how to find inflection points
Find inflection points: Mission Impossible!
28/6/2019
Can we always find the inflection point?
Let' s take the function: \[f (x) = \frac{\frac{1}{8}x+\frac{1}{2}x^2}{1+\frac{1}{8}x+\frac{1}{2}x^2}\] following [3]. What are the inflection points? We must take the first derivative: \[f'(x)= \frac {8+64\,x}{ \left( 4\,{x}^{2}+x+8 \right) ^{2}}\]
and continue to second derivative: \[f''(x)= \frac {-768\,{x}^{2}-192\,x+496}{ \left( 4\,{x}^{2}+x+8 \right) ^{3}}\] Now we must solve above expression for x and find two solutions: \[x_1=-1/8-1/24\,\sqrt {381}=-0.9383008876\] \[x_2=-1/8+1/24\,\sqrt {381}=0.6883008876\] If we study the function we'll see that has a minimum at \(x_{min}=-\frac{1}{8}=-0.125\) and it is concave/convex at interval \(x<x_{min}\) and convex/concave for \(x>x_{min}\).
But what if we do not want to perform such an analytical approach? What if we just wanted the inflection points without more details?
Then we must follow next steps:
- define an interval that encloses an inflection point
- run ESE or EDE to find a first approximation
- choose an interval that encloses inflection with your desired accuracy
- run BESE to find the closest available approximation
Here are the steps for the positive inflection point:
library(inflection) f=function(x){(1 / 8 *x+ 1 / 2 *x^ 2)/(1 + 1 / 8 *x+ 1 / 2 *x^ 2)} x=seq(0,5,0.1) y=f(x) # First approximation cc=check_curve(x,y);cc ## $ctype ## [1] "convex_concave" ## ## $index ## [1] 0 ese(x,y,cc$index) ## j1 j2 chi ## ESE 2 13 0.65 # New interval with equal distances from first approximation x=seq(0,1.3,0.001) y=f(x) # Second approximation cc=check_curve(x,y);cc ## $ctype ## [1] "convex_concave" ## ## $index ## [1] 0 ipbese=bese(x,y,cc$index) ipbese$iplast ## [1] 0.688 plot(x,y,pch= 19,cex= 0.1) abline(v=ipbese$iplast,col= 'blue') 
| n | a | b | ESE |
|---|---|---|---|
| 1301 | 0.000 | 1.300 | 0.8090 |
| 711 | 0.495 | 0.819 | 0.6570 |
| 325 | 0.627 | 0.794 | 0.7105 |
| 168 | 0.638 | 0.720 | 0.6790 |
| 83 | 0.673 | 0.714 | 0.6935 |
| 42 | 0.676 | 0.696 | 0.6860 |
| 21 | 0.684 | 0.694 | 0.6890 |
| 11 | 0.685 | 0.690 | 0.6875 |
| 6 | 0.687 | 0.689 | 0.6880 |
Since our interval was defined with equal size step 0.001 we expect our estimation to have the same accuracy and indeed it is truth.
If we wanted a better accuracy, for example with 4 digits, then we could define another interval with previous approximation inside it and with equal size stpe 0.0001:
# x=seq(0.68,0.69,0.0001) # y=f(x) # cc=check_curve(x,y);cc # ipbese=bese(x,y,cc$index,doparallel = TRUE) # ipbese$iplast # # [1] 0.6883 # ipbese$iters # n a b ESE ## 1 101 0.6800 0.6900 0.68870 ## 2 25 0.6876 0.6887 0.68815 ## 3 12 0.6881 0.6886 0.68835 ## 4 6 0.6882 0.6884 0.68830 What if our data set is noisy?
There exist no problem.
Since ESE and EDE methods both give consistent estimators for the inflection point, we do not need to worry about noise.
