Statistical homogenization under-corrects any station network-wide trend biases

A station of the US Climate Reference Network.


In the last blog post I made the argument that the statistical detection of breaks in climate station data has problems when the noise is larger than the break signal. The post before argued that the best homogenization correction method we have can remove network-wide trend biases perfectly if all breaks are known. In the light of the last post, we naturally would like to know how well this correction method can remove such biases in the more realistic case when the breaks are imperfectly estimated. That should still be studied much better, but it is interesting to discuss a number of other studies on the removal of network-wide trend biases from the perspective of this new understanding.

So this post will argue that it theoretically makes sense that (unavoidable) inaccuracies of break detection lead to network-wide trend biases only being partially corrected by statistical homogenization.

1) We have seen this in our study of the correction method in response to small errors in the break positions (Lindau and Venema, 2018).

2) The benchmarking study of NOAA’s homogenization algorithm shows that if the breaks are big and easy they are largely removed, while in the scenario where breaks are plentiful and small half of the trend bias remains (Williams et al., 2012).

3) Another benchmarking study show that with the network density of Switzerland homogenization can find and remove clear trend biases, while if you thin this network to be similar to Peru the bias cannot be removed (Gubler et al., 2017).

4) Finally, a benchmarking study of relative humidity station observations in Austria could not remove much of the trend bias, which is likely because relative humidity is not correlated well from station to station (Chimani et al., 2018).

Statistical homogenization on a global scale makes warming estimates larger (Lawrimore et al., 2011; Menne et al., 2018). Thus if it can only remove part of any trend bias, this would mean that quite likely the actual warming was larger.

Figure 1: The inserted versus remaining network-mean trend error. Upper panel for perfect breaks. Lower panel for a small perturbation of the break position. The time series are 100 annual values and have 5 break. Figure 10 in Lindau and Venema (2018).

Joint correction method

First, what did our study on the correction method (Lindau and Venema, 2018) say about the importance of errors in the break position? As the paper was mostly about perfect breaks, we assumed that all breaks were known, but that they had a small error in their position. In the example to the right, we perturbed the break position by a normally distributed random number with standard deviation one (lower panel), while for comparison the breaks are perfect (upper panel).

In both cases we inserted a large network-wide trend bias of 0.873 °C over the length of the century long time series. The inserted errors for 1000 simulations is on the x-axis, the average inserted trend bias is denoted by x̅. The remaining error after homogenization is on the y-axis. Its average is denoted by y̅ and basically zero in case the breaks are perfect (top panel). In case of the small perturbation (lower panel) the average remaining error is 0.093 °C, this is 11 % of the inserted trend bias. That is the under-correction for is a quite small perturbation: 38 % of the positions is not changed at all.

If the standard deviation of the position perturbation is increased to 2, the remaining trend bias is 21 % of the inserted bias.

In the upper panel, there is basically no correlation between the inserted and the remaining error. That is, the remaining error does not depend on the break signal, but only on the noise. In the lower panel with the position errors, there is a correlation between the inserted and remaining trend error. So in this more realistic case, it does matter how large the trend bias due to the inhomogeneities is.

This is naturally an idealized case, position errors will be more complicated in reality and there would be spurious and missing breaks. But this idealized case fitted best to the aim of the paper of studying the correction algorithm in isolation.

It helps understand where the problem lies. The correction algorithm is basically a regression that aims to explain the inserted break signal (and the regional climate signal). Errors in the predictors will lead to an explained variance that is less than 100 %. One should thus expect that the estimated break signal is smaller than the actual break signal. It is thus expected that the trend change due to the estimated break signal produces is smaller than the actual trend change due to the inhomogeneities.

NOAA’s benchmark

That statistical homogenization under-corrects when the going gets tough is also found by the benchmarking study of NOAA’s Pairwise Homogenization Algorithm in Williams et al. (2012). They simulated temperature networks like the American USHCN network and added inhomogeneities according to a range of scenarios. (Also with various climate change signals.) Some scenarios were relatively easy, had few and large breaks, while others were hard and contained many small breaks. The easy cases were corrected nearly perfectly with respect to the network-wide trend, while in the hard cases only half of the inserted network-wide trend error was removed.

