MEDIO FÍSICO

4-4

Weighted-SKF and Time-Varying Support

So far, the performance of the KF-LASSO and the Weighted-SKF has been investigated for datasets with a constant support. It was shown that the Weighted-SKF outperforms the other methods in estimatingxtand in estimating the supportSt. Moreover, forN =

1 it was shown that the Weighted-SKF is the only method to give accurate estimates. This section explores the performance of the Weighted-SKF for time-varying support andN = 1. Section 4-4-1 introduces the dataset that is used for this simulation and the goal of Section 4-4-2 is to illustrate the tracking abilities of the Weighted-SKF.

4-4-1 Dataset

A dataset is generated with a sparse, time-varying coefficient vector xt. Furthermore,

P > N ≯S and the support is time-varying,St. Again, model (2-12) is used to generate

the data and the parameters used to generate the dataset are listed in the box below.

Dataset Parameters St= ( {1,2} ift≤40 {1,2,3} ift >40 N = 1, P = 25, T = 100 σ_sys2 = 0.01, σ2= 0.01, Ct=βI,β = 0.99

Addition to the support set is thus simulated at ta = 41 and the resulting nonzero

coefficients are displayed in Figure 4-6.

0 10 20 30 40 50 60 70 80 90 100 −2 −1 0 1 2 time xt,1 xt,2 xt,3

Figure 4-6: Generated nonzero coefficients inxtwith time-varying support.

The tuning parameters of the Weighted-SKF are set to the same values as in the previous section. The complete set of parameter settings is listed in the box below. The simulation results with this dataset and with these parameter settings are discussed in Section 4-4-2.

52 Simulation Results

Tuning Parameters

σ2_sys= 0.01, σ2 = 0.01, β = 0.99

α= 8, λ= 25, µ= 0.1, W = 10, Wd= 10

4-4-2 Discussion Simulation Results

This section discusses the performance of tracking a time-varying support with the Weighted-SKF. The results are evaluated for (i) the Weighted-SKF of Algorithm 3, (ii) the Weighted-SKF of Algorithm 3 with a switching window length and (iii) the Weighted-SKF of Algorithm 4.

Section 3-5-2 discussed that the speed of tracking a change in the support with the Weighted-SKF of Algorithm 3 is probably limited. The red dashed-dotted line in Figure 4-7a confirms that this is true. Since the window length is kept long, it takes until

t=ta+ 7 until the correct support has been found.

If the window length in Algorithm 3 is switched toW = 1, whenever IEN ≥ α, it was argued that coefficients are added more rapidly to the support. It was also discussed that when W = 1, too many coefficients can be added to the support since the zero coefficients are noisy. This is illustrated with the magenta dotted line in Figure 4-7a. The correct support is found att=ta+ 6. Therefore, the Weighted-SKF of Algorithm

4 was proposed.

To prevent too many wrongly added coefficients, only one coefficient is added toCt at

a time and eventually the coefficient that yields minimal prediction error is definitively added. The results confirm that the proposal of Algorithm 4 yields better performance for a time-varying support. It prevents addition of too many coefficients to the support and it finds the correct support at t = ta. Moreover, Figure 4-7b shows the added

coefficient (as a positive value for illustrative purposes) with its estimates. It is shown that, when both methods add the correct coefficient, the Weighted-SKF of Algorithm 4 is faster. This is thanks to the switching window length.

Furthermore, Figure 4-8 shows the tracking of support changes by the Weighted-SKF of Algorithm 4 for a longer time period, including deletions. It is observed that additions to the support are tracked rapidly, since this is based on the IEN and the one-step-ahead prediction error. Detection of deletions is somewhat slower, since a weight is only set to w = 1 when the KF estimates indicate that the mean of the coefficient is below the threshold for W time steps. The speed of the Weighted-SKF thus depends on the KF estimates via the weights in (3-24). For large changes, the KF - and thereby the Weighted-SKF - sometimes needs a few iterations to converge to the new value and therefore it cannot be expected that the correct support is always found immediately. However, the overall performance of the Weighted-SKF is satisfying as illustrated in Table 4-1. This table shows the overall performance of detecting additions for 30 runs with the Weighted-SKF of Algorithm 4. It is shown that a change in the support is detected with the IEN within 1 time step delay in 80% of the time. Moreover, the correct support is in found with a delay of at most 1 time step in 70% of the simulations.

4-4 Weighted-SKF and Time-Varying Support 53 0 20 40 60 80 100 0 2 4 6 8 10 time IEN St Alg3 St Alg3, W = 1 S_t Alg4

(a) Magnitude of the estimated support with the Weighted-SKF of Algorithm 3 and 4, together with the IEN. When IEN≥α, Algorithm 4 rapidly finds the new support.

0 20 40 60 80 100 0 0.5 1 1.5 time xt ˆ xW SKF_t Alg3 ˆ xW SKF_t Alg4

(b) Coefficient becomes nonzero after ta = 41. Fast tracking of the new nonzero coefficient is en- abled for the Weighted-SKF of Algorithm 4 thanks to the switching window length.

Figure 4-7: (a) IEN and the estimated size of the support by the Weighted-SKF of Algo- rithm 3 and 4 (b) Tracking of a coefficient added to the support.

0 50 100 150 200 250 300 0 2 4 6 8 10 time St True S_t Alg 4

Figure 4-8: Additions to and deletions from the support, tracked by the Weighted-SKF. Detection of additions via the IEN is fast. Deletion is somewhat slower since the mean of the estimate needs to be below the threshold forW time steps.

Table 4-1: Speed of detecting a change in the support with the IEN and finding the correct support with Weighted-SKF of Algorithm 4 for 30 runs.

t= ta ta+ 1 ta+ 2 ta+ 3 ta+ 4 ta+ 5 ta+ 6

IEN≥α 63.3% 16.7% 13.3% - 3.3% 3.3% - CorrectSt 46.7% 23.3% 10% 6.7% 6.7% 3.3% 3.3%

54 Simulation Results