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Old 05-05-2016, 08:05 AM
pouramini pouramini is offline
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Default 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!?
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Old 05-05-2016, 10:32 AM
ntvy95 ntvy95 is offline
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Default Re: invalid points in Z transform

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Originally Posted by pouramini View Post
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.
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Old 05-06-2016, 12:03 AM
pouramini pouramini is offline
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Default 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.
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Old 05-06-2016, 08:20 AM
ntvy95 ntvy95 is offline
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Default Re: invalid points in Z transform

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Originally Posted by pouramini View Post
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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