Land snow cover effect on sea ice

[Ned W]

I suspect this might feel like a bit of a piling-on here.  Sorry, Rob, for starting it off -- but actually all these criticisms mean that a whole bunch of us think that your model is interesting and thought-provoking and we want to see more!  That is great. 

Having said that, I reiterate that using the standard error of the model (per Steven's comment up-thread) is technically not sufficient to represent the uncertainty of future predictions based on the model.  The prediction interval will be slightly wider.  Not a lot wider, probably, but slightly.  It doesn't really matter for posts on ASIF.  But if you're publishing predictions from it, it would be more ... elegant ... to use the prediction interval. 

Note also that the width of this will depend on the values of the independent variables that contribute to the prediction.  So the PI will be narrower when you are making a prediction based on "inputs" that are within the range of past values, and it will be wider when making predictions based on conditions that are more extreme than in the past (i.e., if land snow cover is much higher or lower than in the years that were used to develop the model, the prediction for that year will have a wider PI, because there is more uncertainty about model behavior when extrapolating.)

Maybe this suggestion is just me being too finicky.  But having been yelled at by angry statisticians in the past has made me somewhat skittish about stuff like that.

[Ned W]

I think it is great that others are digging into data triggered by Rob Dekkers model but, if you change the graph, you should also change the title.

That's an interesting philosophical problem because it raises the question of what exactly is a model?  Is it the numerical values of the predictions, or is it the process that leads to those predictions? 

What Steven has done is exactly what I would have done myself if he hadn't beat me to it.  He's essentially using the same conceptual model that Rob uses, but with only the data that were available in the past, to see what the "predictions" would have looked like without the omniscience of knowing how that future would turn out.  For example, using only pre-2009 data in the model to make a "prediction" for 2010.

Rob himself does this post-2016 -- he only used 1992-2015 data in developing the model, so 2016 and 2017 were genuinely independent "predictions".  Steven is essentially using the same process to see what that would look like if the training data used to develop the model had stopped in 2014, or 2013, or 2012,  etc.  It's a smart thing to do.

Anyway, people mean a variety of different things when they use the word "model".  It wouldn't be unusual for someone to give a talk in which they said "I took Rob Dekker's model for Arctic sea ice and applied it in the Antarctic ..."  If they also ran a bunch of other models, and then were trying to refer to the output of one vs the other, they could easily say "The Dekker model provided the best results..." meaning it as shorthand for "The model of Antarctic sea ice based on the concepts and processes from Rob Dekker's Arctic sea ice model"

I'm not disagreeing with SH's comment, just taking this as a springboard for (a) complimenting Steven on what he's done, and (b) musing about the meaning of the word "model" and how that affects how people are credited/blamed for use of their model by others.

[cesium62]

I think it is great that others are digging into data triggered by Rob Dekkers model but, if you change the graph, you should also change the title.

What title would you give to each of the two graphs?

[Hyperion]

A few points that may have been missed. More snow cover adjacent to the basin means bigger outflows from the rivers on melt. While this in the past may have acted to refresh the pycnocline, as we enter a regime of more mobile ice, more turbulent weather, and more Pacific and Atlantic waters available for it to mix with over continental shelves, it may have the opposite effect by bridging the halocline differential with saltier waters beneath. Also it supplies more nutrients for algae growth under the thin ice. These algae grow up into the porous young ice, vastly enhancing bottom melt by trapping insolation in the ice base that would otherwise go deep in the clear water, directly melting the ice through their metabolic heat, and releasing antifreeze compounds that reduce the temperature st which the ice melts, and the energy required to do it.
With the washing machine action we are seeing induced by the volatile weather, these nutrients are being spread over a much vaster area.

[Steven]

Overfitting a model based on year, snow area, and ice area to all data from 1979 through 2017, we get the second attached picture.  And a forecast of 4.58 M km2 for the 2018 min extent

I couldn't reproduce that calculation: it gives me 4.99 M km2 rather than 4.58. 

