An Observed Value Consistent Diffusion Model for Imputing Missing Values in Multivariate Time Series

Missing values, which are common in multivariate time series, is most important obstacle towards the utilization and interpretation of those data. Great efforts have been employed on how to accurately impute missing values in multivariate time series, and existing works either use deep learning networks to achieve deterministic imputations or aim at generating different plausible imputations by sampling multiple noises from a same distribution and then denoising them. However, these models either fall short of modeling the uncertainties of imputations due to their deterministic nature or perform poorly in terms of interpretability and imputation accuracy due to their ignorance of the correlations between the latent representations of both observed and missing values which are parts of samples from a same distribution. To this end, in this paper, we explicitly take the correlations between observed and missing values into account, and theoretically re-derive the Evidence Lower BOund (ELBO) of conditional diffusion model in the scenario of multivariate time series imputation. Based on the newly derived ELBO, we further propose a novel multivariate imputation diffusion model (MIDM) which is equipped with novel noise sampling, adding and denoising mechanisms for multivariate time series imputation, and the series of newly designed technologies jointly ensure the involving of the consistency between observed and missing values. Extensive experiments on both the tasks of multivariate time series imputation and forecasting witness the superiority of our proposed MIDM model on generating conditional estimations.