Hypothesised flow of causality by control system element across the hypothesised causal channel.
Control system stepFromFlow of CausalityTo1Candidate leading element and not Candidate controller terms2Candidate controller terms and not Candidate actuators3Candidate actuators and not Outcome
Table 3.
Results of assessments of Granger causality for candidate control elements over Step 1 of the control system. Abbreviations not otherwise explained in text: T-Y – Toda-Yamamoto; AIC – Akaike information criterion. Data series used are monthly.
RowControl system step 1Biosphere leading elementController termVariable AVariable BLags (AIC)T-Y test usedNo autocorrelation out to n lagsProbability that relationship not Granger causalCausality A to BCausality B to A1NDVI_CONTRP_CO2_CONTR74N24 1.58E-050.41232NDVI_CONTRI_CO2_CONTR74N90.00690.23943NDVI_CONTRD_CO2_CONTR49N130.01240.33034NDVI_CONTRDD_CO2_CONTR46N120.0230.355
Table 2.
Candidate control system elements in the combinations to be assessed for connectivity by Granger causality analysis.
Control system step 1Control system step 2Control system step 3Control system step 4Leading element toController termController term toActuatorActuator toPenultimate outcomePenultimate outcome toFinal outcomeNDVI_CONTRP_CO2_CONTRP_CO2_CONTRWind_speed_CONTRWind_speed_FROM_ 2002_CONTROcean_heat_uptake_ REVERSEOcean_cloud_CONTRAt_tempNDVI_CONTRI_CO2_CONTRP_CO2_CONTROcean_cloud_CONTRWind_speed_CONTROLR_REVERSELand_cloud_CONTRAt_tempNDVI_CONTRD_CO2_CONTRP_CO2_CONTRLand_cloud_CONTROcean_cloud_CONTROcean_heat_uptake_ REVERSEOcean_heat_uptake_ REVERSEAt_tempNDVI_CONTRDD_CO2_CONTRP_CO2_CONTRSOI_CONTROcean_cloud_CONTROLR_REVERSEOLR_REVERSEAt_tempI_CO2_CONTRWind_speed_CONTRLand_cloud_CONTROcean_heat_uptake_ REVERSEI_CO2_CONTROcean_cloud_CONTRLand_cloud_CONTROLR_REVERSEI_CO2_CONTRLand_cloud_CONTRSOI_CONTROcean_heat_uptake_ REVERSEI_CO2_CONTRSOI_CONTRSOI_CONTROLR_REVERSED_CO2_CONTRWind_speed_CONTRD_CO2_CONTROcean_cloud_CONTRD_CO2_CONTRLand_cloud_CONTRD_CO2_CONTRSOI_CONTRDD_CO2_CONTRWind_speed_CONTRDD_CO2_CONTROcean_cloud_CONTRDD_CO2_CONTRLand_cloud_CONTRDD_CO2_CONTRSOI_CONTR
Table 4.
Results of assessments of Granger causality for candidate control elements over Step 2 of the control system. Abbreviations not otherwise explained in text: T-Y – Toda-Yamamoto; AIC – Akaike information criterion. Data series used are monthly.
RowControl system step 2Controller termActuatorVariable AVariable BLags (AIC)T-Y test usedNo autocorrelation out to n lagsProbability that relationship not Granger causalCausality A to BCausality B to A1P_CO2_CONTRWind_speed_CONTR41Y120.02680.18472P_CO2_CONTROcean_cloud_CONTR61Y70.07660.65473P_CO2_CONTRLand_cloud_CONTR54Y120.01430.19874P_CO2_CONTRSOI_CONTR67N200.00090.05375I_CO2_CONTRWind_speed_CONTR27Y150.01410.21046I_CO2_CONTROcean_cloud_CONTR15N70.02220.01487I_CO2_CONTRLand_cloud_CONTR27N110.08810.018I_CO2_CONTRSOI_CONTR41N160.01960.00769D_CO2_CONTRWind_speed_CONTR41Y90.0540.116410D_CO2_CONTROcean_cloud_CONTR80N120.03430.485311D_CO2_CONTRLand_cloud_CONTR68N120.04740.808812D_CO2_CONTRSOI_CONTR29N100.0040.000813DD_CO2_CONTRWind_speed_CONTR18Y40.02490.007914DD_CO2_CONTROcean_cloud_CONTR67N90.08090.335415DD_CO2_CONTRLand_cloud_CONTR81N120.02090.694516DD_CO2_CONTRSOI_CONTR80N50.0010.07
Table 5.
Results of assessments of Granger causality for candidate control elements over Step 3 of the control system. Abbreviations not otherwise explained in text: T-Y – Toda-Yamamoto; AIC – Akaike information criterion. Data series used are monthly except where marked with the suffix ‘quar’, which are quarterly. The Granger causality assessment for Wind speed to Ocean_heat_uptake is based on data from 2002 onwards (see text).
