NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-23-307-2016A new estimator of heat periods for decadal climate predictions – a complex network approachWeimerMichaelMieruchSebastianSchädlerGerdKottmeierChristophInstitute of Meteorology and Climate Research, Karlsruhe Institute of Technology, Karlsruhe, GermanyAlfred Wegener Institute, Helmholtz Centre for Polar and Marine Research (AWI), Bremerhaven, GermanyM. Weimer (michael.weimer@kit.edu)24August201623430731714July201513October201520July201621July2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/23/307/2016/npg-23-307-2016.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/23/307/2016/npg-23-307-2016.pdf
Regional decadal predictions have emerged in the past few years as a research
field with high application potential, especially for extremes like heat and
drought periods. However, up to now the prediction skill of decadal
hindcasts, as evaluated with standard methods, is moderate and for extreme
values even rarely investigated. In this study, we use hindcast data from
a regional climate model (CCLM) for eight regions in Europe and quantify the
skill of the model alternatively by constructing time-evolving climate
networks and use the network correlation threshold (link strength) as
a predictor for heat periods. We show that the skill of the network measure
to estimate the low-frequency dynamics of heat periods is superior for
decadal predictions with respect to the typical approach of using a fixed
temperature threshold for estimating the number of heat periods in Europe.
Introduction
Decadal prediction is a relatively new field in climate research. Skillful
prediction of climate from years up to a decade would be beneficial for our
society, economy and for a better adaption to a changing climate. Within the
large international Coupled Model Intercomparison Project Phase 5
(CMIP5) global decadal predictions of climate key
variables like temperature and precipitation have been performed with
state-of-the-art Earth system models. In order to validate the prediction
skill of the models so-called hindcast experiments are conducted. That means
the models are initialized with observations (e.g., in 1961) and then run freely
for 10 years and stop at the end of 1970. In 1971, the models are again
initialized and start to run for another 10 years and so on. More advanced
approaches of initializing every year have followed as well. These hindcasts
can be evaluated against observational data to quantify the prediction skill
of the models depending on the lead time, which is the time range between the
initialization and the forecast datum of interest. In recent years, several
studies on decadal predictions have shown the potential of these initialized
(global) model runs
(e.g., ).
However most studies concentrate on regions like the tropical Pacific or
North Atlantic and on slowly evolving variables like sea-surface temperature.
These regions receive their predictability from large-scale processes like
the Atlantic Meridional Overturning Circulation (AMOC) or Pacific Decadal
Oscillation (PDO) and thus allow the extraction of predictable signals out of the
noise. To be useful for society and climate change adaption, regional
climate predictions are required, which should provide skillful forecasts on
smaller regions and shorter periods, and they include climate extreme events on
populated land areas like the European continent. The European climate is
more connected to short-term processes like the North Atlantic Oscillation
(NAO), which is to a certain extent predictable on seasonal scales, whereas
the decadal predictable signal is weak , which has been
shown also for temperature and precipitation in large projects like ENSEMBLES
. Further, the complex orography with the Alps in the
center contributes to a manifold of general weather situations and hence to
a complex climate e.g., World Climate Research Programme Coordinated
Regional Downscaling Experiment for Europe
(CORDEX-EU),. Nevertheless, the European continent
is influenced by the AMOC and thus this process may yield to a certain
predictability, although the signal-to-noise ratio is most probably small. Up
to now, the prediction skill for Europe is weaker than for regions such as
the South Pacific or North Atlantic. have used
a regional decadal hindcast ensemble for Europe and detected moderate
prediction skill for summer and winter temperature and summer precipitation
anomalies within the lead time of 5 years. analyzed the
predictability of temperature and precipitation extremes in a global model
and found a moderate but significant skill (correlation) for seasonal
extremes. They also found skill beyond the first year, but this skill arose
from external forcing. Thus, compared initialized climate
predictions with uninitialized projections to evaluate the skill gained by
initializing and excluding the external forcing. They found that the
“impact of initialization is disappointing”.
