# Proving that data follows a polynomial function

• gsingh2011
In summary, the conversation discusses the use of log log and natural log transformations to prove that data follows a curve of the form y=Ax^n and y=Ae^x. The speaker also mentions having data that may be more complex, in the form of a polynomial function y=anx^n+an-1x^n-1+...+a1x+a0, and asks for ways to prove that it follows this form and determine the order of the function. The use of multivariate linear regression and model selection criteria is suggested to determine the constants and select the best value for n. However, it is noted that it is not possible to definitively prove that the data follows a certain model, only to show that it is
gsingh2011
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?

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.

Last edited:
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.

## 1. What is a polynomial function?

A polynomial function is a mathematical function that can be expressed as a sum of powers of a variable, with each power multiplied by a constant coefficient. It is written in the form of f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is a non-negative integer and an, an-1, ..., a1, a0 are the coefficients.

## 2. How do you prove that data follows a polynomial function?

To prove that data follows a polynomial function, you can plot the data points on a graph and visually check if the points form a smooth curve. Alternatively, you can use statistical methods such as linear regression to fit a polynomial curve to the data and determine the goodness of fit.

## 3. What is the importance of proving that data follows a polynomial function?

Proving that data follows a polynomial function is important because it helps us understand the relationship between the variables in the data. It allows us to make predictions and draw conclusions about the data, which can be useful in various fields such as economics, physics, and engineering.

## 4. What are some techniques used to prove that data follows a polynomial function?

Some techniques used to prove that data follows a polynomial function include plotting the data points on a graph, using statistical methods such as linear regression, and performing hypothesis tests such as the F-test to determine the significance of the polynomial fit.

## 5. Can data always be represented by a polynomial function?

No, not all data can be represented by a polynomial function. Some data may follow other types of functions, such as exponential or logarithmic functions. It is important to carefully analyze the data and choose the appropriate type of function to represent it accurately.

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