In contrast to precursor methods, extrapolation methods only use the time series of sunspot numbers (or whichever solar activity indicator is considered) but they generally rely on more than one previous point to identify trends that can be used to extrapolate the data into the future. They are therefore also known as time series analysis or, for historic reasons, regression methods.

A cornerstone of time series analysis is the assumption that the time series is homogeneous, i.e., the mathematical regularities underlying its variations are the same at any point of time. This implies that a forecast for, say, three years ahead has equal chance of success in the rising or decaying phase of the sunspot cycle, across the maximum or, in particular, across the minimum. In this case, distinguishing intracycle and intercycle memory effects, as we did in Sections 1.3.2 and 2, would be meaningless. This concept of solar activity variations as a continuous process stands in contrast to that underlying precursor methods, where solar cycles are thought of as individual units lasting essentially from minimum to minimum, correlations within a cycle being considerably stronger than from one cycle to the next. While, as we have seen, there is significant empirical evidence for the latter view, the possibility of time homogeneity cannot be discarded out of hand. Firstly, if we consider the time series of global parameters (e.g., amplitudes) of cycles, homogeneity may indeed be assumed fairly safely. This approach has rarely been used for the directly observed solar cycles as their number is probably too low for meaningful inferences – but the long data sets from cosmogenic radionuclides are excellent candidates for time series analysis.

In addition, there may be good reasons to consider the option of homogeneity of solar activity data even on the scale of the solar cycle. Indeed, in dynamo models the solar magnetic field simply oscillates between (weak) poloidal and (strong) toroidal configuration: there is nothing inherently special about either of the two, i.e., there is no a priori reason to attribute a special significance to solar minimum. While at first glance the butterfly diagram suggests that starting a new cycle at the minimum is the only meaningful way to do it, there may be equally good arguments for starting a new cycle at the time of polar reversal. There is, therefore, plenty of motivation to try and apply standard methods of time series analysis to sunspot data.

Indeed, as the sunspot number series is a uniquely homogeneous and long data set, collected over centuries and generated in a fairly carefully controlled manner, it has become a favorite testbed of time series analysis methods and is routinely used in textbooks and monographs for illustration purposes (Box et al., 2008; Wei, 2005; Tong, 1990). This section will summarize the various approaches, proceeding, by and large, from the simplest towards the most complex.

3.1 Linear regression

3.2 Spectral methods

3.2.1 The 11-year cycle and its harmonics

3.2.2 The even–odd (a.k.a. Gnevyshev–Ohl) rule

3.2.3 The Gleissberg cycle

3.2.4 Supersecular cycles

3.3 Nonlinear methods

3.3.1 Attractor analysis and phase space reconstruction: the pros ...

3.3.2 ... the cons ...

3.3.3 ... and the upshot

3.3.4 Neural networks

3.2 Spectral methods

3.2.1 The 11-year cycle and its harmonics

3.2.2 The even–odd (a.k.a. Gnevyshev–Ohl) rule

3.2.3 The Gleissberg cycle

3.2.4 Supersecular cycles

3.3 Nonlinear methods

3.3.1 Attractor analysis and phase space reconstruction: the pros ...

3.3.2 ... the cons ...

3.3.3 ... and the upshot

3.3.4 Neural networks

Living Rev. Solar Phys. 7, (2010), 6
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