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arXiv:2204.11731v1 [physics.data-an] 25 Apr 2022

Compression-Complexity with Ordinal Patterns for Robust Causal Inference in

Irregularly-Sampled Time Series

Aditi Kathpalia,∗Pouya Manshour, and Milan Paluˇs†

Department of Complex Systems,

Institute of Computer Science of the Czech Academy of Sciences,

Prague, Czech Republic

(Dated: April 26, 2022)

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 es-

timates 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 multi-

dimensional patterns into one-dimensional symbolic sequences, and thus introduce the Permutation

CCC (PCCC), which retains all advantages of the original CCC and can be applied to data from

multidimensional systems with potentially hidden variables. PCCC is tested on numerical simu-

lations and applied to paleoclimate data characterized by irregular and uncertain sampling and

limited numbers of samples.

I. INTRODUCTION

Unraveling systems’ dynamics from the analysis of ob-

served 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 relation-

ships among observables is of particular importance that

can improve our ability to better understand the underly-

ing dynamics and to predict or even control such complex

systems [1, 2].

Around sixty years after the pioneering work of Wiener

and Granger [3, 4] on quantifying linear ‘causality’ from

observations, it has been widely applied not only in eco-

nomics [5–7], for which it was ﬁrst introduced, but also in

various ﬁelds of natural sciences, from neurosciences [8]

to Earth sciences [9–11]. A number of attempts have been

made to generalize Granger Causality (GC) to nonlinear

cases, using, e.g., an estimator based on correlation inte-

gral [6], a non-parametric regression approach [12], local

linear predictors [13], mutual nearest neighbors [14, 15],

kernel estimators [16], to state a few. Several other

causality methods based on the GC principle such as

Partial Directed Coherence [17], Direct Transfer Func-

tion [18] and Modiﬁed Direct Transfer Function [19] have

also been proposed.

Information theory has proved itself as a powerful ap-

proach into causal inference. In this respect, Schreiber

proposed a method for measuring information trans-

fer among observables [20], known as Transfer Entropy

(TE), which is based on Kullback-Leibler distance be-

tween transition probabilities. Paluˇs et al. [21] intro-

duced a causality measure based on mutual information,

∗kathpalia@cs.cas.cz

†mp@cs.cas.cz

called Conditional Mutual Information (CMI). CMI has

been shown to be equivalent to TE [22]. These tools have

been applied in various research studies and have shown

their power in extracting causal relationships between

diﬀerent systems [23–27].

We usually work with time series x(t) and y(t) as real-

izations of mand ndimensional 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 mand ndimensional vectors. In

many cases, only one possible dimension of the phase

space is observable, recordings or knowledge of variables

which may have indirect eﬀects 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 Takens [28],

which reconstructs the dynamics of the entire system

(including other unknown/unmeasurable variables) using

time-delay embedding vectors, as follows: the manifold

of an mdimensional state vector Xcan be reconstructed

as X(t) = {x(t), x(t−η), ..., x(t−(m−1)η)}. Here,

ηis the embedding delay, and can be obtained using

the embedding construction procedure based on the ﬁrst

minimum of the mutual information [29]. Some causal-

ity estimators have applied this phase-space reconstruc-

tion procedure to improve their causal inference power,

such as high dimension CMI [26] and TE [30]. Other

causality measures, such as, Convergent Cross Mapping

(CCM) [31], Topological Causality [32], Predictability

Improvement [33], are based directly on the reconstruc-

tion of dynamical systems.

Vast amounts of data available in the recent years have

pushed some of the above discussed GC extensions, infor-

mation and phase-space reconstruction based approaches

forward as they rely on joint probability density esti-

mations, stationarity, markovianity, topological or linear

modeling. However, still, many temporal observations

2

made in various domains such as climatology [34, 35], ﬁ-

nance [36, 37] and sociology [38] are often short in length,

have missing samples or are irregularly sampled. A signif-

icant challenge arises when we attempt to apply causality

measures in such situations [11]. For instance, CMI or

TE fail when applied to time series which are undersam-

pled or have missing samples [39–41] and also in case

of time series with short lengths [41]. CCM and kernel

based non-linear GC also show poor performance even

in the case of few missing samples in bivariate simulated

data [42].

Kathpalia and Nagaraj recently introduced a causal-

ity measure, called Compression-Complexity Causality

(CCC), which employs ‘complexity’ estimated using loss-

less 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. This has been shown to be the case for

samples which are missing in the two coupled time se-

ries either in a synchronous or asynchronous manner [41].

