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Gregory et al 2019: Unsound claims about bias in local weather comments and local weather sensitivity estimation

Reposted from Dr. Judith Curry’s Local weather And many others.

Posted on October 18, 2019 by means of niclewis |

By way of Nic Lewis

The lately printed open-access paper “How correctly can the local weather sensitivity to CO2 be estimated from ancient local weather trade?” by means of Gregory et al.[i] makes a variety of assertions, many uncontentious however others for my part unjustified, deceptive or without a doubt improper.

Possibly most significantly, they are saying within the Summary that “The actual-world diversifications imply that ancient EffCS [effective climate sensitivity] underestimates CO2 EffCS by means of 30% when taking into account all the ancient duration.” However they don’t point out that this discovering relates most effective to efficient local weather sensitivity in GCMs, after which most effective to when they’re pushed by means of one specific observational sea floor temperature dataset.

Then again, on this article I can center of attention on one specific statistical factor, the place the declare made within the paper can readily be confirmed improper with no need to delve into the main points of GCM simulations.

Gregory et al. believe a regression within the shape R = α T, the place T is the trade in global-mean floor temperature with recognize to an unperturbed (i.e. preindustrial) equilibrium, and R = α T is the radiative reaction of the local weather machine to modify in T. α is thus the local weather comments parameter, and F2xCO2 /α is the EffCS estimate, F2xCO2 being the efficient radiative forcing for a doubling of preindustrial atmospheric carbon dioxide focus.

The paper states that “that estimates of ancient α made by means of OLS [bizarre least squares] regression from real-world R and T are biased low”. OLS regression estimates α because the slope of a instantly line are compatible between R and T records issues (most often with an intercept time period for the reason that unperturbed equilibrium local weather state isn’t recognized precisely), by means of minimising the sum of the squared mistakes in R. Random mistakes in R don’t reason a bias within the OLS slope estimate. Thus within the beneath chart, with R taken as plotted at the y-axis and T at the x-axis, OLS reveals the pink line that minimizes the sum of the squares of the lengths of the vertical strains.

Then again, one of the vital variability in measured T would possibly not produce a proportionate reaction in R. That will happen if, for instance, T is measured with error, which occurs in the actual global. It’s widely known that during such an “error within the explanatory variable” case, the OLS slope estimate is (on reasonable) biased in opposition to 0. This factor has been known as “regression dilution”.

Regression dilution is one explanation why estimates of local weather comments and local weather sensitivity derived from warming over the ancient duration steadily as a substitute use the “distinction means”.[ii] [iii] [iv] [v] The adaptation means comes to taking the ratio of variations, ΔT and ΔR, between T and R values overdue and early within the duration. In apply ΔT and ΔR are most often according to differencing averages over no less than a decade, in an effort to cut back noise.

I can word at this level that after a slope parameter is estimated for the connection between two variables, either one of which might be suffering from random noise, the likelihood distribution for the estimate might be skewed relatively than symmetric. When deriving a very best estimate by means of taking many samples from the mistake distributions of each and every variable, or (if possible) by means of measuring them each and every on many differing events, the right central measure to make use of is the pattern median no longer the pattern imply. Physicists need measures which can be invariant below reparameterization[vi], which is a assets of the median of a likelihood distribution for a parameter however no longer, when the distribution is skewed, of its imply. Regression dilution impacts each the imply and the median estimates of a parameter, even though to a slightly other extent.

Thus far I consider what is alleged by means of Gregory et al. Then again, the paper is going directly to state that “The prejudice [in α estimation] impacts the adaptation means in addition to OLS regression (Appendix D.1).” This statement is improper. If true, this could indicate that observationally-based estimates the use of the adaptation means can be biased reasonably low for local weather comments, and therefore biased reasonably prime for local weather sensitivity. Then again, the declare is no longer true.

The statistical analyses in Appendix D believe estimation by means of OLS regression of the slope m within the linear courting y(t) = m x(t), the place x and y are time collection the to be had records values of which might be suffering from random noise. Appendix D.1 considers the use of the adaptation between the final and primary unmarried time sessions (right here, apparently, of a 12 months), no longer of averages over a decade or extra, and it assumes for comfort that each x and y are recentered to have 0 imply, however neither of those have an effect on the purpose of idea at factor.

