# Options for ACF in R

The autocorrelation function is one of the Niche model basics in the R language. The ACF is the real deal when it comes to diagnosing the internal character of series data, as was done for scale invariance in Scale Invariance for Dummies. Questions of internal character of series are very topical, such as “How Red are My Proxies” where it was claimed that global temperatures are barely AR(1) contrary to views of a colleague discussed in “A Mistake with Repercussions“.

The ACF function in R contains another useful diagnostic option called the partial autocorrelation. Partial autocorrelation is estimated by fitting autoregressive models of successively higher orders up to the lag.max. Partial autocorrelations
are useful in identifying the order of an autoregressive model. The partial autocorrelation
of an AR(p) process is zero at lag p+1 and greater. I thought I would have a look at some data to see what the partial ACF has to say about the possible AR order of temperatures series.

Figure 1 above shows the partial correlation coefficients of the simple series,
from top to bottom,
IID (random independent and identically distributed), MA (moving average of IID), AR (autoregression), and SSS (fractionally differenced FARIMA model). The
partial correlations of the IID and the AR(1) zero at lag 1. The SSS is not zero until
greater than lag 5.

Figure 2 shows partial ACF, from top to bottom, of
the natural series CRU (global average temperatures), MBH99 (Mann’s hockey stick), and spatial precipitation and temperature. The CRU
also has significant correlations at lag 5, as does the MBH99 reconstruction of
The partial correlations of the spatial temperature and precipitation also
have slight partial correlation above lag 1.

These results suggest AR(1) is not adequate for representing global
temperature dynamics.
The partial correlation plots suggest that CRU temperatures should be modeled by an autoregressive process of at least order AR(5), and potentially a
simple scaling process SSS such as described by
Demetris Koutsoyiannis in
Hurst, Joseph, colors and noises.
Together with other evidence of a constant and high Hurst coefficient, there
is a fairly strong case that natural series do have a ‘long tail’ of correlation. Consequently, models based on IID and simple markov processes for modeling global average temperatures should be considered suspect.

This analysis should not be construed as definitive.
The diagnosis of the dynamics of natural series into AR, MA and F components
is very difficult as noted in
Hurst, Joseph, colors and noises.
These results do seem at odds with
outspoken climate scientists and anthropogenic global warming supporters
at realclimate.org that global average temperatures did not differ significantly from a simple markov AR(1), small coefficient process.

R is a great program for ‘doing it yourself’ and getting involved in the debate. However,
for critical policy oriented work on climate
I strongly support the main findings of the recently released report

Ad Hoc Committee Report on the ‘Hockey Stick’ Global Climate Reconstruction
“, namely:

• Independent statistical expertise should be sought and used.
• What is needed is deeper understanding of the physical mechanisms of climate change.

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