Causal Inference for Spatial Data Analytics

Spatial causal inference is still a research area in its infant stage. It deserves a lot of research. This seminar gives a great opportunity to check different opinions from attendees who are from very different disciplines/backgrounds. Some consensus was drawn from the meeting discussion. My overall position statements are:

1. 1.

   Benchmarking: Benchmarking is critical to understanding performance differences among various solutions proposed everyday. [1] provides an overview of related causal discovery and inference applied to Earth science domain, which are mostly spatiotemporal data. It also made efforts to list related synthetic and real-world data used in related research.

2. 2.

   Machine/deep learning + Causality: The integration of machine/deep learning with causal AI could greatly help each other. Integrating machine/deep learning into causal inference could help causal inference’s performance by finding complicated patterns from data. For instance, [2] shows how deep learning can be used to estimate direct and indirect causal effects of spatiotemporal interventions in presence of spatial interference. By integrating causal discovery/inference results could help machine learning models’ explainability.

3. 3.

   Taxonomy: Some primer on the basic taxonomy/terminology will help researchers understand each other’s work.

4. 4.

   Community building: Additional community building efforts including additional rounds of Dagstuhl Seminar, workshops and tutorials would greatly help the community grow.

Intensive discussions among the seminar participants with the background in spatial statistics and/or causal inference have brought up ideas how to define the “spatial causal inference” as well to formulate possible objectives and working directions of this new discipline. Some of the working directions can be:

1. 1.

   In case of multivariate case, it has been proposed to express the spatial distance of the variables in the neighborhood of the target as a moderation variable.

2. 2.

   Predictive causal inference based in Granger [1] and its non-linear versions [2] has been an established field of causal inference, especially for temporal data/observations coming from Earth sciences. Although spatio-temporal Granger causality has been introduced for two variables [3], to our best knowledge, the extension to $p>2$ spatio-temporal variables has not been studied yet.

3. 3.

   Setting the causal discovery graph based on the dependence test (e.g. by PC [4]) and then applying causal inference to this graph (DAG) can provide ambiguous graphs for spatial scenarios, if “new” data (i.e. those with non-zero distance from the data for which was the graph generated) is used. There is a question for which types of data (or for which data distributions) this two-step procedure provides a unique output causal graph. Otherwise these types of causal discovery graphs could be used separately for data in different spatial scenarios and the inference then applied to the resulted graphs.

The Dagstuhl Seminar on causal inference for spatial data analytics aimed to formalize where cause-effect relationships in spatial data exist, how they can be potentially discovered, and how their effect sizes can be quantified.

Briefly, let a causal spatial inference problem be defined as a problem involving an acyclic digraph that denotes statistical (causal) dependencies between random variables $(X,Y,Z)$ and which has a potential spatial component $s$ yielding the spatial graphical model $(X(s),Y(s),Z(s))$.

Intriguingly, in the most general case, the factorization of the joint distribution defined by the spatial graphical model contains both causal dependencies, e.g., $X(s)\to Y(s)$ as well as statistical ones, e.g, encoded via potential functions $\psi (X(s),X(t))$, which can complicate typical operations, such as computation of interventional or counterfactual distributions. In specific cases, e.g., when no correlation structure between spatial variables exists, the spatial causal inference problem seems to be readily reduceable to the conventional causal framework which allows for structure learning with contemporary constraint-based methods and effect estimation using, e.g., the Pearlean identification criteria.

The emerging field of causal spatial inference offers a wide variety of interesting future research directions ranging from discovery to estimation.

Causal inference aims to formalise the investigative query of discovering and quantifying pathways of causation between variables. This query arises within many sciences, such as climate science, neuroscience and geography. In several cases of interest, the variables, or the interactions between them, or the causal structure itself varies over space. A closer look at the several applications cases quickly illustrates the richness of the problem of incorporating spatial statistics into traditional causal inference language. We studied in detail the problem of quantifying the effect of $C{O}_2$ on gross primary production (GPP) of plants. We also investigated disentangling the causal effect of emissions from a power plant at certain space-coordinate from the causal effect of derivative (also called children in causal graphical language) variables around that space-time coordinate also presents several challenges, starting with the complication that the two causal drivers of air quality are non-trivially related to, and influenced by each other. In the course of discussing these and a few other examples, novel problems emerge, for which the causal inference toolbox is found wanting.

