---
title: "Developing methods for identifying the inflection point of a convex/concave curve"
author: "Demetris T. Christopoulos"
date: "25/6/2019"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Developing methods for identifying the inflection point of a convex/concave curve}
  %\VignetteEncoding{UTF-8}
---


```{css, echo=FALSE}
    body .main-container {
      max-width: 1024px !important;
      width: 1024px !important;
    }
    body {
      max-width: 1024px !important;
    }
```

```{r setup, include=FALSE}
library(inflection)
knitr::opts_chunk$set(echo = TRUE)
options(max.width = 1000)
options(max.print = 100000)
```

## The Fisher-Pry sigmoid curve with total symmetry (not noisy)

Let’ s take the function:
$$f (x) = 5 + 5\, tanh (x − 5)$$
after [3], which has p = 5, L = 10, x1 = 2.7024, x99 = 7.2976 and examine
it at the interval [2, 8] in order to have data symmetry w.r.t. inflection point.
The function is also symmetrical around inflection point, i.e. we have total
symmetry. 

From Corollary 1.1 of [1] we compute $x_l = 5.970315941, x_r = 4.029684059$,
$x_{F1} = 3.850750196, x_{F2} = 6.149249804$, all inside [2, 8], thus all methods
are theoretically applicable.

We first take n = 500 sub-intervals equal spaced without error just for checking our estimators. The results are presented at
Table 1 of [1], while here we also present the BESE iterations done by 'bese()'.

```{r, tanhESE, echo=TRUE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
library(inflection)
data("table_01")
x=table_01$x
y=table_01$y
plot(x,y,cex=0.3,pch=19)
grid()
bb=ese(x,y,0);bb
pese=bb[,3];pese
abline(v=pese)
cc=bese(x,y,0)
cc$iplast
abline(v=cc$iplast,col='blue')
knitr::kable(cc$iters, caption = 'BESE')
```


We observe that $\chi_l = 5.9720, \chi_r = 4.0280$, $\chi_{F1} = 3.8480, \chi_{F2} =6.1520$
are very close to the theoretically expected values, so we are on the results of Lemma 1.3 of [1].
The absolutely accuracy from the first apply of all methods confirms our theoretical analysis.

## The Fisher-Pry sigmoid curve with total symmetry (noisy)

We next add the error term $\epsilon_i\sim\,U(−0.05, 0.05)$ via the process 14 of [1] and
run our algorithms again.The results are presented at Table 2 of [1] and here we also present the 
BESE iterations done by 'bese()'.


```{r, tanhESEnoisy, echo=TRUE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
library(inflection)
data("table_02")
x=table_01$x
y=table_01$y
plot(x,y,cex=0.3,pch=19)
grid()
bb=ese(x,y,0);bb
pese=bb[,3];pese
abline(v=pese)
cc=bese(x,y,0)
cc$iplast
abline(v=cc$iplast,col='blue')
knitr::kable(cc$iters, caption = 'BESE')
```


## The Fisher-Pry sigmoid curve with data left asymmetry (not noisy)

We continue with the same sigmoid function, but now we choose a proper [a, b] to show data asymmetry w.r.t. inflection point.

Let’ s take for example [4.2, 8]. If we do our theoretical computations
we find $x_l = 5.974322740, x_r = 4.029684059$, $x_{F1} = 4.025677260$,
$x_{F2} = 5.974322740$. We have that $x_r < a$, so $\chi_r$ has to estimate a = 4.2 
and $\chi_S$ must be close to 4.703504993. Additionally, $x_{F1} < a$, so $\chi_{F1}$ must be also
an estimation of a, thus $\chi_{D}$ must lie near the value 5.087161370. It' s time
to see if our theoretical predictions will be confirmed by experiment.
We use for comparability the same Standard Partition as before and have
the output presented at Table 3 and 4 of [1].

```{r, tanhESEasym, echo=TRUE}
data("table_03_04")
x=table_03_04$x
y=table_03_04$y
tese=ese(x,y,0);tese
pese=tese[,3]
tede=ede(x,y,0);tede
pede=tede[,3]
cc=bese(x,y,0)
cc$iplast
dd=bede(x,y,0)
dd$iplast
```

```{r, tanhESEasymPLOT, echo=FALSE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
plot(x,y,cex=0.3,pch=19)
grid()
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
knitr::kable(cc$iters, caption = 'BESE')
knitr::kable(dd$iters, caption = 'BEDE')
```


