What math level teaches turning data points into equations?

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Discussion Overview

The discussion revolves around the mathematical processes involved in converting data points into equations, particularly in the context of calculus and mathematical modeling. Participants explore various methods of fitting data to curves, including polynomial approximations and interpolation techniques.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants mention that the process of fitting data points to equations is known as Mathematical Modeling or curve fitting, which can be introduced as early as Introductory Algebra.
  • Others suggest that more advanced techniques are typically covered in statistics or numerical analysis courses at the university level.
  • One participant expresses skepticism about the assumption that a simple formula exists for any graph, emphasizing their interest in deriving formulas for complex graphs.
  • Another participant proposes that numerical integration techniques, such as Simpson's rule or Runge-Kutta, may be more suitable for finding integrals from data points without needing a specific formula.
  • Interpolation and extrapolation are mentioned as relevant concepts for deriving equations from data points, with a suggestion to explore cubic splines and Bézier splines for more complex fitting.
  • Basic polynomial curve-fitting was noted as a topic covered in introductory applied mathematics courses.
  • One participant references a specific program for data fitting and interpolation, indicating practical resources for exploring mathematical models.

Areas of Agreement / Disagreement

Participants express a range of views on the methods and levels of mathematics involved in fitting data to equations. There is no consensus on a single approach or the necessity of a simple formula for every graph, indicating ongoing debate and exploration of the topic.

Contextual Notes

Participants highlight limitations in their understanding of complex graphs and the mathematical techniques available for fitting data, suggesting a need for further exploration of topics like differential equations and numerical methods.

willpower101
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And what is the process called?

Say you have a bunch of data you've collected plotted on a graph, what level of math teaches you to convert this data into a formula that you can play with?

I'm in calc 2 right now and we just covered volumes. It's AWESOME! I'm not much of a higher level math person, but this has got me thinking about moving on. But I really don't know what's out there. We are working with such simple equations, but what if I didn't know the equations?

Say I plotted the dimensions of 1/2 beer bottle, lengthwise about the x-axis, and wanted to integrate it from 0 to whatever the height is. But the problem is I don't know what formula represents the shape of these dots. I mean I could estimate and do it in sections, but how do I get an equation that represents it perfectly?

Thanks a lot! Sorry if this is basic :)
 
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willpower101 said:
And what is the process called?

Say you have a bunch of data you've collected plotted on a graph, what level of math teaches you to convert this data into a formula that you can play with?

I'm in calc 2 right now and we just covered volumes. It's AWESOME! I'm not much of a higher level math person, but this has got me thinking about moving on. But I really don't know what's out there. We are working with such simple equations, but what if I didn't know the equations?

Say I plotted the dimensions of 1/2 beer bottle, lengthwise about the x-axis, and wanted to integrate it from 0 to whatever the height is. But the problem is I don't know what formula represents the shape of these dots. I mean I could estimate and do it in sections, but how do I get an equation that represents it perfectly?

Thanks a lot! Sorry if this is basic :)

Why do you think that for any graph there must exist a simple formula that represents it?

What we CAN do is to approximate the graph by piecewise "simple" formulas and sew them together.

Because polynomials are so nice to work with, it is quite popular to approximate difficult curves by piecewise polynomials.
 
Fitting data points to equations is called Mathematical Modeling; also curve fitting. You learn this as early as Introductory Algebra, and you can learn much more of it in Intermediate Algebra and PreCalculus.
 
symbolipoint said:
Fitting data points to equations is called Mathematical Modeling; also curve fitting. You learn this as early as Introductory Algebra, and you can learn much more of it in Intermediate Algebra and PreCalculus.

I think more likely that it is taught more in statistics (around second year of university) and numerical analysis (second or third year of university. The equation to fit data to a curve is quite simple if all you care about is a piece wise approximation. If you wish for derivatives to match up then it may be more complex.
 
arildno said:
Why do you think that for any graph there must exist a simple formula that represents it?
I don't, nor did I say this. Currently most of the work we do is with simple graphs (x^2, x^3, lnx, e^x, 1/x, x^(1/2), x, stretched, shrunk, or shifted accordingly) so that we may focus on calculus. I'm interested in the process of deriving a formula for much more complex graphs. I've been reading the wiki on diffy q's and think this might be the answer, although it's a bit over my head.

actually, after some more searching, I think what I'm looking for is interpolation. That also lead me to info on extrapolation which looks fantastic.
 
If your goal is just to find a mathematical formula to fit the points purely for the purpose of finding the integral then you would be much better off looking at numerical integration techniques like Simpsons rule or Runge-Kutter. These work directly on the data points (you could say they implicitly interpolate the data points).
 
Last edited:
willpower101 said:
I don't, nor did I say this. Currently most of the work we do is with simple graphs (x^2, x^3, lnx, e^x, 1/x, x^(1/2), x, stretched, shrunk, or shifted accordingly) so that we may focus on calculus. I'm interested in the process of deriving a formula for much more complex graphs. I've been reading the wiki on diffy q's and think this might be the answer, although it's a bit over my head.

actually, after some more searching, I think what I'm looking for is interpolation. That also lead me to info on extrapolation which looks fantastic.
You can continue with looking in on cubic splines. :smile:
 
Basic polynomial (3rd degree and lower) curve-fitting I did in first year in applied maths, but it was very introductory stuff.
 
willpower101 said:
actually, after some more searching, I think what I'm looking for is interpolation. That also lead me to info on extrapolation which looks fantastic.

The http://www.digitalCalculus.com/demo/curvfit.html" program has many data-fitting problems solved for ideas about math models. Plus, it does interpolation and/or extrapolation for ones solution. Source code is include if interested.

For periodic data use a Sine, Damped Sine, or Lorentzian series as a math model.

For non-monotonic, but not periodic, data I recommend the Lorentzian series.
 
Last edited by a moderator:
  • #10
I second arildno's suggestion of cubic splines (or, generally, Bézier splines).
 

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