![A symmetric bell curve from f (x) = 1 (1+x 2 ) , x ∈ [−2, 2]](https://www.researchgate.net/profile/Demetris_Christopoulos/publication/334671956/figure/fig1/AS:784490884526082@1564048610111/A-symmetric-bell-curve-from-f-x-1-1-x-2-x-2-2_Q320.jpg)
![A noisy tulip curve from f (x) = 100 − (x − 5) 2 , x ∈ [0, 12]](https://www.researchgate.net/profile/Demetris_Christopoulos/publication/334671956/figure/fig3/AS:784490884509696@1564048610162/A-noisy-tulip-curve-from-f-x-100-x-5-2-x-0-12_Q320.jpg)
![Local maximum of f (x) = 63 8 x 5 − 35 4 x 3 + 15 8 x, x ∈ [0, 0.7]](https://www.researchgate.net/profile/Demetris_Christopoulos/publication/334671956/figure/fig5/AS:784490884497410@1564048610216/Local-maximum-of-f-x-63-8-x-5-35-4-x-3-15-8-x-x-0-07_Q320.jpg)
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New methods for computing extremes and roots of a planar curve: introducing Noisy Numerical Analysis
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Abstract and Figures
By classifying planar curves under their symmetry around extreme points we present the 'tulip' and 'bell' methods for extreme computation. If curve is not symmetric, then we develop the 'integration' method for estimating extremes, which uses BESE or BEDE methods for inflection finding. After observing that absolute value of a function creates a set of local edge minima in the vicinity of roots, we use 'integration' method for estimating roots. All methods are based on the trapezoidal rule for numerical integration which gives consistent estimations for surfaces and consequently makes introduced estimators statistically consistent. New methods can constitute the core of a new discipline called 'Noisy Numerical Analysis'.
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ResearchGate has not been able to resolve any citations for this publication.
- J Howard
J. Howard, II, Computational Methods for Numerical Analysis with R, New York: Chapman and Hall/CRC, 2017.
doi:10.1201/9781315120195.
RootsExtremaInflections: Finds Roots, Extrema and Inflection Points of a Curve, r package version
- D T Christopoulos
D. T. Christopoulos, RootsExtremaInflections: Finds Roots, Extrema and Inflection Points of a Curve, r package
version 1.1 (2017).
Implementation of methods Extremum Surface Estimator (ESE), Extremum Distance Estimator
(EDE) and their iterative versions BESE and BEDE in order to identify the inflection point of a
curve.
In the simplest case we have planar data without noise and what to find the relevant inflection point. This animation shows the two chords, left & right, and the exact procedure that is used for estimating the x-l, x-r points. Then ESE=(x-l+x-r)/2.
Our task is to find time efficient and statistically consistent estimators for revealing the true inflection point of a planar curve when we have only a probably noisy set of points for it, thus we are introducing extremum surface (ESE) and distance estimator (EDE) methods. The analysis is based on the geometric properties of the inflection point for a smooth function. Iterative versions of the methods are also given and tested. Numerical experiments are performed for the class of sigmoid curves and comparison with other available procedures is carried out. It is proven that both methods are quite fast in computational execution. Under a rather common noise type EDE can give a 96% confidence interval, while it always provides estimations for data with more than a million cases at a negligible execution time. An alternative way of mode computation for a distribution by using its CDF is given as a real massive data example.
We are introducing two methods for revealing the true inflection point of
data that contains or not error. The starting point is a set of geometrical
properties that follow the existence of an inflection point p for a smooth
function. These properties connect the concept of convexity/concavity before
and after p respectively with three chords defined properly. Finally a set of
experiments is presented for the class of sigmoid curves and for the third
order polynomials.
We are introducing a new class of regressions called Taylor Regression and we are using it to create a unified method that can estimate a root, an extreme and an inflection point when we have discrete functional data. The method works with good accuracy for both simple and noisy data, while it is not necessary to use equidistant abscissae.