Category: Dendrogram clustering example

Dendrogram clustering example

The dendrogram illustrates how each cluster is composed by drawing a U-shaped link between a non-singleton cluster and its children. The top of the U-link indicates a cluster merge.

The two legs of the U-link indicate which clusters were merged. The length of the two legs of the U-link represents the distance between the child clusters. It is also the cophenetic distance between original observations in the two children clusters. The linkage matrix encoding the hierarchical clustering to render as a dendrogram. See the linkage function for more information on the format of Z.

dendrogram clustering example

The dendrogram can be hard to read when the original observation matrix from which the linkage is derived is large. Truncation is used to condense the dendrogram.

There are several modes:. No truncation is performed default.

StatQuest: Principal Component Analysis (PCA), Step-by-Step

The last p non-singleton clusters formed in the linkage are the only non-leaf nodes in the linkage; they correspond to rows Z[n-pend] in Z. All other non-singleton clusters are contracted into leaf nodes. No more than p levels of the dendrogram tree are displayed. All links connecting nodes with distances greater than or equal to the threshold are colored blue.

dendrogram clustering example

By default labels is None so the index of the original observation is used to label the leaf nodes. When True, the final rendering is not performed. This is useful if only the data structures computed for the rendering are needed or if matplotlib is not available.

Specifies the angle in degrees to rotate the leaf labels. When unspecified, the rotation is based on the number of nodes in the dendrogram default is 0. Specifies the font size in points of the leaf labels.Dendrograms are a convenient way of depicting pair-wise dissimilarity between objectscommonly associated with the topic of cluster analysis.

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This is a complex subject that is best left to experts and textbooksso I won't even attempt to cover it here. I have been frequently using dendrograms as part of my investigations into dissimilarity computed between soil profiles.

Unfortunately the interpretation of dendrograms is not very intuitive, especially when the source data are complex. In addition, pair-wise dissimimlarity computed between soil profiles and visualized via dendrogram should not be confused with the use of dendrograms in the field of cladistics -- where relation to a common ancestor is depicted.

An example is presented below that illustrates the relationship between dendrogram and dissimilarity as evaluated between objects with 2 variables. Essentially, the level at which branches merge relative to the "root" of the tree is related to their similarity. In the example below it is clear that in terms of clay and rock fragment content soils 4 and 5 are more similar to each other than to soil 2.

In addition, soils 1 and 3 are more similar to each other than soils 4 and 5 are to soil 2. Recall that in this case pair-wise dissimilarity is based on the Euclidean distance between soils in terms of their clay content and rock fragment content. Therefore proximity in the scatter plot of frock frags vs. Inline-comments in the code below elaborate further. Image: Data to Dendrogram.

A Graphical Explanation of how to Interpret a Dendrogram Posted: Thursday, March 15th, Dendrograms are a convenient way of depicting pair-wise dissimilarity between objectscommonly associated with the topic of cluster analysis.This is a tutorial on how to use scipy's hierarchical clustering.

One of the benefits of hierarchical clustering is that you don't need to already know the number of clusters k in your data in advance. Sadly, there doesn't seem to be much documentation on how to actually use scipy's hierarchical clustering to make an informed decision and then retrieve the clusters. The only thing you need to make sure is that you convert your data into a matrix X with n samples and m features, so that X.

Well, sure it was, this is python ;but what does the weird 'ward' mean there and how does this actually work? As the scipy linkage docs tell us, 'ward' is one of the methods that can be used to calculate the distance between newly formed clusters.

I think it's a good default choice, but it never hurts to play around with some other common linkage methods like 'single''complete''average'For example, you should have such a weird feeling with long binary feature vectors e. As you can see there's a lot of choice here and while python and scipy make it very easy to do the clustering, it's you who has to understand and make these choices. If i find the time, i might give some more practical advice about this, but for now i'd urge you to at least read up on the mentioned linked methods and metrics to make a somewhat informed choice.

Another thing you can and should definitely do is check the Cophenetic Correlation Coefficient of your clustering with help of the cophenet function. This very very briefly compares correlates the actual pairwise distances of all your samples to those implied by the hierarchical clustering. The closer the value is to 1, the better the clustering preserves the original distances, which in our case is pretty close:.

No matter what method and metric you pick, the linkage function will use that method and metric to calculate the distances of the clusters starting with your n individual samples aka data points as singleton clusters and in each iteration will merge the two clusters which have the smallest distance according the selected method and metric. It will return an array of length n - 1 giving you information about the n - 1 cluster merges which it needs to pairwise merge n clusters.

Z[i] will tell us which clusters were merged in the i -th iteration, let's take a look at the first two points that were merged:. In its first iteration the linkage algorithm decided to merge the two clusters original samples here with indices 52 and 53, as they only had a distance of 0. This created a cluster with a total of 2 samples. In the second iteration the algorithm decided to merge the clusters original samples here as well with indices 14 and 79, which had a distance of 0.

