Causal View of Time Series Imputation: Some Identification Results on Missing Mechanism

TL;DR

This paper proposes a framework for the time series imputation problem by exploring Different Missing Mechanisms (DMM in short) and tailoring solutions accordingly and establishes identifiability results under the nonlinear independent component analysis framework to show that latent variables are identifiable.

Abstract

Time series imputation is one of the most challenge problems and has broad applications in various fields like health care and the Internet of Things. Existing methods mainly aim to model the temporally latent dependencies and the generation process from the observed time series data. In real-world scenarios, different types of missing mechanisms, like MAR (Missing At Random), and MNAR (Missing Not At Random) can occur in time series data. However, existing methods often overlook the difference among the aforementioned missing mechanisms and use a single model for time series imputation, which can easily lead to misleading results due to mechanism mismatching. In this paper, we propose a framework for time series imputation problem by exploring Different Missing Mechanisms (DMM in short) and tailoring solutions accordingly. Specifically, we first analyze the data generation processes with temporal latent states and missing cause variables for different mechanisms. Sequentially, we model these generation processes via variational inference and estimate prior distributions of latent variables via normalizing flow-based neural architecture. Furthermore, we establish identifiability results under the nonlinear independent component analysis framework to show that latent variables are identifiable. Experimental results show that our method surpasses existing time series imputation techniques across various datasets with different missing mechanisms, demonstrating its effectiveness in real-world applications.

Figures (4)

• Figure 1: Data generation processes of time series data under different missing mechanisms. ${\mathbf{z}}_t$ are temporal latent variables that describe the temporal dependencies. ${\mathbf{x}}_t^o$ are the observed variables, ${\mathbf{x}}_t^m$ are the missing data and ${\mathbf{c}}_t$ denotes the missing cause variables. (a) The data generation process under the missing at random mechanism, where missingness is related to the observed data but not the unobserved data. (b) The data generation process under the missing not at random mechanism, where the missingness is influenced by the observed data and missing data in the previous time step. (c) The data generation process under the missing completely at random mechanism, where missing data is led by random issues, and the latent missing variables can be considered as random noises.
• Figure 2: Illustration of the DMM framework. $X^o$ are the observed variables, $X^m$ are the missing data. The latent state variables ${\mathbf{z}}_{1:T}$ and the missing cause variables ${\mathbf{c}}_{1:T}$ are extracted from the encoder. The latent state and missing cause prior networks for DMM-MAR and DMM-MNAR are used to estimate the prior distributions.
• Figure 3: Ablation study on the ETTh1 datasets.
• Figure 4: Computational efficiency of ETTh2 dataset with a missing rate of 0.6 under unsupervised conditions in MNAR

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 1
• proof
• Theorem 2
• proof