# edeci: An improved version of EDE that provides us with a Chebyshev... In inflection: Finds the Inflection Point of a Curve

## Description

It computes except from the common EDE output the Chebyshev confidence interval based on Chebyshev inequality.

## Usage

 `1` ```edeci(x, y, index, k = 5) ```

## Arguments

 `x` The numeric vector of x-abscissas, must be of length at least 4. `y` The numeric vector of the noisy or not y-ordinates, must be of length at least 4. `index` If data is convex/concave then index=0 If data is concave/convex then index=1 `k` According to Chebyshev's inequality we find a relevant Chebyshev confidence interval of the form [mu-k*sigma,mu+k*sigma]

## Details

We define as Chebyshev confidence interval the

[mu-k*sigma,mu+k*sigma]

where usually k=5 because it corresponds to 96%, while an estimator of sigma is given by s_{D}, see Eq. (29) of :

1/n Sum(((y[i]-y[i-1])/2)^2,i=1..n)

## Value

A one row matrix with elements the output of EDE, the given k and the Chebyshev c.i.

## Note

This function works better if the noise is of a "zig-zag"" pattern, see .

## Author(s)

Demetris T. Christopoulos

## References

Demetris T. Christopoulos (2016). On the efficient identification of an inflection point.International Journal of Mathematics and Scientific Computing, (ISSN: 2231-5330), vol. 6(1). https://veltech.edu.in/wp-content/uploads/2016/04/Paper-04-2016.pdf

## See Also

See also the simple `ede` and iterative version `bede`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```# #Gompertz model with noise, unequal spaces #and 1 million cases: N=10^6+1; set.seed(2017-05-11);x=sort(runif(N,0,10));y=10*exp(-exp(5)*exp(-x))+runif(N,-0.05,0.05); #EDE one time only ede(x,y,0) # j1 j2 chi # EDE 372064 720616 5.465584 #Not so close to the exact point #Let's reduce the size using BEDE iters=bede(x,y,0)\$iters;iters; # n a b EDE # 1 1000001 2.273591e-05 9.999994 5.465584 # 2 348553 4.237734e+00 6.017385 5.127559 # 3 177899 4.573986e+00 5.499655 5.036821 #Now we choose last interval, in order for EDE to be applicable on next run ab=apply(iters[dim(iters)-1,c('a','b')],2,function(xx,x){which(x==xx)},x);ab; # a b # 423480 601378 #Apply edeci... edeci(x[ab:ab],y[ab:ab],0) # j1 j2 chi k chi-5*s chi+5*s # EDE 33355 126329 5.036821 5 4.892156 5.181485 #Very close to the true inflection point. # ```

inflection documentation built on June 28, 2019, 5:03 p.m.