RootsExtremaInflections-package: Finds Roots, Extrema and Inflection Points of a Planar Curve
In RootsExtremaInflections: Finds Roots, Extrema and Inflection Points of a Curve
Description
Details
Author(s)
References
Examples
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 {(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)+e_i with a zero mean error, e_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)+e_i, e_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_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).
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.
Author(s)
Demetris T. Christopoulos
Maintainer: Demetris T. Christopoulos <dchristop@econ.uoa.gr>
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
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#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.
Example output
Loading required package: iterators
Loading required package: foreach
Loading required package: parallel
Loading required package: doParallel
2.5 % 97.5 % an
a0 -0.4042465 -0.2625694 -0.3334079
a1 35.6234510 36.8668363 36.2451436
a2 8.2921797 17.8057689 13.0489743
a3 -158.4665008 -136.7277559 -147.5971284
a4 -95.9817375 11.4260406 -42.2778484
a5 31.0401819 220.5978720 125.8190270
[1] 38.0000000 0.6043344
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 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
RootsExtremaInflections documentation built on July 29, 2019, 5:03 p.m.
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Description Details Author(s) References Examples
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 {(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)+e_i with a zero mean error, e_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)+e_i, e_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_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).
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.
Author(s)
Demetris T. Christopoulos
Maintainer: Demetris T. Christopoulos <dchristop@econ.uoa.gr>
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | #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.
|
Example output
Loading required package: iterators
Loading required package: foreach
Loading required package: parallel
Loading required package: doParallel
2.5 % 97.5 % an
a0 -0.4042465 -0.2625694 -0.3334079
a1 35.6234510 36.8668363 36.2451436
a2 8.2921797 17.8057689 13.0489743
a3 -158.4665008 -136.7277559 -147.5971284
a4 -95.9817375 11.4260406 -42.2778484
a5 31.0401819 220.5978720 125.8190270
[1] 38.0000000 0.6043344
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 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
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