What kind of data is best described by a polynomial regression model?

In other words: What kind of data does a polynomial model describe. How will I know if a poynomial regression model is the best model to use for my data?What kind of data is best described by a polynomial regression model?
When trying to discover a relationship between two variables based on observational information, the first step after data collection is viewing a visual relationship of the data.



Many textbook examples will be setups; the data will appear clearly linear, or clearly independent. However, in real-life situations, you have to base your model on what the data looks like TO YOU. Does the data appear to generally tend towards a parabola, or maybe a third or fourth degree curve? For higher degree estimates, you are essentially asking how many times the data trend seems to change directions. It's an "eyeball estimate", really. A polynomial model can describe any particular relationship, if the data curve tends closely enough to that pattern.



Even if data appears to follow a polynomial curve, it may more closely tend towards, say, an exponential or logarithmic curve. There is no instant, easy answer. It's something you do by touch; eventually you get to the point where it's a matter of instinct. However, when you are first learning these methods, it's worth trying several different options and finding which model produces the lowest error.