# Causality for Earth Science - A Review on Time-series and Spatiotemporal
Causality Methods

###### Abstract

This survey paper covers the breadth and depth of time-series and
spatiotemporal causality methods, and their applications in Earth Science.
More specifically, the paper presents an overview of causal discovery and
causal inference, explains the underlying causal assumptions, and enlists
evaluation techniques and key terminologies of the domain area. The paper
elicits the various state-of-the-art methods introduced for time-series and
spatiotemporal causal analysis along with their strengths and limitations. The
paper further describes the existing applications of several methods for
answering specific Earth Science questions such as extreme weather events, sea
level rise, teleconnections etc. This survey paper can serve as a primer for
Data Science researchers interested in data-driven causal study as we share a
list of resources, such as Earth Science datasets (synthetic, simulated and
observational data) and open source tools for causal analysis. It will equally
benefit the Earth Science community interested in taking an AI-driven approach
to study the causality of different dynamic and thermodynamic processes as we
present the open challenges and opportunities in performing causality-based
Earth Science study.

## Significance Statement

Causality is the study of discovering cause-effect relations in data. Earth
Science can benefit greatly from causal methods as researchers have started
utilizing causality to understand the complex interactions leading to climate
change, long term weather patterns, extreme weather events, etc. This paper
provides a primer to causal inference and causal discovery for different Earth
Science domains with some of the existing applications and open questions for
future research. The paper also comprises a holistic list of available tools
and datasets that can help both Data Science and Earth Science communities in
performing causal analysis on Earth data.

##  1 Introduction

Earth science is the study of the Earth’s physical properties, structure, and
processes. It encompasses a wide range of topics including geology,
meteorology, oceanography, and environmental science (Smith and Pun, 2013).
Earth science research relies purely on climate models that replicate complex
dynamical systems. Climate models allow scientists to better understand how
the Earth’s climate system works and how it might change in the future. These
models can be used to make predictions about future climate change and to test
different scenarios, for instance studying the global carbon footprint to
reduce greenhouse gas emissions. However, climate models are computationally
expensive and require a lot of computing power to run. This is because they
need to simulate many different dynamic processes that occur in the
atmosphere, oceans, and on land. For instance a simulation that works on cloud
formation or generating ocean currents.

In recent decades, there has been a significant increase in the availability
of large-scale climate data from various observational sources (such as
satellite remote sensing, station-based or field site measurements) and Earth
system model outputs (Guo et al., 2015). This data, along with the increasing
computational power, has opened up new ways to use data-driven methods for
observational causal discoveries without relying on the correlation and trend
analyses of the data (Rubin, 2005). In Earth system sciences, causal structure
discovery (CSD) models are making inroads into several domains (Melkas et al.,
2021). Climate causal studies pose the changing trend of applying the current
state-of-the-art climate data not only to correlation and regression methods
but also to causal inference methods. For instance, climate researchers have
realized that climate simulations introduce ambiguous values to the datasets
which makes them not applicable in decision-making applications (Ebert-Uphoff
and Deng, 2012). After understanding the importance of data-driven methods in
climate science, scientists have started to explore different causal structure
algorithms and Bayesian networks to get deeper insights into causal hypotheses
and the evaluation of physical models (Ebert-Uphoff and Deng, 2012). In the
past, Bayesian models were used to compute the probability of climate data
where it is needed for forecasting and risk assessment tools. Now, Bayesian
models are significantly used in climate science to test the causal hypothesis
to develop decision-making algorithms. A recent research study (Nowack et al.,
2020) evaluated the ability of the Coupled Model Intercomparison Project Phase
5 (CMIP5) to simulate atmospheric dynamical interactions using the causal
model evaluation (CME) framework. This study highlighted the potential of the
CME framework to offer a pathway to reducing uncertainties in climate change
projections by quantifying differences between models and observations. To sum
it up, causality methods can play a crucial role in Earth Science studies by
helping researchers understand and quantify the cause-and-effect relationships
within complex Earth systems for a variety of usecases. Some of these include:
identifying the primary drivers of environmental changes, attributing specific
climate events like volcanic eruptions, identifying causes of extreme weather
events such as hurricanes, droughts and floods, helping in the prediction of
natural hazards such as earthquakes and tsunamis, designing effective
pollution control policies, understanding El Niño/La Niña events and their
global climate impacts, etc.

This survey paper is therefore an effort to provide an overview of the current
state of knowledge on causality in Earth Science. It will serve as a primer to
help understand key concepts in causality and methods that are commonly used.
In addition to existing survey papers (Table 1), our survey paper will help
identify gaps in current causality-based Earth Science research and highlight
areas where causal methods could be used to improve our understanding of the
Earth’s system. The paper is organized as follows. Section 2 enlists some open
challenges in causality-based study of Earth Science problems. Section 3
explains the concept of causal structure learning or ”Causal Discovery”. Here,
we first explain the foundational concepts of causal discovery such as key
terms, causal assumptions and the relevant evaluation metrics for the
structure learning task. We then enlist several approaches to perform causal
discovery on time-series and spatiotemporal data. Finally, we share the
applications of causal discovery in Earth Science as found in the literature.
Section 4 explains the process of estimating causal effects, also known as
”Causal Inference”. Just like the previous section, we first explain the key
terms, causal assumption and evaluation metrics related to Causal inference.
The subsequent topics elicits the various state-of-the-art methods introduced
for time-series and spatiotemporal causal inference along with their strengths
and limitations. This section further describes the existing applications of
several causal inference methods for answering specific Earth Science
questions such as extreme weather events, sea level rise, etc. In Section 5,
we share a list of resources including Earth Science datasets (synthetic,
simulated and observational data) and open source toolboxes for causal
analysis. Lastly, we conclude our work and share potential research pathways
in Section 6. We hope our paper equally benefits both the Data Science
community interested in data-driven causal study and the Earth Science
community interested in taking an AI-driven approach to study the causality of
different dynamic and thermodynamic processes.

Table 1: Comparison among the existing survey papers. Paper Title | Discovery | Inference | Datasets | Metrics | Software | Time series | Spatial
---|---|---|---|---|---|---|---
Inferring causation from time series in Earth system sciences (Runge et al., 2019b) | $\checkmark$ |  | $\checkmark$ |  |  | $\checkmark$ |
D’ya Like DAGs? A Survey on Structure Learning and Causal Discovery (Vowels et al., 2022) | $\checkmark$ |  | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |
Survey and Evaluation of Causal Discovery Methods for Time Series (Assaad et al., 2022) | $\checkmark$ |  | $\checkmark$ | $\checkmark$ |  | $\checkmark$ |
Causal Structure Learning (Heinze-Deml et al., 2018) | $\checkmark$ |  | $\checkmark$ | $\checkmark$ |  |  |
Review of Causal Discovery Methods Based on Graphical Models (Glymour et al., 2019) | $\checkmark$ |  |  |  |  | $\checkmark$ |
A Survey of Learning Causality with Data: Problems and Methods (Guo et al., 2020) | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |
Causal inference for time series analysis: problems,methods and evaluation (Moraffah et al., 2021) | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  | $\checkmark$ |
A Survey on Causal Discovery Methods for I.I.D. and Time Series Data (Hasan et al., 2023) | $\checkmark$ |  | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |
Causal Inference for Time Series (Runge et al., 2023) | $\checkmark$ | $\checkmark$ |  | $\checkmark$ | $\checkmark$ | $\checkmark$ |
A Primer on Deep Learning for Causal Inference (Koch et al., 2021) |  | $\checkmark$ |  | $\checkmark$ | $\checkmark$ | $\checkmark$ |
Causal Discovery from Temporal Data: An Overview and New Perspectives (Gong et al., 2023) | $\checkmark$ |  | $\checkmark$ | $\checkmark$ |  | $\checkmark$ |
A Survey on Causal Inference (Yao et al., 2021) |  | $\checkmark$ |  | $\checkmark$ | $\checkmark$ |  |
Causal inference for process understanding in Earth sciences (Massmann et al., 2021) | $\checkmark$ | $\checkmark$ |  |  |  |  |
Spatial Causality: A Systematic Review on Spatial Causal Inference (Akbari et al., 2023) |  | $\checkmark$ |  | $\checkmark$ |  |  | $\checkmark$
A review of spatial causal inference methods for environmental and epidemiological applications (Reich et al., 2021) | $\checkmark$ | $\checkmark$ |  | $\checkmark$ |  |  | $\checkmark$
Our Survey | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Table 2: Examples of causality-related open questions in Earth Science domains. Earth Science Domain | Key Questions
---|---
Atmosphere | How do natural and anthropogenic activities influence greenhouse gas concentrations and atmospheric composition?
| How do aerosols influence cloud formation, precipitation patterns, and
regional climate variability?
| How are climate change and global warming influencing patterns of weather
extremes, including heatwaves,
| droughts, extreme rainfall events, and storms?
| What are the mechanisms driving changes in atmospheric circulation patterns,
such as shifts in jet streams
| and tropical cyclones?
Cryosphere | What are the primary drivers of amplified Arctic warming and accelerated loss of sea ice?
| How does Arctic amplification contribute to more frequent extreme heat,
wildfire and increasing precipitation
| at high latitudes?
| What factors drive the loss of glacier mass in both the Arctic and Antarctic
regions, and what implications does
| this have for global weather patterns?
Hydrosphere | What are the climatic and non-climatic drives for water cycle change?
| How does sea level rise impact water-related extreme events (e.g., floods
and droughts)?
| What are the effects of alterations in soil moisture on local weather
patterns, water security, and agricultural systems?
Ocean | How are rising global temperatures affecting ocean temperatures, both at the surface and in deeper layers?
| How is increased atmospheric carbon dioxide leading to ocean acidification?
| How is climate change affecting ocean circulation patterns, including the
Gulf Stream and the Atlantic
| Meridional Overturning Circulation (AMOC)?
| What are the feedback loops between changes in the ocean carbon cycle and
global climate dynamics?
Biosphere | How is rising global temperature influencing the geographic ranges and habitats of species?
| How does amplified Arctic warming affect polar biodiversity?
| What are the consequences of coral reef degradation?
Table 3: Challenges and opportunities in performing causality-based study for Earth Science. Area | Challenges | Opportunities
---|---|---
Causal Discovery | Accurate modeling of complex processes that | Domain Knowledge based Causal Discovery
| occur in the Earth’s system |
Causal Discovery | Lack of ground truth information for validation | Synthetic datasets simulating Earth Science data
| of causal models | distributions
Causal Discovery | No information on true causal frequency | Multi-scale causal discovery
Causal Discovery / Causal Inference | Tackling biases in simulated data | Adjusting for hidden confounders
Causal Discovery / Causal Inference | High dimensional data | Extracting causally relevant variables,
|  | Causal representation learning, Causal discovery
|  | in low dimensional space
Causal Discovery / Causal Inference | Non-linear data distributions | Deep learning based Causal Discovery /
|  | Inference methods
Causal Discovery | Seasonality / Non-stationarity in data | Incorporating periodicity / segmentation,
|  | Sliding window analysis
Causal Inference | Tackling bias due to time-variying confounding | Matching Methods Treatment weighting methods
Causal Inference | Tackling bias due to spatial confounding | Neighboringhood weighting, Adjusting
|  | spillover effects

##  2 Open Challenges in Causality-based Earth Science Study

Due to the multi-way interactions between the atmosphere, hydrosphere,
cryosphere, geosphere, and biosphere, studying causality between them is a
challenging but important task, which makes it an area of high interest within
the Earth Science community. Here, we have identified several fundamental
questions within the Earth Science domain, particularly concerning the rapidly
increasing emissions of greenhouse gases and the consequential impacts of
climate change, which may benefit from causality methods. These inquiries pose
significant challenges due to several factors: 1) Data availability may be
inadequate or insufficient, hindering comprehensive analysis. 2) Climate data
is inherently complex, with multiple dimensions, making analysis and
interpretation challenging. 3) The complex interaction among numerous
variables complicates the disentanglement of causal relationships and feedback
mechanisms. 4) Confounding effects further hinder the clarification of
causality. 5) The inability to conduct controlled laboratory experiments in
climate science limits our understanding of causal relationships. 6) The
current state of climate models may lack the sophistication and precision
necessary to accurately capture the complex interactions among variables. In
Table 2, we enlist some of the causality-related open questions in different
Earth Science domains (Masson-Delmotte et al., 2021; Pörtner et al., 2022). In
Table 3, we identify some of the challenges in performing observational causal
discovery and causal inference in various domains of Earth Science and the
opportunities they pave for researchers.

