Proving that data follows a polynomial function

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SUMMARY

This discussion centers on proving that data follows a polynomial function of the form y=anx^n + an-1x^(n-1) + ... + a1x + a0. The user seeks methods to validate this model and determine its order. Multivariate linear regression is identified as a technique to find the constants a_n, ..., while model selection criteria such as the Bayesian Information Criterion (BIC) are suggested for selecting the best polynomial degree. However, it is noted that BIC is primarily applicable to exponential models, raising the need for alternative methods specific to polynomial fitting.

PREREQUISITES
  • Understanding of polynomial functions and their forms
  • Knowledge of multivariate linear regression techniques
  • Familiarity with model selection criteria, particularly Bayesian Information Criterion
  • Experience with data transformations, including log-log and natural log
NEXT STEPS
  • Research polynomial regression techniques and their applications
  • Explore alternative model selection criteria for polynomial fitting
  • Learn about data transformation methods for polynomial analysis
  • Investigate tools for visual inspection of polynomial fit quality
USEFUL FOR

Data scientists, statisticians, and researchers involved in modeling complex datasets with polynomial functions will benefit from this discussion.

gsingh2011
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I can prove that data follows a curve of the form y=Ax^n and y=Ae^x by using log log and natural log transformations. I have some data that I believe is more complex, something of the form y=anx^n+an-1x^n-1+...+a1x+a0, in other words a polynomial function. Is there any way I can prove that it follows this form and what the order of the function would be?
 
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For fixed n the constants a_n, ... can be found by multivariate linear regression. Two ways to select the "best" n (i.e. to fit but not overfit) are by visual inspection and by use of a model selection criterion such as Bayesian information criterion.

Also a comment on the wording; in working with real data it's never possible to "prove" the data is from a particular model, only to show the model is in good agreement with the data.
 
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Thanks for the reply. According to wikipedia, that model selection criterion you suggested only works if the data follows an exponential curve. Do you know anything for polynomials? I couldn't find anything just from googling, or I might have missed something since this is new to me.
 

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