Compression complexity with ordinal patterns for robust causal inference in irregularly sampled time series

Abstract

Distinguishing cause from effect is a scientific challenge resisting solutions from mathematics, statistics, information theory and computer science. Compression-Complexity Causality (CCC) is a recently proposed interventional measure of causality, inspired by Wiener–Granger’s idea. It estimates causality based on change in dynamical compression-complexity (or compressibility) of the effect variable, given the cause variable. CCC works with minimal assumptions on given data and is robust to irregular-sampling, missing-data and finite-length effects. However, it only works for one-dimensional time series. We propose an ordinal pattern symbolization scheme to encode multidimensional patterns into one-dimensional symbolic sequences, and thus introduce the Permutation CCC (PCCC). We demonstrate that PCCC retains all advantages of the original CCC and can be applied to data from multidimensional systems with potentially unobserved variables which can be reconstructed using the embedding theorem. PCCC is tested on numerical simulations and applied to paleoclimate data characterized by irregular and uncertain sampling and limited numbers of samples.

Full text

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Compression complexity
with ordinal patterns for robust
causal inference in irregularly
sampled time series
Aditi Kathpalia, Pouya Manshour & Milan Paluš*
Distinguishing cause from eect is a scientic challenge resisting solutions from mathematics,
statistics, information theory and computer science. Compression-Complexity Causality (CCC) is a
recently proposed interventional measure of causality, inspired by Wiener–Granger’s idea. It estimates
causality based on change in dynamical compression-complexity (or compressibility) of the eect
variable, given the cause variable. CCC works with minimal assumptions on given data and is robust to
irregular-sampling, missing-data and nite-length eects. However, it only works for one-dimensional
time series. We propose an ordinal pattern symbolization scheme to encode multidimensional
patterns into one-dimensional symbolic sequences, and thus introduce the Permutation CCC (PCCC).
We demonstrate that PCCC retains all advantages of the original CCC and can be applied to data from
multidimensional systems with potentially unobserved variables which can be reconstructed using
the embedding theorem. PCCC is tested on numerical simulations and applied to paleoclimate data
characterized by irregular and uncertain sampling and limited numbers of samples.
Unraveling systems’ dynamics from the analysis of observed data is one of the fundamental goals of many areas of
natural and social sciences. In this respect, detecting the direction of interactions or inferring causal relationships
among observables is of particular importance that can improve our ability to better understand the underlying
dynamics and to predict or even control such complex systems1,2.
Around sixty years aer the pioneering work of Wiener and Granger3,4 on quantifying linear ‘causality’ from
observations, it has been widely applied not only in economics5–7, for which it was rst introduced, but also in
various elds of natural sciences, from neurosciences8 to Earth sciences9–11. A number of attempts have been
made to generalize Granger Causality (GC) to nonlinear cases, using, e.g., an estimator based on correlation
integral6, a non-parametric regression approach12, local linear predictors13, mutual nearest neighbors14,15, kernel
estimators16, to state a few. Several other causality methods based on the GC principle such as Partial Directed
Coherence17, Direct Transfer Function18 and Modied Direct Transfer Function19 have also been proposed.
Information theory has proved itself as a powerful approach into causal inference. In this respect, Schreiber
proposed a method for measuring information transfer among observables20, known as Transfer Entropy (TE),
which is based on Kullback-Leibler distance between transition probabilities. Paluš etal.21 introduced a causality
measure based on mutual information, called Conditional Mutual Information (CMI). CMI has been shown to
be equivalent to TE22. ese tools have been applied in various research studies and have shown their power in
extracting causal relationships between dierent systems23–27.
We usually work with time series x(t) and y(t) as realizations of m and n dimensional dynamical systems,
X(t) and Y(t) respectively, evolving in measurable spaces. It means that x(t) and y(t) can be considered as the
components of these m and n dimensional vectors. In many cases, only one possible dimension of the phase
space is observable, recordings or knowledge of variables which may have indirect eects or play as mediators
in the causal interactions between observables may not be available. In this respect, phase-space reconstruction
is a common useful approach introduced by Takens28, which reconstructs the dynamics of the entire system
(including other unknown/unmeasurable variables) using time-delay embedding vectors, as follows: the manifold
of an m dimensional state vector X can be reconstructed as
X(t)={x(t),x(t−η), ..., x(t−(m−1)η)}
. Here,
η
is the embedding delay, and can be obtained using the embedding construction procedure based on the rst
minimum of the mutual information29. Some causality estimators have applied this phase-space reconstruction
OPEN
Department of Complex Systems, Institute of Computer Science of the Czech Academy of Sciences, 182 07 Prague,
Czech Republic. *email: mp@cs.cas.cz
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procedure to improve their causal inference power, such as high dimension CMI26 and TE30. Other causality
measures, such as, Convergent Cross Mapping (CCM)31, Topological Causality32, Predictability Improvement33,
are based directly on the reconstruction of dynamical systems.
Vast amounts of data available in the recent years have pushed some of the above discussed GC extensions,
information and phase-space reconstruction based approaches forward as they rely on joint probability density
estimations, stationarity, markovianity, topological or linear modeling. However, still, many temporal observa-
tions made in various domains such as climatology34,35, nance36,37 and sociology38 are oen short in length, have
missing samples or are irregularly sampled. A signicant challenge arises when we attempt to apply causality
measures in such situations11. For instance, CMI or TE fail when applied to time series which are undersampled
