Cyclical behavior has long been modeled by a simple sine wave function such as Eqn. 1 that can be applied for temperature as a function of time.
To fit Earth temperature as a function of time to this sine wave, we need amplitude A, period Ƭ, and phase φ. Amplitude is the height of the sine wave above and below a centerline. Period is the length of the wave for a single increase and decrease cycle. Phase is the placement of the entire wave along the time axis. A computerized method called nonlinear regression finds these values by minimizing the differences between the data and model. Nonlinear regression programs try various combinations of the variables in an organized way with the goal of making the deviations between data and model as small as possible. Data points can be given different weights depending on various factors. In this work, all data points were given equal weights except the final fit, where data from 2014-2020 were given increased weights to bring them into better agreement. See Appendix 1 for more details about nonlinear regression.
3.1 Single Sine Fit – 1850-2013 Data
A single sine wave was fit to the measured Earth temperatures and the past five temperature epochs of Table 1. Regression was done for the amplitude, the period and phase. This first regression used only the temperature data through 2013 to be on par with the UNIPCC study . The regression results are shown in Table 2 and Fig. 5.
Every 1,036 years this heating and cooling cycle repeats itself. The last five climate change cycles in recorded history agree reasonably well in timing with this model. To the naked eye, the fit to measured temperatures also looks reasonable. The fit to the Little Ice Age, Medieval Warming, and the three other epochs in historical documents gives confidence that current Earth behavior agrees with its own history. To include these historic cycles, it was assumed that at the midpoint of each epoch (Table 1), the respective maximum or minimum temperature was reached. The single large data point for each epoch is shown in Fig. 5. All these data together yield the Single Sine Cyclical Model shown in Fig. 5 where the next maximum temperature of 14.83° C. is expected about 200 years from now in the year 2220 after a further temperature increase of another 0.24° C. from the 2013 value of 14.59° C. Total increase since 1850 would be about 1.27° C.
If you look closely at the measured temperatures compared to this single sine wave curve fit in Fig. 6, it is apparent that there is another cyclical variant present. The experienced eye of a nonlinear modeler notices that the data in Fig. 6 exhibit an oscillation. This regular trend is a strong indication that even though the fit looks reasonable in Fig. 5, improvements to the model must be made. Another sine wave needs to be added to the primary wave.
3.2 Dual Sine Fit 1850-2013 Data
When a second sine wave is added to the first, the nonlinear regression results are shown in Fig. 7. Only the original temperature data through 2013  were used. Table 3 shows the parameters for this fit where a primary period of 1,100 years was found.
This dual sine model predicts that the Earth’s temperature will increase only about another 0.34° C. which will be achieved in about 190 years in the year 2210 with a maximum temperature then of about 14.93° C. Total temperature increases since 1850 would be about 1.37° C. Additionally, the period of 1,100 years is longer, though not greatly different from the 1,036 years found by the first fit. Fig. 8 shows that Earth temperature oscillations agree much better with the dual sine cyclical model than with the single sine model. The second sine wave oscillation has a period of 68 years according to this fit.
3.3 First Dual Sine Fit 1850-2020 Data
When the same analysis is done including the additional temperatures  from 2014-2020, the nonlinear regression results are shown in Figs. 9 and 10. Each temperature value was equally weighted. The 2014-2020 data do not fit very well. There are different characteristics in these data, as already discussed in Fig. 3. Table 6 shows the resulting parameters for this fit.
3.4 Final Dual Sine Fit 1850-2020 Data
To force the 2014-2020 data into this model, the weights for those years were increased to five compared to one for the rest of the data points. This gives extra importance to recent data assuming it is valid. Weights from 2 to 10 were evaluated with the 5 weights judged to be the best compromise, with deviations for the 2014-2020 data greatly reduced while deviations for all other data not greatly increased. Figs. 11 and 12 and Table 7 show the final best fit for all temperature data. This fit is the best one can do to include the unusual increase in reported temperatures since 2013. If more consistent temperature values are ultimately reported, a final fit may not need to give the 2014-2020 data increased weight.
3.4.1 Regular Downdip and Upswing Temperature Intervals
Each oscillation of the dual sine cyclical model cycles in Figs. 7, 9, and 11 consists of two parts: a downdip followed by an upswing. With the primary sinewave temperature increasing, the temperature declines during a downdip are less than the temperature increases during an upswing. These differences are due to the increasing and decreasing contributions of the secondary sinewave cycle. The time interval between a local maximum and the next minimum is herein defined as a downdip, while the interval between a local minimum and the next maximum is defined as an upswing. Fig. 13 shows the first downdip and upswing since 1850. The slope (derivative) of the dual sine wave curve shown in Fig. 13 as the large-dashed line is used to locate the points of zero slope on the temperature curve. The vertical small-dashed lines designate the points of zero slope showing downdip and upswing.
