2022 · Problem B — CO₂ and Global Warming
Regression Forecasting CorrelationThe problem in one paragraph
Given Mauna Loa annual CO₂ data (1959–2021) and land-ocean temperature anomalies (1958–2021): (1) Fit multiple models to CO₂, evaluate the claim of largest-ever 10-year increase ending in 2004, predict 2100. (2) Model the temperature–CO₂ relationship, predict when temperature anomaly reaches +1.25°C, +1.50°C, +2.0°C. (3) Non-technical article.
Solution outline
1a — Was 2004 really the biggest 10-year jump?
Compute rolling 10-year deltas: $\Delta_{10}(t) = C(t) - C(t-10)$. Plot them. Compare 2004 to subsequent decades. Spoiler: by 2014 the 10-year increase had already exceeded the 2004 value, so the original claim is true only at that snapshot, not in perpetuity.
1b — Fit models
- Linear: $C(t) = a + b(t - 1959)$. Underfits.
- Quadratic: $C(t) = a + b\tau + c\tau^2$ where $\tau = t - 1959$. Strong fit, captures acceleration.
- Exponential excess: $C(t) - 280 = A \cdot e^{r\tau}$ models the "excess over pre-industrial" growing exponentially.
- Logistic: if you believe in eventual saturation.
Report $R^2$, AIC, and 2050/2100 predictions for each.
| Model | 2050 pred. (ppm) | 2100 pred. (ppm) | Caveat |
|---|---|---|---|
| Linear | ~440 | ~510 | Underestimates; ignores acceleration |
| Quadratic | ~480 | ~620 | Reasonable mid-century, careful at 2100 |
| Exp-excess | ~510 | ~770 | Likely overestimates if mitigation kicks in |
Quadratic is usually the best HiMCM-level choice — simple, defensible, fits well.
2 — Temperature vs. CO₂
Step (a): fit $T(t)$ over time using polynomial or piecewise linear. Solve for when $T$ crosses +1.25, +1.50, +2.00. Step (b): regress $T$ on $C$ — strong linear correlation expected (Pearson $r \approx 0.95$). Climate physics says $T \approx S \log_2(C / C_0)$ for some climate sensitivity $S$ (≈ 3 °C per doubling). Using that form fits the data well.
3 — Article
For Scientific Today's general readership. Lead with the most striking finding (e.g., "Our analysis predicts the world will cross the 1.5 °C Paris threshold by 20XX"). End with the policy implication.
Pitfalls
- Reading off a single year (2004) without rolling analysis.
- Comparing models on $R^2$ alone — judges want residual plots and physical reasoning.
- Treating correlation as causation (T-CO₂) without invoking the physics.
- Article opens with "Hello, my name is..." — judges flag this. Lead with the finding.