  Time series analysis of delta-O-18 and delta-C-13 measurements from benthic
foraminifera for purposes of paleoclimatology is challenging. The time series
reach back tens of millions of years, they are relatively sparse in the early
record and relatively dense in the later, the time stamps of the observations
are not evenly spaced, and there are instances of multiple different
observations at the same time stamp. The time series appear non-stationary over
most of the historical record with clearly visible temporary trends of varying
directions. In this paper, we propose a continuous-time state-space framework
to analyze the time series. State space models are uniquely suited for this
purpose, since they can accommodate all the challenging features mentioned
above. We specify univariate models and joint bivariate models for the two time
series of delta-O-18 and delta-C-13. The models are estimated using maximum
likelihood by way of the Kalman filter recursions. The suite of models we
consider has an interpretation as an application of the Butterworth filter. We
propose model specifications that take the origin of the data from different
studies into account and that allow for a partition of the total period into
sub-periods reflecting different climate states. The models can be used, for
example, to impute evenly time-stamped values by way of Kalman filtering. They
can also be used, in future work, to analyze the relation to proxies for CO2
concentrations.
