Editing IPCC:AR6/WGI/Chapter-3 (section)

==== 3.2.1 Methods Based on Regression ====

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Regression-based methods, also known as fingerprinting methods, have been widely used for detection of climate change and attribution of the change to different external drivers. Initially, these methods were applied to detect changes in global surface temperature, and were then extended to other climate variables at different time and spatial scales (e.g., [[#Hegerl--1996|Hegerl et al., 1996]] ; [[#Hasselmann--1997|Hasselmann, 1997]] ; [[#Allen--1999|Allen and Tett, 1999]] ; [[#Gillett--2003b|Gillett et al., 2003b]] ; [[#Zhang--2007|Zhang et al., 2007]] ; [[#Min--2008a|Min et al., 2008a]] , 2011). These approaches are based on multivariate linear regression and assume that the observed change consists of a linear combination of externally forced signals plus internal variability, which generally holds for large-scale variables ( [[#Hegerl--2011|Hegerl and Zwiers, 2011]] ). The regressors are the expected space–time response patterns to different climate forcings (fingerprints), and the residuals represent internal variability. Fingerprints are usually estimated from climate model simulations following spatial and temporal averaging. A regression coefficient which is significantly greater than zero implies that a detectable change is identified in the observations. When the confidence interval of the regression coefficient includes unity and is inconsistent with zero, the magnitude of the model simulated fingerprints is assessed to be consistent with the observations, implying that the observed changes can be attributed in part to a particular forcing. Variants of linear regression have been used to address uncertainty in the fingerprints due to internal variability ( [[#Allen--2003|Allen and Stott, 2003]] ) as well as structural model uncertainty ( [[#Huntingford--2006|Huntingford et al., 2006]] ).

In order to improve the signal-to-noise ratio, observations and model-simulated responses are usually normalized by an estimate of internal variability derived from climate model simulations. This procedure requires an estimate of the inverse covariance matrix of the internal variability, and some approaches have been proposed for more reliable estimation of this ( [[#Ribes--2009|Ribes et al., 2009]] ). A signal can be spuriously detected due to too-small noise, and hence simulated internal variability needs to be evaluated with care. Model-simulated variability is typically checked through comparing modelled variance from unforced simulations with the observed residual variance using a standard residual consistency test ( [[#Allen--1999|Allen and Tett, 1999]] ), or an improved one ( [[#Ribes--2013|Ribes and Terray, 2013]] ). [[#Imbers--2014|Imbers et al. (2014)]] tested the sensitivity of detection and attribution results to different representations of internal variability associated with short-memory and long-memory processes. Their results supported the robustness of previous detection and attribution statements for the global mean temperature change but they also recommended the use of a wider variety of robustness tests.

Some recent studies focused on the improved estimation of the scaling factor (regression coefficient) and its confidence interval. [[#Hannart--2014|Hannart et al. (2014)]] described an inference procedure for scaling factors which avoids making the assumption that model error and internal variability have the same covariance structure. An integrated approach to optimal fingerprinting was further suggested in which all uncertainty sources (i.e., observational error, model error, and internal variability) are treated in one statistical model without a preliminary dimension reduction step ( [[#Hannart--2016|Hannart, 2016]] ). [[#Katzfuss--2017|Katzfuss et al. (2017)]] introduced a similar integrated approach based on a Bayesian model averaging. On the other hand, [[#DelSole--2019|DelSole et al. (2019)]] suggested a bootstrap method to better estimate the confidence intervals of scaling factors even in a weak-signal regime. It is notable that some studies do not optimize fingerprints, as uncertainty in the covariance introduces a further layer of complexity, but results in only a limited improvement in detection ( [[#Polson--2017|Polson and Hegerl, 2017]] ).

Another fingerprinting approach uses pattern similarity between observations and fingerprints, in which the leading empirical orthogonal function obtained from the time-evolving multi-model forced simulation is usually defined as a fingerprint (e.g., [[#Santer--2013|Santer et al., 2013]] ; [[#Marvel--2019|Marvel et al., 2019]] ; [[#Bonfils--2020|Bonfils et al., 2020]] ). Observations and model simulations are then projected onto the fingerprint to measure the degree of spatial pattern similarity with the expected physical response to a given forcing. This projection provides the signal time series, which is in turn tested against internal variability, as estimated from long control simulations. As a way to extend this pattern-based approach to a high-dimensional detection variable at daily time scales, Sippel et al. (2019, 2020) proposed using the relationship pattern with a global climate change metric as a fingerprint. To solve the high-dimensional regression problem which makes regression coefficients not well constrained, they incorporated a statistical learning technique based on a regularized linear regression, which optimizes a global warming signal by giving lower weight to regions with large internal variability.

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