Appendix D.1 displays, appropriately, that after most effective the endpoints of the (noisy) x and y records are utilized in and OLS regression, the slope estimate for m is Δyx, the similar because the slope estimate from the adaptation means. It is going on to assert that taking the slope between the x and y records endpoints is a unique case of OLS regression and that the truth that an OLS regression slope estimate is biased in opposition to 0 when there may be uncorrelated noise within the x variable means that the adaptation means slope estimate is in a similar way so biased.

Then again, this is improper. The median slope estimate isn’t biased on account of mistakes within the x variable when the slope is estimated by means of the adaptation means, nor when there most effective two records issues in an OLS regression. And even though the imply slope estimate is biased, the unfairness is prime, no longer low. Somewhat than going into an in depth theoretical research of why that’s the case, I can display that it’s by means of numerical simulation. I can additionally give an explanation for how in easy phrases regression dilution will also be considered as coming up, and why it does no longer stand up when most effective two records issues are used.

The numerical simulations that I performed are as follows. For simplicity I took the actual slope m as 1, in order that the actual courting is y = x, and that true price of each and every x level is the sum of a linearly trending component operating from zero to 100 in steps of one and a random component uniformly dispensed within the vary -30 to +30, which will also be interpreted as a simulation of a trending “local weather” portion and a non-trending “climate” portion.[vii]  I took each x and y records (measured) values as matter to zero-mean impartial typically dispensed dimension mistakes with an ordinary deviation of 20. I took 10,000 samples of randomly drawn (as to the actual values of x and dimension mistakes in each x and y) units of 101 x and 101 y values.

The usage of OLS regression, each the median and the imply of the ensuing 10,000 slope estimates from regressing y on x the use of OLS have been zero.74 – a 26% downward bias within the slope estimator because of regression dilution.

The median slope estimate according to taking variations between the averages for the primary ten and the final ten x and y records issues used to be 1.00, whilst the imply slope estimate used to be 1.01. When the averaging duration used to be greater to 25 records issues the median bias remained 0 whilst the already tiny imply bias halved.

When variations between simply the primary and final measured values of x and y have been taken,[viii] the median slope estimate used to be once more 1.00 however the imply slope estimate used to be 1.26.

Thus, the slope estimate from the use of the adaptation means used to be median-unbiased, in contrast to for OLS regression, whether or not according to averages over issues at each and every finish of the collection or simply the primary and final issues.

The cause of the upwards imply bias when the use of the adaptation means will also be illustrated merely, if mistakes in y (which on reasonable don’t have any impact at the slope estimate) are disregarded. Think the actual Δx price is 100, in order that Δy is 100, and that two x samples are matter to mistakes of respectively +20 and –20. Then the 2 slope estimates might be 100/120 and 100/80, or zero.833 and 1.25, the imply of which is 1.04, in way over the actual slope of one.

The image stays the similar even if (fractional) mistakes in x are smaller than the ones in y. On decreasing the mistake usual deviation for x to 15 whilst expanding it to 30 for y, the median and imply slope estimates the use of  OLS regression have been each zero.84. The median slope estimates the use of the adaptation means have been once more impartial whether or not the use of 1, 10 or 25 records issues originally and finish, whilst the imply biases remained below zero.01 when the use of 10 or 25 records level averages and decreased to zero.16 when the use of unmarried records issues.

In truth, a second’s idea displays that the slope estimate from 2-point OLS regression will have to be impartial. Since each variables are suffering from error, if OLS regression offers upward thrust to a low bias within the slope estimate when x is regressed on y, it will have to additionally give upward thrust to a low bias within the slope estimate when y is regressed on x. If the slope of the actual courting between y and x is m, that between x and y is 1/m. It follows that if regressing x on y offers a biased low slope estimate, taking the reciprocal of that slope estimate will supply an estimate of the slope of the actual courting between y and x this is biased prime. Then again, when there are 2 records issues the OLS slope estimate from regressing y on x and that from regressing x on y and taking its reciprocal are equivalent (for the reason that are compatible line will pass throughout the 2 records issues in each circumstances). If the y-against-x and x-against-y OLS regression slope estimates have been biased low that might no longer be so.

