RE: [ai-geostats] Probability distribution of a moving object be

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• ... 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 1 of 2 , Feb 21, 2005
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.
>
> Thank you for any help.
> --
> Sunny Elspeth Townsend
> sunnytownsend@...

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@...>
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Date: 21-Feb-05 Time: 10:29:34
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