An important goal
of climate research is to determine the causal contribution of human activity
to observed changes in the climate system. Methodologically speaking, most
climatic causal studies to date have been formulating attribution as a linear
regression inference problem. Under this formulation, the inference is often
obtained by using the generalized least squares (GLS) estimator after
projecting the data on the

An important goal of climate research is to determine the causes of past global
warming in general and the responsibility of human activity in particular

Denoting

The results of the inference on the vector of regression coefficients

Under the assumption that

The GLS estimator of Eq. (

In most applications, the covariance

Under this assumption, the inference on

The solution recalled above in Eq. (

The first objective of this article is to highlight in detail these two
limitations: both will be formalized and discussed immediately below in
Sect. 2. Its second objective is to circumvent these limitations by
building a new estimator of

Section 2 highlights the limitations of the conventional estimator

It is useful to start this discussion by returning for a moment to the case
where

The Sherman–Morrison–Woodbury formula is a well known formula in linear
algebra, which shows that the inverse of a rank-one correction of some
matrix, can be conveniently computed by doing a rank one correction to the
inverse of the original matrix. By applying the Sherman–Morrison–Woodbury
formula to Eq. (

Unsurprisingly, an immediate implication of Eq. (

A similar conclusion can be reached by using a different approach. Consider this
time the transformation

After some algebra, the above expression of

In light of these considerations, the two main limitations of the
conventional estimator

In order to illustrate the second issue more concretely, we use surface
temperature data from

Let us now consider the quantities

As a direct consequence, for

With these preliminary considerations in hand, we return to the situation of
direct interest here: inferring

Like the conventional estimator

Let us denote

Note that infinitely many matrices

It is interesting to note that the orthogonal projection subspace

However, the choice of applying the OLS estimator to the projected
data as implied by Eq. (

The complete likelihood of Eq. (

Our proposed OLS estimator

It is important to underline that the latter approach to uncertainty
quantification relies on the assumption that the projected noise

The signals

It is straightforward to generalize the approach exposed in Sect. 2.3 to
this error-in-variables regression model. Likewise, the data

The so-called total least squares (TLS) estimator results from the
maximization of the above concentrated likelihood in

Summarizing, the proposed solution here to deal with the case of error in
variables thus consists of using Eq. (

The motivation of this subsection is merely to provide a more in-depth mathematical grounding to the proposed estimator, beyond the general considerations of Sect. 2.1. This subsection can safely be skipped without affecting the understanding of the remainder of this article, in so far as it does not provide any additional results or properties regarding the estimator itself.

The idea here is that, in addition to the general considerations of
Sect. 2.1, the proposed estimator

In the present context, we assumed that

The proof of Eq. (

We illustrate the method by applying it to surface temperature data and
climate model simulations over the 1901–2000 period and over the entire
surface of the Earth. For this purpose, we use the data described and studied
in

Performance results on simulated data: MSE of several
estimators of

This section evaluates the performance of the estimators described above, by
applying it to simulated values of

The use of simulated rather than real data aims at verifying that our
proposed estimator performs correctly, and at comparing its performance with
the conventional procedure, a goal which requires the actual values of

Figure 2 shows scatter plots of one realization of the data

Illustration of temperature data with

The conventional estimator

When comparing the performance of

The method is finally illustrated by applying it to actual observations of
surface temperature

We have introduced a new estimator of the vector

When applied on a simulation test bed that is realistic with respect to D&A applications,
we find that the proposed estimator outperforms the conventional estimator by several
orders of magnitude for small values of

Substantial further work is needed to evaluate the performance of the
proposed uncertainty quantification, in particular in an EIV context. Such an
evaluation was beyond the scope of the present work. Its primary focus was to
demonstrate that the choice of the leading eigenvectors as a projection
subspace is a vastly suboptimal one for D&A purposes; and that projecting on
a subspace which is orthogonal to the

The data used for illustration throughout
this article was
first described and studied by Ribes and Terray (2013).
As mentioned in the
latter study, it can be accessed at the following url:

Let

Therefore,

The author declares that there is no conflict of interest.

This work was supported by the ANR grant DADA and by the LEFE grant MESDA. I thank Dáithí Stone, Philippe Naveau and two anonymous reviewers for very useful comments and discussions that helped improve considerably this article. Edited by: Christopher Paciorek Reviewed by: two anonymous referees