Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:04:28Z http://mathoverflow.net/feeds/question/88269 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88269/can-we-approximate-an-arbitrary-function-as-a-probably-infinite-sum-of-bell-sha Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes? ivmarsa 2012-02-12T11:42:07Z 2012-02-13T11:31:46Z <p>The problem is that we want to approximate domain-bounded functions (that is, functions restricted to a domain such as [0,10]) as (probably infinite) series of other functions. We know that, for integrable functions in the interval, we can definitely do that with sin waves (that's the fourier transform), but we wondered if we could use other kind of functions as well. In particular, we are wondering if we could use for the series a bell-shaped function like the following:</p> <p>$fbell(s,c,h,r)=\begin{cases} h-2h\frac{\parallel s-c\parallel^{2}}{r^{2}} &amp; if \parallel s-c\parallel\lt r/2 \end{cases}$</p> <p>$fbell(s,c,h,r)=\begin{cases} \frac{2h}{r^{2}}(\parallel s-c\parallel-r)^{2} &amp; ifr\gt\parallel s-c\parallel\geq r/2 \end{cases}$</p> <p>$fbell(s,c,h,r)= \begin{cases}0 &amp; \parallel s-c\parallel\geq r\end{cases}$</p> <p>where c is the center of the bell, h is the weight and r is the radius. If I'm not mistaken, t is continuous and differentiable in $\parallel s-c\parallel\lt r$.</p> <p>It is useful for us because we work in utility theory and this function makes sense where there is a set of utility peaks and utility fades as we go far from them.</p> <p>I understand that this is related to the Wavelet transform and the orthonormal wavelets, but I'm not sure this is fully applicable to our case (where the domain of the function is bounded). </p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/88269/can-we-approximate-an-arbitrary-function-as-a-probably-infinite-sum-of-bell-sha/88305#88305 Answer by psd for Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes? psd 2012-02-12T22:51:39Z 2012-02-12T22:51:39Z <p>It could be cleaner to use a <a href="http://en.wikipedia.org/wiki/Mixture_model" rel="nofollow">mixture</a> of <a href="http://en.wikipedia.org/wiki/Beta_distribution" rel="nofollow">beta distributions</a> rather than of <a href="http://en.wikipedia.org/wiki/Truncated_normal_distribution" rel="nofollow">truncated normal distributions</a> if you want to approximate an arbitrary distribution on an interval.</p> <p>From <a href="http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5740920" rel="nofollow">this paper</a>:</p> <blockquote> <p>Although the GMM (Gaussian mixture model) can model arbitrary distributions with a proper number of mixture components, a large number of these components are spent on describing the edge when modeling semibounded or bounded support data. Compared to the Gaussian distribution, the beta distribution has a more flexible shape. It has a support range of $[0,1]$, and can be easily generalized to any compact range $[a,b], a, b \in \mathbb{R}$.</p> </blockquote> http://mathoverflow.net/questions/88269/can-we-approximate-an-arbitrary-function-as-a-probably-infinite-sum-of-bell-sha/88319#88319 Answer by Zander for Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes? Zander 2012-02-13T01:06:33Z 2012-02-13T01:06:33Z <p>You can't approximate arbitrary square-integrable functions in the same way as a Fourier series or wavelet series, since your functions cannot make a basis for the function space.</p> <p>But what you described is a quadratic spline (almost but not quite a usual <a href="http://en.wikipedia.org/wiki/B-spline" rel="nofollow">basis B-spline</a>). So you can use <a href="http://en.wikipedia.org/wiki/Spline_interpolation" rel="nofollow">spline interpolation</a> to approximate an arbitrary function $f$ in the sense that you can pick a grid of "knots" and find a set of $h$ weights so that the sum of your $fbell$ splines is a function that equals $f$ at each of the knot points and has a continuous first derivative. If you have almost any kind of smoothness constraint on $f$ (e.g. a bound on some derivative), that will allow you to approximate $f$ to within any specification by choosing a sufficiently fine grid of knots.</p> http://mathoverflow.net/questions/88269/can-we-approximate-an-arbitrary-function-as-a-probably-infinite-sum-of-bell-sha/88340#88340 Answer by Dirk for Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes? Dirk 2012-02-13T11:31:46Z 2012-02-13T11:31:46Z <p>While @Zander is right about the fact that "these functions cannot make a basis for the functions space" there are still a lot of families of translates of "bell shaped functions" which for a dense set in $L^2$. You may consider the paper <a href="http://homepage.univie.ac.at/gerard.ascensi/gaussian.pdf" rel="nofollow">"On aproximations by shifts of the Gaussian function"</a> which treats exactly this problem.</p> <p>Indeed, one knows that for the Gaussians $\phi(x) = \exp(-\pi x^2)$ it holds that the span of the functions $\phi(x-\lambda_k)$ is dense in $L^2(\mathbb{R})$ if the shifts $\lambda_k$ fulfill that the sum $\sum_k \lambda_k^{-2}$ diverges - imho a quite surprising result...</p>