{"id":16652,"date":"2011-08-27T16:25:57","date_gmt":"2011-08-27T22:25:57","guid":{"rendered":"http:\/\/rankexploits.com\/musings\/?p=16652"},"modified":"2011-08-27T16:25:57","modified_gmt":"2011-08-27T22:25:57","slug":"connelly-dekker-bet-actually-robs-got-a-very-good-chance-of-not-losing","status":"publish","type":"post","link":"https:\/\/rankexploits.com\/musings\/2011\/connelly-dekker-bet-actually-robs-got-a-very-good-chance-of-not-losing\/","title":{"rendered":"Connelly-Dekker Bet: Actually…. Rob’s got a very good chance of not losing."},"content":{"rendered":"

Very stupidly, I wrote a whole post on an NH ice extent bet without noticing I’d read in December volume<\/i> data instead of September extent minimum data. This is the rewrite. Without the stupid blunder, Rob has a pretty good chance of winning and an even better chance of not losing.<\/b><\/p>\n

I wasn’t planning on blogging about the Connolley-Dekker Ice Bet. But, the topic came up in comments<\/a>, and Rob Dekker wrote:<\/p>\n

Lucia, regarding the Connolley-Dekker (yep, that\u00e2\u20ac\u2122s me) bet, you write :<\/p>\n

RD\u00e2\u20ac\u2122s bet is well outside my current 95% uncertainty intervals for prediction. Connolley is well inside the uncertainty intervals.<\/p>\n

I\u00e2\u20ac\u2122m curious, did you already present the model that you used that defined these statements, and if not, can you please present it ?<\/p><\/blockquote>\n

Answers: I had not<\/i> explained my method for estimating who was going to win that bet, and I’m not even sure my current<\/I> method will match whatever method I was using at the time I wrote that. I can present the current model for predicting who will win the Connolley-Dekker bet. RD is unlikely to lose. (Why did I say he was in comments earlier? I don’t know. I know it wasn’t the same mistake as the whole post because I hadn’t yet downloaded PIOMAS data at the time I wrote the comment. I may have mis-read the bet thinking it was for 2011. Anyway, this should explain to Rob how I do<\/I> try to estimate odds based on extrapolating data when I actually do it. )
\n
\nWeighted model<\/b>
\nAs some are aware, I have taken to using a “weighted model” approach to obtaining the best estimate for a future outcome based on past data. The general method is discussed here<\/a>. For this post, I am doing a “tweak” which is to ignore deleting models based on lack of statistical significance of the fitting parameters. <\/p>\n

Partial application of weighted prediction method to Connolley-Dekker Bet<\/b><\/p>\n

The Connolley-Dekker Bet is described here:<\/a><\/p>\n

If both NSIDC and IARC-JAXA September 2016 monthly average sea ice extent report are above 4.80 million km^2, RD pays WMC US$ 10,000. If both are below 3.10 million km^2, WMC pays RD US$ 10,000. In all other cases the bet is null and void<\/p><\/blockquote>\n

Note it is based on IARC-JAXA and NSIDC. To shorten discussion, I’m going to present analysis of NSIDC only (particularly as I haven’t done it for JAXA.) Based on the statement of the bet, my understanding is that RD will pay WMC if the NSIDC and JAXA Sept. extent average fall above the upper grey dashed horizontal like in 2016. WMC will pay RD if lower grey dashed horizontal in 2016. If the extent falls between these two lines, neither gentleman looses or wins any money. <\/p>\n

\"\"<\/a><\/p>\n

I’ve highlighted 2016 with a vertical black line which is grey and a bit heavier in the “no win” region between the upper and lower grey dashed lines. Data are shown with open circles.<\/p>\n

To assess who I think is going to win, picked a whole bunch<\/I> of candidate models. In this particular instance, I used the “modified picking out of a hat” method. Based on general physics, I anticipate time<\/I> will be a good proxy for the warming effect of GHG. So, I include time in the regressors. I have no idea what else to pick, so that’s it. I then arbitrarily selected first through 5th order polynomials and a Gompretz fit as candidate models.(Those who think there may be a slowly oscillating component in the data will recognize that their favorite candidate model is missing.) <\/p>\n

