Why linear regression uses “vertical” distance to the best-fit-line, instead of actual distance?What is the difference between linear regression on y with x and x with y?Why does linear regression use a cost function based on the vertical distance between the hypothesis and the input data point?Other ways to find line of “best” fitLine of best fit (Linear regression) over vertical lineOther ways to find line of “best” fitBest method of calculating line of best fit / extrapolate to compensate for delaysCoefficient of determination of a orthogonal regressionWhy is linear regression different from PCA?Visualling results from longitudinal mixed model with subtle time by treatment trendsHow to get the best out of a “bad” set of features for regression?How do I explain the “line of best fit” in this diagram?Why does linear regression use a cost function based on the vertical distance between the hypothesis and the input data point?Can residuals be calculated from N-point moving averages or just the regression line? Also, what is the standard way to determine regression line?

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Why linear regression uses "vertical" distance to the best-fit-line, instead of actual distance?

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Why linear regression uses “vertical” distance to the best-fit-line, instead of actual distance?


What is the difference between linear regression on y with x and x with y?Why does linear regression use a cost function based on the vertical distance between the hypothesis and the input data point?Other ways to find line of “best” fitLine of best fit (Linear regression) over vertical lineOther ways to find line of “best” fitBest method of calculating line of best fit / extrapolate to compensate for delaysCoefficient of determination of a orthogonal regressionWhy is linear regression different from PCA?Visualling results from longitudinal mixed model with subtle time by treatment trendsHow to get the best out of a “bad” set of features for regression?How do I explain the “line of best fit” in this diagram?Why does linear regression use a cost function based on the vertical distance between the hypothesis and the input data point?Can residuals be calculated from N-point moving averages or just the regression line? Also, what is the standard way to determine regression line?






.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;








1












$begingroup$


Linear regression uses the "vertical" (in 2 dimension) distance of (y - ŷ). But this is not the real distance between any point and the best fit line.



i.e. - in the image here:



enter image description here



you use the green lines instead of the purple.



Is this done because the math is simpler? Because the effect of using the real distance is negligible, or equivalent? Because it's actually better to use a "vertical" distance?










share|cite|improve this question







New contributor



David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 6




    $begingroup$
    There is such a thing as minimizing perpendicular distance. It is called Deming Regression. Ordinary linear regression assums the x value are known and the only error is in y. That is often a reasonable assumption.
    $endgroup$
    – Michael Chernick
    9 hours ago






  • 1




    $begingroup$
    Sometimes the ultimate purpose of finding the regression line is to make predictions of $hat Y_i$'s based on future $x_i$'s. (There is a 'prediction interval' formula for that.) Then it is vertical distance that matters.
    $endgroup$
    – BruceET
    9 hours ago










  • $begingroup$
    @MichaelChernick I think your one-liner explained it best, maybe you can elaborate it a bit, and post it as an answer?
    $endgroup$
    – David Refaeli
    9 hours ago










  • $begingroup$
    I think Gung's answer is what I would say elaborating on my comment.
    $endgroup$
    – Michael Chernick
    7 hours ago










  • $begingroup$
    Related: stats.stackexchange.com/questions/63966/…
    $endgroup$
    – Sycorax
    6 hours ago

















1












$begingroup$


Linear regression uses the "vertical" (in 2 dimension) distance of (y - ŷ). But this is not the real distance between any point and the best fit line.



i.e. - in the image here:



enter image description here



you use the green lines instead of the purple.



Is this done because the math is simpler? Because the effect of using the real distance is negligible, or equivalent? Because it's actually better to use a "vertical" distance?










share|cite|improve this question







New contributor



David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 6




    $begingroup$
    There is such a thing as minimizing perpendicular distance. It is called Deming Regression. Ordinary linear regression assums the x value are known and the only error is in y. That is often a reasonable assumption.
    $endgroup$
    – Michael Chernick
    9 hours ago






  • 1




    $begingroup$
    Sometimes the ultimate purpose of finding the regression line is to make predictions of $hat Y_i$'s based on future $x_i$'s. (There is a 'prediction interval' formula for that.) Then it is vertical distance that matters.
    $endgroup$
    – BruceET
    9 hours ago










  • $begingroup$
    @MichaelChernick I think your one-liner explained it best, maybe you can elaborate it a bit, and post it as an answer?
    $endgroup$
    – David Refaeli
    9 hours ago










  • $begingroup$
    I think Gung's answer is what I would say elaborating on my comment.
    $endgroup$
    – Michael Chernick
    7 hours ago










  • $begingroup$
    Related: stats.stackexchange.com/questions/63966/…
    $endgroup$
    – Sycorax
    6 hours ago













1












1








1





$begingroup$


Linear regression uses the "vertical" (in 2 dimension) distance of (y - ŷ). But this is not the real distance between any point and the best fit line.



i.e. - in the image here:



enter image description here



you use the green lines instead of the purple.



Is this done because the math is simpler? Because the effect of using the real distance is negligible, or equivalent? Because it's actually better to use a "vertical" distance?










share|cite|improve this question







New contributor



David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$




Linear regression uses the "vertical" (in 2 dimension) distance of (y - ŷ). But this is not the real distance between any point and the best fit line.



i.e. - in the image here:



enter image description here



you use the green lines instead of the purple.



Is this done because the math is simpler? Because the effect of using the real distance is negligible, or equivalent? Because it's actually better to use a "vertical" distance?







regression linear-model






share|cite|improve this question







New contributor



David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.










share|cite|improve this question







New contributor



David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








share|cite|improve this question




share|cite|improve this question






New contributor



David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








asked 9 hours ago









David RefaeliDavid Refaeli

1043 bronze badges




1043 bronze badges




New contributor



David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




New contributor




David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









  • 6




    $begingroup$
    There is such a thing as minimizing perpendicular distance. It is called Deming Regression. Ordinary linear regression assums the x value are known and the only error is in y. That is often a reasonable assumption.
    $endgroup$
    – Michael Chernick
    9 hours ago






  • 1




    $begingroup$
    Sometimes the ultimate purpose of finding the regression line is to make predictions of $hat Y_i$'s based on future $x_i$'s. (There is a 'prediction interval' formula for that.) Then it is vertical distance that matters.
    $endgroup$
    – BruceET
    9 hours ago










  • $begingroup$
    @MichaelChernick I think your one-liner explained it best, maybe you can elaborate it a bit, and post it as an answer?
    $endgroup$
    – David Refaeli
    9 hours ago










  • $begingroup$
    I think Gung's answer is what I would say elaborating on my comment.
    $endgroup$
    – Michael Chernick
    7 hours ago










  • $begingroup$
    Related: stats.stackexchange.com/questions/63966/…
    $endgroup$
    – Sycorax
    6 hours ago












  • 6




    $begingroup$
    There is such a thing as minimizing perpendicular distance. It is called Deming Regression. Ordinary linear regression assums the x value are known and the only error is in y. That is often a reasonable assumption.
    $endgroup$
    – Michael Chernick
    9 hours ago






  • 1




    $begingroup$
    Sometimes the ultimate purpose of finding the regression line is to make predictions of $hat Y_i$'s based on future $x_i$'s. (There is a 'prediction interval' formula for that.) Then it is vertical distance that matters.
    $endgroup$
    – BruceET
    9 hours ago










  • $begingroup$
    @MichaelChernick I think your one-liner explained it best, maybe you can elaborate it a bit, and post it as an answer?
    $endgroup$
    – David Refaeli
    9 hours ago










  • $begingroup$
    I think Gung's answer is what I would say elaborating on my comment.
    $endgroup$
    – Michael Chernick
    7 hours ago










  • $begingroup$
    Related: stats.stackexchange.com/questions/63966/…
    $endgroup$
    – Sycorax
    6 hours ago







6




6




$begingroup$
There is such a thing as minimizing perpendicular distance. It is called Deming Regression. Ordinary linear regression assums the x value are known and the only error is in y. That is often a reasonable assumption.
$endgroup$
– Michael Chernick
9 hours ago




$begingroup$
There is such a thing as minimizing perpendicular distance. It is called Deming Regression. Ordinary linear regression assums the x value are known and the only error is in y. That is often a reasonable assumption.
$endgroup$
– Michael Chernick
9 hours ago




1




1




$begingroup$
Sometimes the ultimate purpose of finding the regression line is to make predictions of $hat Y_i$'s based on future $x_i$'s. (There is a 'prediction interval' formula for that.) Then it is vertical distance that matters.
$endgroup$
– BruceET
9 hours ago




$begingroup$
Sometimes the ultimate purpose of finding the regression line is to make predictions of $hat Y_i$'s based on future $x_i$'s. (There is a 'prediction interval' formula for that.) Then it is vertical distance that matters.
$endgroup$
– BruceET
9 hours ago












$begingroup$
@MichaelChernick I think your one-liner explained it best, maybe you can elaborate it a bit, and post it as an answer?
$endgroup$
– David Refaeli
9 hours ago




$begingroup$
@MichaelChernick I think your one-liner explained it best, maybe you can elaborate it a bit, and post it as an answer?
$endgroup$
– David Refaeli
9 hours ago












$begingroup$
I think Gung's answer is what I would say elaborating on my comment.
$endgroup$
– Michael Chernick
7 hours ago




$begingroup$
I think Gung's answer is what I would say elaborating on my comment.
$endgroup$
– Michael Chernick
7 hours ago












$begingroup$
Related: stats.stackexchange.com/questions/63966/…
$endgroup$
– Sycorax
6 hours ago




$begingroup$
Related: stats.stackexchange.com/questions/63966/…
$endgroup$
– Sycorax
6 hours ago










2 Answers
2






active

oldest

votes


















5












$begingroup$

Vertical distance is a "real distance". The distance from a given point to any point on the line is a "real distance". The question for how to fit the best regression line is which of the infinite possible distances makes the most sense for how we are thinking about our model. That is, any number of possible loss functions could be right, it depends on our situation, our data, and our goals (it may help you to read my answer to: What is the difference between linear regression on y with x and x with y?).



It is often the case that vertical distances make the most sense, though. This would be the case when we are thinking of $Y$ as a function of $X$, which would make sense in a true experiment where $X$ is randomly assigned and the values are independently manipulated, and $Y$ is measured as a response to that intervention. It can also make sense in a predictive setting, where we want to be able to predict values of $Y$ based on knowledge of $X$ and the predictive relationship that we establish. Then, when we want to make predictions about unknown $Y$ values in the future, we will know and be using $X$. In each of these cases, we are treating $X$ as fixed and known, and that $Y$ is understood to be a function of $X$ in some sense. However, it can be the case that that mental model does not fit your situation, in which case, you would need to use a different loss function. There is no absolute 'correct' distance irrespective of the situation.






share|cite|improve this answer









$endgroup$




















    -1












    $begingroup$

    Summing up Michael Chernick comment and gung answer:



    Both vertical and point distances are "real" - it all depends on the situation.



    Ordinary linear regression assumes the $X$ value are known and the only error is in the $Y$'s. That is often a reasonable assumption.



    If you assume error in the $X$'s as well, you get what is called a Deming regression, which fits a point distance.






    share|cite|improve this answer








    New contributor



    David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.





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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      5












      $begingroup$

      Vertical distance is a "real distance". The distance from a given point to any point on the line is a "real distance". The question for how to fit the best regression line is which of the infinite possible distances makes the most sense for how we are thinking about our model. That is, any number of possible loss functions could be right, it depends on our situation, our data, and our goals (it may help you to read my answer to: What is the difference between linear regression on y with x and x with y?).



      It is often the case that vertical distances make the most sense, though. This would be the case when we are thinking of $Y$ as a function of $X$, which would make sense in a true experiment where $X$ is randomly assigned and the values are independently manipulated, and $Y$ is measured as a response to that intervention. It can also make sense in a predictive setting, where we want to be able to predict values of $Y$ based on knowledge of $X$ and the predictive relationship that we establish. Then, when we want to make predictions about unknown $Y$ values in the future, we will know and be using $X$. In each of these cases, we are treating $X$ as fixed and known, and that $Y$ is understood to be a function of $X$ in some sense. However, it can be the case that that mental model does not fit your situation, in which case, you would need to use a different loss function. There is no absolute 'correct' distance irrespective of the situation.






      share|cite|improve this answer









      $endgroup$

















        5












        $begingroup$

        Vertical distance is a "real distance". The distance from a given point to any point on the line is a "real distance". The question for how to fit the best regression line is which of the infinite possible distances makes the most sense for how we are thinking about our model. That is, any number of possible loss functions could be right, it depends on our situation, our data, and our goals (it may help you to read my answer to: What is the difference between linear regression on y with x and x with y?).



        It is often the case that vertical distances make the most sense, though. This would be the case when we are thinking of $Y$ as a function of $X$, which would make sense in a true experiment where $X$ is randomly assigned and the values are independently manipulated, and $Y$ is measured as a response to that intervention. It can also make sense in a predictive setting, where we want to be able to predict values of $Y$ based on knowledge of $X$ and the predictive relationship that we establish. Then, when we want to make predictions about unknown $Y$ values in the future, we will know and be using $X$. In each of these cases, we are treating $X$ as fixed and known, and that $Y$ is understood to be a function of $X$ in some sense. However, it can be the case that that mental model does not fit your situation, in which case, you would need to use a different loss function. There is no absolute 'correct' distance irrespective of the situation.






        share|cite|improve this answer









        $endgroup$















          5












          5








          5





          $begingroup$

          Vertical distance is a "real distance". The distance from a given point to any point on the line is a "real distance". The question for how to fit the best regression line is which of the infinite possible distances makes the most sense for how we are thinking about our model. That is, any number of possible loss functions could be right, it depends on our situation, our data, and our goals (it may help you to read my answer to: What is the difference between linear regression on y with x and x with y?).