or have missing samples39–41 and also in case of time series with short lengths41. CCM and kernel based non-
linear GC also show poor performance even in the case of few missing samples in bivariate simulated data42.
Kathpalia and Nagaraj recently introduced a causality measure, called Compression-Complexity Causality
(CCC), which employs ‘complexity’ estimated using lossless data-compression algorithms for the purpose of
causality estimation. It has been shown to have the strength to work well in case of missing samples in data for
bivariate systems of coupled autoregressive and tent map processes. is has been shown to be the case for sam-
ples which are missing in the two coupled time series either in a synchronous or asynchronous manner41. Also, it
gives good performance for time series with short lengths41,42. ese strengths of CCC arise from its formulation
as an interventional causality measure based on the evolution of dynamical patterns in time series, independence
from joint probability density functions, making minimal assumptions on the data and use of lossless compres-
sion based complexity approaches which in turn show robust performance on short and noisy time series41,43.
However, as discussed inRef.42, a direct multidimensional extension of CCC is not as straightforward and so a
measure of eective CCC has been formulated and used on multidimensional systems of coupled autoregressive
processes with limited number of variables.
On the other hand, a method for symbolization of phase-space reconstructed (embedded) processes has
been used to improve the ability of info-theoretic causality measures for noisy data, such as symbolic trans‑
fer entropy44,45, partial symbolic transfer entropy46,47, permutation conditional mutual information (PCMI)48 and
multidimensional PCMI49. e symbolization technique used in these works is based on the Bandt and Pompe
scheme for estimation of Permutation Entropy50, and oen referred to as permutation or ordinal patterns coding.
e scheme labels the embedded values of time-series at each time point in ascending order of their magnitude.
Symbols are then assigned at each time point depending on the ordering of values (or the labelling sequence)
at that point. Ordinal patterns have been used extensively in the analysis and prediction of chaotic dynamical
systems and also shown to be robust in applications to real world time series. By construction, this technique
ignores the amplitude information and thus decreases the eect of high uctuations in data on the obtained causal
inference51. Other benets of permutation patterns are: they naturally emerge from the time series and so the
method is almost parameter-free; are invariant to monotonic transformations of the values; keep account of the
causal order of temporal values and the procedure is computationally inexpensive52–55. Ordinal partition has been
shown to have the generating property under specic conditions, implying topological conjugacy between phase
space of dynamical systems and their ordinal symbolic dynamics56. Further, permutation entropy for certain sets
of systems has been shown to have a theoretical relationship to the system’s Lyapunov exponents and Kolmogorov
Sinai Entropy57,58. Because of all these benecial properties of permutation patterns, it is no wonder that the
development of symbolic TE or PCMI helped to make them more robust, giving better performance in the case
of noisy measurements, simplifying the process of parameter selection and making less demands on the data.
In this work, we propose the use of CCC approach with reconstructed dynamical systems which are symbol-
ized using ordinal patterns. e combination of strengths of CCC and ordinal patterns, not only makes CCC
applicable to dynamical systems with multidimensional variables, but we also observe that the proposed Per‑
mutation CCC (PCCC) measure gives great performance on datasets with very short lengths and high levels of
missing samples. e performance of PCCC is compared with that of PCMI (which is identical to symbolic TE),
bivariate CCC and CMI on simulated dynamical systems data. PCCC outperforms the existing approaches and
its estimates are found to be robust for short length time series, and high levels of missing data points.
is development for the rst time opens up avenues for the use of causality estimation tool on real world
datasets from climate and paleoclimate science, nance and other elds where there is prevalence of data with
irregular and/or uncertain sampling times. To determine the major drivers of climate is the need of the hour as
climate change poses a big challenge to humankind and our planet Earth59. Dierent studies have employed either
correlation/coherence, causality methods or modelling approaches to study the interaction between climatic
processes. e results produced by dierent studies are dierent and sometimes contradictory, presenting an
ambiguous situation. We apply PCCC to analyse the causal relationship between the following sets of climatic
processes: greenhouse gas concentrations—atmospheric temperature, El-Niño Southern Oscillation—South
Asian monsoon and North Atlantic Oscillation—European temperatures at dierent time-scales and compare
its performance with bivariate CCC, bivariate and multidimensional CMI, and PCMI. e time series avail-
able for most of these processes are short in length and sometimes have missing samples and (or) are sampled
in irregular intervals of time. We expect our estimates to be reliable and to be helpful to resolve the ambiguity
presented by existing studies.
Results
Simulation experiments. Time series data from a pair of unidirectionally coupled Rössler systems were
generated as per the following equations:
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for the autonomous or master system, and
for the response or slave system. Parameters were set as:
a1=a2=0.15
,
b1=b2=0.2
,
c1=c2=10.0
, and fre-
quencies set as:
ω1=1.015
and
ω2=0.985
. e coupling parameter,
ǫ
, was xed to 0.09. e data were generated
by numerical integration based on the adaptive Bulirsch–Stoer method60 using a sampling interval of 0.314 for
both the master and slave systems. is procedure gives 17–21 samples per one period. 100 realizations of these
systems were simulated and initial 5000 transients were removed before using the data for testing experiments.
As can be seen from the equations, there is a coupling between
x1
and
x2
, with
x1
inuencing
x2
. e analysis
of the causal inuence between the two systems was done using the causality estimation measures: bivariate or