The results of this benchmarking for the three scenarios with a network-wide trend bias are shown below. The three panels are for the three scenarios. Each panel has results (the crosses, ignore the box plots) for three periods over which the trend error was computed. The main message is that the homogenized data (orange crosses) lies between the inhomogeneous data (red crosses) and the homogeneous data (green crosses). Put differently, green is how much the climate actually changed, red is how much the estimate is wrong due to inhomogeneities, orange shows that homogenization moves the estimate towards the truth, but never fully gets there.

If we use the number of breaks and their average size as a proxy for the difficulty of the scenario, the one on the left has 6.4 breaks with an average size of 0.8 °C, the one in the middle 8.4 breaks (size 0.4 °C) and the one on the right 10 breaks (size 0.4 °C). So this suggests there is a clear dose effect relationship; although there surely is more than just the number of breaks.


Figures from Williams et al. (2012) showing the results for three scenarios. This is a figure I created from parts of Figure 7 (left), Figure 5 (middle) and Figure 10 (right; their numbers).

When this study appeared in 2012, I found the scenario with the many small breaks much too pessimistic. However, our recent study estimating the properties of the inhomogeneities of the American network found a surprisingly large number of breaks: more than 17 per century; they were bigger: 0.5 °C. So purely based on the number of breaks the hardest scenario is even optimistic, but also size matters.

Not that I would already like to claim that even in a dense network like the American there is a large remaining trend bias and the actual warming was much larger. There is more to the difficulty of inhomogeneities than their number and size. It sure is worth studying.

Alpine benchmarks

The other two examples in the literature I know of are examples of under-correction in the sense of basically no correction because the problem is simply too hard. Gubler et al. (2017) shows that the raw data of the Swiss temperature network has a clear trend bias, which can be corrected with homogenization of its dense network (together with metadata), but when they thin the network to a network density similar to that of Peru, they are unable to correct this trend bias. For more details see my review of this article in the Grassroots Review Journal on Homogenization.

Finally, Chimani et al. (2018) study the homogenization of daily relative humidity observations in Austria. I made a beautiful daily benchmark dataset, it was a lot of fun: on a daily scale you have autocorrelations and a distribution with an upper and lower limit, which need to be respected by the homogeneous data and the inhomogeneous data. But already the normal homogenization of monthly averages was much too hard.

Austria has quite a dense network, but relative humidity is much influenced by very local circumstances and does not correlate well from station to station. My co-authors of the Austrian weather service wanted to write about the improvements: "an improvement of the data by homogenization was non‐ideal for all methods used". For me the interesting finding was: nearly no improvement was possible. That was unexpected. Had we expected that we could have generated a much simpler monthly or annual benchmark to show no real improvement was possible for humidity data and saved us a lot of (fun) work.

What does this mean for global warming estimates?

When statistical homogenization only partially removes large-scale trend biases what does this mean for global warming estimates? In the global temperature datasets statistical homogenization leads to larger warming estimates. So if we tend to underestimate how much correction is needed, this would mean that the Earth most likely warmed up more than current estimates indicate. How much exactly is hard to tell at the moment and thus needs a nuanced discussion. Let me give you my considerations in the next post.

Other posts in this series

Part 5: Statistical homogenization under-corrects any station network-wide trend biases

Part 4: Break detection is deceptive when the noise is larger than the break signal

Part 3: Correcting inhomogeneities when all breaks are perfectly known

Part 2: Trend errors in raw temperature station data due to inhomogeneities

Part 1: Estimating the statistical properties of inhomogeneities without homogenization

References

Chimani Barbara, Victor Venema, Annermarie Lexer, Konrad Andre, Ingeborg Auer and Johanna Nemec, 2018: Inter-comparison of methods to homogenize daily relative humidity. International Journal Climatology, 38, pp. 3106–3122. https://doi.org/10.1002/joc.5488.

