Confidence Limits of Minimum Roughness Criterion

Here I show more humorous effects of smoothed trend lines with the ‘minimum roughness condition’ (MRC). The confidence limits blow out.

Fitting a straight line to data such as global temperature data is a common linear regression example problem. Linear regression of stock prices tells you your rate of appreciation. Smoothing, (or filtering) is used to give a smooth, curved trend instead of a straight regression line. Instead of applying a linear regression model to data many techniques such as moving averages, splines, or singular spectrum analysis (SSA) can give a smooth trend line. One problem with these methods is what to do at the ends where the data runs out.

One way of handling end points is the MRC. The MRC is referenced in papers including Rahmsdorf et al. 2007 who state the nonlinear trend lines “were computed with an embedding period of 11 years and a minimum roughness criterion at the end (Moore 2006)”. The MRC is described in a paper by Michael Mann (2004) as follows: “[O]ne pads the series with the values within one filter width of the boundary reflected vertically (i.e. about the y axis) relative to the final value.” He states the intent of MRC padding the end of a time series is to ensure a smooth trend line until the end of the series.

But, as was noted in “Mannomatic Smoothing and Pinned End-points” MRC causes the trend line to pass through the value of the final point of the series (the pin). Willis Eschenbach also notes his paper on the pinning property had been twice rejected by GRL.

When I wrote a little routine to implement Mannomatic smoothing, I noticed something really funny. I know that it seems bizarre that there can be humor in smoothing algorithms, but hey, this is the Team. Think about what happens with the Mannomatic smooth: you reflect the series around the final value both horizontally and vertically. Accordingly with a symmetric filter (as these things tend to be), everything cancels out except the final value. The Mannomatic pins the series on the end-point exactly the same as Emanuel’s “incorrect” smoothing.

Well if you take the pinning property a step further and estimate the confidence interval of the trend line, another humorous thing happens.

The figure below shows the confidence limits of the calculated trends in global temperature from GISS (shown as solid black) and the MRC padding (dashed black lines). The upper and lower trend lines (red) were calculated using MRC padding originating at the limits of the 95% confidence intervals at year 2006. The blue line is the linear regression of the GISS trend from 1975 to 2001.

article-001.png

Figure: Confidence intervals of a smooth spline trend line with ‘minimum roughness criterion’ padding of endpoints.

With MRC confidence limits of the trend expand to the width of a single value, rather than a mean value (Solid red lines). This is considerably greater than the uncertainty of a trend line (dashed red line). The only effect of the MRC is to replace a narrow confidence interval of a trend line, with a large confidence interval of a single point!

Why would you want to do this? If you want to emphasize the direction the final point is going in, then MRC provides a strong bias on the trend, but you would only get away with it if you don’t present the full uncertainty. So it would be useful for fraudulent statistical modeling, but such practices are bordering on academic fraud.

To be fair Mann 2004 cautions against the application MRC because it is sensitive to outliers, suggests careful evaluation of goodness of fit, providing a pathological example of MRC padding from climate science. However in general, ad hoc methodologies such as MRC should be avoided, uncertainty limits and formal tests of significance should be performed to support claims.

References

[1]Stefan Rahmstorf, Anny Cazenave, John A. Church, James E. Hansen,
Ralph F. Keeling, David E. Parker, and Richard C. J. Somerville. Recent
Climate Observations Compared to Projections.
Science, 316(5825):709–,
2007.

[2] A. Grinsted Moore, J.C. and S. Jevrejeva. New tools for analyzing time series relationships and trends. Eos, 86(24), 1995.

[3] M. E. Mann. On smoothing potentially non-stationary climate time series.
Geophys. Res. Lett., 31:L07214, doi:10.1029/2004GL019569 2004.

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0 thoughts on “Confidence Limits of Minimum Roughness Criterion

  1. Well, it had struck me that the uncertainty in correct location of the smooth line had to be high, and particularly near the end point. So, it would appear that if Ramhstorf7 had given the slightest consideration to uncertainty, they would have seen that they had no clue where the non-linear fit went through the data!

  2. Well, it had struck me that the uncertainty in correct location of the smooth line had to be high, and particularly near the end point. So, it would appear that if Ramhstorf7 had given the slightest consideration to uncertainty, they would have seen that they had no clue where the non-linear fit went through the data!

  3. That is right. What is frustrating to me is that this is the sort of authoritative paper that convinces world governments we are all going to warming hell in a handbasket.

  4. That is right. What is frustrating to me is that this is the sort of authoritative paper that convinces world governments we are all going to warming hell in a handbasket.

  5. This topic is in my list ‘pictures you won’t see at RC’ ( http://signals.auditblogs.com/2008/01/03/some-interesting-figures/ )

    Indeed, these padding procedures are very sensitive to measurement noise. And the effect near the end-points can be quite easily computed with error propagation formulas (see for example http://www.climateaudit.org/?p=2541#comment-188580 ) . But in these climate papers signal, process noise , measurement noise are not defined well, so they can use any method they prefer.

  6. This topic is in my list ‘pictures you won’t see at RC’ ( http://signals.auditblogs.com/2008/01/03/some-interesting-figures/ )

    Indeed, these padding procedures are very sensitive to measurement noise. And the effect near the end-points can be quite easily computed with error propagation formulas (see for example http://www.climateaudit.org/?p=2541#comment-188580 ) . But in these climate papers signal, process noise , measurement noise are not defined well, so they can use any method they prefer.

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