# Causal inference for process understanding in Earth sciences

Adam Massmann, Pierre Gentine, Jakob Runge Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

There is growing interest in the study of causal methods in the Earth
sciences. However, most applications have focused on causal discovery, i.e.
inferring the causal relationships and causal structure from data. This paper
instead examines causality through the lens of causal inference and how
expert-defined causal graphs, a fundamental from causal theory, can be used to
clarify assumptions, identify tractable problems, and aid interpretation of
results and their causality in Earth science research. We apply causal theory
to generic graphs of the Earth system to identify where causal inference may
be most tractable and useful to address problems in Earth science, and avoid
potentially incorrect conclusions. Specifically, causal inference may be
useful when: (1) the effect of interest is only causally affected by the
observed portion of the state space; or: (2) the cause of interest can be
assumed to be independent of the evolution of the system’s state; or: (3) the
state space of the system is reconstructable from lagged observations of the
system. However, we also highlight through examples that causal graphs can be
used to explicitly define and communicate assumptions and hypotheses, and help
to structure analyses, even if causal inference is ultimately challenging
given the data availability, limitations and uncertainties.

#### Note:

We will update this manuscript as our understanding of causality’s role in
Earth science research evolves. Comments, feedback, and edits are
enthusiastically encouraged, and we will add acknowledgments and/or coauthors
as we receive community contributions. To edit the manuscript directly
(recommended) you can fork the project’s repository and submit a pull request
at https://github.com/massma/causality-earth-science, or you can alternatively
email us comments, questions, and suggestions and we will try to incorporate
them into the manuscript.

##  Introduction

There is growing interest in the study of causal inference methods in the
Earth sciences [e.g., Salvucci et al., 2002, Ebert-Uphoff and Deng, 2012,
Kretschmer et al., 2016, Green et al., 2017, Barnes et al., 2019, Samarasinghe
et al., 2020, Runge, 2018, Runge et al., 2019, Goodwell et al., 2020].
However, most of this work focuses on causal discovery, or the inference
(using data) of causal structure: i.e., the “links” and directions between
variables. In some cases, causal discovery can be used to estimate the
structure of a causal graph and the relationships between variables when the
graph is not known a priori. However in many if not most Earth system
applications, the causal graph is already known based on physical insights.
For instance the impact of El Niño on West American rainfall is known to be
causal and the graph does not need to be discovered (even though using causal
discovery for this problem is a useful sanity check of the method’s ability).

This paper looks at causality through a different, but complementary, lens to
causal discovery and examines how assumed causal graphs [Pearl, 1995], a
fundamental from causal theory, can be used to clarify assumptions, identify
tractable problems, and aid interpretation of results in Earth science
research. Our goal is to distill [e.g., Olah and Carter, 2017] the basics of
the graphical approach to causality in a way that is relatable for Earth
scientists, hopefully motivating more widespread use and adoption of causal
graphs to organize analyses and communicate assumptions. These tools are
relevant now more than ever, as the abundance of new data and novel analysis
methods have inevitably led to more opaque results and barriers to the
communication of assumptions.

Beyond their usefulness as communication tools, if certain conditions are met,
causal graphs can be used to estimate, from data, the generalized functional
form of relationships between Earth science variables [Pearl, 2009b].
Ultimately, deriving generalized functional relationships is a primary goal of
science. While we know the functional relationships between some variables a
priori, there are many relationships we do not know [e.g., ecosystem scale
water and carbon fluxes; Massmann et al., 2019, Zhou et al., 2019b, a,
Grossiord et al., 2020], or that we do know but are computationally
intractable to calculate [e.g., clouds and microphysics at the global scale:
Randall et al., 2003, Gentine et al., 2018, Zadra et al., 2018, Gagne et al.,
2020]. In these types of applications, causal graphs give us a path toward new
scientific knowledge and hypothesis testing: generalized functional
relationships that were inaccessible with traditional tools.

The main contribution of this paper is to demonstrate how causal graphs, a
fundamental tool of causal inference discussed in Section 2, can be used to
communicate assumptions, organize analyses and hypotheses, and ultimately
improve scientific understanding and reproducibility. We want to emphasize
that almost any study could benefit from inclusion of a causal graph in terms
of communication and clarification of hypotheses, even if in the end the
results cannot be interpreted causally. Causal graphs also encourage us to
think deeply in the initial stages of analysis about hypotheses and how the
system is structured, and can identify infeasible studies early in the
research process before time is spent on analysis, acquiring data,
building/running models, etc. These points require some background and
discussion, so the paper is divided into the following sections:

  * •

Section 2: Introduces and discusses causal graphs within the general
philosophy of causality and its application in Earth science.

  * •

Section 3: Using a simple relatable example we explain the problem of
confounders and how causal graphs can be used to isolate the functional
mapping between interventions on some variable(s) to their effect on other
variable(s).

  * •

Section 4: We draw on a real example that benefits from inclusion of a causal
graph, in terms of communicating assumptions, and organizing and justifying
analyses.

  * •

Section 5: We turn to more generic examples of graphs that are generally
consistent with a wide variety of systems in Earth science, to highlight some
of the difficulties we confront when using causal inference in Earth science
and how we may be able to overcome these challenges.

Throughout the discussion key terms will be emphasized in italics.

##  The graphical perspective to causality and its usefulness in the Earth
sciences

Figure 1: A toy graph example to demonstrate basic causal theory, involving
cloud (C), aerosol (A), and surface solar radiation (S).

While there are many different definitions and interpretations of “causality”,
for this manuscript we view causality through the lens of causal graphs, as
introduced in Pearl [1995] and discussed more extensively in Pearl [2009b]. We
take this perspective because we believe causal graphs are useful in Earth
science, rather than because of any particular philosophical argument for
causal graphs as the “true” representation of causality.

Causal graphs are Directed Acyclic Graphs (_DAG_ s) that encode our
assumptions about the causal dependencies of a system. To make a causal graph,
a domain expert simply draws directed edges (i.e. arrows) from variables that
are causes to effects. In other words, to make a causal graph you draw a
diagram summarizing the assumed causal links and directions between variables
(e.g., Figure 1). Causal graphs are useful tools because they can be drawn by
domain experts with no required knowledge of maths or probability, but they
also represent formal mathematical objects. Specifically, underlying each
causal graph are a set of equations called _structural causal models_ : each
node corresponds to a generating function for that variable, and the inputs to
that function are the node’s parents. Parents are the other nodes in the graph
that point to a node of interest (e.g., in the most simple graph $X\to Y$; $X$
is a parent of $Y$). So in reality, drawing arrows from “causes” to “effects”
is synonymous with drawing arrows from function inputs to generating
functions.

In this way, drawing a causal graph is another way to visualize and reason
about a complicated system of equations, which is a very useful tool for the
Earth scientist: we deal with complicated interacting systems of equations and
welcome tools that help us understand and reason about their collective
behavior. In some cases we may know a priori (from physics) the equations for
a given function in a causal graph. However, in practice we often either do
not know all of the functions a priori [e.g., plant stomata response to VPD;
Massmann et al., 2019, Zhou et al., 2019b, a, Grossiord et al., 2020], or some
functions are computationally intractable to compute [e.g., turbulence, moist
convection, and cloud microphysics in large scale models; Zadra et al., 2018,
Gentine et al., 2018]. In these scenarios the benefits of causal graphs are
fully realized: based on the causal graph we can calculate from data, using
the do-calculus [Pearl, 1994], the response of target variables (i.e. effects)
to interventions on any other variables in the graph (i.e. causes). When
combined with statistical modeling (i.e., regression), one can estimate the
functional relationship between interventions on causes and effects [Pearl,
2009a, Shalizi, 2013, Shi et al., 2019, Mao et al., 2020].

