Cluster analysis

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https://en.wikipedia.org/wiki/Cluster_analysis

The result of a cluster analysis shown as the coloring of the squares into three clusters

Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning.

Cluster analysis itself is not one specific algorithm, but the general task to be solved. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small distances between cluster members, dense areas of the data space, intervals or particular statistical distributions. Clustering can therefore be formulated as a multi-objective optimization problem. The appropriate clustering algorithm and parameter settings (including parameters such as the distance function to use, a density threshold or the number of expected clusters) depend on the individual data set and intended use of the results. Cluster analysis as such is not an automatic task, but an iterative process of knowledge discovery or interactive multi-objective optimization that involves trial and failure. It is often necessary to modify data preprocessing and model parameters until the result achieves the desired properties.

Besides the term clustering, there is a number of terms with similar meanings, including automatic classification, numerical taxonomy, botryology (from Greek βότρυς "grape"), typological analysis, and community detection. The subtle differences are often in the use of the results: while in data mining, the resulting groups are the matter of interest, in automatic classification the resulting discriminative power is of interest.

Cluster analysis was originated in anthropology by Driver and Kroeber in 1932 and introduced to psychology by Joseph Zubin in 1938 and Robert Tryon in 1939 and famously used by Cattell beginning in 1943^[4] for trait theory classification in personality psychology.

Definition

The notion of a "cluster" cannot be precisely defined, which is one of the reasons why there are so many clustering algorithms. There is a common denominator: a group of data objects. However, different researchers employ different cluster models, and for each of these cluster models again different algorithms can be given. The notion of a cluster, as found by different algorithms, varies significantly in its properties. Understanding these "cluster models" is key to understanding the differences between the various algorithms. Typical cluster models include:

• Connectivity models: for example, hierarchical clustering builds models based on distance connectivity.
• Centroid models: for example, the k-means algorithm represents each cluster by a single mean vector.
• Distribution models: clusters are modeled using statistical distributions, such as multivariate normal distributions used by the expectation-maximization algorithm.
• Density models: for example, DBSCAN and OPTICS defines clusters as connected dense regions in the data space.
• Subspace models: in biclustering (also known as co-clustering or two-mode-clustering), clusters are modeled with both cluster members and relevant attributes.
• Group models: some algorithms do not provide a refined model for their results and just provide the grouping information.
• Graph-based models: a clique, that is, a subset of nodes in a graph such that every two nodes in the subset are connected by an edge can be considered as a prototypical form of cluster. Relaxations of the complete connectivity requirement (a fraction of the edges can be missing) are known as quasi-cliques, as in the HCS clustering algorithm.
• Signed graph models: Every path in a signed graph has a sign from the product of the signs on the edges. Under the assumptions of balance theory, edges may change sign and result in a bifurcated graph. The weaker "clusterability axiom" (no cycle has exactly one negative edge) yields results with more than two clusters, or subgraphs with only positive edges.
• Neural models: the most well known unsupervised neural network is the self-organizing map and these models can usually be characterized as similar to one or more of the above models, and including subspace models when neural networks implement a form of Principal Component Analysis or Independent Component Analysis.

A "clustering" is essentially a set of such clusters, usually containing all objects in the data set. Additionally, it may specify the relationship of the clusters to each other, for example, a hierarchy of clusters embedded in each other. Clusterings can be roughly distinguished as:

• Hard clustering: each object belongs to a cluster or not
• Soft clustering (also: fuzzy clustering): each object belongs to each cluster to a certain degree (for example, a likelihood of belonging to the cluster)

There are also finer distinctions possible, for example:

• Strict partitioning clustering: each object belongs to exactly one cluster
• Strict partitioning clustering with outliers: objects can also belong to no cluster, in which case they are considered outliers
• Overlapping clustering (also: alternative clustering, multi-view clustering): objects may belong to more than one cluster; usually involving hard clusters
• Hierarchical clustering: objects that belong to a child cluster also belong to the parent cluster
• Subspace clustering: while an overlapping clustering, within a uniquely defined subspace, clusters are not expected to overlap

Algorithms

Main category: Cluster analysis algorithms

As listed above, clustering algorithms can be categorized based on their cluster model. The following overview will only list the most prominent examples of clustering algorithms, as there are possibly over 100 published clustering algorithms. Not all provide models for their clusters and can thus not easily be categorized. An overview of algorithms explained in Wikipedia can be found in the list of statistics algorithms.

There is no objectively "correct" clustering algorithm, but as it was noted, "clustering is in the eye of the beholder." The most appropriate clustering algorithm for a particular problem often needs to be chosen experimentally, unless there is a mathematical reason to prefer one cluster model over another. An algorithm that is designed for one kind of model will generally fail on a data set that contains a radically different kind of model. For example, k-means cannot find non-convex clusters.

Connectivity-based clustering (hierarchical clustering)

Main article: Hierarchical clustering

Connectivity-based clustering, also known as hierarchical clustering, is based on the core idea of objects being more related to nearby objects than to objects farther away. These algorithms connect "objects" to form "clusters" based on their distance. A cluster can be described largely by the maximum distance needed to connect parts of the cluster. At different distances, different clusters will form, which can be represented using a dendrogram, which explains where the common name "hierarchical clustering" comes from: these algorithms do not provide a single partitioning of the data set, but instead provide an extensive hierarchy of clusters that merge with each other at certain distances. In a dendrogram, the y-axis marks the distance at which the clusters merge, while the objects are placed along the x-axis such that the clusters don't mix.

