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When I want to find out what the average difference / error there is between two datasets, such as a predicted output vs. observed output of any system (i.e.: I predict output to be 100V, how does the actual measured output compare?) , intuitively I would do:

\begin{align} \frac{\sum_{i = 1}^{i = n} {|{P(i)-O(i)}|}}{n} \end{align}

where $P$ is a predicted value, $O$ is the observed value, for every instance $i$ up to $n$ instances. This would technically be defined as the MAE or mean average error.

However, I have seen another way to compare observed and predicted values using the RMSE, or root mean square error, defined as,

\begin{align} \sqrt{\frac{\sum_{i = 1}^{i = n} {\{{P(i)-O(i)}\}^2}}{n}} \end{align}

For someone who just wants a good idea of the average differences between two sets of data (in the case of a predicted output vs. observed output scenario), which method would be more useful, RMSE or MAE?

plu
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    They should both be fine. RMS is more common since in many cases squaring/square root operations are easier to deal with than absolute value calculations. Think about derivatives, etc. – Michael Biro Apr 03 '19 at 00:15
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    Also, if you are trying to find an regression model for a data set, defining you error by a RMS is going to penalize higher errors much more in comparison to small errors. – D.B. Apr 03 '19 at 00:20
  • That property of RMS can also be a disadvantage when your data set includes outliers. I think the main appeal of RMS for regression is it is much easier to calculate. (The "predicted" value is part of the output of that process.) – David K Apr 03 '19 at 02:03
  • @DavidK Any example of how a predicted value can be a part of the output of a process? – plu Apr 03 '19 at 23:48
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    A regression model is a estimate of the underlying trend of the data, which is a kind of prediction. You might get a plot with a line (for linear regression) that predicts what $y$ value you would get at each $x$ value if the data followed the model perfectly--but usually the observed $y$ values are not exactly on the line, so now at each of those points you can measure a difference between the predicted value and the observed value. – David K Apr 04 '19 at 01:39
  • Come to think of it, I wonder if there is such a thing as a predicted value that does not come from some sort of process. It could come from a really bad process and just be a wild guess, but even that would be a process. – David K Apr 04 '19 at 01:46
  • @DavidK Thanks. I can wrap my head around that; one can make regression model based on a collection of the actual observed data, and then do something like RSME to find out how closely the model fits that same observed data, so in that sense the prediction (i.e.: line/curve of best fit "prediction") would depend on the output (observed/measured values). – plu Apr 05 '19 at 19:19

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