> Dear list members,

You are asking an undefined question here! Your ellipses are

>

> 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@...

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.

--------------------------------------------------------------------

E-Mail: (Ted Harding) <Ted.Harding@...>

Fax-to-email: +44 (0)870 094 0861

Date: 21-Feb-05 Time: 10:29:34

------------------------------ XFMail ------------------------------