Also, it gives good performance for time series with short

lengths [41, 42]. These 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 loss-

less compression based complexity approaches which in

turn show robust performance on short and noisy time

series [41, 43]. However, as discussed in [42], a direct mul-

tidimensional extension of CCC is not as straightforward

and so a measure of eﬀective CCC has been formulated

and used on multidimensional systems of coupled autore-

gressive 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 causal-

ity measures for noisy data, such as symbolic transfer en-

tropy [44, 45], partial symbolic transfer entropy [46, 47],

permutation conditional mutual information (PCMI) [48]

and multidimensional PCMI [49]. The symbolization

technique used in these works is based on the Bandt

and Pompe scheme for estimation of Permutation En-

tropy [50], and often referred to as permutation or or-

dinal patterns coding. The 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 construc-

tion, this technique ignores the amplitude information

and thus decreases the eﬀect of high ﬂuctuations in data

on the obtained causal inference [51]. Other beneﬁts 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 inexpen-

sive [52–55]. Ordinal partition has been shown to have

the generating property under speciﬁc conditions, imply-

ing topological conjugacy between phase space of dynam-

ical systems and their ordinal symbolic dynamics [56].

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 En-

tropy [57, 58]. Because of all these beneﬁcial properties

of permutation patterns, it is no wonder that the devel-

opment of symbolic TE or PCMI helped to make them

more robust, giving better performance in the case of

noisy measurements, simplifying the process of parame-

ter selection and making less demands on the data.

In this work, we propose the use of CCC approach with

reconstructed dynamical systems which are symbolized

using ordinal patterns. The combination of strengths of

CCC and ordinal patterns, not only makes CCC appli-

cable to dynamical systems with multidimensional vari-

ables, but we also observe that the proposed Permuta-

tion CCC (PCCC) measure gives great performance on

datasets with very short lengths and high levels of miss-

ing samples. The performance of PCCC is compared with

that of PCMI (which is identical to symbolic TE), bivari-

ate CCC and CMI on simulated dynamical systems data.

PCCC outperforms the existing approaches and its esti-

mates are found to be robust for short length time series,

and high levels of missing data points.

This development for the ﬁrst time opens up avenues

for the use of causality estimation tool on real world

datasets from climate and paleoclimate science, ﬁnance

and other ﬁelds where there is prevalence of data with ir-

regular 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 Earth [59]. Diﬀerent studies have employed

either correlation/coherence, causality methods or mod-

elling approaches to study the interaction between cli-

matic processes. The results produced by diﬀerent stud-

ies are diﬀerent and sometimes contradictory, present-

ing an ambiguous situation. We apply PCCC to anal-

yse the causal relationship between the following sets of

climatic processes: greenhouse gas concentrations – at-

mospheric temperature, El-Ni˜no Southern Oscillation –

South Asian monsoon and North Atlantic Oscillation –

European temperatures at diﬀerent time-scales and com-

pare its performance with bivariate CCC, bivariate and

multidimensional CMI, and PCMI. The 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.

3

II. RESULTS

Simulation Experiments: Time series data from a

pair of unidirectionally coupled R¨ossler systems were gen-

erated as per the following equations:

˙x1=−ω1y1−z1,

˙y1=ω1x1+a1y1,(1)

˙z1=b1+z1(x1−c1),

for the autonomous or master system, and

˙x2=−ω2y2−z2+ǫ(x1−x2),

˙y2=ω2x2+a2y2,(2)

˙z2=b2+z2(x2−c2),

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. The coupling

parameter, ǫ, was ﬁxed to 0.09. The data were generated

by numerical integration based on the adaptive Bulirsch-

Stoer method [60] using a sampling interval of 0.314 for

both the master and slave systems. This 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 cou-

pling between x1and x2, with x1inﬂuencing x2. The

analysis of the causal inﬂuence between the two systems

was done using the causality estimation measures: bi-

variate or scalar CCC, CMI, PCCC and PCMI for the

cases outlined in the following paragraphs. The estima-

tion procedure for each of the methods is described in

the ‘Methods’ section. The values of parameters used

for each of the methods are also given in the ‘Methods’

section (Table II).