Another relatively new field in climate research has been established, namely
the complex climate network approach. The general idea of climate networks is
to consider climate time series, for example, at the grid points of a climate model as
nodes of the network and the statistical connection between the time series
as links of the network. A link between two arbitrary time series
(geolocations) exists if the correlation measure between the time series
exceeds a certain threshold.
The climate network community has been very active in recent years.
proposed “A new dynamical mechanism for major climate
shifts” and explained, e.g., decadal shifts in global mean temperature
. discriminated different El Niño
types using the network approach, developed a network
method to improve El Niño forecasting, and revealed
a connection between (paleo-)climate variability and human evolution using
recurrence networks, which are similar to the complex climate networks.
Generally, it has been shown that climate networks contain useful information
for climate applications, e.g., the relation between climate and topography
found by , dynamics of the sun activity using visibility
graphs , and the prediction of extreme floods
.
In this paper, we exploit the idea to use an alternative heat period
estimator, based on complex climate networks, and show that its skill is
superior to the typical approach of using a fixed temperature threshold for
prediction of heat periods on timescales up to a decade.
In Sect. we introduce the daily maximum temperature data used in
this study and the motivation for our approach in Sect. .
Section describes our approach, which includes the preparation
of the data, the definition of heat periods, and the construction of time-evolving climate networks. The results for applying the new approach to
hindcasts are shown in Sect. . Finally, we give the conclusions
and an outlook in Sect. .
Data
We apply the climate network approach to a decadal prediction ensemble
generated within the German research project MiKlip (Mittelfristige
Klimaprognosen, Decadal Climate Prediction; e.g., ) by the
regional COSMO model in CLimate Mode (COSMO-CLM or CCLM) .
CCLM has been used in numerous studies recently (e.g., in
). A comprehensive overview can be found here:
http://www.clm-community.eu. CCLM has been used to downscale global
decadal predictions from the Earth System Model of the Max Planck Institute
for Meteorology MPI-ESM,. From a suite of different
decadal prediction experiments we have selected the so-called regional
baseline 0 ensemble. This ensemble consists of 10 members, each covering the
period 1961–2010 for the European region according to
CORDEX-EU on a 0.22∘ grid. This ensemble
has already been used by .
The regional baseline 0 ensemble (based on the global MPI-ESM model) has been
initialized every 10 years (1961, 1971, 1981, 1991, 2001). Within a decade
the CCLM model runs freely, except for the prescription of the atmospheric
boundary conditions by the global MPI-ESM model.
More details on the development of the ensemble and the initialization can be
found in , , and .
In the study presented here we use daily maximum near-surface temperatures
from the CCLM model and from the E-OBS v8.0 gridded climatology
for the European continent. The E-OBS data basically are
measurements interpolated to a regular latitude–longitude grid.
For our comparison, we use the so-called Prudence regions
http://prudence.dmi.dk/, namely British Isles, Iberian Peninsula,
France, Central Europe, Scandinavia, Alps, the Mediterranean, and Eastern Europe
(shown in Fig. ).
Motivation
Generally, heat periods are maximum temperature values persisting for several
days and occurring on spatially expanded regions. This means that many
temperature time series (grid points) behave in a “cooperative mode” (see,
e.g., ). This cooperative state can be described by
the link strength, i.e., essentially the correlation between time series, of a
climate network. Thus, the link strength of a climate network could turn out
to be the better heat period estimator for model data, because it is
independent of the typically critical thresholds used in classical extreme
value detection.
The standard estimator for heat periods according to the World Meteorological
Organization (WMO) is that the daily maximum temperature is 5K
above the 1961–1990 mean maximum temperature at five consecutive days at
least . Thus, the standard method to compare the prediction
skill of heat periods between observations and model would be to count the
heat periods, e.g., for each year in an observational reference data set and
similarly in the model data, both according to the WMO definition (cf.
Fig. ).
