Friday, March 17, 2017

Temperature residuals and coverage uncertainty.

A few days ago I posted an extensive ANOVA-type analysis of the successive reduction of variance as the spatial behaviour of global temperatures was more finely modelled. This is basically a follow-up to show how the temperature field can be partitioned into a smooth part with known reliable interpolation, and a hopefully small residue. Then the size of the residue puts a limit on the coverage uncertainty.

I wrote about coverage uncertainty in January. It's the uncertainty about what would happen if one could measure in different places, and is the main source of uncertainty in the monthly global indices. A different and useful way of seeing it is as the uncertainty that comes with interpolation. Sometimes you see sceptic articles decrying interpolation as "making up data". But it is the complement of sampling, which is how we measure. You can only measure anything at a finite number of places. You infer what happens elsewhere by interpolation; that can't be avoided. Just about everything we know about the physical world, or economic for that matter, is deduced from a finite number of samples.

The standard way of estimating coverage uncertainty was used by Brohan et al 2006. They took a global reanalysis and sampled at sets of places correponding to possible station distributions. The variability of the resulting averages was the uncertainty estimate. The weakness is that the reanalysis may have different variability to the real world.

I think analysis of residuals gives another way. If you have a temperature anomaly field T, you can try to separate it into a smoothed part s and a residual e:
T = s + e
If s is constructed in such a way that you expect much less uncertainty of interpolation than T, then the uncertainty has been transferred to e. That residual is meor intractable to integrate, but you have an upper bound based on its amplitude, and that is an upper bound to coverage uncertainty.

So below the jump, I'll show how I used a LOESS type smoothing for s. This replaces points but a low-order polynomial weighted regression, and the weighting is by a function decaying with distance, in my case exponentially, with characteristic distance t (ie exp(-|x}/r). With r very high, one can be very sure of interpolation (of s), but the approximation will not be very good, so e will be large, and contains a lot of "signal" - ie what you want to include in the average, which will then be inaccurate. If the distance is very small, the residual will be small too, but there will be a lot of noise still in s. I seek a compromise where s is smooth enough, and e is small enough. I'll show the result of various r values for recent months, focussing on Jan 2017. I'll also show WebGL plots of the smooths and residuals.

I should add that the purpose here is not to get a more accurate integral by this partition. Some of the desired integrand is bound to end up in e. The purpose is to get a handle on the error.

I'll use quadratic LOESS; the reason is that for SST at least there are regions which are otherwise smooth but have curvature on the desired r range which the quadratic can fit. I am forming the integrals with weights from TempLS mesh; these weights depend on geometry only, not on the integrand.

I'll first show the results for January 2017 as a table:

Jan 2017
r kmAve TAve sAve eVar sVar e
1000.7720.76230.00971.3390.809
2000.7720.75440.01760.9910.536
4000.7720.73870.03330.7150.653
8000.7720.71140.06060.5280.942


r is the decay constant of the LOESS weighting; The averages are the respective averages of T and its partitioned components. I have also included the variances of s (not very meaningful, being not random), and e. It shows a minimum variance of e at 200km, with 400km not far behind. This level of smoothing seems to give the best fit. Below that, it just gets noisier; the LOESS makes more noise than it saves.However, the integral ("Ave e") continues to go down, since e becomes more random, so there is more cancellation. But the important thing is that in the mid-range it is uite small, and s does well approximate the integral. This is encouraging, because 400km is quite a good smoothing range - for most of the land plot, you would not expect s to have much interpolation error. This would not be true for some extremes, say mid-Africa.

I'll now show the averages over the 37 months from Jan 2014:

Averages 2014-Jan 2017
r kmAve TAve sAve |e|Var sVar e
100NANA0.031.5090.807
200NANA0.0171.10.515
400NANA0.0180.8440.529
800NANA0.0290.5830.72


I have rubbed out ave T and s, since they aren't meaningful for this analysis. But Ave |e| is the important figure, and says that, if you accept that LOESS smoothing will remove at least a large part of the interpolation error from T, and transfer it to e, then that error is small, of order 0.02°C. This is quite an interesting result, because error on a monthly reading is normally reckoned to be about 0.1°C. That includes other things, but still, on this basis it seem quite a bit lower.

