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-   -   invalid points in Z transform (http://book.caltech.edu/bookforum/showthread.php?t=4670)

pouramini 05-05-2016 09:05 AM

invalid points in Z transform
 
The book says

Quote:

" In general, some points in the Z space may not be valid transforms of any
x E X, and multiple points in X may be transformed to the same z E Z,
depending on the nonlinear transform <I>"
How a point in Z can be not a valid transform of any x?

I suppose any z is a map of a x, not!?

ntvy95 05-05-2016 11:32 AM

Re: invalid points in Z transform
 
Quote:

Originally Posted by pouramini (Post 12335)
The book says



How a point in Z can be not a valid transform of any x?

I suppose any x will be mapped to a z! not!?

Here is my understanding:

Any x should be mapped to a z, but not any z can be mapped to a x: In other words, nonlinear transform \Phi may not be an onto function.

For example the nonlinear transform z = \Phi (x) = [1, x^{2}_{1}, x^{2}_{2}] (given in the book), if z_{1} < 0 or z_{2} < 0 then there is no x can be mapped to such z because there is no x_{1} such that z_{1} = x_{1}^{2} < 0 and no x_{2} such that z_{2} = x_{2}^{2} < 0.

Hope this helps.

pouramini 05-06-2016 01:03 AM

Re: invalid points in Z transform
 
Thank you, yes that seems helpful if we regard all points in Z space, but I thought it speaks about the points in Z which are the mapping of a point in data set D.

ntvy95 05-06-2016 09:20 AM

Re: invalid points in Z transform
 
Quote:

Originally Posted by pouramini (Post 12338)
Thank you, yes that seems helpful if we regard all points in Z space, but I thought it speaks about the points in Z which are the mapping of a point in data set D.

Hm, in my understanding, the quote "some points in the Z space may not be valid transforms of any x E X" regards all the points in the Z space. For example: If z = \Phi (x) = [1, x^{2}_{1}, x^{2}_{2}], then the point z = [1, -3, -5] in the Z space cannot be a valid transform of any x \in X.


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