Are you sure your input data are correct?  For example if you downloaded the monthly sea ice area data from the NSIDC website, those should be corrected for varying "pole hole" sizes:

https://nsidc.org/the-drift/data-update/sea-ice-index-updated-with-a-new-arctic-pole-hole-and-residual-weather-masks/

It is getting late here (Western Europe), but I'll just post the input data that I used in case you want to compare them with your own:  (There's a scroll bar at the right to select all the 1979-2018 values):

Code: [Select]

year rutgers_june nsidc_june_extent nsidc_june_area nsidc_september_extent
1979 12.15 12.53 10.53 7.05
1980 10.9 12.2 10.19 7.67
1981 13.42 12.43 10.22 7.14
1982 9.08 12.48 10.65 7.3
1983 10.35 12.3 10.46 7.39
1984 8.45 12.15 10.28 6.81
1985 13.41 12.22 10.15 6.7
1986 11.03 11.98 10.18 7.41
1987 12.16 12.49 10.58 7.28
1988 8.06 11.94 9.99 7.37
1989 9.19 12.24 10.26 7.01
1990 7.17 11.64 9.51 6.14
1991 10.64 12.11 9.97 6.47
1992 10.26 12.15 10.26 7.47
1993 7.91 11.87 9.57 6.4
1994 9 12.02 9.99 7.14
1995 9.48 11.44 9.25 6.08
1996 11.25 12.08 10.14 7.58
1997 10.11 11.74 9.51 6.69
1998 9.48 11.71 9.49 6.54
1999 8.7 11.78 9.56 6.12
2000 9.72 11.67 9.36 6.25
2001 8.75 11.46 9.4 6.73
2002 9.23 11.58 9.5 5.83
2003 10.25 11.6 9.42 6.12
2004 9.76 11.45 9.55 5.98
2005 9.07 11.16 9.12 5.5
2006 8.41 10.92 8.72 5.86
2007 8.38 11.22 8.52 4.27
2008 6.69 11.21 8.88 4.69
2009 7.13 11.32 9.28 5.26
2010 6.02 10.59 8.4 4.87
2011 6.05 10.75 8.57 4.56
2012 4.92 10.67 8.19 3.57
2013 6.01 11.36 8.98 5.21
2014 6.8 11.03 8.79 5.22
2015 5.43 10.88 8.73 4.62
2016 5.58 10.35 8.16 4.51
2017 9.27 10.72 8.59 4.8
2018 7.84 10.71 8.74

Edit: sorry, apparently you were using the daily minimum rather than the September monthly extent.  Perhaps that explains it.

[cesium62]

Edit: sorry, apparently you were using the daily minimum rather than the September monthly extent.  Perhaps that explains it.

Specifically: I grabbed
http://nsidc.org/arcticseaicenews/sea-ice-tools/
ftp://sidads.colorado.edu/DATASETS/NOAA/G02135/seaice_analysis/Sea_Ice_Index_Min_Max_Rankings_G02135_v3.0.xlsx
Then the nh-annual-5-day-extent tab.

Sounds like you're saying that SIPN is asking for the September average extent to be predicted instead of the minimum extent...

[oren]

Sounds like you're saying that SIPN is asking for the September average extent to be predicted instead of the minimum extent...

Indeed SIPN is asking for predictions of September monthly mean whole-Arctic extent (as well as some other parameters).

[Rob Dekker]

Thanks guys, for your experiments and insight.

It's unfortunate that I don't have access to my data right now, exactly when there is so much discussion about the method I used.
But I promise I will respond to more of your questions and comments on Monday.

Still, there are a few comments general enough to be of addressed right now :

Rob: I think your model is fun and interesting, so I spent some time with it.

Assuming I've correctly re-produced your results...

The question above is, roughly:  what's the probability that the actual result is 0.7M km2 or more away from the predicted result?  This occurred in: 1980, 1982, 1983, 1986, 1988, 1991, 2001, and 2006.  That's 8 out of 39 years, or about a 20% chance.

