Not a process engineer, but I'm guessing that if you're looking at correlation between temperature data vs. AC power, and AC is reacting to changes in temperature (i.e. derivative, D in PID), you'll see AC ramping up power before the temperature meaningfully increases. A naive analysis could then conclude that it's AC that causes high temperature.
Or more simply than that, if you only ever look at "cold" and "hot" rooms (e.g. analysis of cancerous and non-cancerous patients) you're never going to see the dynamics at all.
But more fundamentally, I believe there is a theorem which asserts that it is in general impossible to extract the open loop behavior of a system (i.e. a model of how the system would respond without a controller) solely from closed loop data (i.e. data collected from the system when the controller is on.)
It was several years ago, so I don't remember the details, but I have heard echos of the sentiment in the recent criticism of alzheimers research. I.e. that amyloid plaques may actually be the equivalent of a cellular AC that's been misidentified as the reason the room is hot.
https://www.nature.com/articles/d41586-018-05719-4
>“The current approach is based on the idea that amyloid-β is bad,” says Perry. “My idea is the opposite.” He suggests that amyloid-β and tau accumulation is actually a protective response to age-related metabolic pressures in the cell.
I’d enjoy hearing more about this. Go on, or point to more info?