scalar CCC, CMI, PCCC and PCMI for the cases outlined in the following paragraphs. e estimation procedure
for each of the methods is described in the “Methods” section. e values of parameters used for each of the
methods are also given in the “Methods” section (Table2).
Finite length data. e length of time series, N, of
x1
and
x2
taken from coupled Rössler systems was varied as
shown in Fig.1. e estimation for CMI and PCMI is done up to a higher value of length as CMI did not give
optimal performance until the length became 32,768 samples. Figure1c shows scalar (simple bivariate) CMI or
one-dimensional CMI (CMI1) between
x1
and
x2
(see Paluš and Vejmelka22). is method has high sensitivity
but suers from low specicity. is problem is solved by using conditional CMI or three-dimensional CMI
(CMI3), where the information from other variables (
y1,z1,y2,z2
) is incorporated in the estimation. Its perfor-
mance is depicted in Fig.1e. However, it requires larger length of time series for optimal performance. Figure1a
shows the performance of scalar (or simple bivariate) CCC, which is equivalent to the CMI1 case, considering
dimensionality. Figure1b, d show the performance of PCCC and PCMI respectively. For each length level, all
100 realizations of coupled systems were considered and 100 surrogates generated for each realization in order
to perform signicance analysis of causality estimated (in both directions) from each realization of coupled
processes. ese surrogates were generated for both the processes using the Amplitude Adjusted Fourier Trans-
form method61 and signicance testing done using a standard one-sided z-test with p-value set to 0.05 (this was
justied as the distributions of surrogates for CCC and CMI methods implemented were found to be Gaussian).
Based on this signicance analysis, true positive rate (TPR) and false positive rate (FPR) were computed at each
length level. A true positive is counted for a particular realization of coupled systems when causality estimated
from
x1
to
x2
is found to be signicant and a false positive is counted when causality estimated from
x2
to
x1
is
found to be signicant.
As it can be seen from the plots, direct application of scalar CCC completely fails on multidimensional
dynamical systems data, yielding low true positives and high false positives. Hence the method displays poor
sensitivity as well as specicity. CMI1 also shows poor performance, yielding high false positives. CMI3, which
is appropriate to be applied for multi-dimensional data, only begins to give good performance when the length
of time series is taken to be greater than 32,768 samples. On the other hand, PCCC begins to give high true
positives and low false positives, as the length of time series is increased to 1024 time points, with TPR and FPR
reaching almost 1 and 0 respectively as length is increased to 2048 time points. e use of permutation patterns
also improves the performance of CMI3 for short length data as it can be seen that PCMI begins to show a TPR
of 1 and FPR of 0 for length of time series equal to 2048 time points.
We did further experiments with simulated Rössler data by varying the amount of noise and missing samples
in the data. For these cases, performance of PCCC and PCMI alone were evaluated because it can be seen from
the ‘varying length’ experiments that scalar CCC and CMI1 do not work for multidimensional dynamical systems
data and CMI3 does not perform well for short length data.
Noisy data. White Gaussian noise was added to the simulated Rössler data. e amount of noise added to
the data was relative to the standard deviation of the data. e noise standard deviation (
σn
), is expressed as a
percentage of the standard deviation of the original data (
σs
). For example,
20%
of noise means
σn
=
0.2
σ
s
, and
100% of noise means
σn=σs
. e length of time series taken for this experiment was xed to 2048. For each
realization of noisy data as well, 100 surrogate time series were generated and signicance testing performed as
before using the Amplitude Adjusted Fourier Transform method and z-test respectively. Figure2a,b show the
results for varying noise in the data for the measures PCCC and PCMI respectively.
It can be seen that PCCC performs well for low levels of noise, up to
10%
, but at higher levels of noise, its
performance begins to deteriorate. PCMI, on the other hand, shows high TPR and low FPR even as the noise
level is increased to 50%.
Sparse data. We refer to time-series with missing samples as sparse data. Sparsity or non-uniformly missing
samples were introduced in the data in two ways: (1) Synchronous sparsity and (2) Asynchronous sparsity. In
case of (1), samples were missing from both
x1
and
x2
at randomly chosen time indices and this set of time indi-
ces was the same for both
x1
and
x2
. In case of (2), samples were missing from both
x1
and
x2
based on two dif-
(1)
˙x
1
=−ω
1
y
1
−z
1
,
˙
y
1=ω1x1+a1y1,
˙
z1
=
b1
+
z1(x1
−
c1),
(2)
˙x
2
=−ω
2
y
2
−z
2
+ǫ(x
1
−x
2
),
˙y2=ω2x2+a2y2,
˙
z2
=
b2
+
z2(x2
−
c2)
,
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ferent sets of randomly chosen time indices, that is, the time indices of missing samples were dierent for
x1
and
x2
. e amount of synchronous/ asynchronous sparsity is expressed in terms of percentage of missing samples
relative to the original length of time series taken.
αsync
and
αasync
refer to the level of missing samples for the
cases of synchronous and asynchronous sparsity respectively, and are given by m/N, where m is the number of
missing samples and N is the original length of time series. N was xed to 2048. e length of time series became
shorter as the percentage of missing samples were increased. Causality estimation measures were applied to the
data without any knowledge of whether any samples were missing or the time stamps at which the samples were
missing. Surrogate data generation for each realization in this case was not done post the introduction of miss-
ing samples but prior to that, using the original length time series. Sparsity was then introduced in the surrogate
time series in a manner similar to that for original time series.
Figure2c,d show the results obtained using PCCC and PCMI respectively for synchronous sparsity. Figure2e,f
show the same for asynchronous sparsity. It can be seen that PCCC is robust to the introduction of missing
samples, showing high TPR and low FPR. FPR begins to be greater than 0.2 only when the level of synchronous
sparsity is increased to
25%
and asynchronous sparsity is increased to 20%. PCMI is robust to low levels of
synchronous sparsity but deteriorates beyond 5% of missing samples, giving low true positives. It performs very