I believe there are several physical reasons why the method I use is not very accurate in the 80's :

For starters, in the 80's the ice was much thicker than it is today. That means that it takes more energy to melt, and since my method does not include an ice-thickness variable, it is reasonable to assume it does not work that well on an ice pack that was fundamentally different than it is in recent decades. The albedo effect (on which my method is based) is less profound when the ice is thick.

Also, the ice that melted out in summer of the 80s was mostly outside the Arctic Basin, so there is a topological component that changed. That may very well affect the rate of melting at the ice edge.

And then there is the Rutgers snow lab data ; in the 80's it seems to be quite 'erratic' and I'm not sure we can really trust that (early satellite) data as much as we can in later decades.

That's why I choose the early 90's as a starting date for the regression.

[Rob Dekker]

Also, graphing the trend lines through the minimum and the predicted minimum, suggests that the prediction is diverging from actual (getting larger) as each year passes.  Since both snow cover and minimum extent trend downward year by year, it might be interesting to add 'year' as a parameter to better explore how well snow cover helps explain minimum extent.

Overfitting a model based on year, snow area, and ice area to all data from 1979 through 2017, we get the second attached picture.  And a forecast of 4.58 M km2 for the 2018 min extent.  (With a 360 K km2 geometric mean error.)  (Overfitting Dekker's model gives a forecast of 4.76 M km2 with a 435 K km2 geometric mean error.)  (If I train the year-based model on just 1992 through 2015, the forecast is 4.64 M km2 with a 386 K km2 geometric mean error.)

My simple physical explanation for the year-based model would be: heat is accumulating worldwide year by year due to greenhouse gases; the snow and ice area (or lack thereof) takes into account how much insolation is absorbed in the northern hemisphere in June.  Together, this suggests the amount of heat available for melting ice, subject to the vagaries of weather.

Thanks, cesium. The 'year' variable is indeed very clear in the 1979-2017 record, so it is tempting not to use it. But there are three reasons not to give in to that temptation :

1) One could argue that the 'year' variable is indicative of the increase in greenhouse gases as you suggest. But we did not quantify the relation between greenhouse gas increase to sea ice loss yet, so this relation is kind of speculation at this point.

2) The 'year' variable is adding one more variable, which will increase the risk of "over-fitting". Remember the famous saying by John Von Neumann :
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

3) As pointed out above, the method I use is not very accurate in the 80's, and starting from the 90's the variability can be better explained by the physical measurable variables (snow cover, ice area and ice concentration). Even though each of these physical variable may ultimately be driven by (gradually increasing) greenhouse gasses in the atmosphere...

So that's why I wanted to stay away from including "year" (or "time") as a variable in the prediction method and run the regression with physical variables instead.

[cesium62]

Also, graphing the trend lines through the minimum and the predicted minimum, suggests that the prediction is diverging from actual (getting larger) as each year passes.  Since both snow cover and minimum extent trend downward year by year, it might be interesting to add 'year' as a parameter to better explore how well snow cover helps explain minimum extent.

Overfitting a model based on year, snow area, and ice area to all data from 1979 through 2017, we get the second attached picture.  And a forecast of 4.58 M km2 for the 2018 min extent.  (With a 360 K km2 geometric mean error.)  (Overfitting Dekker's model gives a forecast of 4.76 M km2 with a 435 K km2 geometric mean error.)  (If I train the year-based model on just 1992 through 2015, the forecast is 4.64 M km2 with a 386 K km2 geometric mean error.)

My simple physical explanation for the year-based model would be: heat is accumulating worldwide year by year due to greenhouse gases; the snow and ice area (or lack thereof) takes into account how much insolation is absorbed in the northern hemisphere in June.  Together, this suggests the amount of heat available for melting ice, subject to the vagaries of weather.

Thanks, cesium. The 'year' variable is indeed very clear in the 1979-2017 record, so it is tempting not to use it. But there are three reasons not to give in to that temptation :

1) One could argue that the 'year' variable is indicative of the increase in greenhouse gases as you suggest. But we did not quantify the relation between greenhouse gas increase to sea ice loss yet, so this relation is kind of speculation at this point.

