This post is a follow up to the previous one so I will not be re-iterating information in that post. I’ll also continue with the previous equation numbers. The first equation in this post will be numbered (3); lower number equations are from the previous post.
——

How do the lag 1 correlations of an averaged process relate to the time constant for the AR1 process?

To fully translate the properties of an annual averaged AR(1) process to the monthly averaged properties, we need to know how the lag 1 correlations of the averaged processes (ρ_month and ρ_YEAR) correspond to the time constant for the underlying function, τ.

This can be easily by using the definition of a covariance, the equation describing the integral for the autocorrelation of the continuous function, and integrating. (I’d show it, but as I previously said, I don’t have math functionality installed at the blog yet.)

For integer lags, n, greater than zero, the lag “n” covariance for the process averaged over a window “T” can be show to be:

(3)σ_o,nT²(n)=σ² ( Ï„/T )² [exp^{-(n-1)T/Ï„} ] [ (1-exp^-Ï„/T)² ]

Recall the relation for the variance was given by (1) in the previous post:

(1) σ_T²= σ² [ 2 Ï„/T ] [ 1 + Ï„/T {exp(-Ï„/T) -1} ]

So, we find after averaging, the lag “n” correlation can be obtained by applying the definition which translates to taking the ratio of (3) to (1):
(4)ρ_T(n)=σ_o,nT²(n)/σ_T²

So, what is the τ corresponding to a lag 1 correlation of ρ_YEAR=0.1 for an annual averaged process?

Using the (4) with (3) and (1) it is possible to determine the time constant corresponding to an AR(1) process with a lag1 correlation of ρ_YEAR(1)=0.1 is Ï„= 0.167 years. This is shorter than the time constant of 0.43 years obtained when I ignored the effect of averaging on the lag 1 correlations. (It is the case that if we estimate a time constant of an underlying function using the lag 1 correlation of an averaged AR(1) time series, and don’t account for the effect of averaging, we will always overestimate the time constant of the underlying series. I could go off on other tangents related to the Schwartz paper and the time constant of the earth. But… I’ll try to stick to the topic of this post and instead just explain how I ultimately translated the monthly averaged AR(1) process to an annual averaged AR(1) process. 🙂 )

How do we translate Gavin’s “closer” annual averaged AR(1) process to a monthly process?

Annual averaged process:

1. Recall the “closer” AR(1) function based on Gavin’s model’s has ρ_YEAR(1)=0.1, σ_YEAR=0.1C and a trend of m=2 C/century.
2. ρ_YEAR(1)=0.1 results in τ= 0.167 years
3. For τ= 0.167 years, ρ_month(1)= 0.728.
4. One year gives us a ratio of T/Ï„ = 5.97. So, if the variance of the underlying process is σ², the variance of the 1 year averaged process is σ_YEAR²=0.28 σ²
5. One month gives us a ratio of T/Ï„ = 0.5 and σ_month²=0.28 σ². Using ratios, I get σ_month²=3.06σ_Year². Taking the square root, and using the value for 1 year, I get σ_month=0.175 C.
6. Averaging has no effect on the trends. The continuous, annual averaged and monthly series will all have the same trend of m=2.0

Ready for Monte-Carlo

Having translated the AR(1) series Gavin says is “closer” to the model results to monthly values, I can now run Monte-carlo experiments using the related monthly function. There are a number of quibbles one could advance about the relationship (and we can discuss these in comments.)

However, going forward, I will be testing whether “weather noise” and trend generated using the following AR(1) process is consistent with the 89 months worth of data collected since January 2001:

AR(1) and ρ_month(1)= 0.728 and σ_month=0.175 C and m=2.0 C/century

Logically, if models, based on physics, are supposed to predict the properties of weather and climate data, then we might expect that an AR(1) process that is “closer” to the models should not fall outside the range consistent with actual data. This applies equally to testing the trend, and the whole process.

So, we don’t simply check whether m=2C/century agrees, given the values of the rest of the process. We also must check whether the combination “AR(1) and ρ_month(1)= 0.728 and σ_month=0.175 C” if it is, then further determine whether the full process “AR(1) and ρ_month(1)= 0.728 and σ_month=0.175 C and m=2.0 C/century” is consistent with the data.

Because of the effort involved in running the tests, and documenting, I will be testing incrementally as previously described. So, I’ll first test if AR(1) and ρ_month(1)= 0.728 is consistent with the experimental data for any value of “m and σ_month. If it’s not, I’ll stop further analysis, because obviously, adding the extra requirements won’t “save” that model. If it survives, I’ll do further tests.

What about ρ_month(1)= 0.82?

Recall in the previous comment, I discussed that quick and dirty way of translating from annual to monthly data resulted in ρ_month(1)= 0.82, another quick an dirty way results in ρ_month(1)= 0.88.

Some might wonder if all this involved translation is a tendentious attempt to find something that will falsify. It’s not. Everything above ρ_month(1)= 0.80 pretty well falsifies at the first step when we test whether the autocorrelation exhibited by “model weather” is consistent with real earth weather.

So all this work is to be fair and see whether a more careful treatment prevents falsification! Notice that after doing a bunch of manipulation, I’ve knocked the lag one autocorrelation for the “comparable” autocorrelation to ρ_month= 0.728. This is closer to the lag 1 autocorrelations near 0.4-0.5 we are seeing in the monthly data, and so less likely to falsify based on ρ_month being found inconsistent with the real earth weather data.

I still haven’t run the monte-carlo, so I don’t know what the results will be! 🙂