Global Warming Bent-Line Regression

HadCRUT global near-surface temperatures

HadCRUtemp2lineThis graph, posted with permission, shows a bent line fitted to the HadCRUT annual data series for global near-surface temperature. Professor Thayer Watkins of San Jose State University Department of Economics posted it on his blog about 2009.

HadCRUTsmoothWithout knowing of this work, I constructed the second graph. I used data from the same HadCRUT source, but a data set smoothed by the authors.

In April 2013 I posted it to a forum thread in”weatherzone”.

Next, I added to that graph a logarithmic plot of global carbon emissions, similarly fitted with a series of straight trend lines.

Log from 1850 of world surface air temperature and carbon emissionsThis I included in posts to several forums: in a post to “weatherzone”, in a post to the Alternative Technology Association forum, and finally in a post to this blog.

Both Professor Watkins and I have fitted bent lines to the data. I fitted the lines by eye (for which I was accused of “cherry-picking”). Professor Watkins used an explicit process of Bent-Line Regression, minimising the deviations by the method of least-squares. Like me, he initially chose by eye the dates of the change points where the straight lines meet. But he then adjusted them so as to minimise the least squares deviations.
[See notes below on the method of Bent-Line Regression.]

The trend lines and change points are practically the same in the Thayer Watkins and the “Surly Bond” graphs:
1. (Up to Down) TW: 1881; SB: 1879.
2. (Down to Up) TW: 1911; SB: 1909.
3. (Up to Down) TW: 1940; SB: 1943.
4. (Down to Up) TW: 1970; SB: 1975.
As I said at the time, once straight trend lines are chosen, the dates of change points to fit this data series closely do not allow of much variation.

Relation to the IPO (or PDO) of the Pacific

Not by coincidence, Watkins and I both went on to relate the multi-decadal oscillations of Pacific Ocean temperatures to the global near-surface average temperatures.

My approach

I merely plotted my chosen global temperature change points on to the Pacific graphs (I chose to cite the IPO (Inter-decadal Pacific Oscillation)). In two posts I noted (i) the way the change points in the HadCRUT global temperature series were close to those in the IPO, and (ii) the way the IPO seemed able to explain why the trend in global warming was “bent” in 1943 and 1975 but, in that case, could only sharpen the bends of 1910 and 1880.

Professor Watkins’ approach

AGT_PDO7Professor Watkins did a separate Bent Line Regression Analysis on the Pacific graphs (He chose to cite the earlier-developed PDO (Pacific inter-Decadal Oscillation)). His analysis “A Major Source of the Near-Sixty Year Cycle in Average Global Temperatures is the Pacific (Multi)Decadal Oscillation” is here.

He admits the match is poor, with various lags and a different period. He concludes:
“Thus while the Pacific (Multi)Decadal Oscillation appears to be involved in the cycles of the average global temperature there have to be other factors also involved.”

The significance of the IPO

Multi-decadal oscillations of temperature in the Pacific basin must affect world temperatures. The basin, with almost no internal barriers, has an area greater than all the land in the world. The key influence of Pacific basin sea surface temperatures in global climate is shown in the following research.
A paper by Power et al. (1999) refers to three Empirical Orthogonal Functions of low-frequency sea surface temperature:

(1) the first EOF represents global warming;
(2) the second EOF represents out-of-phase temperature fluctuations between the Northern and Southern Hemispheres;
(3) the third EOF is the Inter-decadal Pacific Oscillation (IPO).

[This EOF analysis originates in an un-linkable paper:
Folland CK, Parker DE, Colman AW, Washington R. (1998) “Largescale
modes of ocean surface temperature since the late nineteenth
century.” Hadley Centre, UK Meteorological Office, Clim Res
Tech Note, CRTN 81, 45 pp]

Notes on Bent-Line Regression

Professor Watkins briefly described his technique for “Estimating the Parameters of a Bent Line in Regression Analysis” here.

The technique of Bent Line Regression was discussed in greater depth by R. Chappell (1989), and by Zhang and Li (2017).

Bent Line Regression has been implemented in the statistical programming language “R”.

Within my own blog, there are data plots in widely different aspects of climate where explicit bent line regression could be applied to add rigor to my inferences.

* This present post refers to trends in global temperature and carbon emissions.

SOI CUSUM plot* The SOI CUSUM plot is found to have a linear up-trend followed by a linear down-trend.

Summer rainfall anomalies and trends* Local summer rainfall is found to have had a very rapid (unsustainable) quasi-linear rise for 81 years from 1896 to 1977, preceded and followed by crashes.

LGandManCusumtb* There are differing successive multi-decadal linear trends (with change point about 1950) in rainfall in eastern NSW when CUSUM values of 7-year rainfall totals are plotted.

Note added 26 May 2017

Jiansong Zhou and Ka-Kit Tung (2013) “Deducing Multidecadal Anthropogenic Global Warming Trends Using Multiple Regression Analysis” Journal of the Atmospheric Sciences 70: 3-8.

This is a mainstream article, published online, that aims to show how various factors contribute to the observed global warming series HadCRUT4. Like this blog post, it is concerned with multi-decadal changes, and it references the work of Folland’s group.

Note added 12 December 2017

Wikipedia has an article on “Segmented regression” that seems to be relevant. These examples are in the category “Segmented linear regression”.

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