Gubler, Stefanie, Stefan Hunziker, Michael Begert, Mischa Croci-Maspoli, Thomas Konzelmann, Stefan Brönnimann, Cornelia Schwierz, Clara Oria and Gabriela Rosas, 2017: The influence of station density on climate data homogenization. International Journal of Climatology, 37, pp. 4670–4683. https://doi.org/10.1002/joc.5114

Lawrimore, Jay H., Matthew J. Menne, Byron E. Gleason, Claude N. Williams, David B. Wuertz, Russell S. Vose and Jared Rennie, 2011: An overview of the Global Historical Climatology Network monthly mean temperature data set, version 3. Journal Geophysical Research, 116, D19121. https://doi.org/10.1029/2011JD016187

Lindau, Ralf and Victor Venema, 2018: On the reduction of trend errors by the ANOVA joint correction scheme used in homogenization of climate station records. International Journal of Climatology, 38, pp. 5255– 5271. Manuscript: https://eartharxiv.org/r57vf/, paywalled article: https://doi.org/10.1002/joc.5728

Menne, Matthew J., Claude N. Williams, Byron E. Gleason, Jared J. Rennie and Jay H. Lawrimore, 2018: The Global Historical Climatology Network Monthly Temperature Dataset, Version 4. Journal of Climate, 31, 9835–9854.
https://doi.org/10.1175/JCLI-D-18-0094.1

Williams, Claude, Matthew Menne and Peter Thorne, 2012: Benchmarking the performance of pairwise homogenization of surface temperatures in the United States. Journal Geophysical Research, 117, D05116. https://doi.org/10.1029/2011JD016761

Trend errors in raw temperature station data due to inhomogeneities

Another serious title to signal this is again one for the nerds. How large is the uncertainty in the temperature trends of raw station data due to inhomogeneities? Too Long, Didn’t Read: they are big and larger in America than in Germany.

We came up with two very different methods (Lindau and Venema, 2020) to estimate this and we got lucky: the estimates match as well as one could expect.

Direct method

Let’s start simple, take pairs of stations and compute their difference time series, that is, we calculate the difference between two raw temperature time series. For each of these difference series you compute the trend and from all these trends you compute the variance.

If you compute these variances for pairs of stations in a range of distance classes you get the thick curves in the figure below for the United States of America (marked as U) and Germany (marked as G). This was computed based on data for the period 1901 to 2000 from the International Surface Temperature Initiative (ISTI) .


Figure 1. The variance of the trend differences computed from temperature data from the United States of America (U) and for Germany (G). Only the thick curves are relevant for this blog post and are used to compute the trend uncertainty. The paper used Kelvin [K], we could also have used degree Celsius [°C] like we do in this post.

When the distance between the pairs is small (near zero on the x-axis) the trend differences are mostly due to inhomogeneities, whereas when the distances get larger also real climate trend differences increase the variance of the trend differences. So we need to extrapolate the thick curves to a distance of zero to get the variance due to inhomogeneities.

For the USA this extrapolation gives about 1 °C² per century². This is the variance due to two stations. If the inhomogeneities are assumed to be independent, one station will have contributed half and the trend variance of one station is 0.5 °C² per century². To get an uncertainty measure that is easier to understand for humans, you can take the square root, which gives the standard deviation of the trends: 0.71 °C per century.

That is a decent size compared to the total warming over the last century of about 1.5 °C over land; see estimates below. This alone is a good reason to homogenize climate data to reduce such errors.


Figure 2. Warming estimates of the land surface air temperature from four different institutions. Figure taken from the last IPCC report.

The same exercise for Germany estimates the variance to be 0.5 °C² per century², the square root of half this variance gives a trend uncertainty of 0.5 °C per century.

In Germany the maximum distance for which we have a sufficient number of pairs (900 km) is naturally smaller than for America (over 1200 km). Interestingly, for America also real trend differences are important, which you can see in the trend variance increasing for pairs of stations that are further apart. In Germany this does not seem to happen even for the largest distances we could compute.

