edeci {inflection}

An improved version of EDE that provides us with a Chebyshev confidence interval for inflection point

Description

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

Usage

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 [2]:

s_{D}^2=\frac{1}{2}\,s^2=\frac{1}{2}\,\sum\limits_{i=1}^{n}{(\frac{y_{i}-y_{i-1}}{2})^2}

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 [1].

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

#
#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.
#