What constrained regression function shuld I use?

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Simon Wang 2015-3-13

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回答： Simon Wang 2015-3-15

I have a regression model log (r(i)) = a + b * log(A(i)) where A(i) is a vector and each element is known. Log is the nature log.

I need to find out a, b, and each element of r(i) such that the sum of r(i) equals to a constant k and the sum of error, i.e. sum(square[log (r(i)) – (a + b * log(A(i)))]) is minimized. Both a and b are scalars.

What regression model can I choose?

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Answer 1

Simon Wang 2015-3-13

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编辑：Simon Wang 2015-3-13

Also, if this really works, do I need to have specific toolboxes? Do I need to an implement specific solver?

Thanks!

Simon

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Answer 2

Torsten 2015-3-13

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Choose a and b such that

exp(a)*(A(1)^b+A(2)^b+...+A(n)^b)=k

Then sum (exp(a)*A(i)^b) = k is satisfied.

Now define r(i) = exp(a) * A(i)^b, and you are done.

Best wishes

Torsten.

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Answer 3

Simon Wang 2015-3-13

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编辑：Simon Wang 2015-3-13

Thanks!

exp(a)*(A(1)^b+A(2)^b+...+A(n)^b)=k and sum (exp(a)*A(i)^b) = k are the same equation.

So there are two unknowns a and b and only one equation, I did not see the way to get the unique a and b.

Note:

1) a and b are scalars

2) Condition is not used: sum(square[log (r(i)) – (a + b * log(A(i)))]) is minimized

Thanks!

Simon

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Torsten 2015-3-13

Choose b=1, a=log(k/(A(1)+A(2)+...+A(n))) and define r(i)=exp(a)*A(i).

Then sum(square[log (r(i)) – (a + b * log(A(i)))]) is minimized (because it equals 0) and sum r(i)=k.

Best wishes

Torsten.

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Answer 4

Simon Wang 2015-3-15

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THank you for the response. I think it should work and this leads me to change my question a little bit because it will make more sense in real world scenarios. I will post another question and close this one. Thank you very much for the help.