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Convert observed time series to state vectors

`XR = phaseSpaceReconstruction(X,lag,dim)`

`[XR,eLag,eDim] = phaseSpaceReconstruction(X)`

`[XR,eLag,eDim] = phaseSpaceReconstruction(X,lag)`

`[XR,eLag,eDim] = phaseSpaceReconstruction(X,[],dim)`

`[___] = phaseSpaceReconstruction(___,Name,Value)`

`phaseSpaceReconstruction(___)`

returns the reconstructed phase space `XR`

= phaseSpaceReconstruction(`X`

,`lag`

,`dim`

)`XR`

of the uniformly
sampled time-domain signal `X`

with time delay
`lag`

and embedding dimension `dim`

as
inputs.

Use `phaseSpaceReconstruction`

to verify the system order and
reconstruct all dynamic system variables, while preserving system properties.
Reconstructing the phase space is useful when limited data is available, or when
the phase space dimension and lag is unknown. The nonlinear features `approximateEntropy`

, `correlationDimension`

, and `lyapunovExponent`

use `phaseSpaceReconstruction`

as the first step of the computation.

`[___] = phaseSpaceReconstruction(___,`

returns the reconstructed phase space `Name,Value`

)`XR`

with additional
options specified by one or more `Name,Value`

pair
arguments.

`phaseSpaceReconstruction(___)`

with no output
arguments creates a matrix of sub-axes of the reconstructed phase space with
histogram plots along the diagonal.

**Phase Space Reconstruction**

For a uniformly sampled univariate time signal $${X}_{1}={\left({x}_{1,1},{x}_{1,2},\mathrm{...},{x}_{1,N}\right)}^{T}$$, `phaseSpaceReconstruction`

computes the delayed
reconstruction

$${X}_{1,i}^{r}=\left({x}_{1,i},{x}_{1,i+{\tau}_{1}},\mathrm{...},{x}_{1,i+\left({m}_{1}-1\right){\tau}_{1}}\right),\text{}i=1,2,\mathrm{...},N-\left({m}_{1}-1\right){\tau}_{1}$$

where, *N* is the length of the time series, *τ _{1}* is the lag, and

Similarly, for a multivariate time series `X`

given by,

$$X=\left[{X}_{1},{X}_{2},\mathrm{...},{X}_{S}\right]=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{l}{x}_{1,1}\\ \vdots \\ {x}_{1,N}\end{array}& \begin{array}{l}\dots \\ \ddots \\ \cdots \end{array}\end{array}& \begin{array}{l}{x}_{S,1}\\ \vdots \\ {x}_{S,N}\end{array}\end{array}\right]$$

`phaseSpaceReconstruction`

computes the
reconstruction for each time series as,

$${X}_{i}^{r}=\left({X}_{1,i}^{r},{X}_{2,i}^{r},\mathrm{...},{X}_{S,i}^{r}\right),\text{}i=1,2,\mathrm{...},N-\left(\mathrm{max}\left\{{m}_{i}\right\}-1\right)\left(\mathrm{max}\left\{{\tau}_{i}\right\}\right)$$

where `S`

is the number of measurements, and
`N`

is the length of the time series.

**Delay Estimation**

The delay for phase space reconstruction is estimated using Average Mutual Information (AMI). For reconstruction, the time delay is set to be the first local minimum of AMI.

Average Mutual Information is computed as,

$$AMI\left(T\right)={\displaystyle \sum _{i=1}^{N}p\left({x}_{i},{x}_{i+T}\right){\mathrm{log}}_{2}\left[\frac{p\left({x}_{i},{x}_{i+T}\right)}{p\left({x}_{i}\right)p\left({x}_{i+T}\right)}\right]}$$

where, *N* is the length of the time series and *Τ* = 1:`MaxLag`

.

**Embedding Dimension Estimation**

The embedding dimension for phase space reconstruction is estimated using False Nearest Neighbor (FNN) algorithm.

For a point

*i*at dimension*d*, the points*X*and its nearest point^{r}_{i}*X*in the reconstructed phase space {^{r*}_{i}*X*},^{r}_{i}*i = 1:N*, are false neighbors if$$\sqrt{\frac{{R}_{i}^{2}\left(d+1\right)-{R}_{i}^{2}\left(d\right)}{{R}_{i}^{2}\left(d\right)}}>DistanceThreshold$$

where, $${R}_{i}^{2}\left(d\right)={\Vert {X}_{i}^{r}-{X}_{i}^{r*}\Vert}^{2}$$ is the distance metric.

The estimated embedding dimension

`d`

is the smallest value that satisfies the condition*p*<_{fnn}`PercentFalseNeighbors`

where,*p*is the ratio of FNN points to total number of points in the reconstructed phase space._{fnn}

[1] Rhodes, Carl & Morari,
Manfred. "False Nearest Neighbors Algorithm and Noise Corrupted Time Series."
*Physical Review. E*. 55.10.1103/PhysRevE.55.6162.

[2] Kliková, B., and Aleš Raidl.
"Reconstruction of phase space of dynamical systems using method of time delay."
*Proceedings of the 20th Annual Conference of Doctoral Students*
WDS 2011.

[3] I. Vlachos, D. Kugiumtzis,
"State Space Reconstruction for Multivariate Time Series Prediction", *Nonlinear Phenomena in Complex Systems*, Vol 11, No 2, pp
241-249, 2008.

[4] Kantz, H., and Schreiber, T.
*Nonlinear Time Series Analysis*. Cambridge:
Cambridge University Press, Vol. 7, 2004.

`approximateEntropy`

| `correlationDimension`

| `lyapunovExponent`