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I have 2 sets of 2-dimensional vectors, one from observations and one produced by a model. I would like to calculate a statistic similar to the RMSE for these. I believe the correct way of doing this is as follows (with thanks to this question):

The observation vectors are $\vec x_i \in \mathbb R^2$, and the model vectors are $\vec y_i$.

The vector RMSE should be calculated as:

$VRMSE = \sqrt{\frac{\sum_i^N (\|\vec x_i - \vec y_i\|_2)^2}{N}} = \sqrt{\frac{\sum_i^N \sum_j^2 (x_{i_j} - y_{i_j})^2}{N}}$.

Is this the correct way of calculating it? Are there suitable citations for this? How do you interpret the magnitude of this value (to make things concrete, take the 2 sets of vectors as being wind fields)?

markmuetz
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  • RMSE is not usually interpreted on its own, rather it's compared to another model's RMSE to see which fares better. Univariate RMSE is convenient to compare since it's in the same measurement scale with the data, but interpreting a single RMSE value is unwieldy. See [Explain Root Mean Square Error to non-technical audience](https://stats.stackexchange.com/questions/407591/explain-root-mean-square-error-to-non-technical-audience) – Nuclear03020704 Aug 09 '20 at 15:57
  • I tried to simplify my question to a minimal working example. I have multiple models (actually, multiple simulations run with the same model and different settings), and would like to be able to compare between these. – markmuetz Aug 18 '20 at 14:24
  • What I wanted was to perform something similar for vector data, where the vector was in R2. This seems like it should be possible, and is not technically a multivariate RMSE as there is only one (vector) quantity, and thus has only one measurement scale (I think). The linked question didn't shed any light on this. – markmuetz Aug 18 '20 at 14:31

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