So the cluster dendogram that's produced by default in R is relatively nice, but it, it is, there is possible to kind of make it a little bit nicer. So this myplclust function which is available on the course website, you can download it straight from there. And once you source it into R. You can get this function which in turn, produces a cluster dendrogram, but then all the, the, the points in the individual clusters are A, labeled by their cluster label. And B, they are colored in separately, so you can see cluster one is labeled with all ones. And it's and it's they're, they're colored black. Cluster two is colored red and cluster three is colored green. So of course, in order to do this, you have to determine, you have to specify the how many clusters there are before you can actually you know, label them one, two, and three, etc. [BLANK_AUDIO] Now you can go to the RGraphGallery to see some kind of more interesting cluster diagrams and, and to look at some software that people have used. To create really nice clustering diagrams. So if you want to look at some, for example, this cluster dendrogram which has labels on the, on the leaf nodes, and it's colored each of the groups. It's added a little bit of other information. You can take a look at this gallery, which a lot of other really nice plots on it too. So, the other i, issue when it comes to using hierarchical clustering is how do you merge points together? And so the question is, you know, when you, when you, when you, when you merge a point together what, what represents its new location. And so one is called average, so average linkage. And the idea is that if you take two points and their new com, coordinate location, it's just the average of their x coordinates and their y coordinates. So it's kind of like the roughly the center of gravity or the middle of tat group of points, now that seems logical and it can, and it can, it will, it will lead to a kind of certain type of clustering result. [NOISE] The other type of, of approach is called complete linkage and there the idea is that if you want to measure the distance between two clusters of points. Then you take the farthest two points from each, from the two clusters, as the distance. So for example, in this picture, if you look at the upper cluster, and the kind of lower left cluster, the distance between those two clusters is the equal to the distance between the two farthest points. And so that gives you the kind of complete linkage approach. The average linkage, of course, gives you the distance between the two centers of gravity. So that's the difference between the two pluses here. So you can see that in the complete linkage example that distance is relatively far. Where as in the average linkage example the distance is somewhat shorter. And so, it's not like there is one right or one wrong way to do it. But the point is to show that each of the, the two different merging approaches can get to, give you very different results and so it's is useful to try, it's often useful to try both approaches. To see what kind of clustering results you get, in the end and whether one set makes more sense than another. The last function I want to highlight here is the heatmap function which is a really nice function for visualizing matrix data. So if you have an extremely large table or a, a large matrix of numbers that are kind of similarly scaled, and you want to, just take a look at them really quickly in an organized way, you can call the heatmap function. And what the heat map function does is essentially runs a hierarchical cluster analysis on the rows of the table and on the columns of the table. So think, on the. So if you can imagine the, the you know, rows of the table are like observations. And then it runs a cluster analysis on the rows of the table. And then, think of the columns of the, of the table as, as sets of observations. Even if they're not actually observations. The columns, for example, might be variables or something like that. But think of them as just different observations. You can write a cluster analysis on that, too. And the idea with the heatmap function is that it uses the hierarchical clustering function to organize the rows and the columns of the tables so that you can visualize them. You can visualize kind of groups or blocks of observations within the table using the image function. So, what it does is it creates a image plot here, and and it reorders the columns and the rows of the table according to the hierarchical clustering algorithm. So here you can see that for example, along the rows I've got this hierarchical, this cluster dendogram which shows that there are probably you know, three clusters. And those clusters, rows are all grouped together. And then there are only two columns in the data frame. So it's not particularly interesting to do a hierarchical cluster analysis on that. But if you had many,many columns, you know, you would reorganize the columns so that we, they would be kind of, the closer ones would be closer together, and the farther ones would be farther apart. And so, the heatmap function is really useful for kind of, or, as a very quickly visualizing this kind of high dimensional table data. So this is summarize what for heirarchical clustering. It's a really useful technique for looking at high-dimensional data. It organizes the data in a kind of logical and intuitive way and partic, in particular, functions like the heatmap function are, are really useful for kind of looking, for quickly looking at table or matrix data. And so, of course, you need to know, you need to define a notion of what it means for two points to be close or far apart. And you have to have a merging strategy. So we talked about complete linkage and average linkage. So given those two things, you can run, the hierarchical clustering algorithm will run, and it will produce a cluster dendrogram, which shows you kind of how the merging is done, or how the, how the points got merged together through the algorithm. So one of, a couple of issues when it comes to, hierarchical clustering. And this is often true for many clustering algorithms. Is that the, the clustering picture may be unstable. And so, for example, if a few points were to change. Or, for example, if you have some outliers. Then the clustering, if you were to remove those points. Or modify them a little bit. Then the clustering dendogram or whatever the final products might be, could change a lot. And so you have to be sensitive to that possibility. So you might, so the, often the use, a useful thing is to try different metrics, distance metrics, to make sure it's not to see how sensitive it is to the different distance metrics. Maybe change emerging strategy to see what kind of picture emer, comes out if you change merging strategy. And also clustering algorithms could be sensitive to the scalings of a given variable, so you might, if, if one variable, for example, is, is measured on with units that are much larger than another variable then that can sometimes throw off the algorithm. So, it may be useful to scale certain variables so that they're more comparable to each other. One nice thing about hierarchical clustering, at least the algorithm as discussed here, is that it is deterministic so there's no random starting point, there's no randomness in it. If you run it once, it will run, and with the same parameters and the same data, it will give you the same picture. Of course, a key question in any clustering approach is the, is where to cut. And the general idea is, you know, determining how many clusters there are. It's not always obvious to figure out how many clusters there are. And, and so, but there are a number of algorithms that has been proposed, to try to figure that out. But I think the best use of hierarchical clustering is probably just for exploratory analysis and just for looking at your data, visualizing your data, getting a sense of what patterns are there, if there are any patterns at all. And then, you can kind of take these ideas and formalize them later in a more sophisticated models. So I give you some links, the kind of more details about these approaches. And I encourage you to take a look at them to explore further.
