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The Generalised Inverse

  1. Oct 31, 2016 #1
    1. The problem statement, all variables and given/known data

    A magnetic data set is believed to be dominated by a strong periodic tidal signal of known tidal period[tex]\Omega[/tex] The field strength [tex]F(t)[/tex] is assumed to follow the relation:

    [tex]F=a+b\cos\Omega t + c\sin\Omega t[/tex]

    If the data were evenly spaced in time, then Fourier analysis would enable simple determination of the three parameters {a, b, c}. For non-uniform data, one technique to obtain the parameters is to calculate a generalized matrix inverse.

    a) Define the model vector m for this problem.
    b) Assume we have three measurements [tex]{F_1, F_2, F_3}[/tex] at times [tex]{t_1, t_2,t_3}[/tex]. Write down the data vector [tex]\gamma[/tex] and matrix A you would derive for these three measurements.

    c) Hence, calculate the normal equations Matrix [tex]A^T A[/tex] and right-hand side vector [tex]A^T \gamma[/tex].

    d) By generalizing your arguement to N data, write down the normal equations matrix.

    f) Imagine you now have many evenly spaced data over one full period of the oscillation. Explain why the off leading-diagonal terms of the matrix are now 0. What are the diagonal terms?

    g) when the data are evenly spaced, explain why the estimates of the parameters {a,b,c} are independent.

    h) What physical properties of the tidal signal could be derived from the values for b and c?

    (20 marks)


    2. Relevant equations

    Given a vector of model parameters m, a data vector [tex]\gamma[/tex] and a matrix A to connect the two vectors, such that [tex]\gamma = Am[/tex]

    a solution for the model parameters can be obtained by solving (inverting) the equation [tex](A^T A)m = A^T \gamma[/tex]

    3. The attempt at a solution

    Starting with a), I'm trying to define my model vector.

    [tex] m = 1/(A^T A) * A^T \gamma [/tex] ??
     
    Last edited: Nov 1, 2016
  2. jcsd
  3. Nov 1, 2016 #2

    Mark44

    Staff: Mentor

    What do you know about the matrix A? Is it a square matrix? If so, is it invertible?

    If A is invertible, then so is AT, so solving the equation ##A^TAm = A^T\nu## involves nothing more than left-multiplying both sides of the equation by ##(A^T)^{-1}##, and then left-multiplying both sides by ##A^{-1}##. There is no division operation for matrices.
     
    Last edited: Nov 1, 2016
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