Gaussian clustering
ArrayMiner 2 : a new clustering model

Different Functional Groups Have Different Variances of Expression Levels

Two groups of genes with different variances of expression levels
    Consider the following situation. The expression levels of a number of genes are measured under two conditions, 1 and 2. The genes belong to two functional groups. Those in the first group are up-regulated in the first condition, but do not show a common tendency in the second condition, their expression levels varying significantly around zero. The genes in the second group are down-regulated in the second condition, but do not show a common tendency in the first, their expression levels varying significantly around zero. Since there are only two conditions, such results are conveniently represented in a two-dimensional drawing, as in the figure on the left, where the first group is drawn in red circles and the second in blue rectangles.

Most Current Clustering Tools Ignore the Fact

Best k-Means clustering of the above data
    Most of the currently popular clustering methods, including k-Means, SOMs, most forms of dendrograms (so-called hierarchical clustering), VxInsight™ by Sandia Labs, etc., ignore the fact that genes in different functional groups are regulated by different biological phenomena, which often gives rise to different variances of expression levels among the clusters, as in the above figure. Indeed, all of those methods rely solely on a measure of distance (similarity) between profiles. This causes them to miss the structure of the data whenever the variances differ among the functional groups, which is most often the case. Thus for instance, the best k-Means clustering of the above data is depicted on the left.

    The flawed clustering supplied by k-Means is due to the fact that a number of genes in the blue cluster above have profiles that are in fact closer to the average of the red cluster.

ArrayMiner2 Takes Variances into Account

    The novel clustering method in ArrayMiner2 is capable of detecting the variances of data in the two clusters along each dimension, and take them into account in deciding the membership of each of the genes. This results in detection of the true structure of the data, leading to the clustering on the left.

    This kind of approach is traditionaly followed by the EM (Expectation Maximization) algorithm. However, EM is extremely slow and known to converge in local optima. Both of these problems are solved in ArrayMiner thanks to its genetic algorithm technology.

ArrayMiner2 clustering of the above data
ArrayMiner2 Detects Outliers

    In addition to different variances of expression levels in different functional groups, real-world data typically feature noise or outliers, expression profiles that cannot be reasonably included in any cluster of the other data. Outliers may be genes with truly unique expression profiles, or may be due to data collection errors. Outliers should be detected and put apart, yet a number of currently available clustering methods, in particular k-Means, nevertheless try to cluster them with the other data at any cost, further deteriorating the quality of the classification. In contrast, ArrayMiner2 has a built-in mechanism for outlier detection.

How Many Clusters ?
Current Methods

k-Means clustering into 2, 3 and 4 clusters, respectively
    Missing the structure of the data has for typical consequence that wildly different "structures" are reported by non-hierarchical clustering methods, such as k-Means or SOMs, when the requested number of clusters changes. For instance, the following figure shows k-Means classifications of a simple set of two-dimensional data into 2, 3 and 4 clusters, respectively.

    The wildly different classifications make it nearly impossible to decide what is the "right" number of clusters in the data.
In contrast, ArrayMiner2's clusters are highly stable and consistent across wide ranges of the number of clusters requested - only the level of detail changes, the reported structure remains stable. This allows the user to obtain the desired level of detail: the "big picture" with a low number of clusters, or minute differences between expression profiles with a high number of clusters. This is illustrated in the figure on the left.

How Many Clusters ?
ArrayMiner2 Method

ArrayMiner2 clustering into 2, 3 and 4 clusters, respectively

    When clustering the same data into two clusters, ArrayMiner2 detected the two large groups of data points as the most salient feature (the "big picture") in the data, as depicted in the figure on the left. The middle group was correctly identified as not being part of either of the two, but since only two clusters were requested, those data points were classified as outliers. When clustering into three clusters, the three groups were correctly identified and no outliers were reported. When clustering into four clusters, a small number of highly similar data points were detected inside the large red group and classified as an extra cluster.

Learn More about the Algorithm

    The above examples illustrate the advantages of ArrayMiner2 in an easy-to-see manner, using simple two-dimensional examples. To learn more about ArrayMiner2's clustering method, and to appreciate its value on real-world data, download the ArrayMiner2 White Paper here.