2) The 'year' variable is adding one more variable, which will increase the risk of "over-fitting". Remember the famous saying by John Von Neumann :
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

3) As pointed out above, the method I use is not very accurate in the 80's, and starting from the 90's the variability can be better explained by the physical measurable variables (snow cover, ice area and ice concentration). Even though each of these physical variable may ultimately be caused by (gradually increasing) greenhouse gasses in the atmosphere...

So that's why I wanted to stay away from including "year" (or "time") as a variable in the prediction method.

You really ought to avoid assuming that people you're talking with are idiots and patronizing them.  Since you've already thoroughly though through all aspects of this problem, I'll stop wasting your time.

[Neven]

You really ought to avoid assuming that people you're talking with are idiots and patronizing them.  Since you've already thoroughly though through all aspects of this problem, I'll stop wasting your time.

I've read Rob's comment twice, but I didn't get the impression that he was being patronizing. But that's just me.

Either way, it's an interesting discussion, and it's about science, so please, don't quarrel. Or at least not like this.

[Peter Ellis]

2) The 'year' variable is adding one more variable, which will increase the risk of "over-fitting". Remember the famous saying by John Von Neumann :
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

This is an unfair comment. Your model has three parameters (June ice area, June ice extent and June snow extent) while cesium62's model also has three parameters (June area, June snow extent and year).  Using standard methods of regression analysis, cesium62 found that there's virtually no benefit to including June extent as a parameter, while including year as a parameter improved the fit.  It's worth looking at why.

[Ned W]

I believe there are several physical reasons why the method I use is not very accurate in the 80's :

For starters, in the 80's the ice was much thicker than it is today. That means that it takes more energy to melt, and since my method does not include an ice-thickness variable, it is reasonable to assume it does not work that well on an ice pack that was fundamentally different than it is in recent decades. The albedo effect (on which my method is based) is less profound when the ice is thick.

Also, the ice that melted out in summer of the 80s was mostly outside the Arctic Basin, so there is a topological component that changed. That may very well affect the rate of melting at the ice edge.

And then there is the Rutgers snow lab data ; in the 80's it seems to be quite 'erratic' and I'm not sure we can really trust that (early satellite) data as much as we can in later decades.

That's why I choose the early 90's as a starting date for the regression.

Those are good points, Rob.  Makes sense. 

I have  some other ideas I'd like to check out, but probably won't have time in the next day or two ... will get back to you.  Anyway, I like the approach you're using, and thanks for putting up with all the unsolicited advice.

[Richard Rathbone]

Peter beat me to it.

I'd stick to two parameters, and use the elephant as a reason not to have three, but since you are using three, you can't use the elephant as a reason to pick one over another.

[crandles]

1) One could argue that the 'year' variable is indicative of the increase in greenhouse gases as you suggest. But we did not quantify the relation between greenhouse gas increase to sea ice loss yet, so this relation is kind of speculation at this point.

So, is it possible to try using either year or CO2 level in Arctic (during melting season? or annual average given that winter values may have affected thickness of ice?) ?

If 'year' works better than 'CO2 level' then the CO2 effect would appear to be just speculation and may well be grounds for sticking with physical variables.

If 'CO2 level' works better than 'year' as a parameter, then this may be a useful physical parameter. Note the 'may'; it may well still not be useful enough to add it as a parameter. Still, knowing this might be useful, if searching for a better set of parameters.

(Aiming to answer clearly seems a desirable feature of explanations, so thank you for taking the time you are to answer.)

The typical size of errors in forecast mode seems like what we should be trying to minimise and judge models by rather than hindcast standard errors?

It isn't easy to do this consistently across models if you exclude 80s where your model performs badly but other models might have taken different choices. So it seems difficult to compare models via error sizes? Your reasoning for excluding 80s seem sensible to me. However, for purpose of comparing model error sizes is there an argument for requiring all teams to use all data and report average/sd/RMSE/geometric mean forecast mode error over a defined set of years like 2000-current year?