## The Fisher-Pry sigmoid curve with data left asymmetry (noisy)

Let’ s add again an same error term $\epsilon_i\sim\,U(−0.05, 0.05)$ and run our 
algorithms. The results at Table 5 of [1] clearly are close enough to the theoretical
expectations. Since ESE method did not estimate the inflection point with
acceptable accuracy, after running BESE and BEDE iterative methods we find Table 6 of [1]
which is a clear improvement of both estimations.

```{r, tanhESEasymNOISE, echo=TRUE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
data("table_05_06")
x=table_05_06$x
y=table_05_06$y
tese=ese(x,y,0);tese
pese=tese[,3]
tede=ede(x,y,0);tede
pede=tede[,3]
cc=bese(x,y,0)
cc$iplast
dd=bede(x,y,0)
dd$iplast
plot(x,y,cex=0.3,pch=19)
grid()
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
knitr::kable(cc$iters, caption = 'BESE')
knitr::kable(dd$iters, caption = 'BEDE')
```


## The Gompertz non symmetric sigmoid curve (not noisy)

Let’ s examine the function:
$$f (x) = 10 e^{-e^{5}e^{−x}}$$
after [4], in the interval [3.5, 8]. It is easy to prove that f is (0.224, 1.0)-asymptotically symmetric around
inflection point, so we can handle it similar to a symmetric sigmoid only for a distance of ±1 from p = 5.

We use, for comparison reasons, the same SP with 500 sub-intervals without error and obtain the Table 8 of [1] which is 
absolutely compatible with theoretical predictions. The ESE & EDE iterations are showed at Table 9 of [1]
where we observe convergence to the real p for both two methods.


```{r, Gomb, echo=TRUE}
data("table_08_09")
x=table_08_09$x
y=table_08_09$y
tese=ese(x,y,0);tese
pese=tese[,3]
tede=ede(x,y,0);tede
pede=tede[,3]
cc=bese(x,y,0)
cc$iplast
dd=bede(x,y,0)
dd$iplast
```

```{r, GombPLOT, echo=FALSE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
plot(x,y,cex=0.3,pch=19)
grid()
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
knitr::kable(cc$iters, caption = 'BESE')
knitr::kable(dd$iters, caption = 'BEDE')
```

## The Gompertz non symmetric sigmoid curve (noisy)

We continue with our familiar SP by adding error uniformly distributed
by U(−0.05, 0.05) and the results are given at Table 10 of [1] while ESE & EDE
iterations are shown at Table 11 of [1].

```{r, GombNOISE, echo=TRUE}
data("table_10_11")
x=table_08_09$x
y=table_08_09$y
tese=ese(x,y,0);tese
pese=tese[,3]
tede=ede(x,y,0);tede
pede=tede[,3]
cc=bese(x,y,0)
cc$iplast
dd=bede(x,y,0)
dd$iplast
```

```{r, GombPLOTNOISE, echo=FALSE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
plot(x,y,cex=0.3,pch=19)
grid()
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
knitr::kable(cc$iters, caption = 'BESE')
knitr::kable(dd$iters, caption = 'BEDE')
```

From these Tables we conclude that convergence to the true value of inflection point p = 5 occurs
from the iterative application of ESE and EDE methods in one or two steps only.