An important reason to homogenize climate data is to remove network-wide trend biases. When these trend biases are due to changes that affected all stations, they will hardly be visible in the difference time series. A good example is a change of the instruments affecting an entire observational network. It is also possible to have such large-scale trend biases due to rare, but big events, such as stations relocating from cities to airports, leading to a greatly reduced urban heat island effect. In such a case, the trend difference would be visible in the difference series and would thus be noticed by the above direct method.

Indirect method

The indirect method to estimate the station trend errors starts with an estimate of the statistical properties of the inhomogeneities and derives a relationship between these properties and the trend error.

Inhomogeneities

How the statistical properties of the inhomogeneities are estimated is described in Lindau and Venema (2019) and my previous blog post. To summarize, we had two statistical models for inhomogeneities. One where the size of the inhomogeneity between two breaks is given by a random number. We called this Random Deviations (RD) from a baseline. The second model is for breaks behaving like Brownian Motion (BM). Here the jump sizes are determined by random numbers. So the difference is whether the levels or the jumps are random numbers.

We found in both countries RD break with a typical jump size of about 0.5 °C. But the frequency was quite different, while in Germany we had one break every 24 years, in America it was once every 5.8 years.

Furthermore, in America there are also breaks that behave like Brownian Motion. For these break we only know the variance of the break multiplied by the frequency of the breaks, this is 0.45 °C² per century². We do not know not whether the value is due to many small breaks or one big one.

Relationship to trend errors

The next step is to relate these properties of the inhomogeneities to trend errors.

For the Random Deviation case, the dependence on the size of the breaks is trivial, it is simply proportional, but the dependence on the number of breaks is quite interesting. The numerical relationship is shown with +-symbols in the graph below.

Clearly when there are no breaks, there is also no trend error. On the other extreme of a large number of breaks, the error is due to a large number of independent random numbers, which to a large part cancel each other out. The largest trend errors are thus found for a moderate number of breaks.

To understand the result we start with the case without any variations in the break frequency. That is, if the break frequency is 5 breaks per century, every single 100-year time series has exactly 5 breaks. For this case we can derive an equation shown below as the thick curve. As expected it starts at zero and the maximum trend error is in case of 2 breaks in the time series.

More realistic is the case when there is a mean break frequency over all stations, but the number of breaks varies randomly per station. In case the breaks are independent of each other one would expect the number of breaks to follow a Poisson distribution. The thin lines in the graph below takes this scatter into account by computing a weighted average over the equation using Poisson weights. This smoothing reduces the height of the maximum and shifts it to a larger average break frequency, about 3 breaks per time series. Especially for more than 5 breaks, the numerical and analytical solutions fit very well.


Figure 3. The relationship between the variance of the trend due to inhomogeneities and the frequency of breaks, expressed as breaks per century. The plus-symbols are the results based on numerical simulation for 100-years time series. The thick line is the equation we found for a fixed break frequency, while the thin line takes into account random variations in the break frequency.

The next graph, shown below, also includes the case of Brownian Motion (BM), as well as the Random Deviation (RD) case. To make the BM and RD cases comparable, they both have jumps following a normal distribution with a variance of 1 °C². Clearly the Brownian Motion case (with O-symbols) produces much larger trend errors than the Random Deviations case (+-symbols).


Figure 4. The variance of the trend as a function of the frequency of breaks for the two statistical models. The O-symbols are for Brownian Motion, the +-symbols for Random Deviations. The variance of the jump sizes was 1 °C² in both cases.

That the variance of the trends due to inhomogeneities is a linear function of the number of breaks can be understood by considering that to a first approximation the trend error for Brownian Motion is given by a line connecting the first and the last segment of the break signal. If k is the number of breaks, the value of the last segment is the sum of k random values. Thus if the variance of one break is σ_β², the variance of the value of the last segment is k*σ_β² and thus a linear function of the number of breaks.

