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There are a lot of ways of deriving and thinking about linear regression, so I'm not sure I can give the One True Explanation. I'll give a couple though:
The practical answer is "whenever there are order-of-magnitude differences, it's a good idea to take the log".
The intuitive answer is that if we're assuming y is a linear function of x, so a fixed change in x should yield (roughly) a fixed change in y. This isn't really sensible if y covers several orders of magnitude but x does not.
Another answer is that it doesn't really make intuitive sense to use L2 loss when your labels vary by orders of magnitude. If I'm predicting the income of a poor person and a rich person, it should probably matter whether I'm $10/hour off on my predictions for the poor person or the rich person. Taking the log of our labels implicitly converts our loss function from
(y - yhat)^2tolog(y/yhat)^2which matches the intuition that a $10 mistake for somebody who makes $1000/hour matters less than it does for somebody who makes $10/hour.Another answer is that if you're going to assume Y = a R + b S + c T then the most sensible distribution for these variables is Gaussian, since the sum of Gaussians is Gaussian. From this philosophy, it's sensible to do some preprocessing on our variables to make them Gaussian. Academia often makes the assumption that income is log-normal, so taking the log of income makes sense. And if we look at the histogram of our data, it indeed looks much more Gaussian after the log transform.
Thanks for the thorough explanation.
I've recently become interested in measuring things, so finding related domains that I'm ignorant about is pretty helpful to keep following the thread.
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