# I Need help predicting inside of the box temperature at a given outside temperature.

### Background

I have a system (also known as a BOX). The BOX is insulated from the outside environment and contains several internal layers, similar to a nesting doll. It is equipped with a temperature sensor inside each layer and one outside the box. I have conducted multiple studies at various outside temperatures, tracking both outside and inside temperatures over time. These studies have resulted in multiple arrays consisting of date-time columns and temperature data columns. I am not interested in date-time data at this time.

The temperature data essentially demonstrates the correlation between outside and inside temperatures.

My goal is to determine the most inside temperature (T3) within the box when the inside temperature of the most outer layer (T1) of the box reaches 19 degrees Celsius, for example.

Example of a data array:

|Datetime      | Outside| InsideLayer1| InsideLayer| InsideLayer|
3/20/24 12:23.    24.      19.           19.         18.

### Question

1. How can I achieve what I need?
2. Do I need to use data from all layers of the box to predict temperature inside of the last (most inner) layer?
3. Which approach/model do I need to use.
4. What should I consider.

### Things to consider

Since the temperature data is basically a contentious data, it might not contain exact temperature at which I need to predict temperature inside of the last layer. It may be close, but not the same.

### My thoughts

I was going to test KNN, GaussianNB, SVC, But I am not sure if I can get a prediction since the temperature that I will use to predict most internal temperature may not be in the array. I would like to determine "exact" temperature with decimal values.

### Quick sketch

• If the problem is as you describe, I would give traditional time-series a try (ARIMA-type), or some mix between those and NN. $T_3$ will be reach $T_2$ with some delay, provided $T_2$ stays constant. It probably will not, so what you will have is $T_3$ being predicted by $T_2$ and $T_1$ with some lag. With some familiarity in heat transfer you could probably reduce this to quasi-analytical model with just few constants that need fitting, but if this is not your field, then IMHO classical time series should be the first port of call.
– Cryo
Mar 21 at 22:33
• If you did try to model it as a heat transfer problem, you would give each box a certain heat capacity and couple the boxes via some heat resistance. Your sensors could also be described as ideal sensors with some noise. This would give a system of first-order coupled ODEs, which can be solved with methods of matrix algebra. The constants of the heat capacity could then be modeled on experimental data, e.g. on max likelihood. I would try this thing first, then reach for more complex models. This model will essentially boil down to something between a Gaussian process and ARIMA-type model.
– Cryo
Mar 21 at 22:39
• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Mar 22 at 5:53
• Cryo, Thank you for your response, highly appreciated. I will try ARIMA. If I understand ARIMA correctly, it will predict a "trend"/future values. I would like to predict T3 at a given value of T1. Example: What is T3 if T1=20C? Mar 23 at 22:05