poorly even with low levels of asynchronous sparsity.
Real data analysis. As discussed in the Introduction, a number of climate datasets are either sampled
at irregular intervals, have missing samples, are sampled aer long intervals of time or have a combination of
two or more of these issues. In addition, their temporal recordings available are short in length. We apply the
Figure1. Specicity and sensitivity of methods with varying length. True positive rate (or rate of signicant
causality estimated from
x1→x2
) and false positive rate (or rate of signicant causality estimated from
x2→x1
), using measures (a) scalar CCC (CCC), (b) permutation CCC (PCCC), (c) scalar CMI (CMI1), (d)
permutation CMI (PCMI) and (e) three-dimensional CMI (CMI3), as the length of time series, N, is varied.
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proposed method, PCCC, to some such datasets described below. We also compare the results obtained with
existing measures: scalar CCC, scalar CMI and PCMI.
Millenial scale CO
2
‑temperature recordings. Mills etal. have compiled independent estimates of global average
surface temperature and atmospheric CO
2
concentration for the Phanerozoic eon. ese paleoclimate proxy
records span the last 424 million years62 and have been used and made available in the study by Wong etal.63.
One data point for both CO
2
and temperature recordings were available for each million year period and was
used in our analysis to check for causal interaction between between the two.
CO
2
, CH
4
and temperature recordings over the last 800,000years. Past Interglacials Working Group of PAGES64
has made available proxy records of atmospheric CO
2
, CH4 and deepwater temperatures over the last 800 ka (1
ka= 1000 years). Each of these time series were reconstructed by separate studies and so the recordings available
are non-synchronous and also irregularly sampled for each variable. Further, some data points are missing in the
Figure2. Specicity and sensitivity of methods with varying noise and sparsity. True positive rate (or rate of
signicant causality estimated from
x1→x2
) and false positive rate (or rate of signicant causality estimated
from
x2→x1
), using measures permutation CCC (PCCC) (le column) and permutation CMI (PCMI) (right
column) as the level of noise: (a, b); level of synchronous sparsity: (c, d); and asynchronous sparsity: (e, f), are
varied.
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temperature time-series. Roughly, single data point is available for each ka for each of the three variables. CO
2
proxy data are based on antarctic ice core composites. is was rst reported by Lüthi etal.65 and the revised val-
ues made available in a study by Bereiter etal.66. Reconstructed atmospheric CH4 concentrations, also based on
ice cores, were as reported by Loulergue etal.67 (on the AICC2012 age scale68). Deepwater temperature record-
ings obtained using shallow-infaunal benthic foraminifera (Mg/Ca ratios) that became available from Ocean
Drilling Program (ODP) site 1123 on the Chatham Rise, east of New Zealand were reported by Eldereld etal.69.
Causal inuence was checked between CO
2
-temperature and separately between CH4-temperature. CO
2
and CH4 data are taken beginning from the 6.5th ka on the AICC2012 scale and temperature data are taken
beginning from the 7th ka. Since the number of data points available for temperature are 792, CO
2
-temperature
analysis was done based on these 792 samples and as the number of samples of CH4 is limited to 756 beginning
from the 6.5th ka, CH4-temperature analysis was done using these 756 data points.
Monthly CO
2
‑temperature dataset. Monthly mean CO
2
data constructed from mean daily CO
2
values as well as
Northern Hemisphere’s combined land and ocean temperature anomalies for the monthly timescale are available
open source on the National Oceanic and Atmospheric Administration (NOAA) website. e CO
2
measure-
ments were made at the Mauna Loa Observatory, Hawaii. A part of the CO
2
dataset (March 1958–April 1974)
were originally obtained by C. David Keeling of the Scripps Institution of Oceanography and are available on the
Scripps website. NOAA started its own CO
2
measurements starting May 1974. e temperature anomaly dataset
is constructed from the Global Historical Climatology Network-Monthly data set70 and International Compre-
hensive Ocean-Atmosphere Data Set, also available on the NOAA website. ese data from March, 1958 to June
2021 (with 760 data points) were used to check for the causal inuence between CO
2
and temperature on the
recent timescale. Both time series were dierenced using consecutive values as they were highly non-stationary.
Yearly ENSO‑SASM dataset. 1100 Year El Niño/Southern Oscillation (ENSO) Index Reconstruction dataset,
made available open source on NOAA website and originally published in Ref.71 was used in this study. South
Asian Summer Monsoon (SASM) Index 1100 Year Reconstruction dataset, also available open source on the
NOAA website and originally published inRef.72, was the second variable used here. e aim of our study was to
check the causal dependence between these two sets of recordings taken from the year 900 AD to 2000 AD (with
one data point being available for each year).
Monthly NINO‑Indian monsoon dataset. Monthly NINO 3.4 SST Index recordings from the year 1870 to 2021
are available open source on the NOAA website. Its details are published in Ref.73. All India monthly rainfall
dataset from 1871 to 2016, available on the ocial website of World Meteorological Organization and originally
acquired from ‘Indian Institute of Tropical Meteorology’, was used for analysis. ese recordings are in the units
of mm/month. Causal inuence was checked between these two recordings using 1752 data points, ranging from
the month January, 1871 to December, 2016.
Monthly NAO‑temperature recordings. Reconstructed monthly North Atlantic Oscillation (NAO) index
recordings from December 1658 to July 2001 are available open source on the NOAA website. e reconstruc-
tions from December 1658 to November 1900 are taken from Refs.74,75 and from December 1900 to July 2001
are derived from Ref.76. Central European 500 year temperature reconstruction dataset, beginning from 1500
AD, is made available open source by NOAA National Centers for Environmental Information, under the World
Data Service for Paleoclimatology. ese were derived in the study77. We took winter only data points (months
December, January and February) starting from the December of 1658 to the February of 2001 as it is known that
