August SEARCH Outlook Contribution

Here is my next contribution:

Pan Arctic Outlook

Christopher Randles

Public outlook, statistical method

1.      Extent Projection  4.438 + (End of July CT area - 4.383) * 0.9176 m km^2

+/- 95% confidence interval of 0.8 m   

2.      Method – single linear, single non-linear regression

A gompertz fit of the NSIDC September extent figures is used as a starting point.

Linear regression is then used to predict the residual from the Gompertz fit using the residual of the end of July 2011 Cryosphere Today area number from a gompertz fit of end of July area numbers.

3.      Rationale

Several contributors have used multiple linear regression. This felt inadequate when there appears to be a curved shape that other contributors have used quadratic, exponential, logistic or gompertz fits to approximate.

For the July outlook I used both area and volume data to predict the residual. For this outlook, PIOMAS volume data does not appear to improve prediction of the residual significantly and has been dropped.

 

Other predictor variables may well exist to further reduce the error but I have not found any significantly useful data sets. Area that has a thickness that could be melted is one obvious likely improvement over just using PIOMAS volume numbers and Lindsay and Zhang use such data.

The following did not appear to be significantly useful:

 
A scatter plot suggests a fairly linear response to area residuals:

 

Hamilton's Contribution used a gompertz fit and yielded an estimate of 4.438m km^2. This prediction updates that prediction with an area based prediction. If area differs from the gompertz fit value of 4.383 then extent is predicted to be different by the regression factor of 0.9176 times the difference.

I have not seen anyone attempting any sort of multiple non-linear regression and this approach which de-trends the extent and area data in a non-linear manner prior to a linear regression to predict the residual in the extent that we are trying to estimate appears to be one way to do that.

4.      Executive Summary

The data appears to have a curved shape which it appears advantageous to recognise and adapt multiple linear regression to predicting the residuals from the curved shape which has been approximated using a Gompertz fit.  – See Hamilton's Contribution. This model yields an average September extent prediction of X.X m km^2 with a 95% confidence interval in the region of +/- 0.8m (though RMSE is as low as 0.28m).

5.      Estimate of Forecast Skill

A 95% confidence interval of +/- 0.8 m is calculated though there are mixed indicators over whether this might understate the uncertainty. This estimate is higher than the inappropriately tuned RMSE figures of as low as 0.28m.

The RMSE of estimates reduces as follows:

Linear regression of September average extent =0.508 m

Gompertz fit of September average extent = 0.438 m

Gompertz fit then linear regression prediction of residual with CT area residual from gompertz fit = 0.282 m

Note however that these RMSE numbers are likely to underestimate the likely error as they have the advantage of the method being tuned with data that cannot be available at the time of making a true prediction.

Removing that advantage

Year	Gompertz	Adjusted Gompertz	Actual	Error
1991	6.8956	6.8076	6.55	-0.2576
1992	6.7332	6.9470	7.55	0.6030
1993	6.8981	6.4133	6.5	0.0867
1994	6.7476	7.1558	7.18	0.0242
1995	6.8010	6.3916	6.13	-0.2616
1996	6.5989	7.6012	7.88	0.2788
1997	6.8246	6.4639	6.74	0.2761
1998	6.7712	6.4440	6.56	0.1160
1999	6.6921	6.7547	6.24	-0.5147
2000	6.4976	6.2863	6.32	0.0337
2001	6.3397	6.4042	6.75	0.3458
2002	6.4390	6.5182	5.96	-0.5582
2003	6.1693	6.2445	6.15	-0.0945
2004	6.0600	6.2112	6.05	-0.1612
2005	5.9505	5.7487	5.57	-0.1787
2006	5.6486	5.4538	5.92	0.4662
2007	5.6232	4.8907	4.3	-0.5907
2008	4.7365	4.8002	4.68	-0.1202
2009	4.3707	4.5991	5.36	0.7609
2010	4.5460	4.3898	4.9	0.5102

Average absolute error                                    0.312 m

RMSE without tuning to unavailable data     0.378 m

A 95% confidence interval is calculated at +/- 0.8 m. All 20 errors are less than this suggesting that 0.8 m may be more than necessary.

However, the average of the absolute errors for the first 10 year is only 0.245 m whereas the average in the last 10 years is higher at 0.378 m. So there may be some growth in the expected size of errors and therefore a 95% credible interval may need to be higher than +/- 0.8 m.