Finite length data: The length of time series, N, of x1

and x2taken from coupled R¨ossler systems was varied

as shown in Fig. 1. The 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. Fig. 1(c) shows scalar (simple bivariate) CMI

or one-dimensional CMI (CMI1) between x1and x2(see

Paluˇs and Vejmelka [22]). This method has high sen-

sitivity but suﬀers from low speciﬁcity. This problem

is solved by using conditional CMI or three-dimensional

CMI (CMI3), where the information from other vari-

ables (y1, z1, y2, z2) is incorporated in the estimation. Its

performance is depicted in Fig. 1(e). However, it re-

quires larger length of time series for optimal perfor-

mance. Fig. 1(a) shows the performance of scalar (or sim-

ple bivariate) CCC, which is equivalent to the CMI1 case,

considering dimensionality. Figs. 1(b) and 1(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 realiza-

tion in order to perform signiﬁcance analysis of causality

estimated (in both directions) from each realization of

coupled processes. These surrogates were generated for

FIG. 1. Speciﬁcity and sensitivity of methods with varying

length. True positive rate (or rate of signiﬁcant causality

estimated from x1→x2) and false positive rate (or rate of

signiﬁcant 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.

both the processes using the Amplitude Adjusted Fourier

Transform method [61] and signiﬁcance testing done us-

ing a standard one-sided z-test with p-value set to 0.05

(this was justiﬁed as the distributions of surrogates for

CCC and CMI methods implemented were found to be

Gaussian). Based on this signiﬁcance analysis, true posi-

tive rate (TPR) and false positive rate (FPR) were com-

puted at each length level. A true positive is counted for

a particular realization of coupled systems when causal-

ity estimated from x1to x2is found to be signiﬁcant and

a false positive is counted when causality estimated from

x2to x1is found to be signiﬁcant.

As it can be seen from the plots, direct application of

scalar CCC completely fails on multidimensional dynam-

ical systems data, yielding low true positives and high

false positives. Hence the method displays poor sensitiv-

ity as well as speciﬁcity. CMI1 also shows poor perfor-

mance, yielding high false positives. CMI3, which is ap-

propriate 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

4

to 1024 time points, with TPR and FPR reaching almost

1 and 0 respectively as length is increased to 2048 time

points. The 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 R¨ossler

data by varying the amount of noise and missing sam-

ples 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 R¨ossler data. The amount of noise added to

the data was relative to the standard deviation of the

data. The 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= 0.2σs,

and 100% of noise means σn=σs. The length of time se-

ries taken for this experiment was ﬁxed to 2048. For each

realization of noisy data as well, 100 surrogate time se-

ries were generated and signiﬁcance testing performed as

before using the Amplitude Adjusted Fourier Transform

method and z-test respectively. Figs. 2(a) and 2(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 sam-

ples 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 x1and x2

at randomly chosen time indices and this set of time in-

dices was the same for both x1and x2. In case of (2),

samples were missing from both x1and x2based on two

diﬀerent sets of randomly chosen time indices, that is, the

time indices of missing samples were diﬀerent for x1and

x2. The 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 respec-

tively, and are given by m/N, where mis the number

of missing samples and Nis the original length of time

series. Nwas ﬁxed to 2048. The length of time series be-

came 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 real-

ization in this case was not done post the introduction

of missing samples but prior to that, using the original

FIG. 2. Speciﬁcity and sensitivity of methods with varying

noise and sparsity. True positive rate (or rate of signiﬁcant

causality estimated from x1→x2) and false positive rate

(or rate of signiﬁcant causality estimated from x2→x1),

using measures permutation CCC (PCCC) (left column) and

permutation CMI (PCMI) (right column) as the level of noise:

(a) and (b); level of synchronous sparsity: (c) and (d); and

asynchronous sparsity: (e) and (f ), are varied.

length time series. Sparsity was then introduced in the

surrogate time series in a manner similar to that for orig-

inal time series.

Figs. 2(c) and 2(d) show the results obtained using

PCCC and PCMI respectively for synchronous sparsity.

Figs. 2(e) and 2(f) show the same for asynchronous spar-

sity. It can be seen that PCCC is robust to the introduc-

tion 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 asyn-

chronous sparsity is increased to 20%. PCMI is robust

to low levels of synchronous sparsity but deteriorates be-

yond 5% of missing samples, giving low true positives.

It performs very poorly even with low levels of asyn-

chronous sparsity.

Real Data Analysis: As discussed in the Introduc-

tion, a number of climate datasets are either sampled at

irregular intervals, have missing samples, are sampled af-

ter long intervals of time or have a combination of two or

more of these issues. In addition, their temporal record-

ings available are short in length. We apply the proposed

5

method, PCCC, to some such datasets described below.

