Let’ s take the function: f(x)=5+5tanh(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 xl=5.970315941,xr=4.029684059, xF1=3.850750196,xF2=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()’.
library(inflection)
data("table_01")
01$x
x=table_01$y
y=table_plot(x,y,cex=0.3,pch=19)
grid()
ese(x,y,0);bb bb=
## j1 j2 chi
## ESE 170 332 5
3];pese pese=bb[,
## [1] 5
abline(v=pese)
bese(x,y,0)
cc=$iplast cc
## [1] 5
abline(v=cc$iplast,col='blue')
::kable(cc$iters, caption = 'BESE') knitr
n | a | b | ESE |
---|---|---|---|
501 | 2.000 | 8.000 | 5 |
163 | 4.556 | 5.444 | 5 |
75 | 4.784 | 5.216 | 5 |
37 | 4.892 | 5.108 | 5 |
19 | 4.952 | 5.048 | 5 |
9 | 4.976 | 5.024 | 5 |
5 | 4.988 | 5.012 | 5 |
We observe that χl=5.9720,χr=4.0280, χF1=3.8480,χF2=6.1520 are very close to the theoretically expected values, so we are on the results of Lemma 1.3 of [1]. The absolutely accuracy from the first apply of all methods confirms our theoretical analysis.
We next add the error term ϵi∼U(−0.05,0.05) via the process 14 of [1] and run our algorithms again.The results are presented at Table 2 of [1] and here we also present the BESE iterations done by ‘bese()’.
library(inflection)
data("table_02")
01$x
x=table_01$y
y=table_plot(x,y,cex=0.3,pch=19)
grid()
ese(x,y,0);bb bb=
## j1 j2 chi
## ESE 170 332 5
3];pese pese=bb[,
## [1] 5
abline(v=pese)
bese(x,y,0)
cc=$iplast cc
## [1] 5
abline(v=cc$iplast,col='blue')
::kable(cc$iters, caption = 'BESE') knitr
n | a | b | ESE |
---|---|---|---|
501 | 2.000 | 8.000 | 5 |
163 | 4.556 | 5.444 | 5 |
75 | 4.784 | 5.216 | 5 |
37 | 4.892 | 5.108 | 5 |
19 | 4.952 | 5.048 | 5 |
9 | 4.976 | 5.024 | 5 |
5 | 4.988 | 5.012 | 5 |
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 xl=5.974322740,xr=4.029684059, xF1=4.025677260, xF2=5.974322740. We have that xr<a, so χr has to estimate a = 4.2 and χS must be close to 4.703504993. Additionally, xF1<a, so χF1 must be also an estimation of a, thus χD 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].
data("table_03_04")
03_04$x
x=table_03_04$y
y=table_ese(x,y,0);tese tese=
## j1 j2 chi
## ESE 2 156 4.7928
3]
pese=tese[,ede(x,y,0);tede tede=
## j1 j2 chi
## EDE 1 234 5.0854
3]
pede=tede[,bese(x,y,0)
cc=$iplast cc
## [1] 5.0018
bede(x,y,0)
dd=$iplast dd
## [1] 4.998
n | a | b | ESE |
---|---|---|---|
501 | 4.2000 | 8.0000 | 4.7928 |
155 | 4.8156 | 5.3704 | 5.0930 |
74 | 4.8232 | 5.0892 | 4.9562 |
36 | 4.9524 | 5.0816 | 5.0170 |
18 | 4.9600 | 5.0208 | 4.9904 |
9 | 4.9904 | 5.0132 | 5.0018 |
n | a | b | EDE |
---|---|---|---|
501 | 4.2000 | 8.0000 | 5.0854 |
234 | 4.5192 | 5.4844 | 5.0018 |
128 | 4.7244 | 5.2716 | 4.9980 |
73 | 4.8460 | 5.1576 | 5.0018 |
42 | 4.9068 | 5.0892 | 4.9980 |
25 | 4.9448 | 5.0512 | 4.9980 |
15 | 4.9676 | 5.0284 | 4.9980 |
9 | 4.9828 | 5.0208 | 5.0018 |
6 | 4.9904 | 5.0132 | 5.0018 |
4 | 4.9904 | 5.0056 | 4.9980 |
Let’ s add again an same error term ϵi∼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.
