invalid points in Z transform

[pouramini]

The book says

" 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]

Originally Posted by pouramini

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 may not be an onto function.

For example the nonlinear transform (given in the book), if or then there is no can be mapped to such because there is no such that and no such that .

Hope this helps.

[pouramini]

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]

Originally Posted by pouramini

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 , then the point in the Z space cannot be a valid transform of any .