A crucial problem of the standard estimator for model predictions is the
inherent static threshold used to detect heat periods. Although this
threshold can be adapted to the model climatology (as we do it in
Sect. ) the problem is that it is still likely
that the model slightly undershoots or otherwise slightly misses the
threshold if a heat period according to the definition at the very beginning
of this section occurs, assuming the model exhibits at least some predictive
skill.
To account for this situation in decadal predictions we propose a new method,
based on complex climate networks, to detect heat periods, which is
independent of a fixed temperature threshold (see Sect. ).
Again we want to emphasize that no new method for the detection of heat
periods is needed, if past observational data or short-term forecasts are
used. The WMO-based definition works well. However, the increased uncertainty
in decadal predictions requires new methods to handle climate extremes like
heat periods.
The eight Prudence regions (topography: ETOPO1;
).
Schematical illustration of our approach
(temperature anomaly on the y axis): (a) model detects correctly
one heat period above the threshold, (b) model underestimates the
number of heat periods, and (c) model overestimates the number of heat
periods (for details see text).
(a) Artificial time series including
three heat periods (dashed lines). (b) Relation between the network link
strength and the number of heat periods, based on 100 artificial time
series.
The following schematic examples in Figs. a–c and
illustrate why the complex network approach is able to
detect heat periods without using a temperature threshold. The black curves
represent (artificially generated) daily maximum temperature model data.
Further we assume that one heat period has actually occurred in
Fig. a–c persisting for 15days from day 11 to
day 25. Accordingly the black curves show different possible model results if
the model exhibits predictive skill to detect a signal out of the noise.
Figure a depicts that using the standard approach the model
correctly detects one heat period above the threshold. In
Fig. b the model detects a signal, but this signal is too
weak to cross the threshold; thus no heat period would have been detected and
the model underestimates the number of heat periods. Overestimation of the
number of heat periods happens in Fig. c, where the model
detects two heat periods (5 days above the threshold at the edges and below
the threshold in between). Now, the key point for our motivation is that
a heat period constitutes an event in space and time; thus in a certain
region, many time series would look like the ones in Fig. .
The link strength of a network would be given by the correlation between
these coherent time series. Since the signals in Fig. a–c
look quite similar, the link strength of the network would thus be very
similar in all three cases. Whereas the standard approach would correctly
estimate the heat period in only one case (Fig. a), the
networks' link strength would correctly estimate it in all three cases, given
a proper relation between link strength and heat periods.
To test the relation in principle, we created 100 artificial time series
(Gaussian noise) and included successively 0–9 heat periods.
Figure a shows such a time series with three artificial
heat periods indicated by the dashed lines. In a following step, we
calculated the mean correlation (link strength) between these 100 coherent
time series dependent on the number of included heat periods depicted in
Fig. b. As can be seen, more heat periods are connected
with a larger link strength. This simplified test shows that a proper
relation between link strength and heat periods could exist. Note that
Fig. b is not a calibration curve for real data, because we
simply used Gaussian noise to create the time series.
It is clear that the argumentation above concerning the link strength as
a heat period estimator is quite simplistic, but it elucidates our approach
and the main idea.
Method
Our hypothesis is that complex network measures may be better estimators for
climate extremes than standard measures like absolute threshold exceedances.
Data pre-processing
Before using the complex networks in general it is necessary to remove
stationary biases and long-term variabilities from the climate time series
.
We remove bias, trend, and the average annual cycle by subtracting a standard
linear regression including a Fourier series from the time series according
to
yi(t)=δi+ωit+∑j=12αi,jsin2πj⋅t365.25+βi,jcos2πj⋅t365.25,
where yi(t) represents daily maximum temperature from 1961 to 2010,
δi is the intercept, ωi is the linear trend, and αi,j and βi,j represent the Fourier coefficients.
Equation () is evaluated individually at each grid point
i=1,…,N.
