The Prob-stack contains all 57 of the records in the LR04 stack plus an additional 123 published benthic δ 18O records. In Section 4, we discuss aspects regarding the stack construction method and describe possible applications of the Prob-stack. In addition, we present one more probabilistic stack, the Prob-LR04-stack, which is constructed using only the 57 δ 18O records used in the LR04 stack. Section 3 presents the Prob-stack and a comparison to the LR04 stack. In the following section, we describe the δ 18O data and the construction method used in the Prob-stack. Thus, we apply this model to obtain a probabilistic stack with the assumption that each δ 18O record is a sample emitted from the probabilistic stack. The estimated parameters of the profile HMM using the expectation–maximization (EM) algorithm successfully characterize the shared information when there is a sufficient amount of data. The profile HMM method has been widely applied to in the field of bioinformatics ( Eddy, 1998) for multiple sequence alignments ( Durbin et al., 1998) and a collection of protein family database ( Bateman et al., 2004) to explain the shared information of data sets. The preceding study ( Lin et al., 2014) proposed the HMM-Match algorithm, which is the pairwise alignment method for δ 18O records based on the HMM and this study extends this algorithm for construction of a probabilistic stack. The Prob-stack is constructed based on the profile hidden Markov model (HMM). To address this limitation, we present a probabilistic stack, called the Prob-stack, which describes changes in the global mean value of δ 18O, including its variability across benthic δ 18O records and uncertainties in the alignments. Furthermore, using such a deterministic stack as a stratigraphic reference provides no information about statistical significance. Because the algorithm finds alignments deterministically and returns only the best solution, the LR04 stack does not reflect uncertainties in the alignments. After aligning the records, the average of all δ 18O measurements within each time interval was used to represent each stack point. This automated algorithm finds the global optimal alignment that minimizes the measures of sequence dissimilarity by comparing stratigraphic features and penalizing unrealistic changes in accumulation rates. The LR04 stack was constructed through pairwise alignment of records using segments of high-resolution records and dynamic programming optimization ( Lisiecki and Lisiecki, 2002). The LR04 stack ( Lisiecki and Raymo, 2005) is one of the most widely used for age model developments and lead/lag relationship analysis ( Clark et al., 2006, 2009 Jouzel et al., 2007 Pollard and DeConto, 2009) because it describes the mean of 57 globally distributed records and continuously spans the Pliocene and Pleistocene (0–5.3 Myr). Due to the usefulness of stacks as measures of global climate change and stratigraphic alignment targets, a progression of stacks ( Imbrie et al., 1984 Pisias et al., 1984 Prell et al., 1986 Williams et al., 1988 Raymo et al., 1990 Bassinot et al., 1994 Shackleton, 1995 Karner et al., 2002 Huybers and Wunsch, 2004 Lisiecki and Raymo, 2005) has been created over the past 30 years as more data have become available. #Stack the states 2 series#A benthic δ 18O stack is a representative time series that describes the global nature of ice volume and deep-water temperature signals by combining benthic δ 18O records from different locations.
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