Stepwise iterative maximum likelihood clustering approach

Abstract

Background: Biological/genetic data is a complex mix of various forms or topologies which makes it quite difficult to analyze. An abundance of such data in this modern era requires the development of sophisticated statistical methods to analyze it in a reasonable amount of time. In many biological/genetic analyses, such as genome-wide association study (GWAS) analysis or multi-omics data analysis, it is required to cluster the plethora of data into sub-categories to understand the subtypes of populations, cancers or any other diseases. Traditionally, the k-means clustering algorithm is a dominant clustering method. This is due to its simplicity and reasonable level of accuracy. Many other clustering methods, including support vector clustering, have been developed in the past, but do not perform well with the biological data, either due to computational reasons or failure to identify clusters.

Results: The proposed SIML clustering algorithm has been tested on microarray datasets and SNP datasets. It has been compared with a number of clustering algorithms. On MLL datasets, SIML achieved highest clustering accuracy and rand score on 4/9 cases; similarly on SRBCT dataset, it got for 3/5 cases; on ALL subtype it got highest clustering accuracy for 5/7 cases and highest rand score for 4/7 cases. In addition, SIML overall clustering accuracy on a 3 cluster problem using SNP data were 97.3, 94.7 and 100 %, respectively, for each of the clusters.

Conclusions: In this paper, considering the nature of biological data, we proposed a maximum likelihood clustering approach using a stepwise iterative procedure. The advantage of this proposed method is that it not only uses the distance information, but also incorporate variance information for clustering. This method is able to cluster when data appeared in overlapping and complex forms. The experimental results illustrate its performance and usefulness over other clustering methods. A Matlab package of this method (SIML) is provided at the web-link http://www.riken.jp/en/research/labs/ims/med_sci_math/ .

Fig. 1

An illustration of stepwise iterative maximum likelihood method using a c = 2 cluster case. In this illustration, two clusters and are given with likelihood functions L₁ and L₂, respectively. The center of clusters are depicted by μ ₁ and μ ₂ (shown as ‘+’ inside two clusters). Initial total likelihood is L_old which is the sum of two likelihood functions (L₁ + L₂). A sample $\mathrm{x}\in$ is checked for grouping. It is advantageous to shift sample $\mathrm{x}$ to cluster only if the new likelihood (L_new = L ₁^* + L ₂^*) is higher than the old likelihood; i.e., L _new > L _old

Fig. 2

An illustration using 3 clusters: a Three cluster data where n = 1500 and d = 2; b k-means clustering, different colors show different clusters; c Support vector clustering (CG method); d Support vector clustering (SEP method); e Stepwise iterative maximum likelihood (SIML) method

Fig. 3

Likelihood plots a L _tot plot, b MaxL _tot plot and c DelL _tot plot

Fig. 4

Processing time of SIML method for n = 3k − 102k and d = 10 − 200

Fig. 5

a Average clustering accuracy on Gaussian data. b Average rand score on Gaussian data

Fig. 6

Clustering by SIML on 2-dimensional BBJ data

Fig. 7

MaxL _tot Plot for 2-dimensional BBJ and HapMap data