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RMSE vs RMSLE

Root Mean Squared Error (RMSE) and Root Mean Squared Logarithmic Error (RMSLE) both are the techniques to find out the difference between the values predicted by the machine learning model and the actual values.

  • But, what is the purpose for RMSLE( "logarithmic")

  • Does a high RMSE imply low RMSLE?

Can somebody explain in-detailed differences between RMSE and RMSLE? And how the metric works under the hood?

  • When would one use RMSE over RMSLE?
  • What are the advantages/disadvantages of using RMSE over RMSLE?
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RMSE will have a drastic effect of outliers on its values. But in case of RMSLE we can reduce the effect of outliers by many magnitudes & their effect is much less.

RMSLE value will only consider the relative error between Predicted and the actual value neglecting the scale of data. But RMSE value will increase in magnitude if the scale of error increases. For e.g.

Actual value = 100
Predicted Value = 90
RMLSE: 0.1053
RMSE: 10

Actual value = 1000
Predicted Value = 900
RMSLE: 0.1053
RMSE : 100

Also in case of under-estimation results from RMSLE are affected greatly. So one can easily understand that it is better than RMSE in certain scenarios but RMSE works better for generalise cases. At last RMSLE is bit better than RMSE but based upon certain cases only.

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  • $\begingroup$ These two terms are not equivalent. Saying that RMSLE is bit better than RMSE but based upon certain cases only - is not correct. $\endgroup$ – Subhash C. Davar Mar 14 at 12:30
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The RMSLE can not be equated with RMSE -root mean squared error. The latter is based on statistical assumptions. And RMSLE is apparently missing a conceptual framework - mathematical or data-science and /or statistical.

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Let's start with the definitions: $$ \text{RMSE} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (y_i - \hat y_i)^2}, $$ $$ \text{RMSLE} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (\log y_i - \log \hat y_i)^2}, $$ where $y_i$ is the target value for example $i$ and $\hat y_i$ is the model's prediction. Both metrics quantify the prediction error, so in general a high RMSE implies a high RMSLE as well.

RMSLE has the meaning of a relative error, while RMSE is an absolute error. Choosing one depends on the nature of your problem. Imagine that the target spans values from around 1 to around 100. Is predicting $\hat y = 1.01$ for true $y = 1$ as bad as $\hat y = 101$ for $y = 100$? Then RMSLE might be a good choice. With RMSE, on the other hand, predicting $\hat y = 2$ for $y = 1$ is as bad as $\hat y = 101$ for $y = 100$.

In general, you would often want to use the logarithm (or maybe the square root or some other power $0 < \alpha < 1$) for strictly positive targets that span a large range.

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