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Least squares

  1. Aug 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Suppose that you are given a set of observations (tk,yk), k = 1,...,M.
    You plot these points on a sheet & it seems that the relationship between (t,y) could be approximated with a second order polynomial.
    a) Write down the model in the form y = Ax + c. Specify the vectors & matrices & give interpretation to all terms.
    b) Write down the least squares estimate x(hat) for x.
    c) Let the elements of x bear a physical interest. How could you assess the accuracy of the estimate x(hat)?
    d) How would you assess the stability of the problem if max(k,j) |tk-tj| is very small? It may help if you draw a picture. Or better still, study the structure of the matrix A.
    2. Relevant equations



    3. The attempt at a solution
    a) Not sure on this one.
    b) Isn't that just using the definition of least squares.
    c) Not sure on this one.
    d) Not sure on this one.
     
  2. jcsd
  3. Aug 11, 2011 #2

    HallsofIvy

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    If the relationship between t and y were a second degree polynomial, we would have [itex]y_k= at_k^2+ bt_k+ c[/tex] for all x and y. That can be written as
    [tex]\begin{bmatrix}y_1 & y_2 & y_3 & \cdot\cdot\cdot\end{bmatrix}= \begin{bmatrix}a & b & c\end{bmatrix}\begin{bmatrix}x_1^2 & x_1 & 1\\ x_2^2 & x_2 & 1 \\ x_3^3 & x_3 & 1\\ \cdot\cdot\cdot & cdot\cdot\cdot & cdot\cdot\cdot \end{bmatrix}[/tex]
     
  4. Aug 11, 2011 #3

    vela

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    Fixed your post up.
     
  5. Aug 14, 2011 #4
    Thanks.
    For b) is it just using the definition of least squares otherwise what do I do?
     
  6. Aug 15, 2011 #5
    Still lost, any ideas.
     
  7. Aug 16, 2011 #6
    What exactly does it mean write down the least squares estimate x(hat)?
     
  8. Aug 16, 2011 #7

    vela

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    Do you know what the least squares method is used for?
     
  9. Aug 17, 2011 #8
    Isn't it to fit a polynomial through a set of points where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.
    For c) does that mean assume x is real world data and if so how do you assess the accuracy of x(hat)?
    For d) How do I assess the stability of the problem if maxk,j |tk - tj| is very small, not too sure on these problems, please help.
     
  10. Aug 17, 2011 #9

    vela

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    So in this problem, the method is used to calculate what exactly?
    Could you clarify what you mean by "real world data"? The data in this problem are the pairs (tk, yk).
    Surely, the topic of stability must have been covered in your book or lecture. What do you know about it?

    So far, all you've done is ask for the answers to the question. You need to show some effort that you've tried to figure out the problems on your own.
     
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