As for a way and why mistakes within the x (explanatory) variable reason the slope estimate in OLS regression to be biased in opposition to 0 (supplied there are greater than two records issues), however mistakes within the y (dependent) variable don’t, the way in which I take a look at it’s this. For simplicity, I take focused (zero-mean) x and y values.  The OLS slope estimate is then Σxy / Σxx, this is to mention the weighted sum of the y records values divided by means of the weighted sum of  the x records values, the weights being the x records values.  An error that strikes a measured x price farther from the imply of 0 no longer most effective reduces the slope y/x for that records level, but additionally will increase the load given to that records level when forming the OLS slope estimate. Therefore such issues are given extra affect when figuring out the slope estimate. However, an error in x that strikes the measured price closer to 0 imply x price, expanding the y/x slope for that records level, reduces the load given to that records level, in order that it’s much less influential in figuring out the slope estimate. The web result’s a bias in opposition to a smaller slope estimate. Then again, for a two-point regression, this impact does no longer happen, as a result of regardless of the indicators of the mistakes affecting the x-values of the 2 issues, each x-values will at all times be equidistant from their imply, and so each records issues can have equivalent affect at the slope estimate whether or not they build up or lower the x-value. Consequently, the median slope estimate might be impartial on this case.  Regardless of the choice of records issues, mistakes within the y records issues is not going to have an effect on the weights given to these records issues when forming the OLS slope estimate, and mistakes within the y-data values will on reasonable cancel out when forming the OLS slope estimate Σxy / Σxx.

So why is the evidence in Gregory et al. AppendixD.1, supposedly appearing that OLS regression with 2 records issues produces a low bias within the slope estimate when there are mistakes within the explanatory (x) records issues, invalid? The solution is understated. The Appendix D.1 evidence depends upon the evidence of low bias within the slope estimate in Appendix D.Three, which is expressed to use to OLS regression with any choice of records issues. But when one works throughout the equations in Appendix D.Three, one reveals that when it comes to most effective 2 records issues no low bias arises – the anticipated price of the OLS slope estimate equals the actual slope.

This can be a little miserable that when a few years of being criticised for his or her insufficiently just right working out of statistics and loss of shut engagement with the statistical group, the local weather science group seems nonetheless to not have solved this factor.

Nicholas Lewis ……………………………………………….. 18 October 2019


[i] Gregory, J.M., Andrews, T., Ceppi, P., Mauritsen, T. and Webb, M.J., 2019. How correctly can the local weather sensitivity to CO₂ be estimated from ancient local weather trade?. Local weather Dynamics.

[ii] Gregory JM, Stouffer RJ, Raper SCB, Stott PA, Rayner NA (2002) An observationally founded estimate of the local weather sensitivity. J Clim 15:3117–3121.

[iii] Otto A, Otto FEL, Boucher O, Church J, Hegerl G, Forster PM, Gillett NP, Gregory J, Johnson GC, Knutti R, Lewis N, Lohmann U, Marotzke J, Myhre G, Shindell D, Stevens B, Allen MR (2013) Power funds constraints on local weather reaction. Nature Geosci 6:415–416

[iv] Lewis, N. and Curry, J.A., 2015. The consequences for local weather sensitivity of AR5 forcing and warmth uptake estimates. Local weather Dynamics, 45(Three-Four), pp.1009-1023.

[v] Lewis, N. and Curry, J., 2018. The affect of new forcing and ocean warmth uptake records on estimates of local weather sensitivity. Magazine of Local weather, 31(15), pp.6051-6071.

[vi] In order that, for instance, the median estimate for the reciprocal of a parameter is the reciprocal of the median estimate for the parameter. This isn’t most often true for the imply estimate. This factor is especially related right here since local weather sensitivity is reciprocally associated with local weather comments.

[vii] There used to be an underlying pattern in T over the ancient duration, and taking it to be linear signifies that, within the absence of noise, linear slope estimated by means of regression and by means of the adaptation means can be equivalent.

[viii] Correcting the small choice of detrimental slope estimates coming up when the x distinction used to be detrimental however the y distinction used to be certain to a good price (see, e.g., Otto et al. 2013). Ahead of that correction the median slope estimate had a 1% low bias. The certain price selected (right here absolutely the price of the detrimental slope estimate concerned) has no impact of the median slope estimate supplied it exceeds the median price of the remainder slope estimates, however does materially have an effect on the imply slope estimate.

