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Math Help - Relationship between mean reversion constant and period of oscillating series

  1. #1
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    Relationship between mean reversion constant and period of oscillating series

    Hi,

    I am creating a series by:

    1) Multiplying the previous value in the series by a negative constant. For example 100.857 * -0.1 = -10.086, where 100.857 is the previous value and -0.1 is the constant.

    2) Calculating the difference between the last value in the series and the value before that. For example, 100.857 - 92.948 = 7.909

    3) Summing the last value in the series with the results from steps 1) and 2) to arrive at the next value in the series. For example, 100.857 -10.086 + 7.909 = 98.680.

    Repeating this process produces an oscillating series that I think is a sine wave. There is a relationship between the constant and the period of the resulting series. For example, if the constant is -0.1 then the period is approximately 19 iterations. My problem is that I would like to know the formula that relates the constant to the period.

    I have attached a spreadsheet that has sample values.

    Sincere Thanks,

    James
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jalsck View Post
    Hi,

    I am creating a series by:

    1) Multiplying the previous value in the series by a negative constant. For example 100.857 * -0.1 = -10.086, where 100.857 is the previous value and -0.1 is the constant.

    2) Calculating the difference between the last value in the series and the value before that. For example, 100.857 - 92.948 = 7.909

    3) Summing the last value in the series with the results from steps 1) and 2) to arrive at the next value in the series. For example, 100.857 -10.086 + 7.909 = 98.680.

    Repeating this process produces an oscillating series that I think is a sine wave. There is a relationship between the constant and the period of the resulting series. For example, if the constant is -0.1 then the period is approximately 19 iterations. My problem is that I would like to know the formula that relates the constant to the period.

    I have attached a spreadsheet that has sample values.

    Sincere Thanks,

    James
    Write out the recurence and post that here.

    CB
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  3. #3
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    The recurrence relation is:

    Y(i) = Y(i-1) - k * Y(i-1) + Y(i-1) - Y(i-2)

    simplified:
    Y(i) = (2 - k) * Y(i - 1) - Y(i - 2)

    where -k is the constant (eg. -0.10)

    The (i), (i - 1) and (i - 2) should be in subscript, but I'm not sure how to do that.

    Thanks,

    James
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  4. #4
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    I have found what appears to be very close to the answer via simple harmonic motion:

    Period = (2 * Pi) / (k^0.5)

    James
    Last edited by jalsck; October 27th 2009 at 03:32 PM.
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