Kalman filtering for Mauna Loa CO₂ time-series

Abstract

The Kalman recursive method plays a significant role in estimating unknown model parameters and hidden states in state-space dynamic linear models, as it effectively estimates all underlying states in dynamic and noisy systems. In this study, the Mauna Loa atmospheric CO₂ time-series data (1958–2024) is modeled and decomposed using a dynamic linear model with a hierarchical state-space approach into the following main components: a changing long-term trend, seasonal variations, irregular fluctuations (noise), and an autoregressive term. The model states and parameters are estimated using Bayesian Kalman recursive algorithm and Adaptive Markov chain Monte Carlo method, due to its ability in handling noisy time-series data. The model residuals are evaluated with model diagnostic tests to assess the uncorrelated noise component.

The results show a slowly changing upward trend with patterns of seasonal variations and moderate noise residuals. This indicates that the dynamic linear model accounts for the underlying uncertainties in the atmospheric CO₂ time-series. The model diagnostic test shows a good model performance, the model residuals exhibit no significant autocorrelation and are normally distributed, with high R² value and low RMSE and MAE respectively. This study indicates that the atmospheric CO₂ emission level has consistently increased since 1958 with an upward trend. This study validates that dynamic linear models are well-suited for modeling complex atmospheric and environmental timeseries data with missing observations, noise, or dominant cyclic variations.