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Fig. 3. Circle-segments of the stroke data Table 5. Eigenvalues of the covariance matrix of the stroke data set Principal PC1 PC2 PC3 PC4 PC5 PC6 Eigenvalue 0.33573 0.15896 0.14386 0.11533 0.09023 0.07917 Proportion 0.262 0.124 0.112 0.090 0.070 0.062 Cumulative 0.262 0.386 0.499 0.589 0.659 0.721
Principal PC7 PC8 PC9 PC10 PC11
PC12 Eigenvalue 0.07403 0.06239 0.05390 0.04268 0.03252 0.03080 Proportion 0.058 0.049 0.042 0.033 0.025 0.024 Cumulative 0.779 0.828 0.870 0.903 0.929 0.953
Principal PC13 PC14 PC15 PC16 PC17 PC18 Eigenvalue 0.02588 0.01546 0.00894 0.00584 0.00293 0.00160 Proportion 0.020 0.012 0.007 0.005 0.002 0.001 Cumulative 0.973 0.985 0.992 0.996 0.999 1.000
Three data sets are produced, i.e., one containing the original data samples, the other two containing the important input features identified by circle-segments and PCA, respectively. Three performance indicators that are commonly used in medical diagnostic systems are computed, i.e., accuracy (ratio of correct diagnoses to total number of patients), sensitivity (ratio of correct positive diagnoses to total number of patients with the disease), and specificity (ratio of correct negative diagnoses to total number of patients without the disease). 632 S.L. Wang et al. Table 6. Eigenvectors of the first six principal components of the stroke data set PC1 PC2 PC3 PC4 PC5 PC6 V1 0.098 0.054 -0.083 0.042 -0.038 0.048 V2
0.611 -0.218 0.364 0.584 -0.316 0.024 V3 0.052 0.362 -0.064 -0.262 -0.741 -0.102 V4
0.520 -0.059 -0.188 -0.252 0.333 -0.410 V5 -0.054 -0.029 0.179 -0.244 -0.300 -0.023 V6
0.569 0.128 -0.027 -0.499 0.088 0.332 V7 0.039 -0.116 -0.389 0.090 -0.163 -0.683 V8 -0.035 0.122 0.407 -0.131 0.027 -0.170 V9 0.017 0.003 0.024 -0.049 -0.050 0.008 V10 -0.011 0.015 0.064 -0.073 -0.066 0.079 V11 0.006 -0.013 -0.023 0.045 0.114 0.136 V12 0.002 -0.011 -0.023 0.019 0.027 0.026 V13 0.018 -0.046 -0.054 0.059 0.082 0.142 V14 0.002 -0.015 -0.029 0.020 0.023 0.017 V15 -0.013 0.011 0.104 -0.072 -0.010 -0.071 V16 0.065 0.064 0.616 -0.217 0.148 -0.328 V17 0.086 0.870 -0.018 0.355 0.214 -0.083 V18 0.056 0.072 -0.267 -0.013 -0.136 0.216
Table 7 summarises the average results of 10 runs for the stroke data set. In general, the classification performances improve with feature selection either using PCA or circle-segments. The circle-segments method yields the best accuracy rates for all the four machine learning systems despite the fact that only three features are used for classification (a reduction of 83% of the number of input features). In terms of sensitivity, the circle-segments method also shows improvements (except FAM) from 17%-35% as compared with those before feature selection. The specificity rates are better than those before feature selection using MLP and FAM, but inferior for SVM and kNN. Table 7. Classification results and the number of input features for the stroke data set Before feature selection (18 input features) Feature selection with PCA (8 input features) Feature selection with circle- segments (3 input features) Methods
A cc u ra cy (%) Se ns itivity (% ) Sp ec ific ity
( %)
A cc u ra cy (%) Se ns itivity (% ) Sp ec ific ity
( %)
A cc u ra cy (%) Se ns itivity (% ) Sp ec ific ity
( %)
MLP 81.79 44.29 91.70 82.39 33.57 95.28 86.57 61.79
93.11 FAM
72.67 57.14 76.79 72.39 49.29 78.49 73.81
51.07 79.81
SVM 80.60 21.43 96.23 82.09 25.00 97.17 86.57 57.14
94.34 kNN 82.09 32.14 95.28 81.42 32.14 94.43 85.45 57.14
92.92
From Table 7, it seems that there is a trade-off between sensitivity and specificity with the use of the circle-segments method, i.e., more substantial improvement in sensitivity with marginal degradation in specificity for SVM and kNN while less substantial improvement in sensitivity with marginal improvement in specificity for MLP and FAM. This observation is interesting as it is important for a medical Use of Circle-Segments as a Data Visualization Technique for Feature Selection 633 diagnostic system to have high sensitivity as well as specificity rates so that patients with and without the diseases can be identified accurately. By comparing the results of circle-segments and PCA, the accuracy and sensitivity rates of circle-segments are better than those of PCA. The trade-off is that PCA generally yields better specificity rates. Again, it is interesting to note the substantial improvement in sensitivity with marginal inferior performance in specificity of both the circle-segment and PCA results. Another observation is that the PCA results do not show substantial improvements in terms of accuracy, sensitivity, and specificity as compared with those from the original data set. 4 Summary and Further Work Feature selection in a data set based on data visualization is an interactive process involving the judgments of domain users. By identifying the patterns in the circle- segments, the important input features are distinguished from other less important ones. The circle-segments method enables the domain users to carry out the necessary filtering of input features, rather than feeding the whole data set, for learning using machine learning systems. With the circle-segments, domain users can visualize the relationships of the input-output data and comprehend the rationale on how the selection of input features is made. The results obtained from the two case studies