Connectivity-based clustering is a whole family of methods that differ by the way distances are computed. Apart from the usual choice of distance functions, the user also needs to decide on the linkage criterion (since a cluster consists of multiple objects, there are multiple candidates to compute the distance) to use. Popular choices are known as single-linkage clustering (the minimum of object distances), complete linkage clustering (the maximum of object distances), and UPGMA or WPGMA ("Unweighted or Weighted Pair Group Method with Arithmetic Mean", also known as average linkage clustering). Furthermore, hierarchical clustering can be agglomerative (starting with single elements and aggregating them into clusters) or divisive (starting with the complete data set and dividing it into partitions).

These methods will not produce a unique partitioning of the data set, but a hierarchy from which the user still needs to choose appropriate clusters. They are not very robust towards outliers, which will either show up as additional clusters or even cause other clusters to merge (known as "chaining phenomenon", in particular with single-linkage clustering). In the general case, the complexity is ${\displaystyle \mathcal{O}(n^3)}$ for agglomerative clustering and ${\displaystyle \mathcal{O}(2^{n-1})}$ for divisive clustering, which makes them too slow for large data sets. For some special cases, optimal efficient methods (of complexity ${\displaystyle \mathcal{O}(n^2)}$) are known: SLINK for single-linkage and CLINK for complete-linkage clustering.

• Linkage clustering examples
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  Single-linkage on Gaussian data. At 35 clusters, the biggest cluster starts fragmenting into smaller parts, while before it was still connected to the second largest due to the single-link effect.

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  Single-linkage on density-based clusters. 20 clusters extracted, most of which contain single elements, since linkage clustering does not have a notion of "noise".

Centroid-based clustering

Main article: k-means clustering

In centroid-based clustering, each cluster is represented by a central vector, which is not necessarily a member of the data set. When the number of clusters is fixed to k, k-means clustering gives a formal definition as an optimization problem: find the k cluster centers and assign the objects to the nearest cluster center, such that the squared distances from the cluster are minimized.

The optimization problem itself is known to be NP-hard, and thus the common approach is to search only for approximate solutions. A particularly well known approximate method is Lloyd's algorithm, often just referred to as "k-means algorithm" (although another algorithm introduced this name). It does however only find a local optimum, and is commonly run multiple times with different random initializations. Variations of k-means often include such optimizations as choosing the best of multiple runs, but also restricting the centroids to members of the data set (k-medoids), choosing medians (k-medians clustering), choosing the initial centers less randomly (k-means++) or allowing a fuzzy cluster assignment (fuzzy c-means).

Most k-means-type algorithms require the number of clusters – k – to be specified in advance, which is considered to be one of the biggest drawbacks of these algorithms. Furthermore, the algorithms prefer clusters of approximately similar size, as they will always assign an object to the nearest centroid. This often leads to incorrectly cut borders of clusters (which is not surprising since the algorithm optimizes cluster centers, not cluster borders).

K-means has a number of interesting theoretical properties. First, it partitions the data space into a structure known as a Voronoi diagram. Second, it is conceptually close to nearest neighbor classification, and as such is popular in machine learning. Third, it can be seen as a variation of model based clustering, and Lloyd's algorithm as a variation of the Expectation-maximization algorithm for this model discussed below.

• k-means clustering examples
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  k-means separates data into Voronoi cells, which assumes equal-sized clusters (not adequate here).

• 

  k-means cannot represent density-based clusters.

Centroid-based clustering problems such as k-means and k-medoids are special cases of the uncapacitated, metric facility location problem, a canonical problem in the operations research and computational geometry communities. In a basic facility location problem (of which there are numerous variants that model more elaborate settings), the task is to find the best warehouse locations to optimally service a given set of consumers. One may view "warehouses" as cluster centroids and "consumer locations" as the data to be clustered. This makes it possible to apply the well-developed algorithmic solutions from the facility location literature to the presently considered centroid-based clustering problem.

Distribution-based clustering

The clustering model most closely related to statistics is based on distribution models. Clusters can then easily be defined as objects belonging most likely to the same distribution. A convenient property of this approach is that this closely resembles the way artificial data sets are generated: by sampling random objects from a distribution.

While the theoretical foundation of these methods is excellent, they suffer from overfitting unless constraints are put on the model complexity. A more complex model will usually be able to explain the data better, which makes choosing the appropriate model complexity inherently difficult.

One prominent method is known as Gaussian mixture models (using the expectation-maximization algorithm). Here, the data set is usually modeled with a fixed (to avoid overfitting) number of Gaussian distributions that are initialized randomly and whose parameters are iteratively optimized to better fit the data set. This will converge to a local optimum, so multiple runs may produce different results. In order to obtain a hard clustering, objects are often then assigned to the Gaussian distribution they most likely belong to; for soft clusterings, this is not necessary.

Distribution-based clustering produces complex models for clusters that can capture correlation and dependence between attributes. However, these algorithms put an extra burden on the user: for many real data sets, there may be no concisely defined mathematical model (e.g. assuming Gaussian distributions is a rather strong assumption on the data).

• Gaussian mixture model clustering examples
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  On Gaussian-distributed data, EM works well, since it uses Gaussians for modelling clusters.

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  Density-based clusters cannot be modeled using Gaussian distributions.

Density-based clustering

In density-based clustering, clusters are defined as areas of higher density than the remainder of the data set. Objects in sparse areas – that are required to separate clusters – are usually considered to be noise and border points.

