Author: Demetris T. Christopoulos

# Introduction to inflection package

#### Demetris T. Christopoulos

#### 25/6/2019

## Inflection Point

When a planar curve is first convex and then concave or vice versa, then there exists an inflection point (ip). We could find that if all our points had no error, but that’s an exception to reality. Since the divided differences of second order are extremely sensitive to the error, it is meaningless to use them as an approximation in order to search for inflection point. For that task Extremum Surface Estimator (ESE), Extremeum Distance Estimator (EDE) and their iterative versions Bisection ESE (BESE), Bisection EDE (BEDE) have been created, see [1] & [2] for definitions and properties.

Lets create a not noisy data set from function \(f(x)=5+5\,tanh(x-5)\) with known inflection point \((ip=5)\), equal x-distances and total symmetry, see [2]. Actually it is the data set “TABLE_2_1” of package.

### Now we compute the results of all available methods for non noisy data sets

#### ESE & BESE

    f=function(x){5+5*tanh(x-5)}
    x=seq(0,10,by=0.05)
    y=f(x)
    plot(x,y,cex=0.3,pch=19)
    grid()
    bb=ese(x,y,0);bb

    ##     j1  j2 chi
    ## ESE 77 125   5

    pese=bb[,3];pese

    ## [1] 5

    abline(v=pese)
    cc=bese(x,y,0)
    cc$iplast

    ## [1] 5

    abline(v=cc$iplast,col='blue')

    knitr::kable(cc$iters, caption = 'BESE')

BESE
n | a | b | ESE
--- | --- | --- | ---
201 | 0.00 | 10.00 | 5
49 | 4.45 | 5.55 | 5
23 | 4.75 | 5.25 | 5
11 | 4.90 | 5.10 | 5
5 | 4.95 | 5.05 | 5

#### EDE & BEDE

    x=seq(0,10,by=0.05)
    y=f(x)
    plot(x,y,cex=0.3,pch=19)
    grid()
    dd=ede(x,y,0);dd

    ##     j1  j2 chi
    ## EDE 72 130   5

    pede=dd[,3];pede

    ## [1] 5

    abline(v=pede)
    ee=bede(x,y,0)
    ee$iplast

    ## [1] 5

    abline(v=cc$iplast,col='red')

    knitr::kable(ee$iters, caption = 'BEDE')

BEDE
n | a | b | EDE
--- | --- | --- | ---
201 | 0.00 | 10.00 | 5
59 | 4.30 | 5.70 | 5
29 | 4.60 | 5.40 | 5
17 | 4.75 | 5.25 | 5
11 | 4.85 | 5.15 | 5
7 | 4.90 | 5.10 | 5
5 | 4.95 | 5.05 | 5

So all methods gave the theoretical expected value. Lets create a data set with data right asymmetry, see [2] and run again:

    x=seq(3,10,by=0.05)
    y=f(x)
    plot(x,y,cex=0.5,pch=19)
    abline(v=5,lty=2)
    grid()
    bb=ese(x,y,0);bb

    ##     j1 j2   chi
    ## ESE 17 56 4.775

    pese=bb[,3];pese

    ## [1] 4.775

    abline(v=pese)
    cc=bese(x,y,0)
    cc$iplast

    ## [1] 5.025

    abline(v=cc$iplast,col='blue')

    knitr::kable(cc$iters, caption = 'BESE')

BESE
n | a | b | ESE
--- | --- | --- | ---
141 | 3.00 | 10.00 | 4.775
40 | 4.65 | 5.55 | 5.100
19 | 4.75 | 5.15 | 4.950
9 | 4.95 | 5.10 | 5.025

    x=seq(3,10,by=0.05)
    y=f(x)
    plot(x,y,cex=0.5,pch=19)
    abline(v=5,lty=2)
    grid()
    dd=ede(x,y,0);dd

    ##     j1 j2 chi
    ## EDE 16 66   5

    pede=dd[,3];pede

    ## [1] 5

    abline(v=pede)
    ee=bede(x,y,0)
    ee$iplast

    ## [1] 5

    abline(v=cc$iplast,col='red')

    knitr::kable(ee$iters, caption = 'BEDE')

BEDE
n | a | b | EDE
--- | --- | --- | ---
141 | 3.00 | 10.00 | 5
51 | 4.35 | 5.65 | 5
27 | 4.65 | 5.35 | 5
15 | 4.80 | 5.20 | 5
9 | 4.90 | 5.10 | 5
5 | 4.95 | 5.05 | 5

## Please send your comments, suggestions or bugs found to dchristop@econ.uoa.gr

## References

[1] Demetris T. Christopoulos, On the Efficient Identification of an Inflection Point, International Journal of Mathematics and Scientific Computing , Volume 6 (1), June 2016, Pages 13-20, ISSN: 2231-5330. Retrieved from https://demovtu.veltech.edu.in/wp-content/uploads/2016/04/Paper-04-2016.pdf

[2] Demetris T. Christopoulos. Developing methods for identifying the inflection point of a convex/concave curve. arXiv.org, 2012. URL: https://doi.org/10.48550/arXiv.1206.5478