positively demonstrate the usefulness of the circle-segments method for feature selection in pattern classification problems using four machine learning systems. Although the circle-segments method is useful, selection of the important features is very dependent on the users’ knowledge, expertise, interpretation, and judgement. As such, it would be beneficial if some objective function could be integrated with the circle-segments method to quantify the information content of the selected features with respect to the original data sets. In addition, the use of the circle-segments method may be difficult when the problem involves high dimensional data, and the users face limitation in analyzing and extracting patterns from massive data sets. It is infeasible to carry out the data analysis process when the dimension of the data is high, which cover hundreds to thousands of variables. Therefore, a better characterization method in reducing the dimensionality of the data set is needed.
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Extraction of Approximate Independent Components from Large Natural Scenes Yoshitatsu Matsuda 1 and Kazunori Yamaguchi 2 1 Department of Integrated Information Technology, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara-shi, Kanagawa, 229-8558, Japan matsuda@it.aoyama.ac.jp http://www-haradalb.it.aoyama.ac.jp/ ∼ matsuda
2 Department of General Systems Studies, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8902, Japan yamaguch@graco.c.u-tokyo.ac.jp Abstract. Linear multilayer ICA (LMICA) is an approximate algo- rithm for independent component analysis (ICA). In LMICA, approx- imate independent components are efficiently estimated by optimizing only highly-dependent pairs of signals. Recently, a new method named “recursive multidimensional scaling (recursive MDS)” has been proposed for the selection of pairs of highly-dependent signals. In recursive MDS, signals are sorted by one-dimensional MDS at first. Then, the sorted sig- nals are divided into two sections and each of them is sorted by MDS recursively. Because recursive MDS is based on adaptive PCA, it does not need the stepsize control and its global optimality is guaranteed. In this paper, the LMICA algorithm with recursive MDS is applied to large natural scenes. Then, the extracted independent components of large scenes are compared with those of small scenes in the four statistics: the positions, the orientations, the lengths, and the length to width ratios of the generated edge detectors. While there are no distinct differences in the positions and the orientations, the lengths and the length to width ratios of the components from large scenes are greater than those from small ones. In other words, longer and sharper edges are extracted from large natural scenes. 1 Introduction Independent component analysis (ICA) is a widely-used method in signal pro- cessing [1,2,3]. It solves blind source separation problems under the assumption that source signals are statistically independent of each other. Though many ef- ficient algorithms of ICA have been proposed [4,5], it nevertheless requires heavy computation for optimizing nonlinear functions. In order to avoid this problem, linear multilayer ICA (LMICA) has been recently proposed [6]. LMICA is a vari- ation of Jacobian methods, where the sources are extracted by maximizing the independency of each pair of signals [7]. The difference is that LMICA optimized M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 635–642, 2008. c Springer-Verlag Berlin Heidelberg 2008 636 Y. Matsuda and K. Yamaguchi only pairs of highly-dependent signals instead of those of all ones. LMICA is based on an intuition that optimizations on highly-dependent pairs probably in- crease the independency of all the signals more than those of low-dependent ones, and its validity was verified by numerical experiments for natural scenes. Be- sides, an additional method named “recursive multidimensional scaling (MDS)” has been proposed for improving the selection of highly-dependent signals [8]. The method is based on the repetition of the simple MDS. It sorts signals in a one-dimensional array by MDS, then divides the array into the former and lat- ter sections and sorts each of them recursively. In consequence, highly-correlated signals are brought into a neighborhood. Because the simple MDS is equivalent to PCA [9], it can be solved efficiently without the stepsize control by adaptive PCA algorithms (e.g. PAST [10]). The global optimality in adaptive PCA is guaranteed if the number of steps is sufficient. In this paper, the above LMICA algorithm with recursive MDS was applied to large natural scenes and the results were compared with those of small natural scenes in the following four statistics: the positions, the orientations, the lengths, and the length to width ratios of the generated edge detectors. As a result, it was observed that the independent components from large natural scenes are longer and sharper edge detectors than those from small ones. This paper is organized as follows. Section 2 gives a brief description of LMICA with recursive MDS. Section 3 shows the results of numerical experiments for small and large natural scenes. This paper is concluded in Sect. 4. 2 LMICA with Recursive MDS Here, LMICA with recursive MDS is described in brief. See [6] and [8] for the details.