The most popular density based clustering method is DBSCAN. In contrast to many newer methods, it features a well-defined cluster model called "density-reachability". Similar to linkage based clustering, it is based on connecting points within certain distance thresholds. However, it only connects points that satisfy a density criterion, in the original variant defined as a minimum number of other objects within this radius. A cluster consists of all density-connected objects (which can form a cluster of an arbitrary shape, in contrast to many other methods) plus all objects that are within these objects' range. Another interesting property of DBSCAN is that its complexity is fairly low – it requires a linear number of range queries on the database – and that it will discover essentially the same results (it is deterministic for core and noise points, but not for border points) in each run, therefore there is no need to run it multiple times. OPTICS is a generalization of DBSCAN that removes the need to choose an appropriate value for the range parameter ${\displaystyle \epsilon }$, and produces a hierarchical result related to that of linkage clustering. DeLi-Clu, Density-Link-Clustering combines ideas from single-linkage clustering and OPTICS, eliminating the ${\displaystyle \epsilon }$ parameter entirely and offering performance improvements over OPTICS by using an R-tree index.

The key drawback of DBSCAN and OPTICS is that they expect some kind of density drop to detect cluster borders. On data sets with, for example, overlapping Gaussian distributions – a common use case in artificial data – the cluster borders produced by these algorithms will often look arbitrary, because the cluster density decreases continuously. On a data set consisting of mixtures of Gaussians, these algorithms are nearly always outperformed by methods such as EM clustering that are able to precisely model this kind of data.

Mean-shift is a clustering approach where each object is moved to the densest area in its vicinity, based on kernel density estimation. Eventually, objects converge to local maxima of density. Similar to k-means clustering, these "density attractors" can serve as representatives for the data set, but mean-shift can detect arbitrary-shaped clusters similar to DBSCAN. Due to the expensive iterative procedure and density estimation, mean-shift is usually slower than DBSCAN or k-Means. Besides that, the applicability of the mean-shift algorithm to multidimensional data is hindered by the unsmooth behaviour of the kernel density estimate, which results in over-fragmentation of cluster tails.

• Density-based clustering examples
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  Density-based clustering with DBSCAN

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  DBSCAN assumes clusters of similar density, and may have problems separating nearby clusters.

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  OPTICS is a DBSCAN variant, improving handling of different densities clusters.

Grid-based clustering

The grid-based technique is used for a multi-dimensional data set. In this technique, we create a grid structure, and the comparison is performed on grids (also known as cells). The grid-based technique is fast and has low computational complexity. There are two types of grid-based clustering methods: STING and CLIQUE. Steps involved in grid-based clustering algorithm are:

1. Divide data space into a finite number of cells.
2. Randomly select a cell ‘c’, where c should not be traversed beforehand.
3. Calculate the density of ‘c’
4. If the density of ‘c’ greater than threshold density
   1. Mark cell ‘c’ as a new cluster
   2. Calculate the density of all the neighbors of ‘c’
   3. If the density of a neighboring cell is greater than threshold density then, add the cell in the cluster and repeat steps 4.2 and 4.3 till there is no neighbor with a density greater than threshold density.
5. Repeat steps 2,3 and 4 till all the cells are traversed.
6. Stop.

Recent developments

In recent years, considerable effort has been put into improving the performance of existing algorithms. Among them are CLARANS, and BIRCH. With the recent need to process larger and larger data sets (also known as big data), the willingness to trade semantic meaning of the generated clusters for performance has been increasing. This led to the development of pre-clustering methods such as canopy clustering, which can process huge data sets efficiently, but the resulting "clusters" are merely a rough pre-partitioning of the data set to then analyze the partitions with existing slower methods such as k-means clustering.

For high-dimensional data, many of the existing methods fail due to the curse of dimensionality, which renders particular distance functions problematic in high-dimensional spaces. This led to new clustering algorithms for high-dimensional data that focus on subspace clustering (where only some attributes are used, and cluster models include the relevant attributes for the cluster) and correlation clustering that also looks for arbitrary rotated ("correlated") subspace clusters that can be modeled by giving a correlation of their attributes. Examples for such clustering algorithms are CLIQUE and SUBCLU.

Ideas from density-based clustering methods (in particular the DBSCAN/OPTICS family of algorithms) have been adapted to subspace clustering (HiSC, hierarchical subspace clustering and DiSH) and correlation clustering (HiCO, hierarchical correlation clustering, 4C using "correlation connectivity" and ERiC exploring hierarchical density-based correlation clusters).

Several different clustering systems based on mutual information have been proposed. One is Marina Meilă's variation of information metric; another provides hierarchical clustering. Using genetic algorithms, a wide range of different fit-functions can be optimized, including mutual information. Also belief propagation, a recent development in computer science and statistical physics, has led to the creation of new types of clustering algorithms.

Evaluation and assessment

Evaluation (or "validation") of clustering results is as difficult as the clustering itself. Popular approaches involve "internal" evaluation, where the clustering is summarized to a single quality score, "external" evaluation, where the clustering is compared to an existing "ground truth" classification, "manual" evaluation by a human expert, and "indirect" evaluation by evaluating the utility of the clustering in its intended application.

Internal evaluation measures suffer from the problem that they represent functions that themselves can be seen as a clustering objective. For example, one could cluster the data set by the Silhouette coefficient; except that there is no known efficient algorithm for this. By using such an internal measure for evaluation, one rather compares the similarity of the optimization problems, and not necessarily how useful the clustering is.

External evaluation has similar problems: if we have such "ground truth" labels, then we would not need to cluster; and in practical applications we usually do not have such labels. On the other hand, the labels only reflect one possible partitioning of the data set, which does not imply that there does not exist a different, and maybe even better, clustering.