In this Dagstuhl Seminar, a fruitful exchange between the spatial statistics and the causal inference community resulted in the perspective that “spatial causal inference” is, in fact, an umbrella term for problems where the space as a dimension plays a role, and either cannot be ignored to ensure soundness, or can be instrumental for certain computations, such as that of de-confounding. In order for this field to progress, a multi-pronged approach is required that is motivated by grouping spatial causal inference tasks into clusters and advancing them individually. The task of establishing a unified framework for any and all causal inference queries that use the space dimension non-trivially would ideally emerge thereafter.

Spatial causality can express itself in various ways; it is not straightforward to represent in directed acyclic graphs, and special care must be taken when establishing equations and governing notation. The following insights from spatial statistical modelling may be useful to bear in mind when constructing spatial causal models:

1. 1.

   Causality is a property of the underlying process: We should resist the temptation to think of a spatial treatment as directly affecting the outcome (or observation); the spatial treatment affects a spatial variable that may only be observed through incomplete and noisy data. Causality is between the spatial treatment and an (unobserved/latent) spatial variable. This distinction is somewhat critical, because

2. 2.

   What you see is not what you want to get (a.k.a. think hierarchically) [1]: Even if the spatial outcome is observed in its entirety (i.e., there are no missing values), it is generally a noisy version of the underlying process. Interest is in the causal effect on the process, and not on the noisy data. Measurement errors, biases, etc. need to be factored in when making inference on the causal effect.

3. 3.

   Think continuously: In spatial causal models, treatments may be point referenced, and data might be areal, or vice versa; this “change of support” problem [2] can be solved by modelling everything on a continuously-indexed spatial domain, and by defining treatments, outcomes, and any confounding variables as integrals over the spatial domain; see also [3, 4].

4. 4.

   Treatments can have far-reaching consequences: A treatment at ${𝐬}_i$ may cause the outcome to change somewhere far away; at ${𝐬}_j$ say. This is clearly the case in environmental problems where polluting rivers affect ecosystems downstream, or where toxic gases from a chemical plant affect people living in villages downwind. One therefore often needs to model the “sensitivity” of the outcome at ${𝐬}_j$ to a treatment at ${𝐬}_i$, potentially for every ${𝐬}_i,{𝐬}_j$ inside the spatial domain of interest. It is from the combination of this sensitivity and the treatment footprint (e.g., their inner product) that the effect on ${𝐬}_j$ can be established.

5. 5.

   Think temporally: Spatial variables are either temporal snapshots of spatio-temporal variables or averages of spatio-temporal variables over time. A legitimate question is: Does averaged spatio-temporal causality lead to spatial causality? I believe the answer to this question is yes (under linearity and some other assumptions): if the outcome at a certain point in space and time is caused by a convolution of the spatio-temporal treatment and a spatio-temporally varying sensitivity, then the temporally-averaged outcome at that spatial location can be obtained from the the temporally averaged treatment and the temporally aggregated sensitivity. This is a consequence of Fubini’s theorem; see [5], Section 4.3, for details.

Causal inference is an important approach for providing useful answers to scientific questions – and solutions to applied problems – in regional science, geography, and urban planning. However, there are a number of challenges to using causal inference in urban-geographic settings: 1) the overlapping correlations inherent in spatial data and data-generating processes often violate the basic assumptions of the potential outcomes model; 2) many of the causal effects of interest in these settings are spatially- and temporally-heterogeneous, and adoption/treatment is often staggered and/or of varying intensity; 3) in many cases true randomized experiments in these settings are not possible to design to answer the research questions of interest, which makes the availability of appropriate secondary data and methodological choices of individual researchers particularly important.

Beyond these challenges, it is also important to note that the theory and methods of causal inference have developed in somewhat distinct literatures and are often used on different kinds of data to answer different kinds of questions. For instance, Rubin’s potential outcomes model originated in statistics [1] and is used in a wide range of domains, often in the social and health sciences. The difference-in-differences method, which was applied seminally in Card and Krueger’s analysis [2] of minimum wage and employment – for which Card was awarded the 2021 Nobel Prize in Economics – is by far the dominant framework for causal inference in economics, and continues to influence approaches to causal inference coming from the spatial econometric perspective using (typically) areal spatial data [3]. Other methods of causal inference, such as the Structural Causal Model (SCM) [4], are used to study earth systems data and spatial-temporal processes. Even more recently, new approaches for estimating heterogeneous treatment effects – such as the “metalearners” [5] and causal forest [6] – have emerged from the machine learning literature.

