I start this blog post with a plot of the sunspot and temperature anomaly data that I have been using. The data is from 1850 to 2013.
The following plots are of the phase spectrum between the sunspot time series and the temperature anomaly time series. Also, the variable tau is plotted next to the spectrum. Tau would give the slope of the phase spectrum curve if the intercept were zero for a line fitting a stretch of curve. Tau is the phase spectrum divided by the frequency. The first set of plots uses all of the data. The second set of plots just uses the first 250 points. There are 1968 months in each time series.
From the plot, with frequencies up to around 275 cycles per 1968 periods (1968 periods divided by 275 cycles equals about 7 periods per cycle) have mainly an increasing phase relationship. From about 300 cycles per 1968 periods to 800 cycles per 1968 periods (about 7 periods per cycle to about 2.5 periods per cycle), the slope is mainly decreasing. For frequencies greater than 800 cycles per 1968 periods, the mainly slope is flat.
Constant positive slopes indicate that sunspots precede temperature anomaly with a constant lag. Constant negative slopes indicate than temperature anomaly precedes sunspots with a constant lag. A flat slope indicates no lagged relationship.
Looking at the plots of tau above, out to around 22 cycles per 1986 periods (about 90 periods per cycle), tau was changing quite a bit. Out beyond around 250 cycles per 1986 periods (about 7 periods per cycle), tau does not change much.