We also compare the results obtained with existing mea-

sures: scalar CCC, scalar CMI and PCMI.

Millenial scale CO2-temperature recordings: Mills et

al. have compiled independent estimates of global aver-

age surface temperature and atmospheric CO2concentra-

tion for the Phanerozoic eon. These paleoclimate proxy

records span the last 424 million years [62] and have been

used and made available in the study by Wong et al. [63].

One data point for both CO2and temperature recordings

were available for each million year period and was used

in our analysis to check for causal interaction between

between the two.

CO2, CH4and temperature recordings over the last

800,000 years: In [64], Past Interglacials Working Group

of PAGES has made available proxy records of atmo-

spheric CO2, CH4and deepwater temperatures over the

last 800 ka (1 ka= 1000 years). Each of these time series

were reconstructed by separate studies and so the record-

ings available are non-synchronous and also irregularly

sampled for each variable. Further, some data points

are missing in the temperature time-series. Roughly, sin-

gle data point is available for each ka for each of the

three variables. CO2proxy data are based on antarctic

ice core composites. This was ﬁrst reported in [65] and

the revised values made available in a study by Bere-

iter et al. [66]. Reconstructed atmospheric CH4con-

centrations, also based on ice cores, were as reported

in [67] (on the AICC2012 age scale [68]). Deepwater

temperature recordings obtained using shallow-infaunal

benthic foraminifera (Mg/Ca ratios) that became avail-

able from Ocean Drilling Program (ODP) site 1123 on

the Chatham Rise, east of New Zealand were reported

in [69].

Causal inﬂuence was checked between CO2-

temperature and separately between CH4-temperature.

CO2and CH4data 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, CO2-

temperature analysis was done based on these 792

samples and as the number of samples of CH4is limited

to 756 beginning from the 6.5th ka, CH4-temperature

analysis was done using these 756 data points.

Monthly CO2-temperature dataset: Monthly mean

CO2data constructed from mean daily CO2values 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 At-

mospheric Administration (NOAA) website. The CO2

measurements were made at the Mauna Loa Observa-

tory, Hawaii. A part of the CO2dataset (March 1958-

April 1974) were originally obtained by C. David Keel-

ing of the Scripps Institution of Oceanography and are

available on the Scripps website. NOAA started its own

CO2measurements starting May 1974. The temperature

anomaly dataset is constructed from the Global Histor-

ical Climatology Network-Monthly data set [70] and In-

ternational Comprehensive Ocean-Atmosphere Data Set,

also available on the NOAA website. These data from

March, 1958 to June 2021 (with 760 data points) were

used to check for the causal inﬂuence between CO2and

temperature on the recent timescale. Both time series

were diﬀerenced using consecutive values as they were

highly non-stationary.

Yearly ENSO-SASM dataset: 1,100 Year El

Ni˜no/Southern Oscillation (ENSO) Index Recon-

struction dataset, made available open source on NOAA

website and originally published in [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 [72], was the second variable used here. The 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 [73]. All India monthly rain-

fall dataset from 1871-2016, available on the oﬃcial web-

site of World Meteorological Organization and originally

acquired from ‘Indian Institute of Tropical Meteorology’,

was used for analysis. These recordings are in the units of

mm/month. Causal inﬂuence 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 record-

ings from December 1658 to July 2001 are available

open source on the NOAA website. The reconstruc-

tions from December 1658 to November 1900 are taken

from [74, 75] and from December 1900 to July 2001 are

derived from [76]. Central European 500 year tempera-

ture 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. These were derived in the

study [77]. We took winter only data points (months

December, January and February) starting from the De-

cember of 1658 to the February of 2001 as it is known that

the NAO inﬂuence is strongest in winter. This 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 inﬂuence 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

published in [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

6

project [81]. This 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-

ﬂuence was checked between daily winter NAO and tem-

perature 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’ sec-

tion. Parameters used for each of the methods are also

given in the ‘Methods’ section (Table II). In order to as-

sess the signiﬁcance of causality value estimated using

each measure, 100 surrogate realizations were generated

using the stationary bootstrap method [82] 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. The length of each block has a geometric

distribution. The probability parameter that determines

the geometric probability distribution for length of each

block was set to 0.1 (as suggested in [82]). Signiﬁcance

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 Ishows whether causal

inﬂuence between the considered variables was found to

be signiﬁcant using each of the causality measures. Fig. 3

depicts the value of the PCCC between original pair of

time series with respect to the distribution of PCCC ob-

tained using surrogate time series for two datasets: kilo-

year scale CO2-temperature (Figs. 3(a) and 3(b)) and

yearly scale ENSO-SASM (Figs. 3(c) and 3(d)) record-

ings. In the tables, Fig. 3and 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.