##  3 Causal Discovery

Causal discovery refers to the process of discovering causal relations and
uncovering underlying data generation process in real world data, helping
researchers make more informed decisions, predict outcomes, and understand how
changes in one variable, state or process may lead to changes in others
(Pearl, 1988). In recent years, several graph-based algorithms are proposed
for discovering the causal structures from empirical data (Spirtes et al.,
2000; Tian and Pearl, 2013). The important causal connections are discovered
purely based on the statistical analyses of observation data. This section
presents an overview of time-series and spatiotemporal causal discovery, the
key concepts in causality, and common approaches for performing causal
structure learning for both temporal and spatial datasets. It further includes
brief overview of renowned causal discovery methods, their limitations and
applications in Earth Science study. To begin with, we have enlisted the key
terminologies in causal discovery in Table 7.

Table 4: Key terminologies in Causal Discovery Terminology | Explanation
---|---
Causal discovery models | It aims to discover causal structures from data
Causal inference | Estimating the strength of the causal relationship between two variables
Undirected Graph | A graph with no direction between the nodes
Probabilistic Graphical Model | It captures conditional independence relationships between interacting random variables.
Bayesian Network Model | A probabilistic graphical model that represents the conditional dependencies of a set of
| random variables via a directed acyclic graph (DAG)
Latent parameter | Unknown / hidden parameters or hypotheses
Conditional independence | Nodes that are not causally related / connected
Conditional dependence | Nodes that are causally related / connected
Markov Network | An undirected graphical model with a set of random variables having a Markov property
Correlation graph | It determines the correlation patterns in the data
Structural Causal Model | It provides a mathematical and graphical representation of causal relationships
Chain | Series-like structure (Figure 1: X1 causes X2 and X2 causes X3)
Fork | This structure shows a common cause of two effects. (Figure 1: X2 causes X1 and X3)
Collider | This structure include two causes of a single effect. (Figure 1: both X1 and X3 are the causes of X2)
Partial correlation | Partial correlation refers to lagging each of several variables on all relevant delays relative to another variable
Mutual independence | Each event is independent of any combination of other events in the collection
Structure learning | The process of inferring the structure of directed acyclic graph (DAG) from data
Moralization | Cognitive enhancement in causal selection process
Markov equivalence class | A set of DAGs that shares the same set of conditional independencies
Mutual Information Network | A measure of mutual information between two variables in the Bayesian network
Probability | Likelihood occurrence of an event
Figure 1: Three structures of causal graphical model. The nodes represents
entities and edges represent direction of causal relation.

###  3.1 Causal Assumptions

There are certain foundational principles and conditions, referred to as
causal assumptions, that underlie the process of inferring causal
relationships between variables. Different causal discovery methods may have
different sets of assumptions. Some of the common causal assumptions are given
below:

##### Causal Sufficiency:

If for every measured variable in the data, all of its common causes are also
measured, then the causal sufficiency assumption holds. As per this condition,
a pair of variables is causally sufficient if all the common causes are
measured, i.e. there are no latent confounders (Nogueira et al., 2021). Causal
sufficiency is a very strong assumption as there is no such thing as a closed
world. In real-world settings, it is not always the case that all possible
causes are measured and hence, this assumption is often violated (Pellet and
Elisseeff, 2008). Causal insufficiency occurs when for a set of variables V,
there exists variables not present in V that are direct causes of more than
one variable in V. In Figure 2, the variables $A,C,B$ are causally
insufficient since another variable D outside of set V is the common cause of
B & C.

Figure 2:  An example to demonstrate causal sufficiency.

##### Causal Markov Condition (CMC) / Causal Markov property:

The causal Markov assumption states that every node in a causal DAG G is
independent of its non-descendants when conditioned on its parents. The idea
behind the CMC is that when a variable’s effects (descendents) are ignored,
all the relevant probabilistic information about that variable can be obtained
from its direct causes (parents) (Scheines, 1997). CMC can also be explained
in terms of d-separation. In a causal DAG G, all variables that are
d-separated will be conditionally independent of each other in the respective
probability distribution (Weinberger, 2018). This also implies that, among
three disjoint nodes, one node must block all paths between the other two
nodes (Nogueira et al., 2021). The causal Markov assumption is true in a DAG
which is causally sufficient i.e. have no latent confounders.

Figure 3:  An example to demonstrate the causal Markov property.

The CMC for the variables in the causal graph of Figure 3 is as follows:

C $\perp$ B $|$ A (C is independent of its non-descendant B conditioned on its
parent A)
B $\perp$ C $|$ A (B is independent of its non-descendant C conditioned on its
parent A)

##### Causal Faithfulness Condition (CFC):

The causal faithfulness assumption can be stated as follows. In a causal DAG
G, no conditional independence between the variables holds unless entailed by
the causal Markov condition (CMC). That is, CFC specifies which variables in a
DAG will be probabilistically dependent (Weinberger, 2018). This is the
converse of the Markov condition since CMC specifies which variables will be
conditionally independent given their parents. With respect to d-separation,
CFC means that except for the d-separated variables in a DAG, all other
variables are dependent. In Figure 3, variables (A,C) and (A,B) are dependent.
Based on CFC, it can be interpreted that no observed interdependency in the
data is accidental, but results from the underlying causal mechanism
(Druzdzel, 2009).

##### Acyclicity:

The acyclicity assumption states that no feedback loops are allowed in a
causal graph. That is, there can be no directed paths from a variable back to
itself similar to the structure of a directed acyclic graph (DAG). This is a
very basic property of a causal graph Figure 4b) that must be fulfilled to
represent the causal relationships accurately. Therefore, a causal graph is a
DAG where no variable is its ancestor or its descendant (Greenland and Pearl,
2007).

(a) (b)

Figure 4:  An example to demonstrate the acyclicity property where the graph
in (a) holds acyclicity and (b) violates acyclicity.

##### Data Assumptions:

Causal discovery approaches heavily rely on the available data and their
assumptions. The data for performing any causal analysis can be either
observational or interventional in nature, or both. To conduct causal
discovery, often different types of assumptions about the data distribution
are considered. They include data being linear or nonlinear, continuously
valued or discrete valued, stationary or non-stationary (for temporal data),
etc. Often assumptions about the varied noise distributions such as Gaussian,
Gumbel, or Exponential noise to which the data may belong are made. Apart from
these, some common assumptions about data are the existence of any sampling or
selection bias, missing data, latent variables or noise, etc.

###  3.2 Evaluation Metrics

Evaluating the performance of causal discovery methods is crucial to assess
their accuracy and reliability in identifying causal relationships from
observational data. Given below are the commonly used evaluation metrics for
causal discovery:

##### True Positive Rate (TPR):

TPR is the probability that an actual positive will test positive. Assuming
the threshold $t$ of probability of an edge $p(a_{ij})$ ranges from $(0,1)$,
TPR is defined as

$TPR_{t}=|\frac{\\{(i,j):p(a_{ij})\geq t\\}\cap S}{S}|$ (1)

where $S$ is the set of ground truth edges (i.e., ${(i,j):a^{*}_{ij}=1})$.

##### False Positive Rate:

False positive rate (FPR) is the probability that an actual negative will test
positive. Assuming the threshold $t$ of probability of an edge $p(a_{ij})$
ranges from $(0,1)$, $FPR$ is defined as

$FPR_{t}=|\frac{\\{(i,j):p(a_{ij})\geq t\\}\cap\hat{S}}{\hat{S}}|$ (2)

where $\hat{S}$ is the set of ground truth missing edges (i.e.,
${(i,j):a^{*}_{ij}=0}$).

##### Structural Hamming Distance (SHD):

Structural Hamming Distance (SHD) is the number of operators required to make
two PDAGs match. The operators include adding or removing an undirected edge,
and adding, removing or reversing the orientation of an edge. Let $\mathbb{P}$
be the space of PDAGs over $p$ variables. SHD is defined as:

$\begin{split}SHD:&\mathbb{P}\times\mathbb{P}\Rightarrow\mathbb{N}\\\
&(\mathcal{G,H})\mapsto{(i,j)\in\mathbf{V^{2}}}\end{split}$ (3)

##### Structural Intervention Distance (SID):

Structural intervention distance (SID) is the number of vertice pairs $(i,j)$
for which the estimate DAG correctly predicts intervention distributions
within the class of distributions that is Markov with respect to another DAG.

##### Area Over Curve (AOC):

Area over curve (AOC) simply area under curve (AUC) with $1-AUC$. The curve is
plotted with $TPR_{t}$ against $FPR_{t}$ as $(FPR_{t},TPR_{t})$, where
$t\in[0,1]$.

###  3.3 Common Approaches

Causal discovery methods can be categorized into following three major
categories or approaches:

##### Constraint based Approach:

One of the most common approaches for performing causal discovery is the
constraint-based approach which relies on testing the conditional
independencies between the variables to identify the causal edges. In the
constraint-based category, one of the oldest approaches is the Peter-Clark
(PC) algorithm proposed by (Spirtes et al., 2000). It assumes faithfulness
which implies that all the interdependencies in a DAG need to be under the
d-separation criterion. Another well-known constraint-based approach is the
Fast Causal Inference (FCI) algorithm (Spirtes et al., 2000) which differs
from the PC algorithm as it violates causal sufficiency. That is, FCI assumes
that some common causes may be unmeasured in real-world data.

##### Score based Approach:

The score-based approach is another common approach for causal discovery.
Score-based approaches try to find the DAG with the maximum likelihood given
the data (Triantafillou and Tsamardinos, 2016). They search over the space of
all possible DAGs aiming to maximize a score function that depicts how well
the graph fits the data. As a score function, the Bayesian Information
Criterion (BIC) and other similar approaches are used to score the produced
graphs and output the DAG with the best score (fits data the best). The score
function can be modified to optimize or reduce the search space of all
possible DAGs. In terms of performance, score-based algorithms can be cost
ineffective as they need to score every possible DAG (Nogueira et al., 2021).
When the number of variables increases even slightly, the number of possible
graphs can grow super-exponentially. One of the famous score-based methods is
the Greedy Equivalence Search (GES) algorithm (Chickering, 2002) that
initially starts with an empty graph, and keeps on adding currently needed
edges, and then deletes unnecessary edges as per the score function.

##### Functional Causal Models-based Approach:

Functional Causal Model (FCM) or also called SCM based approaches express the
causal relationships in a specific functional form (a set of equations) where
the variables are a function ($y=f(pa_{y})+e$) of their direct causes (causal
parents) and an independent noise term $e$. These approaches rely heavily on
the independent noise for the recovery and estimation of the causal relations.
The FCM-based methods try to distinguish different DAGs in the same
equivalence class by imposing additional assumptions on the data distributions
and/or function classes. A wide range of FCM-based approaches exists for both
temporal and non-temporal data that tries to handle both linear and non-linear
causal relations using methods such as independent component analysis (ICA)
(Stone, 2004) and additive noise models (ANMs) (Hoyer et al., 2008)
respectively.

###  3.4 Time-Series Causal Discovery

Timeseries causal discovery refers to the process of identifying causal
relationships between variables in a timeseries data setting. Causal discovery
in timeseries data involves dealing with the specific challenges of temporal
dependencies and potential feedback loops. It has been widely used in
econometrics, neuroscience, and climate science (Spirtes et al., 2000; Zhou et
al., 2014; Van Nes et al., 2015; Ghysels et al., 2016). Identifying time lags
is an important aspect of causal discovery in timeseries data. It involves
determining the delay or time difference between the cause and its effect.
Time lags are crucial for understanding the dynamics of the causal
relationship and for making accurate predictions in timeseries forecasting
tasks. To identify time lags in causal relationships, researchers may use
methods like cross-correlation analysis, autocorrelation, or timeseries cross-
validation techniques (Spirtes et al., 2000). These methods help measure the
strength of the association between variables at different time lags and
assist in determining the optimal lag that maximizes the predictive
performance or the strength of the causal relationship. We explain some of the
widely used causal discovery methods below, along with their variants,
underlying causal assumptions and limitations. A tabulated summary of these
time-series causal discovery methods is given in Table 5.

