To understand the processes driving polar ice coverage it is necessary to identify cyclic variations. Some would wish to trivialise climate into AGW plus random “stochastic” variability. This is clearly unsatisfactory. Much of the variation is more structured than may be apparent from staring at the ups and downs of a time series.
There are many cyclic or pseudo-cyclic repetitions and before attempting to regression fit a linear trend to the data, it is necessary to identify and remove them or include them in the model. Failure to do this will lead to concluding invalid, meaningless “trends” in the data. See cosine warming ref 1.
One means of testing for the presence of periodicity in a dataset is spectral analysis. In particular spectral power distribution can be informative.
The power spectrum can be derived by taking the Fourier transform of the autocorrelation function. The latter is, in its own right, useful for identifying the presence or not of cyclic change in the data.
One condition to get useful results from Fourier analysis is that the data should be stationary (ref 1) . Amongst other things this requires that the mean of the data should be fairly constant over time. FT of a series with a rising/falling trend will produce a whole series of spurious peaks that are a result making an infinite series from a ramping, finite sample.
There are several tests and definitions of stationarity and it is to some degree a subjective question without a black or white answer. A commonly used test is the augmented Dickey-Fuller, unit root test: ADF. (ref 2)
If a time-series is found not to satisfy the condition of stationarity, a common solution is examine instead the rate of change. This is often more desirable than other ‘detrending’ techniques such as subtracting some arbitrary mathematical “trend” such as a linear trend or higher polynomial. Unless there is a specific reason for fitting such a model to remove a known physical phenomenon, such a detrending will introduce non physical changes into the data. Differencing is a linear process whose result is derived purely from the original data and thus avoids injecting arbitrary signals.
The time differential ( as approximated by the first difference of the discrete data ) will often be stationary when the time-series is not. For example a “random walk”, where the data is a sequence of small random variations added to the previous value, will be a series of random values in its differential and hence stationary. This is particularly applicable to climatic data, like temperature, where last year’s or last month’s value will determine to a large extent the next one. This kind of simple autoregressive model is often used to create artificial climate-like series for testing.
To ensure there is no step change, as the end of the data is wrapped around to the beginning, it is usual to also apply a window function that is zero at each extreme and fades the data down at each end. This has the disadvantage of distorting the longer term variations but avoids introducing large spurious signals that can disrupt the whole spectrum.
Most window functions produce some small artificial peaks or ‘ringing’ either side of a real peak. Some do this more than others. The choice of window function depends to some extent on the nature and shape of the data. The choice is often a compromise.
Initial examination of the autocorrelation function of Arctic ice area data revealed the presence of notable periodicity other than the obvious annual cycle. Some recent published work is starting to comment on various aspects of this. (ref 3)
As a first step the annual cycle was removed by a triple running mean filter (ref 4) with a zero at 365 days and designed to avoid the usual distortions caused by simple running mean “smoothers”.
If a Fourier transform were to be done with the presence of the annual cycle, it’s magnitude, at least an order of magnitude greater than anything else, would reduce the accuracy of FFT and also introduce noticeable windowing artefacts in the 0.5 to 2.0 year period range. For this reason it was removed.
The adf.test() function in R package returned values indicating it was not possible to assume that the data was stationary. Contrariwise, the test on the derivative of the time series indicated strongly that it was stationary.
Non-stationarity is probably caused by long term trend or long period cyclic variation (long relative to the duration of the dataset).
Taking rate of change reduces linear trends to a constant and attenuates the amplitude of long periods by an amount proportional to the frequency, making it more amenable to analysis. The 365d filter will also attenuate frequencies less than about 5 years and this needs to be born in mind or corrected for when relative magnitudes are considered in the spectrum.
ref 1 Cosine warming
ref1 stationarity reqt for FFT.
ref3 arctic periodicity
ref 2 (dickey-fuller)