extremexi: Function to Find the Extreme Point of a Planar Curve
In RootsExtremaInflections: Finds Roots, Extrema and Inflection Points of a Curve

Description Usage Arguments Details Value Warnings Author(s) References Examples

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 computed |a_1| coefficients.

1 2	extremexi(x, y, i1, i2, nt, alpha = 5, xlb = "x", ylb = "y", xnd = 3, ynd = 3, plots = TRUE, plotpdf = FALSE, doparallel=FALSE)

`x`	A numeric vector for the independent variable
`y`	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`	The degree of the Taylor polynomial that will be fitted to the data
`alpha`	The level of statistical significance for the confidence intervals of coefficients a_0, a_1,..., a_{nt-1} (default value = 5)
`xlb`	A label for the x-variable (default value = "x")
`ylb`	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`	If plots=TRUE then a plot is created on default monitor (default value = TRUE)
`plotpdf`	If plotpdf=TRUE then a pdf plot is created and stored on working directory (default value = FALSE)
`doparallel`	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}].

It returns an environment with two components:

`an`	a matrix with 3 columns: lower, upper bound of confidence interval and middle value for each coefficient an
`fextr`	a list with 2 members: the position i and the value of the estimated extreme point ρ=x_i

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.

Demetris T. Christopoulos

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

#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
# a0 -3.02708631 -2.94592364 -2.986504975
# a1  0.07660314  0.24706531  0.161834227
# a2  1.42127770  1.58580632  1.503542012
# a3 -0.09037154  0.10377241  0.006700434
# a4 -0.14788899 -0.08719428 -0.117541632
# a5 -0.03822416  0.01425066 -0.011986748
# [1] 22.000000  1.917229
#Compare it with the actual rho_1=1.858407346
#
#The second extreme point is
c2<-extremexi(x,y,50,80,5,5,plots=TRUE);c2$an;c2$fextr;
#          2.5 %       97.5 %         an
# a0  2.89779980  3.064703163  2.9812515
# a1  0.27288720  0.541496278  0.4071917
# a2 -1.81454401 -0.677932480 -1.2462382
# a3 -1.76290384  0.216201349 -0.7733512
# a4  0.02548354  1.269671304  0.6475774
# a5 -0.25156866  0.007565154 -0.1220018
# [1] 7.000000 4.896521
#You have to compare it with the actual value of rho_2=5.0
#
#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
# a0 -3.0637461 -2.9218614 -2.9928037
# a1 -0.2381605  0.2615635  0.0117015
# a2  0.7860259  2.0105383  1.3982821
# a3 -1.4187417  0.7472155 -0.3357631
# a4 -0.7943208  1.0876143  0.1466468
# a5 -0.6677733  1.7628833  0.5475550
# [1] 11.000000  8.137392
#You have to compare it with the actual value of rho_3=8.141592654

Loading required package: iterators
Loading required package: foreach
Loading required package: parallel
Loading required package: doParallel
         2.5 %       97.5 %           an
a0   -6.778082   -6.6276938   -6.7028879
a1   -1.611186    0.1275498   -0.7418182
a2   85.152457   93.4939305   89.3231936
a3  -11.389929   33.5321572   11.0711142
a4 -318.711699 -136.0239437 -227.3678214
a5   31.040182  220.5978720  125.8190270
[1] 18.0000000  0.3101182
         2.5 %      97.5 %           an
a0 -3.02708631 -2.94592364 -2.986504975
a1  0.07660314  0.24706531  0.161834227
a2  1.42127770  1.58580632  1.503542012
a3 -0.09037154  0.10377241  0.006700434
a4 -0.14788899 -0.08719428 -0.117541632
a5 -0.03822416  0.01425066 -0.011986748
[1] 22.000000  1.917229
         2.5 %       97.5 %         an
a0  2.89779980  3.064703163  2.9812515
a1  0.27288720  0.541496278  0.4071917
a2 -1.81454401 -0.677932480 -1.2462382
a3 -1.76290384  0.216201349 -0.7733512
a4  0.02548354  1.269671304  0.6475774
a5 -0.25156866  0.007565154 -0.1220018
[1] 7.000000 4.896521
        2.5 %     97.5 %         an
a0 -3.0637461 -2.9218614 -2.9928037
a1 -0.2381605  0.2615635  0.0117015
a2  0.7860259  2.0105383  1.3982821
a3 -1.4187417  0.7472155 -0.3357631
a4 -0.7943208  1.0876143  0.1466468
a5 -0.6677733  1.7628833  0.5475550
[1] 11.000000  8.137392