According to Eckart and Young (1936) 's theorem, it is possible to dissolve a given observed data matrix Y into Y=U∧V′ (9) Here U′U=V′V=I and ∧ shows the square roots of the eigenvalues. And further, following Tucker (e. g., 1964), Levin (1965) derived Y=(U∧)∧-1(V∧)′=U∧-1V′ =(UT)(P′∧T)-1(VP)′ =A G B′ (10) Here, T and P are the orthogonal rotational matrices. By the least square method, it is possible to estimate G in the following manner: G=(A′A)-1A′YB(B′B)-1=A′YB (11) (cf. Tucker, 1966). G was called the core factor matrix. When X is assumed to be a factor score matrix in the conventional sense, the formula (11) may be G≈A′X (15) This shows the principle of the “points of view” procedure in multiple factor analysis (cf. Cliff, 1968). It seems usual to estimate A and B through the simultaneous factorization of YY′ and Y′Y. However, the factorization sometimes fails to assure salient estimated values of weights (or factor loadings) under the assumed number of components (or factors). The present author propose to estimate the weights through simultaneous factorization of correlational matrices based upon YY′ and Y′Y rather than YY′ and Y′Y, and to substitute the obtained weights into A and B. This is our revised “points of view” procedure in factor analysis, and seems to be a very practical one. Bradford (1965) assumed three components concerning motivation for learning in T-group. They consist of individual defensive motivation moving to individual nondefensive feelings (1 st component), group defensive motivation concerned with survival and loyality moving to stages of group maturity in which there is support and acceptance of deviancy (2 nd component), and release of driving forces of trust and desire for individual growth and to help others (3rd component). First of all, using 23 T-group data which are based on 1 item 7 point. questionaire concerning group cohesiveness (cf. The footnote a) of Table 1), we performed the factorization including the orthogonal rotation of session by session correlational matrices, and found that all the assumed components appeared beautifully in the T-group all the members of which succeeded in learning. On the other hand, in the T-group some of members of which failed to learn, the components did not always appear beautifully (cf. Table 1 & Fig. 1). Here we used the principal component analysis and the varimax rotational method. The results suggested to factorize the space of the members of the T-group, too. In other words, they suggested that our “points of view” procedure could be applicable to the analysis of learning process in T-group. We assumed two factors in the members, and rotated the components of learning process under four factors. Here we used the complete centroid factorization and the orthogonal rotational method by the present author (1965). By the core factor matrices (members by components), it will be possible to know that trainers make their judgements for T-group based on the third component of learning. This was followed from the fact that at least an interrelational value (not normalized) between the divided members and their corresponding third components in G was largest (cf. Table 2). And it may be suggested that reproduced data matrix fits
Natural sleep EEG was recorded monopolarly from the central cortex in 2 adult subjects and was analyzed into 10 frequency bands per 10 sec (Table 1). Means and standard deviations of integrated amplitudes were largest in delta bands and next largest in alpha bands. Correlation between the 2 subjects was fairly high both in mean integrated values (.927) and in standard deviations (.758) across 10 frequency bands. The raw EEG data were classified visually per 10 sec segment into 9 EEG sleep patterns according to the criteria used by Koga and Fujisawa (Fig. 1), which was made prior to processing by the mathematical and statistical methods. Digree of similarity (or dissimilarity) between the 9 sleep EEG patterns was evaluated by Spearman's rank correlation and by a difference measure (formula 1 in text) in integrated values across 10 frequency bands without considering correlations between the bands (Table 2, Fig. 2). Multivariate analyses of variance applied to the data indicated significant differences between 9 visually classified EEG patterns and Mahalanobis' generalized distance functions were calculated and their squareroots are shown in Table 3, indicating statistical differences between the EEG patterns. Inter-subject correlation in these values was again substantial (.753). Based on the correlation matrix in integrated values among 10 frequency bands (Table 4), the first three factors were extracted by Hotelling's method which explained in all about 80% of the total variance in both subjects (Table 5, Fig. 3). Factor I was highly associated with slow frequencies (less than 8cps) having little relation with alpha bands in both subjects who were different in faster frequency bands, although the correlation across the 10 frequency bands was still high (.915). Factor II was heavily loaded in alpha and faster frequency bands with almost no loadings in slower frequency bands. Factor III had a peculiar characteristic in that the 2 subjects showed the opposite profiles across the 10 bands (Fig. 3). Each 10-sec set of 10-band integrated values was transformed into a single ‘component’ score for each of the three factors (components) extracted following Hotelling's principal component analysis (formula 5) and the distributions of these component scores were made as shown in Fig. 5 and Table 6. Mean component scores of the 9 sleep EEG patterns are plotted in Fig. 6 from which the first component (factor I) was assumed to reflect the depth of behavioral sleep. The second component or factor II was associated with the waking state (EEG sleep pattern 1) and slightly negatively related with paradoxical sleep (EEG sleep pattern 3-v). It was hard to interprete the third component or factor III in relation to sleep patterns.
The aim of this paper was to test experimentally the mapping functions in the Luneburg's model. In particular, Eq. (1) and (3) were explored in detail. ψ=φ (1) ϑ=θ (2) ρ=2e-σγ (3) where ψ, ϑ and ρ respectively denote an azimuth angle, an angle of elevation, a radial distance in the Euclidean map. φ, θ and γ respectively denote a bipolar azimuth, an angle of elevation, a bipolar parallax in the physical space. Six Ss participated in the experiment. In a dark room, three light points were presented on the plane of given θ; F was fixed on the line φ=0 and I was fixed at the position of a given φ (Fig. 2). The S was asked to adjust Ij so as to match its azimuth angle to that of I. For a given I, Ij was presented with 3-6 different radial distances. If Eq. (1) holds, then the adjusted positions of Ij are to be on the same straight line of the given angle φ. In so far as 1 is fixed, Ij were found to be on the same straight line irrespective of the radial distance and also of θ. However, the straight line was not the one expected in all the case of I having different values in φ. Either a set of straight lines converging to a point on the y axis or a set of straight lines converging to a point on the x axis can be fitted to the data (Fig. 3). The S was also asked to assess the apparent radial distance to Ij in terms of ratio to the apparent radial distance to I or to F. Let dIj be the scaled radial distance. As a function of γ, dIj was found to be not in the form of Eq. (3) (Fig. 7 (A) (B)). Then, the relation between the apparent radial distance dIj and the radial distance ρ on the Euclidean map was made explicit for the various values of K, that is, Gaussian curvature of the visual space (Fig. 6). Through the relationship, dIj was converted to ρ and still it was found that ρ is not related with γ as is assumed in Eq. (3) (Fig. 7 (C) (D)). It became clear, however, the data are fitted irrespective of φ and θ by the equation ρ-ρ0=2e-σγ (4) as is shown in Fig. 8. The value of σ estimated through Eq. (4) was too large compared with the value obtained in previous studies by the 3- and 4-point experiments and the alley experiments.