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Math Help - Exponential function with espected behavior - removal experiment in ecology

  1. #1
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    Exponential function with espected behavior - removal experiment in ecology

    I'm working with bird fatalities in windfarms and one of the required parameters to estimate mortality is the probability of an animal remain in the field at a given time, in order to be available for detection while a field work session. (which will be another probability to be accounted for).

    Typically bird carcass removal by animals will follow a negative exponential curve. It's very probable that a bird will be removed by a carnivore in the first few days after fatality. There are well described equations to evaluate mortality using there parameters.

    What I'm trying to achieve is a exponential function with two particular behaviors. 1 - be limited by one at day zero.and tend to zero.

    Any help will be appreciated
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  2. #2
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    Quote Originally Posted by pcardoso View Post
    I'm working with bird fatalities in windfarms and one of the required parameters to estimate mortality is the probability of an animal remain in the field at a given time, in order to be available for detection while a field work session. (which will be another probability to be accounted for).

    Typically bird carcass removal by animals will follow a negative exponential curve. It's very probable that a bird will be removed by a carnivore in the first few days after fatality. There are well described equations to evaluate mortality using there parameters.

    What I'm trying to achieve is a exponential function with two particular behaviors. 1 - be limited by one at day zero.and tend to zero.

    Any help will be appreciated
    A nice problem. Are you sure you're qualified for this kind of work? If you can't find the distribution, can you deal with it after you get it?

    You said it was a Negative Exponential. Since you have only two constraints, we get only two paramenters. It might look like this.

    f(x)\;=\;a*e^{-b*x}

    The first constraint: f(0) = 1 leads immediately to a = 1

    The second contstraint, and you tell me why this is so, is \int_{0}^{\infty}f(x)\;dx\;=\;1

    Anyway, this gives b = 1. That couldn't be much simpler.

    f(x)\;=\;e^{-x} valid on [0, \infty)
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    I'm biologist.

    I've done this kind of trials in the field and the removal event in time follow the negative exponential shape. The experiments were done with a number of carcasses distributed in the field and verified daily for 30 days period. By plotting number of remaining carcasses against time, the relation is clearly exponential.

    The mathematical approach is new as exploratory method for me but I think I can understand the process after some input from experts in maths.

    For the second constrain, it seems logical (and it's referred in some bibliography) that the function should tend to zero at \infty. A carcass could remain in the field time enough to disappear naturally without being detected. Aditionally the function must necessarily equals to one at the origin, since the probability at day zero is 100%.

    The intention is to simulate scenarios where searching trials, date of fatality, probability of remain in the field and detection of carcasses probability are all known parameters. I'll do the fatalities vary in time and apply known functions to see how the estimate of mortality behaves when detection sessions intensity vary.

    This is the basic idea.

    You mean that If I have, say, 30 carcasses at day 0, the decay rate could be approached using the simplest equation?
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    1) Approaching zero is trivial. There are many, many ways to do that.

    2) The design of the function provides the constraints. My intent was a statistical distribution. The probability must sum to unity (1) in order to be a proper distribution.

    3) It is possible that your application is not a real good match to the calculus-based derivation. If you're talking about only 30 birds at a time, a continuous distribution, which I showed, may not be appropriate. The probability of finding a carcass is the sum of the areas underneath the curve. A Discrete distribution of some sort may be more appropriate.

    4) Don't worry too much about the concept of "Infinity". It does not have to mean "infinity" It just has to mean "big enough". If you have no expectation that any carcass would survive beyond 30 days, then 31 might be infinity. This may be a conceptual leap.

    5) If one simply evaluates 30*e^{-x}, for x in 0, 1, 2, 3, 4, etc, we get: 30.0, 11.0, 4.1, 1.5, 0.5, 0.2, 0.1, 0.0 for the expected number of birds found still in the field for each day following the initial observation.
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    I understand your point.

    Nonetheless there are a few more considerations. At day 30, there are carcasses not removed. While in field work we detected, a few months latter, that some carcasses still there, and will never be removed by a carnivore due to its condition.

    This leads me to believe that at day 31 cannot be approached as infinity. If I start the experiment at day one, I'll need a probability at day 365.

    The equation I was using was y=7.147^{-exp{(0.007*x)-(3.858/x)}} but I'n not sure if it is under the assumptions that you mentioned.

    How can we graphically express equations in the post?
    Last edited by pcardoso; July 14th 2009 at 02:41 AM. Reason: add comments
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    Quote Originally Posted by pcardoso View Post
    Aditionally the function must necessarily equals to one at the origin, since the probability at day zero is 100%.
    This is wrong, you are confusing a continuous distribution with a discrete, which is why TKH in her first post finds a unique solution with no free parameter to represent the charteristic removal time.

    There should be no constraint on the value at time zero.

    CB
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    I don't understand this statistically.

    Time could be considered a continuos variable. Searching trials is done only dialy, 'discretizing' the data. I don't know if this is a valid argument.

    If the method and data can't match the assumptions, is thare any alternative?

    THis is mandatory that p=1 at time=0
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    Quote Originally Posted by CaptainBlack View Post
    There should be no constraint on the value at time zero.
    why?
    Last edited by CaptainBlack; July 15th 2009 at 01:31 AM. Reason: editing rather than quoting
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  9. #9
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    Quote Originally Posted by pcardoso View Post
    why?

    Because the point value of a pdf has no significance, just it's integral over (to keep things simple I oversimplify here) intervals.

    The probability of any single point when you have a continuous distribution is zero.

    CB
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    I think that I have a lot of theoretical to clarify to myself.

    The relation I'm studying is between the number of carcasses (Nc) that remain not removed at each day (D) in a 30 day interval of observation.

    Te decay rate between Nc and D could also be approached as % of animals that are not removed.

    It was my idea that this percentage could be interpreted as a probability of one carcass remain untouched after a given number of days. This is intentionally done to estimate the probability of one carcass remaining after the 30+N days.

    The first question is that this is a valid statement: Consider the % as a probability.

    If this fails and considering that the probability of a point in a pdf is zero, it means that I cannot approach the decay relation between % of carcasses not removed at day 30+N?
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    It's okay to have a continuous distribution if you are willin gto live with the consequences. Your distribution may suggest a mean number of dead bodies something like 3.75. We may be willing to count 0.75 bodies, but I doubt it.

    As far as infinity goes, you entirely missed my point. I don't really care what infinity is. You should judge what it needs to be based on your experience and judgment.

    I'm still struggling with your design. Are you shutting down the windfarm to conduct your survey? I think not. This mean you do NOT have a Closed System. The implication is that your number of unremoved bodies is likely to INcrease at times. You would do well to rethink. You can't go out and tag the bodies. Your handling will discourage predators and skew your results.

    Remember my first comments? I'm getting back to those. You may need a better background in mathematical statistics in order to wade through this material.
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