RowControl system step 3ActuatorPenultimate outcomeVariable AVariable BLags (AIC)T-Y test usedNo autocor- relation out to n lagsProbability that relationship not Granger causalCausality A to BCausality B to A1Wind_speed_FROM_ 2002_CONTR_QuarOcean_heat_uptake_ REVERSE_quar8Y80.00290.13772Wind_speed_CONTROLR_REVERSE40Y12 0.0055 0.10773Ocean_cloud_ CONTR_quarOcean_heat_uptake_ REVERSE_quar27Y120.70440.6264Ocean_cloud_CONTROLR_REVERSE55Y200.01570.10295Land_cloud_ CONTR_quarOcean_heat_uptake_ REVERSE_quar40Y200.40020.54826Land_cloud_CONTROLR_REVERSE27Y120.00060.13387SOI_CONTR_quarOcean_heat_uptake_ REVERSE_quar41Y200.12830.46548SOI_CONTROLR_REVERSE30Y200.17130.0503
Table 6.
Results of assessments of Granger causality for candidate control elements over Step 4 of the control system. Abbreviations not otherwise explained in text: T-Y – Toda-Yamamoto; AIC – Akaike information criterion. Data series used are monthly except where marked with the suffix ‘quar’, which are quarterly.
RowControl system step 4Actuator or Penultimate outcomeFinal outcomeVariable AVariable BLags (AIC)T-Y test usedNo autocor- relation out to n lagsProbability that relationship not Granger causalCausality A to BCausality B to A1Ocean_cloud_CONTRAt_temp14Y110.01450.31092Land_cloud_CONTR_quarAt_temp12N130.00340.20473Ocean_heat_uptake_REVERSE_ quarAt_temp_quar5Y60.00170.07024OLR_REVERSEAt_temp28Y60.00450.2533
Fig. 1.
Second-difference CO2 (blue curve) and Reverse Southern Oscillation Index (red curve); monthly data.
Fig. 2.
Trends in expected temperature from the output of a mid-range IPCC scenario model (CMIP5, RCP4.5 scenario) (dark blue curve), observed temperature (blue curve), and control system candidate process elements: P_CO2_CONTR (black curve); I_CO2_CONTR (turquoise curve); D_CO2_CONTR (brown curve); and DD_CO2_CONTR (red curve); annual data, 1901 to 2018; all data Z-scored, base period 1901-1987.
Fig. 3.
Trends in expected temperature from the output of a mid-range IPCC scenario model (CMIP5, RCP4.5 scenario) (dark blue curve), observed temperature (blue curve), and control system candidate physical elements: Ocean_heat_uptake_REVERSE (sea green curve); NDVI_CONTR (bright green curve); Wind_speed_CONTR (violet curve); OLR_REVERSE (purple curve); Ocean_cloud_CONTR (black curve); Land_cloud_CONTR (aqua curve); and SOI_CONTR (red curve); annual data, 1901 to 2018; all data Z-scored, base period 1901-1987.
Table 7.
Summary of the results of the scan of Tables 3 to 6 for causal steps connecting to make end-to-end causal chains in line with the control system hypothesis. Results are assembled without the numerical information presented in those tables. Instead, each pair of cells displaying one-way Granger causality at the 0.05 level in accordance with the hypothesis is shown by means of a grey background to the pair of cells.
RowControl system step 1Control system step 2Control system step 3Control system step 4Biosphere leading element toController termController term toActuatorActuator toPenultimate outcomeActuator or Penultimate outcome toFinal outcomeVariable AVariable BVariable AVariable BVariable AVariable BVariable AVariable B1NDVI_CONTRP_CO2_CONTRP_CO2_CONTRWind_speed_CONTRWSPEED_FROM_ 2002_CONTR_QuarOcean heat uptake_ REVERSE_quarOcean_cloud_CONTRAt_temp2NDVI_CONTRI_CO2_CONTRP_CO2_CONTROcean_cloud_CONTRWind_speed_CONTROLR_REVERSELand_cloud_CONTR_quarAt_temp3NDVI_CONTRD_CO2_CONTRP_CO2_CONTRLand_cloud_CONTROcean_cloud_ CONTR_quarOcean heat uptake_ REVERSE_quarOcean heat uptake_ REVERSE_quarAt_temp_quar4NDVI_CONTRDD_CO2_CONTRP_CO2_CONTRSOI_CONTROcean_cloud_CONTROLR_REVERSEOLR_REVERSEAt_temp5I_CO2_CONTRWind_speed_CONTRLand_cloud_ CONTR_quarOcean heat uptake_ REVERSE_quar6I_CO2_CONTROcean_cloud_CONTRLand_cloud_CONTROLR_REVERSE7I_CO2_CONTRLand_cloud_CONTRSOI_CONTR_quarOcean heat uptake_ REVER SE_quar8I_CO2_CONTRSOI_CONTRSOI_CONTROLR_REVERSE9D_CO2_CONTRWind_speed_CONTR10D_CO2_CONTROcean_cloud_CONTR11D_CO2_CONTRLand_cloud_CONTR12D_CO2_CONTRSOI_CONTR13DD_CO2_CONTRWind_speed_CONTTR14DD_CO2_CONTROcean_cloud_CONTR15DD_CO2_CONTRLand_cloud_CONTR16DD_CO2_CONTRSOI_CONTR
Fig. 4.