Table 5: Comparison among the causal discovery methods for time series data. Here () is mentioned against the methods where the assumptions or relation is non-existent or not explicitly mentioned in the paper. Method | Assumptions | Causal Relationship | Produced Edge
---|---|---|---
Faithfulness | Sufficiency |  | Causal Markov
---
Property
Linear | Non-linear | Instantaneous | Lagged
Granger Causality |  | $\checkmark$ |  | $\checkmark$ |  |  |
PCMCI | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  | $\checkmark$
PCMCI+ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
LPCMCI | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  | $\checkmark$
LiNGAM |  | $\checkmark$ |  | $\checkmark$ |  |  |
VarLiNGAM |  | $\checkmark$ |  | $\checkmark$ |  | $\checkmark$ | $\checkmark$
TS-LiNGAM |  | $\checkmark$ |  |  |  |  |
TiMINo |  |  |  |  | $\checkmark$ |  |
NAVAR |  |  |  |  | $\checkmark$ |  | $\checkmark$

####  3.4.1 Granger Causality, V-Granger

Granger causality is a statistical hypothesis test used to determine whether
one time series can predict another Granger (1969). It is named after Clive
Granger, a Nobel laureate economist who developed the concept. Granger
causality is widely used in time series analysis to investigate the causal
relationships between variables. The basic idea behind Granger causality is
that if a time series X ”Granger-causes” another time series Y, then past
values of X contain information that helps predict future values of Y beyond
what can be predicted using past values of Y alone. It is important to note
that Granger causality does not establish a true cause-and-effect relationship
in the sense of a controlled experiment. It only identifies whether one time
series provides useful predictive information for another time series.
Nevertheless, Granger causality is commonly used in various fields, including
economics, finance, engineering, and neuroscience, to explore relationships
between time-varying data. However, it has limitations and assumptions. For
instance, the choice of time lags can influence the results. GC assumes causal
sufficiency condition, stationary and linear relationship and does not predict
the direction of causal relation. The method is data hungry and requires
sufficiently large sample size. Granger causality can be a useful exploratory
tool, and its results should be interpreted carefully in the context of the
specific data and research question at hand.

With the Granger Causality widely used in various fields to understand the
causal relationship between time series, the presence of unobserved
confounders poses a fundamental problem in observational studies, especially
for non-linear Granger causality. V-Granger (Meng, 2019), overcomes the causal
sufficiency limitation by estimating Granger causality in the presence of
unobserved confounders using a variational autoencoder (VAE) to estimate the
intractable posterior distribution and a recurrent neural network to model the
temporal relationship in the data. Through experiments on synthetic and semi-
synthetic datasets, as well as two real datasets: Butter and Temperature, the
authors demonstrated how V-Granger outperforms the traditional Granger test in
some cases. However, the method is not suitable for datasets with high noise
levels.

####  3.4.2 PCMCI, PCMCI+, LPCMCI

PCMCI1 (Runge et al., 2019a) is a method for efficient estimation of causal
graphs from high-dimensional time series datasets, which can be used for
robust forecasting and the estimation and prediction of direct, total, and
mediated effects. A baseline of PCMCI was first introduced by (Runge et al.,
2015) for identifying causal gateways and mediators in complex spatiotemporal
systems. The method combines the PC stable causal discovery algorithm with
momentary conditional independence (MCI) to estimate causal networks from
large-scale time series datasets in a two-step approach. The first step is to
find lagged parents of all individual nodes using the PC stable method. The
second step is to test the momentary conditional independence. As a result of
the second step, if two nodes are independent, then there exists no cause-
effect relationship between them. The outcome of this two-step method includes
a causal adjacency matrix, corresponding lag values and strength of identified
relationships given by p-values. PCMCI supports multiple linear and non-
parametric conditional independence tests depending on the data distribution.
For instance, the partial correlation or $ParCorr()$ test assumes linear
dependencies and Gaussian noise in the data. There are multiple variants of
PCMCI available for different downstream tasks. For instance, PCMCI+ can be
used for discovering contemporaneous and lagged causal relations in
autocorrelated nonlinear time series datasets. For details on PCMCI’s
implementation, independence tests available and different variants of PCMCI,
please refer to their Github111https://github.com/jakobrunge/tigramite
repository.

Similarly, Latent PCMCI or LPCMCI1 (Gerhardus and Runge, 2020) is a
constraint-based approach which is a variant of the PCMCI algorithm for causal
discovery from large-scale time series data in the presence of hidden
confounders. This method is specialized to learn lag-specific causal
relationships for linear and nonlinear, lagged and contemporaneous causal
discovery from observational time series data.

####  3.4.3 LinGAM, VARLiNGAM, TS-LiNGAM

LinGAM222https://lingam.readthedocs.io/en/latest/index.html (Linear Non-
Gaussian Acyclic Model) (Shimizu et al., 2006) is a framework used for causal
discovery and estimation of causal effects from observational data. It focuses
on identifying linear causal relationships among variables while considering
non-Gaussian noise distributions. LinGAM assumes acyclic causal relationships
among variables, meaning it aims to identify a directed acyclic graph (DAG)
that represents the underlying causal structure. VARLiNGAM Hyvärinen et al.
(2010) extends the foundational LiNGAM model to encompass scenarios involving
time series. This augmentation involves integrating the core LiNGAM model with
the conventional vector autoregressive models (VAR). Consequently, it
facilitates the exploration of causal relationships encompassing both delayed
and concurrent (instantaneous) effects. This stands in contrast to the
traditional VAR framework, which solely examines causal relations with a time
delay. VARLiNGAM holds similar causal assumptions as the base LinGAM model,
such as, data linearity, causal sufficiency with no hidden confounders,
acyclicity of contemporaneous causal relations and non-gaussian continuous
error variables.

TS-LiNGAM (Hyvärinen et al., 2008) is a generalization of the LiNGAM algorithm
which uses a non-Gaussianity assumption to estimate both instantaneous and
lagged causal edges. The model is a combination of autoregressive models and
structural equation modeling (SEM). It assumes the external noises to be
mutually independent, temporally uncorrelated, and to be non-Gaussian. TS-
LiNGAM combines classic least-squares estimation of an autoregressive (AR)
model with LiNGAM estimation. It does so in four steps: i) Estimation of a
classic autoregressive model, ii) Residuals computation, iii) Performing
LiNGAM analysis on the residuals, and iv) Computation of the causal effect
estimates. The method has been evaluated on real datasets such as financial
(stock) data and magnetoencephalographic data i.e measurements of the electric
activity in the brain.

####  3.4.4 TiMiNo

TiMINo (Peters et al., 2013) approach provides a new method at that time for
causal inference on time series data that overcomes some of the methodological
issues of previous methods that could not capture nonlinear and instantaneous
effects. Specifically, TiMINo extends Structural Equation Models (SEMs) to the
time series setting, and uses a restricted function class to provide general
identifiability results including lagged and instantaneous effects that can be
nonlinear and unfaithful, and non-instantaneous feedback between the time
series. Given a sample of finite and continuous-distributed multivariate time
series data, and output the summary time graph if the input data satisfies
model assumptions. Otherwise, if the input fails the assumptions, TiMINo
causality remains mostly undecided instead of drawing wrong causal
conclusions. The model is defined as
$X_{t}^{i}=f_{i}((\mathbf{PA_{p}^{i}})_{t-p},\ldots,(\mathbf{PA_{1}^{i}})_{t-1},(\mathbf{PA_{0}^{i}})_{t},N_{t}^{i})$,
where $p$ is the time lag for independence test, $(\mathbf{PA_{1}^{i}})$ is
the set of parent nodes of node $i$, $X_{t}$ is a time-series data with finite
and continuous distribution, and $N_{t}^{i}$ is a noise vector jointly
independent over i and t. The approach assumes that each time series is a
function of all direct causes and some noise variable, and restricts the
function class to provide general identifiability results. Potential
weaknesses of TiMINo are discussed including the possibility of fitting a
model in the wrong direction and the effects of independence testing on
smaller datasets.

####  3.4.5 Tidybench algorithms

(Weichwald et al., 2020) presents four
algorithms333https://github.com/sweichwald/tidybench for causal discovery from
time series data to address some common challenges such as time-aggregation,
time-delays, and time-subsampling. The algorithms also focus on the properties
of realistic weather and climate data and were among the winning algorithms in
the Causality 4 Climate (C4C) competition at the NeurIPS 2019 conference. The
output from these algorithms is an edge score matrix that contains a score for
each pair of variables ($X_{i}$, $X_{j}$) which represents the likelihood that
the edge $X_{i}$ → $X_{j}$ exists. The algorithms are based on the following
concept. Present values are regressed on past values and the regression
coefficients are inspected to decide if one variable is a Granger-cause of
another. Following is a brief description of all the algorithms: (i) SLARAC
(Subsampled Linear Auto-Regression Absolute Coefficients) fits a VAR model on
bootstrapped data samples and each time it randomly selects a lag to include,
(ii) QRBS (Quantiles of Ridge-regressed Bootstrap Samples) also considers
bootstrapped data samples and Ridge-regresses time-delays on the preceding
values, (iii) LASAR (Lasso Auto-Regression) applies Lasso regression on the
residuals of the preceding step onto values one step further in the past and
keeps track of the variable selection at each lag to fit an OLS regression in
the end with the selected variables only, and lastly, (iv) SELVAR (Selective
auto-regressive model) selects edges using a hill-climbing approach based on
the leave-one-out residual sum of squares and then scores the selected edges
with the absolute values of the regression coefficients.

####  3.4.6 Additive Nonlinear Time Series Causal Models

Chu et al. (Chu et al., 2008) presents a feasible procedure for learning a
class of non-linear time series structures, called additive non-linear time
series, and demonstrates its effectiveness in extracting detailed causal
information from stationary models. The authors propose a new model
specification procedure that combines statistical tests for conditional
independence relations and the temporal structure of time series data. Before
this, no previous methods existed for non-linear systems. Based on assumptions
of stationary data, the methods discuss and extend linear methods including PC
and Fast Causal Inference (FCI) with joint Normal distributions. The main
algorithm used is PC, which is known to be consistent only in the absence of
feedback relations and latent common causes. The paper also mentions other
related algorithms such as FCI and an algorithm by Richardson and Spirtes that
allow for latent variables and linear feedback relations, respectively.
However, no algorithm is available that is consistent for searching linear
causal models when both latent variables and feedback may be present. To
extend the PC and related algorithms to a larger class of systems that
includes nonlinear continuous models, a more general conditional independence
test is required. Therefore the paper proposes to use a new definition of the
independence test used for the additive nonlinear time series model and embed
it into a causal inference algorithm. The experiments generate simulation
results and show that the additive non-linear algorithm performs well for
trigonometric lag models and linear lag models, but less satisfactorily for
polynomial lag models. The simulation results demonstrate the effectiveness of
the additive non-linear algorithm in inferring causal structure from time
series data, particularly for trigonometric lag models and linear lag models.

####  3.4.7 Additional Methods

There are several other causal discovery methods introduced for time-series
data in recent years. Temporal Causal Discovery Framework
(TCDF444https://github.com/M-Nauta/TCDF) (Nauta et al., 2019a) is a deep
learning based framework that learns a causal graph structure in observational
time-series data using attention-based convolutional neural networks. The
model takes in a multivariate time-series, whereas the output is a graph
comprising of nodes and the edges with lagged values. TCDF uses attention
mechanism to interpret potential causal edges and Permutation Importance
Validation Method (PIVM) to validate if a potential cause is a true cause.
TCDF has been evaluated on financial data for stock returns and FMRI data to
measure brain blood flow.

