However, there is increasing agreement that both mechanisms are mutually compatible,
as autocorrelation occurs between ego's and alters» behaviors [25 — 27].
For continuously sampled processes, the ACS and ACVS are known
as the autocorrelation function (ACF) and autocovariance function (ACVF) respectively.
You have just described what is known
as an autocorrelation — one of the most useful probabilitiy measures for spatial and temporal analyses.
You and Steve and I can point to the problems, such
as autocorrelation, all we want.
Not exact matches
They measure short term risk
as the average of the worst 1 % of annual returns from 10,000 bootstrapping simulations that randomly draw three months of returns at a time from 20 - year historical pool of returns for these indexes, thereby preserving some monthly return
autocorrelations and cross-correlations.
As a side - note, we prefer a 12 - year horizon when we discuss mean - reversion of valuations, because that's the point where the «
autocorrelation profile» of valuations hits zero (meaning that overvaluation or undervaluation most reliably washes out over that horizon).
Stationarity tests, the white noise test, and the Dickey - Fuller test for unit roots were performed
as well
as examination of
autocorrelation function and partial
autocorrelation function plots17 to help identify appropriately parameterized models for MAR, ED, and lunch participation time series.
To this aim we have screened our electrophysiological recordings of the magnocellular neurons, previously obtained from acute slices, with an analysis of
autocorrelation of action potentials to detect a rhythmic drive
as we recently did for organotypic cultures.
The computer scientist Neil Dodgson investigated whether Bridget Riley's stripe paintings could be characterised mathematically, concluding that while separation distance could «provide some characterisation» and global entropy worked on some paintings,
autocorrelation failed
as Riley's patterns were irregular.
Confidence intervals are corrected for
autocorrelation in the same way
as in Foster & Rahmstorf 2011.
The «detrended» 1880 - 1981 line has a lag 1
autocorrelation for series 60 of 0.62
as opposed to 0.65 for the original series — a trivial change when compared to the spread of lag 1
autocorrelation values for the 10 series you show which vary from 0.65 to nearly 0.80.
The
autocorrelation of the «blade» segment of «hockey - stick» shaped proxies is significantly higher than the rest of the series, this biases their estimates of
autocorrelation parameters because their model assumes a stationary
autocorrelation structure, making their simulated series unrepresentative of most of the length of hockey - stick series,
as can be seen in this graph of lag - 1
autocorrelation coefficients.
Finally the Fig 1 trends do show evidence of
autocorrelation, so successive departures from trend are not
as compelling
as might first seem.
A linear increasing function with time would be described
as first order non-stationary but it would not have an
autocorrelation structure.
A power spectrum such
as used by MM2005, after adding random phase, has an
autocorrelation structure but is deliberately modelled
as stationary.
One can try to deal with stationarity by correcting for
autocorrelation, but
as far
as I know one can not tell definitely whether this actually removes the non-stationarity.
But
as I said, even when I jack up the noise
autocorrelation it still doesn't give a strong enough PC # 1 to come close to that of MBH98.
If the residual series generated by a model show no significant
autocorrelation,
as tested by Ljung - Box portmanteau statistics, then they are not distinguishable from white noise — a covariance stationary process.
Looking at the
autocorrelations there does not appear to be anything left to explain except
as randeom noise.
From what I can tell, the statistical model used (sorry if I'm mistaken about this) doesn't allow for
autocorrelation of residuals and simply treats El Nino and all other forms of internal variability
as white noise.
The more meaningful statistical question, however is this one: Given the «null hypothesis» of red noise with the same statistical attributes (i.e., variance and lag - one
autocorrelation coefficients)
as the actual North American ITRDB series, and applying the MBH98 (non-centered) PCA convention, how likely is one to produce the «Hockey Stick» pattern from chance alone.
However, the use of the
autocorrelation function
as a tool for such comparisons presents a problem.
Re # 78 Let's see a comparative plot of the performance (% correct inferences) of the Kendall test vs. some alternative parametric method, where performance degrades differentially
as a function of p (
autocorrelation) and n (some index of noise variance).
Beyond that, I gave you a quick response
as an aside, Do some frequency analysis,
autocorrelation of something if you want to examine the structure of that record.