In order to minimize the influence of cold periods on the network approach
(details below in Sect. ), we remove the data lower than
the 10 % quantile. This filtering has no influence on the standard
estimator of heat periods. Then, the months from June to September are
selected because we are interested in summer heat periods.
Ensemble mean variation of the temperature threshold
calculated for heat periods with the standard approach in CCLM data (see
Sect. ).
Prudence region12345678Temperature threshold (in K)3.163.382.812.522.662.853.462.79
These summer anomalies are used for both the standard approach, defined in
Sect. , and the new approach illustrated in
Sect. . We introduce a skill measure to compare the number
of heat periods with values of the link strength
(Sect. ). Finally we present a simple approach to apply
the new estimator to real forecasts in Sect. .
The standard approach for determining the number of heat periods
In this study, we define a heat period for E-OBS observational data as a time
range when the anomaly maximum temperature (according to
Eq. ) exceeds a fixed threshold of 3K at 5
consecutive days at least and additionally includes no less than 20 %
of the grid points in the area of interest. This choice has been made to
observe events frequently enough for reliable statistics while simultaneously
ensuring important impacts.
To account for the inherent model bias it is essential to adjust the
temperature threshold to the model climate. Thus, we estimate the percentile
P3K corresponding to the 3K E-OBS
threshold for the complete time from 1961 to 2010 and the area of interest.
Accordingly, we use this percentile as the threshold for heat periods for the
model data, which is nevertheless fixed for the whole area and time range;
the argumentation of Sect. still holds for the model
data. Table shows this threshold in K for the eight Prudence
regions, estimated from the CCLM ensemble means. In the following, we will
refer to this definition as the standard approach.
The new approach
As an alternative heat period estimator, we propose to use the time-varying
link strength Wτ (τ represents the years) of a network, based on
modeled daily maximum temperature time series. The link strength Wτ
is the correlation threshold between time series, which is needed to
construct a network of a given edge density. Accordingly we want to show that
Wτ has the potential to be a better estimator for observational heat
periods than the standard estimator. This approach is similar to that used by
, who forecasted El Niño events using the link
strength of a network and showed the superiority to standard sea surface
temperature predictions by state-of-the-art climate models. By contrast to
, however, we use the predicted 2m maximum
temperature of CCLM to create the networks and to forecast the number of heat
periods.
To apply the method we proceed as follows. Suppose we have initialized our
climate model in the year 2001 with the ocean, soil, ice, and atmospheric
state at that time. Accordingly the climate model runs freely for 10 years,
i.e., a retrospective decadal climate prediction. Now we are interested in the
capability of the model to represent heat periods in summer. Based on the
standard approach of counting heat periods (see
Sect. ) we could determine the prediction skill of
the model in forecasting (hindcasting) the number of heat periods. Our
approach, in contrast, is to create a time-evolving complex network with
fixed edge density
from the
modeled daily maximum temperature time series and to use, as mentioned, the
dynamics of the link strength Wτ as a heat period estimator.
Following our aim to use a network measure as a heat period estimator we
construct a complex network from the daily maximum temperature model data.
Here we use an undirected and unweighted simple graph. Thus, the network
consists of vertices V, which are the spatial grid points of our
temperature data, and edges (connections) E, which are added between
vertices and represent the statistical interdependence between the anomaly
daily maximum temperature time series. This complex climate network can be
represented by the symmetric adjacency matrix A with
Aij=0ifijnot connected1ifijconnected,
where i and j represent the vertices, i.e., time series at grid points i,j=1,…,N. Two grid points are connected if the correlation between their
time series exceeds a predefined threshold. The statistical interdependence
between pairs {ij} (self-loops {ii} are not allowed) of time series
is measured using the Pearson (standard) correlation coefficient
. From sensitivity studies we found that correlations
between time series on the order of 0.7–0.9 yield patterns with not too few
and not too many connections. This is important in order to resolve temporal
dynamics of the network. Correlations on this order of magnitude are
significant on the 5 % level for the here used summer time series with
length of about 120 days. However, since we want to analyze different regions
in Europe and to generate comparable results we decided to alternatively
create our networks with a constant edge density (ratio of number of actual
connections to maximum number of connections) of
ρ=EN2=〈ki〉(N-1)=0.3,
where E is the number of edges and 〈ki〉 is the mean
node degree with
ki=∑j=1NAij,
which gives the number of connections of a vertex i.