Similarly, Neural Additive Vector Autoregression Models
(NAVAR555https://github.com/bartbussmann/NAVAR) is a causal discovery approach
for time series data that can handle highly non-linear relationships between
variables. NAVAR leverages deep neural networks (DNNs) to extract the
(additive) Granger causal influences from the time evolution in multivariate
time series data. NAVAR assumes an additive structure, where the predictions
depend linearly on independent nonlinear functions of the input variables
which are modeled as nonlinear functions using the neural networks. Due to
this additive structure, the causal relations can be scored and ranked. NAVAR
has been evaluated on various benchmark causal discovery datasets available on
the CauseMe666https://causeme.uv.es/ platform for the earth sciences domain.
Another deep learning based causal discovery method is Causal-
HMM777https://github.com/LilJing/causal_hmm (Li et al., 2021a). The method
proposes to predict irreversible diseases at an early stage using time series
data. It is an HMM-based hidden Markov model consisting of a subset of hidden
variables, and a reformulated sequential variational autoencoder (VAE)
framework to learn the proposed causal hidden Markov model. The model is
applied to early prediction of peripapillary atrophy on an in-house dataset of
peripapillary atrophy (PPA), and promising results are achieved on out-of-
distribution test data. An ablation study shows the effectiveness of each
component while visualizing the accurate identification of lesion regions.

###  3.5 Spatiotemporal Causal Discovery

Spatiotemporal causal discovery refers to the process of identifying causal
relationships among variables that vary both in space and time. This field
typically deals with datasets where observations are collected over both
spatial and temporal dimensions, such as data from weather systems, ecological
systems, social networks, or biological processes. The goal of spatiotemporal
causal discovery is to uncover the underlying causal structure that governs
the dynamics of these systems. This involves not only identifying causal
relationships between variables but also understanding how these relationships
change over space and time. Here we present some of the recent methods
proposed to perform causal discovery on spatiotemporal datasets.

####  3.5.1 Mapped-PCMCI

(Tibau et al., 2022) presented a spatiotemporal stochastic climate model SAVAR
which can be used to benchmark causal discovery methods for teleconnections,
providing insights into the strengths and weaknesses of different analysis
methods. The authors also introduced novel causal discovery method named
Mapped-PCMCI that outperforms existing approaches, contributing to the
advancement of process-based understanding and climate model evaluation. The
proposed SAVAR model benchmarks different teleconnection analysis methods
including grid-level causal discovery methods and a combination of dimension-
reduction and causal discovery methods. The experiments conducted in the paper
demonstrate that the grid-level causal method based on the SAVAR model
outperforms baseline causal discovery methods, which do not consider the mode
structure and attempt to directly infer the causal graph at the grid level.
Proposed with the SAVAR is the Mapped-PCMCI method, which is based on the
assumption that the causal dependencies within a gridded dataset have a lower-
dimensional mode representation. The method consists of four steps (Figure 5):
(1) Perform a dimensionality reduction method on the gridded data to extract a
limited number of mode time series variables. (2) Apply a causal discovery
method to the mode time series variables to obtain the parents and the
estimated causal network of the modes. (3) Estimate (lagged) causal effects
for all links to obtain a coefficient matrix. (4) Invert the dimension
reduction by mapping the causal effects among the grid locations using the
modes’ weights. This method aims to overcome the challenges of dealing with
large networks, nonstationary networks, and the high-dimensional and redundant
estimation problem at the grid level. It provides a spatial grid-level
representation of the causal network obtained at the mode level, allowing for
the analysis of causal relationships in complex systems. Python code for both
the SAVAR and Mapped-PCMCI were published on GitHub
888https://github.com/xtibau/.

Figure 5: An application of spatiotemporal causal discovery method (Mapped-
PCMCI) for identifying teleconnections. (Tibau et al., 2022)

####  3.5.2 Interactive Causal Structure Discovery (ICSD)

(Melkas et al., 2021) presented a workflow ICSD (Interactive Causal Structure
Discovery) in Earth System Sciences that takes into account experts’ domain
knowledge during the application of causal discovery (CD) algorithms in Earth
sciences systems. ICSD provides users an interactive way to edit the outputs
of different CD algorithms and iteratively incorporate prior knowledge in the
initial output models. It formulates the structure learning problem as
building a probabilistic model of the data and encodes expert’s prior
knowledge as a prior distribution over all possible causal structures. The
current formulation takes a greedy approach which leads to local optima where
the choice of the initial state affects the final model. A score is associated
with each causal model that reflects the log-likelihood of the model. For
demonstration of the approach, three different CD algorithms (PC, GES and
LiNGAM) were experimented on a forestry dataset that includes daytime
measurements of variables such as shortwave downward radiation, temperature,
latent heat flux vapor, pressure deficit, etc. Experimental results suggest
that even a small amount of prior knowledge is useful in improving the results
of CD algorithms. The experiments also showed how overfitting and concept
drift can occur and be detected by ICSD. At present, the user navigation
begins with the outputs from different CD algorithms which is sub-optimal. A
future work is to find an optimal set of representative starting points for
exploration.

####  3.5.3 Spatio-Temporal Causal Discovery Framework(STCD)

The Spatio-Temporal Causal Discovery Framework (STCD) (Sheth et al., 2022)
aimed to identify causal relationships in the spatio-temporal domain by
enforcing both temporal and spatial constraints. This framework is motivated
by the influence of causal factors in hydrological systems where the
geographical location of a river or a subsidiary is a crucial factor in
deciding the causal parents. STCD primarily extends the TCDF (Nauta et al.,
2019b) framework by adding a component that ensures the enforcement of the
spatial constraint. TCDF is a temporal causal discovery approach based on CNNs
for multivariate time series data that uses an attention mechanism to identify
potential causal parents of the target time-series. Using the attention
scores, it filters the list of potential candidates. The proposed framework
STCD on the other hand eliminates irrelevant candidates by penalizing the
attention scores if the candidates violate the spatial constraints. When the
attention vector for a target time series is obtained, then the spatial
constraint is imposed on it. In hydrological systems, a river located
geographically below the target river can never be a causal parent even if
they satisfy the temporal constraint because the flow of water is always from
top to bottom. Motivated by this idea, the spatial constraint imposes a
direction of the flow. Specifically, it is the product of a distance $d$ and a
spatial coefficient $\lambda$. The distance has either a positive or negative
direction based on the geographical positioning of the locations, and
$\lambda$ controls the effect of the spatial constraint on the attention
score. The framework is tested to discover causal relationships of streamflow
at the mouth of the Brazos basin in Texas from runoff in the basin. A
limitation of STCD is that it models the spatial and temporal interactions
separately which might hinder the discovery process when the spatial and
temporal components influence each other.

####  3.5.4 Group Elastic Net

(Lozano et al., 2009) proposed a data-centric approach to climate change
attribution, using spatial-temporal causal modeling to analyze climate
observations and forcing factors. The authors developed a novel method called
Group Elastic Net to infer causality from the data and incorporate extreme
value modeling to study extreme climate events. With the assumption of spatial
stationarity, the model combines graphical modeling techniques with Granger
Causality to derive effective methods for causal modeling based on the spatio-
temporal structure of the data and enforces sparsity at the group level. The
model also applies extreme-value theory to model extreme events and
incorporates these estimates into the causal modeling and attribution process.
The experiments involve data collection from multiple sources. For real data
collection, the researchers compiled a comprehensive set of relevant variables
for climate modeling in North America. They obtained data from various sources
including the Climate Research Unit (CRU), the National Oceanic and
Atmospheric Administration (NOAA), NASA, the National Climate Data Center
(NCDC), and the Carbon Dioxide Information Analysis Center (CDIAC). These
sources provided data on climate variables such as temperature, precipitation,
solar irradiance, greenhouse gases, and aerosols. In addition to the real
climate data, the researchers also conducted simulation experiments using
synthetic spatial-temporal data. They used a spatial-temporal vector
autoregressive (VAR) model to generate the synthetic data. The experiments
compared the performance of the Group Elastic Net method, which considers
spatial interactions, with a method that neglects such interactions. The
experimental results indicate that changes in temperature are not solely
accounted for by solar radiance but are attributed more significantly to CO2
and other greenhouse gases. The results also show a significant increase in
the intensity of extreme temperatures, and these changes are attributable to
greenhouse gases. The proposed approach offers a useful alternative to
simulation-based climate modeling and provides valuable insights from a fresh
perspective.

####  3.5.5 pg-Causality

Previous methods for identifying causal pathways in air pollution have been
proposed from two perspectives: pattern-based and Bayesian-based. Pattern-
based approaches provide shallow understanding and have limited usability due
to a large number of patterns, while Bayesian-based approaches are limited by
noise, data sparsity, computational cost, and confounding factors. Zhu et al.
(Zhu et al., 2017) present a novel approach, called pg-Causality, which
combines pattern mining and Bayesian learning to efficiently identify
spatiotemporal causal pathways for air pollutants using urban big data. The
approach overcomes the challenges of noise, computational complexity, and
complex causal pathways, and thus outperforming traditional methods in terms
of time efficiency, inference accuracy, and interpretability. The authors use
the FEP Mining Algorithm to mine frequent episode patterns (FEPs) in a
symbolic pollution database. This algorithm considers constraints such as
consecutive symbols being different and a specified temporal constraint
between consecutive records. After discovering the FEPs, the authors extract
candidate causes for each sensor by finding pattern-matched pairs within a
specified time lag threshold. The authors use a Gaussian Bayesian network
(GBN) based graphical model to capture the causal relationships among air
pollutants. They generate initial causal pathways by incorporating the
extracted matched patterns and candidate sensors into the GBN model. Finally,
the authors refine the causal structures using an EM learning phase and a
structure reconstruction phase. The EM learning phase learns the parameters of
the graphical model, while the structure reconstruction phase selects the top
neighborhood sensors based on the newly generated GCscore and updates the Q
matrix. The experiments were conducted using real-world data sets from North
China, Yangtze River Delta, and Pearl River Delta. These data sets included
records of 6 air pollutants and 5 meteorological measurements. The approach
showed scalability in identifying causal pathways for air pollutants at the
sensor level, which is more than ten times larger than at the city level. The
overall experimental results showed that the proposed approach achieved high
accuracy, time efficiency, and scalability in inferring causal relationships
in air pollution data.

###  3.6 Applications of Causal Discovery in Earth Science

We explained some of the renowned causal discovery methods for time-series and
spatiotemporal data in the previous subsection. Here, we will share some of
the applications of those widely used causal discovery methods, specifically
in the Earth Science domain. A summary of these applications is provided in
Table 6.

