Can Multivariable Interpolation Relate Three Sets of Data Points?

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Discussion Overview

The discussion centers around the possibility of establishing a relationship between three sets of data points (x, y, z) through multivariable interpolation. Participants explore methods for deriving a function that can predict z values based on given x and y inputs, with a focus on the mathematical approaches and tools available.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant inquires about the feasibility of interpolating three sets of points to derive a function that outputs z values based on x and y inputs.
  • Another participant suggests that a polynomial of degree 3 in two variables could be constructed using the provided data points, noting that this polynomial would uniquely pass through the specified points.
  • This second participant expresses uncertainty about the interpolation's behavior between the data points, despite confirming that it would interpolate exactly at the given points.
  • A third participant mentions finding equations that yield the desired points without needing interpolation, indicating an alternative approach.
  • A fourth participant introduces the concept of integral transforms in multi-dimensional spaces, potentially suggesting another method for addressing the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for interpolation or the effectiveness of the proposed approaches. Multiple competing views and methods remain under discussion.

Contextual Notes

The discussion includes assumptions about the degree of the polynomial and the nature of the data points, but these assumptions are not universally accepted or verified. The effectiveness of the proposed methods in practical applications remains uncertain.

Who May Find This Useful

This discussion may be of interest to those exploring multivariable interpolation techniques, mathematical modeling, or seeking alternative methods for data analysis in STEM fields.

KV-1
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Hello!

I am wondering if it is possible to establish a relationship between three sets of points (x,y,z) by interpolating.

Basically i need a function that takes x and y and gives me a z that matches the following points:


130 472 5
130 590 6
130 738 7.5
130 944 10
155 563 5
155 704 6
155 880 7.5
155 1126 10
180 654 5
180 817 6
180 1022 7.5
180 1308 10
205 745 5
205 931 6
205 1163 7.5
205 1489 10
240 872 5
240 1472 8

I want the middle column to be yielded by the equation when the outer columns are fed into it.. ex (first point): f(130,5) = 472

is this possible? if so, are there any calculators that you recommend or methods that don't require very complex math? (I only know calculus).

thanks!
 
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Well, let's say you had a function that is just from R to R. That is, it takes in one variable and spits out a number. If you had n points at which you knew the value of the function, then you can construct an n-1 degree polynomial that will pass through each of those points, and this polynomial is unique. Now, you have what looks to be 4 x values and 4 y values. Now, you should be able to construct a polynomial in the two variables x,y that is of degree 3. That is, it has the form: p(x,y) = a_{3,3}x^3y^3 + \cdots + a_{3,0}x^3 + a_{2,3}x^2y^3 + \cdots a_{2,0}x^2 + \cdots a_{0,3}y^3 + \cdots a_{0,0}, where you can find the values of the coeficitnts a_{i,j} by solving a system of 16 equations (using the 16 data points you have.) Now, I can show you how to set this up, but I am not sure how well something like this will interpolate. I know that it will interpolate exactly to each of you data points, but I don't know enough theory to be able to predict how it will behave in between those points.
 
Oh thanks!

Not sure though, I think I found some equations that give me the desired points without having to interpolate though...

It helps to know how this can be done though!
 
Hey KV-1 and welcome to the forums.

Are you aware of integral transforms, especially on multi-dimensional spaces?
 

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