Lets perform the same task with error added:
x=seq(0.0,3.,0.001) set.seed(20190628) y=f(x)+ runif(length(x),- 0.01,0.01) cc=check_curve(x,y) cc ## $ctype ## [1] "convex_concave" ## ## $index ## [1] 0 ipbese=bese(x,y,cc$index) ipbese$iplast ## [1] 0.689 plot(x,y,pch= 19,cex= 0.1) abline(v=ipbese$iplast,col= 'blue') 
| n | a | b | ESE |
|---|---|---|---|
| 3001 | 0.000 | 3.000 | 0.6955 |
| 1034 | 0.473 | 0.993 | 0.7330 |
| 521 | 0.542 | 0.836 | 0.6890 |
What about big data?
If your data set gas million of cases, then EDE and BEDE are your best friends.
Lets see why by using 1000001 cases from a function with inflection point equal to 500:
f=function(x){500 + 500 * tanh(x-500)} x=seq(0,1000,0.001) y=f(x) length(x) ## [1] 1000001 t1=Sys.time();ede(x,y,0);t2=Sys.time();t2-t1 ## j1 j2 chi ## EDE 496201 503801 500 ## Time difference of 0.01399779 secs ipbede=bede(x,y,cc$index) ipbede$iplast ## [1] 500 | n | a | b | EDE |
|---|---|---|---|
| 1000001 | 0.000 | 1000.000 | 500 |
| 7601 | 498.712 | 501.288 | 500 |
| 2577 | 499.341 | 500.659 | 500 |
| 1319 | 499.633 | 500.367 | 500 |
| 735 | 499.791 | 500.209 | 500 |
| 419 | 499.880 | 500.120 | 500 |
| 241 | 499.931 | 500.069 | 500 |
| 139 | 499.960 | 500.040 | 500 |
| 81 | 499.977 | 500.023 | 500 |
| 47 | 499.987 | 500.013 | 500 |
| 27 | 499.993 | 500.007 | 500 |
| 15 | 499.996 | 500.004 | 500 |
| 9 | 499.998 | 500.002 | 500 |
| 5 | 499.999 | 500.001 | 500 |
Situation will not change if we add error \(U(-50,50)\):
f=function(x){500 + 500 * tanh(x-500)} x=seq(0,1000,0.001) set.seed(20190628) y=f(x)+ runif(length(x),- 50,50) length(x) ## [1] 1000001 t1=Sys.time();ede(x,y,0);t2=Sys.time();t2-t1 ## j1 j2 chi ## EDE 496208 503817 500.0115 ## Time difference of 0.01199794 secs ipbede=bede(x,y,0) ipbede$iplast ## [1] 500.046 plot(x[495000 : 505000],y[495000 : 505000],xlab= "x",ylab= "y",pch= '.') abline(v=ipbede$iplast) 
| n | a | b | EDE |
|---|---|---|---|
| 1000001 | 0.000 | 1000.000 | 500.0115 |
| 7610 | 498.815 | 501.277 | 500.0460 |
Even if our data are not symmetric, estimation will not change:
f=function(x){500 + 500 * tanh(x-500)} x=seq(0,700,0.001) set.seed(20190628) y=f(x)+ runif(length(x),- 50,50) length(x) ## [1] 700001 t1=Sys.time();ede(x,y,0);t2=Sys.time();t2-t1 ## j1 j2 chi ## EDE 496208 503646 499.926 ## Time difference of 0.00899601 secs ipbede=bede(x,y,0) ipbede$iplast ## [1] 500.013 plot(x[495000 : 505000],y[495000 : 505000],xlab= "x",ylab= "y",pch= '.') abline(v=ipbede$iplast) 
| n | a | b | EDE |
|---|---|---|---|
| 700001 | 0.000 | 700.000 | 499.926 |
| 7439 | 498.829 | 501.197 | 500.013 |
Just try to solve above problems by using regression or other techniques that involve matrix inversion…
OK, we'll perform that task for you.