Table 6: Applications of time-series and spatiotemporal causal discovery methods in Earth science applications. Data | Method | Application
---|---|---
Time-series | PC | Causal discovery for hydrometeorological systems (Ombadi et al. 2020)
GC | Long-term causal links in climate change events (Smirnov and Mokhov 2009, Kodra et al. 2011)
GC | Causal discovery for hydrometeorological systems (Ombadi et al. 2020)
GC | Causal discovery for teleconnections (Mosedale et al. 2006, Varando et al. 2021)
CEN | Causal discovery for midlatitude winter circulation within the Arctic (Kretschmer et al. 2016)
CEN | Causal discovery for precursors of september Arctic sea-ice extent (Li et al. 2018)
PCMCI | Causal discovery for biosphere–atmosphere interactions (Krich et al. 2020)
PCMCI | Causal discovery to study wildfire impact (Qu et al. 2021)
Causal Feature Learning | Causal discovery for teleconnections (Chalupka et al. 2016)
Tidybench Algorithms | Causal discovery for time-aggregation, time-delays and time-subsampling in
|  | weather data (Weichwald et al. 2020)
Spatiotemporal | PC-stable | Spatiotemporal causal discovery for univariate climate data (Ebert-Uphoff and Deng 2017)
PCMCI | Causal discovery for tropical–extratropical summer interactions (Di Capua et al. 2022)
Mapped-PCMCI | Causal discovery for teleconnections (Tibau et al. 2022)
STCD | Causal discovery for hydrological systems (Sheth et al. 2022)
Group Elastic Net | Causal discovery for climate change attributions ((Lozano et al. 2009)
pg-Causality | Identifying causal pathways for air pollutants (Zhu et al. 2017)

####  3.6.1 Applications of Granger Causality

In climate research, understanding complex phenomena, such as teleconnection
patterns, is important because it links atmospheric changes in one region to
impacts in distant regions. However, the automatic identification of these
patterns from observational data is still unresolved due to nonlinearities,
nonstationarities, and the limitations of correlation analyses. Varando et
al.(Varando et al., 2021) propose a deep learning approach to address these
problems and learn Granger causal feature representations that capture the
true causal effects of the target index, such as El Niño Southern Oscillation
(ENSO) or North Atlantic Oscillation (NAO). The authors propose a method
called the Granger Penalized Autoencoder with the assumptions including the
presence of nonlinearities and nonstationarities in the observational data, as
well as the limitation of correlation analyses in identifying true causal
patterns. By utilizing normalized difference vegetation index (NDVI) data
collected from MODIS reﬂectance data over 11 years, their work identified
clear patterns of the causal footprints of ENSO on vegetation in different
regions. The GitHub repository of this research is
approachable999https://github.com/IPL-UV/LatentGranger. (Mosedale et al.,
2006) used a Granger causality based approach to quantitatively measure the
feedback of daily sea surface temperatures (SSTs) on daily values of the North
Atlantic Oscillation (NAO). This was done by simulating a realistic coupled
general circulation model (GCM). This study is an extension of the work by
Mosedale et al. in 2005. where the Granger causality approach is used to find
the best time series models for modeling the coupled system for greater
flexibility. (Smirnov and Mokhov, 2009) introduced the idea of long-term
causality, which is an extension of Granger causality. Long-term causality was
estimated from data through empirical modeling and analysis of model dynamics
under different conditions. They applied this concept to find out how strongly
the global surface temperature (GST) is affected by variations in carbon
dioxide, atmospheric content, solar activity, and volcanic activity during the
last 150 years. (Kodra et al., 2011) extended the classic Granger causality
test to handle the multisource nature of data. Using a reverse cumulative
Granger causality test, they tested the hypothesis that Granger causality can
be extracted from the bivariate series of globally averaged land surface
temperature (GT) observations and observed CO2 in the atmosphere.

####  3.6.2 Applications of Pearl Causality

Causal discovery algorithms based on probabilistic graphical models have been
applied in geoscience applications to identify and visualize dynamical
processes, but the lack of ground truth and unexplained connections have posed
several challenges. To address these challenges, Ebert-Uphoff et al.(Ebert-
Uphoff and Deng, 2017) developed a simulation framework using synthetic
spatio-temporal data to better understand the physical processes and interpret
the resulting connectivity graphs, ultimately solving the mystery of the
previously unexplained connections. This approach allows for the resolution of
previously unexplained connections and provides a benchmark for other causal
discovery algorithms. The authors used a constraint-based structure learning
method called the PC stable algorithm, which is a modification of the classic
PC algorithm. This algorithm produces graph structures where observed
variables form the nodes, and connections between nodes indicate potential
cause-effect relationships. The PC stable algorithm has advantages such as
increased robustness of results and suitability for parallelization. The
authors dropped the requirement of causal sufficiency and focused on necessary
conditions for cause-effect relationships. The article discusses the results
of three different scenarios in the simulations. In Scenario 1, increasing the
spatial resolution leads to more edges for a specific time interval and fewer
edges for a longer time interval. In Scenario 2, concurrent edges are believed
to be caused by contradictory velocities at the boundaries of the advection
field. In Scenario 3, concurrent edges in the center do not match the typical
diffusion pattern and appear to fill modeling gaps. The velocity estimates in
Scenario 1 with higher speed are weak due to many high-speed interactions
represented as concurrent edges.

####  3.6.3 Applications of PCMCI

(Krich et al., 2020) used the PCMCI algorithm to study the underlying causal
relations in biosphere–atmosphere interactions. Particularly they estimated
the causal graphs from the eddy covariance measurements of land–atmosphere
fluxes and global satellite remote sensing of vegetation greenness datasets.
The causal graphs revealed the gradual shifts that correspond to little
adjustments, such as the relationship between temperature and visible heat as
well as increasing dryness which might not have been discovered merely through
correlation analysis.

(Qu et al., 2021) used PCMCI to recover the causal graphs for 8 vegetation
types which represent the causal relations and time lags between wildfire
burned areas and weather/drought and vegetation conditions. A significant
conclusion they found is that weather and aridity conditions are dominant
indicators to burned areas for grassland. Also, for broadleaf forests,
radiation while for needleleaf forests temperature is the most vital
indicator. To analyze the influence of a set of spatial patterns representing
tropical–extratropical summer interactions, (Di Capua et al., 2022) estimated
causal maps which is an extension of PCMCI to spatial fields of variables.

Lastly, given the challenges in analyzing teleconnections and the lack of
ground truth benchmark datasets, Mapped-PCMCI(Tibau et al., 2022) proposed to
tackle these challenges by presenting a simplified stochastic climate model
that generated gridded data and represents climate modes and their
teleconnections.

####  3.6.4 Applications of other Causal Discovery Methods

One of the predecessors of PCMCI method is the Causal Effect Network
(Kretschmer et al., 2016) which was implemented to identify the time-delayed
causal relationships between different actors of midlatitude winter
circulation within the Arctic. Through experiments on monthly, bi-monthly and
quarter-monthly time-series of seven meteorological variables (See (Kretschmer
et al., 2016) for details), the authors pin-pointed Barents and Kara sea ice
to be important drivers of winter circulation further confirming the
troposphere-stratosphere coupling proposed in previous literature. The causal
graph of this discovery is given in Figure 6. The same approach was utilized
by (Li et al., 2018) to study the precursors of summer (September) Arctic sea-
ice extent.

Figure 6: Causal graph discovered using Causal Effect Network (CEN)
identifying regional actors of Arctic winter circulation based on their
monthly mean time-series data. (Kretschmer et al., 2016)

(Chalupka et al., 2016) learned macro-level causal features from micro-level
variables without supervision using Causal Feature Learning (CFL). These
macro-level features are used to discover causal relationships between them
and target output. Aggregating micro-level variables into the macro-level
variables helps to ignore changes in the micro-level variables that have no
effect on the output variable Y. Here the authors calculated conditional
expectation $E[Y|x]$ using the regression of Y based on x. Then these
conditional expectations are clustered to generate the macro-level variables.
This method was used to find the causal relation between the El Nino and La
Nina events with the eastern Pacific near-surface wind (ZW, zonal wind) and
sea surface temperature (SST) variable.

(Ombadi et al., 2020) aimed to evaluate different causal discovery methods and
their performance in retrieving causal information from synthetic data and
real-world observations for hydrometeorological systems. The methods for
causal discovery in hydrometeorological systems involved in the paper are
Granger causality (GC), Transfer Entropy (TE), graph-based algorithms (PC),
and Convergent Cross Mapping (CCM). The author applies them to examine the
causal drivers of evapotranspiration in a shrubland region during summer and
winter seasons. The study aims to present the fundamentals of these methods
and shed light on their assumptions in the context of hydrometeorological
systems. Specifically, it discusses the assumptions of causal sufficiency,
causal faithfulness, and stationary time series. To evaluate performance of
the four causal discovery methods, the study uses synthetic data generated
from the bucket model and analyzes the causal structures of hydrological
systems. The evaluation includes assessing the asymptotic performance of each
method, investigating the sensitivity to sample size, and assessing the
sensitivity to the presence of noise. The observational data is obtained from
the Santa Rita Mesquite FluxNet site in Arizona to analyze the environmental
controls on evapotranspiration.

##  4 Causal Inference

Causal inference can be defined as the process of estimating the causal
effects (influence) of one event, process, state or object (a cause) on the
another event, process, state or object (an effect). Causal inference has been
applied to study environmental science for several decades, with early
applications dating back to the mid-20th century (Hill, 1965). However, much
of the earlier causal inference based analysis was done on independent and
identically distributed (i.i.d) data utilizing statistical inferential and
regression techniques to estimate the causal effects on potential outcomes
(Pearl, 2009). This section presents the key terminologies (Table 7), common
approaches and causal assumptions required to perform causal inference
techniques. We explain the different time-series and spatiotemporal causal
inference techniques introduced, along with their limitations and applications
in Earth Science.

Table 7: Key terminologies in causal inference Terminology | Explanation
---|---
causal effect | strength or infleunce of a causal relation
instance | a single unit; data sample
treatment | cause; variable that is intervened on
potential outcome | effect; variable exposed to the treatment
confounders | variable influencing both treatment and outcome
covariates | pre-treatment variables; features
intervention | nudging the value of a treatment variable
average treatment effect (ATE) | the average difference in potential outcomes with and without
| undergoing intervention

###  4.1 Common Approaches

For estimation of causal effect, there are two main categories of techniques,
potential outcome framework and do-calculus.

####  4.1.1 Potential Outcome Framework:

The potential outcome framework relies on hypothetical interventions such that
it defines the causal effect as the difference between the outcomes that would
be observed with and without exposure to the intervention. This technique is
widely used in epidemiology where patients are randomly divided into treated
and controlled groups and the effectiveness of treatment is inferred by
observing patients condition with and without undergoing a treatment (Rubin,
2005). The treatment effect can be measured at individual, treated group, sub-
treated group and entire population levels.

####  4.1.2 Do-Calculus

The first method, do-calculus was developed in 1995 to identify causal effects
in non-parametric models using conditional probabilities (Spirtes, 2010). Once
a causal structure is identified, do-calculus can be applied to find
interventional distributions by deriving mathematical representations for a
physical intervention using the $do()$ operator, as shown in Equation 4. Here,
Y represents the outcome, X represents the variable intervened on and Z
represents a set of covariates.

$\begin{multlined}P(Y|do(X=x))=\Sigma_{Z}P(Y=y|X=x,Z=z)\\\
P(Z=z)\end{multlined}P(Y|do(X=x))=\Sigma_{Z}P(Y=y|X=x,Z=z)\\\ P(Z=z)$ (4)

In case of a binary-valued variable X, the average causal effect (ACE) can be
calculated using do-calculus by calculating the difference between do(X=1) and
do(X=0), as shown in Equation 5.

$ACE=P(Y|do(X=1))-P(Y|do(X=0))$ (5)

###  4.2 Assumptions for Causal Inference

For consistent causal effect estimation on observational data, it is important
to hold the following identifiability conditions or causal assumptions:

##### Consistency:

Under the consistency condition, the potential outcome for the treated subject
$Y_{X=1}$ is considered equal to the observed outcome Y. The same goes for the
untreated subject.

##### Positivity:

This assumption implies that the probability of receiving treatment given some
covariates $Z$ is always greater than zero. That is, $Pr(X=x|Z=z)>0$ where
$Pr(Z=z)\neq 0$.

##### Conditional Exchangeability:

Under the conditional exchangeability assumption, also known as ”weak
ignorability”, the condtional probability of receiving treatment depends only
on the covariates $Z$, that is, $Y_{x}$ and treatment $X$ are are
statistically independent given every possible value of $Z$. On the contrary,
unconditional exchangeability implies that treatment group has the same
distribution of outcomes as the untreated control group.

##### SUTVA:

Under the Stable Unit Treatment Value Assumption (SUTVA), the potential
outcome $Y_{i}$ on one unit $i$ is not affected by the treatment effect on
other units and there is no hidden variations of treatment.

###  4.3 Evaluation Metrics

####  4.3.1 Root Mean Square Error (RMSE)

Researchers evaluate the performance of their predictive models using the Root
Mean Square Error (RMSE) which can be only calculated for factual
observational data but cannot be done for counterfactual predictions. Since
the ground truth information is only available for synthetic data, we further
evaluate the causal effect estimation skill of a model using the PEHE metric.

$RMSE=\sqrt{\frac{1}{N}\Sigma_{i=1}^{N}(Y_{i}-\hat{Y_{i}})^{2}}$ (6)

####  4.3.2 Precision in Estimated Heterogeneous Effects (PEHE)

This metric is commonly used in machine learning literature for calculating
the average error across the predicted average treatment effects (ATEs) (Hill,
2011).

