## 4 Results and Discussion
### Pattern Extraction with NMF and Its Spatial Distribution  
In NMF estimation, a previous study [37] suggested the number of patterns; \(k\) should be chosen to observe the plot between NMF model fit (Frobenius distance between the data and NMF model) against \(k\). The \"elbow\" of the graph is \(k\), such that it is an inflection point in the graph regarding \(k\) and the NMF model fit. The NMF model in this study was repeatedly estimated in \(k\) from 1 to 12. From Figure 7a,b, the Frobenius distance constantly decreases starting from \(k\) = 3, hence the \"elbow\" point can be determined to be \(k\) = 3. However, for simplest interpretability, \(k\) in this study is determined by also referring to the similarity in both genders of corresponding cohorts. According to our observation, the patterns with \(k\) = 3 are too complex, abstract, and unique for each gender cohort, making it difficult to interpret in terms of a trend-oriented discussion. Considering the purpose of statistical analysis in this study, the \"elbow\" criteria should be discarded, and the criteria for the similarity between the corresponding male and female cohorts should be referred to. We checked all the patterns starting from \(k\) = 4 to 12 in both models, and we selected the patterns with \(k\) = 8, which almost satisfy the similarity criteria for both models, with some exceptions. Therefore, the patterns in \(k\) = 8 are adopted for the output of the NMF model and then aggregated based on the four different age groups (children, younger age, middle age, and elderly) which we specified in the data sources section. Figures 8 and 9 show the plot of the pattern matrix \(H\) of male and female cohort growth with \(k\) = 8. In Figure 8, Patterns 1 to 8 represent the characteristics of cohort transition from 2010 to 2015. Pattern 1 shows the characteristics of the 20-24 male and female cohorts; Pattern 2 shows the 15-19 female cohort characteristics; Pattern 3 shows the average cohort characteristics; and Pattern 4 represents the 25-29 female cohort characteristics. Pattern 5 shows the 55-59 male and female cohorts; Pattern 6 is about the 10-14 female cohort characteristics; Pattern 7 shows the 45-49 female cohort; and Pattern 8 represents the 50-and-over female cohort characteristics. From Figure 8, we can note that the patterns are common between males and females in Patterns 1, 3, 5 and 8, while in Patterns 2, 4, 6, and 7, it appears that the female cohorts' characteristics appear stronger than the male cohorts'. In Figure 9, Patterns 1 to 8 represent the characteristics of cohort transition from 2015 to 2020. Pattern 1 shows the average cohort characteristics; Pattern 2 shows the 5-9 male and female cohort characteristics; Pattern 3 represents the stronger appearance of the working-age male cohorts between 25 and 54, except for the 45-49 cohort and the 60-64 female cohort characteristics; and Pattern 4 shows the elderly cohort characteristics. Figure 6: Foreign Resident Newly-Born Cohort (0–4) Ratio (**a**) 2015/2010 (**b**) 2020/2015. Pattern 5 shows the 25-29 male and female cohort characteristics; Pattern 6 shows the 55-and-over male and female cohort characteristics; Pattern 7 represents the 25-to-34 male cohorts characteristics along with the appearance of the 45-49 female cohort characteristic; and Pattern 8 shows a type of parent-child household, highlighting the 5-9 and 35-54 for both male and female cohorts characteristics. According to Figure 9, common patterns in male and female cohorts are observed in Patterns 1,2, 4, 6, and 8. Patterns 3 and 7 show a stronger appearance for male working-age cohorts than for female, except in Pattern 5. Figures 10 and 11 show the shares of each pattern obtained from weight matrix \(W\). On a couple of figures, we can observe the dominant patterns are differently distributed among the regions defined in Figure 2.  
Table 2 is a recap of all the outputs of the above analysis.