# What is the correct definition of the root mean square percentage error (RMSPE)?

Göçken et al. define the root mean square percentage error (RMSPE) as $$$$\text{RMSPE} = \sqrt{\frac{100\%}{n} \cdot \sum_{i=1}^n \Delta X^2_{\text{rel},i}}$$$$ with $$$$\Delta X_{\text{rel},i}=\frac{X_i}{T_i}-1,$$$$ where $$T_i$$ is the desired value and $$X_i$$ is the actual value.

However, Göçken et al. and Webber et al. define the root mean square relative error (RMSRE) as:

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

If we express the actual error $$\Delta X_{\text{rel},i}$$ as a percentage and name it $$\Delta X_{\%,i}$$, then we have: $$$$\Delta X_{\%,i}=\left(\frac{X_i}{T_i}-1\right)\cdot 100\%=\Delta X_{\text{rel},i} \cdot 100\%$$$$

From my understanding, RMSPE should be the same as RMSRE, where $$\Delta X_{\text{rel},i}$$ is substituted by $$\Delta X_{\text{%},i}$$. However, this would yield $$$$\text{RMSPE} = \sqrt{\frac{1}{n} \cdot \sum_{i=1}^n \Delta X^2_{\text{rel},i}} \cdot 100\%,$$$$ which differs from the original definition of Göçken et al. by a factor of 10. Are my considerations correct and if so, are there alternative sources for the RMSPE?

• $\sqrt{100\%}=100\%=1$. There is no factor of $10$ difference, though putting it inside the square root is misleading. Jun 8, 2022 at 8:24
• That... makes sense. And it just took 3 years for someone to point it out. :-) Thanks a lot!
– Nos
Jun 8, 2022 at 8:55

There are several alternative sources (Swanson et al., Fomby, Shcherbakov et al.), which agree that the RMSPE is defined as: $$$$\text{RMSPE} = \sqrt{\frac{1}{n} \cdot \sum_{i=1}^n \Delta X^2_{\text{rel},i}} \cdot 100\%$$$$