Understanding Root Mean Squared Logarithmic Error ( RMSLE )
Explaination
A. What is RMSLE?
RMSLE is an evaluation metric used for regression problems. It measures the ratio-based difference between predicted and actual values rather than absolute differences. RMSLE is especially useful when:
You care more about relative error than absolute error.
Your target variable spans multiple orders of magnitude.
You want to penalize under-predictions more than over-predictions (as the log curve is asymmetric).
B. In More Simple Terms
🧠 Imagine this situation:
You’re building a model that predicts house prices. Some houses are worth ₹1 lakh, others ₹1 crore. That’s a huge range!
Now, say your model predicts:
For a ₹1 lakh house → ₹90,000 (so you’re off by ₹10,000)
For a ₹1 crore house → ₹91 lakh (off by ₹9 lakh)
Which is the bigger mistake?
If you just use normal error (like Mean Squared Error), it says the ₹9 lakh mistake is worse — but in reality, being off by 10% is the same in both cases.
📏 That's where RMSLE comes in
Instead of comparing raw values, RMSLE looks at the ratio between the predicted and actual value using logarithms.
Why logs?
Logs compress large numbers — a crore and a lakh become numbers like 12 and 5.
Logs turn ratios into differences, which helps us compare things fairly.
It penalizes underestimates more than overestimates — which can be good when under-predicting is more harmful (e.g., underestimating demand or risk).
Mathematical Derivation
Let’s start from a more common metric and derive our way to RMSLE.
$$y_i = \text{Actual value} $$
$$\hat{y}_i = \text{Predicted value}$$
A. MSE (Mean Squared Error)
$$\text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$
It penalizes absolute error heavily.
B. MSLE (Mean Squared Logarithmic Error)
Instead of direct differences, we take the difference in logarithmic space:
$$\text{MSLE} = \frac{1}{n} \sum_{i=1}^{n} \left( \log_e(1 + y_i) - \log_e(1 + \hat{y}_i) \right)^2$$
The addition of 1 avoids issues with log(0).
It reduces the effect of large values and focuses more on percentage differences.
C. RMSLE (Root Mean Squared Logarithmic Error)
Take the square root of MSLE:
$$\text{RMSLE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} \left( \log_e(1 + y_i) - \log_e(1 + \hat{y}_i) \right)^2}$$