Alternatively you can do a lot of math and at the end find that the problem simplifies like a high school problem and that the actual trend error is 6/5 times the simple approximation from the previous paragraph.

Give me numbers

The variance of the trend error due to BM inhomogeneities in America is thus 6/5 times 0.45 °C² per century², which equals 0.54 °C² per century².

This BM trend error is a lot bigger than the trend error due to the RD inhomogeneities, which for 17.1 breaks per century and a break size distribution with variance 0.12 °C² is 0.13 °C² per century².

One can add these two variances together to get 0.67 °C² per century². The standard deviation of this trend error is thus quite big: 0.82 °C per century and mostly due to the BM component.

In Germany, we found only RD breaks with a frequency of 4.1 breaks per century. Their size is 0.13 °C². If we put this into the equation, the variance of the trends due to inhomogeneities is 0.34 °C² per century². Although the size of the RD breaks is about the same as in America, their influence on the station trend errors is larger, which is somewhat counter-intuitively because their number is lower. The standard deviation of the trend due to inhomogeneities is thus 0.58 °C per century in Germany.

Comparing the two estimates

Finally, we can compare the direct and indirect estimates of the trend errors. For America the direct (empirical) method found a trend error of 0.71 °C per century and the indirect (analytical) method 0.82 °C per century. For Germany the direct method found 0.5 °C per century and the indirect method 0.58 °C per century.

The indirect method thus found slightly larger uncertainties. Our estimates were based on the assumption of random station trend errors, which do not produce a bias in the trend. A difference in sensitivity to such biasing inhomogeneities in the observational data would be a reasonable explanation for these small differences. Also missing data may play a role.

Inhomogeneities can be very complicated. The break frequency does not have to be constant, the break sizes could depend on the year. Random Deviations and Brownian Motion are idealizations. In that light, it is encouraging that the direct and indirect estimates fit that well. These approximations seem to be sufficiently realistic, at least for the computation of station trend errors.

Other posts in this series

Part 5: Statistical homogenization under-corrects any station network-wide trend biases

Part 4: Break detection is deceptive when the noise is larger than the break signal

Part 3: Correcting inhomogeneities when all breaks are perfectly known

Part 2: Trend errors in raw temperature station data due to inhomogeneities

Part 1: Estimating the statistical properties of inhomogeneities without homogenization

References

Lindau, R, Venema, V., 2020: Random trend errors in climate station data due to inhomogeneities. International Journal Climatology, 40, pp. 2393-2402. Open Access. https://doi.org/10.1002/joc.6340

Lindau, R, Venema, V., 2019: A new method to study inhomogeneities in climate records: Brownian motion or random deviations? International Journal Climatology, 39, p. 4769– 4783. Manuscript. https://eartharxiv.org/vjnbd/ https://doi.org/10.1002/joc.6105

On the recent warming surge

"Incidentally, when in the journal Science in 2007 we pointed to the exceptionally large warming trend of the preceding 16 years, which was at the upper end of the [climate] model range, nobody cared, because there is no powerful lobby trying to exaggerate global warming."

"And of course in our paper we named natural intrinsic variability as the most likely reason. But when a trend at the lower end of the model range occurs it suddenly became a big issue of public debate, because that was pushed by the fossil fuel climate sceptics’ lobby. There is an interesting double standard there."

Stefan Rahmstorf

Maybe the comment is deleted in shame. At least I cannot find it any more. Someone on Reddit was making exactly the same errors as the fans of the infamous "hiatus" to argue that global warming since 2011 is exploding and we're soon gonna die. Deleted comments can be freely paraphrased.

The data

So let's have a look at what the data actually says. Below are warming curves of two surface temperature datasets and two upper air temperature satellite retrievals. I shortened the warming curve of Berkeley Earth to match the period to the one of GISTEMP. All the other datasets are shown over their full lengths. Taking this step back, looking at the overview, there clearly is no "hiatus" and no "warming surge".

The red dots are the annual temperature anomalies, which are connected by a thin grey line. The long-term trend is plotted by a thick blue line, which is a [[LOESS]] estimate.

