Fitted EOFs

Abstract

This manuscript is aimed at discussing our thoughts on the use of fitted EOFs for climate studies. Fitted EOF analysis is an extension of traditional EOFs that attempts to extract EOFs and associated PCs encapsulating predictors and response relationship making use of multivariate regression. Fitted EOFs of ENSO and volcanic aerosols are identified by estimating the impact of these factors on grid of surface temperature anomalies in the Tropics. We mapped influence of ENSO and volcanoes on temperature and removed these impacts to provide adjusted reconstructed grid of surface temperature (1856 - 2011). Spatial gaps are filled with ordinary kriging as gappy data are bottleneck for EOF analysis. ENSO accounts for more variability in surface temperature than volcanoes. Adjusted annual average temperature time series indicates warming as does the unadjusted version. However, it plateaus prominently after 2000.

(As it stands, I’ve just taken a LaTeXtemplate for the Journal Frontiers here, but wouldn’t anticipate that we would ever submit it there. I’ve also added in some section outlines. As a primary goal, the abstract should render the general significance and conceptual advance of the work clearly accessible to a broad readership. References should not be cited in the abstract. Refer to http://www.frontiersin.org/ or **Table \ref{Tab:01}** for abstract requirement and length according to article type.)

**Keywords:** EOFs, ENSO, volcanic aerosols, warming, Tropics, surface temperature

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Multivariate regression is employed to characterize the influence of ENSO and volcanoes on temperature. EOF analysis of fitted values of multivariate regression generates fitted EOFs and Principal Components (PCs). Spatial sparsity is dealt by Ordinary Kriging (OK) and 100 gap-filled ensemble members are constructed to account for uncertainty.

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Fitted EOF analysis explores covariance structure of fitted values. Fitted values are obtained by multivariate regression of response matrix (temperature) on independent variables (ENSO, volcanic aerosols). EOF analysis is carried out on fitted covariance matrix to identify fitted EOFs and fitted PCs. This approach differs from (Cook 2007) in a way that instead of inverse regression \((\begin{equation}E(\mathbf{X}/\mathbf{y})\end{equation})\), forward regression \((\begin{equation}E(\mathbf{y}/\mathbf{X})\end{equation})\) is used to obtain fitted EOFs. This is due to the fact that dimensionality of response \((\begin{equation}\mathbf{y}\end{equation})\) instead of predictors \((\begin{equation}\mathbf{X}\end{equation})\) is to be reduced to map leading spatial patterns accounting for the relationship of response and predictors. Let \((\begin{equation}\mathbf{Y}\end{equation})\) be the (\(n \times p\)) matrix of response variables and \((\begin{equation}\mathbf{X}\end{equation})\) be the (\(n \times q\)) vector of independent variables. Then, multivariate regression model(Johnson 2007) of \(\begin{equation}\mathbf{Y}\end{equation}\) on \(\begin{equation}\mathbf{X}\end{equation}\) is,

\[\mathbf{Y} = \mathbf{X B} + \mathbf{E}\]

Error term \(\begin{equation}\mathbf{E}\end{equation}\) is (\(n \times p\)) zero mean noise matrix where each row (ith sample) is assumed to follow multivariate normal distribution \((\begin{equation}N_{q}(\mathbf{0},\mathbf{\Sigma})\end{equation})\) and \(\begin{equation}\mathbf{B}\end{equation}\) is (\(q \times p\)) matrix of regression co - efficients. The ordinary least square estimator of \(\begin{equation}\mathbf{B}\end{equation}\) is, \[\mathbf{B} = \mathbf{X^{t}X}^{-1} \mathbf{X^{t}Y}\]

This is equivalent to performing p-univariate regressions. Principal component analysis is conducted on covariance matrix \(\begin{equation}\mathbf{\Sigma_{f}}\end{equation}\) of fitted values \((\begin{equation}\mathbf{\hat{Y}}\end{equation})\). This is done by calculating eigenvalue-eigenvector pairs \(\begin{equation}(\lambda_{i}, \mathbf{e}_{i})\end{equation}\) associated with \(\begin{equation}\mathbf{\Sigma_{f}}\end{equation}\). Then, \(\begin{equation}\mathbf{e}_{i}\end{equation}\) is called ith fitted EOF and \(\begin{equation}\mathbf{e}_{i}^{t} \mathbf{Y} \end{equation}\) is the ith fitted PC \((i= 1,2,….,p)\).

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