Package RootsExtremaInflections

Transcription

1 Type Package Package RootsExtremaInflections May 10, 2017 Title Finds Roots, Extrema and Inflection Points of a Curve Version 1.1 Date Author Demetris T. Christopoulos Maintainer Demetris T. Christopoulos <dchristop@econ.uoa.gr> Description Implementation of the Taylor Regression Estimator method which is described in Christopoulos (2014,< for finding the root, extreme or inflection point of a curve, when we only have a set of probably noisy xy points for it. The method uses a suitable polynomial regression in order to find the coefficients of the relevant Taylor polynomial for the function that has generated our data. Optional use of parallel computing under request. License GPL-2 Depends iterators, foreach, parallel, doparallel Suggests stats, graphics, grdevices NeedsCompilation no Repository CRAN Date/Publication :36:46 UTC R topics documented: RootsExtremaInflections-package extremexi inflexi rootexinf rootxi xydat Index 15 1

2 2 RootsExtremaInflections-package RootsExtremaInflections-package Finds Roots, Extrema and Inflection Points of a Planar Curve Description Details 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 {(x i, y i ), i = 1, 2,... m}, generated from it by the procedure y i = f(x i ) or for the noisy case by y i = f(x i )+ɛ i with a zero mean error, ɛ i iid(0, σ 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 {(x i, y i ), i = 1, 2,... m}, 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 {(x i, y i ), i = 1, 2,... m}, y i = f(x i ) + ɛ i, ɛ i iid(0, σ 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 ρ, f (x) = a 0 + a 1 (x ρ) + a 2 (x ρ) 2 + a 3 (x ρ) a n (x ρ) n When the coefficients a 0, a 1, a 2, as computed using a polynomial regression, have minimum absolute value, then the corresponding points ρ 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). Package: RootsExtremaInflections Type: Package Version: 1.1 Date: License: GPL 2 Author(s) Demetris T. Christopoulos Maintainer: Demetris T. Christopoulos <dchristop@econ.uoa.gr>

3 RootsExtremaInflections-package 3 References Demetris T. Christopoulos (2014), Roots, extrema and inflection points by using a proper Taylor regression procedure, ResearchGate publications, Examples 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 secs 2.5 % 97.5 % an a a a a a a [1] Compare with exact root = 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 secs 2.5 % 97.5 % an a a a a a a [1] Compare with exact extreme = Find inflection point, plot results, print Taylor coefficients and rho estimation in parallel:

4 4 extremexi d1<-inflexi(x,y,1090,2785,5,5,plots=true,doparallel = TRUE);d1$an;d1$finfl; Available workers are 12 Time difference of secs 2.5 % 97.5 % an a a a a a a [1] Compare with exact inflection = 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 secs 2.5 % 97.5 % an0 a e-05 a e+00 a e-01 a e-01 a e-01 a e % 97.5 % an1 a a a a a a % 97.5 % an2 a a a a a a index value root extreme inflection Here a first plot always is helpful. extremexi Function to Find the Extreme Point of a Planar Curve Description It takes as input the x, y numeric vectors, the indices for the range to be searched plus some other options and finds the extreme point for that interval, while it plots data, Taylor polynomial and the

