Determining causality in correlated time series

New method goes beyond Granger causality to identify only the true causes of a target time series, given some graph constraints.

Given observed time series and a target time series of interest, can we identify the causes of the target, without excluding the presence of hidden time series? This question arises in many fields — such as finance, biology, and supply chain management — where sequences of data constitute partial observations of a system.

Imagine, for instance, that we have time series for the prices of dairy products. From the data alone, can we identify the causes of fluctuations in the price of butter?

Dairy prices.png
The prices of dairy products in Germany are correlated, but do any of those correlations imply causation?

The standard way to represent causal relationships between variables that are associated with each other is with a graph whose nodes represent variables and whose edges represent causal relationships.

In a paper that we presented at the International Conference on Machine Learning (ICML) 2021, coauthored by Bernhard Schölkopf, we described a new technique for detecting all the direct causal features of a target time series — and only the direct or indirect causal features — given some graph constraints. The proposed method yielded false-positive rates of detected causes close to zero.

The constraints we observe refer to the target and the “memory” of some hidden time series (the lack of dependency on their own pasts, in some cases). We wanted to limit our assumptions to those that can be naturally derived from the setting and that could not be avoided otherwise. Therefore, we wanted to avoid strong assumptions made by other methods, such as excluding hidden common causes (unobserved time series that caused multiple observed ones).

We also wanted to avoid other drawbacks of prior methods, such as requiring interventions on the system (to test for particular causal sequences) and requiring large conditioning sets (sets of variables that must be controlled for to detect dependences) or exhaustive conditional-independence tests, which hinder the statistical strength of the outcome.

Our method, by contrast, accounts for hidden common causes, uses only observational data, and constructs conditioning sets that are small and efficient in terms of signal-to-noise ratio, given some graph constraints that seemed hard to avoid.

Conditioning set.gif
The researchers' new method constructs a conditioning set — a set of variables that must be controlled for — that enables tests for conditional dependence and independence in a causal graph.

Conditional independence

As is well known, statistical dependence (i.e., correlation in linear cases) does not imply causation. The graphs we use to represent causal relationships between associated variables are so-called directed acyclic graphs (DAGs), meaning the edges have direction and there are no loops. The direction of the edges (represented by arrows in the graphs below) indicates the direction of causal influence. In the time series case, we use “full time DAGs”, where each node represents a different time step from a time series. 

To analyze whether a third variable, S, explains a statistical dependency (i.e., correlation) between two other variables, one checks whether the dependency disappears after restricting the statistics to data points with fixed values of S. In larger graphs, S can be a whole set of variables, which we call a conditioning set. Controlling for all the variables in a conditioning set is known as conditional independence testing and is the main tool we use in our method. 

Another important notion is that of confounding. If two variables, X and Y, are dependent, not because one causes the other, but because they’re both caused by a third variable, U, we say that they are confounded by U.

Before we get into the complex graphs of time series, let's present the intuition behind our method with simple graphs. 

In the graphs below, we manage to distinguish between causal influence and confounding relationships by searching for different patterns of conditional independence. In both graphs, X and Y are dependent (i.e., they vary together). But in the left-hand graph, Z and Y are independent when we condition on the cause X; i.e., when we control for X, variations in Y become independent from variations in Z

When, however, there is a hidden confounder between X and Y, as in the graph at right, Z and Y become dependent when conditioning on X.

This can seem counterintuitive. When we condition on a variable, we treat it as if we know its outcome. In the graph below, because we know how Z contributes to X, the difference between this contribution and the actual value of X comes from U (with some variation from noise). Since Y varies with U, it reflects that variation as well, and Z and Y become dependent.

simple_iid_case.png
An example of how the presence of a confounder can create causal dependence.

Causality in time series

This idea of finding similar characteristic patterns of conditional independences to distinguish causes from confounders is very relevant to our method. In the time series case, the graph is much more complicated than in the examples above. Here we show such a time series graph:

Baseline causal graph.png
A full time graph with hidden time series (U).

Here, we have a univariate (one-dimensional) target time series, Y, whose causes we want to find. Then we have several observed candidate time series, Xi, which might be causing the target or have different dependencies with it. Finally, we allow for the existence of several hidden time series, U.

We know the directions of some edges from the time order, which is helpful. On the other hand, time series’ dependence on their own pasts complicates the picture, because it creates common-cause schemes between nodes. 

For each candidate time series, we want to isolate the current and previous node and the corresponding target node. We thus extract triplets like the one indicated by green and yellow in the graph below.