[Steven]

Specifically: I grabbed
http://nsidc.org/arcticseaicenews/sea-ice-tools/
ftp://sidads.colorado.edu/DATASETS/NOAA/G02135/seaice_analysis/Sea_Ice_Index_Min_Max_Rankings_G02135_v3.0.xlsx
Then the nh-annual-5-day-extent tab.

Sounds like you're saying that SIPN is asking for the September average extent to be predicted instead of the minimum extent...

Indeed, I had been implicitly assuming that you were using the September monthly extent, as the SIPN does.  My bad.

But even with the 5-day extent annual minimum, I'm still unable to reproduce your calculations.

Edit: I could finally reproduce your calculations and graphs.  It turns out that there is a "pole hole" problem with your June sea ice area data after all.  To correct it, you should add a pole hole correction of 1.19 M km2 to your June sea ice area data for the years 1979-1987,  0.31 M km2 for the years 1988-2007, and 0.03 M km2 for the years 2008-2018.

https://nsidc.org/the-drift/data-update/sea-ice-index-updated-with-a-new-arctic-pole-hole-and-residual-weather-masks/

[Rob Dekker]

I would like to apologize to cesium. I can see that von Neumann's quote was not appropriate, especially since we both use the same number of variables. I'm sorry about that.

Your model with 'time', 'snow cover' and 'area' as a variables looks promising, and in the end may be closer to reality than my model which is only using variables that affect albedo.
After all, warming in the Arctic is not just caused by albedo changes. Greenhouse gas emissions DO affect Arctic temperatures directly too, and greenhouse gas concentration increases (mostly linear) with time.

It's just that we have not quantified that relationship physically yet, to see if the time component can really be caused by greenhouse gas emissions directly, or if it is the result of something else (like albedo changes; which may themselves be a secondary effect of global warming). Attribution of these variables to Arctic sea ice changes is very hard in my opinion.

Maybe we could start with the correlations in the observations : In your model, what is the parameter you obtain for the 'time' variable, as compared to the parameter of the 'snow cover' variable ?

[Rob Dekker]

Steven, I just want to say thank you for all of the verification work you are doing here.

You made me realize that the standard deviation over the residuals (340 k km^2) I used over the past years is incorrect, not just because of the missing variables, but also because I calculated it over the regression period (1992-2015) rather than the actual reported range (1992-2017).

I just submitted my ARCUS SIPN entry (for 5.19 M km^2) with the corrected Standard Deviation of 380 k km^2.

[Rob Dekker]

Regarding the 80's data, and the suggestion to use 'time' as a variable, I ran a number of experiments, using various combinations of variables on two different regression periods :
One on the full 1979 - 2017 record, and one on the more recent 1992 - 2017 record.

I calculated the Adjusted Standard Deviation, as suggested by Steven :
https://forum.arctic-sea-ice.net/index.php/topic,292.msg162485.html#msg162485

This is a much fairer comparison (between different formulas) than the simple SD over the residuals, since it compensates for the number of variables used (prevents over-fitting).

The results are interesting (all results are Adjusted Standard Deviation in k km²) :

Code: [Select]

              1979 - 2017           1992 - 2018
              Adjusted SD           Adjusted SD

k=2 (one variable) :
Time              543               542
Area              420               396
k=3 (two variables) :
snow+area         415               375
k=4 (three variables)
snow+area+extent  413               378
snow+area+time    419               367

Couple of conclusions from these experiments :
1) All these methods perform much better than only using 'time' (which is a linear decline prediction).
2) These methods works much better over the 1992 - 2017 period than over the full 1979-2017 record (this is that 80's data problem I mentioned).
3) Using just 'snow cover' and 'ice area' (just two variables) works very well. Adding another variable is not necessarily causing an improvement in the prediction.
4) Adding 'time' as a variable is improving the results for the 1992-2017 period, but makes it worse for the 1979-2017 period.
5) Adding 'extent' as a variable is improving the result for the 1979-2017 period, but makes it worse for the 1992-2017 period.

I think cesium already mentioned the questionable value of adding 'extent' as a variable, and Steven mentioned the questionable value of adding 'time' as a variable.
These experiments support these opinions.