$\sqrt{PEHE}=\sqrt{\frac{1}{N}\Sigma_{i=1}^{N}(ATE_{i}-\hat{ATE_{i}})^{2}}$
(7)

###  4.4 Time-series Causal Inference

Majority of the causal inference models work in time and space invariant
settings. However, when it comes to time-series data, the question changes to
inferring the effect of treatment $X$ on outcome $Y$ at time $t$ in the
presence of a set of covariates $Z$. Such settings require models that can
estimate time-varying causal or treatment effects. This further leads to the
problem of time-varying confounding, that is the common influence a past
treatment $X_{t}$ or covariate $Z_{t}$ might have on the future treatments
$X_{t+1}$ and the future outcome $Y_{t+1}$. Traditional methods for performing
time-series causal inference typically involve statistical modeling techniques
that aim to identify causal relationships between variables in time-series
data, however, these methods have several limitations, which can affect the
reliability and validity of causal conclusions drawn from the analysis.

In causal inference, confoundedness poses a significant challenge because it
can lead to incorrect conclusions about the causal relationship between the
independent variable and the dependent variable. When confounding variables
are not properly accounted for, the observed association between the
independent and dependent variables may be due to the confounding variable
rather than a true causal effect. Balancing scores that incorporate propensity
score are the most common approaches to debias confounding effects. To
overcome this challenge, Robin’s g-methods have shown to provide promising
results on reducing bias caused by time-varying treatment and covariates on
the potential outcome (Naimi et al., 2017). The prediction models of these
estimators are typically based on linear or logistic regression, the downside
of which is that in case of complex non-linearities in treatment or outcome
variables, these methods will lead to inaccurate results. G-methods provide
metrics to overcome the problem of time-varying confounding through
standardization, g-computation and inverse probability of treatment weighted
(IPTW) estimators (Naimi et al., 2017).

###  4.5 Time Series Causal Inference Methods

So far, we have seen causal inference surveys focusing on CI methods for
i.i.d. data (Table 1), however, real world observations require dynamic or
time-varying causal analysis focusing on calculating the impact of an
intervention on a sequential or time-varying outcome (Moraffah et al., 2021).
For instance, policymakers and climate change activists would be interested in
identifying the impact of lowering CO2 emissions on the rate of ozone
depletion over a specific period of time. Depending upon the underlying
methodology, time-series causal inference methods can be categorized into (i)
time-varying causal inference, and (ii) time-invariant causal inference
methods. We summarize some of the widely-used causal inference methods under
each category below.

###  4.6 Time-varying Causal Inference Methods

Time-varying causal inference methods are approaches used to understand and
analyze causal relationships in situations where the treatment or
intervention, the outcome, and potentially the covariates, change over time.
These methods aim to uncover how a changing treatment influences the outcome
of interest, considering the dynamic nature of both the treatment and the
outcome variables.

In contrast to traditional causal inference, where the focus is on a fixed
intervention and its effect, time-varying causal inference takes into account
the evolving nature of interventions and outcomes. This is particularly
relevant in fields such as epidemiology, economics, and environmental science,
where interventions and exposures can vary over time. Some common time-varying
causal inference methods include:

####  4.6.1 Marginal Structural Models

Marginal Structural Models (MSMs) (Robins et al., 2000) are a class of causal
inference methods designed to estimate the causal effects of time-varying
treatments in the presence of time-dependent confounding. While traditional
methods like ordinary regression models may not properly handle time-varying
treatments and confounders, MSMs were developed to address biases that can
arise when using these traditional regression models to estimate causal
effects in situations where treatments change over time. MSMs provide a
framework to model and adjust for the dynamic nature of treatments,
confounders, and their interdependencies using the IPTW weights. MSMs first
employ IPTW to re-weight the data and emulate a hypothetical time-fixed
treatment scenario. This mitigates confounding by making treated and untreated
groups comparable. A weighted regression model is then employed to estimate
the treatment effect, accounting for the dynamic confounding. MSMs rely on the
assumptions of positivity and no unmeasured confounders. Though MSMs have
become a cornerstone in time-varying causal inference in the fields of
epidemiology (VanderWeele, 2009; Hernán et al., 2000), public health
(Williamson and Ravani, 2017; VanderWeele et al., 2011) and social
sciences(Bacak and Kennedy, 2015), they also have some limitations. Complex
interactions between time-varying treatments and confounders, longitudinal
missing data and information censoring can be challenging to model using MSM
technique.

####  4.6.2 Convergent Cross Mapping

Convergent Cross Mapping (CCM) (Ye et al., 2015) is a nonlinear causal
inference technique used to detect the presence of causal relationships
between variables in time series data. It is particularly useful when the
relationship between variables is complex and non-linear. CCM is based on the
concept of time delay embedding, where time series data is transformed into
higher-dimensional space by embedding time-delayed copies of the series.

####  4.6.3 Instrument Variables

In causal graphs, a variable is called an instrumental variable (IV) if it is
independent of the hidden confounders and related to the effect only through
the cause. For a causal model of the form $Y=\beta X+g(H,\varepsilon^{Y})$
prediction of the Y based on the observation X with the presence of hidden
confounders yields a biased estimation of the $\beta$ coefficient. The
Conditional Instrumental Variable (CIV) (Thams et al., 2022) method used IVs
to identify the $\beta$ coefficient from the time series causal model with the
presence of hidden confounders applying condition on the required number of
previous instances of IVs.

####  4.6.4 Deep Representation Learning based Models

Causal inference methods based on representation learning or deep learning
techniques (Bengio et al., 2013) learn the representation of input data by
extracting features from the covariate space (Koch et al., 2021), where
majority of the existing deep learning based methods are developed for i.i.d
data (Koch et al., 2021). In these deep learning based CI methods, a single
neural network (also called meta learner) can be trained to make predictions
for both treatment and control groups individually to estimate the average
treatment effect (ATE).

For time-series causal inference, researchers have proposed methodologies
based on machine learning and deep learning models that also tackle the
problem of time-varying confounding (Moraffah et al., 2021). Recurrent
Marginal Structural Networks (R-MSN) (Lim, 2018) and Counterfactual Recurrent
Network (CRN) (Bica et al., 2020b) are some of the recent models that claim to
estimate causal effects in the presence of time-varying confounders, however,
contrary to the claim, these methods are healthcare-specific and cannot be
generalized for other domain areas like Earth science because these models
require one-hot encoded treatment flags with multivariate combined dosage.
Time Series Deconfounder - a multi-task method, leverages the assignment of
multiple binary treatments over time to enable the estimation of treatment
effects in the presence of multi-cause hidden confounders (Bica et al.,
2020a). Taking this one step further, G-Net is a recently proposed method for
time-varying dynamic treatment effect estimation (Li et al., 2021b). The
method provides a recurrent-neural network based g-computation technique to
estimate the propensity scores for handling time-varying confoundedness. Most
recently, (Ali et al., 2023) proposed TCINet, a deep learning based
counterfactual prediction model that leverages stabilized weighting (instead
of IPTW weights) using probabilistic modeling. The authors evaluated TCINet on
climate data and presented their findings on the relationship between
Greenland blocking and summer sea ice melt within Barents Sea and Kara Sea.

Though deep representation learning methods are capable of automatically
learning the intrinsic correlations and are also effective in accurate
counterfactual estimation, they often lead to predictions with high variance
or uncertainty estimates.

###  4.7 Time-invariant Causal Inference Methods

Time-invariant treatment effect estimation methods focus on quantifying the
causal impact of a treatment on an outcome when both the treatment and outcome
variables are observed at a single point in time. Whereas, a treatment is
time-invariant or fixed when it occurs at one specific point of time and then
does not change afterwards. We enlist some of the time-invariant causal
inference methods below:

####  4.7.1 Causal-ARIMA

The Causal-ARIMA (C-ARIMA) is a potential outcome framework-based method to
estimate causal effects from time series datasets (Menchetti et al., 2021).
This method is suitable for datasets where it is not possible to separate the
treated and controlled groups to estimate the effects of the applied
treatment. In an observational study, if an intervention event occurs at a
specific time point, the time series divides into two different time horizons,
pre-intervention and post-intervention. Here the intervention is not applied
at each time point, it occurred at a given time and draws the effect on the
rest of the observations of the system. The C-ARIMA method is a three-step
process: first, the relationship between the covariates and target variable is
learned from the pre-intervention data using an ARIMA model. Then using
learned relationships from pre-intervention data the counterfactual outcomes
are predicted for post-intervention data points. Finally, the causal effect is
measured from the factual observations and predicted counterfactual data. The
C-ARIMA method works on the basis of five assumptions: single persistent
intervention, non-anticipating potential outcomes, temporal no-interference,
covariates-treatment independence, and Non-anticipating treatment. The C-ARIMA
method can identify different types of causal effects without making any
structural assumptions about the effect caused due to the intervention.

####  4.7.2 Difference in Difference

The Difference-in-Differences (DID) method (Lechner, 2011) is a statistical
technique used in econometrics and social sciences to estimate the causal
impact of a treatment, intervention, or policy by comparing changes in
outcomes over time between a treatment group and a control group. DID is
particularly useful when randomization of subjects into treatment and control
groups is not possible or ethical, making it challenging to establish
causality through traditional experimental methods. To perform causal
inference using this method, start by selecting two groups: a treatment group
that is exposed to the intervention or policy change and a control group that
is not exposed. These groups should be similar in all relevant aspects except
for the treatment or policy change. Next, data is collected on the outcome of
interest for both groups, typically before and after the treatment or policy
change. Finally, the causal effect is estimated by calculating the difference
in the changes in outcomes in the controlled group and the treated groups. The
DID estimate can be analyzed using regression techniques to control for
potential confounding factors, such as demographics or other trends affecting
the outcome. However, there are some limitations. The method requires adequate
sample size to yield meaningful results and it can handle time-varying
confounders. Overall, Difference-in-Differences is a powerful tool for causal
inference when randomized controlled trials are not feasible or ethical.
However, it relies heavily on the validity of the parallel trends assumption,
and researchers must carefully consider potential sources of bias when
designing and interpreting DID studies. Mathematically, the DID estimate can
be expressed as follows:

$\text{DiD Estimate}=(Y_{T}(\hat{X})-Y_{T}(X))-(Y_{C}(\hat{X})-Y_{C}(X))$ (8)

Here, $Y_{T}$ refers to the outcome in treated group, $Y_{C}$ refers to the
outcome in the controlled group, $X$ refers to the pre-treatment data and
$\hat{X}$ refers to post-treatment data.

####  4.7.3 Interrupted Time Series

Interrupted time series (ITS) analysis (McDowall et al., 1980) is a
statistical method used for assessing the causal impact of an intervention,
treatment, or policy change on a specific outcome by analyzing changes in a
time series data set. Unlike the Difference-in-Differences (DID) method, which
compares two groups (treatment and control) before and after an intervention,
ITS focuses on a single group and tracks changes in that group’s outcomes over
time, before and after the intervention. ITS requires defining a point in time
when the intervention, treatment, or policy change occurred. This is referred
to as the ”interruption point”. A statistical model is built to estimate the
expected trajectory of the outcome in the absence of the intervention. Common
models used for pre-intervention estimation include linear regression,
autoregressive integrated moving average (ARIMA), or segmented regression
models. To estimate the effect of the intervention, observed outcomes after
the intervention are compared with the predicted outcomes from the model. The
difference between the observed and predicted outcomes at each time point
after the intervention represents the intervention effect.

Interrupted time series analysis is particularly valuable for evaluating the
impact of policies, interventions, or treatments that are implemented at a
specific point in time. It provides a rigorous framework for establishing
causal relationships by demonstrating that observed changes in the outcome are
more likely due to the intervention than to other factors. However, as with
any statistical method, the validity of ITS analysis depends on the quality of
data and the appropriate modeling of the pre-intervention trend. ITS can be a
powerful tool for causal inference when experimental designs are not feasible.