By viewing causal graphs through this pragmatic lens of calculating the
functional form of relationships that we do not know a priori, we
simultaneously identify causal graphs’ value for Earth scientists while also
side stepping philosophical arguments about the meaning of causality. Causal
graphs are pragmatic because in the Earth sciences we often need to estimate
how the system responds to _interventions_ (prescribed changes to variables of
interest, or “causes”). For example, sub-grid physical parameterizations in
Earth system models (e.g., turbulence) require estimates of the time
tendencies’ response to _interventions_ on the large scale state and
environment. We also may desire to calculate experiments: for example how
changing land cover from forest to grasslands affects (the statistics of)
surface temperature. Do-calculus is a method to calculate this response to
interventions without relying on approximate numerical models or real world
experimentation, which can be infeasible or unethical [as is the case for
geoengineering; e.g., unilateral decisions to seed the oceans with iron, or
spray aerosols in the atmosphere, Hamilton, 2013]. While we want to maintain
this emphasis on causality as a method for calculating the generalized
response to intervention (possibly using regression to calculate the
functional form of that response), for consistency with the causal literature
we will call the response variables “effects”, and the intervened-upon
variables “causes”.

For some, it may not be clear how the functional response to _interventions_
is different from naive regression between observed variables. We will
demonstrate in Section 3 how uninformed regression is just a functional
mapping of associations between variables, and how this differs from the
response to interventions, i.e. a causal mechanism. This is the problem that
do-calculus solves: it identifies which data are needed and how we can use
those data to calculate the response to _interventions_ , rather than just
associations that may be attributable to other processes entirely. Because we
know the response to the intervention is attributable to the intervention and
not other processes, we have greater confidence in the generalizability of the
response to interventions. This generalizability of the response to
interventions makes do-calculus especially relevant for scientists and
engineers. Working through an example will clarify some of these claims.

##  A Toy Example: the problem of confounding and the necessity of do-calculus
for calculating interventions

To demonstrate the problem of confounding and the necessity of causal
graphs/do-calculus, we use a simple toy example involving clouds, aerosols,
and surface solar radiation/sunlight. As shown in Figure 1, the causal graph
consists of:

  1. 1.

An edge from aerosols to clouds because aerosols serve as cloud condensation
nuclei or ice nucleating particles, which affect the probability of water
vapor conversion to cloud (condensates).

  2. 2.

An edge from aerosols to surface solar radiation, because aerosols can reflect
sunlight back to space and reduce sunlight at the surface.

  3. 3.

An edge from clouds to sunlight, because clouds also reflect sunlight back to
space and can reduce sunlight at the surface.

Causal graphs encode our assumptions about how the system behaves, and the
nodes and edges that are missing from the graph often represent strong
assumptions about the lack of functional dependence. For example, in the
cloud-aerosol-sunlight example, clouds also affect aerosols; e.g., by
increasing the likelihood that aerosol will be scavenged from the atmosphere
during precipitation [e.g., Radke et al., 1980, Jurado et al., 2008, Blanco-
Alegre et al., 2018]. By not including an edge from cloud to aerosol, we are
making the assumption that we are neglecting the effect of clouds on aerosols,
and also preventing the graph from containing any cycles (a path from a
variable to itself) which is a requirement of the theory: graphs must by
acyclic. This acyclic requirement may raise concerns for the reader; many
problems in Earth science contain feedbacks that introduce cycles. However,
any feedback can be represented as an acyclic graph by explicitly resolving
the time evolution of the feedback in the graph (Section 5 contains examples
of such graphs). Considering this example is intended to be pedagogical for
introducing causal theory to the readers, we will continue with the graph as
drawn in Figure 1 (we refer the reader to Gryspeerdt et al. [2019] for a more
realistic treatment of aerosols and clouds).

Even though mathematical reasoning is not required to construct a causal
graph, the resulting graph encodes specific causal meaning based on
qualitative physical understanding of the system. Implicitly, the graph
corresponds to a set of underlying functions, called a structural causal
model, for each variable:

$\displaystyle aerosol$ $\displaystyle\leftarrow f_{aerosol}(U_{aerosol})$ (1)
$\displaystyle cloud$ $\displaystyle\leftarrow f_{cloud}(aerosol,U_{cloud})$
(2) $\displaystyle sunlight$ $\displaystyle\leftarrow
f_{sunlight}(aerosol,cloud,U_{sunlight})$ (3)

where $U$ are random variables due to all the factors not represented
explicitly in the causal graph, and $f$ are deterministic functions that
generate each variable in the graph from their parents and corresponding $U$.

The presence of the random variables $U$ introduces a third meaning to the
causal graph: they induce a factorization of the joint distribution between
variables into conditional and marginal factors:

$P(A,C,S)=P(S\,|\,C,A)\,P(C\,|\,A)\,P(A),$ (4)

where $A$ represents aerosol, $C$ represents cloud, $S$ represents surface
solar radiation, and $P($C$|$A$)$ denotes conditional probability of $C$ given
$A$ (Appendix A describes the notation used in this paper and a brief
introduction to probability theory for unfamiliar readers). The inclusion of
randomness in causal graphs is a key tool: by positing a causal graph, we are
not stating that the variables in the graph are the only processes in the
system nor that the relationships are deterministic. Instead, we are stating
that all other processes not included in the graph induce variations in the
graph’s variables that are _independent_ of each other (e.g., all $U_{\cdot}$
in Equation (1) are independent). For example, sources of aerosol variability
not considered in Figure 1 include anthropogenic aerosol emission, the
biosphere, fires, volcanoes, etc. [e.g., Boucher, 2015]. For cloud, this
includes synoptic forcing or atmospheric humidity, etc. [e.g., Wallace and
Hobbs, 2006]. For radiation, this includes variability of top of atmosphere
radiation, etc. [e.g., Hartmann, 2015]. Figure 1 states that all these
external, or exogenous, sources of variability are independent of each other
[in very technical terms, this means the graph is “Markovian,” Pearl, 2009b]).

We can now apply causal inference theory [e.g., Pearl, 1995, Tian and Pearl,
2002, Shpitser and Pearl, 2006] to the assumptions encoded in our causal graph
to identify which distributions must be estimated from data in order to
calculate the response of effect(s) (e.g. of sunlight) to an experimental
intervention on the cause(s) (e.g. presence or absence of a cloud). The goal
of causal inference is to derive the response to the intervention in terms of
only observed distributions. This process of identifying the necessary
observed distributions is formally termed _causal identification_ [Pearl,
2009b, , Ch. 3]. If a causal effect is not identifiable (_un_ -identifiable),
for example if calculating a causal effect requires distributions of variables
that we do not observe, then we cannot use causal inference to calculate a
causal effect, even with an infinite sample of data.

A necessary condition for _unidentifiability_ is the presence of an unblocked
_backdoor path_ from cause to effect [Pearl, 2009b, , Ch. 3]. Backdoor paths
are any paths going through parents of the cause to the effect. We can block
these paths by selectively observing variables such that no information passes
through them [Geiger et al., 1990]. If observations are not available for the
variables required to block the path, the path will be _unblocked_. However,
if we can observe variables along the backdoor paths such that all backdoor
paths are blocked, then we have satisfied the _back-door criterion_ [Pearl,
2009a] and we can calculate unbiased causal effects from data.

Understanding backdoor paths and the backdoor criterion is helped by an
example. Returning to our toy example (Figure 1), we attempt to calculate the
causal effect of clouds on sunlight. In other words, we want to isolate the
variability of sunlight due to the causal link from cloud to sunlight (Figure
1). However, aerosols affect both cloud and sunlight (i.e., there is a
backdoor path from cloud to aerosol to sunlight), so if we naively calculate a
"causal" effect using correlations between sunlight and cloud, we obtain a
biased estimate. To demonstrate this, consider simulated cloud, aerosol, and
sunlight data from a set of underlying equations consistent with Figure 1 and
Equation (1):

$\displaystyle\begin{split}\text{aerosol}=&\;U_{aerosol};\;U_{aerosol}\sim\text{uniform
(0, 1]}\\\ \text{cloud}=&\;\text{Cloudy if
}U_{cloud}+\text{aerosol}>1;\;U_{cloud}\sim\text{uniform (0, 1]}\\\
\text{sunlight}=&\begin{cases}\text{Cloudy}&:0.6\cdot\text{downwelling clear
sky radiation}\\\ \text{Clear}&:\text{downwelling clear sky
radiation}\end{cases}\end{split}$ (5)

where:

$\text{downwelling clear sky
radiation}=U_{sunlight}\cdot(1-aerosol);\;U_{sunlight}\sim\text{Normal(340 W
m${}^{-2}$, 30 \, W m${}^{-2}$)}$

Figure 2: A naive approach to estimating the “effect” of clouds on sunlight:
bin observations by cloudy and clear day, and compare the values of sunlight.
This approach yields an average difference of 157.74 W m-2 between cloudy and
clear days, and is a large overestimation of the true causal effect of clouds
on sunlight (-68.0 W m-2) in these synthetic data.

