From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Finding inflection point Date: 26 Apr 1999 20:11:23 -0400 Newsgroups: sci.math.num-analysis Keywords: Finding inflection points from numerical data In article <7g28tt$6b3$1@jetsam.uits.indiana.edu>, Raghu R. Gompa wrote: > >Are there any methods to find inflection >points from data, say (x_1,y_1), .. (x_n,y_n)? > >Thanks. .. Raghu This depends on the way your data are or aren't "scattered", and also what they measure. Pure data (no noise) and presumably just one inflection: Assume the x-coordinates are increasing, then either calculate the second divided differences on triples of subsequent points (finding half of the second derivative of the quadratic interpolating polynomial), and find where it changes sign. For a finer location, take the four subsequent points near the sign change, interpolate by a cubic and find the root of its second derivative (which is a linear function, of course). With noise: you might create experimentally a "sliding mask" of k subsequent data points at a time, k>3, and approximate the data using least squares (if the noise can be assumed close to normal) or by other means; again, finding the inflection point of the approximating cubic is routine. And you can estimate the uncertainty of the position by comparing inflection points over several overlapping "masks". If more is known about the meaning of the data: instead of polynomials, match the parameters of the assumed formula. Example: suppose the data are suspected to come from a logistic curve (where the saturation level is twice the inflection level), the form is P(x) = S / (1 + exp(k*(m-x)) with parameters S, k, m to match. Here actually m is the position of the inflection point. If the data can be created "on the run", one point at a time (such as measuring the fuel consumption, given that we can set the velocity as we need), one can work out a scheme similar to Fibonacci search for maxima/minima. Good luck. ZVK(Slavek). ============================================================================== From: "Hamer, Peter (EXCHANGE:HAL02:HK00)" Subject: Re: Finding inflection point Date: Tue, 27 Apr 1999 15:38:42 +0100 Newsgroups: sci.math.num-analysis Zdislav V. Kovarik wrote: a nice posting Can I add: fit a cubic smoothing spline, then look at the derivatives of its piece-wise cubic segments. The spline will help smooth the data. Peter