well, i made some experiments with this type of functions and made the experiences that the maps tends to ''explode'', i.e. the weight vector will get out of every usable range when i used functions with negative values, i.e. where they make the distance between winner neuron and other neurons bigger. i''m often using , that''s this :

@$3.1415rin

The mexican hat is taking twice the derivative of exp(-x^2), so
it is 2*(x^2-1)*exp(-x^2). If you want to make it narrower as function of time you can include the witdh d in it like exp(-(x/d)^2). So the mexhat is (2/d^2)*(x^2-1)*exp(-(x/d)^2). In this way the function always will have the shape of a mexican hat, even when d is decreased over time to make it narrower.
Of course the constant 2 can be removed because it does not affect the shape....

Yes Smilydon, you are correct (well, almost: you''ve lost a factor of 2 in your equation if that is meant to be dxx[e-x^2]).

Note though that expanding out (x-a)(x+a) gives x2-a2. The negative was added to give the correct positive value at zero for the training reinforcement. The equation can be scaled in any way you like. The reason why I presented the factorised version is so that it could easily be seen that the roots of the equation are at (-a,+a), making it easier to fit the function to the task at hand. It''s easy to see that setting a=1 - as I suggested - gives the (-ve) Mexican hat function.

Cheers,

Timkin

This may be interesting to some,

There is an awesome opengl grahing
program , free and small written in delphi
@
http://www.sulaco.co.za/opengl.htm#graphs

check it out!

Mexican Hat is in Utah. Well, actually, there are kind of two Mexican Hats there. The town, and the... Mexican hat, which appears to be upside down in my opinion. Royal Robins was the first guy to put a route up underneath the brim, which incidentally is about 60 feet across.

I''ve been to Mexican Hat twice. Did any of this help you?

___________________________________

Very funny.......

Sorry but I''m not looking for a place to travel. Unfortunately.

use <!-- &nbsp --> to add a space in "normal" text.

According to Teuvo Kohonen who is, as we all know, a God among men when it comes to these matters, there a few different ways to define the "Mexican Hat" formula. Before I attempt to encode this formula using only plain-text, let me first say that any Gaussian formula is a *good* thing, and if you have one already it should probably work fine. However, this is one listed in Kohonen''s "Self Organizing Maps":

H(t) = alpha(t) * exp(-(dist^2)/(2*n(t)^2))

-t is defined to be the number of training steps taken
-alpha(t) is the learning factor with regards to t
-dist is generally considered to be Euclidean, but doesn''t always have to be-n(t) is the "neighborhood" function, which defines what region the given node is in relation to the winning node. Definition of the neighborhood functions is a point that people argue about, but generally it''s just a series of concentric squares. I know this isn''t a very clear explanation, but if you do a search you should find places that have pictures of what I''m trying to describe.

Anyway, don''t worry too much about which function you use, but I would suggest training more than just the two closest neurons when you find the winner. Under-training the net can have a much greater effect than the actual training function on your success.


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The wind shear alone from a pink golfball can take the head off a 90-pound midget from 300 yards.

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