For purposes of discussion, I’ll show how each model fits the data starting with the linear regression:<\/p>\n

\"\"<\/a><\/p>\n

The linear fit is shown in red. The solid line indicates the “best fit” value. The upper and lower dashed curves indicate the upper and the lower 95% confidence intervals assuming the linear model is correct, but including the uncertainty in our fitting parameters. <\/p>\n

Note the mean value for the linear fit (solid red slanted) lines just above the upper<\/i> horizontal dashed grey line 2016. If we assume residuals to the linear fit are gaussian (which they may not be) this would mean that if the linear fit was “true”, Rob would have roughly 1\/2 a chance of having to fork money over the William; the estimated value of 52% is shown in the red message in the lower left of the figure above. Meanwhile the chance of the ice falling in the “no one wins” regions is estimated at approximately 48% and the chance WC would need to fork money over to RD is 0.2%. <\/p>\n

But suppose we evaluate probabilities using a quadratic fit. <\/p>\n

\"\"<\/a><\/p>\n

If we believe the quadratic fit, the most likely outcome in 2016 is for the September ice extent to fall in the “no win” zone. If I assume the residuals to the fit are Gaussian, I estimate the chance Rob will have to pay WC $10,000 is 4%. Meanwhile, the chance WC will have to pay Rob is 23.5%.<\/p>\n

It should be readily apparent now that the choice of model has a huge impact on the estimate of the odds. If someone thinks the linear line is “true”, they will be expecting a 50% chance Rob loses $10,000, a nearly 50% chance no one forks any money over to anyone and a very small chance WC forks over $10,000 to Rob. The chance Rob loses $10,000 is 4% and he’s got a 23% chance of winning the money. It should be readily apparent now that the choice of model has a huge impact on the estimate of the odds. <\/p>\n

But which model is better?<\/p>\n

Using the eyeball method, the quadratic fit appears “better” than the linear fit. The “goodness” is confirmed by noting the quadratic term in the quadratic fit is statistically significant and comparing the corrected Akaike values (AICs), which indicate if the only two possible models are linear or quadratic, the quadratic model much more likely than the linear model. <\/p>\n

Now let’s look at cubic and 4th order polynomials:
\n\"\"<\/a><\/p>\n

The cubic is shown in mint green. If this<\/I> is the correct model, WC has a 61% chance of having to fork money over to Rob while Rob has a 4% chance of having to fork over money to WC. Under the quartic fit, Rob has a 48% chance of winning $10,000 while WC has a 26% chance. <\/p>\n

Supposedly. But it would be reasonable for someone to question the likelyhood that either of these fits is true because the some of the fit parameters for these regressions are not statistically significant. Ordinarily, I filter models with fit coefficients that are not statistically significant out of my weighted model because I think there is risk of introducing too much noise. But today, I’m going to keep them. With respect to evaluating the Dekker\/Connelley bet, the main effect is to over state the probability that money is exchanged and understate the probability that no money will be exchanged. It may also make Rob a bit too confident he will win– but he’s already made the bet. If he wants me to recalculate throwing these out, I can do that. \ud83d\ude42 <\/p>\n

My final fit was Gompretz which at least has the virtue that the best fit curve never falls below zero ice:<\/p>\n

\"\"<\/a><\/p>\n

If this model is “true”, and I assume residuals are Gaussian, Rob has a 3.5% chance of forking money over to WC while WC has a 27.5% chance of forking money over to Rob.<\/p>\n

Which model is best?<\/b>
\nI wish I could suggest which model is best based on phenomenology<\/i>, but the fact is I don’t know. That’s why I picked predictive models based on “modified picking out of a hat”. This method is dangerous. One danger is when used to extrapolate– as I am doing here– candidate models picked this way are likely to make Jeff Id’s head explode.<\/a> However, if I were betting, I’d rather at least look at what extrapolation suggest rather than not looking at it. In that context, we will ignore the head exploding potential of the polynomial fits and decide which looks “best” based on the AICc criteria based on past data. <\/p>\n

