In “the litrachure” some fit proxy values to a regional value (i.e. Northern Hemispheric Temperature, or Southern Hemispheric Temperature) others fit proxy values to an estimate of the temperature near the proxy itself. This choice can affect potential bias and noisiness, so to explores this, my synthetic experiments will use two types of proxies called “global” and “local”. <\/p>\n M “Global” proxies $latex P_{G,i} $ will be generated from a ‘known’ global temperature baselined to the calibration period (which is the item we really wish to reconstruct) using an equation of this form:<\/p>\n where T_{G} represents the ‘global’ temperature which will have an imposed known value and $latex w_{G,i} $ is gaussian white noise with standard deviation 1 and s_T is the sample standard deviation of the temperature over the calibration period. In the current exercise, our goal is to examine how well different methods reproduce this global temperature outside the calibration periods so this temperature will also be referred to as the “Target” temperature. <\/p>\n In the current synthetic experiments all<\/I> global proxies will share the same value of $latex \\lambda_{G,i} = \\lambda_{G} $ which will be 0.25 computed over the calibration period<\/I> of N years. <\/p>\n The second type of proxy will be generated based on “local” temperatures using a very similar equation.<\/p>\n where $latex T_{L,i} $ represents the ‘local’ temperature at proxy location ‘i’ and $latex w_{L,i} $ is gaussian white noise with standard deviation 1. (Note: the sample standard deviation of these proxies will equal the sample standard deviation of the local temperature. However: These s.d. at proxy ‘i’ will not<\/i> match that at proxy ‘j’ This will be important later. I would also suggest that this is the right<\/i> way to weight for what I call “method 1” later on. An alternate way that I think may sometimes be done would be totally c*appy.)<\/p>\n The local temperature $latex T_{L,i} $ will be generated such that on average<\/em>, the correlation between the local<\/i> temperature $latex T_{L,i} $ and the global (or target) temperature $latex T_{G} $ is 0.7. (Note 0.25\/0.7 = 0.357. However, each local temperature will also differ from the global (or target) temperature. I think it’s useful to clarify this with this graph showing the synthetic temperature that will be used:<\/p>\n The global (i.e. target) temperature is illustrated in red. This temperature will be a constant for 80 years and then exhibit a ramp up for 80 years. Each expected value of local temperature at a proxy will have a qualitatively similar shape, but with a different trend during the final 80 years; these are shown with dashed blue lines. The actual temperature at each proxy will be generated by adding noise to the expected value– these actual temperatures are showing in grey. The point of all of this is that the average temperature at proxy ‘i’ $latex T_{L,i} $ will differ from the average temperature at proxy ‘j’ when i\u00e2\u2030\u00a0j.<\/p>\n Before proceeding: In case you haven’t guessed, the final 80 years will be the calibration periods; the first 80 years will be the reconstruction period. <\/p>\n In the current synthetic experiments each proxies, i, will be assigned unique<\/em> values of $latex \\lambda_{L,i} $. These values will be drawn from a Normal distribution with mean $latex E[\\lambda_{L}] = 0.357 $ and standard deviation of $latex \\sigma_{L} =0.1$. <\/p>\n Moreover, to show a potential bias in some of the methods of creating proxies, I will assign the $latex \\lambda_{L,i} $ such that they are negatively correlated with the Given the distribution of $latex \\lambda_{G} $, $latex \\lambda_{L,i} $, $latex T_{G} $, $latex T_{L,i} $ $latex P_{G,i} $ and $latex P_{L,i} $ above, I will use three basic methods of estimating the temperature in during the first 80 ‘years’ based on calibrating proxies to temperature over the final 80 ‘years’ in the figure above. Each basic method will then be ‘tweaked’! <\/p>\n The basic methods are:<\/p>\n Results!<\/b> I am going to let most images speak for themselves. (I’ll admit some of the later ones will require some discussion– but I want to let people look at them and interpret them for themselves first.)<\/p>\n Fit global proxies to global temperatures<\/b> Reconstructions using Methods I-III with trends obtained fitting global proxies to global<\/i> temperature and eliminating those proxies whose “R” values were not significant to at the 95% level. (Note this eliminates roughly 40% of proxies.) Fit to Local Temperatures<\/b> (Note: I think method II is noisier than previously because now the individual proxies display a range<\/i> of responsiveness to temperature.)<\/p>\n There is a lot going on in the graph above. But I’m going to yank my fingers away from the keyboard and let people have fun looking at it and explaining the offsetting biases. That will help people understand just how pesky picking the best method to create the reconstruction might be! <\/p>\n The code: HockeyStick_Lines_July9<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" I continue to be rather interested in the possible ways one might create a reconstruction of historic temperatures based on a time series of temperatures T measured over a finite calibration period with N years and a finite number of proxies M that are thought to respond temperature. There appear to be an assortment of … Continue reading Methods: Suppose we had 5000 proxies.<\/span>
\nwhere $latex \\lambda_{i} $ is a function of proxy ‘i’. T is some temperature. <\/p>\n![\"\"](\"http:\/\/rankexploits.com\/musings\/wp-content\/uploads\/2012\/07\/SyntheticTemperatures.png\" "\\"SyntheticTemperatures\\"")
<\/a><\/p>\naverage<\/del> temperature trend for a at proxy ‘i’ during the calibration period set to -0.5. What this means is that in the figure above, the proxy whose temperature corresponds to the lowest dark blue trace is likely<\/i> to have a higher<\/del> lower than average value of $latex \\lambda_{L,i} $ while the proxy whose temperature corresponds to the highest blue trace is likely<\/i> to have the lowest<\/del> highest value of $latex \\lambda_{L,i} $. Language corrected because I forgot the script sets the correlation based on the trend<\/em>, not the value during the reconstruction period. See blog comments. <\/font><\/p>\n\n
\nI’ll now show reconstructions using M=5000 proxies. This number is selected to demonstrate what we might hypothetically get if we have a a fairly short calibration periods (~80 years) but could somehow– quite miraculously– be able to obtain many, many proxies. (I’d use even more but I don’t want to take 10 minutes to generate a graph!) Reality is we are stuck with very few proxies. But for now, my goal is to merely see whether the results of a method would converge on the correct value if only<\/i> we could get enough decent proxies. (Bo will want me to look at fewer proxies and I can do that afterwards.)<\/p>\n
\nReconstructions using Methods I-III with all trends obtained using fits to global<\/i> temperature (i.e. the Target) and applying no proxies removed for low correlation:<\/p>\n<\/a> <\/p>\n
\n<\/a>
\n(Note: the rms and average difference in the legend are based on comparison between the reconstruction and the target<\/I> during the reconstruction period only.)<\/p>\n
\nReconstructions using Methods I-III with trends obtained fitting local proxies to local<\/i> temperature:
\n<\/a><\/p>\n<\/a> <\/p>\n