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[ai-geostats] Probability distribution of a moving object between 2 points

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  • Sunny Elspeth Townsend
    Dear list members, I would like to know how to find the probability distribution for the position of an object moving between two known points. I am trying to
    Message 1 of 2 , Feb 21, 2005
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      Dear list members,

      I would like to know how to find the probability distribution for the
      position of an object moving between two known points. I am trying to
      reconstruct the tracks of fishing vessel using satellite data which
      gives the ship's position every 2 hours. The ships do not move often
      move in straight lines, and I want to be able to incorporate the
      uncertainty of where the ship coulf be between the known points. I have
      already found the outer limit of movement which takes the form of an
      ellipse with the start and end points as the foci (for this I assumed
      constant speed).

      My question is:
      What is the probability distribution of the object within the ellipse.

      I would like to find the statistical distribution but would be happy if
      anyone knows if any GIS software has a function for this.

      Please reply to sunnytownsend@...
      Thank you for any help.
      --
      Sunny Elspeth Townsend
      sunnytownsend@...
    • Ted Harding
      ... You are asking an undefined question here! Your ellipses are calculated on the basis of constant speed in a straight line from one determined point, out to
      Message 2 of 2 , Feb 21, 2005
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        On 21-Feb-05 Sunny Elspeth Townsend wrote:
        > Dear list members,
        >
        > I would like to know how to find the probability distribution for the
        > position of an object moving between two known points. I am trying to
        > reconstruct the tracks of fishing vessel using satellite data which
        > gives the ship's position every 2 hours. The ships do not move often
        > move in straight lines, and I want to be able to incorporate the
        > uncertainty of where the ship coulf be between the known points. I have
        > already found the outer limit of movement which takes the form of an
        > ellipse with the start and end points as the foci (for this I assumed
        > constant speed).
        >
        > My question is:
        > What is the probability distribution of the object within the ellipse.
        >
        > I would like to find the statistical distribution but would be happy if
        > anyone knows if any GIS software has a function for this.
        >
        > Please reply to sunnytownsend@...
        > Thank you for any help.
        > --
        > Sunny Elspeth Townsend
        > sunnytownsend@...

        You are asking an undefined question here! Your ellipses are
        calculated on the basis of constant speed in a straight line
        from one determined point, out to some unknown point, and then
        at constant speed from there to the next determined point.
        Therefore the sum of the two distances travelled is constant,
        and the result, as you say, is an ellipse with the two determined
        points as foci.

        There is an implicit assumption of what this constant speed is,
        since you need this to work out what total distance is travelled,
        and you do not state what this assumption is based on.

        But in any case there will in real life be, between the two
        determined points, variations in speed (relative to the water)
        and of direction, and further variations in absolute speed and
        direction due to currents. Some of these will be "random" -- due
        to external influences such as wind -- and some will be the result
        of deliberate choices (which are also likely to be influenced
        by external factors, such as locating a shoal of fish which
        could cause the vessel to linger in that area, which themselves
        have a random character).

        The vessel may be following various "policies", e.g.

        a) basically trying to sail in a straight line at constant
        speed in order to get from A to B
        b) zig-zagging haphazardly over an area in order to try to
        locate fish
        c) pursuing a systematic "sweep" over an area on the lines of

        5km E, 200m N, 5km W, 200m N, 5km E, ...

        The real probability distribution will depend on how all these
        factors combine probabilistically.

        You cannot expect any software to have "a function for this"
        unless you are able to supply information about such factors.
        You cannot expect the software to guess it for you.

        One view of how to approach this kind of question would assume
        a kind of "random walk" or "diffusion with drift": There is
        an overall tendency to move in a cerain direction, but at
        frequent moments of time there are random changes of direction
        and possibly also speed. Starting from A, there will (subject
        to explicit assumptions about these random changes) be a
        probability distribution of position after a given time.
        The fact that B is a determined point at a determined time
        will impose a condition on this distribution, from which the
        conditional probability distribution of position at any
        intermediate time can be determined (though not necessarily
        easily). In the case of diffusion according to "Brownian
        motion" this is known to probabilists as the "Brownian
        Bridge".

        Such approaches have been applied to probabilistic study of (e.g.)
        bird migration, on which there is quite a large literature.

        However, no such considerations allow you to escape from the
        necessity of thinking realistically about what variations from
        "uniform motion in a straight line" are likely to occur, and
        about what random laws they may follow.

        Best wishes,
        Ted.


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        E-Mail: (Ted Harding) <Ted.Harding@...>
        Fax-to-email: +44 (0)870 094 0861
        Date: 21-Feb-05 Time: 10:29:34
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