inflexi: Function to Find the Inflection Point of a Planar Curve

In RootsExtremaInflections: Finds Roots, Extrema and Inflection Points of a 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 inflection point for that interval, while it plots data, Taylor polynomial and and the computed |a_2| coefficients.

Usage

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

Arguments

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)

Details

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}].

Value

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 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. SSRN. https://dx.doi.org/10.2139/ssrn.2521403

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;
#         2.5 %     97.5 %           an
# a0  0.1483905  0.2377617  0.193076089
# a1  2.9024852  3.0936024  2.998043835
# a2 -0.2053120  0.2220390  0.008363525
# a3 -0.5845597 -0.3426017 -0.463580702
# a4 -0.2431038  0.1136244 -0.064739689
# a5 -0.0893246  0.0687848 -0.010269897
# [1] 19.000000  3.493296
#Compare it with the actual rho_1=3.429203673
#
#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
# a0 -0.000875677  0.057156356  0.0281403394
# a1 -3.058363342 -2.942026810 -3.0001950762
# a2 -0.056224101  0.044135857 -0.0060441222
# a3  0.433135897  0.528446241  0.4807910691
# a4 -0.011774733  0.012002414  0.0001138404
# a5 -0.026899286 -0.009520899 -0.0182100925
# [1] 23.000000  6.567948
#You have to compare it with the actual value of rho_2=6.570796327

Example output

Loading required package: iterators
Loading required package: foreach
Loading required package: parallel
Loading required package: doParallel
          2.5 %      97.5 %           an
a0    0.6464691    0.784573    0.7155210
a1   35.9261150   37.325631   36.6258731
a2   -4.7633889    5.050487    0.1435489
a3 -162.2798131 -140.556443 -151.4181281
a4  -90.7953845   42.399919  -24.1977330
a5   31.0401819  220.597872  125.8190270
[1] 41.0000000  0.6330743
        2.5 %     97.5 %           an
a0  0.1483905  0.2377617  0.193076089
a1  2.9024852  3.0936024  2.998043835
a2 -0.2053120  0.2220390  0.008363525
a3 -0.5845597 -0.3426017 -0.463580702
a4 -0.2431038  0.1136244 -0.064739689
a5 -0.0893246  0.0687848 -0.010269897
[1] 19.000000  3.493296

RootsExtremaInflections documentation built on July 29, 2019, 5:03 p.m.