# Solar and Anthropogenic Influences on Climate: Regression Analysis and Tentative Predictions

## Abstract

**:**

## 1. Introduction

## 2. Double Regression Analysis

#### 2.1. Data

#### 2.1.1. Temperature Data

#### 2.1.2. Geomagnetic aa Index Data

#### 2.1.3. CO${}_{2}$ Data

#### 2.2. The Most Relevant Data and Their Correlations

#### 2.3. Regression

#### 2.4. Some Plausibility Checks and Comparisons

## 3. Predictions

#### 3.1. Predicting the Solar Dynamo

#### 3.2. Some Scenarios

## 4. Summary and Discussion

## 5. Conclusions and Outlook

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Data of the HadSST sea surface temperature anomaly $\Delta T$ (

**a**), the aa index (

**b**) and ${log}_{2}$ of the ratio of the CO${}_{2}$ concentration to the reference value of 280 ppm (

**c**). The annual data between 1850 and 2018 are complemented by centered moving averages with windows 11 years and 23 years, as also utilized by Mufti and Shah [46]. The sources of the data are described in the text.

**Figure 2.**Comparison between the centered moving averages over 23 years of the four datasets $\Delta T$, aa index, AMO index and PDO index. Note the remarkable parallelity of $\Delta T$ and aa index until 1990, and the divergence thereafter. The AMO index also has some similarity with $\Delta T$, while the PDO index is significantly different. The AMO index data were obtained from https://psl.noaa.gov/data/timeseries/AMO/ (accessed on 17 December 2020), the PDO index data from https://psl.noaa.gov/pdo/ (accessed on 22 December 2020).

**Figure 3.**Correlations between various data, each representing a 23-year centered moving average, in dependence on the time shift $\delta t$ between them. In each case, the first item indicates the earlier dataset, the second the later one. The aa-$\Delta T$ correlation starts at a value $r=0.8$ for $\delta t=0$ and reaches a maximum of $r=0.915$ for $\delta t=10$ years. While this might insinuate a 10-year causal time shift between aa index and $\Delta T$, it is more likely connected with the canceling effect of the latest decade with its poor correlation. In the second curve, for which we completely omitted the last 10 years of both data, we obtain $r=0.95$ at $\delta t=0$, increasing to a maximum of $r=0.962$ at $\delta t=3$ years. Compared to those large r-values, the correlation of $\Delta T$ and AMO is much smaller, viz 0.3 at $\delta t=6$ years (AMO lagging behind temperature). The correlation of aa index and AMO is still smaller, with a maximum of $r=0.129$ for $\delta t=11$ years.

**Figure 4.**Regression analysis for the full data (with end year 2018). (

**a**) $\mathrm{FVU}$ in dependence on ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ for $\mathrm{MAW}=11$ years, with ${\mathrm{FVU}}_{\mathrm{min}}=0.107$ reached for ${w}_{\mathrm{aa}}=0.011$ K/nT and ${w}_{{\mathrm{CO}}_{2}}=1.72$ K. (

**b**) The same for $\mathrm{MAW}=23$ years, with ${\mathrm{FVU}}_{\mathrm{min}}=0.102$ reached for ${w}_{\mathrm{aa}}=0.0162$ K/nT and ${w}_{{\mathrm{CO}}_{2}}=1.54$ K. (

**c**) The 11-year and 23-year moving averages for the original $\Delta T$ (purple) and for the reconstructed $\Delta T$ (red) when using the optimized values of ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ from (

**a**,

**b**).

**Figure 5.**Regression analysis for the reduced data (end year 2013). (

**a**) $\mathrm{FVU}$ in dependence on ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ for $\mathrm{MAW}=11$ years, with ${\mathrm{FVU}}_{\mathrm{min}}=0.134$ reached for ${w}_{\mathrm{aa}}=0.0145$ K/nT and ${w}_{{\mathrm{CO}}_{2}}=1.56$ K. (

**b**) The same for $\mathrm{MAW}=23$ years, with ${\mathrm{FVU}}_{\mathrm{min}}=0.105$ reached for ${w}_{\mathrm{aa}}=0.0232$ K/nT and ${w}_{{\mathrm{CO}}_{2}}=1.21$ K. (

**c**) The 11-year and 23-year moving averages for the original $\Delta T$ (purple) and for the reconstructed $\Delta T$ (red) when using the optimized values of ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ from (

**a**,

**b**).

**Figure 6.**Regression analysis for the reduced data (end year 2008). (

**a**) $\mathrm{FVU}$ in dependence on ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ for $\mathrm{MAW}=11$ years, with ${\mathrm{FVU}}_{\mathrm{min}}=0.149$ reached for ${w}_{\mathrm{aa}}=0.019$ K/nT and ${w}_{{\mathrm{CO}}_{2}}=1.33$ K. (

**b**) The same for $\mathrm{MAW}=23$ years, with ${\mathrm{FVU}}_{\mathrm{min}}=0.099$ reached for ${w}_{\mathrm{aa}}=0.0305$ K/nT and ${w}_{{\mathrm{CO}}_{2}}=0.80$ K. (

