RootsExtremaInflections (version 1.2.1)

RootsExtremaInflections-package: Finds Roots, Extrema and Inflection Points of a Planar Curve

Description

This package contains functions for computing roots, extrema and inflection points of a curve that is the graph of a smooth function when we have only a data set \(\left\{(x_i,y_i),i=1,2,\ldots m\right\}\), generated from it by the procedure \(y_i=f(x_i)\) or for the noisy case by \(y_i=f(x_i)+\epsilon_i\) with a zero mean error, \(\epsilon_i\sim iid(0,\sigma^2)\), by using the Taylor Regression Estimator (TRE) method, which is described briefly here. When we want to find a root for a function \(f\) by using the traditional Numerical Analysis methods (bisection, secant, Newton-Raphson etc), it is necessary to know the exact formula of \(f\). Unfortunately in research problems we do not know that formula and our data are also of a noisy type. In this package we use the Taylor Regression Estimator (TRE) method, which can work when we know the discrete values \(\left\{(x_i,y_i),i=1,2,\ldots m\right\},y_i=f(x_i)\) of our known or unknown smooth function \(f\). Additionally the method works with satisfactory accuracy also for the corresponding noisy values \(\left\{\left(x_i, y_i\right),i=1,2,\ldots m \right\},\,y_i=f(x_i)+\epsilon_i,\,\epsilon_i\sim iid(0,\sigma^2)\). The computation of extrema and inflection points for a smooth \(f\) is merely a problem of root finding for first and second derivative respectively, thus TRE method can also find an extreme or an inflection point. In a few words, the method is referencing to the well known Taylor polynomial of a smooth function \(f\) around a point \(\rho\), $$f \left( x \right)=a_0+a_1\,\left( x-\rho \right)+a_2\,\left( x-\rho \right)^2+a_3\,\left( x-\rho \right)^3+\ldots+a_n\,\left( x-\rho \right)^n$$ When the coefficients \(a_{0},a_{1},a_{2}\), as computed using a polynomial regression, have minimum absolute value, then the corresponding points \(\rho\) are the estimations of the root, extreme or inflection point, respectively. Essentially Taylor Regression (TR) is polynomial regression for Taylor polynomial. For a more rigorous definition of the terms TR, TRE method, further discussion and numerical examples, see Christopoulos (2014). From version 1.2.1 a set of new methods has been added: Tulip Extreme Finding Estimator (TEFE) Bell Extreme Finding Estimator (BEFE) Integration Extreme Finding Estimator (IEFE) Integration Root Finding Estimator (IRFE) All these methods are defined and explained at Christopoulos (2019).

Details

After adding functions for finding roots and extrema without regression, current package constitutes the core of new filed , called 'Noisy Numerical Analysis', which combines Geometry, Calculus, Numerical Analysis and Statistics and treat noisy curves for solving known problems of Numerical Analysis.

References

Demetris T. Christopoulos (2014). Roots, extrema and inflection points by using a proper Taylor regression procedure. SSRN. https://dx.doi.org/10.2139/ssrn.2521403

Demetris T. Christopoulos (2019). New methods for computing extremes and roots of a planar curve: introducing Noisy Numerical Analysis (2019). ResearchGate. http://dx.doi.org/10.13140/RG.2.2.17158.32324