Seems that just using 'snowcover' and 'sea ice area' performs well regardless of which period we run the regression on.

I have not tested other variables, like PIOMAS volume in June, or winter temperatures or anything that may affect the rate of melt during the June-Sept period. But using Steven's adjusted SD formula we could test if adding another variable improves the result even further, or makes it worse...

[Peter Ellis]

Further observation - adding snow cover does little to improve predictions over the whole time course, but is a bit more use in recent years.  That's consistent with the idea that snow cover matters more when the ice pack as a whole is thinner and has less thermal inertia.

It would be interesting to look at extent as an individual variable - is it better or worse than area?  Snow as a single variable is unlikely to be much use :-)

For completeness' sake, the other permutations of the two-variable models might be fun to look at. Formally, there are another 5 pairs that could be checked, namely extent+area,  extent+snow, extent+time or area+time.

Similarly, there are two more three-variable models: snow+extent+time, and area+extent+time, both of which I suspect would be worse; and a four-variable model which would likely be slightly better, but which will fit an elephant if you want it to :-)

[Rob Dekker]

Further observation - adding snow cover does little to improve predictions over the whole time course, but is a bit more use in recent years.  That's consistent with the idea that snow cover matters more when the ice pack as a whole is thinner and has less thermal inertia.

It would be interesting to look at extent as an individual variable - is it better or worse than area?  Snow as a single variable is unlikely to be much use :-)

For completeness' sake, the other permutations of the two-variable models might be fun to look at. Formally, there are another 5 pairs that could be checked, namely extent+area,  extent+snow, extent+time or area+time.

Similarly, there are two more three-variable models: snow+extent+time, and area+extent+time, both of which I suspect would be worse; and a four-variable model which would likely be slightly better, but which will fit an elephant if you want it to :-)

I've added some of these permutations to the table :

Code: [Select]

              1979 - 2017           1992 - 2018
              Adjusted SD           Adjusted SD

k=2 (one variable) :
Time                  543               542
Area                  420               396
Snow                  711               617
Extent                502               522
k=3 (two variables) :
snow+area             415               375
area+time             422               379
snow+extent           495               466
k=4 (three variables) :
snow+area+extent      413               378
snow+area+time        419               367
k=5 (four variables) :
snow+area+extent+time 415         359

Conclusions from these added combinations :

1) 'Extent' is not really a good predictor. 'Area' is much better, as was already noted in the post by Bill Fothergill :
http://neven1.typepad.com/blog/2013/06/problematic-predictions.html

The only reason I included 'extent' in my formula was that (extent - area) seemed to be a good metric for "water that is very close to ice", which I thought may have a physical meaning as an area that would more efficiently turn solar heat into melting ice.
Now there is some correlation with that, but these experiments show that it's basically 'noise' as well (see the "snow+area+extent" line versus the simpler "snow+area" line).

2) Snowcover in June by itself is a terrible predictor of Sept SIE, but that is explainable : snow cover does not 'know' the June state of the ice pack.

To see that it is still useful, let's look at the physics. Since we are calculating how much energy there is in the Arctic system in June, that energy will be melting out ice over the June -> September period. So, we should try to predict the amount of ice that will be melting between June and September.
So let's change the regression formula so that it tries to predict the "June-area minus Sept-extent" variable instead of "Sept-extent" in absolute numbers. Nothing much changed about the regression method itself, since "June-area" is known in June.

When we do that we get these results of the for the individual variables :

Code: [Select]

              1979 - 2017           1992 - 2018
              Adjusted SD           Adjusted SD

k=2 (one variable) :
Time                  421               382
Area                  420               396
Snow                  426               389
Extent                433               428

This shows that 'Snow cover' even all by itself has pretty good predicting value for estimating how much more ice will melt out between June and September, especially over the 1992 - 2017 period. Only the (unphysical) 'time' variable seems to be better than 'snow cover'...

