Results 1 to 3 of 3

Math Help - Normal Equations

  1. #1
    Member
    Joined
    Jan 2011
    Posts
    156

    Normal Equations

    Show that the best least squares fit to a set of measurements y_1,...y_m by a horizontal line y = \lambda (where \lambda is a constant) is their average:

    \lambda = \frac{y_1 + ... + y_m}{m}

    Other than that I need to use normal equations and that this is a problem concerning orthogonality, I am completely stuck and have no idea how to proceed. Can anyone help? Thanks.

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,541
    Thanks
    1394

    Re: Normal Equations

    . That's pretty straight forward if you use the definition isn't it? You want to find the value \lambda such that \sqrt{\sum_{i=1}^m (y_i- \lambda)^2} is a minimum. And because the square root function is is increasing that's the same as finding \lambda such that \sum_{i=1}^m (y_i- \lambda)^2 is a minimum.

    So take the derivative with respect to \lambda and set it equal to 0: 2\sum_{i=1}^m \lambda- y_i= 0. That's the same as \sum_{i=1}^m \lambda- \sum_{i= 1}^m y_i= m\lambda- \sum_{i=1}^m y_i= 0. Solve that for \lambda.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2011
    Posts
    156

    Re: Normal Equations

    Thank you, but no derivatives. I have to approach this as a linear algebra problem using normal equations.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Normal distribution with 2 equations
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 27th 2010, 02:41 PM
  2. normal equations
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: May 8th 2010, 11:32 PM
  3. Normal vector and equations of the line
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 8th 2009, 10:51 AM
  4. Replies: 10
    Last Post: October 26th 2008, 04:37 AM
  5. Equations of tangent and normal to a curve
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 7th 2008, 05:04 PM

Search Tags


/mathhelpforum @mathhelpforum