Some care should be taken since
autocorrelation shows a repetition of form and in no way suggests is it sinusoidal
as VP does in his saw wave.
Spatial
Autocorrelation and the Detection of Non-Climatic Signals in Climate Data was rejected; at least part of the problem was that the paper appeared to add little new to the discussion,
as would be expected from a standalone paper.
Teleconnections is the wrong idea — rather it is measure of
autocorrelation in the global climate system —
as in the Dakos et al paper which you didn't bother to look at.
The hypothesis of
autocorrelation as explained in the Stadium Wave model is therefore invalid.
Parker considered simple
autocorrelation, but the this sort of data also exhibits homoscedasticity,
as well
as spatial and long - term - temporal correlations that likely need to be addressed, too.
The very inconsistency of the series within proxy networks such
as Mann et al 2008 argues forcefully against the interpretation of high empirical
autocorrelation coefficients
as being imported from a climate «signal»,
as opposed to being an inherent feature of the proxies themselves.
Technically — an increase in
autocorrelation as the system approaches bifurcation and non-linear variability at bifurcation.
However, M&M's null proxies were not generated
as AR1 noise
as claimed by Wegman et al, but rather by using the full
autocorrelation structure of the real proxies (in this case, the 70 series of the 1400 NOAMER tree - ring proxy sub-network).
The more linear (lower variance) the average output in two models, the higher the r2, purely
as a result of increased
autocorrelation.
As per Papoulis (for me, 2nd edition, page 233), if a Gaussian process with
autocorrelation function R (t) is squared, the resulting
autocorrelation function, which I will call Q (t), is
Even the modest temp increase recorded in the global temp series is likely to be overstated (and very unlikely to be understated) due to various factors such
as UHI (in its broader definition of stations getting increasingly in close vicinity of concrete, bricks, asphalt, metal and engines, even in rural areas) and also problems with the choice of stations, and other problems with infilling and averaging of gridcells without due consideration of spatial
autocorrelation and other similar problems (cf e.g. the Steig - O'Donnell debate).
I redid the estimate using the magnitude squared because the
autocorrelation function for the square of a Gaussian process is well known,
as I related here:
These results are why I question the use of monthly data with its
autocorrelations (that have to be corrected with methods such
as Cochrane - Orcutt) when the annual data does not require corrections (that could be of uncertain validity — see Steve M remarks on C - O CIs versus those CIs derived using maximum liklihood approach).
If you have Mandelbrot - type
autocorrelation, why are [time] averages or 2nd moments of «climatology» not themselves
as transitory
as the global average temperature or variance from 3 to 4 pm EST on April 11, 1956?
Looking at these results, that are admittedly anecdotal at this point, I see generally better fits to a normal distribution and lower
autocorrelation (AR1) in the residuals
as one goes from monthly to individual months to annual data series and
as one goes to sub periods of a long term temperature anomaly series.
Processes with consistently positive
autocorrelation functions lead to large and long «excursions» from the mean
as shown in Fig. 12 (lower panel), which often tends to be interpreted
as nonstationarity.
Consequently,
as part of the atmospheric continuum, it exhibits high temporal and spatial
autocorrelation.
It's not due to the
autocorrelation structure being more complicated than the e-AR1 series, because the results show interpolation skill improves
as the noise becomes more structured.
In summary there is much ARn
autocorrelation as was the case with RSS SSM / I and the trend is flat and can not be distinguished from 0.
I did the same thing using my method to correct for
autocorrelation and got this (note that my units are deg.C / yr while McKitrick's are deg.C / decade, and that I only tested start years
as late
as 2005 while McKitrick goes all the way to 2009):
We assume that the missing past and future data have similar spectral shape and
autocorrelation as the available data.
Uncertainty is estimated by the variance of the residuals about the fit, and accounts for serial correlation in the residuals
as quantified by the lag - 1
autocorrelation.
The Student's t test was applied to the bins pairwise
as described in the text, adjusting downward the degrees of freedom based on the
autocorrelation in the time series.
So it then occurred to me that if you are comparing the output from a climate model with actual measures at a grid level there was a risk that both datasets would suffer from spatial
autocorrelation, and this should be tested for before using standard statistics
as validation of the model output.