# How to extrapolate partial curve based on other complete curves?

I heat a room to a certain temperature. Then I let it cool over time. I measure the temperature at 6 intervals as it cools. I repeat this process for some other rooms.

From this, I get a set of temperature observations of the same cooling process happening over time in different rooms:

Time interval:  0  1  2  3  4  5
--------------------------------
Room 1:        50 30 25 22 20 18
Room 2:        50 20 10  5  0  0
Room 3:        50 30 20 15 10  0

...

Room 100:      50 35 28 25 23 22


I heat a new room and let it start cooling. I take temperature readings at the first 3 time intervals (see purple line in graph).

How can I predict what the temperature will be at the next 3 time intervals in this room?

(I want to base these predictions on the first 3 temperature measurements I took in this new room, and on the measurements taken in other rooms.)

## 1 Answer

Here is how I solved this problem. I performed an equation search on the three sets of known data in your post to find a single, simple approximating equation that individually gave a good fit to each of the three. I know from my nuclear engineering days that the curve would likely have a logarithmic decrease type of shape. My equation search found that the equation "temperature = a * log(time+b) + offset" (where log is natural log) seemed to work well. Now that I have an approximating equation in hand, I can make another curve fit of the three data points from the remaining room and use that fitted equation for predictions.

Note that it is not usually a good practice to fit an equation with three parameters - here a, b, and offset - to three data points, however model selection is complete at this point and we know the data does in fact lie on this curve.

• The stuff about finding a best-fit equation and using that to extrapolate the points recorded in the new room was extremely helpful. Thank you! – Mary Rose Cook Apr 10 '19 at 14:39