This again formed another cluster with a total of 2 samples. The indices of the clusters until now correspond to our samples. Remember that we had a total of samples, so indices 0 to Let's have a look at the first 20 iterations:.

We can observe that until iteration 13 the algorithm only directly merged original samples. We can also observe the monotonic increase of the distance. In iteration 14 the algorithm decided to merge cluster indices 62 with If you paid attention the should astonish you as we only have original sample indices 0 to for our samples. This means that while idx corresponds to X[] that idx corresponds to the cluster formed in Z[0]idx to Z[1]to Z[2]Hence, the merge iteration 14 merged sample 62 to our samples 33 and 68 that were previously merged in iteration 3 corresponding to Z[2] - We'll later come back to visualizing this, but now let's have a look at what's called a dendrogram of this hierarchical clustering first:.

A dendrogram is a visualization in form of a tree showing the order and distances of merges during the hierarchical clustering. Starting from each label at the bottom, you can see a vertical line up to a horizontal line. The height of that horizontal line tells you about the distance at which this label was merged into another label or cluster. You can find that other cluster by following the other vertical line down again.Clustering starts by computing a distance between every pair of units that you want to cluster.

A distance matrix will be symmetric because the distance between x and y is the same as the distance between y and x and will have zeroes on the diagonal because every item is distance zero from itself.

The table below is an example of a distance matrix.

What is a Dendrogram?

Only the lower triangle is shown, because the upper triangle can be filled in by reflection. Now lets start clustering. The smallest distance is between three and five and they get linked up or merged first into a the cluster '35'.

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To obtain the new distance matrix, we need to remove the 3 and 5 entries, and replace it by an entry "35". Since we are using complete linkage clustering, the distance between "35" and every other item is the maximum of the distance between this item and 3 and this item and 5.

This gives us the new distance matrix. The items with the smallest distance get clustered next. This will be 2 and 4. Continuing in this way, after 6 steps, everything is clustered.

This is summarized below. On this plot, the y-axis shows the distance between the objects at the time they were clustered. This is called the cluster height. Different visualizations use different measures of cluster height.

Complete Linkage. Below is the single linkage dendrogram for the same distance matrix. It starts with cluster "35" but the distance between "35" and each item is now the minimum of d x,3 and d x,5. One of the problems with hierarchical clustering is that there is no objective way to say how many clusters there are.

SciPy Hierarchical Clustering and Dendrogram Tutorial

If we cut the single linkage tree at the point shown below, we would say that there are two clusters. Let's look at some real data. In homework 5 we consider gene expression in 4 regions of 3 human and 3 chimpanzee brains. The RNA was hybridized to Affymetrix human gene expression microarrays.

Here we selected the most significantly differentially expressed genes from the study. We cluster all the differentially expressed genes based on their mean expression in each of the 8 species by brain region treatments.

Here are the clusters based on Euclidean distance and correlation distance, using complete and single linkage clustering. We can see that the clustering pattern for complete linkage distance tends to create compact clusters of clusters, while single linkage tends to add one point at a time to the cluster, creating long stringy clusters.

As we might expect from our discussion of distances, Euclidean distance and correlation distance produce very different dendrograms. Hierarchical clustering does not tell us how many clusters there are, or where to cut the dendrogram to form clusters. However, based on our visualization, we might prefer to cut the long branches at different heights. In any case, there is a fair amount of subjectivity in determining which branches should and should not be cut to form separate clusters.

To understand the clusters we usually plot the log2 expression values of the genes in the cluster, or in other words, plot the gene expressions over the samples.

The numbering in these graphs are totally arbitrary. Even though the treatments are unordered, I usually connect the points coming from a single feature to make the pattern clearer. These are called profile plots.A dendrogram is a network structure. It is constituted of a root node that gives birth to several nodes connected by edges or branches. The last nodes of the hierarchy are called leaves. In the following example, the CEO is the root node.

He manages 2 managers that manage 8 employees the leaves. Note that this kind of matrix can be computed from a multivariate datasetcomputing distance between each pair of individual using correlation or euclidean distance. It is possible to perform hierarchical cluster analysis on this set of dissimilarities.

Basically, this statistical method seeks to build a hierarchy of clusters: it tries to group sample that are close one from another. As expected, cities that are in same geographic area tend to be clusterized together. For example, the yellow cluster is composed by all the Asian cities of the dataset.

Note that the dendrogram provides even more information. For instance, Sydney appears to be a bit further to Calcutta than calcutta is from Tokyo: this can be deduce from the branch size that represents the distance.

A common task consists to compare the result of a clustering with an expected result. For instance, we can check if the countries are indeed grouped in continent using a color bar:. This graphic allows to validate that the clustering indeed grouped cities by continent.

