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Math Help - compact support gaussian function

  1. #1
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    compact support gaussian function

    Hi, I am not mathematician. (I am computer science)
    I need your help to construct a compact support gaussian function

    It is ok for an interval [a, b]?
    [tex]f(x) = \exp(-0.5\frac{(x-\mu)^2}{\sigma^2})[x/tex]
    f(x) = \exp(-0.5\frac{(x-\mu)^2}{\sigma^2})
    if x \in (a, b)
    and
    f(x) = 0 otherwise


    And how I can to prove this
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  2. #2
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    Re: compact support gaussian function

    Quote Originally Posted by jorjasso View Post
    And how I can to prove this
    Prove what?
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  3. #3
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    Re: compact support gaussian function

    is it ok that definition for compact support gaussian function?
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    Re: compact support gaussian function

    I am not familiar with "compact support gaussian function" considered as one term. I found only one occurrence of this phrase on the web. Strictly speaking, this is not a Gaussian function because the latter has infinite support. You can give any definition you like; the question is in the properties of the defined object. Do you need to guarantee any particular properties of this function?
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    Re: compact support gaussian function

    Yes Emakarov, I want to use the gaussian function as membership function of fuzzy numbers.
    An definition of fuzzy numbers from [1] is that
    fuzzy numbers are functions [tex] f:R\to [0,1] [\tex]
    that satisfy four conditions, one condition is that the support has to be closed and bounded (compact).

    [1] Salih Aytar, Serpil Pehlivan, Musa A. Mammadov, The core of a sequence of fuzzy numbers, Fuzzy Sets and Systems, Volume 159, Issue 24, 16 December 2008, Pages 3369-3379, ISSN 0165-0114
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    Re: compact support gaussian function

    Depending on the definition of support, you may need to change the condition x ∈ (a, b) to x ∈ [a, b] because (a, b) is not closed.

    Quote Originally Posted by jorjasso View Post
    An definition of fuzzy numbers from [1] is that
    fuzzy numbers are functions [tex] f:R\to [0,1] [\tex]
    that satisfy four conditions, one condition is that the support has to be closed and bounded (compact).
    Then your function definitely satisfies this condition. Note, however, that if f(x)=e^{- { (x-\mu)^2 \over 2 \sigma^2 } }, then \int_{-\infty}^{\infty}f(x)\,dx=\sqrt{2\pi}\sigma, but \int_{a}^{b}f(x)\,dx<\sqrt{2\pi}\sigma, so f(x)/(\sqrt{2\pi}\sigma) may serve as a probability density on \mathbb{R}, but not on [a, b].
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    Re: compact support gaussian function

    Thanks,
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