the NAO inuence is strongest in winter. is yielded a total of 1029 data points. However, reconstruction based
on embedding was done for each year’s winter separately (with a time delay of 1) and not in a continuous manner
as for the other datasets, reducing the length of ordinal patterns encoded sequence to 343. Causal inuence was
checked between NAO and temperature for the encoded sequences using PCMI and PCCC and directly using
one-dimensional CMI and CCC for the 1029 length sequences.
Daily NAO‑temperature recordings. Daily NAO records are available on the NOAA website and have been pub-
lished inRefs.78–80. Daily mean surface air temperature data from the Frankfurt station in Germany were taken
from the records made available online by the ECA &D project81. is data was taken from 1st January 1950
to 31st April 2021. Once again, daily values from the winter months alone (December, January and February),
comprising of 6390 data points, were extracted for the analysis. While embedding the two time series, care was
taken not to embed the recordings of winter from one year along with that of winter from the next year. Causal
inuence was checked between daily winter NAO and temperature time-series.
For the analysis of causal interaction in each of these datasets, scalar CCC and CMI as well as PCCC and
PCMI were computed as discussed in the “Methods” section. Parameters used for each of the methods are also
given in the “Methods” section (Table2). In order to assess the signicance of causality value estimated using
each measure, 100 surrogate realizations were generated using the stationary bootstrap method82 for both the time
series under consideration. Resampling of blocks of observations of random length from the original time series
is done for obtaining surrogate time series using this method. e length of each block has a geometric distribu-
tion. e probability parameter that determines the geometric probability distribution for length of each block
was set to 0.1 (as suggested inRef.82). Signicance testing of the causal interaction between original time-series
was then done using a standard one-sided z-test, with p-value set to 0.05. Table1 shows whether causal inu-
ence between the considered variables was found to be signicant using each of the causality measures. Figure3
depicts the value of the PCCC between original pair of time series with respect to the distribution of PCCC
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obtained using surrogate time series for two datasets: kilo-year scale CO
2
-temperature (Fig.3a,b) and yearly scale
ENSO-SASM (Fig.3c,d) recordings. In the tables, Fig.3 and in the following text, we use the notation ‘T’ to refer
to temperature generically. Which of the temperature recordings is being referred to, will be clear from context.
Discussion and conclusions
CCC has been proposed as an ‘interventional’ causality measure for time series. It does not require cause-eect
separability in time series samples and is based on dynamical evolution of processes, making it suitable for sub-
sampled time series, time series in which cause and eect are acquired at slightly dierent spatio-temporal scales
than the scales at which they naturally occur and even when there are slight discrepancies in spatio-temporal
scales of the cause and eect time series. is results in its robust performance in the case of missing samples,
non-uniformly sampled, decimated and short length data41. In this work, we have proposed the use of CCC in
combination with ordinal pattern encoding. e latter preserves the dynamics of the systems of observed vari-
ables, allowing for CCC to decipher causal relationships between variables of multi-dimensional systems while
conditioning for the presence ofother variables in these systems which might be unknown or unobserved.
Simulations of coupled Rössler systems illustrate how scalar CCC is a complete failure for observables of
coupled multi-dimensional dynamical systems, while PCCC performs well to determine the correct direction
of coupling. Comparison of PCCC with PCMI for these simulations shows that the former beats the latter by
showing better performance on shorter lengths of time series. Further, while PCMI consistently gave superior
performance for increasing noise in coupled Rössler systems, experiments with sparse data showed that PCCC
outperforms PCMI. is was the case when samples were missing from the driver and response time series either
in a synchronous or asynchronous manner.
As PCCC showed promising results for simulations with high levels of missing samples and short length,
we have applied it to make causal inferences in datasets from climatology and paleoclimatology which suer
from the issues of irregular sampling, missing samples and (or) have limited number of data points available.
Many of these datasets have been analyzed in previous studies. However, dierent studies report dierent results
probably due to the challenging nature of their recordings available or the limitation of the inference methods
applied to work on the data.
For example, the relationship between CO
2
concentrations and temperature of the atmosphere has been
studied from the mid 1800s83,84, beginning when a strong link between the two was recognized. Relatively
recently, with causal inference tools available, a number of studies have begun to look at the directionality of
relationship between the two on dierent temporal scales. To mention a few ndings, Kodra etal.85 found that
CO
2
Granger causes temperature. eir analysis was based on data taken from 1860 to 2008. Atanassio86 found
a clear evidence of GC from CO
2
to temperature using lag-augmented Wald test, for a similar time range. On
the other hand, Stern and Kaufmann87 found bidirectional GC between the two, again for a similar time range.
Kang and Larsson88 also nd bidirectional causation between the two using GC, however, by using data from
ice cores for the last 800,000 years. Many of these latter studies criticize the former. Also, the drawbacks of one
or more of these studies are explicitly mentioned inRefs.87,89,90 and highlight the issues with the data and/ or the
methodology employed. Other than GC and its extensions, a couple of other measures have also been used to
study CO
2
-T relationship. Stips etal.91 have applied a measure called Liang’s Information ow on CO
2
-T record-
ings, both on recent (1850–2005) and paleoclimate (800ka ice core reconstructions) time-scales. e study nds
unidirectional causation from CO
2
→
T on the recent time-scale and from T
→
CO
2
on the paleoclimatic scale.
Table 1. Causal inference obtained for real datasets using dierent causality measures.