While the underlying philosophy of these approaches are arguably the same, they currently speak different languages, use different notation, and focus on different assumptions. Thus I think that any “spatial-causal” project must first acknowledge the unique development of the various strands of causal inference in different disciplines. It should also attempt to build a more holistic theory – or at least a more holistic accounting – of causal inference as applied to spatial data, starting from general principles and moving to more specific assumptions and approaches that can be applied to different kinds of spatial data in different substantive domains. In my view, spatial causal inference should be a “broad church” that includes any work dealing with problems of causal inference where the underlying process of causation – including treatment, outcome, susceptibility, or confounding – varies in space and is accounted for somehow in the analysis.

Spatial DAGS

It is quite challenging to understand how to appropriately represent contextual effects in directed acylic graph (DAG) representations of models. These are the bread and butter of contemporary causal inference, yet it is challenging to understand how to represent spatial concepts within them. After this seminar, it seems that one useful way forward may be through chain graph concepts, which require us to specify process-specific forms of spatial dependence within the DAG itself.

Attributing Spatial Context

Contextual effects are a very important component of spatial planning and program evaluation. Broadly speaking, this refers to the effect that surrounding conditions have on an outcome. In an interventionist case, it refers to the effect that surrounding treatment may have on your outcome. Distinct from spatial interference (where surrounding treatment interferes with your treatment), this is an important novel component for spatial causal analysis which is difficult to represent in classical causal analytical frameworks. This is distinct from the much more difficult example of spatial endogeneity within $Y$, as might happen when outcomes that are near one another influence one another.

General Spatial Causal Model

Placing contextual effects alongside other well-studied spatial causal issues (such as spatial confounding or interference), it becomes important to define a so-called general spatial causal model that can be used to simluate data according to multiple different processes. It seems that only by combining these various processes can we actually identify treatment effect estimators that are robust to these processes.

Spatial Targeting

In classical causal inference, I have learned this week that targeting is the practice of identifying which subset would most benefit from treatment. Classical targeting approaches assume that individuals’ treatments can be administered independently, but this is not so when treatment has spatial components. In a spatial targeting problem, applying a treatment in a given location (or with a given distance decay effect) may not be beneficial to the surroundings of the treatment, even though it is beneficial at the site of treatment. In GIScience, it is important to try to identify where an intervention might maximize the post-intervention difference. This has mathematical similarities to a maximal covering location problem (MCLP), a kind of set covering problem, where it would be important to identify specific target locations that maximize benefits as a function both of the direct and spillover treatment. Solving this remains an open question.

Regardless, this seminar has been quite effective in stimulating cross-community collaboration between spatial statisticians and computer scientists studying spatial causal analysis (or any subset of those terms). I believe this was immensely valuable, and it will undoubtedly influence my thinking and future work.

Spatial causal inference focuses on estimating the effect of treatments, interventions, exposures, or policies, and inferring causal relationships using spatial data. The Dagstuhl Seminar on spatial causal inference is both timely and important for fostering discussions among experts from diverse fields and backgrounds.

The importance of spatial causal inference is underscored by the emergence of numerous scientific questions that are inherently causal in nature [1], and the increasing availability of large studies containing spatial data, such as those in environmental health, epidemiology, geoscience, economics, urban planning, and earth science. Despite its potential, spatial causal inference remains in its early stages and encounters significant challenges.

A fundamental characteristic of spatial data is that variables located closer together tend to be more similar than those further apart, as per Tobler’s First Law of Geography. This spatial correlation can violate classic causal assumptions, such as the independence and identical distribution of observations, and the stable unit treatment value assumption, where an outcome at one location may depend not only on the treatment at that location but also on treatments at nearby locations. Additionally, causal relationships may vary spatially due to differing environmental conditions across large areas. Addressing these complexities in spatial causal inference is challenging but essential.

Nevertheless, spatially structured data can be an asset rather than a drawback. The inherent structure can be leveraged to enhance causal inference. For example, different spatial patterns in outcomes and confounders can be utilized to mitigate biases resulting from missing spatial confounders [2]. This potential makes me optimistic about the future of spatial causal inference.

