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Math Help - Write down the normal eq for estmiating the constant A and B

  1. #1
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    Write down the normal eq for estmiating the constant A and B

    Question :

    Write down the normal equations for estimating the constant A and B in the least-square fit of the model

     y = A + Bx,

    using n data points i.e. (x,y),(x_2,y_2),.....,(x_n,y_n).
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  2. #2
    MHF Contributor matheagle's Avatar
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    take the sum of squares and differentiate wrt A and B.
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  3. #3
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    Is this correct?

    Quote Originally Posted by matheagle View Post
    take the sum of squares and differentiate wrt A and B.

    y \ = \ A + Bx

    q  \ = \ \sum_{i} [y - (A+Bx)]^2

    Differentiating partuall wrt to A and B weget

    \frac{\partial q}{\partial A} \ = \ (-2)[y_i -(A + Bx_i)] \ = \ 0

    and
    \frac{\partial q}{\partial B} \ = \ (-2) x_i[y_i -(A + Bx_i)] \ = \ 0

    normal equation is

    \sum_{i} y_i \ = \ An + B \sum_{i} x_i

    \sum_{i} x_i y_i \ = A \sum_{i} x_i + B \sum_{i} x_i ^2

    Is this correct??
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  4. #4
    MHF Contributor matheagle's Avatar
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    Impressive, we got you to do some decent work.
    The normal equations look good,
    but you left out the sums in the partial derivatives
    in the previous set of equations.
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  5. #5
    MHF Contributor matheagle's Avatar
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    a few adjustments....

    Quote Originally Posted by zorro View Post

    y_i \ = \ A + Bx_i

    q \ = \ \sum_{i} [y_i - (A+Bx_i)]^2

    Differentiating wrt to A and B we get

    \frac{\partial q}{\partial A} \ = \ (-2)\sum_i[y_i -(A + Bx_i)] \ = \ 0

    and
    \frac{\partial q}{\partial B} \ = \ (-2)\sum_i x_i[y_i -(A + Bx_i)] \ = \ 0

    the normal equations are

    \sum_{i} y_i \ = \ An + B \sum_{i} x_i

    \sum_{i} x_i y_i \ = A \sum_{i} x_i + B \sum_{i} x_i ^2
    Last edited by matheagle; December 13th 2009 at 06:24 PM.
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