I have some pairs of data ${(x_1,y_1),..., (x_n,y_n)}$ genereated by some process and would like to fit it with a function so that $y_i \approx \hat{f}(x_i)$.

By plotting the $(X,Y)$ on a 2D plot, and eyeballing, we find the relationship of data is monotonically decreasing, and the shape is similar to $y=x^{-\alpha}$ where $0<\alpha<1$.

The idea then is to use the sum of a series of basis functions to fit the data. In other words, let $\hat{f}(x)=\sum_{i=1}^m \beta_i h_i(x)$ where $h_i(x)= x^{-\alpha_i}$ for a set of predefined $\alpha$'s - ${\alpha_1, \alpha_2..., \alpha_m}$, where $0<\alpha_i<1$. We can then fit the data and find the $\beta_1,.., \beta_m$ with least squares.

My question is then is if the basis functions $h(x)=x^{-\alpha_i}$ are okay to use? Do I need to somehow make the basis function better? Is there any better ideas?

  • $\begingroup$ The brevity of your post is laudable, but it raises many questions about what it means. How is the function represented? What is the purpose of approximation? How should the accuracy of the approximation be measured? How is this seemingly mathematical question related to statistics? $\endgroup$ – whuber Feb 7 '19 at 21:02
  • $\begingroup$ Since you think that the shape is similar to $x^{-\alpha},$ have you tried fitting the log-log data to a linear model and observing the goodness of fit and slope? $\endgroup$ – Bridgeburners Feb 7 '19 at 21:23
  • $\begingroup$ Would you please post a link to the data? $\endgroup$ – James Phillips Feb 7 '19 at 21:30
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    $\begingroup$ Such models are notoriously difficult to fit unless you have collected extremely accurate data at just the right values. Thus, the details of your data might matter. In particular, how do you determine $m$ and the $\alpha_i$? $\endgroup$ – whuber Feb 7 '19 at 22:01
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    $\begingroup$ Then you could also try a monotone spline. There are examples on this site. $\endgroup$ – kjetil b halvorsen Feb 8 '19 at 10:40

I am not sure to well understand your question. The given data is $$(x_1,y_1),...,(x_k,y_k),...,(x_n,y_n)$$ You want to approximately fit the function : $$y(x)=\sum_{i=1}^m\beta_i x^{\alpha_i}$$ that is : $$y_k\simeq\sum_{i=1}^m\beta_i x_k^{\alpha_i}+\epsilon_k$$

and you wrote " for a set of predefined $\alpha_1,\alpha_2,..,\alpha_m\:$".

This makes think that $\alpha_1,\alpha_2,..,\alpha_m$ are known (given or a-priori chosen). Is it true ?

If NOT TRUE, the $\alpha_i$ have to be optimized. The problem is not easy. They are some methods that we could eventually discuss later.

If TRUE, the solution is very easy because the regression is linear (with respect to the unknowns $\beta_i$ ) :

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