3) For the fun of it I also included a four-variable formula with "snow+area+extent+time". You can see that it makes things worse for the 1979-2017 period, and better for the 1992-2017 period.
With the increased risk that we are fitting an elephant here

Overall, I really like the simplicity (only 2 variables) and performance of the "snow+area" formula.
And it makes perfect physical sense as well : "snow+area" is how "white" the Arctic is in June.
The adjusted SD of 375 k km² is excellent, and beats most prediction methods in the SIPN. So I think I'm going to switch over to that simple "snow+area" model in my future predictions.

P.S. Almost all of these regression formulas end up projecting into the 4.8 - 5.2 range for Sept 2018 SIE.

[Tor Bejnar]

I have little to offer here, but the 'unphysical' time variable can be replace, I suspect, with the very physical CO2 or 'CO2e in the Arctic'. (Do we even know what it is? Does it matter if it's rate of increase is different from Mona Loa?)  I wonder about the H2O component in Arctic air: especially about the amount that falls as rain onto ice/snow.  Clouds, of course, are an intriguing factor: does the June experience tell us anything about September ASIE?  Finally (or similarly), does some metric associated with average atmospheric pressure or gradients or rate of change of pressure regimes (in June) help with September forecasting?

All this will make a fine drawing of an elephant, or maybe .

[Rob Dekker]

I have little to offer here, but the 'unphysical' time variable can be replace, I suspect, with the very physical CO2 or 'CO2e in the Arctic'. (Do we even know what it is? Does it matter if it's rate of increase is different from Mona Loa?)

Thanks Tor. I think crandles also made the link to CO2.
Now, there is little doubt in my mind that CO2 is ultimately the driver of Arctic sea ice decline, but it is very hard to prove this scientifically.

A statistician can probably explain this much better detail than I can, but if you have two variables which both correlate well with Arctic sea ice decline, then how do you know which one is the cause of the decline ? And how do we know if one variable is caused by the other ?

So what I am trying to do clarify one step in this process : to look at variables that we KNOW affect the energy input into the Arctic because the albedo changes, and I found out that these variables correlate very well with the observed sea ice decline (especially land snow cover and sea ice area).

If we can somehow find evidence that land snow cover and June sea ice area are going down because of CO2 increases, then we may finally close the link to CO2 as a driver of Arctic sea ice loss.

[crandles]

I have little to offer here, but the 'unphysical' time variable can be replace, I suspect, with the very physical CO2 or 'CO2e in the Arctic'. (Do we even know what it is? Does it matter if it's rate of increase is different from Mona Loa?)

Thanks Tor. I think crandles also made the link to CO2.
Now, there is little doubt in my mind that CO2 is ultimately the driver of Arctic sea ice decline, but it is very hard to prove this scientifically.

A statistician can probably explain this much better detail than I can, but if you have two variables which both correlate well with Arctic sea ice decline, then how do you know which one is the cause of the decline ? And how do we know if one variable is caused by the other ?

So what I am trying to do clarify one step in this process : to look at variables that we KNOW affect the energy input into the Arctic because the albedo changes, and I found out that these variables correlate very well with the observed sea ice decline (especially land snow cover and sea ice area).

If we can somehow find evidence that land snow cover and June sea ice area are going down because of CO2 increases, then we may finally close the link to CO2 as a driver of Arctic sea ice loss.

Thanks for continuing the discussion.

What you want is not two variables that do the same thing, but an extra variable that might explain some of the wriggles that your other variables do not explain.

One approach might be: After choosing best variable, work out unexplained changes and look for next variable that best correlates with that. However, this doesn't guarantee you will find the best combination. An alternative, brute force try everything might find better correlation but risks overfitting effect - ie if you try enough spurious relationships and you will find one that appears to work well but is actually just spurious. Sticking with physically linked variables does seem like a good plan.

Ice Area and land snow do some of the area, albedo and energy input effects but there is little about thickness of the ice. CO2 affects rate of heat loss and hence the equilibrium thickness of FY ice. Yes this is not about energy input, but maybe energy input is not the only thing that matters, state and thickness of ice seems likely to also matter? Thus I tend to think CO2 is a sensible physically linked variable even if it is hard to show that.

