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\begin{equation} \text{RMSRE} = \sqrt{\frac{1}{n}\cdot\sum_{i=1}^n \Delta X^2_{\text{rel},i}} \end{equation} with \begin{equation} \Delta X_{\text{rel},i} = \frac{X_i}{T_i}-1, \end{equation} where $T_i$ is the desired value and $X_i$ is the actual value.

If this error actually exists, then a citable source for it would be greatly appreciated. If it does not exist, then why so?

Thanks in advance.

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    $\begingroup$ This question is related to link. $\endgroup$
    – Nos
    Jun 16, 2019 at 0:07

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Short answer : It exists and here is the reference and also there is a similar post regarding difference between RRMSE & RMSRE I would like to draw your attention to. It depends where you apply division to make the error relative!

Mathematically, when you divide the difference between the predicted output and actual(expected) output $T_i-X_i$ by $T_i$ then the error will be considered Relatively which means that each residual is scaled against the actual value or normalized by that. \begin{equation} \Delta X_{\text{rel},i}=\frac{X_i-T_i}{T_i} \end{equation}

\begin{equation} \text{RMSRE} = \sqrt{\frac{1}{n}\cdot\sum_{i=1}^{n}\Delta X^2_{\text{rel},i}} \end{equation}

\begin{equation} \text{RRMSE} = \sqrt{\frac{\frac{1}{n}\cdot\sum_{i=1}^{n}(X_i-T_i)^2}{\sum_{i=1}^{n}T_i^2}} \end{equation}

\begin{equation} \text{PRMSE} = \sqrt{\frac{\frac{1}{n}\cdot\sum_{i=1}^{n}(X_i-T_i)^2}{\sum_{i=1}^{n}T_i^2}}\cdot 100 \% \end{equation}

Another reference for using RRMSE could be found here and the last form is known as rRMSE as well as shown here

Basically the same approach is used in MAPE to express error either relatively or in form of percentage as the last form. Obviously expression in form of percentage itself includes relative error.

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  • $\begingroup$ Thank you very much for your quick and detailed reply. $\endgroup$
    – Nos
    Jun 16, 2019 at 8:06
  • $\begingroup$ The primary sources for the $\text{RMSRE}$, according to the paper you referenced are Göçken et al. and Webber et al.. For completeness and to make this question a real duplicate, you should add $\text{RMSPE}$. $\endgroup$
    – Nos
    Jun 16, 2019 at 8:10
  • $\begingroup$ Swanson et al., Fomby and Shcherbakov et al.) define the RMSPE as: \begin{equation} \text{RMSPE} = \sqrt{\frac{1}{n} \cdot \sum_{i=1}^n \Delta X^2_{\text{rel},i}} \cdot 100\% \end{equation} $\endgroup$
    – Nos
    Jun 16, 2019 at 10:54
  • $\begingroup$ Frankly, I wouldn't use this. I would argue for the related measure $$\frac{1}{\sqrt{n}}\bigg\|\mathbf{\frac{T}{X}}-1\bigg\|$$ for several reasons. That form is more literally proportional modelling error: It allows for direct comparison of different models $\mathbf{T_1}$ and $\mathbf{T_2}$ using $\mathbf{X}$ as the reference values. In that form it is $\frac{1}{\sqrt{n}}$ times the $1/\mathbf{X}^2$ weighted ordinary-least-squares regression value, and can be parsed into modelling error and noise. $\endgroup$
    – Carl
    May 20, 2022 at 23:15
  • $\begingroup$ waveclimate.com/clams/redesign/help/docs/… has a different formula for RRMSE than the one you cited, @Mario $\endgroup$
    – quant_dev
    Feb 27, 2023 at 21:28

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