Description
Finds Roots, Extrema and Inflection Points of a Curve.
Description
Implementation of Taylor Regression Estimator (TRE), Tulip Extreme Finding Estimator (TEFE), Bell Extreme Finding Estimator (BEFE), Integration Extreme Finding Estimator (IEFE) and Integration Root Finding Estimator (IRFE) for roots, extrema and inflections of a curve . Christopoulos, DT (2019) <doi:10.13140/RG.2.2.17158.32324> . Christopoulos, DT (2016) <doi:10.2139/ssrn.3043076> . Christopoulos, DT (2016) <https://veltech.edu.in/wp-content/uploads/2016/04/Paper-04-2016.pdf> . Christopoulos, DT (2014) <arXiv:1206.5478v2 [math.NA]> .
README.md
Overview
RootsExtremaInflection is a package that finds roots, extrema and inflection points of a planar curve which is given as a data frame of discrete (xi,yi) points.
Basic functions are:
rootxi()Taylor Regression Estimator for rootsextremexi()Taylor Regression Estimator for extremainflexi()Taylor Regression Estimator for inflectionsclassify_curve()curve classification by convexity and shape typefindmaxbell()Bell Extreme Finding Estimator for symmetric extremafindmaxtulip()Tulip Extreme Finding Estimator for symmetric extremafindextreme()Integration Extreme Finding Estimator for all types of extremafindroot()Integration Root Finding Estimator for rootsscan_curve()scans a not noisy curve and finds all roots and extrema, inflections between themscan_noisy_curve()scans a noisy curve and finds all roots and extrema, inflections between them
Installation
# Install with dependencies:
install.packages('RootsExtremaInflection',dependencies=TRUE)
Usage
library(RootsExtremaInflection)
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
#
## Next examples are for the
## Legendre polynomial of 5th order:
#
f=function(x){(63/8)*x^5-(35/4)*x^3+(15/8)*x}
#
### findextreme()
#
## True extreme point p=0.2852315165, y=0.3466277
x=seq(0,0.7,0.001);y=f(x)
plot(x,y,pch=19,cex=0.5)
a=findextreme(x,y)
a
## x1 x2 chi yvalue
## 0.2840000 0.2860000 0.2850000 0.3466274
sol=a['chi']
abline(h=0)
abline(v=sol)
abline(v=a[1:2],lty=2)
abline(h=f(sol),lty=2)
points(sol,f(sol),pch=17,cex=2)
#
## The same function with noise from U(-0.05,0.05)
set.seed(2019-07-26);r=0.05;y=f(x)+runif(length(x),-r,r)
plot(x,y,pch=19,cex=0.5)
a=findextreme(x,y)
a
## x1 x2 chi yvalue
## 0.2890000 0.2910000 0.2900000 0.3895484
sol=a['chi']
abline(h=0)
abline(v=sol)
abline(v=a[1:2],lty=2)
abline(h=f(sol),lty=2)
points(sol,f(sol),pch=17,cex=2)
#
### findroot()
#
x=seq(0.2,0.8,0.001);y=f(x);ya=abs(y)
plot(x,y,pch=19,cex=0.5,ylim=c(min(y),max(ya)))
abline(h=0);
lines(x,ya,lwd=4,col='blue')
rt=findroot(x,y)
rt
## x1 x2 chi yvalue
## 5.370000e-01 5.400000e-01 5.385000e-01 -7.442574e-05
abline(v=rt['chi'])
abline(v=rt[1:2],lty=2);abline(h=rt['yvalue'],lty=2)
points(rt[3],rt[4],pch=17,col='blue',cex=2)
