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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!

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  • $\begingroup$ 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? $\endgroup$ – whuber Sep 14 '15 at 14:02
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    $\begingroup$ Thanks for the reply and comments. I find this post has covered my question. $\endgroup$ – Lijie Xu Sep 14 '15 at 14:19
  • $\begingroup$ Good point whuber. Be careful with the spread of the errors, that is a common pitfall. $\endgroup$ – Gumeo Sep 14 '15 at 14:26
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This is only trial and error. Plot your data and look if it has a shape that you recognize. Maybe there is some theory on this kind of growth, maybe there are already functions in the literature that you can start with to model this.

After plotting it, you can try different functions and look at the residuals. If there is some obvious pattern in them, then your model is missing something.

Also, it would be a nice addition if you could add a plot to your post to help people see what you are dealing with :)

Edit: One way to do this automatically is to fit a bunch of functions you select and find which one is best at minimizing your RSS.

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