As mentioned above we removed the data lower than the 10 % quantile, to
avoid that the link strength Wτ is influenced by possible cold
periods in the data. We tested smaller quantiles (5 %) and larger
quantiles (20 %) and found that the results are robust: they
changed only slightly. The above used parameters (like the density of 0.3)
and the 10 % filtering turned out to be optimal for our data. For other
data, these parameters most probably have to be adjusted. Additionally, by
removing the data lower than the 10 % quantile, gaps in the time series
are generated. To ensure significance, we take into account only correlation
coefficients where the two underlying time series exhibit 60 common data
points (days). An effective way to estimate the link strength of a network
with an edge density of 0.3 is to calculate the 70 % quantile of all
correlation coefficients involved in the network.
In a similar way as we analyze the temporal variation of
the link strength Wτ, i.e the correlation threshold between time
series (grid points) for a single year τ (summer) from 1961 to 2010.
Thus, instead of using the node degree as an estimator of heat
periods we use the link strength Wτ.
Using the definitions above, we finally construct a network for the summer
months of each year based on anomaly maximum temperature model data. The
quantity whose year-to-year variation we are interested in is the link
strength Wτ; however, since we are interested in decadal
variability, and since we do not expect the model to represent the year-to-year fluctuations, we applied a 10-year moving average filter to both
link strength and number of heat periods, subsequently. Since the CCLM model
has been initialized every decade (1961, 1971,…, 2001) we apply the filter
only within a decade in order to avoid transferring information between
decades. At the boundaries of the decades, the time range for the running
average is shortened: for instance at the beginning of the decade, we use
only the 6-year mean (e.g., from 2001 to 2005), in the second year a 7-year mean,
and so on.
Comparison of the different quantities
To quantify the prediction skill of the model, we calculate the absolute mean
difference (see Eqs. and ) between the number of heat
periods in E-OBS (o) and CCLM (m) and the CCLM link strength
(Wτ). To be comparable we normalized the time series to the range
{0,1} by a subtraction of the minimum of the time series and accordingly
a division by the maximum for the whole time span, e.g., for the number of
heat periods in CCLM:
μd,τr=md,τr-minτ=150md,τrmaxτ=150md,τr-minτ=150md,τr,
where r denotes the European region, d stands for the decade and τ
represents the years. The similarly rescaled E-OBS number of heat periods
will be denoted as Ω and the rescaled CCLM link strength as ψ.
Thus the absolute mean difference (based on normalized data) between
observation and model heat periods for a region r and a decade d is given
by
Mdr(μ)=110∑τ=110Ωd,τr-μd,τr=Ω‾dr-μ‾dr,
and the mean difference between observation heat periods and model link
strength is
Mdr(ψ)=110∑τ=110Ωd,τr-ψd,τr=Ω‾dr-ψ‾dr,
where the bars in the above equations denote temporal averages. Therefore, if
the absolute mean difference is about 0, observations and model agree well,
whereas a difference of about 1 denotes the maximum discrepancy.
Usage of the new estimator in predictions
For a real application of our method to estimate the number of heat periods
in forecasts, a calibration step using observational data o is needed to
convert the link strength of the model to the number of heat periods my
(the index y stands for year in the future). Therefore long hindcast data
are needed. Based on our analysis we suggest as a first attempt to apply a
linear conversion from link strength Wy to the number of heat periods
my, which is also supported by our tests shown in Fig. :
my,Wr=Wyr-minτ=150(Wτr)maxτ=150(Wτr)-minτ=150(Wτr)⋅maxτ=150(oτr)-minτ=150(oτr)+minτ=150(oτr).