First task
# library(nlme) # x=seq(0,5,0.1) # f=function(x){(1/8*x+1/2*x^2)/(1+1/8*x+1/2*x^2)} # y=f(x) # df=data.frame("x"=x,"y"=y) # Asym=1;xmid=1;scal=1; # fmla=as.formula("y~SSlogis(x,Asym,xmid,scal)");fmla # try(nls(fmla,df)) # est=try(nls(fmla,df)) # coef(est) When it works it gives the result xmid=1.2516 which is away from true inflcetion (0.6883008876), altough all results seem to be statistically 'super':
# y ~ SSlogis(x, Asym, xmid, scal) # Nonlinear regression model # model: y ~ SSlogis(x, Asym, xmid, scal) # data: df # Asym xmid scal # 0.8909 1.2516 0.5478 # residual sum-of-squares: 0.05242 # # Number of iterations to convergence: 0 # Achieved convergence tolerance: 1.401e-06 ### # est=try(nls(fmla,df)) # coef(est) # Asym xmid scal # 0.8908913 1.2516410 0.5478263 # summary(est) # # Formula: y ~ SSlogis(x, Asym, xmid, scal) # # Parameters: # Estimate Std. Error t value Pr(>|t|) # Asym 0.890891 0.007872 113.18 <2e-16 *** # xmid 1.251641 0.025638 48.82 <2e-16 *** # scal 0.547826 0.023718 23.10 <2e-16 *** # --- # Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 # # Residual standard error: 0.03305 on 48 degrees of freedom # # Number of iterations to convergence: 0 # Achieved convergence tolerance: 1.401e-06 Second task
# x=seq(0.0,3.,0.001) # f=function(x){(1/8*x+1/2*x^2)/(1+1/8*x+1/2*x^2)} # set.seed(20190628) # y=f(x)+runif(length(x),-0.01,0.01) # df=data.frame("x"=x,"y"=y) # Asym=1;xmid=1;scal=1; # fmla=as.formula("y~SSlogis(x,Asym,xmid,scal)");fmla # try(nls(fmla,df)) When it works it gives next result xmid=1.0938 which is away from true inflcetion (0.6883008876) and again rsults seem to be 'excellent':
# Nonlinear regression model # model: y ~ SSlogis(x, Asym, xmid, scal) # data: df # Asym xmid scal # 0.8018 1.0938 0.4318 # residual sum-of-squares: 1.884 # # Number of iterations to convergence: 0 # Achieved convergence tolerance: 1.768e-07 ### # est=try(nls(fmla,df)) # coef(est) # Asym xmid scal # 0.8018405 1.0937758 0.4318175 # summary(est) # # Formula: y ~ SSlogis(x, Asym, xmid, scal) # # Parameters: # Estimate Std. Error t value Pr(>|t|) # Asym 0.801840 0.001175 682.7 <2e-16 *** # xmid 1.093776 0.002417 452.5 <2e-16 *** # scal 0.431817 0.002063 209.3 <2e-16 *** # --- # Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 # # Residual standard error: 0.02507 on 2998 degrees of freedom # # Number of iterations to convergence: 0 # Achieved convergence tolerance: 1.768e-07 Third task
We set as initial parameter (xmid) the true solution (500), but it failed to proceed:
# f=function(x){500+500*tanh(x-500)} # x=seq(0,1000,0.001) # y=f(x) # length(x) # df=data.frame("x"=x,"y"=y) # Asym=1000;xmid=500;scal=1; # fmla=as.formula("y~SSlogis(x,Asym,xmid,scal)");fmla # # t1=Sys.time(); # try(nls(fmla,df)) # Error in nls(y ~ 1/(1 + exp((xmid - x)/scal)), data = xy, start = list(xmid = aux[[1L]], : # step factor 0.000488281 reduced below 'minFactor' of 0.000976562 # t2=Sys.time();t2-t1 # Time difference of 19.29784 secs So R package inflection can give you an answer for all extremely difficult cases.
References
[1] Demetris T. Christopoulos (2014), Developing methods for identifying the inflection point of a convex/concave curve. arXiv:1206.5478v2 [math.NA]. URL: http://arxiv.org/abs/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] Bardsley, W. G. & Childs, R. E. Sigmoid curves, non-linear double-reciprocal plots and allosterism. Biochemical Journal, Portland Press Limited, 1975, 149, 313-328
how to find inflection points
Source: https://cran.r-project.org/web/packages/inflection/vignettes/inflectionMissionImpossible.html
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