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## BOE: Gamma PDF Estimate of Equilibrium Climate Sensitivity

2012 May 25

Estimating the probability distribution for equilibrium climate sensitivity with a Gamma distribution visually fitted to Meinshausen 2009 Fig 3a” (top). The gamma distribution for the 2 to 4.5C spread for equilibrium climate sensitivity has a cumulative 85% probability.

The gamma distribution is coded as follows:

#gamma distro
x = seq(0.1,10,by=0.1)
alpha = 12# width of dist, higher is narrower
beta = alpha/3.2 # shift, higher is right shift
ts_pdf = beta^alpha/gamma(alpha) * x^(alpha-1) * exp(-beta*x)
ts_pdf = ts_pdf/sum(ts_pdf)
sum(ts_pdf[x>=2&x<=4.5])/sum(ts_pdf)
#[1] 0.8490118

The Gamma Distribution was not my first attempt at a fit. My first fit was a Poisson distribution, rescaled and shifted to fit the Meinshausen ECD PDF. The Poisson function for the 2 to 4.5 spread for climate sensitivity also integrates to an 85% probability.

The code for the Poisson fit is as follows:

lambda = 3 # ECS mean = 3
xmax = 14 # long enough tail to go to zero
xx = 0:xmax
xx2 = xx/2 # scale the Poisson back down to 0:7
xxtck = 0:7 # labels for the chart
yy = lambda^xx * exp(-lambda) / factorial(xx) # Poisson
Ts = approx(xx2,yy,n=71) # need more points, so interpolate to dx = 0.1
shift = 17 # shift peak back to ECS = 3
yy2 = rep(0,71)
yy2[shift:71] = Ts\$y[1:(71-shift+1)]
Ts\$y = yy2 / 5 # shift complete
sum(Ts\$y[Ts\$x>=2 & Ts\$x<=4.5])
#[1] 0.8526035

The similarity between the two is even more obvious when plotted together. It can’t be accidental, but I haven’t studied the two yet to derive their relationship.

1. 2012 June 19 at 8:43 am

Thanks for illustrating this. I’ve long thought there was too much focus, explicitly and implicitly, on gaussians.

Q1: What’s the source for your H.P. Lovecraft quote? (I have a couple of his books, so would as soon get to the right story.)

Comment 2: I’ll offer the guess on your last question that the relationship between poisson and gamma is like that between binomial and gaussian distributions. In some appropriate limit of continuity vs. discrete, one maps to the other.

2. 2012 June 19 at 7:12 pm

Can’t say it’s a great fit, but definitely better than a Gaussian.

Schmittner’s PDF, otoh, isn’t vaguely similar to any simple eqn. Maybe some combinations. I took a discrete version to play with in a future post.

I agree with your thought on the two distributions. But still want more meat on those bones.

The quote is from the opening paragraph of the “Call of Cthulu”

1. 2012 May 28 at 1:24 pm