# Comparison of Multivariate Analysis Methods as Applied to English Speech
## 3 Results
### Factor Scores of the Three Spectral Factors  
Following the analysis of the factor loadings, the factor scores were analyzed. In Figures 2-5 below, panels (a), (b), and (c) show the distributions of combinations of the three spectral-factor scores, i.e., between the \"low & mid-high factor\" (around 300 and 2300 Hz), the \"mid-low factor\" (around 1100 Hz), and the \"high factor\" (above 3000 Hz). Panel (d) in Figures 2-5 shows the distribution in the three-dimensional configuration as viewed in a direction from above-right to below-left in panel (a). In panel (d), the horizontal axis represents the combination of the \"mid-low factor\" and the \"high factor\", by calculating \((x-y)/\sqrt{2}\), where \(x\) indicates the factor score of the \"mid-low factor\", and \(y\) the factor score of the \"high factor\", following [PERSON] et al. (2017) [8]. Figure 2: The distribution of the factor scores by origin-shifted factor analysis with cepstral liftering, following [PERSON] et al. (2016) [12]. (**a–c**) represent all possible combinations of two factors out of the three factors, and (**d**) shows a two-dimensional mapping of the three factors. The distribution of the factor scores showed a similar tendency in each of the four factor analyses (Figures 2-5). Although the cepstral liftering had some influence on the factor loadings of the second peak of the bimodal \"low & mid-high factor\", overall the factor scores showed very similar distributions in the factor space. The same can be said for the use of the origin shift. Whether the origin shift was applied or not, the factor loadings (Figure 1a-d) and the factor scores (Figures 2-5) showed very similar tendencies. As can be seen, all phonemes were distributed into three fairly distinctive areas in the three-dimensional space. On the \"mid-low factor\", the highest factor scores were obtained by vowels, then by sonorant consonants, while the lowest factor scores were obtained by obstruents. Most of the obstruents occupied a position very near to or below zero on the \"mid-low factor\". By contrast, they occupied a position above zero on the \"low & mid-high factor\", while on the \"high factor\" they even reached the highest factor scores. In the three-factor factor analysis, [PERSON] et al. (2017) [8] showed that the distribution of the three English-phoneme categories (i.e., vowels, sonorant consonants, and obstruents) constituted an L-shape in the two-dimensional factor space of the \"high factor\" and the \"mid-low factor\". That is, the distributions of the phonemes linearly extended along both axes from the origin into the positive directions. Although not as clearly pronounced and narrowly shaped as in [PERSON] et al. (2017) [8], also in the present results, a rather similar L-shaped distribution was  
Figure 3: The distribution of factor scores by normal factor analysis with cepstral liftering. (**a-c**) represent all possible combinations of two factors out of the three factors, and (**d**) shows a two-dimensional mapping of the three factors. found as in Figures 2, 3, 4 and 5a. Interestingly, the distribution of the obstruents also showed an L-shape in the two-dimensional factor space of Figures 2, 3, 4 and 5c. The factor scores of obstruents were mutually exclusive between the \"low & mid-high factor\" and the \"high factor\": If the factor score of the \"low & mid-high factor\" was higher, then the factor score of the \"high factor\" was very low, and vice versa. Generally, the L-shaped distribution of the factor scores of the obstruents across the \"low & mid-high factor\" and the \"high factor\" as analyzed here also appeared in [PERSON] et al. (2017) [PERSON] et al. (2017), but hardly in a noticeable way. In the present analyses, the L-shaped distribution appeared far more prominently. Figure 4: The distribution of factor scores by origin-shifted factor analysis without cepstral liftering. (**a–c**) represent all possible combinations of two factors out of the three factors, and (**d**) shows a two-dimensional mapping of the three factors.