Now, consider not knowing the underlying generative processes, but instead
just passively observing cloud and sunlight. If one were interested in
calculating the effect of cloud on sunlight, and aerosol data were not
available or one were not aware that aerosol could have an impact on clouds
and aerosol, one direct but incorrect approach would be to bin the data by
cloudy and clear conditions and compare the amount of sunlight between cloudy
and clear observations (Figure 2). This approach would suggest that clouds
reduce sunlight by, on average, 158 W m-2; this is is a strong overestimation
of the true average effect of clouds (-68 W m-2), derived from Equation (5).
This overestimation is due to aerosol-induced co-variability between cloud and
sunlight that is unrelated to the causal link from cloud to sunlight. However
if aerosols were constant (e.g. observed or not varying), any co-variability
between cloud and sunlight would be attributable to the causal edge between
cloud and sunlight (Figure 1). In other words, conditional on aerosol, all co-
variability between cloud and sunlight is only due to the causal effect of
cloud on sunlight. We can mathematically encode this requirement that we must
condition on aerosol to isolate the causal effect of cloud on radiation, and
doing so derives the causal effect of cloud on sunlight by satisfying the
backdoor criterion with adjustment on aerosol:

$P(S|do(C=c))=\int_{a}P(S\,|\,C=c,A=a)\,P(A=a)\;da,$ (6)

where the do-expression ($P(S\,|\,do(C\,=\,c))$) represents the probability of
sunlight if we did an experiment where we intervened and set cloud to a value
of our choosing (in this case $c$, which could be “True” for the presence of a
cloud, or “False” for no cloud). In the case that observations of aerosols are
not available, our causal effect is not identifiable and we cannot generally
use causal inference without further assumptions, no matter how large the
sample size is of our data.

Causal graphs are therefore powerful analysis tools: after encoding our domain
knowledge in a causal graph, we can analyze the available observations to
determine whether a causal calculation is possible, without needing to
collect, download, or manipulate any data. For more complicated graphs, causal
identification can be automated [Tian and Pearl, 2002, Shpitser and Pearl,
2006, Huang and Valtorta, 2006, Bareinboim and Pearl, 2016, Textor et al.,
2017, and http://www.dagitty.net/, https://causalfusion.net]. We later use
this theory to assess assumptions that lead to tractable causal analyses for
generic Earth science scenarios (Section 5).

Once we have established that a causal effect is identifiable from data, we
must estimate the required observational distributions (Equation (6)) from
data. Often it may be more computationally tractable to calculate an average
causal effect, rather than the full causal distribution $P(S|do(C=c))$, which
might be difficult to estimate. Returning to our toy example (Figure 1), the
average effect is defined as:

$\mathbb{E}(S|do(C=c))=\int_{s}s\,P(S=s|do(C=c))\,ds,$ (7)

where $\mathbb{E}$ is the expected value. Substituting Equation (6) into
Equation (7), and rearranging gives:

$\mathbb{E}(S|do(C=c))=\int_{a}P(A=a)\;\mathbb{E}(S\,|\,C=c,A=a)\,da,$ (8)

Where $\mathbb{E}(S\,|\,C=c,A=a)$ is just a regression of sunlight on cloud
and aerosol. Estimating the marginal $P(A)$ is difficult, but if we assume
that our observations are independent and identically distributed (IID) and we
have a large enough sample, we can use the law of large numbers to approximate
Equation (8) as [Shalizi, 2013] :

$\mathbb{E}(S|do(C=c))\approx\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}(S\,|\,C=c,A=a_{i}).$
(9)

Data or prior knowledge can inform the estimate of
$\mathbb{E}(S|C=c,A=a_{i})$, but whatever regression method is used, it should
be checked to ensure it is representative of the data and there is sufficient
signal to noise ratio to robustly estimate the regression. It is important to
note how this estimate is different from the naive association of
$\mathbb{E}(S|C=c)$; in $\mathbb{E}(S|C=c,A=a_{i})$ we are controlling for the
impact of aerosol on sunlight by conditioning on $A$ and including aerosol in
the regression. As we will see, using the do-calculus estimate results in an
estimate of the effect of cloud on aerosol that is very different from the
naive association, and close to the true effect.

Figure 3: A linear relationship between aerosol and sunlight, conditional on
cloud. If we use this regression to calculate the average causal effect of
cloud on sunlight, as in Equation (9), our result is very close to the true
causal effect of -68.0 W m-2.

In our simple example, a linear model conditional on cloud is a suitable
choice for the regression function $\mathbb{E}(S|C=c,A=a_{i})$ (Figure 3).
However, many problems in Earth science require non-linear approximation
methods like neural networks and/or advanced machine learning methods; for
examples of such machine learning methods, we refer readers to [Bishop, 2006].

The causal effect of clouds on sunlight as calculated using Equation (9))
(e.g. $\mathbb{E}(S|do(C=\text{cloudy}))-\mathbb{E}(S|do(C=\text{clear}))$) is
-67.69 W m-2, which closely matches the true causal effect from Equation (5)
of -68 W m-2. This example demonstrates how causal inference and theory can be
used to calculate unbiased average effects using regression, subject to the
assumptions clearly encoded in the causal graph. Further, causal inference can
be used to justify and communicate assumptions in any observational analyses
employing regression. In the best case, the causal effect is identifiable from
the available observations, and the regression analysis can be framed as an
average causal effect. In the worst case that identification is not possible
from the available observations, one may present the regression as observed
associations between variables. However, presentation of a causal graph still
aids the reader: the reader can see from the causal graph what the confounders
and unobserved sources of covariability are between the predictors and the
output. In all cases, the presentation of a causal graph makes explicit the
assumptions about the causal dependencies of the system. Wherever possible, we
recommend including causal graphs with any observation-based analyses.

In summary of the main points of this introduction to causal graphical models
and do-calculus:

  * •

Graphical causal models encode our assumptions about causal dependencies in a
system (edges are drawn _from_ causes _to_ effects). “Causal dependencies”
really just refer to functional dependencies between inputs (causes) and
outputs (effect), which are useful in the Earth sciences to reason about
graphically .

  * •

In order to calculate an unbiased causal effect from data, we must isolate the
covariability between cause and effect that is due to the directed causal path
from cause to effect. The presence of non-causal dependencies between the
cause and effect can be deduced from the causal graph: the presence of an
unblocked backdoor path from the cause to the effect leads to non-causal
dependencies (and co-variation).

  * •

The _backdoor criterion_ identifies the variables that we must condition on in
order to block all backdoor paths, remove non-causal dependence between the
cause and effect, and calculate an unbiased causal effect from data.

  * •

The _average_ causal effect can be reliably approximated with a regression
(Equation (9)) derived from the backdoor criterion. In this scenario, causal
theory and graphs identify the variables that should be included in the
regression in order to calculate an unbiased causal effect (however
researchers should still ensure their choice of regression model is
appropriate for the data). While not demonstrated by our example, causal
theory and graphs also identify the variables that should not be included in
the regression [Pearl, 2009a]: we can also bias a causal effect by including
too many variables in a regression .

  * •

The do-calculus and identification theory provide a flexible tool to determine
whether an effect is identifiable and, if so, which distributions should be
estimated from data, while making no assumptions about the forms of the
underlying functions and distributions. However, parametric assumptions can be
applied to make the calculation of those distributions from data more
computationally tractable.

Here we focused on the _backdoor criterion_ to block backdoor paths. An
unblocked backdoor path from the cause to the effect is a necessary condition
for unidentifiability. However, an unblocked backdoor path from the cause to
the effect is not a sufficient condition for unidentifiability: there are
other identification strategies like the front door criterion [see Section
3.5.2 in Pearl, 2009b] and instrumental variables [see Chapter 8 in Pearl,
2009b] that do not rely on observing variables along the backdoor path, and
can be used in some cases where observations are not available to satisfy the
backdoor criterion (also see Tian and Pearl [2002] for more discussion on
sufficient conditions for unidentifiability). These are examples of how the
do-calculus admits any strategy that frames the response to interventions in
terms of observed distributions. We focus on the backdoor criterion because it
is the most fundamental and direct method for adjusting for confounding, the
most intuitive for an introduction to causality, and is the most relevant for
the generic temporal systems present in the Earth sciences (Figure 5 in
Section 5). However, causal identification through other methods like
instrumental variables and the front door criterion can also be automated; we
refer the reader to Pearl [2009b] for further discussion and software tools
like http://www.dagitty.net/ and http://www.causalfusion.net for interactive
exploration.