####  4.7.4 Causal Impact

The Causal Impact method (Brodersen et al., 2015) is a statistical approach
used for estimating the causal effect of an intervention, event, or treatment
on a time series data set. It was developed by Google’s Research and Data
Science teams and is implemented in the R package
(http://google.github.io/CausalImpact/CausalImpact.html). This method is
particularly useful for businesses and researchers to assess the impact of
marketing campaigns, policy changes, or any other interventions on metrics
such as sales, website traffic, or user engagement. The Causal Impact method
builds a Bayesian structural time series (BSTS) model to represent the
expected behavior of the time series data in the absence of the intervention.
The BSTS model captures trends, seasonality, and other patterns in the pre-
intervention data. The difference between the observed post-intervention data
and the counterfactual prediction represents the causal impact of the
intervention. This difference is used to estimate the treatment effect and
assess its statistical significance. Causal Impact is particularly valuable
when you want to assess the impact of a specific event or intervention on a
time series metric while accounting for the underlying patterns and variations
in the data. It offers a probabilistic and data-driven approach to causal
inference, which can be beneficial for decision-making in various domains.

####  4.7.5 Regression Discontinuity Design

The Regression Discontinuity Design (RDD) method (Hahn et al., 2001) is a
quasi-experimental research design used in causal inference to estimate the
causal effect of a treatment, intervention, or policy change when the
assignment to the treatment or control group is determined by a cutoff or
threshold in a continuous or ordinal variable. RDD is particularly useful when
random assignment to treatment and control groups is not feasible, but there
exists a natural ”discontinuity” in the assignment process that can be
exploited to make causal inferences. RDD is a powerful method for causal
inference when the assignment to treatment or control groups is determined by
a discontinuity in a continuous or ordinal variable. However, it relies on the
assumption of a smooth discontinuity and requires careful design and analysis
to ensure valid causal inferences.

RDD estimates the treatment effect using regression analysis. The method
requires defining a threshold $c$ in the data $X$. The ATE is estimated by
comparing the average outcomes for individuals just below the threshold
$(X<c)$ with those just above the threshold $(X>=c)$. The difference in
outcomes between the two groups can be attributed to the treatment or
intervention. Mathematically, the ATE estimate can be expressed as follows:

$ATE=E(Y|X>=c)-E(Y|X<c)$ (9)

Table 8: Comparison of existing time-series causal inference methods and their relevance to Earth Science. Method |  | Binary/
---
fixed treatment
| Continuous
---
treatment
| Time varying
---
treatment
| Time varying
---
covariates
| Applicable on
---
Earth Science
Difference in Difference |  |  |  |  |
Causal Impact |  |  |  |  |
Instrument Variables |  |  |  |  |
CRN |  |  |  |  |
MSM |  |  |  |  |
R-MSN |  |  |  |  |
Time-series Deconfounder |  |  |  |  |

###  4.8 Spatiotemporal Causal Inference

Methods for spatiotemporal causal inference often integrate techniques from
both causal inference and spatiotemporal analysis. These methods may include
statistical modeling, machine learning algorithms, and deep learning
approaches to estimate causal effects in observational data while considering
spatial and temporal dependencies. Reich et al. provided a comprehensive
review of the challenges and limitations of performing causal inference in
spatial settings and enlisted methods that exploit spatial structure and those
that account for spatial interference in spatiotemporal settings (Reich et
al., 2021). Here, we present some of the recent methods introduced to perform
causal inference on spatiotemporal data.

####  4.8.1 Spatially Interrupted Time-Series (SITS)

(Zhang and Ning, 2023) investigated the spatiotemporal heterogeneities in the
causal effects of mobility intervention policies during the COVID-19 outbreak
using a spatially interrupted time-series (SITS) analysis. The study aimed to
understand the spatiotemporal heterogeneity in the causal effectiveness of
these policies, particularly at finer-grained spatial and temporal
resolutions, and provide insights for policymakers on when and where to
implement specific policies to control the spread of the virus. The SITS
system involves the application of the Interrupted Time-Series (ITS) design
combined with spatial analysis. The ITS design is a quasi-experimental design
for policy evaluation that records and measures behavioral outcomes before and
after implementing a policy intervention. The SITS analysis requires a long
time series with multiple observations over time, especially before the
treatment period, to establish a credible estimation of counterfactuals. It
compares mobility outcomes over time within a single group of individuals or
neighborhoods, rather than between a treatment group and a control group. This
helps to control for potential biases arising from between-group differences.
To incorporate spatial heterogeneities, the SITS analysis considers the
spatial contexts of neighborhoods, including socioeconomic and built
environment settings. The effects of the policy intervention, such as the
level and slope changes, are moderated by these spatial contexts. The SITS
model estimates the spatiotemporally varying causal effects of the policy
interventions by incorporating these spatial moderating effects. The SITS
analysis in this study used a timeseries dataset of neighborhood-based
mobility levels extracted from a large set of mobile phone signaling data in
Shenzhen, China. The dataset allowed for tracking the daily mobility trend at
the neighborhood level before and after the COVID-19 outbreak. Two mobility
control policies were focused on: the first-level response to public health
emergencies (FLR) declared on January 23, 2020, and the lockdown of
residential communities (COM) implemented on February 7, 2020. By revealing
the disparities in the ability to alter normal activity patterns in response
to NPIs, the study also provides insights into sociospatial equality and
resilient neighborhood planning in the post-epidemic era. The study analyzed
the effects of two mobility intervention policies: the first-level response
(FLR) policy and the closed-off management (COM) policy. The spatial
distributions of the decay rates of the two policies were different, with
neighborhoods with larger decay rates primarily located in compact areas near
city centers for the FLR and at the urban fringe for the COM. The study also
found that the spatial variations in mobility change became increasingly small
or homogenous over time.

####  4.8.2 Propensity Score based Spatiotemporal Causal Inference Methods

Propensity scoring techniques are extended in spatiotemporal causal inference
setting to account for confounding caused by spatial confounders.
Papadogeorgou et. al. proposed a distance adjusted propensity score method to
incorporate spatial proximity in causal inference techniques in the presence
of observed confounders (Papadogeorgou et al., 2019). Later, they proposed a
spatiotemporal causal inference method for stochastic interventions assuming
both the treatment and outcome variables are coming from the spatiotemporal
point processes (Papadogeorgou et al., 2022). In the stochastic intervention
setting the treatments are specified through a probability distribution
instead of fixed values in the potential outcome framework. This model makes
two assumptions on the treatment: unconfoundedness and bounded relative
overlap of the treatment, which ensures that the intervention with high
probability does not become a fixed treatment. This method allows the
intervention can be applied to any spatial location, an infinite number of
interventions at the same time and measures the outcome for all applied
interventions using an intervention distribution pattern. So it is possible to
apply and measure the outcomes for any spatial and temporal intervention
patterns.

Figure 7: A schematic flow-diagram of spatiotemporal causal inference method
proposed by (Li et al., 2023)

(Li et al., 2023) proposed a method for recognizing spatiotemporal causality
and effect estimation, integrating a spatiotemporal propensity score matching
(STPSM) and Bayesian spatiotemporal LASSO regression model (BST-LASSO-RM).
This three-step method is illustrated in Figure 7.

####  4.8.3 Additional Spatiotemporal Causal Inference Methods

Christiansen et al. proposed a latent spatial confounder model to study
relationship between conflict and the forest loss in Columbia (Christiansen et
al., 2022). Papadogeorgou et al. recently proposed a Bayesian modeling
approach to causal inference in the presence of latent confounders and spatial
interference (Papadogeorgou and Samanta, 2023). Overall, spatiotemporal causal
inference still remains a lesser tapped area with much focus on spatiotemporal
causal structure learning (part of spatiotemporal causal discovery). One
reason behind this is the challenging task of inferring causation in both
space and time-varying setting in the presence of latent confounders and
spatial interference, two of the major challenges in this research area.

###  4.9 Applications of Causal Inference in Earth Science

Applications of causal inference in Earth Science is constrained by the
inability of validating the findings as observational data cannot be
controlled, randomized or intervened on. Nevertheless, we enlist some of the
recent inroads in causal inference for Earth Science domains. (Yang et al.,
2022) applied an advanced data-driven causal inference method based on the
convergent cross mapping (CCM) algorithm that provides valuable insights to
identify thermal influence and feedback among the different climate regions in
the contiguous U.S using the long-term (1901–2018) near surface air
temperature observations. (Gao et al., 2022) studied some widely applied
causal inference approaches to study the extent of the influence of different
vegetation structures on net primary production (NPP) from both temporal and
spatial perspective. Granger Causality Test (GCT), CCM, etc. are some of the
used temporal causal models in this study. The authors conclude that in
scenarios where it is difficult to infer causation from a temporal
perspective, it may be feasible to use spatial causal inference there. (Wu et
al., 2023) used a CCM method to study the the driving effects of various
environmental factors such as air temperature, vegetation index, soil
moisture, net surface radiation, precipitation, water vapor, etc. on land
surface temperature in China based on the remote-sensing and reanalysis data
from 2003–2018.

##  5 Datasets and Toolbox for Causality

Datasets play an important role when it comes to validating and analyzing the
effectiveness of any new model or method. The ground truth knowledge of the
dataset helps to approve or reject the result produced through any model. In
this section, we provide reference to multiple synthetic, simulation and real-
world datasets used for benchmarking causal discovery and inference methods
for Earth Science domain. A summary of the publicly available toolboxes for
causal study is given in Table LABEL:tab:tools.

###  5.1 Synthetic datasets

This type of data is generated using self-defined processes/equations by
introducing different statistical and distribution features. So we know the
structure of the data before applying the causal analysis method which is
helpful to computer different evaluation matrices through ground truth.

##### Harvard Dataverse:

Six synthetic datasets are available at Harvard Dataverse101010
https://dataverse.harvard.edu/dataverse/basic_causal_structures_additive_noise,
where these datasets represent different causal structures such as fork,
v-structure, mediator, diamond, and seven variables, etc. The time series of
each dataset is generated using a nonlinear function of cause variables,
linear self-causation, and additive Gaussian noise. The number of time steps
of all time series is set to 1000.

##### FLAIRS:

(Huang and Kleinberg, 2015) is a set of simulated time series datasets for
causal analysis. This resource contains 22 simulated datasets, two datasets
with a common cause and common effect, and 20 datasets each with a different
random causal structure. All datasets contain 20 variables, 1000 time points,
lags of [1,3] time units and all variables have continuous values.

##### FinanceCPT:

(Kleinberg, 2013) is a simulated financial time series dataset collection that
contains 25 variables and 10 causal structures. For each causal structure two
different observation time periods of 4000 time steps are generated, so this
collection has 20 different datasets. The two observation periods of the same
causal structure help to test any causal model that can find the same
structure in both periods. These datasets have [20, 40] random relationships
between variables with 1-3 time lags

##### PROMO dataset:

(Guyon et al., 2011) is a simple causal effect time series dataset of product
sales and different promotions applied for each product. This dataset contains
about 1000 promotion types applied on 100 different products. The daily sales
value for each product is recorded for three years (i.e., 1095 days). So the
daily sales and active promotions of each day for each product are available
in the dataset. The goal is to predict the effect of each promotion on daily
sales of each product. The ground truth causal relationship matrix is
available in the dataset to evaluate the predicted result.

###  5.2 Realistic Simulated Data

In physical systems like the Earth system, is it not possible to perform
randomized control experiments to collect data for studying causal properties.
So the acceptable alternative is the computerized simulation of the physical
systems which implements almost all dynamic processes to generate data
resembling the actual system. Though these simulations are sometimes very
time-consuming and expensive but help to produce data by changing different
factors and are possible to generate counterfactual data.

##### North American Mesoscale:

(NAM) dataset (National Centers for Environmental Prediction, National Weather
Service, NOAA, U.S. Department of Commerce, 2015) is generated by the National
Centers for Environmental Prediction (NCEP) using the WRF Non Hydrostatic
Mesoscale Model. This is a spatiotemporal dataset of 12km resolution covering
the continental United States and the data is generated every 6 hours from
2012-01-01 00:00 to 2023-10-15 18:00. Different properties of Air Temperature,
Geopotential Height, Humidity, Sea Level Pressure, Snow, Surface Pressure,
Upper Air Temperature, and Upper Level Winds are available in this simulation.