Neither of these approaches can therefore ultimately judge the actual quality of a clustering, but this needs human evaluation, which is highly subjective. Nevertheless, such statistics can be quite informative in identifying bad clusterings, but one should not dismiss subjective human evaluation.

Internal evaluation

See also: Determining the number of clusters in a data set

When a clustering result is evaluated based on the data that was clustered itself, this is called internal evaluation. These methods usually assign the best score to the algorithm that produces clusters with high similarity within a cluster and low similarity between clusters. One drawback of using internal criteria in cluster evaluation is that high scores on an internal measure do not necessarily result in effective information retrieval applications. Additionally, this evaluation is biased towards algorithms that use the same cluster model. For example, k-means clustering naturally optimizes object distances, and a distance-based internal criterion will likely overrate the resulting clustering.

Therefore, the internal evaluation measures are best suited to get some insight into situations where one algorithm performs better than another, but this shall not imply that one algorithm produces more valid results than another. Validity as measured by such an index depends on the claim that this kind of structure exists in the data set. An algorithm designed for some kind of models has no chance if the data set contains a radically different set of models, or if the evaluation measures a radically different criterion. For example, k-means clustering can only find convex clusters, and many evaluation indexes assume convex clusters. On a data set with non-convex clusters neither the use of k-means, nor of an evaluation criterion that assumes convexity, is sound.

More than a dozen of internal evaluation measures exist, usually based on the intuition that items in the same cluster should be more similar than items in different clusters. For example, the following methods can be used to assess the quality of clustering algorithms based on internal criterion:

• Davies–Bouldin index

The Davies–Bouldin index can be calculated by the following formula:

${\displaystyle DB=\frac{1}{n}\sum _{i=1}^n\underset{j\ne i}{max}\left(\frac{{\sigma }_i+{\sigma }_j}{d(c_i,c_j)}\right)}$

where n is the number of clusters, ${\displaystyle c_i}$ is the centroid of cluster ${\displaystyle i}$, ${\displaystyle {\sigma }_i}$ is the average distance of all elements in cluster ${\displaystyle i}$ to centroid ${\displaystyle c_i}$, and ${\displaystyle d(c_i,c_j)}$ is the distance between centroids ${\displaystyle c_i}$ and ${\displaystyle c_j}$. Since algorithms that produce clusters with low intra-cluster distances (high intra-cluster similarity) and high inter-cluster distances (low inter-cluster similarity) will have a low Davies–Bouldin index, the clustering algorithm that produces a collection of clusters with the smallest Davies–Bouldin index is considered the best algorithm based on this criterion.

• Dunn index

The Dunn index aims to identify dense and well-separated clusters. It is defined as the ratio between the minimal inter-cluster distance to maximal intra-cluster distance. For each cluster partition, the Dunn index can be calculated by the following formula:

${\displaystyle D=\frac{\underset{1\le i<j\le n}{min}d(i,j)}{\underset{1\le k\le n}{max}{d}^{\mathrm{\prime }}(k)},}$

where d(i,j) represents the distance between clusters i and j, and d '(k) measures the intra-cluster distance of cluster k. The inter-cluster distance d(i,j) between two clusters may be any number of distance measures, such as the distance between the centroids of the clusters. Similarly, the intra-cluster distance d '(k) may be measured in a variety ways, such as the maximal distance between any pair of elements in cluster k. Since internal criterion seek clusters with high intra-cluster similarity and low inter-cluster similarity, algorithms that produce clusters with high Dunn index are more desirable.

• Silhouette coefficient

The silhouette coefficient contrasts the average distance to elements in the same cluster with the average distance to elements in other clusters. Objects with a high silhouette value are considered well clustered, objects with a low value may be outliers. This index works well with k-means clustering, and is also used to determine the optimal number of clusters.

External evaluation

In external evaluation, clustering results are evaluated based on data that was not used for clustering, such as known class labels and external benchmarks. Such benchmarks consist of a set of pre-classified items, and these sets are often created by (expert) humans. Thus, the benchmark sets can be thought of as a gold standard for evaluation. These types of evaluation methods measure how close the clustering is to the predetermined benchmark classes. However, it has recently been discussed whether this is adequate for real data, or only on synthetic data sets with a factual ground truth, since classes can contain internal structure, the attributes present may not allow separation of clusters or the classes may contain anomalies. Additionally, from a knowledge discovery point of view, the reproduction of known knowledge may not necessarily be the intended result. In the special scenario of constrained clustering, where meta information (such as class labels) is used already in the clustering process, the hold-out of information for evaluation purposes is non-trivial.

A number of measures are adapted from variants used to evaluate classification tasks. In place of counting the number of times a class was correctly assigned to a single data point (known as true positives), such pair counting metrics assess whether each pair of data points that is truly in the same cluster is predicted to be in the same cluster.

As with internal evaluation, several external evaluation measures exist, for example:

• Purity: Purity is a measure of the extent to which clusters contain a single class. Its calculation can be thought of as follows: For each cluster, count the number of data points from the most common class in said cluster. Now take the sum over all clusters and divide by the total number of data points. Formally, given some set of clusters ${\displaystyle M}$ and some set of classes ${\displaystyle D}$, both partitioning ${\displaystyle N}$ data points, purity can be defined as:

${\displaystyle \frac{1}{N}\sum _{m\in M}\underset{d\in D}{max}|m\cap d|}$

This measure doesn't penalize having many clusters, and more clusters will make it easier to produce a high purity. A purity score of 1 is always possible by putting each data point in its own cluster. Also, purity doesn't work well for imbalanced data, where even poorly performing clustering algorithms will give a high purity value. For example, if a size 1000 dataset consists of two classes, one containing 999 points and the other containing 1 point, then every possible partition will have a purity of at least 99.9%.