##  Beyond toys: causal graphs as communicators, organizers, and time-savers

In Section 3 we used a toy example to demonstrate a causal analysis starting
with drawing a graph and ending with the successful calculation of the average
response of sunlight (the effect) to an intervention on cloud (the cause).
However, often we may not be able to ultimately estimate the causal effects
from the available data. In many cases there are serious challenges due to
unobserved confounding in generic Earth science problems (Section 5), and for
other cases we may lack enough samples, or samples could be too systematically
biased, to estimate the necessary distributions (or regressions; e.g.,
Equation (9)) with sufficient certainty. However, we want to emphasize that
even if calculating a causal effect might in some cases be impossible, drawing
a causal graph at the beginning of an analysis still offers tremendous
benefits in terms of organization and scientific communication. Investing time
to reason about the functional structure of the problem at the outset can save
scientists time in the long term, forcing us to clarify our thinking early,
expose potential challenges, and identify intractable approaches.

Additionally, once the causal graph is drawn, we can use it as a communication
tool and include it in presentations, papers, and discussions of our results.
Making our assumptions about dependencies in the system explicit greatly
improves the interpretability and reproducibility of our results. Perhaps our
analysis and graph meet the standards for a causal interpretation, but even if
they do not, the causal graph helps the rest of the community asses the
sources of confounding in the graph that were not controlled for, and
understand if their conceptualization of the graph structure matches the
authors’ hypotheses. Often in research there are more assumptions being made
than are communicated, and even when they are communicated, the assumptions do
not always get discussed in a precise way. Including a causal graph allows the
assumptions to be clearly known, and discussed in a precise and rigorous way
[e.g., Hannart et al., 2016].

To support the idea that many analyses would benefit from a causal graph, we
will detail how a past project benefits from a causal graph. This example also
moves beyond the toy example of Section 3, and demonstrates causal graphs’
applicability to real problems in Earth science.

###  Causal graphs’ utility in a real example

In Massmann et al. [2017], the lead author of this manuscript participated in
a field campaign designed to study the impact of microphysical rain regime
(specifically the presence of ice from aloft falling into orographic clouds)
on orographic enhancement of precipitation. This field campaign and analysis
benefits from a causal graph and is a real-world example argument for the more
common use of causal graphs as research tools. Our retrospective causal graph
of orographic enhancement in the Nahuelbuta mountains under steady conditions
clearly communicates our assumptions about the system (Figure 4).

Figure 4: A graph representing steady conditions for orographic enhancement
during the Chilean Coastal Orographic Precipitation Experiment [CCOPE,
Massmann et al., 2017]. The field campaign attempted to quantify the effect of
“rain regime” on “orographic enhancement.” Observed quantities are represented
by solid nodes, while unobserved quantities are represented by dashed nodes.
Variables that must be observed to block all backdoor paths are shaded. The
effect of rain regime on orographic enhancement is identifiable through
adjustment on these shaded variables.

Many of the variables in the graph, such as “synoptic forcing”, “wind”, and
“orographic flow”, are quite general quantities. Keeping quantities general
can lead to more intuitive and interpretable graphs by limiting the number of
details and nodes that one must consider. However, if logic needs clarifying,
the graph can become more explicit (e.g., differentiating wind into speed,
direction, and spatial distribution, both horizontally and vertically). We
also include unobserved variables explicitly in the graph, so we can reason
about processes’ impact on the system even if they are not observed. As graphs
become more complicated, one can leverage interactive visualization software,
or use static graph abstractions like plates [Bishop, 2006], which are
particularly well suited to representing repeated structure common to
spatiotemporal systems.

While the exact details of the graph and the assumptions it encodes are
interesting (e.g., “wind”, “stability”, and “atmospheric moisture” all refer
to upwind conditions, and we assume that these upwind conditions are the
relevant “boundary condition” for the downwind orographic clouds and
precipitation), the noteworthy feature of the graph is that the field
campaign’s effect of interest, rain regime on orographic enhancement, is
identifiable from the field campaign’s observations. This is subject to the
assumptions encoded in the graph, but those assumptions are explicitly
represented and communicated by the graph. The causal graph helps interpret
the field campaign’s results, and in some sense proves that the design of the
field campaign is sound.

Therefore, for field campaigns, causal graphs are particularly useful at the
planning and proposal stage. Such a causal graph could be included in any
field campaign proposal, improving communication about the system and also
rigorously justifying the campaign’s observations as necessary for calculating
the desired effect(s). Even before the proposal, one could start with a causal
graph, and then analyze it to determine which observations are needed to meet
the campaign’s goals. Building on this idea, one could attach costs associated
with observing each variable in the graph, and automatically determine the set
of observations that minimizes cost while still allowing us to calculate our
effect(s) of interest.

While this is just one example, it demonstrates that causal graphs are useful
beyond toy problems in the Earth system. Additionally, as we will see in
Section 5, we can draw quite general graphs that are representative of many
problems in Earth science. We hope the reader considers drawing a causal graph
as a first step in their next project; they help structure, organize, and
clarify our analysis and its assumptions.

##  Overcoming unobserved confounding and partial observation of Earth system
state

So far we have focused on toy (Section 3) and specific (Section 4) examples.
We now turn our attention to more general and generic problems in Earth
science systems, the common challenges we may encounter when attempting causal
inference, and how we can overcome these challenges.

Earth science systems and their components are (typically) dynamical systems
evolving through time according to an underlying system state [Lorenz, 1963,
1996, Majda, 2012]. This offers both opportunities and challenges for causal
inference. When constructing causal graphs we may benefit from the temporal
ordering of events [Runge et al., 2019]: we know that future events can have
no causal effect on the past. We can also use the time dimension to explicitly
resolve feedbacks in the system, and transform cyclic graphs with feedbacks
into directed acyclic graphs (DAGs) required for causal inference. While
handling feedbacks and avoiding cyclic graphs is a challenge of causal
inference in Earth science, resolving the time dimension is a generic path to
overcome this challenge when observations of sufficient resolution are
available.

However, confounding due to incomplete observation of the system’s state
variables also introduces challenges; challenges that are not unique to this
paper’s causal lens: incomplete observation of the system precludes the use of
many “causal discovery” algorithms as well (see Runge et al. [2019] for a
detailed review). Causal identification and tractable causal inference in
Earth science requires assumptions about the unobserved portions of the state
space that introduce this confounding, and how the unobserved portions of the
state space affect observed variables. Without such assumptions the unobserved
portions of the state space will introduce confounding for any causal effect
of interest (Figure 5). For example, we generally do not observe the state
space at every time (e.g. $S(t-1/2)$, Figure 5), and at any given time, we do
not observe the state space at all locations and for all state variables (e.g.
$S(t)$ and $S(t-1)$ in Figure 5). In other words, despite our impressive and
growing array of satellite, remote sensing, and in situ observation systems,
we are still very far from observing every relevant state variable at every
location in time and space. So, if we are interested in the causal effect of
any state variable at time $t$ on some variable at time $t+1$ (e.g., $E$ in
Figure 5), then the causal effect will be confounded by the unobserved
portions of the state space, and calculating a causal effect will be
impossible (un-identifiable) without additional assumptions.

However, there are assumptions we can make that may be reasonable for many
generic applications, which remove this problem of unobserved confounding due
to partial observation of the state space. We elaborate upon these assumptions
in the following sections, and for each assumption, we draw a graph, briefly
discuss the scenario and assumptions, present the identification formula and
how average effects can be estimated (for example, using regression to
estimate $\mathbb{E}(\cdot)$), and include some strategies for testing the
assumption(s) with data.