Using exponential fits instead of Gompertz fits yielded a minor improvement in the fits giving a RMSE of 0.2814 instead of 0.282 but I prefer to stick with the Gompertz fit in case this is needed for a levelling off in the rate of decrease which could occur.

In the format

 

The regression factors and data are

Multiple Regression Factors – Area
0.917599	0.00023
0.140987	0.051496
0.585401	0.291307
42.35909	30
3.594582	2.545792

6.      Projection based on August data

Unless improvements are found, Average September Extent projection =

 = 4.438 + (End August CT Area – 3.124 ) * 1.1284 + (NSIDC Aug Extent – 5.916) * 0.323

m km^2 +/- unfairly tuned RMSE of 0.23 m.

7.      Invitation to discuss

Comments on this method or the error estimate or comparing different methods or error estimates between different contributions are welcome. I suggest such discussion could be useful be done at Neven’s blog. The latest appropriate post being

http://neven1.typepad.com/blog/2011/07/august-search-outlook-contribution.html

Comments

[crandles]

With the century break reported earlier for the 26th, area at 4.452 is not much above the gompertz fit for end of the month of 4.383. So there seems time for the area to not only get below the gompertz fit but also lose another 250k or so area so that the prediction comes in just below the 2007 record.

This method does not account for weather so there is still a large margin for error.

[Neven]

Great stuff, crandles!

[Artful Dodger]

CT SIA was 4.452 M km^2 for July 25, 2011 (year.frac = 2011.5643)

SIA for July 26 has not been reported yet.

[crandles]

I am using the data for the day ending .5781 so there are 5 more days to report. How do you get .5643 to be 25th Jul?

[Jon Torrance]

Lodger,

So the CT SIA value associated with 2011.0000 in their table is the one for December 31, 2010 and CT typically reports SIA not the day after they measure it but approximately two days later, while posting their concentration imagery with only a one day delay? Not utterly ridiculous but certainly counterintuitive. Are you basing this on personal communication with the CT people or is all this explained somewhere online I haven't noticed yet?

[L. Hamilton]

Regarding conversion from the decimal dates that CT uses to more comprehensible elapsed dates, here's a piece of code using Stata's date functions that might be adapted to other systems.

In this snippet, "date" refers to decimal date, and all other variables are defined from that. "yearlength" figures out whether it's a leap year or not (based on what doy, day of year, Dec 31 calculates to be), "year" is just year, and "edate" is the elapsed date -- number of days since Jan 1 1960, then formatted so that 11894 (=2011.5643) shows up as 25jul2011.

gen year = floor(date)
gen yearlength = doy(mdy(12,31,year))
gen doy = max(round((date-year*yearlength,1),1)
gen edate= mdy(1,1,year) + doy -1
format edate %d
label variable edate "Elapsed date"

[Artful Dodger]

Gas Glow, Jon, Lord Soth, and other Citizens of the Blog...

First, look at the top of the CT SIA table. The very first line begins with a year.frac value: 1979.0027

Understand that year.frac does not specify a day, it specifies an instant in time, in this example 00:00 hrs on Jan 2, 1979.

The time period covered by each line in the CT table is bounded by the time given on the previous line. So the previous line is 2011.5616 which is Jul 25 2011 00:00

Then the line beginning 2011.5644 is for data collected during the 24-hr period we commonly refer to as July 25, 2011.

Thanks Larry for the Stata code snippet. This can also be calculated in Excel with something like:

=ROUND(YEARFRAC(A2;A3;1);4)
where
A2 = Jan 01 2011 00:00
A3 = Jul 26 2011 00:00
the result is .5644

I believe FrankD came to this realization earlier this year by comparing daily SI concentration charts between CT and IJIS. Perhaps Frank, Peter Ellis, or Neven can explain it better than me. I them to comment.

In addition (and separate from this explanation), CT uses NASA AMSR-E level 3 data, which has not been released yet for July 26, 2011. This is Public information provided through their website. I have explained in private communications to Neven why I decline to link to the data here, but there is already ample evidence showing how CT data uses the year.frac function.

Cheers,
Lodger

[crandles]

OK thanks for the explanations.

[Jon Torrance]

"I believe FrankD came to this realization earlier this year by comparing daily SI concentration charts between CT and IJIS. Perhaps Frank, Peter Ellis, or Neven can explain it better than me. I them to comment."

