How Do You Derive a Vector Function for Taylor Series Expansion?

[alnoy]

Hey,
Can somebody help me on this one. I feel out of my depth and have to solve it somehow.

I have a variable vector v=[v1 v2]^T, a constant vector vc = [vc1 vc2]^T, a scalar variable d and a vector function:

s= d/(Vs/V-1)

I need the first derivative ds/dv at a point of the mean of v to use in Taylor series expansion.

Any sugestions

 

[HallsofIvy]

Hey,
Can somebody help me on this one. I feel out of my depth and have to solve it somehow.

I have a variable vector v=[v1 v2]^T, a constant vector vc = [vc1 vc2]^T, a scalar variable d and a vector function:

s= d/(Vs/V-1)

I don't understand what this means. It looks like you are dividing vectors.

I need the first derivative ds/dv at a point of the mean of v to use in Taylor series expansion.

Any sugestions

What is "the mean of v"?
s maps v, in R², to what? What space is s(v) in?

 

[alnoy]

I will explain a bit more hope it clarifies the problem,

Everything is in x,y (euclidian space), v and vc are speeds of two bodies,i.e v1, vc1 are the x components and v2,vc2 are y components. d is some distance.
from the Kalman filter that tracks the body the spead is estimated as v but also has uncertainty. The uncertainty is given by the 2x2 variance-covariance matrix Pv.

s gives the error(distortion) in the seen image when measured with a certain sensor.
I want to know the var to be able to deside whether the image that the sensor prodices comes from a certain shape or not (with certain probability).

Var[S(v)], can be approximated through the delta method (Oehlert 1992) that uses second-order Taylor expansion in matrix form which calls for the expected value of v E(v) so I assumed that this is the mean (but maybe I am wrong in this)

Var[S(V)]≈S' (E[V])Var[V](S' (E[V]))^T
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http://en.wikipedia.org/wiki/Delta_method

Oehlert, G.W., 1992. A Note on the Delta Method. American Statistician, 46(1), pp.27–29. Available at: http://www.jstor.org/stable/2684406?origin=crossref.