Computational Intelligence - August 2015 - 17

2) Experimental Comparisons with Different percentages
of missing Entries
Furthermore, we investigate the performance of the various
algorithms on different percentages ( p) of missing entries. In
order to clearly observe the performance differences of the various methods, Fig. 9 shows the logarithm values of mean estimation errors of six methods when p varies from 10% to 50% for
13 sequences. Note that if p is larger than 50%, the algorithms
LF, WLS and LF-WLS cannot work any longer for some
sequences. Generally speaking, CSF-SWE-LP and CSF-LP
achieve the best and the second-best performances, respectively.
Although the estimation errors of WLS and LF-WLS are the
smallest for the sequences drink, face2 and pickup, they are very
close to those of CSF-SWE-LP. Moreover, the estimation errors
of WLS and LF-WLS are obviously larger than those of CSFSWE-LP and CSF-LP for the other sequences. Furthermore, it
can be seen that the estimation errors of CSF-LP are smaller
than those of CSF. Thus, this again verifies that the strategy of the
LP weighting model is effective for missing-data estimation. In
addition, the estimation errors of CSF-SWE-LP are lower than
those of CSF-LP. Therefore, the strategy of selecting weaker estimators can further improve the performance of CSF-LP.
3) Experimental Comparisons on the FErEt Database
For the FERET database, as there are many distinct individuals,
and the face images of each individual constitute one sequence,
we cannot present the experimental results of all the sequences.
In the experiments, the face images of the first 10 subjects of
the database are used to evaluate the performances of the various algorithms.
Table 5 shows the estimation errors of the six methods for
the first 10 subjects of the FERET database when the percentage of missing entries is set at 30%. It can be seen that
the estimation errors of WLS are less than those of LF except
Table 4 The mean ( n ), standard deviation ( v ), and their ratio
^ v/n h of the estimation errors of six methods when the starting
frames of the small-size sequences are randomly selected ten
times from the original sequence cubes.

LF
WLS
LF-WLS
CSF
CSF-LP
CSF-SWE-LP

0.2078
21.4961
0.1596
0.0884
0.0817
0.0790

0.0354
4.9698
0.0381
0.0105
0.0084
0.0072

v/n
0.1705
0.2312
0.2388
0.1188
0.1027
0.0909

LF-WLS

CSF-LP

WLS

CSF

CSF-SWE-LP

3
Mean Estimation Errors (Log)

( n ), standard deviation ( v ), and their ratio ( v/n ) of the estimation errors for these ten trials. We can see that both the mean
and the standard deviation of CSF-SWE-LP are the lowest
among the six methods. Therefore, the proposed CSF-SWELP method not only has the highest estimation accuracy but
also has the highest robustness. Moreover, compared to other
algorithms, CSF-LP has the second-best performance in terms
of the estimation accuracy and the robustness.

2
1
0
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6 7 8 9 10 11 12 13
Sequence Number

Figure 9 The logarithm values of mean estimation errors of six
methods when the percentage of missing entries varies from 10% to
50% for 13 sequences.

for Subjects 9 and 10. This indicates again that WLS indeed
has a relatively good estimation performance, but it sometimes depends on the initialization. Nevertheless, we can note
that the estimation errors of LF-WLS are far lower than those
of WLS for Subjects 9 and 10. Thus, LF can provide a good
initialization for WLS. In conclusion, LF-WLS is a relatively
more accurate and robust missing-data estimation approach
compared to LF and WLS. As the face image sequences
include non-rigid deformation caused by the expression variation, we can see that the estimation performance of CSF is
obviously better than those of LF, WLS, and LF-WLS.
From Table 5, it can be seen that the estimation errors of
CSF-LP are lower than those of CSF. This indicates that the
LP weighting model can effectively reduce the estimation
errors of CSF for the FERET database. Moreover, we can
see that the estimation errors of CSF-SWE-LP are lower
than those of CSF-LP. This demonstrates that the strategy of

Table 5 The estimation errors of six methods for the first ten
subjects of the fereT database when the percentage of missing
entries is set at 30%.
SubjecTS lF

WlS

lF-WlS cSF

cSF-lP cSF-SWe-lP

1
2
3
4
5
6
7
8
9
10

175.41
164.68
182.33
190.10
187.92
176.86
184.96
177.57
860.75
960.27

175.41
164.68
182.33
190.10
187.92
176.86
184.96
177.57
193.30
173.51

37.35
37.48
23.88
20.88
26.76
19.54
11.75
29.50
24.38
24.35

258.08
328.55
283.81
278.85
311.01
190.06
193.28
325.39
230.40
207.15

42.86
44.32
27.69
25.10
31.35
25.06
15.78
33.61
27.61
26.77

35.09
35.28
22.88
19.47
25.63
17.40
10.42
28.11
23.21
23.46

august 2015 | IEEE ComputatIonal IntEllIgEnCE magazInE

Table of Contents for the Digital Edition of Computational Intelligence - August 2015