2.1 MaxKurt Algorithm LMICA is an algorithm for extracting approximate independent components from signals. It is based on MaxKurt [7], which minimizes the contrast function of kurtoses by optimally “rotating pairs” of signals. The observed signals x = (x i ) are assumed to be prewhitened. Then, one iteration of MaxKurt is given as follows: – Pick up every pair (i, j) of signals, and find the optimal rotation ˆ θkurt given as ˆ θkurt = argmin θ − E{(x i ) 4 + (x j ) 4 }, (1) where E {} is the expectation operator, x i = cos θ
· x i + sin θ · x j , and x j = − sin θ · x i + cos θ
· x j . ˆ θkurt is determined analytically and calculated easily. E {x
i } can generalize to E {G (x
i ) } where G (u) is any suitable function such as log (cosh (u)). In this case, ˆ θkurt is initially set to θ because it is expected to give a good initial value, Extraction of Approximate Independent Components 637
then θ is incrementally updated by Newton’s method for minimizing E {G (x
i ) +
G (x j ) } w.r.t θ. Though it is much more time-consuming, it is expected to be much more robust to outliers. 2.2 Recursive MDS In MaxKurt, all the pairs are optimized. On the other hand, LMICA optimizes only highly-correlated pairs in higher-order statistics so that approximate com- ponents can be extracted quite efficiently. In order to select such pairs, LMICA forms a one-dimensional array where highly-correlated signals are near to each other by recursive MDS. Then, LMICA optimized only the nearest-neighbor pairs of signals in the array. In recursive MDS, first, a one-dimensional mapping is formed by a simple MDS method [9], where the original distance D ij between the i-th and j-th signals is defined by D ij = E { x
2 i − x 2 j 2 }. (2)
Because D ij is greater if x i and x
j are more independent of each other in higher-order statistics, highly-dependent signals are globally close together in the formed map. Such a one-dimensional map is easily transformed into a dis- crete array by sorting it. Because the simple MDS utilizes all the distances among signals instead of only neighbor relations, the nearest-neighbor pair in the formed array do not always correspond to the one with the smallest distance D ij . In order to avoid this problem, the array is divided into the former and latter parts and MDS is applied to each part recursively. If a part includes only two signals, the recursion is terminated and pair optimization is applied. If a part include three signals, the pair of the first and second signals and that of the second and third ones are optimized after MDS is applied to the three signals. The whole algorithm of RMDS(signals) is described as follows: RMDS(signals) If the number of signals (N ) is 2, optimize the signals. Otherwise, 1. Sort the signals in a one-dimensional array by the simple MDS. 2. If N is 3, optimize the pairs of the first and second signals, and then do the second and third ones. Otherwise, do RMDS(the former section of signals) and RMDS(the latter section of signals). Note that recursive MDS also does not confirm that D ij of the nearest-neighbor pair is the smallest. But, numerical experiments has verified the validity of this method in [8]. Regarding the algorithm of the simple MDS in the one-dimensional space, its solution is equivalent to the first principal component of a covariance matrix of 638 Y. Matsuda and K. Yamaguchi transformed signals z, each component of which is given as z i = x 2 i − i x 2 i N where N is the number of signals [9]. Therefore, MDS is solved efficiently by applying an adaptive PCA algorithm to a sequence of signals z. Here, well-known PAST [10] is employed. It is fast and does not need the stepsize control. PAST repeats the following procedure for T times where y = (y i ) is the coordinates of signals in the one-dimensional space: 1. Pick up a z randomly, 2. Calculate α = i y i z i and β := β + α, 3. Calculate e = (e i ) as e
i = z
i − αy