In my research, I work on theory and methods for causal inference as well as the application of these methods to real-world data. My main focus lies on time series data, but I am also increasingly dealing with applications to spatio-temporal data. This is why I am enthusiastic to participate in and contribute to a week of discussions on causal analysis for spatial data. In my view, it is particularly important to discuss the importance of specifying the respective targets of estimation and discovery before proceeding to derive estimands and estimates for these targets. Moreover, I am looking forward to discussions on the different roles that space can play in the analysis. For example, in one case one might be interested in the cause-and-effect relationships between variables that by themselves are spatial, whereas in another case one might be interested in the relationships between the variables at individual space points and to this end need to take care of the confounding effect by variables at the other spatial locations. From the seminar, I hope to take away thoughts for future work and to lay the foundation for potential collaborations in the future.

As many examples brought forward during this seminar aptly demonstrated, many questions on cause and effect in the sciences involve data that is inherently spatial. On the one hand, geoscientists, econometricians and statisticians have developed practical techniques to deal with spatially structured data (many of which were discussed and presented at this seminar), while causal inference researchers have formalized the notion of causal effects and interventions with a focus on non-spatial data. In my opinion, there is a need for models that clearly delineate between causally induced and spatially induced relationships, and on which the concept of an intervention is unambiguously defined. These models should be close to actual scientific practice. Therefore, equipping existing models with a notion of intervention that implies a definition of causal effect which matches applied researchers’ intuition would be a useful step forward. Hierarchical spatial process model [1] seem to be a particularly fitting model class for this goal as they enable to formulate causal notions on the level of the underlying process instead of directly on the measured data. Another reference that has crystallized as a useful starting point for incorporating the causal notion explicitly into spatio-temporal models is [2]. In addition, tools for generating data with both causal and spatial components and literature that reviews existing ideas in combination with practical examples and code would help the community significantly.

The Dagstuhl Seminar has done a particularly great job in making researchers explain their methodology to experts from other fields, spurning interactions that would have been rare otherwise. To keep the community going, future meetings, whether at Dagstuhl or in other venues, would be crucial and first steps have been taken towards that goal.

Spatial causal inference has the potential to be highly relevant for data scientists working in industry. One immediate application is A/B testing, where spatial information can be used to understand treatment effect heterogeneity and the regional targeting of interventions. Ideas discussed in the seminar include spatial versions of existing causal machine learning algorithms and using spatial features as covariates. Spatial causal inference can also help with more well-known problems such as spatial confounding and spillover effects.

Industry can also benefit from the conceptual ideas discussed in the seminar, including ways to reason about and represent space in the context of inference. Because there is no systematic ways to represent space in data science, assumptions about its influence are often left unstated. Developing frameworks for spatial causal reasoning would therefore plausibility improve decision-making in industry.

To move spatial causal inference forward in industry, we need case studies that show its successful application in concrete problems with non-curated data. Working on such case studies will likely also surface interesting new methodological challenges.

Both spatial dependence and causality are still often ignored in Earth system data analysis and machine learning, leading to potentially biased results and misleading conclusions about environmental effects on vegetation and ecosystems. That’s why this seminar on Causal inference for Spatial Data Analysis is double-important.

Spatial dependence can be curse or blessing and I find it most interesting to see how it can help with adjusting for confounding effects. For instance if we want to quantify the effect of CO2 on vegetation photosynthesis from observations, we have to consider confounders such as nitrogen deposition and should exploit those have a different spatial structure than CO2 concentrations.

Another question is how to identify causal effects in a hybrid modelling framework, which combines a process-based model with a machine-learning approach.

Other ideas inspired from the seminar:

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  Can we identify temporally or spatially varying DAGs from a spatio-temporal data set?

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  Can we use administrative boundaries to adjust for human confounders when we want to find effects of climate variables?

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  How can we identify causal spatial context effects (aka convolutions in machine learning), for instance on ecosystem responses to drought stress?

A major endeavour of the social sciences is to explain social process and therefore to generate causal rather than associational claims. However, the identification of causal effects can prove to be challenging if not impossible in the presence of social and spatial complexity. Namely, interaction (direct or indirect) between individuals or events within a geographic environment can result in spatial mechanisms that is difficult to completely disentangle in the form of clear directed causal relationships. Having acknowledged such dynamics, what is essentially put into question is the validity of the causal claims that stem from spatial prediction models as well as from causal models that do not explicitly address the spatial nature, or the “spatiality” of the problem. From this observation emerges the importance but also the challenges of spatial causal inference. Defining this term and describing the holistic landscape of this burgeoning field have been the focal points of this seminar. If one were to think of spatial causality, as we have, as the more general form of causality, then one may also consider what it takes to cement spatial considerations in the causal framework. From the very interesting discussions I have noted several aspects that I see as “directions” for this:

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  Methods for identification (or falsification) of “spatial effects” and learning relevant spatial representations: this problem relates to one of invariance search under spatial representations and transformations (permutation, aggregation…) for the identification (or falsification) of “spatial effects” in the problem.