E(t+1)S(t)S(t-1/2)S(t-1) Figure 5: A generic graph of the Earth system state
sequence. Unobserved nodes are outlined by dashed lines. We only observe the
state space at certain times (e.g., no observations at $S(t-1/2)$), and at
times with observations, we only partially observe the full state ($S(t)$,
$S(t-1)$). In the scenario that we are interested in calculating the causal
effect of any portion of the state space at time $t$ on some effect ($E$) at
time $t+1$, the causal effect will be confounded by the unobserved portions of
the state space, and calculating the causal effect is impossible (un-
identifiable) without additional assumptions.

#### Assumption: we observe all state variables that impact our effect(s) of
interest

Figure 6: A generic graph for the assumption that we observe all state
variables that impact our effect(s) of interest. $C(t)$ is the cause of
interest, $E(t+1)$ is the future effect of interest, and $O(t)$ are all
observations not including $C(t)$. The full state $S(t-1)$ is not observed
(dashed node). Observing $O(t)$ (grey shading) blocks the backdoor path from
$C(t)$ to $E(t+1)$.

  * •

Discussion: While we may not observe the entire state of the system, sometimes
it is reasonable to assume that we observe the portion of the state space that
affects our specific effect of interest ($E(t+1)$ in Figure 6). In this case,
we can calculate causal effects by blocking backdoor paths with the observed
portion of the state space.

  * •

Identification formula:

$P(E(t+1)\,|\,do(C(t)=c))=\int_{o}P(E(t+1)\,|\,C(t)=c,O(t)=o)\,P(O(t)=o)\;do,$

  * •

Average causal effect:

$\mathbb{E}(E(t+1)\,|\,do(C(t)=c))\approx\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}(E(t+1)\,|\,C(t)=c,O(t)=o_{i}),$

where $C(t)$ is the cause of interest, $E(t+1)$ is the effect of interest, and
$O(t)$ are all observed variables not including $C(t)$.

  * •

Check: There is no way to check this assumption with data. Therefore, this
assumption requires strong physical justification well supported by the
literature. Care is also required to insure that there are no interactions
between $C(t)$ and $O(t)$; e.g., the observations at a given time are truly
“simultaneous” and cannot causally affect each other.

#### Assumption: We can reconstruct the state at any given time using lagged
observations of the system

Figure 7: A generic graph for the assumption that we can reconstruct the state
at any given time using lagged observations of the system [Takens, 1981].
$C(t)$ is the cause of interest, $E(t+1)$ is the future effect of interest,
$O(t)$ are all observations not including $C(t)$, and $S^{\prime}(t-1)$ is the
reconstructed state. Observing $S^{\prime}(t-1)$ (grey shading) blocks all
backdoor paths from $C(t)$ to $E(t+1)$.

  * •

Discussion: While we may not observe the entire state space, in some cases we
may be able to reconstruct the state at any given time using lagged
observations of the system [see Takens’ theorem, Takens, 1981]. In this case,
we can use the reconstructed state to block backdoor paths and examine the
effect of any observed variable ($C(t)$) on future variables ($E(t+1)$ in
Figure 7). Note that additionally controlling for $O(t)$ can also make the
effect estimate more reliable. More generally, there may be more than one
suitable set of adjustment co-variates and current research is targeted at
finding optimal ones that yield the lowest estimation error [e.g., Witte et
al., 2020].

  * •

Identification formula:

$P(E(t+1)\,|\,do(C(t)=c))=\int_{s}P(E(t+1)\,|\,C(t)=c,S^{\prime}(t-1)=s)\,P(S^{\prime}(t-1)=s)\;ds.$

  * •

Average causal effect:

$\mathbb{E}(E(t+1)\,|\,do(C(t)=c))\approx\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}(E(t+1)\,|\,C(t)=c,S^{\prime}(t-1)=s_{i}),$

where $C(t)$ is the cause of interest, $E(t+1)$ is the effect of interest, and
$S^{\prime}(t-1)$ is the reconstructed state using lagged observations.

  * •

Check: One check on the success of the state space reconstruction is to test
whether the observed variables are conditionally independent given the
reconstructed state variable. This approach bears similarity to the
deconfounder approach introduced by Wang and Blei [2019], which argues that
causal effects can be calculated for many problems when we can infer a latent
variable that renders the (multiple) causes conditionally independent given
the latent variable.

#### Assumption: the cause of interest is independent of the systems’ state
evolution

Figure 8: A generic graph asserting an assumption that there are forcings
external to the evolution of the state-space.

  * •

Discussion: In some cases, we may assume that the cause of interest is
independent of the systems’ state evolution (Figure 8). While not generally
true, in some cases a variable may behave independent of the system’s state
while still causally affecting that state. For example, some human behavior
may be approximated as independent of the climate state (e.g., city planning
and land use decisions).

  * •

Identification formula:

$P(E(t+1)\,|\,do(C=c))=P(E(t+1)\,|\,C=c)$

  * •

Average causal effect:

$\mathbb{E}(E(t+1)\,|\,do(C=c))=\mathbb{E}(E(t+1)\,|\,C=c)$

  * •

Check: To check this assumption, we can test if the cause/forcing is
independent of the past state. If the cause is independent of the past state,
we have stronger confidence that the assumption holds.

While this list of assumptions is certainly not exhaustive, it presents some
approaches that may apply to many scenarios in Earth science. Also while the
provided checks might identify when assumptions break down, there is no
general way to “validate” assumptions using data. Physical and science-based
justification and reasoning are ultimately always required.

##  Conclusions

In summary, causal graphs and causal inference are powerful tools to reason
about problems in the Earth system, whether in models or observations.
Specifically, this review aimed at showing that:

  * •

Causal graphs concisely and clearly encode physical assumptions about
causal/functional dependencies between processes. Including a causal graph
benefits any observational or modeling analysis, including those that use
regression.

  * •

Whether a causal effect can be calculated from data is determined by the
causal graph. Thus the tractability of a causal analysis, or the strength of
assumptions necessary to make the analysis tractable, is determined and
assessed before collecting, generating, or manipulating data (which can cost a
tremendous amount in terms of researchers’ time or computational resources).
We recommend early causal analysis to determine tractability during a
project’s conception, before resources are spent obtaining or analyzing data.

  * •

Calculated causal effects measure the response of target variables (i.e.
effects) to interventions on other variables in the system (i.e. causes). With
statistical modeling (i.e., regression) one can estimate the functional
relationship between interventions on causes and effects. These functional
relationships generalize because they map interventions onto the response:
unlike observed associations (e.g., naive regression), we know the response is
attributable to the function’s input, and not to some other process in the
system (e.g., an unobserved common cause). This causal approach opens up a new
path to calculate generalized functional relationships when we either do not
know the functional form a priori, or it is too computationally intractable to
calculate from models.

  * •

Because the Earth system and its constituents are dynamical systems evolving
through time, we can construct broadly applicable, generic Earth system causal
graphs. We can use these graphs to calculate generalized functional
relationships between processes of interest, which would not be possible with
associations, correlations or simple regressions. However, causal inference in
the Earth sciences also presents challenges as we only partially observe the
state space of the system.

  * •

These challenges can be alleviated by applying causal theory to generic causal
graphs of the Earth system and identifying the assumptions that allow for
causal inference from data (Section 5).

Here we focus on the fundamentals of calculating causal effects from data.
However, causal inference is a thriving, active area of research, and there
are many other causal inference techniques and abstractions that could benefit
the Earth system research community. For example, there are techniques for
representing variables observed under selection bias in the causal graph and
analyzing whether a causal effect can be calculated (i.e. identified) given
that selection bias [e.g., Bareinboim et al., 2014, Correa et al., 2018].
Selection bias, defined as a preferential sampling of data according to some
underlying mechanism, is very relevant in Earth sciences. For example,
satellite observations are almost always collected under selection bias (e.g.
clouds obscure surface data, satellites sample at certain local times of the
day which is connected to top of atmosphere solar forcing, or the sensors
themselves could have a bias). Additionally, transportability [e.g.,
Bareinboim and Pearl, 2012, 2016, Lee et al., 2019] identifies whether one can
calculate a causal effect in a passively observed system called the “target
domain”, by merging experiments from other systems, called “source domains”,
that may differ from the target domain. A potential application for
transportability in Earth sciences would be to merge numerical model
experiments (e.g., Earth system models) and formally transport their results
to the real world. In this case, numerical models are the source domains that
differ from the target domain (“real world”) due to approximations and
different resolutions.