If you want to see the "hiatus" you have to think the last two data points away and start in the temperature peak of 1998. Don't worry, I'll wait while you search for it.

A hiatus state of mind

After seeing reality, let's now get into the mindset of a climate "sceptic" claiming to have evidence for a "hiatus", but do this differently by looking at the data since 2011.

Naturally we only plot the recent part of the data, so that context is lost. I am sure the climate "sceptics" do not mind, they also prefer to make their "hiatus" plots start around the huge 1998 El Nino warming peak and not show the fast warming before 1998 for context.




The thick blue lines are quadratic functions fitted to the data. They fit snugly. As you can see both the linear and quadratic coefficients are statistically significant. So clearly the recent warming is very fast, faster than linear and we can soon expect to cross the 2 °C limit, right?

I am sure the climate "sceptics" do not mind what I did. They also cherry picked a period for their "hiatus" and applied a naive statistical test as if they had not cherry picked the period at all. The climate "sceptics" agreeing with Christopher Monckton doing this on WUWT will surely not object to our Redditor doing the same and will conclude with him that the end is nigh.

By the way, I also cherry picked the dataset. The curves of the upper air temperature retrievals are not as smooth and the quadratic terms were not statistically significant. But in case of the "hiatus" debate our "sceptical" friends also only showed data for the datasets showing the least warming. So I am sure they do not object to this now.

If you look at the full period you can see that the variability of the temperature signal is much larger than the variability around the quadratic fit. It is thus clearly a complete coincidence that the curve is so smooth. But, well, the "hiatus" proponents also just look statistically at their cherry picked period and ignore the actual uncertainties (including slowly varying ones.)

Some more knowledgeable catastrophists may worry that it looks as if 2017 may not be another record scorching year. No worries. Also no worries if 2018 is colder again. The Global Warming Policy Foundation thinks it is perfectly acceptable to ignore a few politically inconvenient years and claims that we just have to think 2015 and 2016 away and that thus the death of the "hiatus" has been greatly exaggerated. I kid you not. I have not seen any climate "sceptic" or any "luckwarmer" complaining about this novel statistical analysis method, so surely our Redditor can do the same. Catastrophic warming surge claims are save till at least 2019.

Don't pick cherries

Back to reality. What to do against cherry picking periods? My advice would be: don't cherry pick periods. Not for the global mean temperature, not for the temperature of the Antarctic Peninsula, not for any other climate variable. Just don't do it.

If you have a physical reason to expect a trend change, by all means use that date as start of the period to compute a trend. But for 1998 or 2011 there is no reason to expect a trend change.

Our cheerful Redditor had even a bit more physics in his claim. He said that the Arctic was warming and releasing more carbon dioxide and methane. However, these emissions were not large enough to make the increase in the global greenhouse concentrations speed up. And they count. From our good-natured climate "sceptics" I have never heard a physical explanation why global warming would have stopped in 1998. But maybe I have missed their interest in what happens with the climate system.

If you have no reason to expect a trend change the appropriate test would be for a trend change test at an unknown date. Such a test "knows" that cherry picked periods can have hugely different trends and thus correctly only sees larger trend changes over longer periods as statistically significant. Applied to temperature data the result of such as test does not see any "hiatus" nor any "warming surge".

This test gave the right result during the entire "hiatus" madness period. Don't fool yourself. Use good stats.

Related reading

Stefan Rahmstorf Grant Foster and Niamh Cahill: Global temperature evolution: recent trends and some pitfalls. Environmental Research Letters, 2017. See also "Change points of global temperature".

Cranberry picking short-term temperature trends

Statistically significant trends - Short-term temperature trend are more uncertain than you probably think

How can the pause be both ‘false’ and caused by something?

Atmospheric warming hiatus: The peculiar debate about the 2% of the 2%

Statistically significant trends - Short-term temperature trend are more uncertain than you probably think

Yellowknife, Canada, where the annual mean temperature is zero degrees Celsius.