The AICc criteria and associated weights determined based on the AIC criteria are shown below: <\/p>\n

<\/p>\n\n\n\n\n\n\n\n
name <\/td>\n AICc <\/td>\n weights<\/td>\n<\/tr>\n
linear<\/td>\n 53.93372 <\/td>\n 1.6%<\/td>\n<\/tr>\n
quadratic<\/td>\n 47.70328 <\/td>\n 35.5% <\/td>\n<\/tr>\n
cubic <\/td>\n 49.36520 <\/td>\n 15.4%<\/td>\n<\/tr>\n
fourth <\/td>\n 49.36520 <\/td>\n 3.9%<\/td>\n<\/tr>\n
Gompretz <\/td>\n 49.36520 <\/td>\n 43.6%<\/td>\n<\/tr>\n<\/table>\n

<\/center><\/p>\n

According to the weights, if we assume this set of models contains the full set that “might” be “true”, the Gompretz<\/I> fit is the most likely model and has a 43.6% chance of being “true”.<\/p>\n

Applying my weights, I obtain a best fit model illustrated by the solid black curve below:<\/p>\n

\"\"<\/a><\/p>\n

The dashed black curves show the \u00c2\u00b195% confidence intervals assuming probability distribution of errors around the mean curve are gaussian. Under this assumption, the probability Rob will ow WC $10,000 is 8.5% Bear in mind: This lower interval was computed assuming that probability distribution function for the weighted<\/i> model is normally distributed. This makes some sort of sense if we think of the candidate models as being samples drawn from a population of all possible “sane” models. <\/p>\n

We could also interpret the weighted model literally: that is, we could assume there really are only 5 possible candidate models. In this case, even if the pdf of each candidate model is normally, the pdf for the weighted model is not<\/I> normal. However, under the assumption the residuals from each candidate model are nomally, I can find the probability Rob must pay WC $10,000 by weighting the probability he looses under each candidate by the weight of that model. Doing it this way, Rob has a 5.4% chance of losing $10,000, meanwhile he has a 31.5% chance of winning. <\/p>\n

Either way, it seems Rob has a very good chance of not losing any money. WC has enough of a chance of losing he should be a bit worried. The most likely outcome is no one wins.<\/p>\n

Should anyone be contemplating side bets, they should bear in mind: though I presented this, I have not identified all possible plausible “candidate models”. Including outer models might change the outcome, but I can’t include them unless someone suggest a model form that might make sense. Using models with phenomenological support would be preferable to just looking at a range of polynomial fits. It may also be that considering modeling residuals with ARMA would make a difference. I haven’t looked, so I don’t know. <\/p>\n

Another point to consider: we are approaching the 2011 extent minimum, and that value is likely to fall below level that would have been predicted by the best fit curve shown here. If this calculation is repeated in October, Rob’s chance of winning will look better than it does today. <\/p>\n

Owing to my blunder, many reasonable comments are already on the other thread. But I think this post shows more detail, and may clarify my general procedure to people. <\/p>\n

Now, it’s Saturday night. We’re having pizza. Hope your doing something fun too! <\/p>\n","protected":false},"excerpt":{"rendered":"

Very stupidly, I wrote a whole post on an NH ice extent bet without noticing I’d read in December volume data instead of September extent minimum data. This is the rewrite. Without the stupid blunder, Rob has a pretty good chance of winning and an even better chance of not losing. I wasn’t planning on … Continue reading Connelly-Dekker Bet: Actually…. Rob’s got a very good chance of not losing.<\/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":[49,15],"tags":[],"class_list":["post-16652","post","type-post","status-publish","format-standard","hentry","category-betting","category-data-comparisons"],"_links":{"self":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/16652","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=16652"}],"version-history":[{"count":0,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/16652\/revisions"}],"wp:attachment":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/media?parent=16652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/categories?post=16652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/tags?post=16652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}