LFD Book Forum - invalid points in Z transform

LFD Book Forum (http://book.caltech.edu/bookforum/index.php)

- Chapter 3 - The Linear Model (http://book.caltech.edu/bookforum/forumdisplay.php?f=110)

- - 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

should be mapped to a

, but not any

can be mapped to a

: 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

can be mapped to such

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

in the Z space cannot be a valid transform of any $x \in X$ .

All times are GMT -7. The time now is 01:13 AM.

Powered by vBulletin® Version 3.8.3
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.
The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. Abu-Mostafa, Malik Magdon-Ismail, and Hsuan-Tien Lin, and participants in the Learning From Data MOOC by Yaser S. Abu-Mostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.