Trends in NDVI (red line) and volcanic aerosols (sign reversed – blue line). Also shown is the linear trend for volcanic aerosols (black line); annual data, 1947 to 2018; all data Z-scored.
Fig. 5.
Evidence for the existence of a physical control system: end-to-end sequences of candidate control system elements that display one-way Granger causality at 0.05 probability level across each step of the sequence (derived from Table 7). Note that for clarity in this Figure all terms are expressed without their _CONTR or _REVERSE suffixes.
Table 8.
Results of Granger causality tests from leading element to control system final outcome.
Variable AVariable BLagsT-Y test usedNo autocorrelation out to n lagsProbability that relationship not Granger causalCausality A to BCausality B to ANDVI_CONTRAt_temp40N90.01130.1593
Table A1.
Eviews ARDL estimation output for period May 1960 to July 2018 for SOI as a function of second-difference CO2: overall statistics for short-run model dynamic relationship.
Model dependent variableNumber of models evaluatedSelected modelAutocorrelation out to 12 lagsAdjusted R-squaredF-statisticp-valueSOI_CONTR156ARDL (4,2)Nil0.9871217610.5440.00E+100
Table A2.
Eviews ARDL estimation output for period May 1960 to July 2018 for SOI as a function of second-difference CO2: statistics for each dependent variable in short-run model dynamic relationship.
Model independent variablesCoefficientStd. Errort-StatisticProb.*SOI_CONTR(-1)1.3371570.03786535.313360.00E+00SOI_CONTR(-2)−0.11110.06333−1.754367.98E-02SOI_CONTR(-3)−0.168880.062944−2.6830057.50E-03SOI_CONTR(-4)−0.084940.037848−2.2442430.0251DD_CO2_CONTR0.0823160.0271143.0359410.0025DD_CO2_CONTR(-1)−0.012750.047181−0.2702720.787DD_CO2_CONTR(-2)−0.055430.02721−2.0371530.042C−0.001360.004421−0.3085680.7577
Table A3.
Eviews ARDL estimation output for period May 1960 to July 2018 for atmospheric surface temperature as a function of putative control system P_CONTR, I_CONTR and D_CONTR terms: overall statistics for short-run model dynamic relationship.
Model dependent variableNumber of models evaluatedSelected model (SIC)Autocorrelation out to 36 lagsAdjusted R-squaredF-statisticp-valueAkaike Information CriterionAt_temp3,584ARDL(4, 0, 0, 5)Nil0.902242539.37780.00E+000.525009
Table A4.
Eviews ARDL estimation output for period May 1960 to July 2018 for temperature as a function of putative control system P_CONTR, I_CONTR and D_CONTR terms: statistics for long-run model independent variables.
Model independent variablesCoefficientStd. Errort-StatisticProb.*P_CO2_CONTR−1.6877910.596673−2.8286680.0048I_CO2_CONTR1.0130320.5980241.6939650.0907D_CO2_CONTR−0.2903990.087033−3.3366610.0009C0.1813490.0516873.5086260.0005
Table A5.
Eviews ARDL estimation output for period May 1960 to July 2018 for atmospheric surface temperature as a function of putative control system P_CONTR, I_CONTR, D_CONTR and DD_CONTR terms: overall statistics for short-run model dynamic relationship.
Model dependent variableNumber of models evaluatedSelected modelAutocorrelation out to 36 lagsAdjusted R-squaredF-statisticp-valueAkaike Information CriterionAt_temp28,672ARDL(4, 1, 0, 2, 0)Nil0.903408594.48070.00E+000.507034
Table A6.
Eviews ARDL estimation output for period May 1960 to July 2018 for temperature as a function of putative control system P_CONTR, I_CONTR, D_CONTR and DD_CONTR terms: statistics for long-run model independent variables.
Model independent variablesCoefficientStd. Errort-StatisticProb.*P_CO2_CONTR−1.94080.6487−2.991830.0029I_CO2_CONTR1.40160.6546592.1409610.0326D_CO2_CONTR−0.5193310.104663−4.9619128.81E-07DD_CO2_CONTR−0.259820.098192−2.6460330.0083C0.3705060.1030633.5949280.0003
Candidate controller terms2Candidate controller terms
