regression
How many lags, and who's driving whom?
It is 1968 at a Japanese cement plant, and the rotary kiln is fighting back: fuel-feed rate and burning-zone temperature sit in a closed loop, each one driving the other, and the operators cannot say which lever causes what. You have two coupled time series and two questions that must be answered together. First, how many past lags does an autoregressive model of this system need — too few and the real dynamics go unmodeled, too many and you are fitting sensor noise that collapses the moment it forecasts. Second, harder: does temperature respond to fuel, or fuel to temperature, or both — a control scheme built on the wrong direction of influence amplifies oscillations instead of damping them. No textbook order exists to consult. Derive the criterion that trades fit against forecast error honestly, then read the direction of influence out of the fitted model. Guess wrong on either count and the kiln runs hot, the fuel bill climbs, and the cement comes out wrong.
Who this problem belongs to
The two figures whose methods fit it best, out of 8 in contention.
This is Akaike's own problem, transplanted from his own working life. Through the 1960s at Tokyo's Institute of Statistical Mathematics he worked directly on cement-kiln and thermal-plant control, where the practical question was never abstract: how many lags does the autoregressive model of a coupled process need, and which channel is actually driving which. His 1969 Final Prediction Error criterion gave the order-selection answer directly from one-step-ahead forecast error rather than in-sample fit, and his multivariate AR feedback-identification work — reformulated through a Kalman-filter state-space representation for efficient likelihood computation across candidate orders — gave the relative-power-contribution machinery that reads causal direction out of a fitted closed-loop model. The 1973 AIC generalizes the same instinct beyond AR order. There is no daylight between this problem and the one he actually solved.
Kalman supplies half the machine this problem needs without ever asking Akaike's question. His 1960 paper gave the recursive filter for optimal state estimation in a known linear dynamical system, and it is exactly this state-space reformulation that Akaike leaned on from the mid-1960s onward to compute likelihoods for AR models efficiently across many candidate orders — the practical trick that makes trying five, ten, or twenty lag lengths computationally tractable instead of prohibitive. Kalman's toolkit is unmatched for fusing noisy measurements into a state estimate once the model and its order are already fixed. What he never built is the order-selection criterion itself, or any notion of reading feedback direction out of a fitted model — his filter assumes the system structure, it does not choose it. He is the engine under the hood, not the driver deciding which road to take.
8 figures are scored on this problem. Draw it in a battle to see where you land.