How I use numerical methods to calculate roots in R-CodePudding

A machine learning model predicted probability p using input x. It is unknown how model calculates the probability.

In the example below,

We have 100 xand p values.

Can someone please show an algorithm to find all values of x for which p is 0.5.

There are two challenges

I don't know the function p = f(x). I don't wish to fit some smooth polynomial curves which will remove the noise. The noises are important.
x values are discrete. So, we need to interpolate to find the desired values of x.

library(tidyverse)

x <- c(0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2.0, 2.1,2.2,2.3,2.4,2.5,2.6,2.7,2.8,2.9,3.0,3.1,3.2,3.3,3.4,3.5,3.6,3.7,3.8,3.9,4.0,4.1, 4.2,4.3,4.4,4.5,4.6,4.7,4.8,4.9,5.0,5.1,5.2,5.3,5.4,5.5,5.6,5.7,5.8,5.9,6.0,6.1,6.2, 6.3,6.4,6.5,6.6,6.7,6.8,6.9,7.0,7.1,7.2,7.3,7.4,7.5,7.6,7.7,7.8,7.9,8.0,8.1,8.2,8.3, 8.4,8.5,8.6,8.7,8.8,8.9,9.0,9.1,9.2,9.3,9.4,9.5,9.6,9.7,9.8,9.9, 10.0)
p <- c(0.69385203,0.67153592,0.64868391,0.72205029,0.64917218,0.66818861,0.55532616,0.58631660,0.65013198,0.53695673,0.57401464,0.57812980,0.39889101,0.41922821,0.44022287,0.48610191,0.34235438,0.30877592,0.20408235,0.17221558,0.23667792,0.29237938,0.10278049,0.20981142,0.08563396,0.12080935,0.03266140,0.12362265,0.11210208,0.08364931,0.04746024,0.14754152,0.09865584,0.16588175,0.16581508,0.14036209,0.20431540,0.19971309,0.23336415,0.12444293,0.14120138,0.21566896,0.18490258,0.34261082,0.38338941,0.41828079,0.34217964,0.38137610,0.41641546,0.58767796,0.45473784,0.60015956,0.63484702,0.55080768,0.60981219,0.71217369,0.60736818,0.78073246,0.68643671,0.79230105,0.76443958,0.74410139,0.63418201,0.64126278,0.63164615,0.68326471,0.68154362,0.75890922,0.72917978,0.55839943,0.55452549,0.69419777,0.64160572,0.63205751,0.60118916,0.40162340,0.38523375,0.39309260,0.47021037,0.33391614,0.22400555,0.20929558,0.20003229,0.15848124,0.11589228,0.13326047,0.11848593,0.17024106,0.11184393,0.12506915,0.07740497,0.02548386,0.07381765,0.02610759,0.13271803,0.07034573,0.02549706,0.02503864,0.11621910,0.08636754)


tbl <- tibble(x, p)


# plot for visualization
ggplot(data = tbl,
       aes(x = x,
           y = p))   
  geom_line()   
  geom_point()   
  geom_hline(yintercept = 0.5)  
  theme_bw()   
  theme(aspect.ratio = 0.4)

The figure below shows that there are five roots.

CodePudding user response：

You can use function RootSpline1 defined in my answer: