We here present additional results used for deciding which model parameters (delay time τ for time delay embedding and number of neighbours p used to estimate derivatives for the discrete Legendre polynomials) and which type of sampling (non-uniform, uniform using linear/ cubic spline/ moving Taylor Bayesian regression (MoTaBaR) interpolation) performs best for each approach. As for all those scenarios, we vary the embedding dimension, the noise level, the shape of the non-uniform sampling interval distribution, and additionally consider the network transitivity and the average shortest path length, we only show selected results.
Time delay embedding
For time delay embedding, we first have to choose the delay time τ. Figure A.1 exemplarily compares the relative differences to the reference network transitivity for time delay embedding with linear and cubic spline interpolation when varying the delay times for embedding dimension m = 3, standard deviation of the noise σ = 1.0, and shape of the sampling interval distribution k = 1.0. We choose delay times τ = 2 ⟨dt⟩ for the Lorenz and τ = ⟨dt⟩ for the Rössler system which is on the one hand in accordance with the first root of the autocorrelation function and on the other hand performs reasonably well with respect to the reference solutions for all dimensions and measures.
After having chosen the delay time, we have to decide whether to use the linearly or the cubic spline interpolated data. Figures A.2 and A.3 show the performance of the two interpolation methods when varying the noise level and the shape of the sampling distribution for the Lorenz and the Rössler system, respectively. We see that depending on the system and the parameters, the approaches perform differently well.
This is also the case when taking into account the results for the other dimensions and the average shortest path length. We decide to use the linear interpolation to compare time delay embedding to derivative embedding because this is the most
A.2. Parameter choices for the phase space reconstruction appraoches
Figure A.1.: Mean (points) and standard deviation (errorbars) of the relative difference to the reference transitivity for the Lorenz (left) and the Rössler (right) system using time delay embedding for varying delay times τ and embedding dimension m = 3. The dark violet line corresponds to cubic spline interpolation and the light violet line to linear interpolation. The standard deviation of the noise is σ = 1.0and the shape parameter of the sampling distribution is k = 1.0.
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 σ of noise
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 shape k
Figure A.2.: Mean (points) and standard deviation (errorbars) of the relative difference to the reference transitivity for the Lorenz system using time delay embedding for delay time τ = 2 ⟨dt⟩ and embedding dimension m = 3. The dark violet line corresponds to cubic spline interpolation and the light violet line to linear interpolation.
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 σ of noise
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 shape k
Figure A.3.: Same as in figure A.2 but for the Rössler system with delay time τ = ⟨dt⟩.
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 shape k
Figure A.4.: Mean (points) and standard deviation (errorbars) of the relative difference to the reference transitivity for the Lorenz system using derivative embedding with central differences and embedding dimension m = 3. The dark cyan line corresponds to non-uniform sampling, the medium cyan line to linear interpolation, and the light cyan line to cubic spline interpolation.
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 σ of noise
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 shape k
Figure A.5.: Same as in figure A.4 but for the Rössler system.
commonly used approach in applications. But we note that in some cases, using cubic spline interpolation instead of linear interpolation can be of advantage.
Derivative embedding using central differences
For derivative embedding with central differences, we do not have to choose any model parameters, instead, we have to choose whether to use the results obtained directly from the non-uniformly sampled data or from uniformly sampled data obtained by linear or cubic spline interpolation. As the results are generally much better when scaling the reconstructed coordinates to unit variance, we here only show the corresponding results.
Figures A.4 and A.5 show the performance of the different samplings for the network transitivity when varying the standard deviation of the noise σ and the shape parameter k of the sampling distribution for the two model systems. As expected, the results for the non-uniformly sampled data show a particular dependence on the noise level and the shape of the sampling distribution. We observe that cubic spline interpolation performs best in almost all cases and thus, use this interpolation
A.2. Parameter choices for the phase space reconstruction appraoches
Figure A.6.: Mean (points) and standard deviation (errorbars) of the relative difference to the reference transitivity for the Lorenz (left) and the Rössler (right) system using derivative embedding with discrete Legendre polynomials for varying values of p and embedding dimension m = 3. The dark cyan line corresponds to linear interpolation, the medium cyan line to cubic spline interpolation, and the light cyan line to non-uniform sampling. The standard deviation of the noise is σ = 1.0and the shape parameter of the sampling distribution is k = 1.0.
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 σ of noise
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 shape k
Figure A.7.: Mean (points) and standard deviation (errorbars) of the relative difference to the reference transitivity for the Lorenz system using derivative embedding with discrete Legendre polynomials for p = 4 and embedding dimension m = 3.
The dark cyan line corresponds to linear interpolation, the medium cyan line to cubic spline interpolation, and the light cyan line to non-uniform sampling.
method for comparing derivative embedding with central differences to the other phase space reconstruction approaches.
Derivative embedding using discrete Legendre polynomials
For derivative embedding with discrete Legendre polynomials, we first note that scaling the coordinates to unit variance is of advantage for the Lorenz and of disadvantage for the Rössler system, such that we use the scaling only for the Lorenz system. We then have to choose the value of p, that is, the number of neighbours to each side that is taken into account for estimating the derivatives. Figure A.6 shows the performance of the network transitivity for varying values of p for embedding dimension m = 3, noise standard deviation σ = 1.0, and shape parameter k = 1.0. For the Lorenz system, the performance is best for small values of p, while for the Rössler system
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 shape k
Figure A.8.: Same as in figure A.7 but for the Rössler system.
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 σ of noise
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 shape k
Figure A.9.: Mean (points) and standard deviation (errorbars) of the relative difference to the reference transitivity for the Lorenz system using derivative embedding with MoTaBaR and embedding dimension m = 3. The grey line corresponds to non-uniform sampling, the dark blue line to cubic spline interpolation, the medium blue line to the internal MoTaBaR interpolation, and the light blue line to linear interpolation.
the difference to the reference transitivity of the interpolated solutions has minima at p = 4 and p = 9. When considering all results, we choose to use the value of p dependent on the embedding dimension as for higher derivatives, more points should be taken into account. Here, we choose p = 4 of dimension m = 3 and p = 6 for dimensions m = 4 and m = 5, but in general, we think that the choice of p ≈ m + 1 is well justified by the results.
Next, we have a look at the performance of the non-uniformly sampled reconstruc-tions and the interpolated reconstrucreconstruc-tions. Figures A.7 and A.8 show the difference to the reference transitivity depending on the noise level and the sampling distribution of the time intervals between observations. We see that the results for non-uniform sampling show small values of ∆T for the Lorenz but large values for the Rössler system. The two interpolated solutions behave very similarly but the cubic spline interpolation mostly outperforms the linear interpolation. Thus, for the comparison between the approaches, we use the results obtained with cubic spline interpolation.
A.2. Parameter choices for the phase space reconstruction appraoches
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 σ of noise
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
∆T
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∆T
Figure A.10.: Same as in figure A.9 but for the Rössler system.
Derivative embedding using MoTaBaR
For derivative embedding with MoTaBaR, we also note that scaling the coordinates to unit variance generally improves the results. To compare the results for the non-uniformly sampled case to the differently interpolated cases, figures A.9 and A.10 show the performance of the network transitivity for varying standard deviation of the noise and varying shape parameter of the gamma distribution for all those cases.
We see that for the Lorenz system, cubic interpolation performs best, while for the Rössler system, the internal interpolation routine of MoTaBaR performs best. Taking into account the other dimensions and the results for the average path length, we decide to use the internal MoTaBaR interpolation routine to compare the results of the MoTaBaR phase space reconstruction to the other reconstruction approaches.