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# 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
All article types: you may provide up to 8 keywords; at least 5 are mandatory.

# Introduction

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# Material & Methods

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 Empirical Orthogonal Functions

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 $$($$E(\mathbf{X}/\mathbf{y})$$)$$, forward regression $$($$E(\mathbf{y}/\mathbf{X})$$)$$ is used to obtain fitted EOFs. This is due to the fact that dimensionality of response $$($$\mathbf{y}$$)$$ instead of predictors $$($$\mathbf{X}$$)$$ is to be reduced to map leading spatial patterns accounting for the relationship of response and predictors. Let $$($$\mathbf{Y}$$)$$ be the ($$n \times p$$) matrix of response variables and $$($$\mathbf{X}$$)$$ be the ($$n \times q$$) vector of independent variables. Then, multivariate regression model(Johnson 2007) of $$$$\mathbf{Y}$$$$ on $$$$\mathbf{X}$$$$ is,
$\mathbf{Y} = \mathbf{X B} + \mathbf{E}$
Error term $$$$\mathbf{E}$$$$ is ($$n \times p$$) zero mean noise matrix where each row (ith sample) is assumed to follow multivariate normal distribution $$($$N_{q}(\mathbf{0},\mathbf{\Sigma})$$)$$ and $$$$\mathbf{B}$$$$ is ($$q \times p$$) matrix of regression co - efficients. The ordinary least square estimator of $$$$\mathbf{B}$$$$ is, $\mathbf{B} = \mathbf{X^{t}X}^{-1} \mathbf{X^{t}Y}$
This is equivalent to performing p-univariate regressions. Principal component analysis is conducted on covariance matrix $$$$\mathbf{\Sigma_{f}}$$$$ of fitted values $$($$\mathbf{\hat{Y}}$$)$$. This is done by calculating eigenvalue-eigenvector pairs $$$$(\lambda_{i}, \mathbf{e}_{i})$$$$ associated with $$$$\mathbf{\Sigma_{f}}$$$$. Then, $$$$\mathbf{e}_{i}$$$$ is called ith fitted EOF and $$$$\mathbf{e}_{i}^{t} \mathbf{Y}$$$$ is the ith fitted PC $$(i= 1,2,….,p)$$.

Maximum size of the Manuscript\label{Tab:01}
Abstract max. legth (incl. spaces) Figures or tables Manuscript max. length Final PDF length
Clinical Case Study
Clinical Trial
Hypothesis and Theory
Methods 2000 characters 15 12000 words 12 pages
Original Research
Review
Technology Report
Focused Review 2000 characters 5 5000 words 5 pages
CPC 1250 characters 6 2500 words 4 pages
Perspective 1250 characters 2 3000 words 3 pages
Mini Review
Classification 1250 characters 10 2000 words 12 pages
Editorial none none 1000 words 1 page
Book review
Frontiers Commentary none 1 1000 words 1 page
General Commentary
Field Grand Challenge
Opinion none 1 2000 words 2 pages
Specialty Grand Challenge

Please note that very large tables (covering several pages) cannot be included in the final PDF for reasons of space. These tables will be published as supplementary material on the online article abstract page at the time of acceptance. The author will notified during the typesetting of the final article if this is the case. A link in the final PDF will direct to the online material.

## Data Sources

For Original Research Articles, Clinical Trial Articles, and Technology Reports the section headings should be those appropriate for your field and the research itself. It is recommended to organize your manuscript in the following sections or their equivalents for your field:

• Introduction: Succinct, with no subheadings.

• Materials and Methods: This section may be divided by subheadings. This section should contain sufficient detail so that when read in conjunction with cited references, all procedures can be repeated.

• Results: This section may be divided by subheadings. Footnotes should not be used and have to be transferred into the main text.

• Discussion: This section may be divided by subheadings. Discussions should cover the key findings of the study: discuss any prior art related to the subject so to place the novelty of the discovery in the appropriate context; discuss the potential short-comings and limitations on their interpretations; discuss their integration into the current understanding of the problem and how this advances the current views; speculate on the future direction of the research and freely postulate theories that could be tested in the future.