In times of publish or perish, it can be tempting to put "hiatus" in your title and publish an average article on climate variability in one of the prestigious Nature journals. But my impression is that this does not explain all of the enthusiasm for short-term trends. Humans are greedy pattern detectors: it is better to see a tiger, a conspiracy or trend change one time too much than one time too little. Thus maybe humans have a tendency to see significant trends where statistics keeps a cooler head.

Whatever the case, I expect that also many scientists will be surprised to see how large the difference in uncertainty is between long-term and short-term trends. However, I will start with the basics, hoping that everyone can understand the argument.

Statistically significant

That something is statistically significant means that it is unlikely to happen due to chance alone. When we call a trend statistically significant, it means that it is unlikely that there was no trend, but that the trend you see is due to chance. Thus to study whether a trend is statistically significant, we need to study how large a trend can be when we draw random numbers.

For each of the four plots below, I drew ten random numbers and then computed the trend. This could be 10 years of the yearly average temperature in [[Yellowknife]]*. Random numbers do not have a trend, but as you can see, a realisation of 10 random numbers appears to have one. These trends may be non-zero, but they are not significant.



If you draw 10 numbers and compute their trends many times, you can see the range of trends that are possible in the left panel below. On average these trends are zero, but a single realisation can easily have a trend of 0.2. Even higher values are possible with a very small probability. The statistical uncertainty is typically expressed as a confidence interval that contains 95% of all points. Thus even when there is no trend, there is a 5% chance that the data has a trend that is wrongly seen as significant.**

If you draw 20 numbers, 20 years of data, the right panel shows that those trends are already quite a lot more accurate, there is much less scatter.



To have a look at the trend errors for a range of different lengths of the series, the above procedure was repeated for lengths between 5 and 140 random numbers (or years) in steps of 5 years. The confidence interval of the trend for each of these lengths is plotted below. For short periods the uncertainty in the trend is enormous. It shoots up.



In fact, the confidence range for short periods shoots up so fast that it is hard to read the plot. Thus let's show the same data with different (double-logarithmic) axis in the graph below. Then the relationship look like a line. That shows that size of the confidence interval is a power law function of the number of years.

The exponent is -1.5. As an example that means that the confidence interval of a ten year trend is 32 (10^1.5) times as large as the one of a hundred year trend.



Some people looking at the global mean temperature increase plotted below claim to see a hiatus between the years 1998 and 2013. A few years ago I could imagine people thinking: that looks funny, let's make a statistical test whether there is a change in the trend. But when the answer then clearly is "No, no way", and the evidence shows it is "mostly just short-term fluctuations from El Nino", I find it hard to understand why people believe in this idea so strongly that they defend it against this evidence.

Especially now it is so clear, without any need for statistics, that there never was anything like an "hiatus". But still some people claim there was one, but it stopped. I have no words. Really, I am not faking this dear colleagues. I am at a loss.

Maybe people look at the graph below and think, well that "hiatus" is ten percent of the data and intuit that the uncertainty of the trend is only 10 times as large, not realising that it is 32 times.



Maybe people use their intuition from computing averages; the uncertainty of a ten year average is only 3 times as large that of a 100 year average. That is a completely different game.

The plots below for the uncertainty in the average are made in the same way as the above plots for the trend uncertainty. Also here more data is better, but the function is much less steep. Plots of power laws always look very similar, you need to compare the axis or the computed exponent, which in this case is only -0.5.





It is typical to use 30 year periods to study the climate. These so-called climate normals were introduced around 1900 in a time the climate was more or less stable and the climate needed to be described for agriculture, geography and the like. Sometimes it is argued that to compute climate trends you need at least 30 years of data, that is not a bad rule of thumb and would avoid a lot of nonsense, but the 30 year periods were not intended as a period on which to compute trends. Given how bad the intuition of people apparently is there seems to be no alternative to formally computing the confidence interval.

That short-term trends have such a large uncertainty also provides some insight into the importance of homogenisation. The typical time between two inhomogeneities is 15 to 20 years for temperature. The trend over the homogeneous subperiods between two inhomogeneities is thus very uncertain and not that important for the long-term trend. What counts is the trend of the averages of the homogeneous subperiods.

That insight makes you want to be sure you do a good job when homogenising your data rather than mindlessly assume everything will be alright and raw data good enough. Neville Nicholls wrote about how he started working on homogenisation:

When this work began 25 years or more ago, not even our scientist colleagues were very interested. At the first seminar I presented about our attempts to identify the biases in Australian weather data, one colleague told me I was wasting my time. He reckoned that the raw weather data were sufficiently accurate for any possible use people might make of them.

Sad.

[UPDATE: In part 2 of this series, I show how these large trend uncertainties in combination with the deceptive strategy of "cherry-picking" a specific period very easily produces a so-called "hiatus".]

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Related reading

How can the pause be both ‘false’ and caused by something?

Atmospheric warming hiatus: The peculiar debate about the 2% of the 2%

Sad that for Lamar Smith the "hiatus" has far-reaching policy implications

Temperature trend over last 15 years is twice as large as previously thought

Why raw temperatures show too little global warming

Notes

* In Yellowknife the annual mean temperature is about zero degrees Celsius. Locally the standard deviation of annual temperatures is about 1°C. Thus I could conveniently use the normal distribution with zero mean and standard deviation one. The global mean temperature has a much smaller standard deviation of its fluctuations around the long-term trend.
** Rather than calling something statistically significant and thus only communicating whether the probability was below 5% or not, it fortunately becomes more common to simply give the probability (p-value). In the past this was hard to compute and people compared their computation to the 5% levels given in statistical tables in books. With modern numerical software it is easy to compute the p-value itself.
*** Here is the cleaned R code to generated the plots of this post.

The photo of YellowKnife at the top is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Introduction to series on weather variability and extreme events

This is the introduction to a series on changes in the daily weather and extreme weather. The series discusses how much we know about whether and to what extent the climate system experiences changes in the variability of the weather. Variability here denotes the the changes of the shape of probability distribution around the mean. The most basic variable to denote variability would be the variance, but many other measures could be used.

Dimensions of variability

Studying weather variability adds more dimensions to our apprehension of climate change and also complexities. This series is mainly aimed at other scientists, but I hope it will be clear enough for everyone interested. If not, just complain and I will try to explain it better. At least if that is possible, we do not have much solid results on changes in the weather variability yet.

The quantification of weather variability requires the specification of the length of periods and the size of regions considered (extent, the scope or domain of the data). Different from studying averages is that the consideration of variability adds the dimension of the spatial and temporal averaging scale (grain, the minimum spatial resolution of the data); thus variability requires the definition of an upper and lower scale. This is important in climate and weather as specific climatic mechanisms may influence variability at certain scale ranges. For instance, observations suggest that near-surface temperature variability is decreasing in the range between 1 year and decades, while its variability in the range of days to months is likely increasing.

Similar to extremes, which can be studied on a range from moderate (soft) extremes to extreme (hard) extremes, variability can be analysed by measures which range from describing the bulk of the probability distribution to ones that focus more on the tails. Considering the complete probability distribution adds another dimension to anthropogenic climate change. Such a soft measure of variability could be the variance, or the interquartile range. A harder measure of variability could be the kurtosis (4th moment) or the distance between the first and the 99th percentile. A hard variability measure would be the difference between the maximum and minimum 10-year return periods.

Another complexity to the problem is added by the data: climate models and observations typically have very different averaging scales. Thus any comparisons require upscaling (averaging) or downscaling, which in turn needs a thorough understanding of variability at all involved scales.

A final complexity is added by the need to distinguish between the variability of the weather and the variability added due to measurement and modelling uncertainties, sampling and errors. This can even affect trend estimates of the observed weather variability because improvements in climate observations have likely caused apparent, but non-climatic, reductions in the weather variability. As a consequence, data homogenization is central in the analysis of observed changes in weather variability.