The rate of change of Tmax was approximated as the first difference of the time series. This produces data point offset by 0.5 months when correctly centred.
1928.083333 0.000000 1928.041667 42.800000
1928.166667 0.000000 1928.125000 59.200000
1928.250000 0.000000 1928.208333 96.400000
1928.333333 0.000000 1928.291667 183.600000
1928.416667 0.000000 1928.375000 249.600000
1928.500000 0.000000 1928.458333 258.300000
1928.583333 0.000000 1928.541667 105.700000
1928.666667 0.000000 1928.625000 173.000000
1928.750000 0.000000 1928.708333 145.200000
Since the diff amplifies the noise, sun-hours was taken for the abscissa. The plot and NNLS regression were done with a data lag of two intervals on the temperature data, leaving a net lag of 1.5 months. with rate of change of temperature leading sun-hours.
This brings into question the idea of sun-hours having a causal link on Tmax. It rather suggests common causation. This can readily be suggested to be the annual increase in insolation both warming ground and air temps as well as affecting the weather systems encroaching from the North Atlantic.
There is slight necking in the middle of the data so there is still a slightly different phase relationship between winter and summer, though fairly well balanced with the lag of 1.5 months.
Though the conventional sense of regression is clearly best, there is some curvature in the data showing it is not a simple linear relationship.
[The title of the graph shows 2 mo lag. This is the lag of one line in the file. The correct analytical lag is 1.5mo as stated above.]
The same processing with 0.5 mo lag is shown here: