set S of coordinates ( x , y ), and am estimating f ( x ) = a x + b where a &gt; 0. I also happen to know that ∀ x , y ( ( x , y ) ∈ S ⟹ y &lt; f ( x ) ).The question is how I can utilize this knowledge of the upper bound on values to improve the regression result?My intuition is to run a "normal" linear regression on all coordinates in S giving g ( x ) and then construct g ′ ( x ) = g ( x ) + c, with c being the lowest number such that ∀ x , y ( ( x , y ) ∈ S ⟹ y ≤ g ′ ( x ) ), e.g. such that g ′ ( x ) lies as high as it can whilst still touching at least point in S. I do, however, have absolutely no idea if this is the best way to do it, nor how to devise an algorithm that does this efficiently.

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set   S of coordinates   (  x  ,  y  ), and am estimating   f  (  x  )  =  a  x  +  b where   a  &amp;gt;  0. I also happen to know that   ∀  x  ,  y  (  (  x  ,  y  )  ∈  S    ⟹    y  &amp;lt;  f  (  x  )  ).The question is how I can utilize this knowledge of the upper bound on values to improve the regression result?My intuition is to run a &quot;normal&quot; linear regression on all coordinates in   S giving   g  (  x  ) and then construct       g    ′    (  x  )  =  g  (  x  )  +  c, with   c being the lowest number such that   ∀  x  ,  y  (  (  x  ,  y  )  ∈  S    ⟹    y  ≤      g    ′    (  x  )  ), e.g. such that       g    ′    (  x  ) lies as high as it can whilst still touching at least point in   S. I do, however, have absolutely no idea if this is the best way to do it, nor how to devise an algorithm that does this efficiently.

rideonthebussp · Accepted Answer

Your suggestion to do an ordinary regression and then move it up is a fine way to go about it. This is pretty easy, especially if you have a decent statistics library:1. Fit the regression.2. Calculate the residuals, the difference between the y coordinate of each point and the y coordinate of the line at that point, given by               β      ^        0    +              β      ^        1        x    i  , where       x    i   is the       x   coordinate of the point. There may be a built-in way to do this. There are matrix-algebra representations as well.3. Find the largest residual. Not largest in absolute value, just straight-up largest.4. Add the value of the largest residual to the intercept               β      ^        0   of your regression model.Your regression line now passes through the highest point and is above all the other points.

Set S of coordinates (x,y), and am estimating f(x)=ax+b where a>0. ∀x,y((x,y)∈S⟹y<f(x)). ∑

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