How to predict Long term time series? I am working on a time series problem where I am trying to predict the temperature of a product in a refrigerator by using the information about the temperature of the air in the refrigerator with respect to time. So in our case temperature of the air is the independent variable and temperature of product is the dependent variable. We are getting the temperature of air and product every one minute (for training the model).
Lets say for example the temperature of the air in the refrigerator fluctuates between 2°C to 6°C, then the temperature of a particular product may fluctuate from 3°C to 5°C and the problem statement is to predict accurately these fluctuation in the temperature of the product by using the fluctuation in the air.
If I get the temperature of both the air and product every minute, I am able to predict the fluctuations in the temperature of the product by using the fluctuations of air with an error of 0.25°C using ARIMA (as I have the latest temperature of both, air and product and by using the historical values I am able to get very good accuracy).
But the challenge here is to make the future point in time predictions just by looking at the temperature of air at the particular moment for let say next 3 months. Because the temperature of the product will not be available in future. So in this case we will have the historical values of air and product temperature for some time but the future predictions have to be made using the temperature of air only.
So currently we are stuck in making long time series predictions, if anyone has any expertise in working on long time series prediction problem. Any suggestions/help is really appreciated.

Of course, you won't know the temperature of the air in the next three months, either.
So you basically have two possibilities:

*

*Predict product temperatures based on historical product temperatures alone

*Predict product temperatures based on historical product temperatures and forecasts of air temperatures as a driver

Unless you have knowledge of external drivers (maybe a cooling unit reliably reduces the air temperature in the first 15 out of every 60 minutes? If so, use seasonal methods with season length 60), both forecasts involved (air and product) will likely converge to a long-term average. Which is pretty much what you should expect for a long-term forecast, and probably the best you can do.
Assuming you have enough data, it would make sense to try the two approaches described above with a holdout sample of the same length as you plan on using in your production forecast. For the second approach above, note that prediction intervals for product temperatures will be too narrow if you use air temperature as a fixed forecast. Consider simulating multiple trajectories of future air temperatures and feeding each one into the product temperature forecast, then simulating product temperature trajectories and finally taking pointwise quantiles to get prediction intervals.
If the accuracy is not good enough (which you should assess based on the costs of forecast error - what will the forecast be used for?), it may make sense to invest in making the series more forecastable, instead of chasing accuracy that may be hard to impossible to achieve. For instance, you could work on how the cooling unit operates.
