🚀 go-pugleaf

For linear regression   y^ = a + bx,  b = Sxy/Sxx, a = y_ - b x_
and the prediction interval formula for y is
                 _____________________
     ^          /      1    (x0-x_)^2           2   SxxSyy-(Sxy)^2
     y  +  S   /  1 + --- + ---------    where S  = --------------
        -    \/        n      Sxx                     (n-2)Sxx

This is in many stats texts.  Also for polynomial regression, say
quartic
regression, y^ = b0 + b1.x + b2.x^2 + b3.x^3 b4.x^4, to get the point
estimate, we solve the normal equations
            [$y    ]   [ n    $x   $x^2 $x^3 $x^4 ] [b0]
            [$xy   ]   [ $x   $x^2 $x^3 $x^4 $x^4 ] [b1]
            [$x^2.y] = [ $x^2 $x^3 $x^4 $x^5 $x^6 ] [b2]
            [$x^3.y]   [ $x^3 $x^4 $x^5 $x^6 $x^7 ] [b3]
            [$x^4.y]   [ $x^4 $x^5 $x^6 $x^7 $x^8 ] [b4]
where $ denotes 'sigma' for summation.  Can anyone tell me the formula
for the prediction interval in this case?

[One could use the General Linear Model i.e. treating x, x^2, x^3, x^4
as
independent variables.  but they are not independent variables.  so that
could affect the formula somewhat.]
            
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