(p = 50, h = 5) (p = 50, h **= 8)**

![0_image_1.png](0_image_1.png)

(p = 100, h = 5) (p = 100, h **= 8)**
The graphs in Fig. 3 and Fig. 4 show the scatter-plots of the elements of the estimated and generating adjacency matrix B **(for randomly generated 10 datasets** in the respective case). GroupLiNGAM (Alg. 1) and L-GroupLiNGAM (Alg. 2) were respectively applied to relatively small and large sized graphs (p=5, **10 (Fig. 3)**
and p=50, 100 (Fig. 4)). The parameters δ and kneig were set as 1.0 × 10−2 and 0.05 × n, respectively. For Alg. 2, the number of subsets in a covering, **i.e.**, N, was set as 50 in the experiment. Although the estimation seems to fail sometimes depending on dimensionality p, the number of samples n **or coverage cardinality** h, the estimation seems to work reasonably well.

Next, for the purpose (ii), we created datasets as in the same manner with the procedure described in [11], which is same with the above procedure except that each block contains only one element. As described above, GroupLiNGAM is specialized to the DAG case by setting δ as +∞ **and thus, in this experiment,**
δ was set as 1 × 106**for Alg. 1. The graphs in Fig. 5 show the medians of the**
numbers of errors, i.e.**, the numbers of elements in the strictly upper triangular** part when the true connection strength matrix B **is permuted according to the** estimated orders by the algorithms. The median errors by the algorithms are similar for almost experimental conditions and, hence, we can say that the estimation of GroupLiNGAM works reasonably well in the DAG case too. Here, we should note again that GroupLiNGAM can be applied not only to DAGs but also to chain graphs while the existing algorithms (ICA-LiNGAM and DirectLiNGAM) cannot.

6. Application to real data To evaluate the applicability of the proposed algorithm (GroupLiNGAM), we analyzed a dataset taken from a sociological data repository on the Internet called 10

![0_image_0.png](0_image_0.png)

