How to symbolically solve the minimization problem in simple linear regression

I want to use mathematica to symbolically solve the minimization problem in simple linear regression:

Find argminα,βQ(α,β),for Q(α,β)=n∑i=1(yi−α−βxi)2,{\text{Find }}\text{arg}\min _{\alpha ,\,\beta }Q(\alpha ,\beta ),\qquad {\text{for }}Q(\alpha ,\beta ) =\sum _{i=1}^{n}(y_{i}-\alpha -\beta x_{i})^{2},

where yiy_i, xix_i, and nn are symbolic but not specific numbers.The expected answer would be something like

ˆβ=∑ni=1(xi−ˉx)(yi−ˉy)∑ni=1(xi−ˉx)2,ˆα=ˉy−ˆβˉx\begin{align}
\hat {\beta }&={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}},\\
{\hat {\alpha }}&={\bar {y}}-{\hat {\beta }}\,{\bar {x}}
\end{align}

Can someone show some code example for doing this? Thanks in advance!

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– Michael E2
Jul 9 at 21:48

  

 

Thanks Michael for your suggestions!
– CrossD
Jul 9 at 21:49

  

 

See (107426).
– MarcoB
Jul 9 at 21:54

  

 

What have you done so far? People are more likely to help if you show effort.
– Feyre
Jul 9 at 21:56

1

 

You’re welcome. Maybe someone else can help with your question. My experience has been that Mathematica’s facility with algebraic/symbolic sums is limited.
– Michael E2
Jul 9 at 21:56

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1 Answer
1

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The following is an adaptation of my answer to this question, which focused on the 3D linear least-squares problem.

(* Rules to get constants out of sums (or integrals etc) *)
outrules = {
Sum[f_ + g_, it : {x_Symbol, __}] :> Sum[f, it] + Sum[g, it],
Sum[c_ f_, it : {x_Symbol, __}] :> c Sum[f, it] /; FreeQ[c, x],
Sum[c_, it : {x_Symbol, __}] :> c Sum[1, it] /; FreeQ[c, x]
};

(*Generate the sum of squares*)
Sum[Expand[(y[i] – a x[i] – b)^2], {i, 1, n}];

(*Calculate the derivatives*)
Grad[%, {a, b}];

(*Use the linearity property of sums*)
Distribute /@ %;

(*Pull out any constants from summations*)
% //. outrules;

(*Set the derivatives equal to zero to generate a system of equations*)
Simplify[Thread[% == 0]];

(*Solve for the a, b parameters*)
Solve[%, {a, b}] // FullSimplify

These sums are not yet expressed as a function of (ˉx,ˉy)(\bar{x},\bar{y}), i.e. the average values of the (xi,yi)(x_i,y_i), respectively, as you have them in your question, but that should be a question of some algebraic transformations. Some such transformations are reported e.g. in this MathWorld document of least-squares fitting.