I was reading Tamino's latest post, and one of the commenters wrote about losses in Arctic sea ice volume, and how some other people had said it could be as high as 75% over the last 30 years. So I thought I'd check that using the latest PIOMAS data.
But when I do calculate something like this, I always wonder/worry about what if the first few years in the data series were high or low relative to the overall trend, or what if the last year was an unusual fluctuation -- that would skew the results and give a number that wasn't really indicative of the long-term trend.
But if you assume the trend is linear, you can, with just a little algebra, get a useful result for the percentage loss (or gain) based on just the linear trend. (I'm sure this isn't original, but I haven't encountered it before.)
The long-term percentage loss L, in the linear model, is
L = (y2-y1)/y1
where y1 and y2 are the values from the linear fit: y1=mx1+b and y2=mx2+b, where m is the slope of the trend line as determined by linear regression, b is the intercept, and x1 and x2 the endpoints of the line.
With a little algebra you find, defining the data's average value as A [= (y1+y2)/2]
where Δx (= x2 - x1) is just the length of the data record. [If you need help deriving this, just note that you have two equations with two variables, y1 and y2. Solve as usual.] So you only have to calculate "bulk properties" of the data: m, A and Δx. Pretty neat.
The PIOMAS data (Δx = 33.4 yrs) gives A = 20.64 Kkm3, m = -0.30 Kkm3/yr, so
L = -39.0%
or -1.2% per year.
(A linear fit probably isn't the "best" fit -- my spreadsheet finds a slightly higher R2 for a power law fit -- but this is just blog-work, not rocket science.)
Like I said, I'm sure this isn't original, but it's a little useful (at least to numbers geeks like me), and it was fun to work out.
(Mostly it was fun to work out.)