5 extremexi 5 computed a 1 coefficients. Usage extremexi(x, y, i1, i2, nt, alpha = 5, xlb = "x", ylb = "y", xnd = 3, ynd = 3, plots = TRUE, plotpdf = FALSE, doparallel=false) Arguments x y A numeric vector for the independent variable A numeric vector for the dependent variable i1 The first index for choosing a specific interval [a, b] = [x i1, x i2 ] i2 The second index for choosing a specific interval [a, b] = [x i1, x i2 ] nt alpha xlb ylb Details The degree of the Taylor polynomial that will be fitted to the data The level of statistical significance for the confidence intervals of coefficients a 0, a 1,..., a nt 1 (default value = 5) A label for the x-variable (default value = "x") A label for the y-variable (default value = "y") xnd The number of digits for plotting the x-axis (default value = 3) ynd The number of digits for plotting the y-axis (default value = 3) plots plotpdf doparallel If plots=true then a plot is created on default monitor (default value = TRUE) If plotpdf=true then a pdf plot is created and stored on working directory (default value = FALSE) If doparallel=true then parallel computing is applied, based on the available workers of current machine (default value = FALSE) The point x i which makes the relevant a 1 minimum is the estimation for the function s extreme point at the interval [x i1, x i2 ]. Value It returns an environment with two components: an fextr a matrix with 3 columns: lower, upper bound of confidence interval and middle value for each coefficient an a list with 2 members: the position i and the value of the estimated extreme point ρ = x i Warnings When you are using RStudio it is necessary to leave enough space for the plot window in order for the plots to appear normally. The data should come from a function at least C (1) in order to be able to find an extreme point, if exists.

6 6 extremexi Author(s) Demetris T. Christopoulos References Demetris T. Christopoulos (2014), Roots, extrema and inflection points by using a proper Taylor regression procedure, ResearchGate publications, Examples Load data: data(xydat) Extract x and y variables: x=xydat$x;y=xydat$y Find extreme point, plot results, print Taylor coefficients and rho estimation: c<-extremexi(x,y,1,length(x),5,5,plots=true);c$an;c$fextr; Find multiple extrema. Let's create some data: f=function(x){3*cos(x-5)};xa=0.;xb=9; set.seed(12345);x=sort(runif(101,xa,xb));r=0.1;y=f(x)+2*r*(runif(length(x))-0.5);plot(x,y) The first extreme point is c1<-extremexi(x,y,1,40,5,5,plots=true);c1$an;c1$fextr; 2.5 % 97.5 % an a a a a a a [1] Compare it with the actual rho_1= The second extreme point is c2<-extremexi(x,y,50,80,5,5,plots=true);c2$an;c2$fextr; 2.5 % 97.5 % an a a a a a a [1] You have to compare it with the actual value of rho_2=5.0

7 inflexi 7 Finally the third extreme point is c3<-extremexi(x,y,80,length(x),5,5,plots=true);c3$an;c3$fextr; 2.5 % 97.5 % an a a a a a a [1] You have to compare it with the actual value of rho_3= inflexi Function to Find the Inflection Point of a Planar Curve Description Usage It takes as input the x, y numeric vectors, the indices for the range to be searched plus some other options and finds the inflection point for that interval, while it plots data, Taylor polynomial and and the computed a 2 coefficients. inflexi(x, y, i1, i2, nt, alpha = 5, xlb = "x", ylb = "y", xnd = 3, ynd = 3, plots = TRUE, plotpdf = FALSE, doparallel=false) Arguments x y A numeric vector for the independent variable A numeric vector for the dependent variable i1 The first index for choosing a specific interval [a, b] = [x i1, x i2 ] i2 The second index for choosing a specific interval [a, b] = [x i1, x i2 ] nt alpha xlb ylb The degree of the Taylor polynomial that will be fitted to the data The level of statistical significance for the confidence intervals of coefficients a 0, a 1,..., a nt 1 (default value = 5) A label for the x-variable (default value = "x") A label for the y-variable (default value = "y") xnd The number of digits for plotting the x-axis (default value = 3) ynd The number of digits for plotting the y-axis (default value = 3) plots plotpdf doparallel If plots=true then a plot is created on default monitor (default value = TRUE) If plotpdf=true then a pdf plot is created and stored on working directory (default value = FALSE) If doparallel=true then parallel computing is applied, based on the available workers of current machine (default value = FALSE)