I'll show the WebGL plot of smooths and residuals for Jan 2017, and then the full table of the 37 months



Finally, here is the big table of results for each of the 37 months:

Jan 2014
r kmAve TAve sAve eVar sVar e
1000.52980.5466-0.01681.9630.91
2000.52980.5270.00281.5770.697
4000.52980.52080.0091.1990.794
8000.52980.5230.00680.8461.171
Feb 2014
r kmAve TAve sAve eVar sVar e
1000.34370.33160.01212.1070.958
2000.34370.32930.01451.7790.743
4000.34370.32380.021.3850.912
8000.34370.31450.02920.9581.345
Mar 2014
r kmAve TAve sAve eVar sVar e
1000.54310.51030.03282.2950.719
2000.54310.54246e-041.9110.54
4000.54310.5487-0.00561.6140.561
8000.54310.53160.01151.2740.746
Apr 2014
r kmAve TAve sAve eVar sVar e
1000.65740.686-0.02861.7480.793
2000.65740.667-0.00961.1440.502
4000.65740.6649-0.00750.8840.508
8000.65740.6607-0.00330.5650.574
May 2014
r kmAve TAve sAve eVar sVar e
1000.69470.67910.01560.9780.65
2000.69470.67630.01840.6280.336
4000.69470.66090.03380.4150.356
8000.69470.6190.07570.1810.566
Jun 2014
r kmAve TAve sAve eVar sVar e
1000.62060.56210.05850.9420.908
2000.62060.57730.04330.6040.638
4000.62060.59810.02250.3030.524
8000.62060.60.02060.1350.651
Jul 2014
r kmAve TAve sAve eVar sVar e
1000.51490.5288-0.01391.0490.554
2000.51490.5325-0.01760.7340.311
4000.51490.5293-0.01440.5190.3
8000.51490.5448-0.02990.2780.468
Aug 2014
r kmAve TAve sAve eVar sVar e
1000.64720.5620.08521.1211.255
2000.64720.59820.0490.6340.829
4000.64720.62740.01980.3580.63
8000.64720.61170.03550.2140.762
Sep 2014
r kmAve TAve sAve eVar sVar e
1000.70230.690.01230.6690.696
2000.70230.68360.01860.5140.586
4000.70230.68770.01460.3850.446
8000.70230.65380.04850.1820.547
Oct 2014
r kmAve TAve sAve eVar sVar e
1000.66760.65740.01011.0510.631
2000.66760.65650.01110.7180.38
4000.66760.6610.00660.5170.354
8000.66760.6781-0.01050.2950.512
Nov 2014
r kmAve TAve sAve eVar sVar e
1000.56620.6156-0.04941.5330.76
2000.56620.586-0.01971.160.506
4000.56620.56566e-040.9290.47
8000.56620.55490.01140.7660.608
Dec 2014
r kmAve TAve sAve eVar sVar e
1000.68090.62710.05381.2170.605
2000.68090.65280.02810.9720.458
4000.68090.65610.02480.7860.467
8000.68090.65970.02110.5560.609
Jan 2015
r kmAve TAve sAve eVar sVar e
1000.63480.6584-0.02371.6560.778
2000.63480.6441-0.00931.20.479
4000.63480.6494-0.01470.9220.526
8000.63480.6526-0.01790.7420.733
Feb 2015
r kmAve TAve sAve eVar sVar e
1000.68920.6984-0.00922.6010.645
2000.68920.6934-0.00422.3850.457
4000.68920.68210.0072.1760.52
8000.68920.65960.02961.9980.807
Mar 2015
r kmAve TAve sAve eVar sVar e
1000.70190.65820.04371.5050.734
2000.70190.67680.02521.1750.502
4000.70190.69210.00980.9140.519
8000.70190.68490.0170.6990.664
Apr 2015
r kmAve TAve sAve eVar sVar e
1000.62920.6742-0.04511.4391.18
2000.62920.6337-0.00450.6510.559
4000.62920.6298-6e-040.4030.581
8000.62920.63-9e-040.250.699
May 2015
r kmAve TAve sAve eVar sVar e
1000.61880.6683-0.04951.2150.937
2000.61880.6268-0.00810.8410.463
4000.61880.6285-0.00980.5740.499
8000.61880.6639-0.04510.2980.693
Jun 2015
r kmAve TAve sAve eVar sVar e
1000.6640.7307-0.06671.231.002
2000.6640.6838-0.01980.6610.377
4000.6640.6764-0.01240.5120.369
8000.6640.6853-0.02120.3430.467
Jul 2015
r kmAve TAve sAve eVar sVar e
1000.6130.60990.0031.1060.527
2000.6130.6324-0.01940.8110.434
4000.6130.6489-0.03590.5270.408
8000.6130.6589-0.04590.3010.517
Aug 2015