This linear approach corresponds to our skill analysis, where a linear
connection between the link strength and the number of heat periods is
assumed as well. Again we note that this study presents only the skill
analysis of hindcast data and Eq. () is actually not used
now.
Number of heat periods (1961–2010) in France
(Prudence 3) in summer from E-OBS o (solid line) and corresponding E-OBS
link strength W (dashed line). The M's denote the absolute mean
difference within a decade between E-OBS standard approach and the E-OBS link
strength after normalization (cf. Eq. ).
Results
Figure depicts the number of observed heat periods (solid
line) and the corresponding link strength (dashed line) retrieved from the
complex evolving network, both from E-OBS data for France (Prudence region 3),
and shows that the link strength Wτ is a suitable estimator of
heat periods. It shows that the network contains climate information in the
sense that the dynamics of the link strength Wτ is similar to the
dynamics of heat periods, both based on the same data. So, the link strength
can here be considered as an estimator for heat periods, which is comparable
to the standard heat period estimator. Jumps between the decades occur as the
running mean filter is only applied within the decades (see
Sect. ). The corresponding figures for the seven other
Prudence regions can be found in the supplementary material.
As an example, Prudence region 8 (Eastern Europe) is a region where the
network method performs better than the standard approach
(Figs. and ). Figure
shows the E-OBS number of heat periods o (black) and the CCLM ensemble mean
number of heat periods m (blue) for Eastern Europe together with the
interquartile range (25th and 75th percentiles), and
Fig. shows again the E-OBS number of heat periods now
compared to the CCLM link strength. Comparing the absolute mean differences,
denoted as M in the two figures, reveals that our network approach enhances
the skill in four decades, namely 1970s, 1980s, 1990s, and 2000s. Especially the
1970s, 1980s, and 1990s show a clear improvement and our network approach better
reflects the low-frequency dynamics of the heat periods. The 2000s seem to be
off in both model cases, the number of heat periods, and the link strength,
which indicates a failed model initialization.
Number of heat periods (1961–2010) in Eastern
Europe (Prudence 8) in summer from E-OBS o (black) and CCLM number of heat
periods (blue: ensemble mean and interquartile range). The M's denote the
absolute mean difference within a decade between E-OBS and the CCLM ensemble
mean after normalization (see Eq. ).
Number of heat periods (1961–2010) in Eastern
Europe (Prudence 8) in summer from E-OBS o (black) and CCLM link strength
or correlation threshold W (red: ensemble mean and interquartile range).
The M's denote the absolute mean difference within a decade between E-OBS
and the CCLM ensemble mean after normalization (see Eq. ).
Absolute mean differences like in
Figs. to of all Prudence regions for
E-OBS standard approach compared to E-OBS link strength (left), to CCLM
standard approach (middle), and to CCLM link strength (right).
In order to see how the prediction skill of the standard as well as the
network heat period estimators vary with the considered region, we performed
the same analysis as above for the eight Prudence regions in Europe and for
the 1960s, 1970s, 1980s, 1990s, and 2000s. The corresponding figures for the
other regions can be found in the Supplement.
To summarize the results we calculated the absolute mean differences (Eqs. 6
and 7) for all the Prudence regions, see Fig. . Blue colors
in the panels stand for low values (high skill) whereas red colors depict
high values (low skill) in the absolute mean difference.
The left panel of Fig. shows how the network method
performs using only E-OBS data similar to Fig. . Therefore
we estimated the number of heat periods in the E-OBS data and the link
strength (from the complex network) of the E-OBS data and accordingly
calculated the differences (after normalization, cf. Eq. )
between these two estimators. As can be seen blue colors dominate the plot,
i.e., low differences and hence high skill. Thus, this reference test shows
that the link strength is coupled to the number of heat periods in maximum
daily summer temperature data and so can be used as an alternative, possibly
better, heat period estimator. There are some exceptions like the 1990s and
2000s of Prudence region 6. Further investigation on the reasons of these
cases has to be performed.