indicates signicant
causality and
×
indicates non-signicant causality.
System
Measure
Direction CCC PCCC CMI PCMI
Millenial scale CO
2
-T CO
2
→
T

×
×
×
T
→
CO
2


×
×
Kilo-year scale CO
2
-T CO
2
→
T
×
×
×
×
T
→
CO
2
×

×
×
Kilo-year scale CH4-T CH4
→
T
×

×
×
T
→
CH4
×
×
×
×
Monthly scale CO
2
-T CO
2
→
T
×

×
×
T
→
CO
2
×
×
×
×
Yearly ENSO-SASM ENSO
→
SASM
×

×
×
SASM
→
ENSO


×
×
Monthly NINO-Indian monsoon NINO
→
Monsoon
×



Monsoon
→
NINO

×


Monthly NAO-European T NAO
→
T


×
×
T
→
NAO
×
×
×
×
Daily NAO-Frankfurt T NAO
→
T

×

×
T
→
NAO
×
×
×
×
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ey have also analysed the CH4-T relationship and found T to drive CH4 on the paleoclimate scale. is study
has been criticized by Goulet etal.92. ey show that an assumption of ‘linearity’ made by Liang’s information
ow is nearly always rejected by the data. Convergent cross mapping, which is applied to the 800 ka recordings
in another study, nds a bidirectional causal inuence between both CO
2
- T and CH4-T93. Another recent
study, that infers causation using lagged cross-correlations between monthly CO
2
and temperature, taken from
the period 1980–2019, has found a bidirectional relationship on the recent monthly scale, with the dominant
inuence being from T
→
CO
2
94. In the light of the limitations of CCM95,96, especially for irregularly sampled or
missing data42, and of the widely known pitfalls of correlation coecient97, it is dicult to rely on the inferences
of the latter two studies.
PCCC indicates unidirectional causality from T
→
CO
2
on the paleoclimatic scale, using both millenial
and kilo-year scale recordings. On the recent monthly scale, the situation is reversed with CO
2
driving T. ese
results are in line with some of the existing CO
2
-T causal analysis studies and clearly PCCC does not suer the
limitations of existing approaches. On the kilo-year scale, PCCC suggests that CH4 drives T. While none of the
above discussed causality studies have found this result, other works have suggested that methane concentra-
tions modulate millenial-scale climate variability because of the sensitivity of methane to insolation98,99. Other
Figure3. PCCC surrogate analysis results. PCCC surrogate analysis results for: (a) Kilo-year scale CO
2
→
T,
(b) Kilo-year scale T
→
CO
2
, (c) Yearly ENSO
→
SASM, (d) SASM
→
ENSO. Dashed line indicates PCCC value
obtained for original series. Its position is indicated with respect to Gaussian curve tted normalized histogram
of surrogate PCCC values. PCCC for cases (b)–(d) is found to be signicant.
Table 2. Parameters corresponding to each method, used for dierent datasets.
Dataset Embedding CCC PCCC CMI/ PCMI
Rössler
η
x
1
=
5
,
η
x
2
=
5
,
m=3
L=300
,
w=30
,
δ=30
,
B=8
L=25
,
w=15
,
δ=20
τ=20
Millenial CO
2
-T
ηCO2
=
11
,
ηT
=
16
,
m
=
3
L
=
60
,
w
=
15
,
δ
=
20
,
B=4
L
=
60
,
w
=
30
,
δ
=
20
τ
=
1
−
30
Kilo-year CO
2
-T
η
CO
2
=
24
,
ηT
=
8
,
m=3
L=60
,
w=15
,
δ=20
,
B=4
L=30
,
w=15
,
δ=20
τ=1−30
Kilo-year CH4-T
η
CH
4
=
10
,
ηT
=
8
,
m=3
L=60
,
w=15
,
δ=20
,
B=4
L=30
,
w=15
,
δ=20
τ=1−30
Monthly CO
2
-T
ηCO2
=
3
,
ηT
=
2
,
m=3
L=60
,
w=15
,
δ=20
,
B=4
L=30
,
w=15
,
δ=20
τ=1−30
Yearly ENSO-SASM
η
ENSO =
1
,
η
SASM =
4
,
m=3
L=60
,
w=15
,
δ=20
,
B=4
L=60
,
w=30
,
δ=30
τ=1−30
Monthly NINO-India Monsoon
ηNINO
=
10
,
ηmon
=
3
,
m=3
L=60
,
w=15
,
δ=20
,
B=4
L=30
,
w=15
,
δ=20
τ=1−30
Monthly NAO-T
ηNAO
=
1
,
ηT
=
1
,
m=3
L=60
,
w=15
,
δ=20
,
B=4
L=30
,
w=15
,
δ=10
τ=1−30
Daily NAO-T
η
NAO =
15
,
η
T=
15
,
m=3
L=40
,
w=15
,
δ=20
,
B=4
L=30
,
w=15
,
δ=20
τ=1−30
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approaches implemented in this study – CCC, CMI, PCMI also do not duplicate the results obtained by PCCC
because of their specic limitations such as the inability to work on multi-dimensional, short length or irregu-
larly sampled data.
ENSO events and the Indian monsoon are other major climatic processes of global importance59. e relation-
ship between the two has been studied extensively, especially using correlation and coherence approaches100–105.
While ENSO is normally expected to play a driving role, there is no clear consensus on the directionality of the
relationship between the two processes. More recently, causal inference approaches have been used to study the
nature of their coupling. InRefs.106,107, both linear and non-linear GC versions were implemented on monthly
mean ENSO-Indian monsoon time series, ranging from the period 1871–2006 and bidirectional coupling was
inferred between the two processes. Other studies have studied the causal relationship indirectly by analyzing the
ENSO-Indian Ocean Dipole link. For example, inRef.108, this connection was studied by applying GC on yearly
reanalysis as well as model data ranging from 1950–2014. e study found robust causal inuence of Indian
Ocean Dipole on ENSO while the inuence in opposite direction had lower condence. Using PCCC, we nd
a bidirectional causal inuence between yearly recordings of ENSO-SASM. However, on the shorter monthly
scales, NINO is found to drive Indian Monsoon and there is insignicant eect in the opposite direction.
Although the NAO is known to be a leading mode of winter climate variability over Europe109–111, the direc-
tionality or feedback in NAO related climate eects has been studied by a few causality analysis studies9,112,113.
We investigate the NAO-European temperatures relationship on both monthly and daily time scales using winter
only data. While PCCC indicates that NAO drives central European temperatures with no signicant feedback
on the longer monthly scale, on the daily scale it shows no signicant causation in either direction. On the other
hand, CCC and CMI, based on one dimensional time series, indicate a strong inuence from NAO to Frankfurt
daily mean temperatures. is result indicates that the NAO inuence on European winter temperature on the
daily scale can be explained as a simple time-delayed transfer of information between scalar time series in which
no role is played by higher-dimensional patterns, potentially reected in ordinal coding. Such an information
transfer in the atmosphere is tied to the transfer of mass and energy as indicated in the study of climate networks
by Hlinka etal.114. CMI and PCMI estimates can be considered to be reliable for this analysis as the time-series
analyzed are long, close to 6000 time points.
CCC is free of the assumptions of linearity, requirement of long-term stationarity, extremely robust to miss-
ing samples, irregular sampling and short length data; and its combination with permutation patterns allows it
to make reliable inferences for coupled systems with multiple variables. us, we can expect our analysis and
inferences presented here on some highly-researched and long-debated climatic interactions to be highly robust
and reliable. We also expect that the use of PCCC on other challenging datasets from climatology and other elds
will be helpful to shed light on the causal linkages in considered systems.
Methods
Compression‑complexity causality (CCC) is dened as the change in the dynamical compression-complexity of
time series y when
y
is seen to be generated jointly by the dynamical evolution of both
ypast
and
xpast
as opposed
to by the reality of the dynamical evolution of
ypast
alone.
ypast ,xpast
are windows of a particular length L taken
