Two series can vary together without acting on each other: one driver pushes both (A ← C → B), or one acts only through a third (A → C → B). The links matrix tests each pair alone and cannot tell these apart. Here, each link is tested given the past of all other variables (PCMCI, Runge et al., 2019): a link produced by a common driver or by an intermediary disappears, and the direction of a delayed link follows from time order. A link of the graph is still not a demonstrated cause: the method assumes that no common driver is missing among the variables, and that links are linear and stable over time.
Variables
19
1984-07 to 2026-02, 476 usable months
Links retained
10
1 directed, 9 within the month, out of 3,699 tests
Matrix links confirmed as direct
7 / 20
4 weakened but present
False links per trial
0.00
on 10 sets of independent series (at most 0)
Causal graph of the monthly series
Modes of variability Climate Solid Earth Sun and space same direction
opposite directions
same month, no direction
Links retained
| Link | Partial correlation | q | Variables conditioned on | Stability |
|---|---|---|---|---|
| Solar wind speed – Geomagnetism (ap) (same month) | 0.67 | 0.0000 | 6 | 100 % |
| NAO – Arctic Oscillation (AO) (same month) | 0.54 | 0.0000 | 14 | 100 % |
| Global temperature – Snow (N. Hemisphere) (same month) | -0.30 | 0.0000 | 17 | 100 % |
| Southern Annular Mode – Global temperature (same month) | -0.28 | 0.0000 | 13 | 100 % |
| Arctic Oscillation (AO) – Global temperature (same month) | 0.23 | 0.0005 | 15 | 100 % |
| Arctic Oscillation (AO) – Snow (N. Hemisphere) (same month) | -0.23 | 0.0006 | 15 | 100 % |
| PDO – Arctic Oscillation (AO) (same month) | -0.22 | 0.0015 | 15 | 100 % |
| Southern Annular Mode – Length of day (same month) | 0.20 | 0.0092 | 14 | 100 % |
| Arctic Oscillation (AO) → CO₂ growth (2 months) | -0.18 | 0.0269 | 15 | 67 % |
| ENSO (Niño 3.4) – Indian dipole (same month) | 0.18 | 0.0395 | 10 | 67 % |
What becomes of the links of the matrix?
Each significant link of the matrix, and its strongest test once the other variables are accounted for (smallest p-value over the lags and both directions).
| Pair | Matrix (pair alone) | Strongest conditional test | Verdict |
|---|---|---|---|
| Solar wind speed – Geomagnetism (ap) | r = 0.71 (lag 0 months) | Solar wind speed – Geomagnetism (ap) (same month) : r = 0.67, p = 0.0000 | direct link |
| NAO – Arctic Oscillation (AO) | r = 0.60 (lag 0 months) | NAO – Arctic Oscillation (AO) (same month) : r = 0.54, p = 0.0000 | direct link |
| Global temperature – Sea level | r = 0.52 (lag -4 months) | — | not tested |
| CO₂ growth – Reefs (T anomaly) | r = 0.51 (lag -4 months) | — | not tested |
| Global temperature – Reefs (T anomaly) | r = 0.49 (lag 1 months) | — | not tested |
| North Atlantic (AMO) – Global temperature | r = 0.49 (lag 0 months) | North Atlantic (AMO) → Global temperature (1 months) : r = 0.17, p = 0.0003 | weakened |
| ENSO (Niño 3.4) – Global temperature | r = 0.49 (lag 2 months) | ENSO (Niño 3.4) → Global temperature (1 months) : r = 0.16, p = 0.0005 | weakened |
| ENSO (Niño 3.4) – Reefs (T anomaly) | r = 0.48 (lag 6 months) | — | not tested |
| North Atlantic (AMO) – Reefs (T anomaly) | r = 0.47 (lag -2 months) | — | not tested |
| ENSO (Niño 3.4) – CO₂ growth | r = 0.47 (lag 9 months) | CO₂ growth → ENSO (Niño 3.4) (3 months) : r = -0.09, p = 0.0503 | not retained |
| ENSO (Niño 3.4) – PDO | r = 0.43 (lag 3 months) | ENSO (Niño 3.4) → PDO (2 months) : r = 0.13, p = 0.0042 | weakened |
| Global temperature – CO₂ growth | r = 0.40 (lag 5 months) | Global temperature → CO₂ growth (2 months) : r = 0.12, p = 0.0087 | weakened |
| ENSO (Niño 3.4) – Sea level | r = 0.40 (lag 2 months) | — | not tested |
| ENSO (Niño 3.4) – Length of day | r = 0.39 (lag 1 months) | ENSO (Niño 3.4) → Length of day (7 months) : r = -0.17, p = 0.0003 | not retained |
| ENSO (Niño 3.4) – Indian dipole | r = 0.37 (lag -2 months) | ENSO (Niño 3.4) – Indian dipole (same month) : r = 0.18, p = 0.0001 | direct link |
| Southern Annular Mode – Global temperature | r = -0.26 (lag 0 months) | Southern Annular Mode – Global temperature (same month) : r = -0.28, p = 0.0000 | direct link |
