Temperature timeseries agreement significance test

Question

I hope my question wasn't answered before, but after extensive research I wasn't able to find something similar, so I'm asking for help (this is not my field). I have air temperatures timeseries observed by identical sensors deployed in near locations, like the following:

|------------|-----------------------|-----------|
| TEMP       | DATE                  | SENSOR_ID |
|------------|-----------------------|-----------|
|25.2        | 2015-03-01 14:20:00   | S1        |
|------------|-----------------------|-----------|
|25.3        | 2015-03-01 14:30:00   | S1        |
|------------|-----------------------|-----------|
|25.2        | 2015-03-01 14:40:00   | S1        |
|------------|-----------------------|-----------|

and

|------------|-----------------------|-----------|
| TEMP       | DATE                  | SENSOR_ID |
|------------|-----------------------|-----------|
|25.3        | 2015-03-01 14:20:00   | S2        |
|------------|-----------------------|-----------|
|25.2        | 2015-03-01 14:30:00   | S2        |
|------------|-----------------------|-----------|
|25.4        | 2015-03-01 14:40:00   | S2        |
|------------|-----------------------|-----------|

and

|------------|-----------------------|-----------|
| TEMP       | DATE                  | SENSOR_ID |
|------------|-----------------------|-----------|
|25.2        | 2015-03-01 14:20:00   | S3        |
|------------|-----------------------|-----------|
|25.1        | 2015-03-01 14:30:00   | S3        |
|------------|-----------------------|-----------|
|25.2        | 2015-03-01 14:40:00   | S3        |
|------------|-----------------------|-----------|

..and so on for many other sensor (about 15).

I also have the distance between each pair of sensors.

I would like to perform a test to see if the difference in the timeseries is related with the sensor's distance, and check at what distance this difference becomes significative. My null hypotesis is that the distance isn't related to the differences in the temperature timeseries.

Reading some paper, I was considering the use of the index of agreement (d), or the coefficient of determination (R^2) but I'm a bit lost.

Thank you in advance!

Fabio

Answer 1

########## measuring temperature difference

First of all, it is rather unclear how you will measure temperature differences, namely, how to choose the first and second term of the subtraction. If you swap them, the differences will change signs... But if you know how to do that, follow my advise below.

There are several possible solutions, but at first I would go for linear modelling.

Let's denote the temperature difference as dif, sensor id as id, and distance as dis.

Fit a model: dif = a * dis + b, using OLS regression.

What you are interested in is the t-statistic T associated with the coefficient a.

If T is larger then 2, you can decline the null hypothesis of no relation between dif and dis with confidence level approx. 0.95.

Now about the details: if you have 15 sensors, you want to take (15^2-15)/2 dis measurements as independent variable, and for each measurement take all dif measurements as dependent variable.

R code:

lmm <- 
     stats::lm(
          dif ~ dis
          , data = your_data
     )

summary(lm)

########## measuring the variance of the temperature difference

If you don't know how to choose the first and second term of the subtraction, I suppose you can take the variance OR standard deviation of those differences and measure whether it increases/decreases as a result of varying distance between sensors.

In this sense, you will take (15^2-15)/2 values of the variance of temperature difference as a dependent variable. Go on with linear model as above.