Evaluation of proxy-based millennial reconstruction methods

Abstract

A range of existing statistical approaches for reconstructing historical temperature variations from proxy data are compared using both climate model data and real-world paleoclimate proxy data. We also propose a new method for reconstruction that is based on a state-space time series model and Kalman filter algorithm. The state-space modelling approach and the recently developed RegEM method generally perform better than their competitors when reconstructing interannual variations in Northern Hemispheric mean surface air temperature. On the other hand, a variety of methods are seen to perform well when reconstructing surface air temperature variability on decadal time scales. An advantage of the new method is that it can incorporate additional, non-temperature, information into the reconstruction, such as the estimated response to external forcing, thereby permitting a simultaneous reconstruction and detection analysis as well as future projection. An application of these extensions is also demonstrated in the paper.

Appendix

1.1 Expectation maximization algorithm

Here we present the steps required to estimate the unknown parameters that specify the state-space model (Eq. 6). We use ${\Theta}= \{\zeta,R,\phi,\Upsilon, Q, {\mu,}\Sigma\}$ to represent the vector of unknown parameters. The derivations presented here are expanded from Shumway and Stoffer (1982, 2000, pp. 321–325). Detail derivations of the following procedure are given in Lee (2008). We first write down the likelihood function for {P _t ; t = 1, 2, ..., n} as in Shumway and Stoffer (2000). Ignoring a constant, the likelihood function L _P(Θ) can be expressed as:

$$ -2 \ln L_{\rm P}(\Theta)= \sum_{t=1}^{n}\ln(\Omega_t)+\sum_{t=1}^{n}(\varepsilon_t^2/\Omega_t) $$

where Ω _t  = ζ² S ^t-1 _t  + R > 0 and ɛ _t  = P _t  − ζT ^t-1 _t for t = 1, 2, ..., n. The calibration period hemispheric mean temperatures {T _t ; t = n + 1, ..., n + m } are not involved in the above likelihood function. To better utilize the available information, we have modified the likelihood function to include {T _t ; t = n + 1, ..., n + m}. Ignoring a constant, one can express the likelihood function for the observation data set {P _t ,T _s ; t = 1, 2, ..., n; s = n + 1, ..., n + m}, namely L _P,T(Θ), as

$$ \begin{aligned} -2 \ln L_{\rm P,T}(\Theta)&= \sum_{t=1}^{n+m}\ln(\Omega_t) +\sum_{t=1}^{n+m}(\varepsilon_t^2/\Omega_t)+(m-1)\ln(Q) +\ln(\phi^2 S_n^n+Q)\\&\quad + ({{\mathbf{T}}}_{n+1}-\phi{{\mathbf{T}}}_n^n-\Upsilon {{\mathbf{F}}}_{n+1})^2/(\phi^2 S_n^n+Q)+ \sum_{t=n+2}^{n+m}({{\mathbf{T}}}_t- \phi{{\mathbf{T}}}_{t-1}- \Upsilon {{\mathbf{F}}}_t)^2/Q \end{aligned} $$

where Ω _t and ɛ _t is defined as before for t = 1, 2, ..., n. For t = n + 1, ..., n + m, Ω _t  = R and ɛ _t  =  P _t  − ζT _t . The goal here is to find the Θ values that maximize the likelihood function L _P,T(Θ). Using the EM algorithm, such Θ values can be found through an iterative procedure. The estimation procedure is summarized as follows.

• 1. Select the starting values for the parameters $\Theta^{(0)}= \{\zeta^{(0)},R^{(0)},\phi^{(0)}, \Upsilon^{(0)},Q^{(0)}, \mu^{(0)}\}$ while fixing Σ, one of the initial condition for the Kalman recursion. (In our analysis, we set Σ = 0.05. Results were almost identical when different Σ values were used.) On iteration j, (j  = 1, 2, ...), do steps 2–4.

• 2. Let N = n + m. Using Θ^(j-1), compute the Kalman filter and smoother estimates using Eqs. (7) and (8) for t = 1, 2, ..., N and the likelihood function lnL _P,T(Θ^(j-1)). Also calculate, for t = N − 1, N − 2, ..., 0

  $$ S_t^{N} = S_t^t + J_t^2(S_{t+1}^{N}-S_{t+1}^t) $$

  and for t = n, n − 1, ..., 0,

  $$ \begin{aligned} C_t &= J_{t-1}S_{t}^{N}\\ {\widetilde{{\mathbf{T}}}}_t^{N}&={{\mathbf{T}}}_t^{N}+(J_tJ_{t+1}\ldots J_n)({{\mathbf{T}}}_{n+1}-{{\mathbf{T}}}_{n+1}^N)\\ {\widetilde{S}}_t^{N}&=S_t^{N}- (J_t J_{t+1}\ldots J_n)^2S_{n+1}^N \end{aligned} $$

  and the following quantities:

  $$ \begin{aligned} Z_{11}&=\sum_{t= 1}^{n}[({\widetilde{{\mathbf{T}}}}_t^{N})^2+{\widetilde{S}}_t^{N}]+\sum_{t= n+1}^{N}({{\mathbf{T}}}_t)^2\\ Z_{00}&=\sum_{t= 0}^{n}[({\widetilde{{\mathbf{T}}}}_t^{N})^2+{\widetilde{S}}_t^{N}]+\sum_{t= n+1}^{N- 1}({{\mathbf{T}}}_t)^2\\ Z_{10}&=\sum_{t= 1}^{n}({\widetilde{{\mathbf{T}}}}_t^{N} {\widetilde{{\mathbf{T}}}}_{t-1}^{N}+C_t) + {{\mathbf{T}}}_{n+1} {\widetilde{{\mathbf{T}}}}_{n}^{N}+\sum_{t=n+2}^{N}{{\mathbf{T}}}_t {{\mathbf{T}}}_{t-1}\\ F_{11}&=\sum_{t=1}^{n} {{\mathbf{F}}}_t{\widetilde{{\mathbf{T}}}}_t^{N}+\sum_{t= n+1}^{N}{{\mathbf{F}}}_t {{\mathbf{T}}}_t\\ F_{10}&=\sum_{t=1}^{n+1} {{\mathbf{F}}}_{t}{\widetilde{{\mathbf{T}}}}_{t-1}^{N}+\sum_{t=n+2}^{N}{{\mathbf{F}}}_{t} {{\mathbf{T}}}_{t-1}\\ F_{00}&=\sum_{t= 1}^{N}{{\mathbf{F}}}_t{{\mathbf{F}}}_t^{\rm T}. \end{aligned} $$

• 3. Obtain Θ^(j) using the following:

  $$ \begin{aligned} \zeta^{(j)}&= \left[\sum_{t= 1}^n[({\widetilde{{\mathbf{T}}}}_t^N)^2+{\widetilde{S}}_t^N]+\sum_{t= n+1}^N{{\mathbf{T}}}_t^2\right]^{-1} \left[\sum_{t=1}^n{\widetilde{{\mathbf{T}}}}_{t|N}{{\mathbf{P}}}_t+\sum_{t= n+1}^N {{\mathbf{T}}}_t{{\mathbf{P}}}_t\right]\\ R^{(j)}&=N^{-1}\sum_{t= 1}^n\left[{\widetilde{S}}_t^N(\zeta^{(j)})^2+({{\mathbf{P}}}_t-\zeta^{(j)} {\widetilde{{\mathbf{T}}}}_t^N)^2\right] +N^{-1}\sum_{t=n+1}^N\left({{\mathbf{P}}}_t- \zeta^{(j)}{{\mathbf{T}}}_t\right)^2\\ \Upsilon^{(j)}&=\left(F_{11}^{\rm T}-F_{10}^{\rm T}Z_{10}/Z_{00}\right) \left(F_{00}-F_{10}F_{10}^{\rm T}/Z_{00}\right)^{-1}\\ \phi^{(j)}&=\left(Z_{10}-\Upsilon^{(j)}F_{10}\right)Z_{00}^{-1}\\ Q^{(j)}&=N^{-1}\left(Z_{11}-\phi^{(j)}Z_{10}- \Upsilon^{(j)}F_{11}\right)\\ \mu_0^{(j)}&={\widetilde{{\mathbf{T}}}}_0^N. \end{aligned} $$

• 4. Repeat steps 2 and 3 until convergence. For the analysis presented in this paper, the algorithm is stopped when lnL _P,T(Θ^(j)) − ln L _P,T(Θ^(j-1)) < 0.0005.

To provide a final estimate of T _t , the Kalman filter and smoother estimates are recalculated using Eqs. (7) and (8) with the estimate parameters ${\hat{\Theta}},$ for t = 1, 2, ..., n + m. At the time of convergence, one can also calculate the standard errors for ${\hat{\Theta}}.$ For parameters estimated using ln L _P(Θ), the asymptotic variance of ${\hat{\Theta}}$ is defined as (Caines 1988, Chap. 7; Jensen and Petersen 1999):

$$ \hbox{Var}({\hat{\Theta}})=-\left[\left.\frac{\partial^2} {\partial\Theta}\ln L_{\rm P}(\Theta)\right|_{\hat{\Theta}}\right]^{-1} $$

where ∂²/∂Θ denotes the second derivatives with respect to Θ. In our application, we have used L _P,T(Θ) in the estimation procedure and hence a reasonable estimate of the asymptotic variance can be obtained by replacing L _P(Θ) with L _P,T(Θ) in the above formula. More discussions of this can be found in Lee (2008). An analytical form of the asymptotic variance is generally hard to find and hence we calculated it numerically. The asymptotic variance can be used to provide confidence bounds for the estimated parameters. In particular, the 95% confidence bound for Θ is defined as ${\hat{\Theta}} \pm 1.96 \sqrt{\hbox{Var}({\hat{\Theta}})}.$ In our application, we are interested in the confidence bounds for the parameters δ _GS , δ _VOL and δ _SOL . These bounds can provide a detection assessment of the importance of the response to the GS, VOL and SOL forcing on the hemispheric mean temperature.