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  An appropriate formalism for the problem, as both a tool to reflect on the role of space and to describe it. A formalisation should translate the relevant spatial (and temporal) situation of the problem while being as actionable as possible. In the case of a DAG-based descriptions, a spatiotemporal process may blur the distinctions between nodes and edges (or what happens between two nodes). Solutions relate to the embedding of spatial nodes in the graph or to the embedding of nodes in space (implying often a change of scale or change of support). An interesting prospect would be to also consider the “embedding” of edges in space: how do spatial relations relate to causal ones? (ex: perhaps considering the use of chain graphs)

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  Inference methods: One of the objectives of developing community around this topic is to bring together the tools that have been developed in various strands of research. This also requires understanding their differences in order to build a structured set of methods. I am interested in how representations of space and spatially formalising the causal problem can support the modelling choices (ex: how may a causal forest method integrate spatial information?)

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  Evaluation methods: In the absence of empirical ground truth, I believe one way forward is to integrate our assumptions on the underlying spatial processes in a simulation framework. By distinguishing between types of spatial (and temporal) mechanism one can combine to generate a space of assumptions that would serve as a set (collection of artificial datasets or a simulation protocol) on which to tests spatial causal methods.

I am interesting in exploring the synergies between empirical and simulation-based methods in the context of spatial causal inference, particularly under complexity assumptions. In that perspective, a complex systems approach such as Agent Based Models could serve the development of statistical tests and inference methods. Furthermore, I believe there is ground for developing multi-agent simulation-based inference methods for spatial causal analysis, harnessing their ability to describe the transition from micro-level processes to emergent properties. On the condition of course that the questions of uncertainty and validation be systematically addressed.

There has been a growing research interest in reasoning causality in spatial data analysis, aiming to answer “what if” questions beyond “what is” queries. A fundamental challenge in spatial causal inference is to understand, identify, and quantify the influence space exerts in causal inference processes.

This research direction involves two key components. On the theoretical side, we need to develop fundamental analytical frameworks for spatial causal inference, for example, extending the potential outcome framework or the structural causal models to explicitly account for the spatial dimension. On the methodological side, such as data-driven causal discovery and causal effect estimation, we need robust approaches to distill causal relationships from data. It is also important to understand in what situations these causal relationships will be infeasible to identify or quantify due to spatial interference and spatial confounding.

The spatial factor also poses challenges to causal machine learning research [1]. Previous studies have highlighted that integrating causality can enhance the interpretability and robustness of machine learning models. This is particularly appealing for spatial data science, where the lack of transparency and generalizability of machine learning models hinders their adoption in various spatial applications to support decision-making [2]. In recent years, some causal machine learning approaches have been developed in this direction [3]. However, accounting for the spatial dimension remains challenging. For example, in causal representation learning or causal generative modeling, the spatial dimension can either be considered as a factor to group other features or it can be considered as a separate feature. How the formulation influences the analysis and interpretation of results deserves further investigation.

How should we evaluate all these causal discovery methods out there? Datasets with reliable ground truth are rare, and evaluation on simulated data are questionable [1]. To enable significant progress in the field we need extensive benchmarking rather than discussing plausibility of results for a few datasets. To this end, we need a new theory of falsification of causal explanations that tells us the testable implications entailed by a causal explanation, following the spirit of Popper’s theory of science. To this end, we are working on “self-compatibility” and test whether outputs of causal discovery algorithms are compatible across different subsets of variables [2]. This way, algorithms can be falsified without ground truth. Although not contradicting itself is not a guarantee for being true, we have discussed notions of compatibility that are so strong that they can at least provide some evidence for the output of the algorithms. However, one of the most challenging questions raised by this approach is the definition of a good measure of compatibility together with a notion of calibration that tells us whether the observed inconsistencies are “many” or “few”. After all, perfect compatibility will never be achieved – how do we decide whether the number of contradictions is small enough to ensure that the algorithms are still useful? This is ongoing work!