Guest essay by M.S.Hodgart
(Visiting Reader Surrey Space Centre University of Surrey)
The figure presented here is a new graph of the story of global warming – and cooling. The graph makes no predictions and should be used only to see what has been happening historically.
The boxed points in the figure are the ‘raw data’ – the annualised global average surface temperature known as HadCRUT4 as released by the UK Meteorological Office. Strictly these are ‘temperature anomalies’. The plot runs from 1870 up to the last complete calendar year 2012. The raw data cannot of course be treated as absolutely true – but let us give the Met Office the benefit of the doubt – this is hopefully their best effort so far.
It is a difficult statistical problem to estimate the historical trend in these kind of time series. The solution requires some kind of smoothing of the data but how exactly? There are an unlimited number of ways of drawing some curve through the data.
A popular method – much used in the climate science literature – is by a moving average. One trouble with it is that quite different looking curves obtain depending on the width of the smoothing window used in that average – also on the choice of window. Another difficulty is its poor dynamic tracking capability
The other popular method is to fit a selection of straight lines (least square estimate) to a selected span of years. The notorious difficulty here is the quite different impression one gets depending on the choice of start and stop years.
The difficulty is finding for a best estimate – some curve which is most likely to be closest to the truth. There is an outstanding problem in what the statistical literature identifies as model selection.
The source of the problem is what the telecommunication and control engineers call noise in the data – a random-looking variation from one year to the next.
As a conspicuous example of this random variation: in recent years, according to the record, the global temperature (anomaly) was 0.18 deg in 1996 ; had jumped to 0.52 deg in 1998 but had fallen again to 0.29 deg by 2000.
Respecting normal linguistic usage and common-sense we would not want to describe a jump of 0.34 deg in only two years as a phenomenon of ‘global warming’; nor a drop by 0.23 deg over the next two years as ‘global cooling’. Ordinary language, when expressed in mathematics, envisages some smooth slow-varying curve which passes on a middle course through the scattered data, ignoring these rapid changes, but responsive over a longer term to general movement . There needs to be an explicit decomposition
HadCRUT4 annual data = trend in the data + temperature noise
The problem is to estimate that trend in the data when it is corrupted by the presence of this significant noise.
HadCRUT4 global annual averaged temperature anomaly 1870 – 2012 (connected brown box points). Brown curve 26-year span cubic loess estimate. Dashed brown curve 10th degree PR estimate. Red curve is a mean trend. Blue curve is the offset cyclic component of loess. The red circled points identify coincident years of trend and mean trend: in years 1870, 1891, 1927, 1959, 1992, & 2012. Blue circled points delineate alternating cooling and warming in cyclic variation: 1877, 1911, 1943, 1976, & 2005.
A novel principle of joint estimation is proposed here – using two relatively simple methods of smoothing.
In the figure the continuous brown curve is an estimate by locally weighted regression (loess) – using a locally-fitting cubic polynomial and the standard ‘tri-cube’ weighting. Loess is greatly superior generalisation of the moving average . Professor Mills deserves credit for first pointing out the superiority of a cubic over the usual linear or quadratic local polynomial . Unfortunately the standard statistical tools seem not to have caught up with him here – nor with his ‘natural’ solution to the end-point problem (where data runs out after 2012 and before 1870 on this graph.
The dashed brown curve is a standard (unweighted) polynomial regression. The principle of joint estimation is to look for span of years in loess and a degree in the polynomial regression where the two curves most closely resemble each other. There is a least disparity
Empirical search finds for a span of 26 years for the loess and a 10th degree for the polynomial. No other combination of loess span and polynomial degree gives such a close agreement. The condition is unique and therefore automatically solves the problem of model selection
In the author’s view this joint estimate is really the best that can be done in finding for the trend of global surface temperature. For various reasons the loess estimate should be prioritised.
The optimal estimate identifies alternating cooling and warming intervals from 1877 to 2005. Two cooling intervals alternated with two warming intervals. These two cycles of alternating cooling and warming were barely conceded, and certainly not discussed let alone explained, in the influential IPCC 4th report (AR4) published in 2007 and based on data available to 2005.
But this property conflicts with a different requirement: that a trend should be a “smooth broad movement non-oscillatory in nature” (see 1.22 in Kendall and Ord’s classic text  ). To reconcile these different requirements the estimated trend must be further decomposed into a non-oscillatory mean trend (red curve) and a quasi-periodic oscillation (blue curve).
trend in the data = mean trend in the data + quasi-periodic oscillation
A unique decomposition is achieved by computer-assisted iterative adjustment of four intersecting common years (red circled points). The mean trend is a cubic spline interpolation which deviates least from a straight line while the oscillatory component has a zero average over the record.
The strong oscillating component – the blue curve – is seen to be contributing more than half of the rate of increase when global warming was at a peak in the early 1990s.
What goes up may come down. This oscillating component looks to be continuing. Assessment is increasingly uncertain the closer one gets to the last data year of 2012. But despite this difficulty the probability that there is again global cooling in recent years can be stated with high confidence (IPCC terminology – better than 80%).
In the author’s view the whole climate debate has been muddled – and continues to be muddled – by not differentiating between this trend in the data (which oscillates) and the mean trend (which does not).
So yes – global warming looks to have stopped (if you believe in HadCRUT4) when one defines global surface temperature in terms of that trend – the brown curve. In fact it has more than stopped – it looks very much to have gone into reverse.
But no – average global warming continues ever upwards (still believing in HadCRUT4) when one defines an average global surface temperature in terms of that mean trend – the red curve.
A non-ambiguous computation of the rate of temperature increase is achieved by working from those common years (red circled points) when the two estimates coincide. The increase for HadCRUT4 from 1870 to 2012 of 0.75 ± 0.24 deg is equivalent to an average rate of 0.053 ± 0.017 deg/decade. From 1959 to 2012 this average rate looks to have increased to 0.090 ± 0.034 deg/decade. The error limits here are the usual ± 2 standard deviations or 95% confidence limits)
If this faster trend were to continue then we would be looking at an average rise from now of 0.8 deg by the end of this century (not choosing to set controversial error limits into the future). It is not however safe to make any predictions on the basis of the plot and the methodology adopted here.
It should not need to be stressed that there is no contradiction between these results and finding that regional warming may be continuing – particularly in high Northern latitudes and the Arctic.
There is a great deal more than can and needs to be said to justify these results. Interested readers can apply to the author for longer treatments and in particular a full and detailed mathematical justification.
 “Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting” W. S. Cleveland, S.J. Devlin Journal of the American Statistical Association, Vol. 83, No. 403 (Sep., 1988), pp. 596- 610.
 “Modelling Current Trends in Northern Hemisphere temperatures” T.C.Mills International Journal of Climatology 26 p 867- 884 (2006)
 “Time Series” Kendall and Ord (1990) 3rd edition Edward Arnold