# Automatically find the best growth (linear, quadratic, exponential, etc.) to fit the points [duplicate]

Suppose there is a set of points $[(x_1, y_1),(x_2, y_2), ..., (x_n, y_n)]$.

I want to figure out whether $y$ will grow with $x$. If yes, I want to automatically find the best growth (linear growth, quadratic growth, exponential growth, or others) to fit it.

I know that pearson correlation can be used to judge whether $y$ and $x$ have a linear correlation. However, is there a threshold $t$ ? For example, if $PearsonCorr(x,y) > t$, we can conclude that $y$ and $x$ have a linear correlation.

Linear/polynomial regression can also be used to fit the points, and there are measurements such as $R^2$ and $F$ test. However, I do not know how to use them to determine the best fit, especially when $n$ is small (say 10).

Any suggestions will be appreciated!

• It is usually the case that questions of growth concern positive variables $y_i$. Is that true here? It is also usually true that in such circumstances, the spreads of the errors are not fixed: they tend to increase with the value of $y_i$. Is that true in your application? Finally, "..., quadratic growth, exponential growth, or others" is both vague and suggestive: it includes more than polynomials and more than exponentials. Could you be more specific about what family of growth models you do wish to use? – whuber Sep 14 '15 at 14:02
• Thanks for the reply and comments. I find this post has covered my question. – Lijie Xu Sep 14 '15 at 14:19
• Good point whuber. Be careful with the spread of the errors, that is a common pitfall. – Gumeo Sep 14 '15 at 14:26