Examples

# NOT RUN {
#Load data:
data(xydat)
#
#Extract x and y variables:
x=xydat$x;y=xydat$y
#
#Find root, plot results, print Taylor coefficients and rho estimation:
b<-rootxi(x,y,1,length(x),5,5,plots=TRUE);b$an;b$froot;
#
#Find extreme, plot results, print Taylor coefficients and rho estimation:
c<-extremexi(x,y,1,length(x),5,5,plots=TRUE);c$an;c$fextr;
#
#Find inflection point, plot results, print Taylor coefficients and rho estimation:
d<-inflexi(x,y,1,length(x),5,5,plots=TRUE);d$an;d$finfl;
# Create a relative big data set...
f=function(x){3*cos(x-5)};xa=0.;xb=9;
set.seed(12345);x=sort(runif(5001,xa,xb));r=0.1;y=f(x)+2*r*(runif(length(x))-0.5);
#
#Find root, plot results, print Taylor coefficients and rho estimation in parallel:
#b1<-rootxi(x,y,1,round(length(x)/2),5,5,plots=TRUE,doparallel = TRUE);b1$an;b1$froot;
# Available workers are 12 
# Time difference of 5.838743 secs
#           2.5 %       97.5 %           an
# a0 -0.006960052  0.004414505 -0.001272774
# a1 -2.982715739 -2.933308292 -2.958012016
# a2 -0.308844145 -0.213011162 -0.260927654
# a3  0.806555336  0.874000586  0.840277961
# a4 -0.180720951 -0.161344935 -0.171032943
# a5  0.007140500  0.009083859  0.008112180
# [1] 177.0000000   0.2924279
# Compare with exact root = 0.2876110196
#Find extreme, plot results, print Taylor coefficients and rho estimation in parallel:
#c1<-extremexi(x,y,1,round(length(x)/2),5,5,plots=TRUE,doparallel = TRUE);c1$an;c1$fextr;
# Available workers are 12 
# Time difference of 5.822514 secs
#            2.5 %       97.5 %           an
# a0 -3.0032740050 -2.994123850 -2.998698927
# a1 -0.0006883998  0.012218393  0.005764997
# a2  1.4745326519  1.489836668  1.482184660
# a3 -0.0340626683 -0.025094859 -0.029578763
# a4 -0.1100798736 -0.105430525 -0.107755199
# a5  0.0071405003  0.009083859  0.008112180
# [1] 1022.000000    1.852496
# Compare with exact extreme = 1.858407346
#Find inflection point, plot results, print Taylor coefficients and rho estimation in parallel:
#d1<-inflexi(x,y,1090,2785,5,5,plots=TRUE,doparallel = TRUE);d1$an;d1$finfl;
# Available workers are 12 
# Time difference of 4.343851 secs
#           2.5 %       97.5 %            an
# a0 -0.008238016  0.002091071 -0.0030734725
# a1  2.995813560  3.023198534  3.0095060468
# a2 -0.014591465  0.015326175  0.0003673549
# a3 -0.531029710 -0.484131902 -0.5075808056
# a4 -0.008253975  0.007556465 -0.0003487551
# a5  0.016126428  0.034688019  0.0254072236
# [1] 800.000000   3.427705
# Compare with exact inflection = 3.429203673
# Or execute rootexinf() and find a set of them at once and in same time:
#a<-rootexinf(x,y,100,round(length(x)/2),5,plots = TRUE,doparallel = TRUE);
#a$an0;a$an1;a$an2;a$frexinf;
# Available workers are 12 
# Time difference of 5.565372 secs
#           2.5 %      97.5 %           an0
# a0 -0.008244278  0.00836885  6.228596e-05
# a1 -2.927764078 -2.84035634 -2.884060e+00
# a2 -0.447136449 -0.30473094 -3.759337e-01
# a3  0.857290490  0.94794071  9.026156e-01
# a4 -0.198104383 -0.17360676 -1.858556e-01
# a5  0.008239609  0.01059792  9.418764e-03
#           2.5 %      97.5 %          an1
# a0 -3.005668018 -2.99623116 -3.000949590
# a1 -0.003173501  0.00991921  0.003372854
# a2  1.482600580  1.50077450  1.491687542
# a3 -0.034503271 -0.02551597 -0.030009618
# a4 -0.115396537 -0.10894117 -0.112168855
# a5  0.008239609  0.01059792  0.009418764
#           2.5 %       97.5 %          an2
# a0  0.083429390  0.092578772  0.088004081
# a1  3.007115452  3.027343849  3.017229650
# a2 -0.009867779  0.006590042 -0.001638868
# a3 -0.517993955 -0.497886933 -0.507940444
# a4 -0.043096158 -0.029788902 -0.036442530
# a5  0.008239609  0.010597918  0.009418764
#            index     value
# root          74 0.2878164
# extreme      923 1.8524956
# inflection  1803 3.4604842
#Here a first plot always is helpful.
# }