However, because these developments in causal theory are relatively new,
applied domains have yet to establish these recent theoretical developments’
utility for applied analysis. While we encourage applied scientists to explore
how these developments may apply to their domains, we recognize that many
scientists prefer tools with established utility. To that end, we believe that
the use of causal graphs to organize and structure analyses is mature and
directly applicable to many projects, and can serve as a gateway to applying
these more recent causal developments. We hope that drawing and including
causal graphs in Earth science research becomes more common in our field, and
that this manuscript provides some of the necessary foundation for readers to
attempt using causal graphs in their future research.

#### Acknowledgments

The authors want to thank Elias Bareinboim, Beth Tellman, James Doss-Gollin,
David Farnham, and Masa Haraguchi for thoughtful feedback and comments that
greatly improved an earlier version of this manuscript.

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##  Appendix A Basic probability and syntax

In this paper we use capital letters to represent random variables (e.g.,
“$X$”). For example, $P(X)$ is the marginal probability distribution of a
random variable $X$. $P(X)$ is a function of one variable that outputs a
probability (or density, in the case of continuous variables) given a specific
value for $X$. We represent specific values that a random variable can take
with lowercase letters (e.g., $x$ in the case of $X$). $P(X)$ is shorthand; a
more descriptive but less concise way to write $P(X)$ is $P(X=x)$ which
represents the fact that $P(X)$ is a function of a specific value of $X$,
represented by $x$. We use both notations, and $P(X)$ has the same meaning as
$P(X=x)$.

For the unfamiliar reader, there are a few basic rules and definitions in
probability that provide relatively complete foundations for building deeper
understanding of probability. These are the sum rule:

$P(X=x)=\sum_{Y}P(X=x,\,Y=y)$ (10)

and the product rule:

$P(X=x,\,Y=y)=P(X=x\,|\,Y=y)P(Y=y)=P(Y=y\,|\,X=x)P(X=x)$ (11)

The joint probability distribution ($P(X=x,Y=y)$) is the probability that the
random variable $X$ equals some value $x$ _and_ the random variable $Y$ equals
$y$. The joint distribution is a function of two variables, $x$ and $y$ which
are values in the domains of the random variables $X$ and $Y$ respectively.
The conditional probability distribution ($p(X=x\,|\,Y=y)$) is also a function
of two variables $x$ and $y$, but it is the probability of observing $X$ equal
to $x$, given that we have observed $Y$ equal to $y$. In other words, if we
filter our domain to only values where $Y=y$, then $p(X=x\,|\,Y=y)$ is the
probability of observing $X=x$ in this sub-domain where $Y=y$. The marginal
probability distribution ($P(Y=y)$) is just the probability that $Y$ equals
some value $y$, and is a function of only $y$. We can calculate the marginal
probability from the joint distribution by summing over all possible values
values of the other random variables in the joint (the “sum rule” - Equation
(10)). Additionally, the joint distribution can factorize into a product of
conditional and marginal distributions (“the product rule” - Equation (11)).
These two simple rules can be used to build much of the theory and
applications of probability theory (e.g., Bayes’ theorem
$P(Y|X)=\frac{P(X|Y)P(Y)}{P(X)}$). While Equations (10) deals with probability
distributions of discrete random variables, there is also a sum rule analog
for continuous random variables and probability density functions (the syntax
of the product rule is the same):

$P(X=x)=\int_{Y}P(X=x,\,Y=y)\,dy$

where $\int_{Y}$ represents an integral over the domain of $Y$ (e.g.,
$\int_{-\infty}^{\infty}$ if $Y$ is a Gaussian random variable).


# Approximation by a special de la Vallée Poussin type matrix transform mean
of Walsh-Fourier series

István Blahota  Institute of Mathematics and Computer Sciences
University of Nyíregyháza
H-4400 Nyíregyháza, Sóstói street 31/b
Hungary<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider norm convergence for a special matrix-based de la
Vallée Poussin-like mean of Fourier series for the Walsh system. We estimate
the difference between the named mean above and the corresponding function in
norm, and the upper estimation is given by the modulus of continuity of the
function.

###### Key words and phrases:

character system, Fourier series, Walsh-Paley system, rate of approximation,
modulus of continuity, matrix transform

###### 1991 Mathematics Subject Classification:

42C10

##  1\. Definitions and notations

We follow the standard notions of dyadic analysis introduced by F. Schipp, W.
R. Wade, P. Simon, and J. Pál [17] and others.

##  2\. Definitions and notation

Let $\mathbb{P}$ be the set of positive natural numbers and
${\mathbb{N}}:={\mathbb{P}}\cup\\{0\\}$. Let denote by ${\mathbb{Z}}_{2}$ the
discrete cyclic group of order 2, the group operation is the modulo $2$
addition. Let be every subset open. The normalized Haar measure $\mu$ on
${\mathbb{Z}}_{2}$ is given in the way that $\mu(\\{0\\}):=\mu(\\{1\\}):=1/2$.
$G:=\overset{\infty}{\underset{k=0}{{\times}}}{\mathbb{Z}}_{2},\ G$ is called
the Walsh group. The elements of Walsh group $G$ are sequences of numbers 0
and 1, that is $x=(x_{0},x_{1},\dots,x_{k},\dots)$ with $x_{k}\in\\{0,1\\}\
(k\in{\mathbb{N}}).$

The group operation on $G$ is the coordinate-wise addition (denoted by $+$),
the normalized Haar measure $\mu$ is the product measure and the topology is
the product topology. Dyadic intervals are defined in the usual way

$I_{0}(x):=G,\ I_{n}(x):=\\{y\in
G:y=(x_{0},\dots,x_{n-1},y_{n},y_{n+1},\dots)\\}$

for $x\in G,n\in{\mathbb{P}}$. They form a base for the neighbourhoods of $G$.
Let $0:=(0:i\in{\mathbb{N}})\in G$ denote the null element of $G$ and
$I_{n}:=I_{n}(0)$ for $n\in{\mathbb{N}}$.

Let $L_{p}(G)$ denote the usual Lebesgue spaces on $G$ (with the corresponding
norm $\|.\|_{p}$), where $1\leq p<\infty$.

For the sake of brevity in notation, we agree to write $L_{\infty}(G)$ instead
of $C(G)$ and set $\|f\|_{\infty}:=\sup\\{\lvert f(x)\rvert:x\in G\\}.$ Of
course, it is clear that the space $L_{\infty}(G)$ is not the same as the
space of continuous functions, i.e. it is a proper subspace of it. But since
in the case of continuous functions the supremum norm and the $L_{\infty}(G)$
norm are the same, for convenience we hope the reader will be able to tolerate
this simplification in notation.

Next, we define the modulus of continuity in $L_{p}(G),1\leq p\leq\infty,$ of
a function $f\in L_{p}(G)$ by

$\omega_{p}(f,\delta):=\sup_{\lvert
t\rvert<\delta}\|f(.+t)-f(.)\|_{p},\quad\delta>0,$

with the notation

$\lvert x\rvert:=\sum_{i=0}^{\infty}\frac{x_{i}}{2^{i+1}}\quad\textrm{for all
}x\in G.$

The Lipschitz classes in $L_{p}(G)$ (for each $\alpha>0$) are defined as

$\textrm{Lip}(\alpha,p,G):=\\{f\in
L_{p}(G):\omega_{p}(f,\delta)=O(\delta^{\alpha})\textrm{ as }\delta\to 0\\}.$

We introduce some concepts of Walsh-Fourier analysis. The Rademacher functions
are defined as

$r_{k}(x):=(-1)^{x_{k}}\ (x\in G,k\in{\mathbb{N}}).$

The Walsh-Paley functions are the product functions of the Rademacher
functions. Namely, each natural number $n$ can be uniquely expressed in the
number system based 2, in the form

$n=\sum_{k=0}^{\infty}n_{k}2^{k},\ n_{k}\in\\{0,1\\}\ (k\in{\mathbb{N}}),$

where only a finite number of $n_{k}$’s different from zero. Let the order of
$n\in\mathbb{P}$ be denoted by $|n|:=\max\\{j\in{\mathbb{N}}:n_{j}\neq 0\\}$.
Walsh-Paley functions are $w_{0}:=1$ and for $n\in\mathbb{P}$

$w_{n}(x):=\prod_{k=0}^{\infty}r_{k}^{n_{k}}(x)=(-1)^{\sum_{k=0}^{|n|}n_{k}x_{k}}.$

Let ${\mathcal{P}}_{n}$ be the collection of Walsh polynomials of order less
than $n$, that is, functions of the form

$P(x)=\sum_{k=0}^{n-1}a_{k}w_{k}(x),$

where $n\in\mathbb{P}$ and $\\{a_{k}\\}$ is a sequence of complex numbers.