          It is often the case that vertical distances make the most sense, though. This would be the case when we are thinking of $Y$ as a function of $X$, which would make sense in a true experiment where $X$ is randomly assigned and the values are independently manipulated, and $Y$ is measured as a response to that intervention. It can also make sense in a predictive setting, where we want to be able to predict values of $Y$ based on knowledge of $X$ and the predictive relationship that we establish. Then, when we want to make predictions about unknown $Y$ values in the future, we will know and be using $X$. In each of these cases, we are treating $X$ as fixed and known, and that $Y$ is understood to be a function of $X$ in some sense. However, it can be the case that that mental model does not fit your situation, in which case, you would need to use a different loss function. There is no absolute 'correct' distance irrespective of the situation.






          share|cite|improve this answer









          $endgroup$



          Vertical distance is a "real distance". The distance from a given point to any point on the line is a "real distance". The question for how to fit the best regression line is which of the infinite possible distances makes the most sense for how we are thinking about our model. That is, any number of possible loss functions could be right, it depends on our situation, our data, and our goals (it may help you to read my answer to: What is the difference between linear regression on y with x and x with y?).



          It is often the case that vertical distances make the most sense, though. This would be the case when we are thinking of $Y$ as a function of $X$, which would make sense in a true experiment where $X$ is randomly assigned and the values are independently manipulated, and $Y$ is measured as a response to that intervention. It can also make sense in a predictive setting, where we want to be able to predict values of $Y$ based on knowledge of $X$ and the predictive relationship that we establish. Then, when we want to make predictions about unknown $Y$ values in the future, we will know and be using $X$. In each of these cases, we are treating $X$ as fixed and known, and that $Y$ is understood to be a function of $X$ in some sense. However, it can be the case that that mental model does not fit your situation, in which case, you would need to use a different loss function. There is no absolute 'correct' distance irrespective of the situation.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 9 hours ago









          gunggung

          111k34 gold badges272 silver badges543 bronze badges




          111k34 gold badges272 silver badges543 bronze badges























              -1












              $begingroup$

              Summing up Michael Chernick comment and gung answer:



              Both vertical and point distances are "real" - it all depends on the situation.



              Ordinary linear regression assumes the $X$ value are known and the only error is in the $Y$'s. That is often a reasonable assumption.



              If you assume error in the $X$'s as well, you get what is called a Deming regression, which fits a point distance.






              share|cite|improve this answer








              New contributor



              David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.





              $endgroup$

















                -1












                $begingroup$

                Summing up Michael Chernick comment and gung answer:



                Both vertical and point distances are "real" - it all depends on the situation.



                Ordinary linear regression assumes the $X$ value are known and the only error is in the $Y$'s. That is often a reasonable assumption.



                If you assume error in the $X$'s as well, you get what is called a Deming regression, which fits a point distance.






                share|cite|improve this answer








                New contributor



                David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.





                $endgroup$















                  -1












                  -1








                  -1





                  $begingroup$

                  Summing up Michael Chernick comment and gung answer:



                  Both vertical and point distances are "real" - it all depends on the situation.



                  Ordinary linear regression assumes the $X$ value are known and the only error is in the $Y$'s. That is often a reasonable assumption.



                  If you assume error in the $X$'s as well, you get what is called a Deming regression, which fits a point distance.






                  share|cite|improve this answer








                  New contributor



                  David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.





                  $endgroup$



                  Summing up Michael Chernick comment and gung answer:



                  Both vertical and point distances are "real" - it all depends on the situation.



                  Ordinary linear regression assumes the $X$ value are known and the only error is in the $Y$'s. That is often a reasonable assumption.



                  If you assume error in the $X$'s as well, you get what is called a Deming regression, which fits a point distance.







                  share|cite|improve this answer








                  New contributor



                  David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.








                  share|cite|improve this answer



                  share|cite|improve this answer






                  New contributor



                  David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.








                  answered 7 hours ago









                  David RefaeliDavid Refaeli

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                  1043 bronze badges




                  New contributor



                  David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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                  New contributor




                  David Refaeli is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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