from contemporary time points of time series y and x respectively and
y
is a window of length w following
ypast
41. Dynamical compression-complexity (CC) is estimated using the measure eort-to-compress (ETC)115
and given by:
Equation(3) computes the dynamical compression-complexity of
y
as a dynamical evolution of
ypast
alone.
Equation(4) computes the dynamical compression-complexity of
y
as a dynamical evolution of both
ypast
and
xpast
.
CCCxpast
→
y
is then estimated as:
Averaged CCC from x to y over the entire length of time series with the window
y
being slided by a step-size
of
δ
is estimated as:
If
CC
(�y|y
past
)≈CC(�y|x
past
,y
past )
, there is no causality from x to y. Surrogate time series are generated for
both x and y and the
CCCx→y
values of the original and surrogate time series compared. If the CCC computed
for original time series is statistically dierent from that of surrogate time series, we can infer the presence of
causal relation from
x→y
42.
CCCx→y
can be both < or
>0
depending upon the nature or quality of the causal
relationship41. e magnitude indicates the strength of causation.
Selection of parameters:
L,w,δ
and the number of bins, B, for symbolizing the time series using equidistant
binning (ETC is applied to symbolic sequences) is done using parameter selection criteria given in the supple-
mentary text ofRef.41.
Permutation compression‑complexity causality is the causal inference technique proposed and imple-
mented in this work. Given a pair of time series
x1
and
x2
from dynamical systems in which causation is to be
checked from
x1
to
x2
, we rst embed the time series of the potential driver (
x1
here) in the following manner:
(3)
CC(�y|ypast )=ETC(ypast +�y)−ETC(ypast ),
(4)
CC(�y|ypast ,xpast )=ETC(ypast +�y,xpast +�y)−ETC(xpast ,ypast ),
(5)
CCCxpast
→�
y=CC(�y|ypast )−CC(�y|ypast ,xpast ).
(6)
CCC
x→y=CCCxpast →�y
=CC(�y|y
past
)−CC(�y|x
past
,y
past
)
,
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x1(t),x1(t+η),x1(t+2η),...x1(t+(m−1)η)
, where
η
is the time delay and m is the embedding dimension
of
x1
.
η
is computed as the rst minimum of auto mutual information function. e embedded time-series at
each time-point is then symbolized using permutation or ordinal patterns binning. For example, if
m=3
, the
embedding at time point t is given as
ˆx1(t)=(x1(t),x1(t+η),x1(t+2η))
. Symbols 0,1,2 are then used for
labelling the pattern for
ˆx1(t)
at each time point by sorting the embedded values in ascending order, with 2 being
used for the highest value and 0 for the lowest. If two or more values are exactly same in
ˆx1(t)
, they are labelled
dierently depending on the order of their occurrence, where the same value takes a smaller symbol at its rst (or
earlier) occurrence. However, this may lead to two or more dierent embedded vectors having the same ordinal
representation. For example, the embeddings, (3,5,5), (3,3,5) and (3,3,3), all have an ordinal representation
of (0,1,2). is limits the total number of possible patterns at time t to
m!=3!
. us,
ˆx1(t)
is symbolized to a
one dimensional sequence consisting of m! possible symbols or bins. CCC is then estimated from
ˆx1(t)
to
x2(t)
,
using Eq.(6) aer symbolizing
x2(t)
using standard equidistant binning with m! bins. us,
Permutation binning is not done for the potential driver series as it was found from simulation experiments
(Rössler data) that embedding the ‘cause’ alone works better for the CCC measure. Full dimensionality of the
cause is necessary to predict the eect. Hence, embedding only the cause helps to recover the causal relation-
ship. PCCC helps to take into account the multidimensional nature of the coupled systems. Parameter selection
for PCCC is done in the same manner as for the case of CCC, using the symbolic sequences,
ˆx1(t)
and
x2(t)
, for
selection of the parameters. When PCCC is to be estimated from
x2→x1
,
x2
is embedded and
x1
remains as it
is. Just like CCC, the PCCC measure can also take negative values.
Conditional mutual information (CMI) of the variables X and Y given the variable Z is a common information-
theoretic functional used for the causality detection, and can be obtained as
where
H(X1,X2, ...|Z)=H(X1,X2, ...)−H(Z)
is the conditional entropy, and the joint Shannon entropy
H(X1,X2, ...)
is dened as:
where
p(x1,x2, ...)=Pr[X1=x1,X2=x2, ...]
is the joint probability distribution function of the amplitude of var-
iables
{X1,X2, ...}
. In order to detect the coupling direction among two dynamical variables of X and Y, Paluš etal.21
used the conditional mutual information
I(X(t);Y(t+τ)|Y(t))
, that captures the net information about the
τ
-future of the process Y contained in the process X. As mentioned in the Introduction, to estimate other unknown
variables, an m-dimensional state vector X can be reconstructed as
X(t)={x(t),x(t−η), ..., x(t−(m−1)η)}
.
Accordingly, CMI dened above can be represented by its reconstructed version for all variables of X(t),
Y(t+τ)
and Y(t). However, extensive numerical studies22 demonstrated that CMI in the form
is sucient to infer direction of coupling among dynamical variables of X(t) and Y(t). In this respect, we use this
measure to detect causality relationships in this article.
Permutation conditional mutual information (PCMI) can be obtained based on the permutation analysis
described earlier in the PCCC denition. In this approach, all marginal, joint or conditional probability distri-
bution functions of the amplitude of the variables are replaced by their symbolized versions, thus Eq.(9) should
be replaced by
where
p
(ˆ
x1,
ˆ
x2, ...
)=
Pr
[
ˆ
X1
=ˆ
x1,ˆ
X2
=ˆ
x2, ...]
is the joint probability distribution function of the symbolized
variables
ˆ
Xi(t)={Xi(t),Xi(t+η), ..., Xi(t+(m−1)η)}
. By using Eqs.(8) and (11), permutation CMI can be
obtained as
I(ˆ
X(t);ˆ
Y(t+τ)|ˆ
Y(t))
. Finally, one should replace
τ
with
τ+(m−1)η
in order to avoid any over-
lapping between the past and future of the symbolized variable
ˆ
Y
.
Parameters of the methods used were set as shown in Table2 for dierent datasets.
Data availability
e millenial scale CO
2
and temperature datasets are freely available at h t t ps:// zenodo. org/ record/ 45629 96#. YiD-
bTN_ ML3A. Kilo-year scale CO
2
, CH
4
and temperature datasets are available as supplementary les forRef.64
at https:// agupu bs. onlin elibr ary. wiley. com/ doi/ full/ 10. 1002/ 2015R G0004 82. Monthly CO
2
recordings are taken
from the NOAA repository and are available at h t t ps:// gml. noaa. gov/ ccgg/ trends/. Monthly Northern hemisphere
temperature anomaly recordings are taken from the NOAA repository and are available at https:// www. ncdc.
noaa. gov/ cag/ global/ time- series. e yearly El Niño/Southern Oscillation Index Reconstruction dataset is taken
from the NOAA repository, ht t ps:// www. ncei. noaa. gov/ acces s/ paleo- searc h/ study/ 11194. e yearly South Asian
Summer Monsoon Index Reconstruction dataset is taken from the NOAA repository, https:// www. ncei. noaa.
gov/ access/ paleo- search/ study/ 17369. Monthly Niño 3.4 SST Index dataset is taken from the NOAA reposi-
tory, available at https:// psl. noaa. g ov/ gcos_ wgsp/ Times eries/ Nino34/. Monthly all India rainfall dataset is made
available by the World Metereological Organization at http:// c lime xp. knmi. nl/ data/ pALLIN. dat. Reconstructed
(7)
PCCCx1→x2=CCCˆx1→x2.
(8)
I(X;Y|Z)=H(X|Z)+H(Y|Z)−H(X,Y|Z)
(9)
H
(X1,X2, ...)=−

