currently unknown how combining
multivariate EWS indicators influences prediction reliability.