{"id":22929,"date":"2013-07-01T14:23:31","date_gmt":"2013-07-01T20:23:31","guid":{"rendered":"http:\/\/rankexploits.com\/musings\/?p=22929"},"modified":"2013-07-04T09:45:14","modified_gmt":"2013-07-04T15:45:14","slug":"bayesian-projection-of-rcp4-5","status":"publish","type":"post","link":"https:\/\/rankexploits.com\/musings\/2013\/bayesian-projection-of-rcp4-5\/","title":{"rendered":"Bayesian Projection of RCP4.5."},"content":{"rendered":"

I’ve been threatening<\/del> promising to post a Bayesian weighted projection based on the rcp4.5 models I’d downloaded from The Climate Explorer. Today, I’m going to explain how I concocted a projections and compare it to the projection I would obtain if I simply averaged over model means from models with more than 1 run in the forecast period. <\/p>\n

A model mean projection created by computing the multi-model mean over from models with more than 1 run in the rcp4.5 scenario is shown in the R color ‘tomato’ below. The solid line is the mean projection. The dashed line is the 2-sigma spread in the combined uncertainty of the model mean (of hypothetical models that might have been included in the set), uncertainty in which model is correct and the ‘model weather’ creating the spread in the runs for each model. Model runs are show in various ‘rainbow’ colors, with the higher trends in ‘warmer’ colors and lower trend in cooler colors.
\n\"MeanProjection\"<\/a><\/p>\n

One of the difficulties with the projections above is that we already know that:<\/p>\n

    \n
  1. Observed temperatures are lagging projections after 2000. This suggests the models may be running to warm but does not provide an explanation why. <\/li>\n
  2. The residuals to trends fit over 30 years are consistently high for models compared to observations. This suggests that at frequencies less than 30 years ‘weather noise’ in models is too large<\/i>. <\/li>\n<\/ol>\n

    Given this I wondered: What projections would I obtain if I assigned a weight by initially giving all models above equal weights, then computing relative probabilities of each models in the following way:<\/p>\n

      \n
    1. Assume that one of the models is “correct” and initially all models are assumed equally probable. This is my prior and will be written: Po<\/sub>(j=correct model) where the o subscript will indicate ‘prior’.<\/li>\n
    2. Divide the 90 year period ending in Dec. 2012 into three 30 year periods. These will be denoted with subscripts ‘k’ one for each period.<\/li>\n
    3. For period (a) the linear trend, mobs,k<\/sub> and (b) rmsobs,k<\/sub> of residuals for each of the ‘n’ runs for an observational data set. (I chose GISTemp here.) <\/li>\n
    4. For each model, j, compute the (a) rms of residuals for each of the ‘n’ runs. Compute the mean and standard deviation of the rms of residuals over these ‘n’ runs rmsmean,j<\/sub>, sdrms,j<\/sub>. Then assuming rms of residuals are normally distributed about the model mean, find the probability the that the observed<\/i> value would occur given<\/i> model ‘j’ is assumed correct. That is P(rmsobs,k<\/sub>|j=correct model) is the probability we would observed trend rmsobs,k<\/sub> during period ‘k’ if ‘j’ is the correct model. This probability is defined by my assumption the distributions would be Gaussian, the model mean rms for the periods, and the standard deviation of the rms values for runs (plus a slight addition to account for the finite size effect to to having a finite number of runs.)\n

      Note that given the prior<\/i>, one would expect the probability of observing rmsobs,k<\/sub> to have been the sum over all ‘j’ models of the product of observing this trend given<\/i> model ‘j’ is assumed true and the probability that model ‘j’ is thought to be true: <\/p>\n

      Po<\/sub>(rmsobs,k<\/sub>)= P(rmsobs,k<\/sub>|j=correct model) * Po<\/sub>(j=correct model)<\/indent><\/p>\n