8 8 inflexi Details Value The point x i which makes the relevant a 2 minimum is the estimation for the function s inflection point at the interval [x i1, x i2 ]. It returns an environment with two components: an fextr a matrix with 3 columns: lower, upper bound of confidence interval and middle value for each coefficient an a list with 2 members: the position i and the value of the estimated inflection point ρ = x i Warnings When you are using RStudio it is necessary to leave enough space for the plot window in order for the plots to appear normally. The data should come from a function at least C (2) in order to be able to find an inflection point, if exists. Author(s) Demetris T. Christopoulos References Demetris T. Christopoulos (2014), Roots, extrema and inflection points by using a proper Taylor regression procedure, ResearchGate publications, Examples Load data: data(xydat) Extract x and y variables: x=xydat$x;y=xydat$y 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; Find multiple inflection points. Let's create some data: f=function(x){3*cos(x-5)};xa=0.;xb=9; set.seed(12345);x=sort(runif(101,xa,xb));r=0.1;y=f(x)+2*r*(runif(length(x))-0.5);plot(x,y) The first inflection point is d1<-inflexi(x,y,20,50,5,5,plots=true);d1$an;d1$finfl;

9 rootexinf % 97.5 % an a a a a a a [1] Compare it with the actual rho_1= The second inflection point is d2<-inflexi(x,y,50,length(x),5,5,plots=true);d2$an;d2$finfl; 2.5 % 97.5 % an a a a a a a [1] You have to compare it with the actual value of rho_2= rootexinf Function to Find the Root, Extreme and Inflection of a Planar Curve Description Usage It takes as input the x, y numeric vectors, the indices for the range to be searched plus some other options and finds the root, extreme and inflection for that interval, while it plots data, Taylor polynomial and and the computed a 0, a 1, a 2 coefficients. rootexinf(x, y, i1, i2, nt, alpha = 5, xlb = "x", ylb = "y", xnd = 3, ynd = 3, plots = TRUE, plotpdf = FALSE, doparallel=false) Arguments x y A numeric vector for the independent variable A numeric vector for the dependent variable i1 The first index for choosing a specific interval [a, b] = [x i1, x i2 ] i2 The second index for choosing a specific interval [a, b] = [x i1, x i2 ] nt alpha xlb ylb The degree of the Taylor polynomial that will be fitted to the data The level of statistical significance for the confidence intervals of coefficients a 0, a 1,..., a nt 1 (default value = 5) A label for the x-variable (default value = "x") A label for the y-variable (default value = "y")

10 10 rootexinf Details Value xnd The number of digits for plotting the x-axis (default value = 3) ynd The number of digits for plotting the y-axis (default value = 3) plots plotpdf doparallel If plots=true then a plot is created on default monitor (default value = TRUE) If plotpdf=true then a pdf plot is created and stored on working directory (default value = FALSE) If doparallel=true then parallel computing is applied, based on the available workers of current machine (default value = FALSE) The points x i that make the relevant a 0, a 1, a 2 minimum are the estimations for the function s root, etreme and inflection point at the interval [x i1, x i2 ]. It returns an environment with four components: an0 an1 an2 frexinf a matrix with 3 columns: lower, upper bound of confidence interval and middle value for each coefficient a_n at the best choice in root searching a matrix with 3 columns: lower, upper bound of confidence interval and middle value for each coefficient a_n at the best choice in extreme searching a matrix with 3 columns: lower, upper bound of confidence interval and middle value for each coefficient a_n at the best choice in inflection searching a 3 x 3 matrix: for each row (root, extreme, inflection) the position i and the value of the estimated root, extreme and inflection ρ = x i Warnings When you are using RStudio it is necessary to leave enough space for the plot window in order for the plots to appear normally. The data should come from a function at least C (2) in order to find the root, extreme and inflection point, provided those points exist. Author(s) Demetris T. Christopoulos References Demetris T. Christopoulos (2014), Roots, extrema and inflection points by using a proper Taylor regression procedure, ResearchGate publicationss, Examples Load data: Let's create some data: f=function(x){3*cos(x-5)+1.5};xa=1.;xb=5; set.seed(12345);x=sort(runif(5001,xa,xb));

