# Regression RMSE when dependent variable is log transformed

I want to predict the duration a trip would take. For this I transformed my dependent variable (trip time in sec) to log transformed.

When I do regression on this variable with some other features,

I get this:

The score on held out data is: 0.08395386395024673
Hyper-Parameters for Best Score : {'l1_ratio': 0.15, 'alpha': 0.01}

The R2 Score of sgd_regressor on test data is: 0.0864573982691922

The mse of sgd_regressor on test data is: 0.5503753581
The mean absolute error of sgd_regressor on test data is: 0.566328128068


This is the code which does the above calculation:

from sklearn.metrics import mean_squared_error, mean_absolute_error

#
print("The R2 Score of "+ name + " on test data is: {}\n".format(self.g_cv.best_estimator_.score(self.test_X,self.test_Y)))

print ("The mse of "+ name + " on test data is:",\
mean_squared_error(test_Y, self.g_cv.best_estimator_.predict(self.test_X)))

print ("The mean absolute error of "+ name + " on test data is:",\
mean_absolute_error(test_Y, self.g_cv.best_estimator_.predict(self.test_X)))


Problem is R2 as you see is very bad. 0.08, but RMSE and Mean Absolute error seem to be very low. If I look at Mean Absolute Error, its just 0.56 sec. Which means on an average my predicted time is only half a second different from true time.

Something doesn't look right. Do I need to convert the predicted and original time variable back to linear scale from log scale before I calculate the above metrics (RMSE and MAE)?.

• Once you take logs, your response is not in seconds. In effect it's unit free. Are you calculating mean absolute error on the log scale? If so that's telling you something about the typical size of percentage error on the original scale. – Glen_b Nov 19 '17 at 3:28
• Mean absolute error here is taken of the log transformed values. So it is then not correct? – Baktaawar Nov 19 '17 at 3:48
• It depends on what you mean by "it": there's nothing wrong with calculating a MAE on the log scale as long as you don't misinterpret what it is. It's not a measurement in seconds. – Glen_b Nov 19 '17 at 5:16
• I've posted an answer because I couldn't locate a duplicate reasonably quickly -- however, this probably is a duplicate and may eventually close on that basis. – Glen_b Nov 19 '17 at 5:27
• Calculate precision on the original scale of the outcome! – AdamO Nov 20 '17 at 5:25

Once you take logs, your response is not in seconds. In effect it's unit free.

When you calculate mean absolute error on the log scale, it, too, is not a measurement in seconds.

It's (roughly-speaking) telling you something about the typical size of percentage error on the original scale.

An MAE(-of-the-logs) of 0.01 would tell you that typically your original values deviate by about 1% from the geometric mean.

Let $z_i=\log(y_i)$. Then an MAE of 0.01 in the logs means that $\frac{_1}{^n}|z_i-\bar{z}|=0.01$. Now on the original scale $\exp(\bar{z})$ is the geometric mean of the $y$-values, $\text{GM}(y)$.

Now consider observations sitting as far away from the mean as the MAE: $z_i=\bar{z}+ 0.01$ and $z_j = \bar{z}- 0.01$. Then

$y_i=\exp(z_i) = \exp(\bar{y}) \times \exp(0.01)$ $= 1.01005 \text{ GM}(y)\approx 1.01 \text{ GM}(y)$

or about 1% above the geometric mean. Similarly

$y_j=\exp(z_j)$ $= \exp(\bar{y}) \times \exp(-0.01)$ $= 0.99005 \text{ GM}(y)$ $\approx 0.99 \text{ GM}(y)$

or about 1% below the geometric mean.

Similarly an MAE (log scale) of 0.10 would tell you that typically your original values deviate by about 10.5% from the geometric mean. As you move further away (as MAE gets bigger) this convenient approximate-percentage relationship changes.

There's nothing wrong with calculating a MAE on the log scale as long as you don't misinterpret what it is. If you want an MAE on the original scale you'd need to compute it on that scale (but the fact that you're working with modelling the logs suggests that perhaps it may not actually be especially useful on the original scale)

• So do you think, if my MAE is 0.56 here of the log transformed variable, then it's a decent MAE ? I am not sure how you got those 1% deviation from geometric mean? Isn't MAE just the absolute deviation of predicted value with true value? I am trying to understand the interpretation of this MAE with log values. How do I say if MAE is good enough and model is doing decent in terms of MAE? – Baktaawar Nov 19 '17 at 6:02
• I can't judge what's a suitable MAE of logs for your purposes, nor even whether MAE on the log scale is what you want to look at. On the next part I've made some edits but that's really a new question (though one likely already answered); on the last part you need to figure out what it is you want to find out. – Glen_b Nov 19 '17 at 8:18
• MAE in regression is between true value and predicted value. And not with respect to mean of prediction. Similarly the case with RMSE. We try to check the error between predicted value and true value. So this geometric mean of y values is correct for case where you are finding MAE with respect to mean of the sample. – Baktaawar Nov 20 '17 at 0:08
• MAE in regression is between true value and predicted value. And not with respect to mean of prediction. Similarly the case with RMSE. We try to check the error between predicted value and true value. So this geometric mean of y values is correct for case where you are finding MAE with respect to mean of the sample. I have added the same question problem but for another question here: pls see if you can provide some thought to that. Thanks for your help! stats.stackexchange.com/questions/314607/… – Baktaawar Nov 20 '17 at 0:17