Couldn't hurt. I'm unclear as to what the daily concentration charts from CT and IJIS have to do with this, assuming that means what can currently be seen at http://arctic.atmos.uiuc.edu/cryosphere/NEWIMAGES/arctic.seaice.color.000.png and http://www.ijis.iarc.uaf.edu/cgi-bin/seaice-monitor.cgi?lang=e&mode=img&size=L&date=set&y=2011&m=07&d=26.

[Artful Dodger]

The ice doesn't match up between CT and IJIS unless you adjust the CT dates. Frank noticed this earlier this year.

[Artful Dodger]

I dug out Frank's original comment, which was posted May 21, 2011 at 01:09

I think there's something not-quite-right with Cryosphere Today's map. I don't see this: http://rapidfire.sci.gsfc.nasa.gov/realtime/single.php?A111391740 from May 19th reflected on their site at all. The break up in HB is clearly more advanced than their maps or graphs show. I imagine we'll catch up soon.

You can see effect when comparing CT to IJIS charts as well. So the CT date issue applies to their ice concentration charts and SIA graphs/tables.

[Artful Dodger]

Peter doesn't seem to be around today, but he had this to say on April 06, 2011 at 09:31

Ah hang on, they start their numbering at zero, so you have to add a day. That makes it the April 3rd value, as you say.

[Jon Torrance]

Peter's problem of not having initially noticed that the table contained a row starting with 2011.0000 isn't really relevant to the issue here. It's somewhat unfortunate, if your interpretation is correct on all points, that CT labels an image based on observations taken on July 25th "07/26/2011". Even more unfortunate that they don't publish more metadata generally.

That said, given that JAXA's extent numbers are based on observations from two days averaged together, it's not clear to me that it much matters which of the two days in question the SIA input to a CAPIE calculation comes from. Though ideally, assuming you're correct to think the CT SIA numbers are each based on one day's worth of observations, I suppose one would use the average SIA for the two days in question in the calculation.

[Neven]

After that last discussion I redid my CAPIE spreadsheet, and I guess it's fine now. I don't think it's a huge mistake if it's one day off. Just make sure you make the same mistake for every year so you're still comparing rotten apples to rotten apples, right?

I once ran a restaurant and one of the waitresses would put the fork on the right side of one plate, and on the left side of another. The sharp edges of the knives pointed to the left here, to the right there. I said to her that she didn't have to do it my way (which happened to be the right way), as long as she did it the same everywhere.

It's important to be consistent. Even when you make mistakes. :-)

[Artful Dodger]

It's possible to use 2-day avg CT data for CAPIE, but even more interesting to use 1-day data for IJIS SIE.

[Neven]

In addition (and separate from this explanation), CT uses NASA AMSR-E level 3 data, which has not been released yet for July 26, 2011. This is Public information provided through their website. I have explained in private communications to Neven why I decline to link to the data here, but there is already ample evidence showing how CT data uses the year.frac function.

I have indeed seen the site, and Lodger is right. CT is a bit behind IJIS when it comes to releasing data. So for instance, the +26K that was reported today by CT was for July 26th, corresponding to the -55K from IJIS, and not today's -73K.

[crandles]

The .5781 year-fraction data is now available as 4.271 m km^2. This is .112 below the gompertz fit of 4.383. So the prediction works out to 0.9176*0.112 below the gompertz fit of 4.438 which calculates to 4.34 rounded to 4.3 m km^2.

[crandles]

If you want the next week data to form some sort of target: The Gompertz fit for 0.6 year-fraction data (8 days' data time = 7th August) is 3.965 m km^2.

The regression factor is 1.093.

[L. Hamilton]

Regarding the SEARCH SIO, this year I'm going with a "let it ride" theory. My August prediction will be the same one I had in April, 4.4.
http://neven1.typepad.com/blog/2011/04/trends-in-arctic-sea-ice-extent.html

[crandles]

>"let it ride"

But that backtracking on what your executive summary said
"Refinements to this predict should be possible when we have further information, as the 2011 melt season progresses."

Only kidding ;o) Sorry if I have pinched your intended refinements.

4.4 may well be more likely to be more accurate than my 4.3. I just don't have a data set that I think I can use for expected weather.

It did vaguely cross my mind that we could try to start one by counting comments on this blog that indicate weather will be good for small extent reductions minus posts on this blog indicating weather will be good for large extent reductions. Unfortunately I think it may take a long time time to see if this was any use.