11 rootxi 11 r=0.1;y=f(x)+2*r*(runif(length(x))-0.5);plot(x,y);abline(h=0) a<-rootexinf(x,y,1,length(x),5,plotpdf = TRUE,doparallel = TRUE);a$an0;a$an1;a$an2;a$frexinf; Available workers are 12 Time difference of secs File 'root_extreme_inflection_plot.pdf' has been created 2.5 % 97.5 % an0 a a a a a a % 97.5 % an1 a a a a a a % 97.5 % an2 a a a a a a index value root extreme inflection You have to compare with the exact values root= extreme= inflection= rootxi Function to Find the Root of a Planar Curve Description It takes as input the x, y numeric vectors, the indices for the range to be searched plus some other options and finds the root for that interval, while it plots data, Taylor polynomial and and the computed a 0 coefficients. Usage rootxi(x, y, i1, i2, nt, alpha = 5, xlb = "x", ylb = "y", xnd = 3, ynd = 3, plots = TRUE, plotpdf = FALSE, doparallel=false)

12 12 rootxi Arguments x y A numeric vector for the independent variable A numeric vector for the dependent variable i1 The first index for choosing a specific interval [a, b] = [x i1, x i2 ] i2 The second index for choosing a specific interval [a, b] = [x i1, x i2 ] nt alpha xlb ylb Details Value The degree of the Taylor polynomial that will be fitted to the data The level of statistical significance for the confidence intervals of coefficients a 0, a 1,..., a nt 1 (default value = 5) A label for the x-variable (default value = "x") A label for the y-variable (default value = "y") xnd The number of digits for plotting the x-axis (default value = 3) ynd The number of digits for plotting the y-axis (default value = 3) plots plotpdf doparallel If plots=true then a plot is created on default monitor (default value = TRUE) If plotpdf=true then a pdf plot is created and stored on working directory (default value = FALSE) If doparallel=true then parallel computing is applied, based on the available workers of current machine (default value = FALSE) The point x i which makes the relevant a 0 minimum is the estimation for the function s root at the interval [x i1, x i2 ]. It returns an environment with two components: an froot a matrix with 3 columns: lower, upper bound of confidence interval and middle value for each coefficient a_n a list with 2 members: the position i and the value of the estimated root ρ = x i Warnings When you are using RStudio it is necessary to leave enough space for the plot window in order for the plots to appear normally. The data should come from a function at least C (0) in order to find the root, provided that such a root exists. Author(s) Demetris T. Christopoulos References Demetris T. Christopoulos (2014), Roots, extrema and inflection points by using a proper Taylor regression procedure, ResearchGate publicationss,

13 rootxi 13 Examples 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 multiple roots. Let's create some data: f=function(x){3*cos(x-5)};xa=0.;xb=9; set.seed(12345);x=sort(runif(101,xa,xb));r=0.1;y=f(x)+2*r*(runif(length(x))-0.5);plot(x,y) The first root is b1<-rootxi(x,y,1,20,5,5,plots=true);b1$an;b1$froot; 2.5 % 97.5 % an a a a a a a [1] Compare it with the actual rho_1= The second root is b2<-rootxi(x,y,20,50,5,5,plots=true);b2$an;b2$froot; 2.5 % 97.5 % an a a a a a a [1] You have to compare it with the actual value of rho_2= Finally the third root is b3<-rootxi(x,y,50,90,5,5,plots=true);b3$an;b3$froot; 2.5 % 97.5 % an a a a a

14 14 xydat a a [1] You have to compare it with the actual value of rho_3= xydat xydat Description A dataset containing 61 xy-points Usage data("xydat") Format A data frame with 61 observations on the following 2 variables. x a numeric vector y a numeric vector Examples data(xydat)

15 Index Topic datasets xydat, 14 Topic extremexi extremexi, 4 inflexi, 7 RootsExtremaInflections-package, 2 Topic inflexi RootsExtremaInflections-package, 2 Topic rootexinf rootexinf, 9 Topic rootxi RootsExtremaInflections-package, 2 rootxi, 11 extremexi, 4 inflexi, 7 rootexinf, 9 RootsExtremaInflections (RootsExtremaInflections-package), 2 RootsExtremaInflections-package, 2 rootxi, 11 xydat, 14 15