III. DISCUSSION AND CONCLUSIONS

CCC has been proposed as an ‘interventional’ causal-

ity measure for time series. It does not require cause-

eﬀect separability in time series samples and is based on

dynamical evolution of processes, making it suitable for

subsampled time series, time series in which cause and

eﬀect are acquired at slightly diﬀerent 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 eﬀect time series. This

results in its robust performance in the case of missing

samples, non-uniformly sampled, decimated and short

length data [41]. In this work, we have proposed the

use of CCC in combination with ordinal pattern encod-

ing. The latter preserves the dynamics of the systems of

observed variables, allowing for CCC to decipher causal

relationships between variables of multi-dimensional sys-

-0.02 0 0.02 0.04

0

0.05

0.1

Probability

(a)

0 0.02 0.04 0.06

0

0.05

0.1

Probability

(b)

0.02 0.03 0.04 0.05

0

0.05

0.1

Probability

(c)

0.02 0.04 0.06

0

0.05

0.1

Probability

(d)

FIG. 3. PCCC surrogate analysis results. PCCC surrogate

analysis results for: (a) Kilo-year scale CO2→T, (b) Kilo-

year scale T →CO2, (c) Yearly ENSO →SASM, (d) SASM →

ENSO. Dashed line indicates PCCC value obtained for orig-

inal series. Its position is indicated with respect to Gaussian

curve ﬁtted normalized histogram of surrogate PCCC values.

PCCC for cases (b), (c), (d) is found to be signiﬁcant.

tems while conditioning for the presence other variables

in these systems which might be unknown or unobserved.

Simulations of coupled R¨ossler systems illustrate how

scalar CCC is a complete failure for observables of cou-

pled multi-dimensional dynamical systems, while PCCC

performs well to determine the correct direction of cou-

pling. Comparison of PCCC with PCMI for these simu-

lations shows that the former beats the latter by show-

ing better performance on shorter lengths of time series.

Further, while PCMI consistently gave superior perfor-

mance for increasing noise in coupled R¨ossler systems,

experiments with sparse data showed that PCCC out-

performs PCMI. This 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 suﬀer 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, diﬀerent studies report diﬀerent results prob-

ably due to the challenging nature of their recordings

available or the limitation of the inference methods ap-

plied to work on the data.

For example, the relationship between CO2concentra-

tions and temperature of the atmosphere has been stud-

ied from the mid 1800s [83, 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

7

TABLE I. Causal inference obtained for real datasets using diﬀerent causality measures. Xindicates signiﬁcant causality and

✗indicates non-signiﬁcant causality.

System

Measure Direction CCC PCCC CMI PCMI

Millenial scale CO2-T CO2→TX✗ ✗ ✗

T→CO2X X ✗ ✗

Kilo-year scale CO2-T CO2→T✗ ✗ ✗ ✗

T→CO2✗X✗ ✗

Kilo-year scale CH4-T CH4→T✗X✗ ✗

T→CH4✗ ✗ ✗ ✗

Monthly scale CO2-T CO2→T✗X✗ ✗

T→CO2✗ ✗ ✗ ✗

Yearly ENSO-SASM ENSO →SASM ✗X✗ ✗

SASM →ENSO X X ✗ ✗

Monthly NINO-Indian monsoon NINO →Monsoon ✗XXX

Monsoon →NINO X✗X X

Monthly NAO-European T NAO →TX X ✗ ✗

T→NAO ✗ ✗ ✗ ✗

Daily NAO-Frankfurt T NAO →TX✗X✗

T→NAO ✗ ✗ ✗ ✗

between the two on diﬀerent temporal scales. To men-

tion a few ﬁndings, Kodra et al. [85] found that CO2

Granger causes temperature. Their analysis was based

on data taken from 1860 to 2008. Atanassio [86] found

a clear evidence of GC from CO2to temperature using

lag-augmented Wald test, for a similar time range. On

the other hand, Stern and Kaufmann [87] found bidi-

rectional GC between the two, again for a similar time

range. Kang and Larsson [88] also ﬁnd bidirectional cau-

sation 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 [87, 89, 90] and highlight the issues with the data and/