data("table_05_06")
05_06$x
x=table_05_06$y
y=table_ese(x,y,0);tese tese=
## j1 j2 chi
## ESE 3 149 4.77
3]
pese=tese[,ede(x,y,0);tede tede=
## j1 j2 chi
## EDE 3 231 5.0816
3]
pede=tede[,bese(x,y,0)
cc=$iplast cc
## [1] 5.036
bede(x,y,0)
dd=$iplast dd
## [1] 4.9828
plot(x,y,cex=0.3,pch=19)
grid()
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
::kable(cc$iters, caption = 'BESE') knitr
n | a | b | ESE |
---|---|---|---|
501 | 4.2000 | 8.0000 | 4.7700 |
147 | 4.8156 | 5.2792 | 5.0474 |
62 | 4.9144 | 5.1576 | 5.0360 |
::kable(dd$iters, caption = 'BEDE') knitr
n | a | b | EDE |
---|---|---|---|
501 | 4.2000 | 8.0000 | 5.0816 |
229 | 4.5268 | 5.5148 | 5.0208 |
131 | 4.7244 | 5.2412 | 4.9828 |
Let’ s examine the function: f(x)=10e−e5e−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.
data("table_08_09")
08_09$x
x=table_08_09$y
y=table_ese(x,y,0);tese tese=
## j1 j2 chi
## ESE 72 266 5.012
3]
pese=tese[,ede(x,y,0);tede tede=
## j1 j2 chi
## EDE 67 311 5.192
3]
pede=tede[,bese(x,y,0)
cc=$iplast cc
## [1] 4.9985
bede(x,y,0)
dd=$iplast dd
## [1] 4.9985
n | a | b | ESE |
---|---|---|---|
501 | 3.500 | 8.000 | 5.0120 |
195 | 4.625 | 5.489 | 5.0570 |
97 | 4.778 | 5.201 | 4.9895 |
48 | 4.904 | 5.120 | 5.0120 |
25 | 4.940 | 5.048 | 4.9940 |
13 | 4.976 | 5.030 | 5.0030 |
7 | 4.985 | 5.012 | 4.9985 |
n | a | b | EDE |
---|---|---|---|
501 | 3.500 | 8.000 | 5.1920 |
245 | 4.454 | 5.669 | 5.0615 |
136 | 4.670 | 5.363 | 5.0165 |
78 | 4.805 | 5.210 | 5.0075 |
46 | 4.886 | 5.120 | 5.0030 |
27 | 4.931 | 5.066 | 4.9985 |
16 | 4.958 | 5.039 | 4.9985 |
10 | 4.976 | 5.021 | 4.9985 |
6 | 4.985 | 5.012 | 4.9985 |
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].
data("table_10_11")
08_09$x
x=table_08_09$y
y=table_ese(x,y,0);tese tese=
## j1 j2 chi
## ESE 72 266 5.012
3]
pese=tese[,ede(x,y,0);tede tede=
## j1 j2 chi
## EDE 67 311 5.192
3]
pede=tede[,bese(x,y,0)
cc=$iplast cc
## [1] 4.9985
bede(x,y,0)
dd=$iplast dd
## [1] 4.9985
n | a | b | ESE |
---|---|---|---|
501 | 3.500 | 8.000 | 5.0120 |
195 | 4.625 | 5.489 | 5.0570 |
97 | 4.778 | 5.201 | 4.9895 |
48 | 4.904 | 5.120 | 5.0120 |
25 | 4.940 | 5.048 | 4.9940 |
13 | 4.976 | 5.030 | 5.0030 |
7 | 4.985 | 5.012 | 4.9985 |
n | a | b | EDE |
---|---|---|---|
501 | 3.500 | 8.000 | 5.1920 |
245 | 4.454 | 5.669 | 5.0615 |
136 | 4.670 | 5.363 | 5.0165 |
78 | 4.805 | 5.210 | 5.0075 |
46 | 4.886 | 5.120 | 5.0030 |
27 | 4.931 | 5.066 | 4.9985 |
16 | 4.958 | 5.039 | 4.9985 |
10 | 4.976 | 5.021 | 4.9985 |
6 | 4.985 | 5.012 | 4.9985 |
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.
