Calculating a rate using non-continuos time steps

[Bahramiarmy]

I want to calculate a rate for my data, n time-1 or n/time. However the data I am using does not have continuous time steps, there I wonder what the rate time should be plotted on the y-axis.

For example, I am trying to calculate the rate of energy in a number of earthquakes as a function of temperature. Earthquake 1 happened in minute 1 and had an energy (E) of 5, but earthquake two was not until minute 12 (E= 20)...so on with different time gaps and energies.

So Energy rate (E.) = (E2-E1)/(t2-t1),I then copy this formula in a row (in excel), to provide the rates throughout my data set.

So now my y-axis is E. and x-axis is temperature. But of course the time interval is not constant, so it is not Energy per minute or second. My question: is this acceptable? Or mathematically correct? Are there any other ways to tackle this problem?


Thank you for your help.
JB

 

[Ishuda]

I want to calculate a rate for my data, n time-1 or n/time. However the data I am using does not have continuous time steps, there I wonder what the rate time should be plotted on the y-axis.

For example, I am trying to calculate the rate of energy in a number of earthquakes as a function of temperature. Earthquake 1 happened in minute 1 and had an energy (E) of 5, but earthquake two was not until minute 12 (E= 20)...so on with different time gaps and energies.

So Energy rate (E.) = (E2-E1)/(t2-t1),I then copy this formula in a row (in excel), to provide the rates throughout my data set.

So now my y-axis is E. and x-axis is temperature. But of course the time interval is not constant, so it is not Energy per minute or second. My question: is this acceptable? Or mathematically correct? Are there any other ways to tackle this problem?


Thank you for your help.
JB

What you have is a Joules per minute assuming you are measuring energy in Joules and time in minutes. Even if the time intervals are not the same, the units are still those of, say Joules per minute. The difference is not is what it is but in the possible accuracy of the actual rate. As an example, assume a simple model of
E(t) = 1000 - t³
where t is in minutes. We know the rate is
dE(t)/dt = -3 t²
So at 1 minute, the rate is -3. Suppose we have measurements at .99 and 1.01, then we approximate dE/dt at t=1 by
dE(1)/dt ~ - \(\displaystyle \frac{1.01^2 - 0.99^2}{.02} = -3.0001\)
which is a pretty good approximation. But suppose we have a measurement at 2 minutes and 4 minutes, then
dE(3)/dt ~ - \(\displaystyle \frac{4^3 - 2^3}{2} = -28\)
which is not quite as good an approximation to the correct answer of -27 but still pretty good.

There are other models which would lead to different methods depending on the data you have.