• Rand index

The Rand index computes how similar the clusters (returned by the clustering algorithm) are to the benchmark classifications. It can be computed using the following formula:

${\displaystyle RI=\frac{TP+TN}{TP+FP+FN+TN}}$

where ${\displaystyle TP}$ is the number of true positives, ${\displaystyle TN}$ is the number of true negatives, ${\displaystyle FP}$ is the number of false positives, and ${\displaystyle FN}$ is the number of false negatives. The instances being counted here are the number of correct pairwise assignments. That is, ${\displaystyle TP}$ is the number of pairs of points that are clustered together in the predicted partition and in the ground truth partition, ${\displaystyle FP}$ is the number of pairs of points that are clustered together in the predicted partition but not in the ground truth partition etc. If the dataset is of size N, then ${\displaystyle TP+TN+FP+FN=(\genfrac{}{}{0ex}{}{N}{2})}$.

One issue with the Rand index is that false positives and false negatives are equally weighted. This may be an undesirable characteristic for some clustering applications. The F-measure addresses this concern, as does the chance-corrected adjusted Rand index.

• F-measure

The F-measure can be used to balance the contribution of false negatives by weighting recall through a parameter ${\displaystyle \beta \ge 0}$. Let precision and recall (both external evaluation measures in themselves) be defined as follows:

${\displaystyle P=\frac{TP}{TP+FP}}$

${\displaystyle R=\frac{TP}{TP+FN}}$

where ${\displaystyle P}$ is the precision rate and ${\displaystyle R}$ is the recall rate. We can calculate the F-measure by using the following formula:

${\displaystyle F_{\beta }=\frac{({\beta }^2+1)\cdot P\cdot R}{{\beta }^2\cdot P+R}}$

When ${\displaystyle \beta =0}$, ${\displaystyle F_0=P}$. In other words, recall has no impact on the F-measure when ${\displaystyle \beta =0}$, and increasing ${\displaystyle \beta }$ allocates an increasing amount of weight to recall in the final F-measure.

Also ${\displaystyle TN}$ is not taken into account and can vary from 0 upward without bound.

• Jaccard index

The Jaccard index is used to quantify the similarity between two datasets. The Jaccard index takes on a value between 0 and 1. An index of 1 means that the two dataset are identical, and an index of 0 indicates that the datasets have no common elements. The Jaccard index is defined by the following formula:

${\displaystyle J(A,B)=\frac{|A\cap B|}{|A\cup B|}=\frac{TP}{TP+FP+FN}}$

This is simply the number of unique elements common to both sets divided by the total number of unique elements in both sets.

Note that ${\displaystyle TN}$ is not taken into account.

• Dice index

The Dice symmetric measure doubles the weight on ${\displaystyle TP}$ while still ignoring ${\displaystyle TN}$:

${\displaystyle DSC=\frac{2TP}{2TP+FP+FN}}$

• Fowlkes–Mallows index

The Fowlkes–Mallows index computes the similarity between the clusters returned by the clustering algorithm and the benchmark classifications. The higher the value of the Fowlkes–Mallows index the more similar the clusters and the benchmark classifications are. It can be computed using the following formula:

${\displaystyle FM=\sqrt{\frac{TP}{TP+FP}\cdot \frac{TP}{TP+FN}}}$

where ${\displaystyle TP}$ is the number of true positives, ${\displaystyle FP}$ is the number of false positives, and ${\displaystyle FN}$ is the number of false negatives. The ${\displaystyle FM}$ index is the geometric mean of the precision and recall ${\displaystyle P}$ and ${\displaystyle R}$, and is thus also known as the G-measure, while the F-measure is their harmonic mean. Moreover, precision and recall are also known as Wallace's indices ${\displaystyle B^I}$ and ${\displaystyle B^{II}}$.^[45] Chance normalized versions of recall, precision and G-measure correspond to Informedness, Markedness and Matthews Correlation and relate strongly to Kappa.

• Chi Index is a external validation index that measure the clustering results by applying the chi-squared statistic. This index scores positively the fact that the labels are as sparse as possible across the clusters, i.e., that each cluster has as few different labels as possible. The higher the value of the Chi Index the greater the relationship between the resulting clusters and the label used.
• The mutual information is an information theoretic measure of how much information is shared between a clustering and a ground-truth classification that can detect a non-linear similarity between two clusterings. Normalized mutual information is a family of corrected-for-chance variants of this that has a reduced bias for varying cluster numbers.
• Confusion matrix

A confusion matrix can be used to quickly visualize the results of a classification (or clustering) algorithm. It shows how different a cluster is from the gold standard cluster.

Cluster tendency

To measure cluster tendency is to measure to what degree clusters exist in the data to be clustered, and may be performed as an initial test, before attempting clustering. One way to do this is to compare the data against random data. On average, random data should not have clusters.