or the methodology employed. Other than GC and its ex-

tensions, a couple of other measures have also been used

to study CO2-T relationship. Stips et al. [91] have ap-

plied a measure called Liang’s Information ﬂow on CO2-T

recordings, both on recent (1850-2005) and paleoclimate

(800 ka ice core reconstructions) time-scales. The study

ﬁnds unidirectional causation from CO2→T on the re-

cent time-scale and from T →CO2on the paleoclimatic

scale. They have also analysed the CH4-T relationship

and found T to drive CH4on the paleoclimate scale. This

study has been criticized by Goulet et al. [92]. They show

that an assumption of ‘linearity’ made by Liang’s infor-

mation ﬂow is nearly always rejected by the data. Con-

vergent cross mapping, which is applied to the 800 ka

recordings in another study, ﬁnds a bidirectional causal

inﬂuence between both CO2- T and CH4-T [93]. An-

other recent study, that infers causation using lagged

cross-correlations between monthly CO2and tempera-

ture, taken from the period 1980-2019, has found a bidi-

rectional relationship on the recent monthly scale, with

the dominant inﬂuence being from T →CO2[94]. In the

light of the limitations of CCM [95, 96], especially for ir-

regularly sampled or missing data [42], and of the widely

known pitfalls of correlation coeﬃcient [97], it is diﬃcult

to rely on the inferences of the latter two studies.

PCCC indicates unidirectional causality from T →

CO2on the paleoclimatic scale, using both millenial and

kilo-year scale recordings. On the recent monthly scale,

the situation is reversed with CO2driving T. These re-

sults are in line with some of the existing CO2-T causal

analysis studies and clearly PCCC does not suﬀer the

limitations of existing approaches. On the kilo-year scale,

8

PCCC suggests that CH4drives T. While none of the

above discussed causality studies have found this re-

sult, other works have suggested that methane concen-

trations modulate millenial-scale climate variability be-

cause of the sensitivity of methane to insolation [98, 99].

Other approaches implemented in this study – CCC,

CMI, PCMI also do not duplicate the results obtained

by PCCC because of their speciﬁc limitations such as

the inability to work on multi-dimensional, short length

or irregularly sampled data.

ENSO events and the Indian monsoon are other ma jor

climatic processes of global importance [59]. The rela-

tionship between the two has been studied extensively, es-

pecially using correlation and coherence approaches [100–

105]. While ENSO is normally expected to play a driv-

ing role, there is no clear consensus on the directional-

ity of the relationship between the two processes. More

recently, causal inference approaches have been used to

study the nature of their coupling. In [106] and [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 cou-

pling 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 [108], this connection was studied by ap-

plying GC on yearly reanalysis as well as model data

ranging from 1950-2014. The study found robust causal

inﬂuence of Indian Ocean Dipole on ENSO while the in-

ﬂuence in opposite direction had lower conﬁdence. Us-

ing PCCC, we ﬁnd a bidirectional causal inﬂuence be-

tween yearly recordings of ENSO-SASM. However, on

the shorter monthly scales, NINO is found to drive Indian

Monsoon and there is insigniﬁcant eﬀect in the opposite

direction.

Although the NAO is known to be a leading mode

of winter climate variability over Europe [109–111], the

directionality or feedback in NAO related climate ef-

fects has been studied by a few causality analysis stud-

ies [9, 112, 113]. We investigate the NAO-European tem-

peratures relationship on both monthly and daily time

scales using winter only data. While PCCC indicates

that NAO drives central European temperatures with no

signiﬁcant feedback on the longer monthly scale, on the

daily scale it shows no signiﬁcant causation in either di-

rection. On the other hand, CCC and CMI, based on one

dimensional time series, indicate a strong inﬂuence from

NAO to Frankfurt daily mean temperatures. This re-

sult indicates that the NAO inﬂuence on European win-

ter 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 reﬂected in ordinal cod-

ing. 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, require-

ment 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. Thus, we can expect our analysis

and inferences presented here on some highly-researched

and long-debated climatic interactions to be highly ro-

bust and reliable. We also expect that the use of PCCC

on other challenging datasets from climatology and other

ﬁelds will be helpful to shed light on the causal linkages

in considered systems.

