## 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 km Ave T Ave s Ave e Var s Var e 100 0.772 0.7623 0.0097 1.339 0.809 200 0.772 0.7544 0.0176 0.991 0.536 400 0.772 0.7387 0.0333 0.715 0.653 800 0.772 0.7114 0.0606 0.528 0.942

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 km Ave T Ave s Ave |e| Var s Var e 100 NA NA 0.03 1.509 0.807 200 NA NA 0.017 1.1 0.515 400 NA NA 0.018 0.844 0.529 800 NA NA 0.029 0.583 0.72

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 km Ave T Ave s Ave e Var s Var e 100 0.5298 0.5466 -0.0168 1.963 0.91 200 0.5298 0.527 0.0028 1.577 0.697 400 0.5298 0.5208 0.009 1.199 0.794 800 0.5298 0.523 0.0068 0.846 1.171
Feb 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.3437 0.3316 0.0121 2.107 0.958 200 0.3437 0.3293 0.0145 1.779 0.743 400 0.3437 0.3238 0.02 1.385 0.912 800 0.3437 0.3145 0.0292 0.958 1.345
Mar 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.5431 0.5103 0.0328 2.295 0.719 200 0.5431 0.5424 6e-04 1.911 0.54 400 0.5431 0.5487 -0.0056 1.614 0.561 800 0.5431 0.5316 0.0115 1.274 0.746
Apr 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.6574 0.686 -0.0286 1.748 0.793 200 0.6574 0.667 -0.0096 1.144 0.502 400 0.6574 0.6649 -0.0075 0.884 0.508 800 0.6574 0.6607 -0.0033 0.565 0.574
May 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.6947 0.6791 0.0156 0.978 0.65 200 0.6947 0.6763 0.0184 0.628 0.336 400 0.6947 0.6609 0.0338 0.415 0.356 800 0.6947 0.619 0.0757 0.181 0.566
Jun 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.6206 0.5621 0.0585 0.942 0.908 200 0.6206 0.5773 0.0433 0.604 0.638 400 0.6206 0.5981 0.0225 0.303 0.524 800 0.6206 0.6 0.0206 0.135 0.651
Jul 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.5149 0.5288 -0.0139 1.049 0.554 200 0.5149 0.5325 -0.0176 0.734 0.311 400 0.5149 0.5293 -0.0144 0.519 0.3 800 0.5149 0.5448 -0.0299 0.278 0.468
Aug 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.6472 0.562 0.0852 1.121 1.255 200 0.6472 0.5982 0.049 0.634 0.829 400 0.6472 0.6274 0.0198 0.358 0.63 800 0.6472 0.6117 0.0355 0.214 0.762
Sep 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.7023 0.69 0.0123 0.669 0.696 200 0.7023 0.6836 0.0186 0.514 0.586 400 0.7023 0.6877 0.0146 0.385 0.446 800 0.7023 0.6538 0.0485 0.182 0.547
Oct 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.6676 0.6574 0.0101 1.051 0.631 200 0.6676 0.6565 0.0111 0.718 0.38 400 0.6676 0.661 0.0066 0.517 0.354 800 0.6676 0.6781 -0.0105 0.295 0.512
Nov 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.5662 0.6156 -0.0494 1.533 0.76 200 0.5662 0.586 -0.0197 1.16 0.506 400 0.5662 0.5656 6e-04 0.929 0.47 800 0.5662 0.5549 0.0114 0.766 0.608
Dec 2014
 r km Ave T Ave s Ave e Var s Var e 100 0.6809 0.6271 0.0538 1.217 0.605 200 0.6809 0.6528 0.0281 0.972 0.458 400 0.6809 0.6561 0.0248 0.786 0.467 800 0.6809 0.6597 0.0211 0.556 0.609
Jan 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.6348 0.6584 -0.0237 1.656 0.778 200 0.6348 0.6441 -0.0093 1.2 0.479 400 0.6348 0.6494 -0.0147 0.922 0.526 800 0.6348 0.6526 -0.0179 0.742 0.733
Feb 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.6892 0.6984 -0.0092 2.601 0.645 200 0.6892 0.6934 -0.0042 2.385 0.457 400 0.6892 0.6821 0.007 2.176 0.52 800 0.6892 0.6596 0.0296 1.998 0.807
Mar 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.7019 0.6582 0.0437 1.505 0.734 200 0.7019 0.6768 0.0252 1.175 0.502 400 0.7019 0.6921 0.0098 0.914 0.519 800 0.7019 0.6849 0.017 0.699 0.664
Apr 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.6292 0.6742 -0.0451 1.439 1.18 200 0.6292 0.6337 -0.0045 0.651 0.559 400 0.6292 0.6298 -6e-04 0.403 0.581 800 0.6292 0.63 -9e-04 0.25 0.699
May 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.6188 0.6683 -0.0495 1.215 0.937 200 0.6188 0.6268 -0.0081 0.841 0.463 400 0.6188 0.6285 -0.0098 0.574 0.499 800 0.6188 0.6639 -0.0451 0.298 0.693
