LA INSTRUCCIÓN PENAL EN EL DERECHO COMPARADO

Spectral clustering will partition the domain into coherent clusters and an incoherent mixing region. Once the coherent clusters have been identified, there is no standard means of knowing how coherent they are relative to each other or relative to the background. The eigenvalues of L cannot be used to quantify the coherence of the output clusters because the k eigenvectors of L do not generally correspond to the K = k + 1 identified clusters. To determine how coherent the detected clusters are, we define a metrics for coherence: the robustness of the cluster to random noise perturbation. This is similar, in spirit, to the approach by Froyland [2013], who investigated the role of noise amplitude on coherent sets. The robustness to noise is computed as follows. For a trajectory set initially located

inside a given cluster and advected over [t0; tf], take the final positions at time tf. Estimate

the average distance between particles D and add random noise between [0; D/100] to these final positions. Advect the ‘perturbed’ positions in backward time to the initial time t0.

Then count how many perturbed particles fall within the original clusters’ boundaries. The percentage of particles that will return within the boundaries of their assigned cluster is the proposed coherence metrics.

An illustration of the proposed procedure is provided in figure 2-10 on 72. For a flow within a domain Lx-by-Ly over time interval [t0; tf], carry-on the spectral clustering protocol

by, first, sweeping through values of r while keeping the offset fixed; second, for each of the local maxima in gap ratios, verifying convergence in the offset coefficient. The example assumes K = 3 clusters identified above and below a jet-like feature and a corresponding eigengraph showing three eigenvalues before the maximum eigengap. The clusters are shown in green, pink and turquoise. The coherence metrics is computed for the three clusters and the fourth incoherent background cluster. The positions at tf are slightly perturbed, then

advected backward to t0. For each of the clusters, the percentage of particles that return

within the original boundaries extracted at t0 is computed. In this example, the coherence

values were 100% for the green cluster, 87.5% for the pink cluster, 90% for the turquoise cluster and 90% for the incoherent background. Note that the incoherent background has a coherence percentage similar to the coherent clusters: it is a direct measure of the number of particles that return within the boundaries. The purpose of the metrics is to compare the detected coherent clusters: for example, to evaluate the coherence of the vortices in the Bickley jet example. Generally, with this coherence metrics, large sets tend to be more coherent because of the large volume-to-perimeter ratio.

Figure 2-10: Illustration of the spectral clustering protocol with coherence metrics. For a toy model flow from t0 to tf, the parameter-free spectral clustering protocol sweeps through

values of r, with r < Lx and r < Ly, and through the power of offset coefficients until

convergence in the gap ratio, representative of the spectral decomposition. Here, for example, there are k = 3 eigenvalues before the eigengap, which means 3 connected components detected. For the noise metrics, the particle positions at the final position tf are perturbed

with random noise, then advected back to t0. The percentage of particles that fall back

within the clusters boundaries at t0 is the coherence metrics. This is done for all 4 clusters:

the 3 connected components (in colors) and the incoherent background (in gray).

A key component of the coherence metric calculations is finding the boundaries of the clusters. Extracting boundaries is a notoriously hard problem to implement numerically and it becomes especially arduous when clusters have filaments or lobes. The numerical resolution will typically be the limitation for the extraction of appropriate boundaries around the clusters, so outliers at the extremities of the lobes or filaments may need to be removed. Keeping in mind the purpose of the protocol, which is to carry on an objective clustering

analysis for LCS detection with as few user-input parameters as possible, boundaries must be extracted in an automatic manner without user input.

One solution to this problem of filamenting boundaries is to automatically remove these distant outliers by filtering particles that are too distant from the bulk of the cluster. To proceed, calculate the mean pairwise distances for all particles within a cluster and remove particles that are statistical outliers, such as, for example, particles that are more than 4 standard deviations above the mean. Figure 2-11 on 74 shows an example for two Bickley jet vortices and for the a cluster in the asymmetric Duffing oscillator, described in section 2.4.3. For the vortex example with r = 0.5, the area to perimeter ratio is much smaller than for r = 2.0, as the detected vortices are much smaller and the number of numerical nodes within the boundaries are much fewer. Therefore, even small changes in the number of particles will affect the results. Applying the 4σ threshold to the mean pairwise distances yields appropriate boundaries in both cases. The filaments detected with r = 2.0 are still included in the boundary. The last example is a very filamented cluster, in figure 2-11c; nevertheless, the procedure using the 4σ filtering extracted boundaries that were faithfully following the filaments.