IV. METHODS

Compression Complexity Causality (CCC) is

deﬁned as the change in the dynamical compression-

complexity of time series ywhen ∆yis seen to be gen-

erated jointly by the dynamical evolution of both ypast

and xpast as opposed to by the reality of the dynami-

cal evolution of ypast alone. ypast, xpast are windows of a

particular length Ltaken from contemporary time points

of time series yand xrespectively and ∆yis a window

of length wfollowing ypast [41]. Dynamical compression-

complexity (CC) is estimated using the measure eﬀort-

to-compress (ETC) [115] and given by:

CC (∆y|ypast ) = E T C(ypast + ∆y)−E T C(ypast),(3)

CC (∆y|ypast , xpast ) =

ET C (ypast + ∆y, xpast + ∆y)−ET C (xpast, ypast),(4)

Eq. (3) computes the dynamical compression-

complexity of ∆yas a dynamical evolution of ypast alone.

Eq. (4) computes the dynamical compression-complexity

of ∆yas a dynamical evolution of both ypast and xpast.

CCCxpast→yis then estimated as:

CCCxpast→∆y=CC(∆y|ypast )−CC(∆y|ypast, xpast ).

(5)

Averaged CCC from xto yover the entire length of

time series with the window ∆ybeing slided by a step-

size of δis estimated as —

CCCx→y=CCCxpast→∆y

=CC (∆y|ypast )−CC(∆y|xpast , ypast),(6)

If CC (∆y|ypast )≈CC(∆y|xpast , ypast), there is no

causality from xto y. Surrogate time series are gen-

erated for both xand yand the CCCx→yvalues of the

original and surrogate time series compared. If the CCC

computed for original time series is statistically diﬀerent

9

from that of surrogate time series, we can infer the pres-

ence of causal relation from x→y[42]. CCCx→ycan

be both <or >0 depending upon the nature or quality

of the causal relationship [41]. The magnitude indicates

the strength of causation.

Selection of parameters: L, w, δ and the number of

bins, B, for symbolizing the time series using equidis-

tant binning (ETC is applied to symbolic sequences) is

done using parameter selection criteria given in the sup-

plementary text of [41].

Permutation Compression-Complexity Causal-

ity is the causal inference technique proposed and im-

plemented in this work. Given a pair of time series x1

and x2from dynamical systems in which causation is

to be checked from x1to x2, we ﬁrst embed the time

series of the potential driver (x1here) in the following

manner: x1(t), x1(t+η), x1(t+ 2η),...x1(t+ (m−1)η),

where ηis the time delay and mis the embedding di-

mension of x1.ηis computed as the ﬁrst minimum

of auto mutual information function. The 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 tis given as

ˆx1(t) = (x1(t), x1(t+η), x1(t+ 2η)). Symbols 0,1,2 are

then used for labelling the pattern for ˆx(t) at each time

point by sorting the embedded values in ascending or-

der, with 2 being used for the highest value and 0 for the

lowest. If two or more values are exactly same, they are

labelled diﬀerently depending on the order of their occur-

rence. A total of m! = 3! patterns at time tare possible in

this case. Thus, ˆx(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) af-

ter symbolizing x2(t) using standard equidistant binning

with m! bins. Thus,

P CC Cx1→x2=CCCˆx1→x2.(7)

Permutation binning is not done for the potential

driver series as it was found from simulation experiments

(R¨ossler data) that embedding the ‘cause’ alone works

better for the CCC measure. Full dimensionality of the

cause is necessary to predict the eﬀect. Hence, embed-

ding only the cause helps to recover the causal relation-

ship. PCCC helps to take into account the multidimen-

sional 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 esti-

mated from x2→x1,x2is embedded and x1remains as

it is. Just like CCC, the PCCC measure can also take

negative values.

Conditional Mutual Information (CMI) of the

variables Xand Ygiven the variable Zis a common

information-theoretic functional used for the causality

detection, and can be obtained as

I(X;Y|Z) = H(X|Z) + H(Y|Z)−H(X, Y |Z) (8)

where H(X1, X2, ...|Z) = H(X1, X2, ...)−H(Z) is

the conditional entropy, and the joint Shannon entropy

H(X1, X2, ...) is deﬁned as:

H(X1, X2, ...) = −X

x1,x2,...

p(x1, x2, ...) log p(x1, x2, ...)