x1,x2,...
p(x1,x2, ...)log p(x1,x2, ...
)
(10)
I(X(t);Y(t+τ)|Y(t),Y(t−η), ..., Y(t−(m−1)η))
(11)
H
(ˆ
X1,ˆ
X2, ...)=− 
ˆx1,ˆx2,...
p(ˆx1,ˆx2, ...)log p(ˆx1,ˆx2, ...
)
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monthly North Atlantic Oscillation Index is available at the NOAA repository, https:// psl. noaa. gov/ gcos_ wgsp/
Times eries/ RNAO/. Monthly Central European 500 Year Temperature Reconstructions are available at the NOAA
repository, https:// www. ncei. noaa. gov/ access/ metad ata/ landi ng- pag e/ bin/ iso? id= noaa- recon- 9970. Daily North
Atlantic Oscillation Index is available at the NOAA repository, https:// www . cpc. ncep. noaa. gov/ produ cts/ preci p/
CWlink/ pna/ nao. shtml. Daily Frankfurt air temperatures are made available by the ECA &D project at https://
www. ecad. eu/ daily data/ prede ned series. php.
Code availability
e computer codes used in this study are freely available at ht t ps:// github. com/ Adi ti K athp alia/ Permu tatio nCCC
under the Apache 2.0 Open-source license.
Received: 14 May 2022; Accepted: 9 August 2022
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Acknowledgements
is study is supported by the Czech Science Foundation, Project No.GA19-16066S and by the Czech Academy
of Sciences, Praemium Academiae awarded to M. Paluš.
Author contributions
A.K. performed the research, implementation and computations of CCC and PCCC, wrote the manuscript dra;
P.M. implemented and computed CMI and PCMI; M.P. proposed and led the project. All authors contributed
to the nal version of the manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to M.P.
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