Continuous-time state-space methods for delta-O-18 and delta-C-13

Time series analysis of d18O and d13C from benthic foraminifera for paleoclimatology poses significant challenges. The data span tens of millions of years, with sparse early records, dense later ones, uneven time stamps, and occasional multiples. These time series are largely non-stationary, exhibiting temporary, varying trends. We propose a continuous-time state space framework that handles these irregularities effectively. Univariate signal-plus-noise models are specified for d18O and d13C, with parameters estimated via maximum likelihood using Kalman filter recursions for signal extraction and likelihood evaluation. The framework interprets state space models as time-domain Butterworth filters. Measurement-error variances are differentiated by deep-sea drill site, including site-specific level offsets, and the record is partitioned into sub-periods reflecting the distinct climate states that drive the transition variance. Two extensions of the univariate model are explored: (i) modifying the signal specification for the Kalman filter to approximate a Butterworth filter of any order, and (ii) specifying a bivariate signal-plus-noise model for joint analysis. Results reveal substantial signal changes during the ``icehouse'' period (3.3 to 0.0006 Ma); the correlation between d18O and d13C signals is generally positive but turns negative during this period.