Causal graph conditional tests.png
Tests for conditional dependence and independence in the full time graph.

If we manage to do that, then it is enough to check whether the green nodes become independent when we simultaneously condition on the yellow node and all the purple ones. 

If there is a hidden confounder between the yellow node and the target’s green node, then, conditioning on the yellow node will force a dependence between the two green nodes, as in the first example above. But to perform that test, we need to isolate our triplet from the causal influences of other time series. 

To do that, we construct a conditioning set, S, that includes at most one node from each time series that is dependent on the target. This node corresponds to the one that enters the previous time stamp of the target (Yt in the graph above). And of course, we also need to include the previous time stamp of the target node itself (Yt, above) to remove the target's past dependency, as well as the yellow node.

Here we see that indeed the relationship between Xj and Y is confounded (Xj does not cause Y, although they appear to be related). We see that the second condition of our method is violated, and consequently, Xj is correctly rejected (as it is not a cause of Y).

Given some restrictions on the graph, which we do not consider extreme given the hardness of hidden confounding, we propose and prove two theorems for the identification of direct and indirect causes in single-lag graphs — that is, graphs in which a node in a candidate time series shares only one edge with nodes in the target time series. These theorems result in an algorithm with only two conditional-independence tests and well-defined conditioning sets, which scales linearly with the number of candidate time series. 

dairy_experiments_graphs.PNG
Graphs of the causal relationships between dairy-product prices in Germany, Ireland, and the UK, with the true-positive rates (TPR) and true-negative rates (TNR) achieved by the researchers' new method.

We now return to our original motivational example, predicting the price of butter. The real-world data we used to test our approach included the price of raw milk, the price of butter, and, depending on the country, the prices of other dairy products, such as cheese and whey powder. Our method correctly deduced that the price of butter was caused by the price of raw milk but not by the prices of other dairy products, although they were strongly dependent on it. In one dataset, where the data did not include the price of raw milk, our method correctly deduced that the dependencies between the price of butter and the prices of other dairy products did not imply causation. 