| Arctic Oscillation (AO) – Snow (N. Hemisphere) | r = -0.25 (lag 0 months) | Arctic Oscillation (AO) – Snow (N. Hemisphere) (same month) : r = -0.23, p = 0.0000 | direct link |
| PDO – Arctic Oscillation (AO) | r = -0.24 (lag -1 months) | PDO – Arctic Oscillation (AO) (same month) : r = -0.22, p = 0.0000 | direct link |
| ENSO (Niño 3.4) – MJO (amplitude) | r = 0.23 (lag -8 months) | — | not tested |
| Global temperature – Snow (N. Hemisphere) | r = -0.22 (lag 0 months) | Global temperature – Snow (N. Hemisphere) (same month) : r = -0.30, p = 0.0000 | direct link |
“Direct link”: PCMCI keeps a link between the two variables. “Weakened”: it survives the conditioning alone (p < 0.05, same sign) but not the correction for thousands of tests. “Explained by others”: it disappears, and the graph shows a variable that can carry the correlation. “Not retained”: it disappears without an explanation in the graph; this is not proof of absence, but possibly a lack of power (a slow forcing whose effect spreads over many months, series defined as twelve-month growth, almost redundant with their own past). “Not tested”: a variable is outside the common window.
Window and choice of variables
PCMCI needs every variable at every month: the window is the longest run of months common to them, here 1984-07 to 2026-02 (476 usable months once the lags are accounted for). A variable that would shorten it below forty years is left out.
Left out for lack of common years: Florida Current, Sea level, MJO (amplitude).
Left out because it is part of the global temperature (same measurement, not a separate process): Reefs (T anomaly). Physical knowledge imposed: no Earth variable can act on Sunspots, Solar wind speed, Geomagnetism (ap).
39 memory links (a variable and its own past) are retained; they are not drawn on the graph but serve as conditions in every test.
Method and sources
PCMCI (Runge et al., 2019), in two steps. Condition selection (PC1): for each variable, the candidates Xi(t − τ), τ = 1 to 12 months, are tested given the 0, 1, 2… strongest candidates, and dropped as soon as p > 0.2; a small set of probable “parents” remains. MCI test: each link Xi(t − τ) → Xj(t) is tested given the parents of Xj and the three strongest parents of Xi shifted by τ, which removes both indirect links and the effect of autocorrelation; same-month links are tested in both orders and the larger p-value is kept, without orientation. Partial correlation on least-squares residuals, Student test with n − 2 − |Z| degrees of freedom. False discovery rate (Benjamini & Hochberg, 1995) at 5 %, computed separately for links between variables and for each variable’s memory: pooled, the many memory links raise the threshold and let false links through (seen on surrogate series).
Data. The series of the links matrix, prepared the same way (monthly anomalies, variations longer than ten years removed, standardisation), with the same maximum lag of twelve months. Physical knowledge: no link from an Earth variable to sunspots, the solar wind or the Ap index. Variable left out: the surface temperature of the reefs, part of the global temperature.
Checks. Implementation checked against the reference library tigramite: p-values identical to 10⁻¹² on synthetic data. Calibration: PCMCI applied to independent surrogate series (random phases, Ebisuzaki, 1997), where any link found is false. Robustness: selection threshold 0.05, 0.2 and 0.4; stability is the share of the three in which the link is retained.
Reading these data
A link of the graph is causal only if the assumptions of the method hold: no common driver outside the variables (CO₂ from human emissions, volcanic aerosols, the Sun on the decadal scales removed by the filter are not all there), linear links, stable over forty years, and slow enough to show from one month to the next. Same-month links, the most numerous here, say nothing about direction: NAO and AO are two close measures of the same circulation, the solar wind and the Ap index are linked within hours. The absence of a known link, such as El Niño and CO₂ growth, recalls the limits of power: a slow forcing whose effect spreads over a year cannot be tied to one precise lag once all the others are accounted for.
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