It wouldn't surprise me greatly if it turned out CO2 level in Arctic didn't turn out to be better than time: the wriggles in CO2 have certain delay to ENSO and ENSO doesn't seem well correlated with arctic sea ice. It could easily be that there are ocean heat effects and CO2 effects of ENSO that have different effects with different delays such that it is hard to see any correlation effect. Also time may do more than just CO2 effects.

.

When you include 'snow+area+extent' in your table, is this really 'snow' + 'area' + '(extent-area)'? Should you show (extent-area) as a separate variable to extent?

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Barrow has CO2 data from 1971. Alert is further north but only has data from 1985. Only found graph rather than monthly average table so far. CO2e may be better but I am less sure about the data being findable.

[crandles]

Is it possible to approximate the energy input down to one variable something like

( Ice Area + 0.5(Extent-area) + 0.25(Land snow) )

then use CO2 or time as a second variable?

[Peter Ellis]

So let's change the regression formula so that it tries to predict the "June-area minus Sept-extent" variable instead of "Sept-extent" in absolute numbers.

That's mathematically meaningless.  All you're doing is including area as one of your parameters while pretending you aren't. 

[crandles]

So let's change the regression formula so that it tries to predict the "June-area minus Sept-extent" variable instead of "Sept-extent" in absolute numbers.

That's mathematically meaningless.  All you're doing is including area as one of your parameters while pretending you aren't.

I don't believe it is. (Though I am not enough of a expert to be sure.)

This is targeting a different metric. With area as a parameter it gets fine tuned as to how much weight is given to this parameter. That isn't happening with setting a different target like that.

Above I suggested using ( Ice Area + 0.5(Extent-area) + 0.25(Land snow) ).
If that was fine tuned to something like ( Ice Area + 0.53(Extent-area) + 0.29(Land snow) ) then this would still be 3 parameters. However if the 0.5 and 0.25 are fixed and come from something objective before you start your analysis, such that it can only be tuned in one way by the weighting the linear regression applies then I suggest it is only one parameter.

Proportion of horizontal directions that heat could travel in and reach sea ice is what is immediately occurring to me with this.

[Rob Dekker]

So let's change the regression formula so that it tries to predict the "June-area minus Sept-extent" variable instead of "Sept-extent" in absolute numbers.

That's mathematically meaningless.  All you're doing is including area as one of your parameters while pretending you aren't.

It is true that 'area' is included in the equation, but as a constant, not as a parameter.

It is basically just a mathematical re-write of the regression equation.
The regular regression equation to estimate Sept SIE is this one :

  Sept_extent = alpha + beta * ( June_formula )

Now let's rewrite that to estimate ice loss between June and September :

  June-area - Sept_extent = -alpha - beta * (June_formula - June-area/beta)

See, it's still the same equation.
The factor "June-area/beta" is a constant, not a parameter.
If you run this regression, everything stays the same (even the SD of the residuals).
Only thing is that alpha and beta flip polarity.
 

[Peter Ellis]

Now let's rewrite that to estimate ice loss between June and September :

  June-area - Sept_extent = -alpha - beta * (June_formula - June-area/beta)

See, it's still the same equation.
The factor "June-area/beta" is a constant, not a parameter.

Depends whether June-area is also included as a variable in June_formula.  If it is, then it's a parameter. Note that when you do use area as a parameter, you get exactly the same result whether you're predicting Sept_extent or predicting (June-area - Sept_extent_.

I've added some of these permutations to the table :

Code: [Select]

              1979 - 2017           1992 - 2018
              Adjusted SD           Adjusted SD

k=2 (one variable) :
[...]
Area                  420               396
[...]

So let's change the regression formula so that it tries to predict the "June-area minus Sept-extent" variable instead of "Sept-extent" in absolute numbers.

Code: [Select]

              1979 - 2017           1992 - 2018
              Adjusted SD           Adjusted SD

k=2 (one variable) :
[...]
Area                  420               396
[...]


In cases where you're not actually using June-area as a parameter, it's justifiable to include it as a constant.