There are a few discrepencies that are logical. Indeed, Mexico city has been considered as a city of South America here, altough it is probably closer from North America as suggested by the clustering. Many variations exist for dendrogram. It can be horizontal or vertical as shown before. It can also be linear or circular. The advantage of the circular verion being that it uses the graphic space more efficiently:.

dendrogram clustering example

Another common variation is to display a heatmap at the bottom of the dendrogram. Indeed, it allows to visualize the distance between each sample and thus to understand why the clustering algorythm put 2 samples next to each other. Representation of data where the individual values contained in a matrix are represented as colors.

Displays hierarchical data as a set of nested rectangles. Perfect to show how the whole is divided. The R and Python graph galleries are 2 websites providing hundreds of chart example, always providing the reproducible code. Click the button below to see how to build the chart you need with your favorite programing language.

R graph gallery. Any thoughts on this? Found any mistake? Please drop me a word on twitter or in the comment section below:. A work by Yan Holtz for data-to-viz. Definition A dendrogram is a network structure.

Dendrogram from hierarchic data. Perform hierarchical cluster analysis. Variation Many variations exist for dendrogram. The advantage of the circular verion being that it uses the graphic space more efficiently: Libraries library ggraph library igraph library tidyverse library RColorBrewer set.A dendrogram is a diagram representing a tree.

This diagrammatic representation is frequently used in different contexts:. The hierarchical clustering dendrogram would show a column of five nodes representing the initial data here individual taxaand the remaining nodes represent the clusters to which the data belong, with the arrows representing the distance dissimilarity.

The distance between merged clusters is monotone, increasing with the level of the merger: the height of each node in the plot is proportional to the value of the intergroup dissimilarity between its two daughters the nodes on the right representing individual observations all plotted at zero height. From Wikipedia, the free encyclopedia. A tree-shaped diagram showing the arrangement of various elements.

Not to be confused with Dendrogramma. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources.

Unsourced material may be challenged and removed. Molecular Systematics, 2nd edition. Sunderland, MA: Sinauer. PLOS One. Bibcode : PNAS Dictionary of Statistics. The American Statistician.

Dendrograms and Clustering

Encyclopedia Britannica. Retrieved Paris: Hachette. Retrieved October 20, A dendrogram is a tree-structured graph used in heat maps to visualize the result of a hierarchical clustering calculation.

The result of a clustering is presented either as the distance or the similarity between the clustered rows or columns depending on the selected distance measure.

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See Distance Measures Overview and the detailed description for each measure for further information about the available distance measures. You can perform hierarchical clustering on an existing heat map by opening the Dendrograms page of the Visualization Properties.

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You can also use the Hierarchical Clustering tool to cluster with a data table as the input. Note that only numeric columns will be included when clustering. The row dendrogram shows the distance or similarity between rows and which nodes each row belongs to, as a result of clustering. An example of a row dendrogram is shown below. The individual rows in the clustered data are represented by the right-most nodes, the leaf nodes, in the row dendrogram. Each node in the dendrogram represents a cluster of all rows that lie to the right of it in the dendrogram.

The left-most node in the dendrogram is therefore a cluster that contains all rows. The vertical dotted line is the pruning line, which can be dragged sideways in the dendrogram. The values next to the pruning line indicate the number of clusters starting from the current position of the line, as well as the calculated distance or similarity at that position.

In the example above, the calculated distance is 1. The upper two, indicated by pink circles, contain two or more rows, while the lower cluster contains only one individual row. The column dendrogram is drawn in the same way as the row dendrogram, but shows the distance or similarity between the variables the cell value columns.

At the position of the pruning line in the above example, there are two clusters. The left-most cluster contains two columns, while the right-most cluster contains only one individual column. The calculated distance is 6. The dendrogram makes it easy to highlight and mark in the heat map.

You can mouseover the dendrogram to highlight clusters and their corresponding cells in the heat map. You can click to mark a cluster. This will also mark the corresponding cells in the heat map, as in the example below. The tooltip displays information about the cluster. As mentioned, a dendrogram is added to the heat map when clustering is performed. A new column is also added to the data table, and made available in the filters panel.

A Graphical Explanation of how to Interpret a Dendrogram

The cluster column is dynamic, and the position of the pruning line decides its content. The example below shows what the cluster column and cluster filter would look like for the row dendrogram above.

The cluster column contains unique identifiers for the cluster nodes corresponding to the position of the pruning line. In the example above, two cluster nodes are identified. The cluster column makes it possible to filter out entire clusters at a time. You can also use it to color or trellis other visualizations by.

Note: If you add a column dendrogram to a heat map that is set up with multiple cell value columns, then the cluster column cannot show any cluster IDs. This means that the cluster column cannot be used for filtering, or to color or trellis other visualizations by.


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