• Hopkins statistic

There are multiple formulations of the Hopkins statistic. A typical one is as follows. Let ${\displaystyle X}$ be the set of ${\displaystyle n}$ data points in ${\displaystyle d}$ dimensional space. Consider a random sample (without replacement) of ${\displaystyle m\ll n}$ data points with members ${\displaystyle x_i}$. Also generate a set ${\displaystyle Y}$ of ${\displaystyle m}$ uniformly randomly distributed data points. Now define two distance measures, ${\displaystyle u_i}$ to be the distance of ${\displaystyle y_i\in Y}$ from its nearest neighbor in X and ${\displaystyle w_i}$ to be the distance of ${\displaystyle x_i\in X}$ from its nearest neighbor in X. We then define the Hopkins statistic as:

${\displaystyle H=\frac{\sum _{i=1}^m{u}_i^d}{\sum _{i=1}^m{u}_i^d+\sum _{i=1}^m{w}_i^d},}$

With this definition, uniform random data should tend to have values near to 0.5, and clustered data should tend to have values nearer to 1.

However, data containing just a single Gaussian will also score close to 1, as this statistic measures deviation from a uniform distribution, not multimodality, making this statistic largely useless in application (as real data never is remotely uniform).

Applications

Biology, computational biology and bioinformatics

See also: Distance matrices in phylogeny

Plant and animal ecology

Cluster analysis is used to describe and to make spatial and temporal comparisons of communities (assemblages) of organisms in heterogeneous environments. It is also used in plant systematics to generate artificial phylogenies or clusters of organisms (individuals) at the species, genus or higher level that share a number of attributes.

Transcriptomics

Clustering is used to build groups of genes with related expression patterns (also known as coexpressed genes) as in HCS clustering algorithm. Often such groups contain functionally related proteins, such as enzymes for a specific pathway, or genes that are co-regulated. High throughput experiments using expressed sequence tags (ESTs) or DNA microarrays can be a powerful tool for genome annotation – a general aspect of genomics.

Sequence analysis

Sequence clustering is used to group homologous sequences into gene families.

 This is a very important concept in bioinformatics, and evolutionary biology in general. See evolution by gene duplication.

High-throughput genotyping platforms

Clustering algorithms are used to automatically assign genotypes.

Human genetic clustering

The similarity of genetic data is used in clustering to infer population structures.

Medicine

Medical imaging

On PET scans, cluster analysis can be used to differentiate between different types of tissue in a three-dimensional image for many different purposes.

Analysis of antimicrobial activity

Cluster analysis can be used to analyse patterns of antibiotic resistance, to classify antimicrobial compounds according to their mechanism of action, to classify antibiotics according to their antibacterial activity.

IMRT segmentation

Clustering can be used to divide a fluence map into distinct regions for conversion into deliverable fields in MLC-based Radiation Therapy.

Business and marketing

Market research

Cluster analysis is widely used in market research when working with multivariate data from surveys and test panels. Market researchers use cluster analysis to partition the general population of consumers into market segments and to better understand the relationships between different groups of consumers/potential customers, and for use in market segmentation, product positioning, new product development and selecting test markets.

Grouping of shopping items

Clustering can be used to group all the shopping items available on the web into a set of unique products. For example, all the items on eBay can be grouped into unique products (eBay does not have the concept of a SKU).

World Wide Web

Social network analysis

In the study of social networks, clustering may be used to recognize communities within large groups of people.

Search result grouping

In the process of intelligent grouping of the files and websites, clustering may be used to create a more relevant set of search results compared to normal search engines like Google. There are currently a number of web-based clustering tools such as Clusty. It also may be used to return a more comprehensive set of results in cases where a search term could refer to vastly different things. Each distinct use of the term corresponds to a unique cluster of results, allowing a ranking algorithm to return comprehensive results by picking the top result from each cluster.

Slippy map optimization

Flickr's map of photos and other map sites use clustering to reduce the number of markers on a map. This makes it both faster and reduces the amount of visual clutter.

Computer science

Software evolution

Clustering is useful in software evolution as it helps to reduce legacy properties in code by reforming functionality that has become dispersed. It is a form of restructuring and hence is a way of direct preventative maintenance.

Image segmentation

Clustering can be used to divide a digital image into distinct regions for border detection or object recognition.

Evolutionary algorithms

Clustering may be used to identify different niches within the population of an evolutionary algorithm so that reproductive opportunity can be distributed more evenly amongst the evolving species or subspecies.

Recommender systems

Recommender systems are designed to recommend new items based on a user's tastes. They sometimes use clustering algorithms to predict a user's preferences based on the preferences of other users in the user's cluster.

Markov chain Monte Carlo methods

Clustering is often utilized to locate and characterize extrema in the target distribution.

Anomaly detection

Anomalies/outliers are typically – be it explicitly or implicitly – defined with respect to clustering structure in data.

Natural language processing

Clustering can be used to resolve lexical ambiguity.

DevOps

Clustering has been used to analyse the effectiveness of DevOps teams.

Social science

Sequence analysis in social sciences

Cluster analysis is used to identify patterns of family life trajectories, professional careers, and daily or weekly time use for example.

Crime analysis

Cluster analysis can be used to identify areas where there are greater incidences of particular types of crime. By identifying these distinct areas or "hot spots" where a similar crime has happened over a period of time, it is possible to manage law enforcement resources more effectively.

Educational data mining

Cluster analysis is for example used to identify groups of schools or students with similar properties.

Typologies

From poll data, projects such as those undertaken by the Pew Research Center use cluster analysis to discern typologies of opinions, habits, and demographics that may be useful in politics and marketing.

Others

Field robotics

Clustering algorithms are used for robotic situational awareness to track objects and detect outliers in sensor data.

Mathematical chemistry

To find structural similarity, etc., for example, 3000 chemical compounds were clustered in the space of 90 topological indices.

Climatology

To find weather regimes or preferred sea level pressure atmospheric patterns.