Reposted from Dr. Judith Curry’s Local weather And many others.

Posted on October 18, 2019 by means of niclewis |

By way of Nic Lewis

The lately printed open-access paper “How correctly can the local weather sensitivity to CO2 be estimated from ancient local weather trade?” by means of Gregory et al.[i] makes a variety of assertions, many uncontentious however others for my part unjustified, deceptive or without a doubt improper.

Possibly most significantly, they are saying within the Summary that “The actual-world diversifications imply that ancient EffCS [effective climate sensitivity] underestimates CO2 EffCS by means of 30% when taking into account all the ancient duration.” However they don’t point out that this discovering relates most effective to efficient local weather sensitivity in GCMs, after which most effective to when they’re pushed by means of one specific observational sea floor temperature dataset.

Then again, on this article I can center of attention on one specific statistical factor, the place the declare made within the paper can readily be confirmed improper with no need to delve into the main points of GCM simulations.

Gregory et al. believe a regression within the shape R = α T, the place T is the trade in global-mean floor temperature with recognize to an unperturbed (i.e. preindustrial) equilibrium, and R = α T is the radiative reaction of the local weather machine to modify in T. α is thus the local weather comments parameter, and F2xCO2 /α is the EffCS estimate, F2xCO2 being the efficient radiative forcing for a doubling of preindustrial atmospheric carbon dioxide focus.

The paper states that “that estimates of ancient α made by means of OLS [bizarre least squares] regression from real-world R and T are biased low”. OLS regression estimates α because the slope of a instantly line are compatible between R and T records issues (most often with an intercept time period for the reason that unperturbed equilibrium local weather state isn’t recognized precisely), by means of minimising the sum of the squared mistakes in R. Random mistakes in R don’t reason a bias within the OLS slope estimate. Thus within the beneath chart, with R taken as plotted at the y-axis and T at the x-axis, OLS reveals the pink line that minimizes the sum of the squares of the lengths of the vertical strains.

Then again, one of the vital variability in measured T would possibly not produce a proportionate reaction in R. That will happen if, for instance, T is measured with error, which occurs in the actual global. It’s widely known that during such an “error within the explanatory variable” case, the OLS slope estimate is (on reasonable) biased in opposition to 0. This factor has been known as “regression dilution”.

Regression dilution is one explanation why estimates of local weather comments and local weather sensitivity derived from warming over the ancient duration steadily as a substitute use the “distinction means”.[ii] [iii] [iv] [v] The adaptation means comes to taking the ratio of variations, ΔT and ΔR, between T and R values overdue and early within the duration. In apply ΔT and ΔR are most often according to differencing averages over no less than a decade, in an effort to cut back noise.

I can word at this level that after a slope parameter is estimated for the connection between two variables, either one of which might be suffering from random noise, the likelihood distribution for the estimate might be skewed relatively than symmetric. When deriving a very best estimate by means of taking many samples from the mistake distributions of each and every variable, or (if possible) by means of measuring them each and every on many differing events, the right central measure to make use of is the pattern median no longer the pattern imply. Physicists need measures which can be invariant below reparameterization[vi], which is a assets of the median of a likelihood distribution for a parameter however no longer, when the distribution is skewed, of its imply. Regression dilution impacts each the imply and the median estimates of a parameter, even though to a slightly other extent.

Thus far I consider what is alleged by means of Gregory et al. Then again, the paper is going directly to state that “The prejudice [in α estimation] impacts the adaptation means in addition to OLS regression (Appendix D.1).” This statement is improper. If true, this could indicate that observationally-based estimates the use of the adaptation means can be biased reasonably low for local weather comments, and therefore biased reasonably prime for local weather sensitivity. Then again, the declare is no longer true.

The statistical analyses in Appendix D believe estimation by means of OLS regression of the slope m within the linear courting y(t) = m x(t), the place x and y are time collection the to be had records values of which might be suffering from random noise. Appendix D.1 considers the use of the adaptation between the final and primary unmarried time sessions (right here, apparently, of a 12 months), no longer of averages over a decade or extra, and it assumes for comfort that each x and y are recentered to have 0 imply, however neither of those have an effect on the purpose of idea at factor.

Appendix D.1 displays, appropriately, that after most effective the endpoints of the (noisy) x and y records are utilized in and OLS regression, the slope estimate for m is Δyx, the similar because the slope estimate from the adaptation means. It is going on to assert that taking the slope between the x and y records endpoints is a unique case of OLS regression and that the truth that an OLS regression slope estimate is biased in opposition to 0 when there may be uncorrelated noise within the x variable means that the adaptation means slope estimate is in a similar way so biased.

Then again, this is improper. The median slope estimate isn’t biased on account of mistakes within the x variable when the slope is estimated by means of the adaptation means, nor when there most effective two records issues in an OLS regression. And even though the imply slope estimate is biased, the unfairness is prime, no longer low. Somewhat than going into an in depth theoretical research of why that’s the case, I can display that it’s by means of numerical simulation. I can additionally give an explanation for how in easy phrases regression dilution will also be considered as coming up, and why it does no longer stand up when most effective two records issues are used.

The numerical simulations that I performed are as follows. For simplicity I took the actual slope m as 1, in order that the actual courting is y = x, and that true price of each and every x level is the sum of a linearly trending component operating from zero to 100 in steps of one and a random component uniformly dispensed within the vary -30 to +30, which will also be interpreted as a simulation of a trending “local weather” portion and a non-trending “climate” portion.[vii]  I took each x and y records (measured) values as matter to zero-mean impartial typically dispensed dimension mistakes with an ordinary deviation of 20. I took 10,000 samples of randomly drawn (as to the actual values of x and dimension mistakes in each x and y) units of 101 x and 101 y values.

The usage of OLS regression, each the median and the imply of the ensuing 10,000 slope estimates from regressing y on x the use of OLS have been zero.74 – a 26% downward bias within the slope estimator because of regression dilution.

The median slope estimate according to taking variations between the averages for the primary ten and the final ten x and y records issues used to be 1.00, whilst the imply slope estimate used to be 1.01. When the averaging duration used to be greater to 25 records issues the median bias remained 0 whilst the already tiny imply bias halved.

When variations between simply the primary and final measured values of x and y have been taken,[viii] the median slope estimate used to be once more 1.00 however the imply slope estimate used to be 1.26.

Thus, the slope estimate from the use of the adaptation means used to be median-unbiased, in contrast to for OLS regression, whether or not according to averages over issues at each and every finish of the collection or simply the primary and final issues.

The cause of the upwards imply bias when the use of the adaptation means will also be illustrated merely, if mistakes in y (which on reasonable don’t have any impact at the slope estimate) are disregarded. Think the actual Δx price is 100, in order that Δy is 100, and that two x samples are matter to mistakes of respectively +20 and –20. Then the 2 slope estimates might be 100/120 and 100/80, or zero.833 and 1.25, the imply of which is 1.04, in way over the actual slope of one.

The image stays the similar even if (fractional) mistakes in x are smaller than the ones in y. On decreasing the mistake usual deviation for x to 15 whilst expanding it to 30 for y, the median and imply slope estimates the use of  OLS regression have been each zero.84. The median slope estimates the use of the adaptation means have been once more impartial whether or not the use of 1, 10 or 25 records issues originally and finish, whilst the imply biases remained below zero.01 when the use of 10 or 25 records level averages and decreased to zero.16 when the use of unmarried records issues.

In truth, a second’s idea displays that the slope estimate from 2-point OLS regression will have to be impartial. Since each variables are suffering from error, if OLS regression offers upward thrust to a low bias within the slope estimate when x is regressed on y, it will have to additionally give upward thrust to a low bias within the slope estimate when y is regressed on x. If the slope of the actual courting between y and x is m, that between x and y is 1/m. It follows that if regressing x on y offers a biased low slope estimate, taking the reciprocal of that slope estimate will supply an estimate of the slope of the actual courting between y and x this is biased prime. Then again, when there are 2 records issues the OLS slope estimate from regressing y on x and that from regressing x on y and taking its reciprocal are equivalent (for the reason that are compatible line will pass throughout the 2 records issues in each circumstances). If the y-against-x and x-against-y OLS regression slope estimates have been biased low that might no longer be so.

As for a way and why mistakes within the x (explanatory) variable reason the slope estimate in OLS regression to be biased in opposition to 0 (supplied there are greater than two records issues), however mistakes within the y (dependent) variable don’t, the way in which I take a look at it’s this. For simplicity, I take focused (zero-mean) x and y values.  The OLS slope estimate is then Σxy / Σxx, this is to mention the weighted sum of the y records values divided by means of the weighted sum of  the x records values, the weights being the x records values.  An error that strikes a measured x price farther from the imply of 0 no longer most effective reduces the slope y/x for that records level, but additionally will increase the load given to that records level when forming the OLS slope estimate. Therefore such issues are given extra affect when figuring out the slope estimate. However, an error in x that strikes the measured price closer to 0 imply x price, expanding the y/x slope for that records level, reduces the load given to that records level, in order that it’s much less influential in figuring out the slope estimate. The web result’s a bias in opposition to a smaller slope estimate. Then again, for a two-point regression, this impact does no longer happen, as a result of regardless of the indicators of the mistakes affecting the x-values of the 2 issues, each x-values will at all times be equidistant from their imply, and so each records issues can have equivalent affect at the slope estimate whether or not they build up or lower the x-value. Consequently, the median slope estimate might be impartial on this case.  Regardless of the choice of records issues, mistakes within the y records issues is not going to have an effect on the weights given to these records issues when forming the OLS slope estimate, and mistakes within the y-data values will on reasonable cancel out when forming the OLS slope estimate Σxy / Σxx.

So why is the evidence in Gregory et al. AppendixD.1, supposedly appearing that OLS regression with 2 records issues produces a low bias within the slope estimate when there are mistakes within the explanatory (x) records issues, invalid? The solution is understated. The Appendix D.1 evidence depends upon the evidence of low bias within the slope estimate in Appendix D.Three, which is expressed to use to OLS regression with any choice of records issues. But when one works throughout the equations in Appendix D.Three, one reveals that when it comes to most effective 2 records issues no low bias arises – the anticipated price of the OLS slope estimate equals the actual slope.

This can be a little miserable that when a few years of being criticised for his or her insufficiently just right working out of statistics and loss of shut engagement with the statistical group, the local weather science group seems nonetheless to not have solved this factor.

Nicholas Lewis ……………………………………………….. 18 October 2019


[i] Gregory, J.M., Andrews, T., Ceppi, P., Mauritsen, T. and Webb, M.J., 2019. How correctly can the local weather sensitivity to CO₂ be estimated from ancient local weather trade?. Local weather Dynamics.

[ii] Gregory JM, Stouffer RJ, Raper SCB, Stott PA, Rayner NA (2002) An observationally founded estimate of the local weather sensitivity. J Clim 15:3117–3121.

[iii] Otto A, Otto FEL, Boucher O, Church J, Hegerl G, Forster PM, Gillett NP, Gregory J, Johnson GC, Knutti R, Lewis N, Lohmann U, Marotzke J, Myhre G, Shindell D, Stevens B, Allen MR (2013) Power funds constraints on local weather reaction. Nature Geosci 6:415–416

[iv] Lewis, N. and Curry, J.A., 2015. The consequences for local weather sensitivity of AR5 forcing and warmth uptake estimates. Local weather Dynamics, 45(Three-Four), pp.1009-1023.

[v] Lewis, N. and Curry, J., 2018. The affect of new forcing and ocean warmth uptake records on estimates of local weather sensitivity. Magazine of Local weather, 31(15), pp.6051-6071.

[vi] In order that, for instance, the median estimate for the reciprocal of a parameter is the reciprocal of the median estimate for the parameter. This isn’t most often true for the imply estimate. This factor is especially related right here since local weather sensitivity is reciprocally associated with local weather comments.

[vii] There used to be an underlying pattern in T over the ancient duration, and taking it to be linear signifies that, within the absence of noise, linear slope estimated by means of regression and by means of the adaptation means can be equivalent.

[viii] Correcting the small choice of detrimental slope estimates coming up when the x distinction used to be detrimental however the y distinction used to be certain to a good price (see, e.g., Otto et al. 2013). Ahead of that correction the median slope estimate had a 1% low bias. The certain price selected (right here absolutely the price of the detrimental slope estimate concerned) has no impact of the median slope estimate supplied it exceeds the median price of the remainder slope estimates, however does materially have an effect on the imply slope estimate.

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