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Math Help - Rain Gage Network Linear Algebra?

  1. #1
    turten
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    Question Linear Algebra?

    Hi Math Experts,

    I have a 22 Rain-Gauge network, I have bulit 22 nominal models of each rain gauge from correlation analysis, and I have something like that:

    RG1 = a1*RG2+a2*RG7+a3*RG15
    RG2 = b1*RG1+b2*RG9+b3*RG19
    ...
    RG22 = v1*RG3+v2*RG5+v3*RG9

    where RGn is the rain income in the last 5 minutes in the RainGauge number n.

    now I have to generate a "simulated consistent rain" with a computer program that I have to build. This means that I can fix a rain in RG1, and I have to obtain the rain in the 21 other RainGauges according to the relations.

    For me this problem is similar to solving a Linear Equation System like Ax=b, but this system is something like Ax=x and I don't know how to deal with. I've read Gaussian Reduction and LU Decompossition, but I think this is a wrong way.

    Can anyone give me some light to see the correct path to solve this.

    Thank you in advance,

    Jaume.
    Last edited by Math Help; July 26th 2005 at 10:32 PM.
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  2. #2
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    'straya
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    It looks like you have a linear regression model.

    It doesn't look like you have taken auto (serial)correlation into account there.

    I'm not the top statistician in the world, but i bet you a tenner there is correlation in rainfall data.

    Depending on your purpose and the information you have, i'd say there is a better way to do it. Find a statistician or do a google search for simulated rainfall data or the like.

    Cheers,
    TF
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  3. #3
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    Briefly, to turn an equation of the form Ax = x into one of the form Ax = b, consider the matrix A-I: we have Ax = x iff (A-I)x = 0.
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  4. #4
    hpe
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    Quote Originally Posted by turten
    Hi Math Experts,

    I have a 22 Rain-Gauge network, I have bulit 22 nominal models of each rain gauge from correlation analysis, and I have something like that:

    RG1 = a1*RG2+a2*RG7+a3*RG15
    RG2 = b1*RG1+b2*RG9+b3*RG19
    ...
    RG22 = v1*RG3+v2*RG5+v3*RG9
    This is (in matrix notation) a model of the form Ax = x (see the other posts), where A is a 22 x 22 matrix. It appears from what you write that the matrix is quite sparse (it has only about three non-zero entries in each row).

    For this to be self-consistent, the matrix A should have the eigenvalue  \lambda = 1, that is there should be a vector y such that Ay = y. Find that vector y (it should have all positive entries assuming all a1, ..., v3 are positive). It contains the ratios of rainfall data that is consistent with your model, i.e. y_k/y_l = RGk/RGl for all k,l. You can therefore predict all rain data from RG1 with the formula  RGk = RG1 \cdot y_k/y_1.
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