The middle panel of Fig. shows how well the standard
method performs in predicting heat periods using E-OBS observations and CCLM
model data. The right panel indicates the performance of the new network
method in estimating heat periods using E-OBS and CCLM data. In contrast to
the relatively low values in the left panel, the values in the middle and
right panels on the one hand are higher for many decades and Prudence
regions. On the other hand, the visual impression is that the absolute
differences of the right panel are slightly smaller than those of the middle
panel, especially during the 1990s and 2000s.
Thus, we can conclude with Fig. that the method works in
principle but that the uncertainties in the model simulations lead to
increased differences between observations and model simulations. In
addition, the link strength seems to work better than the standard approach
for the model simulations with respect to the observations.
Differences between the two right panels of
Fig. . Blue: network approach performs better, red:
standard approach performs better, white: tie.
To quantify this last statement, we calculated the difference between the
middle and right panels of Fig. (see
Fig. ). This basically shows which method performs better
regarding the eight regions (columns) and five decades (rows). Blue color in
Fig. indicates that the network approach performs better
(Mdr(ψ)<Mdr(μ)) and red color stands for a better
performance of the standard approach (Mdr(ψ)>Mdr(μ)).
White boxes in Fig. denote a tie between the methods in the
case of too small differences (|Mdr(ψ)-Mdr(μ)|≤0.05).
The matrix of Fig. shows that the network method is clearly
superior in three regions (5, 7, 8) and slightly superior in two regions (4,
6), the standard approach is superior in two regions (1, 3), and in region 2
we observed a tie, i.e., no clear result.
The crucial question is whether this result indicates that the network method
performs significantly better than the standard approach or not. However,
testing for statistical significance bears serious problems. There are so
many factors involved in the analysis, i.e., the models themselves, the
downscaling, the ensemble, the initialization, the different regions, the
filtering, etc. that any null hypothesis would be not well-posed and any test
would be questionable. This issue is discussed in detail in a 2013 paper
entitled “Testing ensembles of climate change scenarios for `statistical
significance”' by climate statistics instances of Hans von Storch and Francis
Zwiers , who claim that “a statistical
null hypothesis may not be a well-posed problem…” and “Even if
statistical testing were completely appropriate, the dependency of the power
of statistical tests on the sample size n remains a limitation on
interpretation.” and finally “propose to employ instead a simple
descriptive approach for characterising the information in an ensemble…”.
Although we totally agree with the argumentation by
that a “classical” significance test would most probably fail in our
analysis, we think that alternative significance tests, based on
bootstrapping or surrogate data, could definitely help obtain a better
interpretation of the results. Thus, we construct the following significance
test based on surrogate data to answer the following question: what is the
probability of getting a rank matrix like the one in Fig.
by chance?
Probability that blue matrix elements in
Fig. dominate in n regions by chance. Half of the
elements are colored blue and red, respectively, and eight white elements are
randomly added subsequently.
Number of regions n12345678Probability in %10099823550.100
First, we have to define what is the possibly “significant” characteristic
of the matrix in Fig. . It is, as we concluded above, that
the network method is superior in five regions. Thus the question is the following: what
is the probability to observe at least five regions in which we have in each at
least one blue matrix element more than a red one by chance?
Accordingly we constructed matrices like in Fig. by
randomly coloring 20 matrix elements blue and 20 red. Afterwards we colored
8 matrix elements white as in Fig. . Finally, we repeated
this surrogate procedure 1000 times and counted the cases (regions) where the
blue matrix elements dominate. Table shows the probabilities that
blue matrix elements dominate in n regions. Since we have 16 blue elements
and 16 red, it is sure that blue dominates in n=1 region, and it is impossible to
dominate in n=7 and n=8 regions. As can be seen from Table the
probability of dominating in n=5 regions by chance is only about 5 %;
thus the results of our network approach have to be stated significant. Due
to the symmetry of the test, the same argumentation is valid for red matrix
elements. Dominating in n=2 regions, as achieved by the standard approach
(Fig. ), can be realized easily by chance with a probability
of approx. 99 %. Page 1 in the Supplement shows an example of 12 of
these randomly generated matrices, where one matrix, depicted by a black
frame, fulfills the “significance” criterion.