[L. Hamilton]

It's true, earlier I had expected that I would refine that April prediction. Last year I was fiddling with new models right up to the deadline. Tried some of that this year and didn't come up with anything that made me think "Wow!" On the other hand, the Gompertz idea had appealing simplicity. So I figure, what the heck, let's see how close or how wrong that first guess will be.

[crandles]

OK we now have 7 Aug area at 3.695. As mentioned earlier, the Gompertz fit for 7 Aug (yearfraction 0.6) is 3.965 so area is .27 below the gompertz fit.

Using only this CT area data the prediction would be 4.438-1.093*.27 = 4.142

However, I now also have the GFSC-IJIS melded daily extent data which I can use in addition to area.

The gompertz fit for 7th August is 6,187,305. This is suprisingly higher than the 7 Aug 2007 value.

The 7 Aug IJIS extent is 6,174,375 which is marginally below the gompertz fit. So using this record alone would predict less than 4.438 by .894*.0109 = 4.427

Using both the prediction is 4.438-.602*.27-.053*.0109 =4.271 m km^2

So all three of these predictions are well within the uncertainty range of the 2007 minimum. It could very easily go either way.

[crandles]

In the report due soon, Paragraph 6 should be changed to:

6. Projection based on August data

Unless improvements are found, Average September Extent projection =
= 4.438+(IJIS JAXA End Aug Extent–4.824)*.9083+(End Aug CT Area–3.124)*0.3785
m km^2 +/- unfairly tuned RMSE of 0.14 m.
The IJIS JAXA daily extent record is only short starting in 2002. Thanks to Lucia and blog readers the NASA GFSC daily extent record here was made known to me. I have attempted to create a homogenous daily extent record by applying a step change to the GFSC data. The data and calculations are made available here. This appears to work well and is a better guide to NSIDC average extent than CT daily area at the end of August. At the end of July, both CT area and GFSC-JAXA daily extent are significant but CT area is a better guide than this daily extent record. There is a large overlap of information between these two as using both only performed better than area and 8 out of 10 sets of random numbers. So there does not appear to be a significant advantage to using this GFSC-JAXA daily extent record at 31 July and it has not been used.

[crandles]

Target information for 15th August:

Gompertz fit of CT area for 15th Aug is 3.623; we are nearly down to that already!

Gompertz fit of GFSC-IJIS is 5.824.

Prediction is 4.438+(CT Area for 15/8 - 3.623)*0.9787 + (IJIS for 15/8 - 5.824)*0.296

[crandles]

The August report is out

http://www.arcus.org/search/seaiceoutlook/2011/august

[Seke Rob]

Re: crandles | August 09, 2011 at 16:39

Hey there,

GFSC or GSFC... got data from latter. Was there a link meant where you say

The data and calculations are made available here

Thanks

[crandles]

The meld data and calcs should be at

https://docs.google.com/spreadsheet/ccc?key=0AjpGniYbi4andElTczhVS2t2VW5Ka0sySnFrcndTTkE&hl=en_US#gid=0

Sorry about that and delay in answering.

[Seke Rob]

Good stuff, thx Chris R.

[crandles]

Following my recent comments at Lucia's blog

Lucia's vunerable ice = extent-area may not be a useful predictor.

I am now thinking of calculating a weighted average of extent and area for the last few available days. Deducting the September minimum and this is what I will try to predict. I might predict it using area or the weighted average of extent and area or I might try some sort of weighted average cumulative area since May 1 or perhaps later.

Just wondering what weightings to use for the cumulative area. A sine wave that peaks and troughs on the soltices or is there some more complex formula to weight the energy capture by ocean rather than ice?

Or should it be more weighted to recent areas?

Thoughts anyone?

[crandles]

So it is now clear that the 3.437 is a revision to 15/8 area. Using that and extent of 5.588 for 15/8, the gompertz fit adjustment method comes out predicting 4.318 m km^2.

If I predict the fall from a weighted average of area and extent for the 3 days to 15/8 using the weighted average as the predictor then I get a prediction of 4.538+/-0.2

If I wanted to average these I would want to give more weight to the 4.538 prediction so I end up with about 4.45.

This averaged prediction is indistinuishable from Larry's 4.438. 'Let it ride' seems to be working well so far.

[L. Hamilton]

One cute thing about my black box approach is that it applies with equal simplicity (simplistic-ness?) to any of the ice indexes. So I've also got estimates made in April for September volume, area, and 3 kinds of extent ... it will be interesting to see which if any of those turn out close.