Linearization with Polynomials

[MikeSv]

Hi everyone.

I started to look at different linearization techniques like:

-linear interpolation
- spline interpolation
- curve fitting...

Now Iam wondering (and I guess its very stupid) : As polynomials with a degree > 1 are not linear, why can I use them for linearization?

With the method of piecewise linear interpolation its clear. You connect two datapoints with a linear function.

But what about quadratic and cubic splines which are not linear?

Thanks in advance for any help,

Kind regards,

Mike

 

[BvU]

Hi,

Can you give an example where someone claims an interpolation by means of a cubic spline or a polynomial is a linear interpolation ?

 

[MikeSv]

Hi and thanks for your reply.

Iam trying to learn more about how to linearize outputs of a nonlinear sensor and I found a list in a Master thesis on the web which desribes different techniques for linearization.
The list says "Piecewise polynomial or spline interpolation"...

Or is linearization of nonlinear sensors is done with linear polynomials only?

Regards,

Mike

 

[MikeSv]

Hi again.

I found a section in a scientific paper which mention techniques like:
- look up table
- polygonal approximation
- polynomial approximation
- cubic spline interpolation

So I guess that answers one part of my question. Polynomials of higher order can be used?

But then another question came up... Whats the difference between plynomial approximation and spline interpolation?

Thanks again,

Mike

 

[fresh_42]

I found a list in a Master thesis on the web which desribes different techniques for linearization.

I found a section in a scientific paper

I found somewhere ... tell me what is meant, is not an acceptable quotation. So the demand

Can you give an example where someone claims an interpolation by means of a cubic spline or a polynomial is a linear interpolation ?

is still valid.

E.g. polynomials could be the elements (points) of the phase space and not the object of the linearization process. In this case, we might talk about a linearization between polynomials of any degree by linear polynomials of polynomials. A bit constructed, I admit, but it shows you the difficulties we face, if you say things like "I have read / found / heard ... somewhere / on the web / in an article".

 

[MikeSv]

Hi and thanks for your reply!

Please take into consideration that I have just started with the topic and Iam just confused what is meant by "linearization" of data... Maybe its the same as interpolation in that context?

Regards,

Mike

 

[Stephen Tashi]

Maybe its the same as interpolation in that context?

What context? You haven't revealed the title of the scientific paper or quoted an abstract from it.

In mathematics, some nouns like "linearization" can be used with a variety of meanings. In spite of the fact that mathematics frowns on ambiguous terminology, mathematics is cultural phenomenon subject to the habits of human beings. So it frowns on ambiguous terminology and also uses it.

 

[MikeSv]

Hi again.

As I mentioned earlier I would like to take raw data of a non linear sensor (i.e. a thermocouple) and learn how to linearize these.
But Iam not sure which techniques are used for that.

Regards,

Mike

 

[fresh_42]

Here's a standard method and a pretty extensive explanation.
https://en.wikipedia.org/wiki/Linear_regression

But you can probably find more on the subject (LR) on the web.

 

[zmth]

Difference between spline and polynomial approximation is that that to some order the splines are only piecewise continuous in derivatives usually some order is discontinuous at the fitting points while a polynomial has continuous derivatives of all orders generally.

 

[WWGD]

In some Statistical treatments a model or expression are considered linear if they are linear in the coefficients. Maybe this is the case with your book