Jun 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.664 0.7307 -0.0667 1.23 1.002 200 0.664 0.6838 -0.0198 0.661 0.377 400 0.664 0.6764 -0.0124 0.512 0.369 800 0.664 0.6853 -0.0212 0.343 0.467
Jul 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.613 0.6099 0.003 1.106 0.527 200 0.613 0.6324 -0.0194 0.811 0.434 400 0.613 0.6489 -0.0359 0.527 0.408 800 0.613 0.6589 -0.0459 0.301 0.517
Aug 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.7039 0.6745 0.0295 0.964 0.411 200 0.7039 0.6898 0.0141 0.686 0.286 400 0.7039 0.6957 0.0082 0.54 0.311 800 0.7039 0.7158 -0.0119 0.363 0.408
Sep 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.6776 0.6985 -0.0209 1.115 0.36 200 0.6776 0.7039 -0.0263 0.949 0.272 400 0.6776 0.7056 -0.028 0.785 0.327 800 0.6776 0.7099 -0.0323 0.554 0.454
Oct 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.9294 0.9234 0.006 1.161 0.552 200 0.9294 0.886 0.0434 0.784 0.438 400 0.9294 0.8829 0.0466 0.588 0.416 800 0.9294 0.8587 0.0707 0.421 0.626
Nov 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.8963 0.9049 -0.0086 1.017 0.653 200 0.8963 0.8886 0.0077 0.748 0.522 400 0.8963 0.8737 0.0226 0.559 0.553 800 0.8963 0.8393 0.057 0.389 0.719
Dec 2015
 r km Ave T Ave s Ave e Var s Var e 100 0.9697 1.0136 -0.0439 2.748 0.844 200 0.9697 0.9941 -0.0244 2.401 0.605 400 0.9697 0.9805 -0.0108 2.088 0.629 800 0.9697 0.9731 -0.0034 1.642 0.938
Jan 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.8963 0.8685 0.0279 2.518 1.273 200 0.8963 0.8857 0.0106 1.932 0.868 400 0.8963 0.8805 0.0158 1.407 0.904 800 0.8963 0.8494 0.0469 0.734 1.318
Feb 2016
 r km Ave T Ave s Ave e Var s Var e 100 1.0767 1.1111 -0.0344 2.9 1.191 200 1.0767 1.0993 -0.0226 2.366 0.801 400 1.0767 1.0896 -0.0129 1.918 0.774 800 1.0767 1.0796 -0.0029 1.398 1.048
Mar 2016
 r km Ave T Ave s Ave e Var s Var e 100 1.0271 1.0178 0.0094 2.249 0.996 200 1.0271 1.0424 -0.0152 1.585 0.563 400 1.0271 1.0514 -0.0243 1.291 0.569 800 1.0271 1.0528 -0.0257 1.049 0.763
Apr 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.9384 0.9661 -0.0277 1.569 0.655 200 0.9384 0.9496 -0.0113 1.155 0.435 400 0.9384 0.9416 -0.0032 0.874 0.494 800 0.9384 0.9334 0.0049 0.624 0.683
May 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.7556 0.8262 -0.0706 1.48 1.28 200 0.7556 0.7504 0.0052 0.709 0.498 400 0.7556 0.7277 0.0279 0.503 0.466 800 0.7556 0.7011 0.0545 0.302 0.559
Jun 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.6923 0.7515 -0.0591 2.176 1.242 200 0.6923 0.6904 0.0019 1.095 0.464 400 0.6923 0.6597 0.0326 0.801 0.68 800 0.6923 0.6961 -0.0038 0.349 0.817
Jul 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.6621 0.6974 -0.0353 0.704 0.508 200 0.6621 0.7011 -0.039 0.515 0.368 400 0.6621 0.6934 -0.0313 0.42 0.331 800 0.6621 0.7153 -0.0531 0.196 0.515
Aug 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.8616 0.855 0.0066 0.83 0.389 200 0.8616 0.8639 -0.0023 0.676 0.293 400 0.8616 0.8654 -0.0038 0.482 0.361 800 0.8616 0.8292 0.0324 0.226 0.626
Sep 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.7263 0.7026 0.0236 1.008 0.452 200 0.7263 0.7165 0.0097 0.754 0.371 400 0.7263 0.7229 0.0034 0.495 0.398 800 0.7263 0.7336 -0.0073 0.273 0.523
Oct 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.6964 0.7353 -0.039 1.651 1.195 200 0.6964 0.6821 0.0143 0.887 0.524 400 0.6964 0.666 0.0304 0.62 0.563 800 0.6964 0.6753 0.0211 0.368 0.77
Nov 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.6914 0.685 0.0065 1.826 1.037 200 0.6914 0.6719 0.0196 1.48 0.785 400 0.6914 0.6507 0.0407 1.168 0.729 800 0.6914 0.6293 0.0622 0.814 0.916
Dec 2016
 r km Ave T Ave s Ave e Var s Var e 100 0.6598 0.6391 0.0207 1.135 0.758 200 0.6598 0.6368 0.0231 0.89 0.617 400 0.6598 0.6343 0.0256 0.663 0.661 800 0.6598 0.6322 0.0276 0.451 0.876
Jan 2017
 r km Ave T Ave s Ave e Var s Var e 100 0.772 0.7623 0.0097 1.339 0.809 200 0.772 0.7544 0.0176 0.991 0.536 400 0.772 0.7387 0.0333 0.715 0.653 800 0.772 0.7114 0.0606 0.528 0.942