i , 4. Update y := y + α β e. The initial y is given randomly and the initial β is set to 1.0. This algorithm is guaranteed to converge to the global optimum. The value of T is the only parameter to set empirically. 3 Results 3.1 Experimental Settings Two experiments were carried out for comparing independent components of small natural scenes and those of large ones. In the first experiment, simple fast ICA [4] is applied to 10 sets of 30000 small natural scenes of 12 × 12 pixels with the symmetrical approach and G (u) = log (cosh (u)). Because no dimension reduction method was applied, the total number of extracted components is 1440 (144 ∗ 10). Second, LMICA with re- cursive MDS was applied to 100000 large images of 64 × 64 pixels for 1000 layers with G (u) = log (cosh (u)) The original images were downloaded from http://www.cis.hut.fi/projects/ica/data/images/. The large images were prewhitened by ZCA. Note that an image of 64 × 64 pixels is not regarded as “large” in general. Since they are enough large in comparison with image patches usually used in other ICA algorithms, they are referred as large images in this paper. 3.2
Experimental Results Figure 1 shows that the decreasing curve of the contrast function nearly con- verged around 1000 layers. It means that LMICA reached to an approximate optimal solution at the 1000th layer. Figure 2 displays extracted independent components in the two experiments. It shows that edge detectors were gener- ated in both cases. Regarding the other comparative experiments of efficiency between our method and other ICA algorithms, see [8]. In order to examine the statistical properties of the generated edge detectors, they are analyzed in the similar way as in [11]. First, the envelope of each detector was calculated by Hilbert transformation. Next, the dominant elements were strengthened by raising each element to the fourth power. Then, the means and
Extraction of Approximate Independent Components 639
0.314 0.316
0.318 0.32
0.322 0.324
0 200
400 600
800 1000
E[log cosh x] layers
Decrease of Contrast Function for LMICA LMICA
Fig. 1. Decreasing curves of the contrast function E (log (cosh (x i ))) along the layers by LMICA for large natural scenes (a) From small natural scenes. (b) From large natural scenes. Fig. 2. Independent components extracted from small and large natural scenes covariances on each detector was calculated by regarding the values of elements as a probability distribution on the discrete two-dimensional space ([0.5, 11.5] × [0.5, 11.5] for small scenes or [0.5, 63.5] × [0.5, 63.5] for large scenes because they exist only in the ranges). By approximating each edge as a Gaussian distribution with the same means and covariances on the continuous two-dimensional space, its position (means), its orientation (the angle of the principal axis), its length (the full width at half maximum (FWHM) along the principal one), and its width (FWHM perpendicular to the principal one) were calculated. The results are shown in Figs. 3-6 and Table 1. The scatter diagrams of the positions of edges in Fig. 3 show that they are uni- formly distributed in both cases and there are no distinct differences. Figure 4 displays the histograms of orientations of edges from 0 to π. It shows that edges with the horizontal (0 and π) and vertical (0.5π) orientations are dominant as re- ported in [11] and there are no distinct differences between small and large scenes. 640 Y. Matsuda and K. Yamaguchi Table 1. Means and medians of the lengths and the length to width ratios for small and large scenes small scenes large scenes mean of lengths 1.84 2.15
median of lengths 1.53
1.23 mean of lw ratios 1.69 2.53
median of lw ratios 1.55
1.70 0.5
6 11.5
0.5 6 11.5 y-axis x-axis
scatter diagram of places (small) (a) small scenes. 0.5 32
0.5 32 63.5 y-axis x-axis
scatter diagram of places (large) (b) large scenes. Fig. 3. Plot diagrams of the positions of edges over the two-dimensional spaces: (a). Distribution for small scenes over [0.5, 11.5] × [0.5, 11.5]. (b). Distribution for large scenes over [0.5, 63.5] × [0.5, 63.5]. 0 5 10 15 20 0 0.5
π π percentage angles of edges histgram of orientations (small) (a) small scenes. 0 5 10 15 20 0 0.5
π π percentage angles of edges histgram of orientations (large) (b) large scenes. Fig. 4. Histograms of the orientations of edges from 0 to π On the contrary, Figs. 5 and 6 show that the statistical properties of edges for large scenes obviously differ from those for small ones. In Fig. 5, short edges of 1-1.5 length are much more dominant for large scenes than for small ones. But, the rate of the long edges over 3 in length is more for large scenes than for small scenes. Table 1 also shows this strangeness, where edges for large scenes are shorter on the median and longer on the mean than those for small ones. In Fig. 6, the length to width ratios for large scenes are greater than those for
