The CSALT model is able to pick out the TSI component of the average global temperature, matching the predicted value of about 0.05C thermal forcing over the last 130 years.
As an interesting experiment, I replaced the empirically observed and measured TSI profile with a set of harmonics representing the Hale cycle of solar activity. This essentially described a pure sine wave with period of 22 years and five harmonics with periods of 11, 7.3, 5.5, 4.4, and 3.7 years. Each of these has an arbitrary phase and amplitude that is fitted by the CSALT multiple linear regression algorithm.
The result is shown in Figure 1 below.
What is significant about alignment is that the GISS temperature profile does not clearly show the Hale cycle, yet the CSALT regression model is able to dig the primary peaks out of the signal, showing the correct phase and relative amplitude corresponding to the TSI. Note that the TSI does not provide any hints as that component was removed. The actual assistance was provided by the compensating removal of the other forcing components consisting of CO2, SOI, volcanic aerosols, and LOI corrections. In addition, the orbital and tidal parameters recently added to CSALT provided further discrimination. Significantly, the CSALT tidal factors also showed an exact phase registration with the observed diurnal tides observed.
A critical harmonic in the series is the 1/3 period of the Hale cycle of approximately 7.3 years. This appears to be an important factor in interpreting the North Atlantic oscillation dynamics, with the Hale "family" of harmonics as described in . The 7.3 year period also has a relationship to the tidal precession, as it describes the duration of time it takes for the spring tides to realign with the calendar date -- which is close to the perigee cycle of 7.7 years. As  states, "whole year multiples are important because, ultimately, climate cycles have to be tied to seasons". So this may in fact be a resonance that ties the tides to the solar cycle.
The 6 Hale frequencies also match those observed from solar neutrino experiments , as Mandal and Rachauduri find from the Homestake solar neutrino flux data:
"Wavelet amplitude versus periodicities graphs have been presented and it is observed that clear peaks are arising around the 22 years, 11 years, 7.3 years, 5.5 years, 4.4 years & 3.7 years periodicities. Among the height of the peaks shows that the periodicities of 5.5 years, 7.3 years, 11 years, 22 years are above the 95% confidence level, which strongly favours the existence of periodicities in the Homestake solar neutrino flux data. These type of periodicities are observed in the other forms of solar activities (i.e. sunspot number data, solar flare data etc.)."
Higher-order harmonics are evidently important in periodically pulsed dynamos as they provide the high-frequency content necessary to create the spikes while maintaining the observed periodic content.
What is fascinating about the CSALT model approach is that as the correlation between temperature data and model nears unity (nearing 0.995 now with limited filtering) the ability to discriminate fine features in the factoring increases markedly. This is not about being able to model a system given enough parameters, but closer to the idea that all these minor factors play a role and we are simply exposing them with the CSALT approach.
It also appears that the ongoing work presented by Judith Lean at the recent AGU work parallels this approach. The NRL statistical climate model shown below and described by David Appel here builds on her previous work . The truth is uncovered by looking at the statistics and mean value of the compensating natural factors, and not by GCMs and other analyses that allow the non-determinism of individual runs to suggest the outcome.
All these factors need to be taken together in a statistical fashion, and the truth will continue to emerge.