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Use of Prior Knowledge to Discover Causal Additive Models with Unobserved Variables and its Application to Time Series Data

Takashi Nicholas Maeda, Shohei Shimizu

TL;DR

This work addresses causal discovery in the presence of latent confounders for i.i.d. and time-series data by proposing CAM-UV, a generalized additive model (GAM)-based causal framework that accommodates unobserved variables, and its time-series extension TS-CAM-UV. It introduces CAM-UV-PK to inject prior knowledge (e.g., prohibiting certain causal directions) and TS-CAM-UV to exploit time-ordering constraints, leveraging a CAM-UV-PK backbone. The identifiability analysis clarifies when direct causal relations can be recovered under nonlinearity and latent confounding, and when only the presence of unobserved paths can be detected. Empirical results on simulated data show that prior knowledge improves precision and F-measure, while TS-CAM-UV demonstrates favorable performance relative to LPCMCI and VarLiNGAM on both simulated and real foreign-exchange time-series data, highlighting its ability to handle nonlinearities and latent confounding in time-series contexts. The proposed methods enable robust causal inference in complex systems where unobserved variables and nonlinear interactions are present, with practical applicability to time-series analysis.

Abstract

This paper proposes two methods for causal additive models with unobserved variables (CAM-UV). CAM-UV assumes that the causal functions take the form of generalized additive models and that latent confounders are present. First, we propose a method that leverages prior knowledge for efficient causal discovery. Then, we propose an extension of this method for inferring causality in time series data. The original CAM-UV algorithm differs from other existing causal function models in that it does not seek the causal order between observed variables, but rather aims to identify the causes for each observed variable. Therefore, the first proposed method in this paper utilizes prior knowledge, such as understanding that certain variables cannot be causes of specific others. Moreover, by incorporating the prior knowledge that causes precedes their effects in time, we extend the first algorithm to the second method for causal discovery in time series data. We validate the first proposed method by using simulated data to demonstrate that the accuracy of causal discovery increases as more prior knowledge is accumulated. Additionally, we test the second proposed method by comparing it with existing time series causal discovery methods, using both simulated data and real-world data.

Use of Prior Knowledge to Discover Causal Additive Models with Unobserved Variables and its Application to Time Series Data

TL;DR

This work addresses causal discovery in the presence of latent confounders for i.i.d. and time-series data by proposing CAM-UV, a generalized additive model (GAM)-based causal framework that accommodates unobserved variables, and its time-series extension TS-CAM-UV. It introduces CAM-UV-PK to inject prior knowledge (e.g., prohibiting certain causal directions) and TS-CAM-UV to exploit time-ordering constraints, leveraging a CAM-UV-PK backbone. The identifiability analysis clarifies when direct causal relations can be recovered under nonlinearity and latent confounding, and when only the presence of unobserved paths can be detected. Empirical results on simulated data show that prior knowledge improves precision and F-measure, while TS-CAM-UV demonstrates favorable performance relative to LPCMCI and VarLiNGAM on both simulated and real foreign-exchange time-series data, highlighting its ability to handle nonlinearities and latent confounding in time-series contexts. The proposed methods enable robust causal inference in complex systems where unobserved variables and nonlinear interactions are present, with practical applicability to time-series analysis.

Abstract

This paper proposes two methods for causal additive models with unobserved variables (CAM-UV). CAM-UV assumes that the causal functions take the form of generalized additive models and that latent confounders are present. First, we propose a method that leverages prior knowledge for efficient causal discovery. Then, we propose an extension of this method for inferring causality in time series data. The original CAM-UV algorithm differs from other existing causal function models in that it does not seek the causal order between observed variables, but rather aims to identify the causes for each observed variable. Therefore, the first proposed method in this paper utilizes prior knowledge, such as understanding that certain variables cannot be causes of specific others. Moreover, by incorporating the prior knowledge that causes precedes their effects in time, we extend the first algorithm to the second method for causal discovery in time series data. We validate the first proposed method by using simulated data to demonstrate that the accuracy of causal discovery increases as more prior knowledge is accumulated. Additionally, we test the second proposed method by comparing it with existing time series causal discovery methods, using both simulated data and real-world data.
Paper Structure (17 sections, 6 theorems, 10 equations, 11 figures, 1 algorithm)

This paper contains 17 sections, 6 theorems, 10 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related studies
  3. Models
  4. CAM-UV: Causal additive models with unobserved variables
  5. TS-CAM-UV: Time series causal additive models with unobserved variables
  6. Identifiability
  7. CAM-UV
  8. TS-CAM-UV
  9. Methods
  10. CAM-UV-PK: Causal additive models with unobserved variables using prior knowledge
  11. TS-CAM-UV: Time series causal additive models with unobserved variables
  12. Experiments
  13. CAM-UV-PK: Causal additive models with unobserved variables using prior knowledge
  14. TS-CAM-UV: Time series causal additive models with unobserved variables
  15. Simulated data
  16. ...and 2 more sections

Key Result

Lemma 1

Assume the data generation process of the variables is CAM-UV as defined in Section section:modelCAM-UV. If and only if Equation eq:lem1 is satisfied, there is a UCP or UBP between $X_i$ and $X_j$ where $G_1$ and $G_2$ denote regression functions satisfying Assumption assumption:ass2. Equation eq:lem1 indicates that the residual of $X_i$ regressed on any subset of $X\setminus\{X_i\}$ and the resi

Figures (11)

  • Figure 1: Definitions of an unobserved causal path (UCP) and an unobserved backdoor path (UBP).
  • Figure 2: (a): True causal graph. (b): Causal graph generated by the CAM-UV algorithm.
  • Figure 3: The performance of the CAM-UV-PK and CAM-UV algorithms: The CAM-UV algorithm is equivalent to the CAM-UV-PK algorithm with no prior knowledge.
  • Figure 4: The performance of the TS-CAM-UV compared to LPCMCI and VarLiNGAM.
  • Figure 5: The causal graph only with directed edges generated from TS-CAM-UV.
  • ...and 6 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 1 more