##### FMRI benchmark:

The realistic simulated Functional Magnetic Resonance Imaging dataset contains
BOLD (Blood-oxygen-level dependent) datasets for 28 different simulations for
different brain networks (Smith et al., 2011). Each dataset contains the
neural activity, based on the blood flow change. Each dataset contains up to
50 time series which contains between 50 and 5000 time points. Note that these
datasets are considered causally sufficient.

##### DREAM4:

(Marbach et al., 2009) is a synthetic gene expression dataset, generated by
software, and based on patterns found in model organisms. This dataset
consisted of both time-series and steady-state data and the objective is to
learn gene regulatory networks from gene expression data. DREAM4 consists of 5
independent datasets, each with 10 different time-series recordings for 100
genes across 21 time steps.

##### NCEP-DOE Reanalysis 2 product:

NCEP-DOE Reanalysis 2 product (Kanamitsu et al., 2002) for the time period
1979 to current, is a data product provided by the US National Centers for
Environmental Protection (NCEP) and the Department of Energy (DOE). Reanalysis
data sets are generated by fitting a complex climate model to all available
data for a given period of time, thus generating estimates for times and
locations that were not originally observed. A large set of almost 40
atmospheric variables are measured in the reanalysis dataset. The dataset
covers 90N - 90S, 0E - 357.5E with a 2.5-degree latitude x 2.5-degree
longitude global grid (144x73).

##### CausalWorld:

(Ahmed et al., 2021) is an open-source benchmarking for causal structure
learning. This platform contains a robotic environment manipulation dataset
for different tasks. These datasets represent different causal structures of
interacting objects like robot and object masses, colors, sizes, etc.
Different causal studies like do-interventions, counterfactual situations,
structure learning, inference, etc. can be performed and evaluated using this
platform.

##### Advection Diffusion:

The Advection Diffusion simulation (Ebert-Uphoff and Deng, 2015) mimics two
important dynamic processes of many geoscience systems addiction and diffusion
of a substance or property by a fluid or air. This is a spatiotemporal dataset
of different properties of these processes in two grid sizes of 10x10 and
20x20. This dataset contains different scenarios ranging from simple to
complex controlled by advection velocity field, advection and diffusion
parameters, numerical parameters, spatial and temporal resolution, etc.
parameters of these processes, and multiple runs of each scenario.

###  5.3 Real-world datasets

These are real observational datasets from the physical systems. So we don’t
know the ground truth effect for these datasets, but we can generate
counterfactuals from the knowledge learned from these datasets and measure
future effects.

##### Beijing Multi-Site Air-Quality Dataset:

The Beijing Multi-Site Air-Quality Dataset (Chen, 2019) is an observational
dataset collected by the Beijing Municipal Environmental Monitoring Center
from different monitoring stations. This dataset contains hourly observation
of 6 pollutants in the air: PM2.5, PM10, SO2, NO2, CO, and O3, and 6
meteorological variables: air temperature, wind direction and speed, pressure,
dew point temperature, and precipitation. These data were collected from March
1st, 2013 to February 28th, 2017.

##### ERA5:

ERA5 (Hersbach et al., 2023) is a global climate and weather dataset
maintained by the European Centre for Medium-Range Weather Forecasts (ECMWF).
This dataset contains hourly observations of the different atmospheric, ocean-
wave, and land-surface variables from 1940 to the present day. This dataset
covers the observations from the whole globe and the repository is updated
daily for new data.

##### Sea Ice Data:

The NSIDC DAAC SMMR and SSM/I-SSMIS data collection (DiGirolamo et al., 2022)
is a polar sea ice observational dataset maintained by the National Snow and
Ice Data Center (NSIDC). This dataset is collected from the Scanning
Multichannel Microwave Radiometer (SMMR) instrument on the Nimbus-7 satellite
and the Special Sensor Microwave/Imager (SSM/I) and Special Sensor. The data
collection consists of observational measurements of sea ice concentration and
extent, sea surface temperatures, wind stress, snow cover, rainfall rates, and
more variables from 1978 to the present.

##### Metropolit Cohort:

The Metropolit Cohort dataset contains data from 11532 men born in 1953 in the
Copenhagen Metropolitan area who were living in Denmark in 1968 (Osler et al.,
2006). This is a longitudinal dataset of these men starting from their birth
and data from different stages of their life is recorded. This dataset
comprises physical, medical, mental, social, and diagnosis information
collected from nationwide social and health registers. This is a high-quality
dataset with good validity and a low degree of measurement error.

##### YahooFinancials:

Yahoo Financials 111111Yahoo Finance datasets.
https://pypi.org/project/yahoofinancials/ is a data module consisting of
financial data of daily stock prices of companies from the stock market. Using
this dataset module we can get financial data for stock, cryptocurrency,
forex, mutual fund, commodity futures, ETF, and US Treasury from Yahoo Finance
for various time intervals and time ranges.

##### Lalonde:

The Lalonde dataset (LaLonde, 1986; Dehejia and Wahba, 1999) is a popular
observational dataset collected from the National Supported Work
Demonstration. The study looked at the effectiveness of a job training program
(the treatment) on the real earnings of an individual, a couple of years after
completion of the program. The dataset contains 445 observations,
corresponding to 185 treated and 260 control subjects, a number of demographic
variables (age, race, academic background, and previous real earnings), as
well as a treatment indicator, and the real earnings in the year 1978 (the
response). The treatment assignment indicator is the first variable of the
data frame: treatment (1 = treated; 0 = control).

Besides the dataset discussed here, CauseMe (Muñoz-Marí et al., 2020) is a
public platform that contains a large number of datasets with ground truth
causal graphs to validate the causal discovery methods. This platform contains
synthetic datasets with a wide range of properties like extreme events,
complex dynamics, different errors, time delay, etc. resembling real dynamical
systems. Real-world datasets with multiple dimensions are also included in
this benchmarking platform where the causal graph is accepted by domain
experts. It is also possible to submit new time series datasets from different
domains to this platform.

###  5.4 Toolboxes for Causal Analysis

Many software tools have been developed in recent years for analyzing causal
relationships and effects from the dataset. We summarize some open-source
causality tools with the list of available methods in Table LABEL:tab:tools.

Table 9: Toolbox and APIs available for performing Causal Discovery and Causal Inference. Methods with (*) are suitable for time-series data. Tools |  Methods |  Language |  Causal Discovery (CD), Causal Inference (CI)
---|---|---|---
pcalg (Markus Kalisch et al., 2012; Alain Hauser and Peter Bühlmann, 2012) |  PC, stable PC, CPC, GES, GIES, ARGES, GDS, AGES, FCI, FCI–JCI, FCI+, RFCI, LINGAM, IDA algorithm, Generalized Backdoor Criterion (GBC), Generalized Adjustment Criterion (GAC) |  R |  CD, CI
Tetrad (Ramsey et al., 2018) |  FCI, RFCI-BSC, FGES, GFCI, PC, PCStable, CPC, PcMax, RFCI, MBFS, GLASSO, FOFC, FTFC, LINGAM, TsFCI*, TsGFCI*, TsIMaGES*, MultiFASK*. |  Java |  CD
Causal-cmd (Bui et al., 2022) |  BPC, EB, Fang-Concatenated, FAS, Fask-Concatenated, FCI, FGES, FGES-MB, FOFC, FTFC, GFCI, GLASSO, IMGS_CONT, IMGS_DISC, MBFS, MGM, PC-ALL, PC-STABLE-MAX, R-SKEW, R-SKEW-E, R1, R2, R3, R4, RFCI, SKEW, SKEW-E, TS-FCI*, TS-GFCI*, TS-IMGS* |  Java |  CD
causal-learn (Zheng et al., 2023) |  PC, FCI, CD-NOD, GES, ICA-based LiNGAM, DirectLiNGAM, VAR-LiNGAM*, RCD, CAM-UV, Additive noise models, Generalized Independence Noise (GIN) condition-based method, GRaSP, Granger causality |  Python |  CD
CausalNex (Beaumont et al., 2021) |  Bayesian Networks, Counterfactual analysis, Identify the right intervention |  Python |  CD, CI
LiNGAM (Ikeuchi et al., 2023) |  DirectLiNGAM, MultiGroupDirectLiNGAM, VARLiNGAM*, VARMALiNGAM*, Bootstrap, RCD, RESIT, LiM, CAM-UV, MultiGroupRCD |  Python |  CD
Tigramite 121212https://github.com/jakobrunge/tigramite |  PCMCI*, PCMCI+*, LPCMCI*, RPCMCI*, J-PCMCI+* |  Python |  CD
Causal ML (Chen et al., 2020) |  Uplift Random Forests, Interaction Tree, Causal Inference Tree, S-Learner, T-Learner, X-Learner, R-Learner, Doubly Robust (DR) learner, 2-Stage Least Squares (2SLS), Doubly Robust Instrumental Variable (DRIV) learner, CEVAE, DragonNet |  Python |  CI
EconML (Keith et al., 2019) |  Double Machine Learning (RLearner), Dynamic Double Machine Learning, Causal Forests, Orthogonal Random Forests, T-Learner, S-Learner, X-Learner, Domain Adaptation Learner, Doubly Robust Learner, Non-Parametric Double Machine Learning, Deep Instrumental Variables, Dynamic Double Machine Learning |  Python |  CI
DoWhy (Sharma and Kiciman, 2020; Blöbaum et al., 2022) |  Average causal effect for backdoor, frontdoor, instrumental variable, and other identification methods |  Python |  CI
linearmodels (Sheppard et al., 2023) |  Fixed effects, First difference regression, Between estimator for panel data, Pooled regression for panel data, Fama-MacBeth estimation of panel models, Absorbing Least Squares, Two-stage Least Squares, Limited Information Maximum Likelihood, k-class Estimators, Generalized Method of Moments, 2- and 3-step estimation, Time-series estimation, GMM estimation, Seemingly Unrelated Regression (SUR/SURE), Three-Stage Least Squares (3SLS), Generalized Method of Moments (GMM) System Estimation |  Python |  CI
Causal Discovery Toolbox (Kalainathan and Goudet, 2019) |  CGNN, PC, GES, GIES, LiNGAM, CAM, GS, IAMB, MMPC, SAM, CCDr, ANM, IGCI, RCC, NCC, GNN, Bivariate fit, Jarfo, CDS, RECI |  Python |  CD, CI

##  6 Conclusion

In this survey paper, we have explored the landscape of causal discovery and
causal inference methods for time-series and spatiotemporal data. We
summarized common approaches, discussed limitations and assumptions, and
showcased various applications across different domains of Earth Science.
Owing to the critical role of causal inference methods in advancing Earth
Science research, the review paper highlights open challenges and
opportunities for causality to unravel environmental processes and ecosystem
dynamics. By leveraging the methods explored in this paper, researchers can
better understand the underlying mechanisms shaping Earth’s systems and make
informed decisions to address pressing environmental challenges. Despite their
potential, we acknowledge several challenges and limitations associated with
causal discovery and inference methods, including issues related to
confounding, endogeneity, data stationarity and non-linearity. Addressing
these challenges will require interdisciplinary collaboration, methodological
innovations, and improved data integration techniques. Looking ahead, several
promising avenues for future research in causal inference for Earth Science
emerge. Such as extending the application of causal inference methods to
emerging areas of Earth system modeling, enabling methods for more robust
decision-making under uncertainty, improving model interpretability and
development of novel algorithms tailored to the unique characteristics of
Earth Science data, such as spatiotemporal dependencies, spatial interference
and non-linear interactions. By pursuing these research directions, we can
further enhance the capabilities of causal discovery and inference methods to
address the pressing environmental challenges facing our planet and pave the
way for more sustainable and resilient Earth systems.

## Acknowledgement

This work is supported by NSF grants: CAREER: Big Data Climate Causality
(OAC-1942714) and HDR Institute: HARP - Harnessing Data and Model Revolution
in the Polar Regions (OAC-2118285).

## Availability Statement

No datasets were generated or analyzed during the current study. Software
(other than for typesetting) was not used for this research.

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