Research areas

Related content

US, MA, Boston
The Artificial General Intelligence (AGI) team is looking for a passionate, talented, and inventive Applied Scientist with a strong deep learning background, to build industry-leading technology with Large Language Models (LLMs) and multi-modal systems. You will support projects that work on technologies including multi-modal model alignment, moderation systems and evaluation. Key job responsibilities As an Applied Scientist with the AGI team, you will support the development of novel algorithms and modeling techniques, to advance the state of the art with LLMs. Your work will directly impact our customers in the form of products and services that make use of speech and language technology. You will leverage Amazon’s heterogeneous data sources and large-scale computing resources to accelerate advances in generative artificial intelligence (GenAI). You are also expected to publish in top tier conferences. About the team The AGI team has a mission to push the envelope in LLMs and multimodal systems. Specifically, we focus on model alignment with an aim to maintain safety while not denting utility, in order to provide the best-possible experience for our customers.
IN, HR, Gurugram
Our customers have immense faith in our ability to deliver packages timely and as expected. A well planned network seamlessly scales to handle millions of package movements a day. It has monitoring mechanisms that detect failures before they even happen (such as predicting network congestion, operations breakdown), and perform proactive corrective actions. When failures do happen, it has inbuilt redundancies to mitigate impact (such as determine other routes or service providers that can handle the extra load), and avoids relying on single points of failure (service provider, node, or arc). Finally, it is cost optimal, so that customers can be passed the benefit from an efficiently set up network. Amazon Shipping is hiring Applied Scientists to help improve our ability to plan and execute package movements. As an Applied Scientist in Amazon Shipping, you will work on multiple challenging machine learning problems spread across a wide spectrum of business problems. You will build ML models to help our transportation cost auditing platforms effectively audit off-manifest (discrepancies between planned and actual shipping cost). You will build models to improve the quality of financial and planning data by accurately predicting ship cost at a package level. Your models will help forecast the packages required to be pick from shipper warehouses to reduce First Mile shipping cost. Using signals from within the transportation network (such as network load, and velocity of movements derived from package scan events) and outside (such as weather signals), you will build models that predict delivery delay for every package. These models will help improve buyer experience by triggering early corrective actions, and generating proactive customer notifications. Your role will require you to demonstrate Think Big and Invent and Simplify, by refining and translating Transportation domain-related business problems into one or more Machine Learning problems. You will use techniques from a wide array of machine learning paradigms, such as supervised, unsupervised, semi-supervised and reinforcement learning. Your model choices will include, but not be limited to, linear/logistic models, tree based models, deep learning models, ensemble models, and Q-learning models. You will use techniques such as LIME and SHAP to make your models interpretable for your customers. You will employ a family of reusable modelling solutions to ensure that your ML solution scales across multiple regions (such as North America, Europe, Asia) and package movement types (such as small parcel movements and truck movements). You will partner with Applied Scientists and Research Scientists from other teams in US and India working on related business domains. Your models are expected to be of production quality, and will be directly used in production services. You will work as part of a diverse data science and engineering team comprising of other Applied Scientists, Software Development Engineers and Business Intelligence Engineers. You will participate in the Amazon ML community by authoring scientific papers and submitting them to Machine Learning conferences. You will mentor Applied Scientists and Software Development Engineers having a strong interest in ML. You will also be called upon to provide ML consultation outside your team for other problem statements. If you are excited by this charter, come join us!
US, WA, Seattle
Do you want to re-invent how millions of people consume video content on their TVs, Tablets and Alexa? We are building a free to watch streaming service called Fire TV Channels (https://techcrunch.com/2023/08/21/amazon-launches-fire-tv-channels-app-400-fast-channels/). Our goal is to provide customers with a delightful and personalized experience for consuming content across News, Sports, Cooking, Gaming, Entertainment, Lifestyle and more. You will work closely with engineering and product stakeholders to realize our ambitious product vision. You will get to work with Generative AI and other state of the art technologies to help build personalization and recommendation solutions from the ground up. You will be in the driver's seat to present customers with content they will love. Using Amazon’s large-scale computing resources, you will ask research questions about customer behavior, build state-of-the-art models to generate recommendations and run these models to enhance the customer experience. You will participate in the Amazon ML community and mentor Applied Scientists and Software Engineers with a strong interest in and knowledge of ML. Your work will directly benefit customers and you will measure the impact using scientific tools.
US, MA, Boston
The Artificial General Intelligence (AGI) team is looking for a passionate, talented, and inventive Senior Applied Scientist with a strong deep learning background, to build industry-leading technology with Large Language Models (LLMs) and multimodal systems. Key job responsibilities As a Senior Applied Scientist with the AGI team, you will work with talented peers to lead the development of novel algorithms and modeling techniques, to advance the state of the art with LLMs. Your work will directly impact our customers in the form of products and services that make use of speech and language technology. You will leverage Amazon’s heterogeneous data sources and large-scale computing resources to accelerate advances in generative artificial intelligence (GenAI). About the team The AGI team has a mission to push the envelope in LLMs and multimodal systems, in order to provide the best-possible experience for our customers.
IN, KA, Bengaluru