Conclusions and outlook
We presented a novel approach examining heat periods using a complex network
analysis. We have investigated the predictability of the slow dynamics of the
occurrence of heat periods in Europe based on daily maximum near-surface
temperature data.
We found that the network approach is superior (significance is ≈ 5 %)
to the standard approach in estimating heat periods in Europe,
hence highlighting the potential of network methods to improve the skill
estimation in decadal prediction experiments. Picking up our hypothesis and
simplified argumentation from Sect. , the crucial point
why we detect heat periods with the network link strength is that heat
periods are cooperative events in space and time. Thus, the link strength can
be used as an estimator of heat periods. The drawback of the standard
approach is most probably the inflexible threshold for the detection of heat
periods (cf. Fig. ). If the climate model contains the
signal of a heat period, but with a slightly too small amplitude, the
threshold will not be crossed and no heat period will be detected. In
contrast, the complex climate network does not depend on such fixed
thresholds and can use this information, which makes it the more robust
estimator of heat periods.
The general prediction skill of climate in Europe using standard measures is
still moderate. In this sense our work adds new aspects to our previous study
and also the work of , who found
a strong variation of skill with region and decade. In essence, we found
regions and decades in Europe where our climate model output, or more
specifically the used network estimator, follows the slowly evolving dynamics
of observed heat periods. We also found regions and decades in which the
network estimator is not able to represent the observational reference.
Understanding of this variability in prediction skill is one of the future
challenges of decadal predictions.
Concluding, our approach shows that the complex climate networks approach
yields meaningful climate information and has the potential to improve skill
measures within the framework of climate prediction. It is the first time
that such network techniques have been used in climate predictions. Since
climate or decadal predictions aim to predict natural variability on the
order of years, suitable statistics are needed. Natural variability on the
order of years evolves highly dynamically and often nonlinearly. Thus, the
complex climate networks could bear the potential to be very useful in
climate predictions. Our approach, which is even based on the most simple
network measure, the node degree (or as we used it the link strength), yields
optimistic results. So, we think that our analysis could be the starting
point for using the complex networks in climate predictions. Using other
measures and/or multivariate data could turn out to be the better way of
analyzing predictions of natural variability years ahead than using methods
from short- or medium-range forecasting. Further, from the network
perspective it would be interesting to analyze other network measures like
clustering, similarities, or path lengths and how they are connected to
climate evolution. The incorporation of other relevant variables like
precipitation, wind, or soil moisture into the network is an appealing aspect.
From a physical or climatological point of view it is important to understand
why the network measures are able to represent climate dynamics, which could
also contribute to a better understanding of the sources of decadal
predictability. Thus, the incorporation and investigation of processes like
the AMOC, PDO, or NAO together with complex networks and climate prediction
might be an option for the future.
Data availability
E-OBS can be downloaded via the website www.ecad.eu.
The Supplement related to this article is available online at doi:10.5194/npg-23-307-2016-supplement.
Acknowledgements
We acknowledge the E-OBS data set from the EU-FP6 project ENSEMBLES
(http://ensembles-eu.metoffice.com) and the data providers in
the ECA&D project (http://www.ecad.eu).
The research programme MiKlip is funded by the German Ministry of
Education and Research (BMBF).
We also acknowledge the ETOPO1 data set from the
NGDC (National Geophysical Data Center, Boulder, Colorado).The
article processing charges for this open-access
publication were covered by a Research Centre of the Helmholtz
Association.Edited by: A. M. Mancho
Reviewed by: four anonymous referees
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