# Average of two fits vs fit of a combined data

Lets say I have data from two independent simulations. One of them look like this: The fitted curve is done by minimizing absolute difference between each data point and the curve. The fit information of interest are y-values at x=0 and x=90. The fit takes a form specified by me.

Lets say I have another data set with different x and y-values. I make a similar fit. Now I average the fit information: average y_fit(x=0) from two simulations and similarly y_fit(x=90) from two simulations.

Now I make a combined fit using data from two simulations. Will the y_fit(x=0) of combined data equal to the averaged y_fit(x=0) from the two different data sets?

Which method is better in terms of accuracy?

• The simulations are independent of one and other, does that mean that the data points within a simulation are not independent of one and other? Are the red crosses ordered in time? If the y value of one red cross is quite far above the line, does this impact where any other red cross is likely to sit relative to the line? – gazza89 Sep 28 '18 at 11:19
• The intervals in x axis are not uniform and also are different for different simulations. I observed that the fit of combined data is different than the average of fits. Will be nice if someone can prove though... – Sathish Sanjeevi Sep 28 '18 at 12:06
• You wouldn't expect the fit of combined data to be the same as the average of separate fits, they're different optimisation problems and the onus would be to prove that they should be the same, otherwise you'd expect them to be numerically different. If however, they are materially different, then you need to decide which one is more "correct". I don't know enough about your simulation, but if y points are independent (within a simulation) given x, you should combine all data, if they're potentially correlated, you should fit individually (and run more simulations) – gazza89 Sep 28 '18 at 12:38
• Yes, the y-data is correlated to x with a known scaling law. Would you recommend combining the data and make a single fit? – Sathish Sanjeevi Sep 28 '18 at 13:29
• Sure, but given x, you don't know y definitively, there's a distribution on y (as seen that there's spread in the vertical direction for fixed x position), but are different measurements of y independent if x is known? It's another way of asking are your errors i.i.d ? If your points are ordered in time, I'd suspect that's not the case – gazza89 Sep 28 '18 at 16:24