## A symmetric 3rd order polynomial with total symmetry

Let the polynomial function:
$$f(x)=-\frac{1}{3}\,x^3+\frac{5}{2}\,x^2-4x+\frac{1}{2}$$
We study it at [-2, 7], it has inflection point at p = 2.5 and we have total symmetry.
The SP with 500 sub-intervals without error gives Table 13 of [1] which is absolutely compatible with theoretical predictions.
There is no need for any kind of iteration, because both methods agree with the true value.


```{r, poly3sym, echo=TRUE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
data("table_13")
x=table_13$x
y=table_13$y
plot(x,y,cex=0.3,pch=19)
grid()
bb=ese(x,y,0);bb
pese=bb[,3];pese
abline(v=pese)
```








The same SP with uniform error distributed by U(−2, 2) gives the results of Table 14 of [1] 
and two ESE iterations are presented at Table 15 of [1].


```{r, poly3symNOISE, echo=TRUE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
data("table_14_15")
x=table_14_15$x
y=table_14_15$y
plot(x,y,cex=0.3,pch=19)
grid()
bb=ese(x,y,0);bb
pese=bb[,3];pese
abline(v=pese)
cc=bese(x,y,0)
cc$iplast
abline(v=cc$iplast,col='blue')
knitr::kable(cc$iters, caption = 'BESE')
```


## A symmetric 3rd order polynomial with data right asymmetry

For the same symmetric 3rd order polynomial as above we change the interval to
[-2, 8], thus we have data right asymmetry now. The case of SP with 500 sub-intervals 
and no error gives Table 17 of [1], while ESE and EDE iterations are presented at Table 18 of [1].
First results are absolutely compatible with theoretical predictions for ESE method.

```{r, poly3asym, echo=TRUE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
data("table_17_18")
x=table_17_18$x
y=table_17_18$y
bb=ese(x,y,0);bb
pese=bb[,3];pese
plot(x,y,cex=0.3,pch=19)
grid()
cc=bese(x,y,0)
cc$iplast
dd=bede(x,y,0)
dd$iplast
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
knitr::kable(cc$iters, caption = 'BESE')
knitr::kable(dd$iters, caption = 'BEDE')
```

We add uniform error distributed by U(-2, 2) and we have the results of Table 19 of [1], 
while one ESE & one EDE iteration are given at Table 20 of [1].


```{r, poly3asymNOISE, echo=TRUE,out.width='75%', fig.align='center',fig.width=10, fig.height=6}
data("table_19_20")
x=table_19_20$x
y=table_19_20$y
bb=ese(x,y,0);bb
pese=bb[,3];pese
plot(x,y,cex=0.3,pch=19)
grid()
cc=bese(x,y,0)
cc$iplast
dd=bede(x,y,0)
dd$iplast
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
knitr::kable(cc$iters, caption = 'BESE')
knitr::kable(dd$iters, caption = 'BEDE')
```

There exist a problem here. Although we have a symmetric polyno-
mial, the TESE is not equal to the true inflection point. A remedy for this
problem for the class of 3rd order polynomials is given with Lemma 2.1 of [1].
Lets apply it here. We have that a = −2, b = 8 and from Table 19 of [1]
is $\chi_{r} = −0.26, \chi_{l} = 4.74$, so we have that:
$$\hat{p}=\frac{1}{3}\,\chi_{l} + \frac{1}{3}\,\chi_{r}+\frac{1}{6}\,a+\frac{1}{6}\,b=2.493333333$$
which is much closer to the true value of 2.5.


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


## References

[1] Demetris T. Christopoulos (2014), Developing methods for identifying the inflection point of a convex/concave curve. arXiv:1206.5478v2 [math.NA]. 
URL: https://doi.org/10.48550/arXiv.1206.5478


[2] Demetris T. Christopoulos (2016), 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. 
URL: https://demovtu.veltech.edu.in/wp-content/uploads/2016/04/Paper-04-2016.pdf


[3] J.C. Fisher and R.H. Pry (1971), A Simple Substitution Model of Technological Change, Technological Forecasting and Social Change, 3, pp. 5–88. 
URL: https://doi.org/10.1016/S0040-1625(71)80005-7


[4] B. Gompertz (1825), On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of
Life Contingencies, Philosophical Transactions of the Royal Society of London, 115, pp. 513–585.