It is known [10] that the system $(w_{n},n\in{\mathbb{N}})$ is the character
system of $(G,+)$. The $n$th Fourier-coefficient, the $n$th partial sum of the
Fourier series and the $n$th Dirichlet kernel is defined by

$\hat{f}(n):=\int_{G}fw_{n}d\mu,\ S_{n}(f):=\sum_{k=0}^{n-1}\hat{f}(k)w_{k},\
D_{n}:=\sum_{k=0}^{n-1}w_{k},\ D_{0}:=0.$

Fejér kernels are defined as the arithmetical means of Dirichlet kernels, that
is,

$K_{n}:=\frac{1}{n}\sum_{k=1}^{n}D_{k}.$

Let $T:=\left(t_{i,j}\right)_{i,j=1}^{\infty}$ be a doubly infinite matrix of
numbers. It is always supposed that matrix $T$ is lower triangular.

Let us define the $(m,n)$th matrix transform de La Vallée Poussin mean
determined by the matrix $T$ as

$\sigma_{m,n}^{T}(f):=\sum_{k=m}^{n}t_{k,n}S_{k}(f),$

where $m,n\in\mathbb{P}$ and $m\leq n$.

The $(m,n)$th matrix transform de La Vallée Poussin kernel is defined as

$K_{m,n}^{T}:=\sum_{k=m}^{n}t_{k,n}D_{k}.$

It is very easy to verify that

$\sigma_{m,n}^{T}(f;x)=\int_{G}f(u)K_{m,n}^{T}(u+x)d\mu(u).$

We introduce the notation $\Delta t_{k,n}:=t_{k,n}-t_{k+1,n},$ where
$k\in\\{1,\ldots,n\\}$ and $t_{n+1,n}:=0$.

##  3\. Historical overview

Matrix transforms means are common generalizations of several well-known
summation methods. It follows by simple consideration that the Nörlund means,
the Fejér (or the $(C,1)$) and the $(C,\alpha)$ means are special cases of the
matrix transform summation method introduced above.

Our paper is motivated by the work of Móricz, Siddiqi [14] on the Walsh-
Nörlund summation method and the result of Móricz and Rhoades [13] on the
Walsh weighted mean method. As special cases, Móricz and Siddiqi obtained the
earlier results given by Yano [23], Jastrebova [11] and Skvortsov [19] on the
rate of the approximation by Cesàro means. The approximation properties of the
Walsh-Cesàro means of negative order were studied by Goginava [9], the
Vilenkin case was investigated by Shavardenidze [18] and Tepnadze [20]. A
common generalization of these two results of Móricz and Siddiqi [14] and
Móricz and Rhoades [13] was given by Nagy and the author [2].

In 2008, Fridli, Manchanda and Siddiqi generalized the result of Móricz and
Siddiqi for homogeneous Banach spaces and dyadic Hardy spaces [8]. Recently,
the author, Baramidze, Memić, Nagy, Persson, Tephnadze and Wall presented some
results with respect to this topic [1],[3], [5],[12]. See [7, 22], as well.
For the two-dimensional results see [4, 16, 15].

It is important to note that in the paper of Chripkó [6] some methods and
results with respect to Jacobi-Fourier series gave us some ideas and used in
this paper.

##  4\. Auxiliary results

To prove Theorem 1 we need the following Lemmas.

######  Lemma 1 (Paley’s Lemma [17], p. 7.).

For $n\in\mathbb{N}$

$D_{2^{n}}(x)=\begin{cases}2^{n},&\textrm{ if }x\in I_{n},\\\ 0,&\textrm{ if
}x\notin I_{n}.\end{cases}$

######  Lemma 2 ([17], p. 34.).

For $j,n\in\mathbb{N},\ j<2^{n}$ we have

$D_{2^{n}+j}=D_{2^{n}}+r_{n}D_{j}.$

######  Lemma 3 (Yano’s Lemma [24]).

The norm of the Fejér kernel is bounded uniformly. That is, for all
$n\in\mathbb{P}$

$\|K_{n}\|_{1}\leq 2.$

In 2018, Toledo improved this result.

######  Lemma 4.

[21]

$\sup_{n\in\mathbb{P}}\|K_{n}\|_{1}=\frac{17}{15}.$

######  Lemma 5.

[14] Let $n\in\mathbb{P},\ g\in{\mathcal{P}}_{2^{n}}$, $f\in L_{p}(G)$ ($1\leq
p\leq\infty$). Then

$\left\|\int_{G}r_{n}(t)g(t)(f(\cdot+t)-f(\cdot))d\mu(t)\right\|_{p}\leq\frac{1}{2}\|g\|_{1}\omega_{p}\left(f,2^{-n}\right)$

holds.

In the next lemma, we give a decomposition of the kernels
$K_{2^{n},2^{n+1}-1}^{T}$.

######  Lemma 6.

Let $n$ be a positive integer, then we have

$\displaystyle K_{2^{n},2^{n+1}-1}^{T}$
$\displaystyle=\sum_{k=0}^{2^{n}-1}t_{2^{n}+k,2^{n+1}-1}D_{2^{n}}+r_{n}\sum_{k=1}^{2^{n}-2}\Delta
t_{2^{n}+k,2^{n+1}-1}kK_{k}$
$\displaystyle\quad+r_{n}t_{2^{n+1}-1,2^{n+1}-1}(2^{n}-1)K_{2^{n}-1}$
$\displaystyle=:\sum_{j=1}^{3}K_{j,n}.$

###### Proof.

We write

$\displaystyle
K_{2^{n},2^{n+1}-1}^{T}=\sum_{l=2^{n}}^{2^{n+1}-1}t_{l,2^{n+1}-1}D_{l}.$

Now, we apply Lemma 2. We get

$\displaystyle\sum_{l=2^{n}}^{2^{n+1}-1}t_{l,2^{n+1}-1}D_{l}$
$\displaystyle=\sum_{k=0}^{2^{n}-1}t_{2^{n}+k,2^{n+1}-1}D_{2^{n}+k}$
$\displaystyle=\sum_{k=0}^{2^{n}-1}t_{2^{n}+k,2^{n+1}-1}D_{2^{n}}+r_{n}\sum_{k=1}^{2^{n}-1}t_{2^{n}+k,2^{n+1}-1}D_{k}.$

Using Abel-transform

$\displaystyle\sum_{k=1}^{2^{n}-1}t_{2^{n}+k,2^{n+1}-1}D_{k}$
$\displaystyle=\sum_{k=1}^{2^{n}-2}\Delta t_{2^{n}+k,2^{n+1}-1}kK_{k}$
$\displaystyle\quad+\ t_{2^{n+1}-1,2^{n+1}-1}(2^{n}-1)K_{2^{n}-1}.$

Summarizing these it completes the proof of Lemma 6. ∎

##  5\. The rate of the approximation

######  Theorem 1.

Let $f\in L_{p}(G)\ (1\leq p\leq\infty)$. For every $n\in\mathbb{P},\
\\{t_{k,2^{n+1}-1}:2^{n}\leq k\leq 2^{n+1}-1\\}$ be a finite sequence of non-
negative numbers such that

(5.1) $\sum_{k=2^{n}}^{2^{n+1}-1}t_{k,2^{n+1}-1}=1$

be satisfied.

a) If the finite sequence $\\{t_{k,2^{n+1}-1}:2^{n}\leq k\leq 2^{n+1}-1\\}$ is
non-decreasing for a fixed $n$ and

(5.2) $t_{2^{n+1}-1,2^{n+1}-1}=O\left(\frac{1}{2^{n+1}-1}\right),$

or

b) if the finite sequence $\\{t_{k,2^{n+1}-1}:2^{n}\leq k\leq 2^{n+1}-1\\}$ is
non-increasing for a fixed $n$, then

$\left\|\sigma_{2^{n},2^{n+1}-1}^{T}(f)-f\right\|_{p}\leq
c\omega_{p}\left(f,2^{-n}\right)$

holds.

