I was a TPMS developer for 7 years.
1) No matter what gas you put in a tire, if it gets cold the pressure goes down. PV = nRT works also for tires, and it is pretty accurate.
Want some numbers? 100mBar drop/10C works in most cases.
if you drive the tire gets hot and pressure goes up, become flat after 15min drive normally.
Yes altitude play a role too, if you fill up the tire on top of the mountain, when you go down in the valley the tire have lost pressure. Yes, even if no air escaped.
2) DIRECT SYSTEMS (normally with metal valve stem): The sensors have an RFID and sent temp, pressure (plus other stuff) to the car. The challenge it to let the car know which are ist own 4 RFIDs and the positions (I could write pages and pages on this, I make it short). Expensive system.
3) INDIRECT SYSTEMS (normally with rubber/plastic valve stem): Compare the ABS wheel speed sensors and find out the bad (low pressure) tires. Works 100% only with tires approved/tested by RnD as you have to map in the SW how a particular tires change its rotations properties when pressure drop. If all 4 tires drop about the same pressure [we call it "temperature diffusion"] (for example because its gets cold) there is no way to catch it with an ind. system. Cheap system, no hardware just some SW lines.
By doing a FFT you can catch a diffusion and warn the driver, but you need a lot of computational power: $$ goes up, make no sense for industrialization.
4) No tire is perfect, some air always escape. About 0.4-0.6 Bar/year is what I saw in good tires. This is called "tire diffusion".
Customer loves indirect systems as they throw a warning only when there is a puncture in the tire. On the other hands you should NOT install an unknown tire, even if it fits.
Customer hates direct system because when it get cold the TPMS light turns on, plus they need to buy new sensors every 10 years. Moreover there are some cases of interference messages popping up if you are close to some radio antennas (normally harbors). The f used is normally 433MHz or 315MHz.
Why we have direct systems on the market? I make it short, people are lazy and never check the tire pressure. So they put in place a law to enforce a warning when the pressure is low to save gas and reduce CO2. In the specs, only an direct system could fulfill the diffusion detection requirements.
Ah, do not attempt to replace the battery in a TPMS Sensors. Those bastard have implemented a count down variable in FW that basically kills the sensors after 10 years.
Just do me a favor, every 6 months check your tires pressure and refill when they are cold. Reset the TPMS system after that if necessary.