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Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
The centerGroup!centre of a group G consists of those elements of G each of which commutes with all the elements of G. It is denoted by C(G). Thus, C(G)={x∈G:xa=axfor eacha∈G}
Monitoring proportions with two components of common cause variation
Published in Journal of Quality Technology, 2022
Rob Goedhart, William H. Woodall
In Montgomery (2020) it is shown that when applying the method of Laney (2002) for this dataset, the estimated standard deviation for each subgroup in the p-chart has to be multiplied by a factor 3.87 to account for the variability not explained by the binomial distribution alone. In terms of the variance, this means a multiplication by 14.98 (). In other words, with the method of Laney (2002) it is estimated that the total variance is 14.98 times larger than the intra-subgroup (or within-subgroup) variance. That means that the intra-subgroup variance is estimated to be just 6.7% of the total variance. Thus, it would be expected that the differences between control limits for different subgroups would be very small as well. However, when taking a closer look at the estimated control limits in Figure 2, we observe substantial variation in the limits still. In particular, consider for each subgroup the distance between the upper control limit and the center value of the chart. It turns out that the largest distance (subgroup 14) is 44% larger than the smallest distance (subgroup 16). This is rather surprising given the small impact (6.7%) of the intra-subgroup variation estimated earlier. In the next section we elaborate in more detail on the causes for this phenomenon with the method of Laney (2002), and the problems it can cause for the control chart performance when sample sizes vary.
Optimal sensors placement for structural health monitoring based on system identification and interpolation methods
Published in Journal of the Chinese Institute of Engineers, 2021
The simplified K-means clustering is performed as following: All data are divided into n groups, as shown in Figure 3(a).Randomly distribute n points to be the center of group, as shown in Figure 3(b).Calculate the distance from each data point to the center of group, as shown in Figure 3(c).Classify each point into the nearest center of group, as shown in Figure 3(d).Set the center of each group to be the new center of group.Repeat steps 3 to 5 until converged, as shown in Figure 3(e,f).Obtain the center of group to be the modal frequency.
Automatic generation of landslide profile for complementing landslide inventory
Published in Geomatics, Natural Hazards and Risk, 2020
Langping Li, Hengxing Lan, Alexander Strom
The other type of group anchors is associated with the initial and distal groups, which have to be treated as exceptions, because they each have only one group boundary. Therefore, their group centers cannot be defined using their group boundary centers. For the initial group, a minimum bounding box is placed around the group. The center of the box is then connected to the very first inter-group center. This line segment is then extrapolated back up the slope to its intersection with the landslide edge. This point on the edge becomes the initial group anchor (Figure 4(c)). In the rare situation that no intersection can be found, the middle of the edge points of the initial group will be adopted as the initial group anchor. A z-value for this anchor is obtained from the plane fitted to the points of the entire initial group. Another anchor is placed at the midpoint of the line segment, between the initial group anchor and the very first inter-group center. This procedure is repeated for the distal group, except the line is extrapolated forward to intersect with the landslide edge (Figure 4(c)). The final count of group anchors will be 2·n – 3 + 4, which is equal to 2·n + 1.