###### Proof of Theorem 1.

The proof is carried out in cases where $1\leq p<\infty$, while the proof of
case $p=\infty$ is similar. Recall that by the case $p=\infty$ we mean that we
are considering the space of continuous functions.

During our proofs $c$ denotes a positive constant, which may vary at different
appearances.

We use condition (5.1), the usual Minkowski inequality and Lemma 6

$\displaystyle\left\|\sigma_{2^{n},2^{n+1}-1}^{T}(f)-f\right\|_{p}$
$\displaystyle=\left(\int_{G}|\sigma_{2^{n},2^{n+1}-1}^{T}(f;x)-f(x)|^{p}d\mu(x)\right)^{\frac{1}{p}}$
$\displaystyle=\left(\int_{G}\left|\int_{G}K_{2^{n},2^{n+1}-1}^{T}(u)F(x,u)d\mu(u)\right|^{p}d\mu(x)\right)^{\frac{1}{p}}$
$\displaystyle\leq\sum_{j=1}^{3}\left(\int_{G}\left|\int_{G}K_{j,n}(u)F(x,u)d\mu(u)\right|^{p}d\mu(x)\right)^{\frac{1}{p}}$
$\displaystyle=:\sum_{j=1}^{3}I_{j,n}$

with notation $F(x,u):=f(x+u)-f(x)$.

Using generalized Minkowski inequality, Lemma 1 and condition (5.1) for the
expressions $I_{1,n}$, we obtain

$\displaystyle I_{1,n}$
$\displaystyle\leq\sum_{k=0}^{2^{n}-1}t_{2^{n}+k,2^{n+1}-1}\int_{G}D_{2^{n}}(u)\left(\int_{G}\left|F(x,u)\right|^{p}d\mu(x)\right)^{\frac{1}{p}}d\mu(u)$
$\displaystyle\leq\sum_{k=2^{n}}^{2^{n+1}-1}t_{k,2^{n+1}-1}\omega_{p}\left(f,2^{-n}\right)$
$\displaystyle\leq\omega_{p}\left(f,2^{-n}\right).$

Now, applying Lemma 4 and Lemma 5 we get

$\displaystyle I_{2,n}$ $\displaystyle\leq\sum_{k=1}^{2^{n}-2}\left|\Delta
t_{2^{n}+k,2^{n+1}-1}\right|k$
$\displaystyle\quad\times\left(\int_{G}\left|\int_{G}r_{n}(u)K_{k}(u)F(x,u)d\mu(u)\right|^{p}d\mu(x)\right)^{\frac{1}{p}}$
$\displaystyle\leq\sum_{k=1}^{2^{n}-2}\left|\Delta
t_{2^{n}+k,2^{n+1}-1}\right|k\frac{1}{2}\|K_{k}\|_{1}\omega_{p}\left(f,2^{-n}\right)$
$\displaystyle\leq\sum_{k=1}^{2^{n}-2}\left|\Delta
t_{2^{n}+k,2^{n+1}-1}\right|k\omega_{p}\left(f,2^{-n}\right).$

We write in case a)

$\displaystyle\sum_{k=1}^{2^{n}-2}|\Delta t_{2^{n}+k,2^{n+1}-1}|k$
$\displaystyle=\sum_{k=1}^{2^{n}-2}(t_{2^{n}+k+1,2^{n+1}-1}-t_{2^{n}+k,2^{n+1}-1})k$
$\displaystyle=(2^{n}-2)t_{2^{n+1}-1,2^{n+1}-1}-\sum_{k=1}^{2^{n}-2}t_{2^{n}+k,2^{n+1}-1}$
$\displaystyle\leq(2^{n+1}-1)t_{2^{n+1}-1,2^{n+1}-1}$

and using condition (5.2)

$\displaystyle I_{2,n}$
$\displaystyle\leq(2^{n+1}-1)t_{2^{n+1}-1,2^{n+1}-1}\omega_{p}\left(f,2^{-n}\right)$
$\displaystyle\leq c\omega_{p}\left(f,2^{-n}\right).$

We estimate the expression $I_{3,n}$ in case a). Lemma 4, Lemma 5 and
condition (5.2) yield

$\displaystyle I_{3,n}$ $\displaystyle\leq(2^{n}-1)t_{2^{n+1}-1,2^{n+1}-1}$
$\displaystyle\quad\times\left(\int_{G}\left|\int_{G}r_{n}(u)K_{2^{n}-1}(u)F(x,u)d\mu(u)\right|^{p}d\mu(x)\right)^{\frac{1}{p}}$
$\displaystyle\leq(2^{n+1}-1)t_{2^{n+1}-1,2^{n+1}-1}\frac{1}{2}\|K_{2^{n}-1}\|_{1}\omega_{p}\left(f,2^{-n}\right)$
$\displaystyle\leq(2^{n+1}-1)t_{2^{n+1}-1,2^{n+1}-1}\omega_{p}\left(f,2^{-n}\right)$
$\displaystyle\leq c\omega_{p}\left(f,2^{-n}\right).$

In case b) we estimate $I_{2,n}+I_{3,n}$. In this situation

$\displaystyle\sum_{k=1}^{2^{n}-2}\left|\Delta t_{2^{n}+k,2^{n+1}-1}\right|k$
$\displaystyle=\sum_{k=1}^{2^{n}-2}t_{2^{n}+k,2^{n+1}-1}-(2^{n}-2)t_{2^{n+1}-1,2^{n+1}-1},$

so Lemma 4, Lemma 5 and condition (5.1) imply

$\displaystyle I_{2,n}+I_{3,n}$
$\displaystyle\leq\sum_{k=1}^{2^{n}-2}\left|\Delta
t_{2^{n}+k,2^{n+1}-1}\right|k\times$
$\displaystyle\quad\quad\times\left(\int_{G}\left|\int_{G}r_{n}(u)K_{k}(u)F(x,u)d\mu(u)\right|^{p}d\mu(x)\right)^{\frac{1}{p}}$
$\displaystyle\quad+(2^{n}-1)t_{2^{n+1}-1,2^{n+1}-1}\times$
$\displaystyle\quad\quad\times\left(\int_{G}\left|\int_{G}r_{n}(u)K_{2^{n}-1}(u)F(x,u)d\mu(u)\right|^{p}d\mu(x)\right)^{\frac{1}{p}}$
$\displaystyle\leq\left(\sum_{k=1}^{2^{n}-2}t_{2^{n}+k,2^{n+1}-1}+(2^{n}-1)t_{2^{n+1}-1,2^{n+1}-1}-(2^{n}-2)t_{2^{n+1}-1,2^{n+1}-1}\right)$
$\displaystyle\quad\times\frac{1}{2}\cdot\frac{17}{15}\omega_{p}\left(f,2^{-n}\right)$
$\displaystyle=\sum_{k=1}^{2^{n}-1}t_{2^{n}+k,2^{n+1}-1}\frac{17}{30}\omega_{p}\left(f,2^{-n}\right)$
$\displaystyle\leq\frac{17}{30}\omega_{p}\left(f,2^{-n}\right).$

This completes the proof of our Theorem 1. ∎

######  Remark 1.

We mention, that assuming (5.1) is natural, because many well-known means
satisfy it and this equality is a part of regularity conditions [25, page
74.].

######  Corollary 1.

Let us suppose that the conditions in Theorem 1 are satisfied. If
$f\in\textrm{Lip}(\alpha,p,G)$, then

$\|\sigma_{2^{n},2^{n+1}-1}(f)-f\|_{p}=O\left(2^{-n\alpha}\right).$

######  Remark 2.

In case b) we can formulate the statement of Theorem 1 in following form

$\left\|\sigma_{2^{n},2^{n+1}-1}^{T}(f)-f\right\|_{p}\leq\frac{47}{30}\omega_{p}\left(f,2^{-n}\right).$

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