Finance

Cluster analysis has been used to cluster stocks into sectors.

Petroleum geology

Cluster analysis is used to reconstruct missing bottom hole core data or missing log curves in order to evaluate reservoir properties.

Geochemistry

The clustering of chemical properties in different sample locations.

DNA digital data storage

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/DNA_digital_data_storage

DNA digital data storage is the process of encoding and decoding binary data to and from synthesized strands of DNA.

While DNA as a storage medium has enormous potential because of its high storage density, its practical use is currently severely limited because of its high cost and very slow read and write times.

In June 2019, scientists reported that all 16 GB of text from the English Wikipedia had been encoded into synthetic DNA. In 2021, scientists reported that a custom DNA data writer had been developed that was capable of writing data into DNA at 18 Mbps.

Encoding methods

Countless methods for encoding data in DNA are possible. The optimal methods are those that make economical use of DNA and protect against errors. If the message DNA is intended to be stored for a long period of time, for example, 1,000 years, it is also helpful if the sequence is obviously artificial and the reading frame is easy to identify.

Encoding text

Several simple methods for encoding text have been proposed. Most of these involve translating each letter into a corresponding "codon", consisting of a unique small sequence of nucleotides in a lookup table. Some examples of these encoding schemes include Huffman codes, comma codes, and alternating codes.

Encoding arbitrary data

To encode arbitrary data in DNA, the data is typically first converted into ternary (base 3) data rather than binary (base 2) data. Each digit (or "trit") is then converted to a nucleotide using a lookup table. To prevent homopolymers (repeating nucleotides), which can cause problems with accurate sequencing, the result of the lookup also depends on the preceding nucleotide. Using the example lookup table below, if the previous nucleotide in the sequence is T (thymine), and the trit is 2, the next nucleotide will be G (guanine).

Trits to nucleotides (example)

Previous	0	1	2
T	A	C	G
G	T	A	C
C	G	T	A
A	C	G	T

Various systems may be incorporated to partition and address the data, as well as to protect it from errors. One approach to error correction is to regularly intersperse synchronization nucleotides between the information-encoding nucleotides. These synchronization nucleotides can act as scaffolds when reconstructing the sequence from multiple overlapping strands.

In vivo

The genetic code within living organisms can potentially be co-opted to store information. Furthermore synthetic biology can be used to engineer cells with "molecular recorders" to allow the storage and retrieval of information stored in the cell's genetic material. CRISPR gene editing can also be used to insert artificial DNA sequences into the genome of the cell. For encoding developmental lineage data (molecular flight recorder), roughly 30 trillion cell nuclei per mouse * 60 recording sites per nucleus * 7-15 bits per site yields about 2 TeraBytes per mouse written (but only very selectively read).

In-vivo light-based direct image and data recording

A proof-of-concept in-vivo direct DNA data recording system was demonstrated through incorporation of optogenetically regulated recombinases as part of an engineered "molecular recorder" allows for direct encoding of light-based stimuli into engineered E.coli cells. This approach can also be parallelized to store and write text or data in 8-bit form through the use of physically separated individual cell cultures in cell-culture plates.

This approach leverages the editing of a "recorder plasmid" by the light-regulated recombinases, allowing for identification of cell populations exposed to different stimuli. This approach allows for the physical stimulus to be directly encoded into the "recorder plasmid" through recombinase action. Unlike other approaches, this approach does not require manual design, insertion and cloning of artificial sequences to record the data into the genetic code. In this recording process, each individual cell population in each cell-culture plate culture well can be treated as a digital "bit", functioning as a biological transistor capable of recording a single bit of data.

History

The idea of DNA digital data storage dates back to 1959, when the physicist Richard P. Feynman, in "There's Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics" outlined the general prospects for the creation of artificial objects similar to objects of the microcosm (including biological) and having similar or even more extensive capabilities. In 1964–65, Mikhail Samoilovich Neiman, the Soviet physicist, published 3 articles about microminiaturization in electronics at the molecular-atomic level, which independently presented general considerations and some calculations regarding the possibility of recording, storage, and retrieval of information on synthesized DNA and RNA molecules. After the publication of the first M.S. Neiman's paper and after receiving by Editor the manuscript of his second paper (January, the 8th, 1964, as indicated in that paper) the interview with cybernetician Norbert Wiener was published. N. Wiener expressed ideas about miniaturization of computer memory, close to the ideas, proposed by M. S. Neiman independently. These Wiener's ideas M. S. Neiman mentioned in the third of his papers. This story is described in details.

One of the earliest uses of DNA storage occurred in a 1988 collaboration between artist Joe Davis and researchers from Harvard University. The image, stored in a DNA sequence in E.coli, was organized in a 5 x 7 matrix that, once decoded, formed a picture of an ancient Germanic rune representing life and the female Earth. In the matrix, ones corresponded to dark pixels while zeros corresponded to light pixels.

In 2007 a device was created at the University of Arizona using addressing molecules to encode mismatch sites within a DNA strand. These mismatches were then able to be read out by performing a restriction digest, thereby recovering the data.

In 2011, George Church, Sri Kosuri, and Yuan Gao carried out an experiment that would encode a 659 kb book that was co-authored by Church. To do this, the research team did a two-to-one correspondence where a binary zero was represented by either an adenine or cytosine and a binary one was represented by a guanine or thymine. After examination, 22 errors were found in the DNA.

