Global temperatures and sunspot numbers. Are they related? Yes, but non linearly. A reply to Gil-Alana et al. (2014)

Gil-Alana et al. (Physica A: 396, 42-50, 2014) compared the sunspot number record and the temperature record and found that they differ: the sunspot number record is characterized by a dominant 11-year cycle while the temperature record appears to be characterized by a singularity or pole in the spectral density function at the zero frequency. Consequently, they claimed that the two records are characterized by substantially different statistical fractional models and rejected the hypothesis that sun influences significantly global temperatures. I show that: (1) the singularity or pole in the spectral density function of the global surface temperature at the zero frequency does not exist - it is a typical misinterpretation that discrete power spectra of non-stationary signals can suggest; (2) appropriate continuous periodograms clarify the issue and also show a signature of the 11-year solar cycle (amplitude <0.1 K), which since 1850 has an average period of about 10.4 year, and of many other natural oscillations; (3) the solar signature in the surface temperature record can be recognized only using specific techniques of analysis that take into account non-linearity and filtering of the multiple climate change contributions; (4) the post 1880-year temperature warming trend cannot be compared or studied against the sunspot record and its 11-year cycle, but requires solar proxy models showing short and long scale oscillations plus the contribution of anthropogenic forcings, as done in the literature. Multiple evidences suggest that global temperatures and sunspot numbers are quite related to each other at multiple time scales through complex and non-linear processes. Finally, I show that the prediction of a semi-empirical model for the global temperature based on astronomical oscillations and anthropogenic forcing proposed by Scafetta since 2009 has up to date been successful.