#
## Same curve but with noise from U(-0.5,0.5)
#
set.seed(2019-07-24);r=0.05;y=f(x)+runif(length(x),-r,r)
ya=abs(y)
plot(x,y,pch=19,cex=0.5,ylim=c(min(y),max(ya)))
abline(h=0)
points(x,ya,pch=19,cex=0.5,col='blue')
rt=findroot(x,y)
rt
## x1 x2 chi yvalue
## 0.53400000 0.53700000 0.53550000 -0.01762159
abline(v=rt['chi'])
abline(v=rt[1:2],lty=2);abline(h=rt['yvalue'],lty=2)
points(rt[3],rt[4],pch=17,col='blue',cex=2)
#
### scan_curve()
#
x=seq(-1,1,0.001);y=f(x)
plot(x,y,pch=19,cex=0.5)
abline(h=0)
rall=scan_curve(x,y)
rall$study
rall$roots
## x1 x2 chi yvalue
## [1,] -0.907 -0.905 -9.060000e-01 1.234476e-03
## [2,] -0.540 -0.537 -5.385000e-01 7.447856e-05
## [3,] -0.001 0.001 5.551115e-17 1.040844e-16
## [4,] 0.537 0.540 5.385000e-01 -7.444324e-05
## [5,] 0.905 0.907 9.060000e-01 -1.234476e-03
rall$extremes
## x1 x2 chi yvalue
## [1,] -0.766 -0.764 -0.765 0.4196969
## [2,] -0.286 -0.284 -0.285 -0.3466274
## [3,] 0.284 0.286 0.285 0.3466274
## [4,] 0.764 0.766 0.765 -0.4196969
rall$inflections
## x1 x2 chi yvalue
## [1,] -0.579 -0.576 -5.775000e-01 9.659939e-02
## [2,] -0.001 0.001 5.551115e-17 1.040829e-16
## [3,] 0.576 0.579 5.775000e-01 -9.659935e-02
#
### scan_noisy_curve()
#
x=seq(-1,1,0.001)
set.seed(2019-07-26);r=0.05;y=f(x)+runif(length(x),-r,r)
plot(x,y,pch=19,cex=0.5)
rn=scan_noisy_curve(x,y)
rn
## $study
## j dj interval i1 i2 root
## 3 97 351 TRUE 97 448 FALSE
## 18 477 502 TRUE 477 979 FALSE
## 39 1021 505 TRUE 1021 1526 FALSE
## 54 1558 343 TRUE 1558 1901 FALSE
##
## $roots_average
## x1 x2 chi yvalue
## 1 -0.906 -0.904 -0.9050 -0.002342389
## 2 -0.553 -0.524 -0.5385 0.005003069
## 3 -0.022 0.020 -0.0010 0.003260937
## 4 0.525 0.557 0.5410 -0.007956680
## 5 0.900 0.911 0.9055 -0.008015683
##
## $roots_optim
## x1 x2 chi yvalue
## 1 -0.909 -0.901 -0.9050 -0.023334404
## 2 -0.531 -0.527 -0.5290 0.029256059
## 3 0.001 0.003 0.0020 0.001990572
## 4 0.530 0.565 0.5475 0.019616283
## 5 0.909 0.912 0.9105 0.009288338
##
## $extremes
## x1 x2 chi yvalue
## [1,] -0.773 -0.766 -0.7695 0.4102010
## [2,] -0.280 -0.274 -0.2770 -0.3804006
## [3,] 0.308 0.316 0.3120 0.3372764
## [4,] 0.741 0.744 0.7425 -0.4414494
##
## $inflections
## x1 x2 chi yvalue
## [1,] -0.772 -0.275 -0.5235 -0.076483193
## [2,] -0.275 0.281 0.0030 -0.007558037
## [3,] 0.301 0.776 0.5385 0.018958334
#
Why should I use RootsExtremaInflection package in R?
- Because it can give you a reliable estimation by using Taylor Regression, if you prefer non parametric statistical methods
- Because it can find the desired point(s) by using geometric and analytic methods, without any kind of model adoption
- Because it is the core of a new field called ‘Noisy Numerical Analysis’ which combines Geometry, Calculus, Numerical Analysis and Statistics in order to solve known problems of Numerical Analysis
Contact
Please send comments, suggestions or bug breports to dchristop$econ.uoa.gr.