Let the polynomial function: f(x)=−13x3+52x2−4x+12 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.
data("table_13")
13$x
x=table_13$y
y=table_plot(x,y,cex=0.3,pch=19)
grid()
ese(x,y,0);bb bb=
## j1 j2 chi
## ESE 126 376 2.5
3];pese pese=bb[,
## [1] 2.5
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].
data("table_14_15")
14_15$x
x=table_14_15$y
y=table_plot(x,y,cex=0.3,pch=19)
grid()
ese(x,y,0);bb bb=
## j1 j2 chi
## ESE 115 375 2.392
3];pese pese=bb[,
## [1] 2.392
abline(v=pese)
bese(x,y,0)
cc=$iplast cc
## [1] 2.473
abline(v=cc$iplast,col='blue')
::kable(cc$iters, caption = 'BESE') knitr
n | a | b | ESE |
---|---|---|---|
501 | -2.000 | 7.000 | 2.392 |
261 | 1.222 | 3.688 | 2.455 |
138 | 1.564 | 3.382 | 2.473 |
For the same symmetric 3rd order polynomial as above we change the interval to [-2, 8], thus we have data right asymmetry now. The case of SP with 500 sub-intervals and no error gives Table 17 of [1], while ESE and EDE iterations are presented at Table 18 of [1]. First results are absolutely compatible with theoretical predictions for ESE method.
data("table_17_18")
17_18$x
x=table_17_18$y
y=table_ese(x,y,0);bb bb=
## j1 j2 chi
## ESE 88 338 2.24
3];pese pese=bb[,
## [1] 2.24
plot(x,y,cex=0.3,pch=19)
grid()
bese(x,y,0)
cc=$iplast cc
## [1] 2.5
bede(x,y,0)
dd=$iplast dd
## [1] 2.5
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
::kable(cc$iters, caption = 'BESE') knitr
n | a | b | ESE |
---|---|---|---|
501 | -2.00 | 8.00 | 2.24 |
251 | 1.38 | 3.88 | 2.63 |
126 | 1.80 | 3.06 | 2.43 |
64 | 2.22 | 2.86 | 2.54 |
33 | 2.32 | 2.64 | 2.48 |
17 | 2.42 | 2.58 | 2.50 |
9 | 2.46 | 2.54 | 2.50 |
5 | 2.48 | 2.52 | 2.50 |
::kable(dd$iters, caption = 'BEDE') knitr
n | a | b | EDE |
---|---|---|---|
501 | -2.00 | 8.00 | 2.5 |
293 | 0.82 | 4.18 | 2.5 |
169 | 1.54 | 3.46 | 2.5 |
97 | 1.94 | 3.06 | 2.5 |
57 | 2.18 | 2.82 | 2.5 |
33 | 2.32 | 2.68 | 2.5 |
19 | 2.40 | 2.60 | 2.5 |
11 | 2.44 | 2.56 | 2.5 |
7 | 2.46 | 2.54 | 2.5 |
5 | 2.48 | 2.52 | 2.5 |
We add uniform error distributed by U(-2, 2) and we have the results of Table 19 of [1], while one ESE & one EDE iteration are given at Table 20 of [1].
data("table_19_20")
19_20$x
x=table_19_20$y
y=table_ese(x,y,0);bb bb=
## j1 j2 chi
## ESE 88 338 2.24
3];pese pese=bb[,
## [1] 2.24
plot(x,y,cex=0.3,pch=19)
grid()
bese(x,y,0)
cc=$iplast cc
## [1] 2.65
bede(x,y,0)
dd=$iplast dd
## [1] 2.35
abline(v=pese)
abline(v=cc$iplast,col='blue')
abline(v=dd$iplast,col='red')
::kable(cc$iters, caption = 'BESE') knitr
n | a | b | ESE |
---|---|---|---|
501 | -2.00 | 8.00 | 2.24 |
251 | 1.46 | 3.84 | 2.65 |
::kable(dd$iters, caption = 'BEDE') knitr
n | a | b | EDE |
---|---|---|---|
501 | -2.00 | 8.00 | 2.70 |
297 | 0.86 | 3.84 | 2.35 |
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 χr=−0.26,χl=4.74, so we have that: ˆp=13χl+13χr+16a+16b=2.493333333 which is much closer to the true value of 2.5.
[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://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.