**c**) The 11-year and 23-year moving averages for the original $\Delta T$ (purple) and for the reconstructed $\Delta T$ (red) when using the optimized values of ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ from (

**a**,

**b**).

**Figure 7.**Results of the regression in dependence on the width of the moving average window ($\mathrm{MAW}$). (

**a**) ${R}^{2}$ and its adjusted version ${\overline{R}}^{2}$, each for the three time intervals ending in 2018, 2013, 2008. The (shallow) local maxima of ${\overline{R}}^{2}$ around $\mathrm{MAW}=25$ years are indicated by full symbols. (

**b**) ${w}_{\mathrm{aa}}$ in dependence on $\mathrm{MAW}$. The full symbols are the values corresponding to the local maxima in (

**a**). (

**c**) Same as (

**b**), but for ${w}_{{\mathrm{CO}}_{2}}$. (

**d**) Regression result in the two-dimensional parameter space of ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$. Note the universal shape of the solution resulting in a slightly bent, but nearly linear function connecting the extremal values ${w}_{{\mathrm{CO}}_{2}}\approx 1.9$ K on the ordinate axis and ${w}_{\mathrm{aa}}\approx 0.04$ K/nT on the abscissa.

**Figure 8.**Original $\Delta T$ data, averaged over 25 years and reconstructions based on the optimal combinations of ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ from Figure 7. The dashed segments of the green and blue curves indicate the time intervals that did not enter in the respective regressions. From the red over green to blue curve, we see an ever improving reconstruction of the oscillatory behaviour, and an ever increasing divergence with the observed data at later years.

**Figure 9.**Climate predictions until 2150. (

**a**) The 23-year moving average of the aa index, and three 3-frequency fitted to it, extrapolated until 2150. (

**b**) Three scenarios for CO${}_{2}$ concentration. (

**c**) Temperature forecasts for the first CO${}_{2}$ scenario (dashed curve in (

**b**)), for the three pairs of ${w}_{\mathrm{aa}}$ and ${w}_{{\mathrm{CO}}_{2}}$ resulting from regression with end year 2018 (red), 2013 (green), 2008 (blue). Each of the coloured bundles comprise the three different 3-frequency fits from (

**a**). (

**d**) Same as (

**c**), but for the second CO${}_{2}$ scenario with a mild decarbonization scheme (dotted line in (

**b**)). (

**e**) Same as (

**c**), but for a third CO${}_{2}$ scenario with a radical decarbonization (dash-dotted line in (

**b**)).

**Table 1.**Empirical correlation coefficients r between different datasets, each of which represents a centered moving average over 11 or 23 years. The values in parentheses indicate the (one-sided) p-values, based on the “honest” number of degrees of freedom, i.e., ${N}_{\mathrm{y}}/11$ or ${N}_{\mathrm{y}}/23$, wherein ${N}_{\mathrm{y}}$ is the number of years. Note the large values for the correlation both between CO${}_{2}$ and the aa index with $\Delta T$, compared to weak or barely existing correlations of the AMO and PDO index with $\Delta T$.

MAW 11 Years: 1855 till | MAW 23 Years: 1867 till | |||||
---|---|---|---|---|---|---|

Correlated Data | 2013 | 2008 | 2003 | 2008 | 2003 | 1998 |

CO${}_{2}$ with aa | 0.440 | 0.606 | 0.720 | 0.703 | 0.783 | 0.803 |

(one-sided p) | (0.053) | (0.0099) | (0.0017) | (0.045) | (0.023) | (0.02) |

CO${}_{2}$ with $\Delta T$ | 0.926 | 0.911 | 0.891 | 0.916 | 0.894 | 0.869 |

(one-sided p) | (0) | (0) | (0) | (0.0017) | (0.0037) | (0.0073) |

aa with $\Delta T$ | 0.595 | 0.738 | 0.819 | 0.806 | 0.900 | 0.950 |

(one-sided p) | (0.010) | (0.001) | (0.0001) | (0.016) | (0.0032) | (0.0005) |

AMO with $\Delta T$ | 0.260 | 0.164 | 0.106 | |||

(one-sided p) | (0.30) | (0.38) | (0.42) | |||

aa with AMO | −0.015 | −0.013 | −0.028 | |||

(one-sided p) | (0.49) | (0.40) | (0.48) | |||

PDO with $\Delta T$ | −0.131 | −0.001 | 0.126 | |||

(one-sided p) | (0.40) | (0.50) | (0.41) | |||

aa with PDO | 0.050 | 0.049 | 0.074 | |||

(one-sided p) | (0.46) | (0.46) | (0.44) |

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Stefani, F. Solar and Anthropogenic Influences on Climate: Regression Analysis and Tentative Predictions. *Climate* **2021**, *9*, 163.
https://doi.org/10.3390/cli9110163

**AMA Style**

Stefani F. Solar and Anthropogenic Influences on Climate: Regression Analysis and Tentative Predictions. *Climate*. 2021; 9(11):163.
https://doi.org/10.3390/cli9110163

**Chicago/Turabian Style**

Stefani, Frank. 2021. "Solar and Anthropogenic Influences on Climate: Regression Analysis and Tentative Predictions" *Climate* 9, no. 11: 163.
https://doi.org/10.3390/cli9110163