Thursday, March 19, 2009

Gradient vector calculation for non parametric data

Gradient vector points to the nearst position where a high rate of change can b e seen.

 Normally for a function it is obtained by applying the partial detivative operator. For example if we could represent the temperature inside a room by a function ( with parameter hight or time any conceiveing things )  , we could find the the gradient vector by applying partial derivative operator on that function. Using this technique we can go to the position where the temperature is highest than the current position.  We just need to simply go along the direction of gradient vector.

Okay .. Things are simple when we can paramterize things. Practically parametrization of physical things is not easy. In such situation we can find the gradient density vector using kernal density gradiant estimator.  In the follwing figure illustrates this mathematic technique. You can see that the circle is moved from it orginal position to the densest region (where more points present) .  I calculated gradient vector and moved the circle through that direction , thus finally the circle reached at the densest region . 

Used Epanechnikov Kernel for density estimation.

1 comment:

Anonymous said...

You made a few excellent points there. I did a search about the topic and almost not found any specific details on other sites, but then happy to be here, really, appreciate that.

- Lucas