Fig. 14 shows all five downdips and upswings between 1850 and the next overall maximum temperature predicted to be in 2232. Tables 8 and 9 contain the detailed values where downdips become longer as upswings become shorter. Fig. 15 shows the measured Earth temperature data with downdip interval temperatures shown as open circles and upswing intervals as solid circles. Downdips and upswings are clearly seen, with upswings demonstrating increased temperatures, while the two downdips show more irregularity. Note that the period from 1945-1975 when another ice age was feared  possibly due to aerosols in the atmosphere fits closely (1947-1976) with the 29-year Downdip 2 shown in the figure. There is also a prior Downdip 1 from 1875-1903 that did not receive notoriety as a nonincreasing temperature interval. The final dual sinewave cyclical model predicts that we will be in Downdip 3 from 2019 until 2049.
At this point all results presented were generated only from Earth temperature data since 1850 and the midpoint dates for the last five historical epochs. The only assumption was that the current and past five epochs reached the same high and low temperatures. Nonlinear regression alone determined the model parameters to make model and measured temperatures fit best. The next step is to compare the final model (Fig. 11) with various solar and climatological correlations.
3.4.2 Model Comparison to Solar Cycles
The primary cycle of 1,071 years found in the final dual sine model is supported by 14C/12C results using wavelet analyses where a 1,000 year solar cycle  was found. Additional support is given from Atlantic drift ice cycles found to exhibit 1,470 Earth temperature cycles . All the other proxy data  having found cycles between 1,000 and 2,000 years also are consistent with 1,071 years.
The final dual sine cyclical model correlates closely with solar sunspot history . Fig. 16 shows that downdips occur during sunspot minima while upswings occur during maxima. This agreement affirms the counterintuitive observation that “Irradiance is greatest during sunspot maxima and lowest during sunspot minima” . Fig. 17 shows that the 24 sunspot data tops from 1749-2019 exhibit an approximate secondary cyclicity of 71 years in addition to the primary Schwabe 11 year cycles. The 71 year cycle is nearly equal to the 73 year cyclicity within the Earth’s temperature data itself. It is apparent that sunspots with their effect on solar irradiance are the direct cause for the Earth secondary temperature cycles.
Future sunspot behavior that would be consistent with the final dual sine cyclical model is shown in Fig 18. Tables 8 and 9 contain the specific timing predictions for future sunspot activity. Current sunspot 25 is predicted to be “feeble” just like 24 with a value of 115 in 2025 . The future sunspot maxima values shown are very approximate with total sunspot numbers increasing until 2232 and afterward decreasing. Such behavior would be consistent with slightly increasing solar irradiance followed by reductions after the next maximum Earth temperature in 2232. Sunspots should be minimum for cycles 25-26 during the next downdip from 2019-2049. Fig. 19 is a plot like Fig. 18 using temperature data only from 1850-2013. It is seen that the agreement here is slightly better with solar cycle 24 just at the beginning of the downdip whereas in Fig. 18 it is slightly before the end of an upswing. This sheds a little bit of doubt on the Earth temperature data from 2014-2020.
In summary, the 1,071 year primary cycle period of the final dual sine cyclical model is believed to be driven by solar activity  and is consistent with proxy results .
3.4.3 Model Comparison with Climatological Correlations
The 73-year secondary sine cycle found within the temperature data also correlates closely with the Atlantic Multidecadal Oscillation (AMO) and the Pacific Decadal Oscillation (PDO). Fig. 20 shows this close correlation with the AMO , with a 69-year oscillation between 1926 and 1999. Not only does the AMO have a similar period, but it is also in close sync with the increases and decreases of the temperature fluctuations. From Fig. 20 the expectation is that there would be a decreasing AMO index and cooling during the next 20 to 30 years. The PDO index  also has a good correlation with the secondary temperature variations, as shown in Fig. 21. The PDO period was approximately 65 years compared to the 73 years found in the secondary temperature oscillation. It also says that in the next 20 or 30 years, cooling is expected.
In summary, the secondary temperature oscillations within the Earth temperature data agree with the AMO and PDO surface temperature oscillations and all three are caused by sunspot cycles.