r kmAve TAve sAve eVar sVar e
1000.70390.67450.02950.9640.411
2000.70390.68980.01410.6860.286
4000.70390.69570.00820.540.311
8000.70390.7158-0.01190.3630.408
Sep 2015
r kmAve TAve sAve eVar sVar e
1000.67760.6985-0.02091.1150.36
2000.67760.7039-0.02630.9490.272
4000.67760.7056-0.0280.7850.327
8000.67760.7099-0.03230.5540.454
Oct 2015
r kmAve TAve sAve eVar sVar e
1000.92940.92340.0061.1610.552
2000.92940.8860.04340.7840.438
4000.92940.88290.04660.5880.416
8000.92940.85870.07070.4210.626
Nov 2015
r kmAve TAve sAve eVar sVar e
1000.89630.9049-0.00861.0170.653
2000.89630.88860.00770.7480.522
4000.89630.87370.02260.5590.553
8000.89630.83930.0570.3890.719
Dec 2015
r kmAve TAve sAve eVar sVar e
1000.96971.0136-0.04392.7480.844
2000.96970.9941-0.02442.4010.605
4000.96970.9805-0.01082.0880.629
8000.96970.9731-0.00341.6420.938
Jan 2016
r kmAve TAve sAve eVar sVar e
1000.89630.86850.02792.5181.273
2000.89630.88570.01061.9320.868
4000.89630.88050.01581.4070.904
8000.89630.84940.04690.7341.318
Feb 2016
r kmAve TAve sAve eVar sVar e
1001.07671.1111-0.03442.91.191
2001.07671.0993-0.02262.3660.801
4001.07671.0896-0.01291.9180.774
8001.07671.0796-0.00291.3981.048
Mar 2016
r kmAve TAve sAve eVar sVar e
1001.02711.01780.00942.2490.996
2001.02711.0424-0.01521.5850.563
4001.02711.0514-0.02431.2910.569
8001.02711.0528-0.02571.0490.763
Apr 2016
r kmAve TAve sAve eVar sVar e
1000.93840.9661-0.02771.5690.655
2000.93840.9496-0.01131.1550.435
4000.93840.9416-0.00320.8740.494
8000.93840.93340.00490.6240.683
May 2016
r kmAve TAve sAve eVar sVar e
1000.75560.8262-0.07061.481.28
2000.75560.75040.00520.7090.498
4000.75560.72770.02790.5030.466
8000.75560.70110.05450.3020.559
Jun 2016
r kmAve TAve sAve eVar sVar e
1000.69230.7515-0.05912.1761.242
2000.69230.69040.00191.0950.464
4000.69230.65970.03260.8010.68
8000.69230.6961-0.00380.3490.817
Jul 2016
r kmAve TAve sAve eVar sVar e
1000.66210.6974-0.03530.7040.508
2000.66210.7011-0.0390.5150.368
4000.66210.6934-0.03130.420.331
8000.66210.7153-0.05310.1960.515
Aug 2016
r kmAve TAve sAve eVar sVar e
1000.86160.8550.00660.830.389
2000.86160.8639-0.00230.6760.293
4000.86160.8654-0.00380.4820.361
8000.86160.82920.03240.2260.626
Sep 2016
r kmAve TAve sAve eVar sVar e
1000.72630.70260.02361.0080.452
2000.72630.71650.00970.7540.371
4000.72630.72290.00340.4950.398
8000.72630.7336-0.00730.2730.523
Oct 2016
r kmAve TAve sAve eVar sVar e
1000.69640.7353-0.0391.6511.195
2000.69640.68210.01430.8870.524
4000.69640.6660.03040.620.563
8000.69640.67530.02110.3680.77
Nov 2016
r kmAve TAve sAve eVar sVar e
1000.69140.6850.00651.8261.037
2000.69140.67190.01961.480.785
4000.69140.65070.04071.1680.729
8000.69140.62930.06220.8140.916
Dec 2016
r kmAve TAve sAve eVar sVar e
1000.65980.63910.02071.1350.758
2000.65980.63680.02310.890.617
4000.65980.63430.02560.6630.661
8000.65980.63220.02760.4510.876
Jan 2017
r kmAve TAve sAve eVar sVar e
1000.7720.76230.00971.3390.809
2000.7720.75440.01760.9910.536
4000.7720.73870.03330.7150.653
8000.7720.71140.06060.5280.942




2 comments:

  1. How do I subscribe to your blog? The subscribe to (atom) link isn't working.

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    1. I don't know. It's a Blogger gadget, and I have no apparent way to influence it. I think the RSS address is "http://feeds.feedburner.com/Moyhu".

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