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I have $(x,\ y)$ pairs with a strongly suspected linear correlation. So I want to fit the "best" linear function in order to make predictions for unknown $x$'s. These pairs don't represent a function, i.e. there are many different values of $y$ for some $x$. I want to know if there is an issue with that considering this as a machine learning problem. Should I use some unique "representative" (average, maximum, minimum, most frequent, etc.) and convert the input data in a function or it is OK to work with the data as it is?

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If you perform linear regression on your raw data using Ordinary Least Squares to estimate the model parameters you will get a model that estimates the conditional mean of $Y$ given the observed value of $X$. I.e. $\hat{f}(x)=E[Y|X=x].$ Since your model outputs the conditional expectation of the response it wouldn't be necessary to convert your $Y|X=x$ into a descriptive statistic prior to your analysis.

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  • $\begingroup$ OK. I see that. But now suppose that I have to evaluate how good are the estimated parameters of my linear model with, let's say, MAE or MSE. I want to do that with cross-validation. So I take my evaluation set and do all the predictions with my model and I want to compare it with the "real" one. Which value should I take as the "real" one given that the input data is not a function? Thanks for your answer! $\endgroup$ – Martin Chaia Nov 8 '13 at 13:30
  • $\begingroup$ You're right that your validation method wouldn't make much sense. Since you said, "there are many different values of y for some x," you know that there is no "real" value of $Y$ at a given level of $X$, i.e. the relationship isn't deterministic. Using linear regression with OLS estimators you can predict $E[Y|X=x]$, so you could compare these predictions to the average Y value at a given level of X to determine how accurate your model is. The regression output should provide you with diagnostics to make checks similar to this. $\endgroup$ – tjnel Nov 8 '13 at 21:42
  • $\begingroup$ You may also find prediction intervals useful to determine the range within which the model assumes most observations would fall at a given level of X. $\endgroup$ – tjnel Nov 9 '13 at 0:49
  • $\begingroup$ Thanks tjnel! I really appreciate your answer. Can you please suggest me a technique to "find prediction intervals"? $\endgroup$ – Martin Chaia Nov 11 '13 at 19:26
  • $\begingroup$ @MartinChaia if you are using statistical computing software to perform your analysis you can check if it supports prediction intervals for regression. $\endgroup$ – tjnel Nov 11 '13 at 23:04

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