      This provides a relative weighting for the model at this step. Using the relative weighting and Bayes law, I computed the posterior<\/I> probability that model “j=correct model” based on it’s agreement with ‘rms’ data during period ‘k’ as:<\/p>\n

      P1<\/sub>(j=correct model)
      \n= Po<\/sub>(j=correct model) * [ P(rmsobs,k<\/sub>|j=correct model) \/ Po<\/sub>(rmsobs,k<\/sub>) ] <\/indent><\/p>\n

      I then repeated this for all ‘k’ periods each time using the posterior from the previous trend as the current prior.<\/p>\n

      This step tended to pick out models with higher-than-30 year frequency ‘weather noise’ was closest to that in the observations. (As a practical matter, it tended to pick out models with runs that had less<\/i> ‘noise’ than the typical model.)\n <\/li>\n

    5. After weighting by ‘noise, I repeated the above procedure but using trends<\/i> during each of the three periods. I cpmputed (a) the linear trend ( mmean,j<\/sub>, sdm,j<\/sub>) over these ‘n’ runs, assumed the distribution of trends was Gaussian. <\/li>\n<\/ol>\n

      When this process was completed, I had a posterior distribution of models giving the probability<\/i> each model was “the” correct model with the weighting based on agreement with GISTemp based on the criterion defined above. <\/p>\n

      Once completed, I plotted out the new projection, which is shown in blue below:<\/p>\n

      \"ReweightedProjection\"<\/a>
      \nThe more probable models with weights given on a scale of (1-100) are indicated. Others have been dropped off the graph. As you can see, the rescaled trends are lower; I think this is principally due to the fact that the final observed 30 year trends are much lower than the data so models with the lowest trend are given higher probabilities. <\/p>\n

      I’m not going to discuss this too much because I suspect people are going to want to note key features of the analysis more than the actual results. Two very important features are:<\/p>\n

        \n
      1. The Bayesian projection is clearly affected by the prior<\/i> chosen to describe which models we thought probable in the first place. In this case, one prior assumption is that the models I am including in the analysis do<\/i> contain the “truth”. If you do not believe the models in anyway shape of form, the Bayesian projections ought not to be seen as “better” (even if you ‘like’ it ‘better’).<\/li>\n
      2. The shift in the posterior is clearly affected by which observational data sets we used to compute transition probabilities for the priors. My choice was dictated by wanting to use metrics based on surface temperature — which is what I want to predict. Of these, I wanted to select easy to compute metrics. I used the agreement with the observations of a metric whose magnitude is affected by ‘weather noise at frequencies less than 30 years’ and the agreement with ’30 year trends’ in surface temperatures. Other people could pick other metrics (e.g. ability to predict tropical cloudiness). There are many metrics one could pick and I might note there is a vast choice to pick ( or cherry pick ) from! <\/li>\n<\/ol>\n

        For now: The blue curve is “Lucia’s Bayesian for rcp4.5”. Will it pan out? Beats me! <\/p>\n

        Update<\/b>: (July 4, 2013. Place holder so I remember relevance! )Saw this paper on Twitter<\/a>: Climate Models, Calibration and Confi\frmation<\/a>. I need to read further in light of thoughts I had while doing above calculation and thinking about comparing my “Bayesian” optimum from the models to “no warming” models and so on. <\/p>\n","protected":false},"excerpt":{"rendered":"

        I’ve been threatening promising to post a Bayesian weighted projection based on the rcp4.5 models I’d downloaded from The Climate Explorer. Today, I’m going to explain how I concocted a projections and compare it to the projection I would obtain if I simply averaged over model means from models with more than 1 run in … Continue reading Bayesian Projection of RCP4.5.<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"class_list":["post-22929","post","type-post","status-publish","format-standard","hentry","category-data-comparisons"],"_links":{"self":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/22929","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/comments?post=22929"}],"version-history":[{"count":0,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/22929\/revisions"}],"wp:attachment":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/media?parent=22929"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/categories?post=22929"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/tags?post=22929"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}