Estimate the acceptable minimum embedding dimension

Estimate the acceptable minimum embedding dimension.

`est_dimension_w_fnn`

dev

Estimate the acceptable minimum embedding dimension using False Nearest Neighbors Algorithm [Kennel_1992].

Select $xk$ appropriately from the embedded data with dimension $d$ , find the closest tangent vector $xn$ , and set the distance to $Rkn$ . It is embedded from 1 to D_max. Calculate the closest distance for all data. The points that satisfy the following two conditions are counted. $Atol$ (threshold_A) and $Rtol$ (threshold_R) are thresholds.

$\Biggl( \frac{R_{d+1}^2(k, n)-R_d^2(k, n)}{R_d^2(k, n)} \Biggr)^\frac{1}{2}>R_{tol}$

$\frac{R_{d+1}(k, n)}{R_A}>A_{tol}$

$RA$ is the standard deviation obtained from the time series.
Let the acceptable minimum embedding dimension be the first dimension below 1% (threshold_percent) in the percentages obtained for each embedded dimension.

By default, the actual calculation result is displayed. The point actually adopted is red. If you want to turn off the display, give plot=False.

It is necessary to set the threshold appropriately, and it is difficult to adapt to unknown data. Different value of $Rtol$ and $Atol$ may lead to different result (dimension). AFF Algorithm (est_dimension_w_afn) is devised.

from hundun.exploration import est_dimension_w_fnn

Parameters

u_seq
T
D_max=10
threshold_R=10
threshold_A=2
threshold_percent=1
plot=True
path_save_plot=None

Returns

dimension
percentages: numpy.ndarray

Example

Estimate the acceptable minimum embedding dimension for x in the Lorenz time series.

from hundun.equations import Lorenz
from hundun.exploration import est_dimension_w_fnn


x_seq = Lorenz.get_u_seq(25000)[::5, 0]
D, percentages = est_dimension_w_fnn(x_seq, T=10)
print(D)

img:est_dimension_w_fnn

Reference

Determining embedding dimension for phase-space reconstruction using a geometrical construction

(1992) Matthew B. Kennel, Reggie Brown, and Henry D. I. Abarbanel
DOI: 10.1103/PhysRevA.45.3403

The analysis of observed chaotic data in physical systems

(1993) Abarbanel, Henry D. I. and Brown, Reggie and Sidorowich, John J. and Tsimring, Lev Sh
DOI: 10.1103/RevModPhys.65.1331

`est_dimension_w_afn`

dev

Estimate the acceptable minimum embedding dimension using Averaged False Neighbors Algorithm [Cao_1997].

Calculate the distance $Rkn$ in the same way as FNN. However, it uses maximum norm instead of Euclidean distance.
Estimate the acceptable minimum embedding dimensions with reference to E1 and E2 obtained by calculation.
The first value at which E1 and E2 each exceed the threshold (threshold_E1=0.9, threshold_E2=1) is retrieved as the dimension.

from hundun.exploration import est_dimension_w_afn

Parameters

u_seq
T
D_max=10
threshold_E1=0.9
threshold_E2=1
plot=True
path_save_plot=None

Returns

dimension_list: List[int]
E_list: List[numpy.ndarray]

Example

from hundun.equations import Lorenz
from hundun.exploration import est_dimension_w_afn
import numpy as np


x_seq = Lorenz.get_u_seq(25000)[::5, 0]
D_list, E_list = est_dimension_w_afn(x_seq, T=1)
print(D_list)

[4, 3]

img:est_dimension_w_afn

Reference

Practical method for determining the minimum embedding dimension of a scalar time series

(1997) Liangyue Cao
DOI: 10.1016/S0167-2789(97)00118-8

`est_dimension_w_wayland`

dev

Estimate the dimension based on Wayland Algorithm Wayland_1993.

Find $K$ (K) closest tangent vectors for $xt0$ at a given time $t0$ . These vectors are represented as $xti$ $irange$ . Using the vector after the $tau$ (tau) step elapses, $v$ is defined as follows.

$\mathbf{v}(t_i)=\mathbf{x}(t_i+\tau T) - \mathbf{x}(t_i)$

Calculate the variance of $v$ with the following equation.

$E_{trans}=\frac{1}{K+1}\sum_{i=0}^{K}\frac{|\mathbf{v}(t_i)-\bar{\mathbf{v}}|}{|\bar{\mathbf{v}}|}$

$\mathbf{\bar{v}}=\frac{1}{K+1}\sum_{i=0}^{K}\mathbf{v}(t_i)$

Randomly calculate $Etrans$ from M=51 $xt0$ to find the median. Do this Q=10 times.
$medianE$ is calculated in each embedded dimension and the minimum value is adopted as the dimension.

from hundun.exploration import est_dimension_w_afn

Parameters

u_seq
T
D_max=10
tau=5
K=4
Q=10
M=51
reshape=True
plot=True
path_save_plot=None

Returns

D
median_e_trans_ave

Example

from hundun.equations import Lorenz
from hundun.exploration import est_dimension_w_wayland


x_seq = Lorenz.get_u_seq(25000)[::5, 0]
D, median_e_trans_ave = est_dimension_w_wayland(x_seq, T=1)
print(D)