(9)

where p(x1, x2, ...) = P r[X1=x1, X2=x2, ...] is the

joint probability distribution function of the amplitude

of variables {X1, X2, ...}. In order to detect the cou-

pling direction among two dynamical variables of X

and Y, Paluˇs et al. [21] used the conditional mu-

tual information I(X(t); Y(t+τ)|Y(t)), that captures

the net information about the τ-future of the process

Ycontained in the process X. As mentioned in the

Introduction, to estimate other unknown variables, an

m-dimensional state vector Xcan be reconstructed as

X(t) = {x(t), x(t−η), ..., x(t−(m−1)η)}. Accord-

ingly, CMI deﬁned above can be represented by its re-

constructed version for all variables of X(t), Y(t+τ) and

Y(t). However, extensive numerical studies [22] demon-

strated that CMI in the form

I(X(t); Y(t+τ)|Y(t), Y (t−η), ..., Y (t−(m−1)η)) (10)

is suﬃcient to infer direction of coupling among dy-

namical 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 anal-

ysis described earlier in the PCCC deﬁnition. In this

approach, all marginal, joint or conditional probability

distribution functions of the amplitude of the variables

are replaced by their symbolized versions, thus Eq. (9)

should be replaced by

H(ˆ

X1,ˆ

X2, ...) = −X

ˆx1,ˆx2,...

p(ˆx1,ˆx2, ...) log p(ˆx1,ˆx2, ...)

(11)

where p(ˆx1,ˆx2, ...) = P r [ˆ

X1= ˆx1,ˆ

X2= ˆx2, ...] 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 τwith τ+ (m−1)ηin 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 II for diﬀerent datasets.

DATA AVAILABILITY

The millenial scale CO2and temper-

ature datasets are freely available at

https://zenodo.org/record/4562996#.YiDbTN_ML3A.

10

TABLE II. Parameters corresponding to each method, used

for diﬀerent datasets.

Dataset Embedding CCC PCCC

CMI/

PCMI

R¨ossler

ηx1= 5

ηx2= 5

m= 3

L= 300

w= 30

δ= 30

B= 8

L= 25

w= 15

δ= 20 τ= 20

Millenial

CO2-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

CO2-T

ηCO2= 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

ηCH4= 10

ηT= 8

m= 3

L= 60

w= 15

δ= 20

B= 4

L= 30

w= 15

δ= 20 τ= 1 −30

Monthly

CO2-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

ηEN SO = 1

ηSAS M = 4

m= 3

L= 60

w= 15

δ= 20

B= 4

L= 60

w= 30

δ= 30 τ= 1 −30

Monthly

NINO-

India

Monsoon

ηNI N O = 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

ηNAO = 15

ηT= 15

m= 3

L= 40

w= 15

δ= 20

B= 4

L= 30

w= 15

δ= 20 τ= 1 −30

Kilo-year scale CO2, CH4and temperature datasets

are available as supplementary ﬁles for [64] at

https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2015RG000482.

Monthly CO2recordings are taken from

the NOAA repository and are available at

https://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.

The yearly El Ni˜no/Southern Oscillation Index Recon-

struction dataset is taken from the NOAA repository,

https://www.ncei.noaa.gov/access/paleo-search/study/11194.

The yearly South Asian Summer Monsoon Index Recon-

struction dataset is taken from the NOAA repository,

https://www.ncei.noaa.gov/access/paleo-search/study/17369.

Monthly Ni˜no 3.4 SST Index dataset is

taken from the NOAA repository, available at

https://psl.noaa.gov/gcos_wgsp/Timeseries/Nino34/.

Monthly all India rainfall dataset is made avail-

able by the World Metereological Organization

at http://climexp.knmi.nl/data/pALLIN.dat.

Reconstructed monthly North Atlantic Oscilla-

tion Index is available at the NOAA repository,

https://psl.noaa.gov/gcos_wgsp/Timeseries/RNAO/.

Monthly Central European 500 Year Temperature Re-

constructions are available at the NOAA repository,

https://www.ncei.noaa.gov/access/metadata/landing-page/bin/iso?id=noaa-recon-9970.

Daily North Atlantic Oscillation Index

is available at the NOAA repository,

https://www.cpc.ncep.noaa.gov/products/precip/CWlink/pna/nao.shtml.

Daily Frankfurt air temperatures are

made available by the ECA&D project at

https://www.ecad.eu/dailydata/predefinedseries.php.

CODE AVAILABILITY

The computer codes used in

this study are freely available at

https://github.com/AditiKathpalia/PermutationCCC

under the Apache 2.0 Open-source license.

ACKNOWLEDGMENTS

This study is supported by the Czech Science Foun-

dation, Pro ject No. GA19-16066S and by the Czech

Academy of Sciences, Praemium Academiae awarded to

M. Paluˇs.

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