Extraction of Approximate Independent Components 641
small scenes. Table 1 also shows both mean and median of the ratios for large scenes are greater than those for small scenes. 3.3 Discussion The results show that there are significant differences in the distributions of the length and width of edges. First, a few long edges were observed for large scenes in Fig. 5. It shows that large images includes some intrinsic components of long edges and the division into small images hides them. Second, it was also observed in Fig. 5 that the rate of short edges for large images was greater than that for small ones. This seemingly strange phenomenon may be caused by the approximation of LMICA. Because LMICA optimizes only nearest neighbor pairs, it is expected to be biased in favor of locally short edges. Third, Fig. 6 shows that the length to width ratios for large natural scenes were greater than those for small ones. In other words, the edges from large scenes were “sharper” than those from small ones. The utilization of large scenes without any division 0 20 40 60 80 0 2 4 6 8 10 percentage lengths of edges histgram of lengths (small) (a) small scenes. 0 20
60 80 0 2 4 6 8 10 percentage lengths of edges histgram of lengths (large) (b) large scenes. Fig. 5. Histograms of the lengths of edges from 0 to 10. Edges longer than 10 are counted in the rightmost bar. 0 20 40 60 1 3 5 7 9 11 percentage lw ratios of edges histgram of lw ratios (small) (a) small scenes. 0 20 40 60 1 3 5 7 9 11 percentage lw ratios of edges histgram of lw ratios (large) (b) large scenes. Fig. 6. Histograms of the length to width ratios of edges from 1 to 11. Edges beyond 11 in ratio are counted in the rightmost bar.
642 Y. Matsuda and K. Yamaguchi drastically weakens the effect of constraints that edges can not exist beyond the borders. It may be the reason why many sharper edges were generated. 4 Conclusion In this paper, the method of LMICA with recursive MDS was described first. Then, the method was applied to two datasets of small natural scenes and large natural scenes and the generated edge detectors were compared in some statisti- cal properties. Consequently, it was observed in the experiment for large scenes that there are a few long edges and many edges are shaper than those generated from small scenes. We are now planning to do additional experiments for verifying the specu- lations in this paper. Besides, we are planning to compare our results with the statistical properties observed in real brains in the similar way to [11]. In addi- tion, we are planning to apply this algorithm to a movie where a sequence of large natural scenes is given as a sample. This work is supported by Grant-in-Aid for Young Scientists (KAKENHI) 19700267. References 1. Jutten, C., Herault, J.: Blind separation of sources (part I): An adaptive algorithm based on neuromimetic architecture. Signal Processing 24(1), 1–10 (1991) 2. Comon, P.: Independent component analysis - a new concept? Signal Processing 36, 287–314 (1994) 3. Bell, A.J., Sejnowski, T.J.: An information-maximization approach to blind sepa- ration and blind deconvolution. Neural Computation 7, 1129–1159 (1995) 4. Hyv¨ arinen, A.: Fast and robust fixed-point algorithms for independent component analysis. IEEE Transactions on Neural Networks 10(3), 626–634 (1999) 5. Cardoso, J.F., Laheld, B.: Equivariant adaptive source separation. IEEE Transac- tions on Signal Processing 44(12), 3017–3030 (1996) 6. Matsuda, Y., Yamaguchi, K.: Linear multilayer ICA generating hierarchical edge detectors. Neural Computation 19, 218–230 (2007) 7. Cardoso, J.F.: High-order contrasts for independent component analysis. Neural Computation 11(1), 157–192 (1999) 8. Matsuda, Y., Yamaguchi, K.: Linear multilayer ICA with recursive MDS (preprint, 2007) 9. Cox, T.F., Cox, M.A.A.: Multidimensional scaling. Chapman & Hall, London (1994) 10. Yang, B.: Projection approximation subspace tracking. IEEE Transactions on Sig- nal Processing 43(1), 95–107 (1995) 11. van Hateren, J.H., van der Schaaf, A.: Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings of the Royal Society of London: B 265, 359–366 (1998) M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 643–652, 2008. © Springer-Verlag Berlin Heidelberg 2008 Download 12.42 Mb. Do'stlaringiz bilan baham: 
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