The Amazon Alexa AI team in India is seeking a talented, self-driven Applied Scientist to work on prototyping, optimizing, and deploying ML algorithms within the realm of Generative AI. Key responsibilities include: - Research, experiment and build Proof Of Concepts advancing the state of the art in AI & ML for GenAI. - Collaborate with cross-functional teams to architect and execute technically rigorous AI projects. - Thrive in dynamic environments, adapting quickly to evolving technical requirements and deadlines. - Engage in effective technical communication (written & spoken) with coordination across teams. - Conduct thorough documentation of algorithms, methodologies, and findings for transparency and reproducibility. - Publish research papers in internal and external venues of repute - Support on-call activities for critical issues Basic Qualifications: - Master’s or PhD in computer science, statistics or a related field - 2-7 years experience in deep learning, machine learning, and data science. - Proficiency in coding and software development, with a strong focus on machine learning frameworks. - Experience in Python, or another language; command line usage; familiarity with Linux and AWS ecosystems. - Understanding of relevant statistical measures such as confidence intervals, significance of error measurements, development and evaluation data sets, etc. - Excellent communication skills (written & spoken) and ability to collaborate effectively in a distributed, cross-functional team setting. - Papers published in AI/ML venues of repute Preferred Qualifications: - Track record of diving into data to discover hidden patterns and conducting error/deviation analysis - Ability to develop experimental and analytic plans for data modeling processes, use of strong baselines, ability to accurately determine cause and effect relations - The motivation to achieve results in a fast-paced environment. - Exceptional level of organization and strong attention to detail - Comfortable working in a fast paced, highly collaborative, dynamic work environment
IN, KA, Bengaluru
Amazon is investing heavily in building a world class advertising business and we are responsible for defining and delivering a collection of self-service performance advertising products that drive discovery and sales. Our products are strategically important to our Retail and Marketplace businesses driving long term growth. We deliver billions of ad impressions and millions of clicks daily and are breaking fresh ground to create world-class products. We are highly motivated, collaborative and fun-loving with an entrepreneurial spirit and bias for action. With a broad mandate to experiment and innovate, we are growing at an unprecedented rate with a seemingly endless range of new opportunities. The ATT team, based in Bangalore, is responsible for ensuring that ads are relevant and is of good quality, leading to higher conversion for the sellers and providing a great experience for the customers. We deal with one of the world’s largest product catalog, handle billions of requests a day with plans to grow it by order of magnitude and use automated systems to validate tens of millions of offers submitted by thousands of merchants in multiple countries and languages. In this role, you will build and develop ML models to address content understanding problems in Ads. These models will rely on a variety of visual and textual features requiring expertise in both domains. These models need to scale to multiple languages and countries. You will collaborate with engineers and other scientists to build, train and deploy these models. As part of these activities, you will develop production level code that enables moderation of millions of ads submitted each day.
US, WA, Seattle
The Search Supply & Experiences team, within Sponsored Products, is seeking an Applied Scientist to solve challenging problems in natural language understanding, personalization, and other areas using the latest techniques in machine learning. In our team, you will have the opportunity to create new ads experiences that elevate the shopping experience for our hundreds of millions customers worldwide. As an Applied Scientist, you will partner with other talented scientists and engineers to design, train, test, and deploy machine learning models. You will be responsible for translating business and engineering requirements into deliverables, and performing detailed experiment analysis to determine how shoppers and advertisers are responding to your changes. We are looking for candidates who thrive in an exciting, fast-paced environment and who have a strong personal interest in learning, researching, and creating new technologies with high customer impact. Key job responsibilities As an Applied Scientist on the Search Supply & Experiences team you will: - Perform hands-on analysis and modeling of enormous datasets to develop insights that increase traffic monetization and merchandise sales, without compromising the shopper experience. - Drive end-to-end machine learning projects that have a high degree of ambiguity, scale, and complexity. - Build machine learning models, perform proof-of-concept, experiment, optimize, and deploy your models into production; work closely with software engineers to assist in productionizing your ML models. - Design and run experiments, gather data, and perform statistical analysis. - Establish scalable, efficient, automated processes for large-scale data analysis, machine-learning model development, model validation and serving. - Stay up to date on the latest advances in machine learning. About the team We are a customer-obsessed team of engineers, technologists, product leaders, and scientists. We are focused on continuous exploration of contexts and creatives where advertising delivers value to shoppers and advertisers. We specifically work on new ads experiences globally with the goal of helping shoppers make the most informed purchase decision. We obsess about our customers and we are continuously innovating on their behalf to enrich their shopping experience on Amazon
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. About the team The International Seller Services (ISS) Economics team is a dynamic group at the forefront of shaping Amazon's global seller ecosystem. As part of ISS, we drive innovation and growth through sophisticated economic analysis and data-driven insights. Our mission is critical: we're transforming how Amazon empowers millions of international sellers to succeed in the digital marketplace. Our team stands at the intersection of innovative technology and practical business solutions. We're leading Amazon's transformation in seller services through work with Large Language Models (LLMs) and generative AI, while tackling fundamental questions about seller growth, marketplace dynamics, and operational efficiency. What sets us apart is our unique blend of rigorous economic methodology and practical business impact. We're not just analyzing data – we're building the frameworks and measurement systems that will define the future of Amazon's seller services. Whether we're optimizing the seller journey, evaluating new technologies, or designing innovative service models, our team transforms complex economic challenges into actionable insights that drive real-world results. Join us in shaping how millions of businesses worldwide succeed on Amazon's marketplace, while working on problems that combine economic theory, advanced analytics, and innovative technology.
