Description Usage Arguments Details Value Note Author(s) References See Also Examples

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

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

`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 |

`k` |
According to Chebyshev's inequality we find a relevant Chebyshev confidence interval of
the form |

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 [2]:

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

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

This function works better if the noise is of a "zig-zag"" pattern, see [1].

Demetris T. Christopoulos

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 the simple `ede`

and iterative version `bede`

.

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]-1,c('a','b')],2,function(xx,x){which(x==xx)},x);ab;
# a b
# 423480 601378
#Apply edeci...
edeci(x[ab[1]:ab[2]],y[ab[1]:ab[2]],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.
#
``` |

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