3.1 Linear regression

Linear (auto)regression means representing the value of a time series at time t by a linear combination of values at times t − Δt, t − 2 Δt, ..., t − pΔt. Admitting some random error 𝜖n, the value of R in point n is
p ∑ Rn = R0 + cn−iRn −i + 𝜖n , i=1

where p is the order of the autoregression and the ci’s are weight parameters. A further twist on the model admits a propagation of errors from the previous q points:

p q ∑ ∑ Rn = R0 + cn−iRn− i + 𝜖n + dn −i𝜖n− i. i=1 i=1

This is known as the ARMA (AutoRegressive Moving Average) model.

Linear regression techniques have been widely used for solar activity prediction during the course of an ongoing cycle. Their application for cycle-to-cycle prediction has been less common and successful (Lomb and Andersen, 1980Jump To The Next Citation PointBox et al., 2008Wei, 2005).

Brajša et al. (2009Jump To The Next Citation Point) applied an ARMA model to the series of annual values of R. A successful fit was found for p = 6, q = 6. Using this fit, the next solar maximum was predicted to take place around 2012.0 with an amplitude 90 ± 27, and the following minimum occurring in 2017.

Instead of applying an autoregression model directly to SSN data, Hiremath (2008Jump To The Next Citation Point) applied it to a forced and damped harmonic oscillator model claimed to well represent the SSN series. This resulted in a predicted amplitude of 110 ± 10 for solar cycle 24, with the cycle starting in mid-2008 and lasting 9.34 years.

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