US, WA, Seattle
Have you ever wondered how Amazon launches and maintains a consistent customer experience across hundreds of countries and languages it serves its customers? Are you passionate about data and mathematics, and hope to impact the experience of millions of customers? Are you obsessed with designing simple algorithmic solutions to very challenging problems? If so, we look forward to hearing from you! At Amazon, we strive to be Earth's most customer-centric company, where both internal and external customers can find and discover anything they want in their own language of preference. Our Translations Services (TS) team plays a pivotal role in expanding the reach of our marketplace worldwide and enables thousands of developers and other stakeholders (Product Managers, Program Managers, Linguists) in developing locale specific solutions. Amazon Translations Services (TS) is seeking an Applied Scientist to be based in our Seattle office. As a key member of the Science and Engineering team of TS, this person will be responsible for designing algorithmic solutions based on data and mathematics for translating billions of words annually across 130+ and expanding set of locales. The successful applicant will ensure that there is minimal human touch involved in any language translation and accurate translated text is available to our worldwide customers in a streamlined and optimized manner. With access to vast amounts of data, cutting-edge technology, and a diverse community of talented individuals, you will have the opportunity to make a meaningful impact on the way customers and stakeholders engage with Amazon and our platform worldwide. Together, we will drive innovation, solve complex problems, and shape the future of e-commerce. Key job responsibilities * Apply your expertise in LLM models to design, develop, and implement scalable machine learning solutions that address complex language translation-related challenges in the eCommerce space. * Collaborate with cross-functional teams, including software engineers, data scientists, and product managers, to define project requirements, establish success metrics, and deliver high-quality solutions. * Conduct thorough data analysis to gain insights, identify patterns, and drive actionable recommendations that enhance seller performance and customer experiences across various international marketplaces. * Continuously explore and evaluate state-of-the-art modeling techniques and methodologies to improve the accuracy and efficiency of language translation-related systems. * Communicate complex technical concepts effectively to both technical and non-technical stakeholders, providing clear explanations and guidance on proposed solutions and their potential impact. About the team We are a start-up mindset team. As the long-term technical strategy is still taking shape, there is a lot of opportunity for this fresh Science team to innovate by leveraging Gen AI technoligies to build scalable solutions from scratch. Our Vision: Language will not stand in the way of anyone on earth using Amazon products and services. Our Mission: We are the enablers and guardians of translation for Amazon's customers. We do this by offering hands-off-the-wheel service to all Amazon teams, optimizing translation quality and speed at the lowest cost possible.
US, CA, Santa Clara
Amazon Q Business is an AI assistant powered by generative technology. It provides capabilities such as answering queries, summarizing information, generating content, and executing tasks based on enterprise data. We are seeking a Language Data Scientist II to join our data team. Our mission is to engineer high-quality datasets that are essential to the success of Amazon Q Business. From human evaluations and Responsible AI safeguards to Retrieval-Augmented Generation and beyond, our work ensures that Generative AI is enterprise-ready, safe, and effective for users. As part of our diverse team—including language engineers, linguists, data scientists, data engineers, and program managers—you will collaborate closely with science, engineering, and product teams. We are driven by customer obsession and a commitment to excellence. In this role, you will leverage data-centric AI principles to assess the impact of data on model performance and the broader machine learning pipeline. You will apply Generative AI techniques to evaluate how well our data represents human language and conduct experiments to measure downstream interactions. Key job responsibilities * oversee end-to-end evaluation data pipeline and propose evaluation metrics and methods * incorporate your knowledge of linguistic fundamentals, NLU, NLP to the data pipeline * process and analyze diverse media formats including audio recordings, video, images and text * perform statistical analysis of the data * write intuitive data generation & annotation guidelines * write advanced and nuanced prompts to optimize LLM outputs * write python scripts for data wrangling * automate repetitive workflows and improve existing processes * perform background research and vet available public datasets on topics such as long text retrieval, text generation, summarization, question-answering, and reasoning * leverage and integrate AWS services to optimize data collection workflows * collaborate with scientists, engineers, and product managers in defining data quality metrics and guidelines. * lead dive deep sessions with data annotators About the team About AWS Diverse Experiences AWS values diverse experiences. Even if you do not meet all of the preferred qualifications and skills listed in the job description, we encourage candidates to apply. If your career is just starting, hasn’t followed a traditional path, or includes alternative experiences, don’t let it stop you from applying. Why AWS? Amazon Web Services (AWS) is the world’s most comprehensive and broadly adopted cloud platform. We pioneered cloud computing and never stopped innovating — that’s why customers from the most successful startups to Global 500 companies trust our robust suite of products and services to power their businesses. Inclusive Team Culture AWS values curiosity and connection. Our employee-led and company-sponsored affinity groups promote inclusion and empower our people to take pride in what makes us unique. Our inclusion events foster stronger, more collaborative teams. Our continual innovation is fueled by the bold ideas, fresh perspectives, and passionate voices our teams bring to everything we do. Mentorship & Career Growth We’re continuously raising our performance bar as we strive to become Earth’s Best Employer. That’s why you’ll find endless knowledge-sharing, mentorship and other career-advancing resources here to help you develop into a better-rounded professional. Work/Life Balance We value work-life harmony. Achieving success at work should never come at the expense of sacrifices at home, which is why we strive for flexibility as part of our working culture. When we feel supported in the workplace and at home, there’s nothing we can’t achieve.