In 2012, George Church and colleagues at Harvard University published an article in which DNA was encoded with digital information that included an HTML draft of a 53,400 word book written by the lead researcher, eleven JPEG images and one JavaScript program. Multiple copies for redundancy were added and 5.5 petabits can be stored in each cubic millimeter of DNA. The researchers used a simple code where bits were mapped one-to-one with bases, which had the shortcoming that it led to long runs of the same base, the sequencing of which is error-prone. This result showed that besides its other functions, DNA can also be another type of storage medium such as hard disk drives and magnetic tapes.

In 2013, an article led by researchers from the European Bioinformatics Institute (EBI) and submitted at around the same time as the paper of Church and colleagues detailed the storage, retrieval, and reproduction of over five million bits of data. All the DNA files reproduced the information with an accuracy between 99.99% and 100%. The main innovations in this research were the use of an error-correcting encoding scheme to ensure the extremely low data-loss rate, as well as the idea of encoding the data in a series of overlapping short oligonucleotides identifiable through a sequence-based indexing scheme. Also, the sequences of the individual strands of DNA overlapped in such a way that each region of data was repeated four times to avoid errors. Two of these four strands were constructed backwards, also with the goal of eliminating errors. The costs per megabyte were estimated at $12,400 to encode data and $220 for retrieval. However, it was noted that the exponential decrease in DNA synthesis and sequencing costs, if it continues into the future, should make the technology cost-effective for long-term data storage by 2023.

In 2013, a software called DNACloud was developed by Manish K. Gupta and co-workers to encode computer files to their DNA representation. It implements a memory efficiency version of the algorithm proposed by Goldman et al. to encode (and decode) data to DNA (.dnac files).

The long-term stability of data encoded in DNA was reported in February 2015, in an article by researchers from ETH Zurich. The team added redundancy via Reed–Solomon error correction coding and by encapsulating the DNA within silica glass spheres via Sol-gel chemistry.

In 2016 research by Church and Technicolor Research and Innovation was published in which, 22 MB of a MPEG compressed movie sequence were stored and recovered from DNA. The recovery of the sequence was found to have zero errors.

In March 2017, Yaniv Erlich and Dina Zielinski of Columbia University and the New York Genome Center published a method known as DNA Fountain that stored data at a density of 215 petabytes per gram of DNA. The technique approaches the Shannon capacity of DNA storage, achieving 85% of the theoretical limit. The method was not ready for large-scale use, as it costs $7000 to synthesize 2 megabytes of data and another $2000 to read it.

In March 2018, University of Washington and Microsoft published results demonstrating storage and retrieval of approximately 200MB of data. The research also proposed and evaluated a method for random access of data items stored in DNA. In March 2019, the same team announced they have demonstrated a fully automated system to encode and decode data in DNA.

Research published by Eurecom and Imperial College in January 2019, demonstrated the ability to store structured data in synthetic DNA. The research showed how to encode structured or, more specifically, relational data in synthetic DNA and also demonstrated how to perform data processing operations (similar to SQL) directly on the DNA as chemical processes.

In April 2019, due to a collaboration with TurboBeads Labs in Switzerland, Mezzanine by Massive Attack was encoded into synthetic DNA, making it the first album to be stored in this way.

In June 2019, scientists reported that all 16 GB of Wikipedia have been encoded into synthetic DNA. In 2021, CATALOG reported that they had developed a custom DNA writer capable of writing data at 18 Mbps into DNA.

The first article describing data storage on native DNA sequences via enzymatic nicking was published in April 2020. In the paper, scientists demonstrate a new method of recording information in DNA backbone which enables bit-wise random access and in-memory computing.

Davos Bitcoin Challenge

On January 21, 2015, Nick Goldman from the European Bioinformatics Institute (EBI), one of the original authors of the 2013 Nature paper, announced the Davos Bitcoin Challenge at the World Economic Forum annual meeting in Davos. During his presentation, DNA tubes were handed out to the audience, with the message that each tube contained the private key of exactly one bitcoin, all coded in DNA. The first one to sequence and decode the DNA could claim the bitcoin and win the challenge. The challenge was set for three years and would close if nobody claimed the prize before January 21, 2018.

Almost three years later on January 19, 2018, the EBI announced that a Belgian PhD student, Sander Wuyts, of the University of Antwerp and Vrije Universiteit Brussel, was the first one to complete the challenge. Next to the instructions on how to claim the bitcoin (stored as a plain text and PDF file), the logo of the EBI, the logo of the company that printed the DNA (CustomArray), and a sketch of James Joyce were retrieved from the DNA.

The Lunar Library

The Lunar Library, launched on the Beresheet Lander by the Arch Mission Foundation, carries information encoded in DNA, which includes 20 famous books and 10,000 images. This was one of the optimal choices of storage, as DNA can last a long time. The Arch Mission Foundation suggests that it can still be read after billions of years.

DNA of things

The concept of the DNA of Things (DoT) was introduced in 2019 by a team of researchers from Israel and Switzerland, including Yaniv Erlich and Robert Grass. DoT encodes digital data into DNA molecules, which are then embedded into objects. This gives the ability to create objects that carry their own blueprint, similar to biological organisms. In contrast to Internet of things, which is a system of interrelated computing devices, DoT creates objects which are independent storage objects, completely off-grid.

As a proof of concept for DoT, the researcher 3D-printed a Stanford bunny which contains its blueprint in the plastic filament used for printing. By clipping off a tiny bit of the ear of the bunny, they were able to read out the blueprint, multiply it and produce a next generation of bunnies. In addition, the ability of DoT to serve for steganographic purposes was shown by producing non-distinguishable lenses which contain a YouTube video integrated into the material.