Order-preserving Submatrix Research Articles

Mining Order-preserving Submatrices under Data Uncertainty: A Possible-world Approach and Efficient Approximation Methods

Given a data matrix\( D \), a submatrix\( S \)of\( D \)is an order-preserving submatrix (OPSM) if there is a permutation of the columns of\( S \), under which the entry values of each row in\( S \)are strictly increasing. OPSM mining is widely used in real-life applications such as identifying coexpressed genes and finding customers with similar preference. However, noise is ubiquitous in real data matrices due to variable experimental conditions and measurement errors, which makes conventional OPSM mining algorithms inapplicable. No previous work on OPSM has ever considered uncertain value intervals using the well-established possible world semantics.We establish two different definitions of significant OPSMs based on thepossible world semantics: (1) expected support-based and (2) probabilistic frequentness-based. An optimized dynamic programming approach is proposed to compute the probability that a row supports a particular column permutation, with a closed-form formula derived to efficiently handle the special case of uniform value distribution and an accurate cubic spline approximation approach that works well with any uncertain value distributions. To efficiently check the probabilistic frequentness, several effective pruning rules are designed to efficiently prune insignificant OPSMs; two approximation techniques based on the Poisson and Gaussian distributions, respectively, are proposed for further speedup. These techniques are integrated into our two OPSM mining algorithms, based on prefix-projection and Apriori, respectively. We further parallelize our prefix-projection-based mining algorithm using PrefixFPM, a recently proposed framework for parallel frequent pattern mining, and we achieve a good speedup with the number of CPU cores. Extensive experiments on real microarray data demonstrate that the OPSMs found by our algorithms have a much higher quality than those found by existing approaches.

Gradual representation of shadowed set for clustering gene expression data

Micro array has been a widely used microscopic measurement that accumulates the expression levels of a large number of genes varying over different time points. Cluster analysis more over the concept of bi-clustering provides insight into meaningful information from the correlation of a subset of genes with a subset of conditions. This eventually helps in discovering biologically meaningful clusters over analyzing missing values, imprecision and noise present in micro array data set. Although the concept of fuzzy set is enough to deal with the overlapping nature of the bi-clusters but the use of shadowed set helps in identifying and analyzing the nature of the genes lying in the confusion area of the clusters. In this article, we have suggested a bi-clustering model of the shadowed set with gradual representation of cardinality and named it as Gradual shadowed set for gene expression (GSS-GE) clustering. It identifies the bi-clusters in the core and in the shadowed region and evaluates their biological significance. The excellence of the proposed GSS-GE has been demonstrated by considering three real data sets, namely yeast data, serum data and mouse data set. The performance is compared with Ching Church’s algorithm (CC), Bimax, order preserving sub matrix (OPSM), Large Average Sub matrices (LAS), statistical plaid model and a modified fuzzy co-clustering (MFCC) algorithm. For the mouse data set there is no cluster level analysis of the micro array has been done so far. We have also provided the statistical and biological significance to prove the superiority of the proposed GSS-GE.

Indexing and Search of Order-Preserving Submatrix for Gene Expression Data

Bicluster pattern discovery plays a key role in analysis of gene expression data. One vital model of bicluster mining is Order-Preserving SubMatrix (OPSM), which finds similar tendency of some genes on some conditions. Most of the OPSM discovery methods are batch mining techniques and not suitable for low latency data query. To make data analysis efficient and effective, in this paper, we first propose a prefix-tree based indexing method <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pfTree</i> , then give an optimization technique <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pIndex</i> that employs row and column header tables to search the positive, negative and time-delayed OPSMs. Meanwhile, we present an online sharing query technique to accelerate the frequent searches. Finally, we conduct extensive experiments and compare our methods with the existing approaches. Experimental results demonstrate the efficiency and effectiveness of the proposed methods.

Mining of inland water traffic accident data using a biclustering algorithm: A case study of the Yangtze River

Analysis of maritime accident data is important for improving safety management. Clustering is the favoured method of mining marine accident data. However, traditional one-way clustering methods are limited by their focus on global patterns, which does not account for the contingent characteristics of accidents. In this study, biclustering algorithms (BAs) typically used for gene expressions are introduced for analysis of inland water traffic accident data. BAs are good for discovering local patterns (LPs), which represent the similarities between partial accidents and partial attributes. LPs are the more likely modes in accident data, which are difficult to discern using who is traditional one-way clustering. During biclustering of original accident data, six LPs involving replicative accidents are uncovered, thereby suggesting a high risk in similar scenarios. With biclustering of accident attribute factors, the interrelationships among factors are discovered. According to the LPs explored using BAs, high-risk scenarios should gain the attention of shipping companies and safety management departments. Two recommendations are presented: raising awareness of the need for immediate accident reporting and disseminating rescue knowledge. After comparing their applications, the order-preserving submatrix (OPSM) and conserved gene expression motif (xMotifs) algorithms are regarded as the most suitable BAs for analysing maritime accident data.

Discovery of deep order-preserving submatrix in DNA microarray data based on sequential pattern mining

In recent years, order-preserving submatrix (OPSM) model has been widely used in gene expression data analysis. Since it focuses on the changes between the elements rather than the real value, it shows better robustness and statistical significance among results than other models do. Currently, many OPSM algorithms are heuristic. They cannot mine all OPSMs as well as the deep OPSMs which are of biological significance in gene expression data. In this paper, an exact algorithm is proposed to find OPSMs by using frequent sequential pattern mining method. Firstly, we find out all common subsequences (ACS) between any two rows through dynamic programming. Then, we store them into a suffix tree. After that, we can get all OPSMs in this suffix tree, including deep OPSMs. Verified by the real gene data and artificially synthesised data, it is proved that our algorithm is efficient and meaningful.

Constrained query of order-preserving submatrix in gene expression data

Order-preserving submatrix (OPSM) has become important in modelling biologically meaningful subspace cluster, capturing the general tendency of gene expressions across a subset of conditions. With the advance of microarray and analysis techniques, big volume of gene expression datasets and OPSM mining results are produced. OPSM query can efficiently retrieve relevant OPSMs from the huge amount of OPSM datasets. However, improving OPSM query relevancy remains a difficult task in real life exploratory data analysis processing. First, it is hard to capture subjective interestingness aspects, e.g., the analyst's expectation given her/his domain knowledge. Second, when these expectations can be declaratively specified, it is still challenging to use them during the computational process of OPSM queries. With the best of our knowledge, existing methods mainly focus on batch OPSM mining, while few works involve OPSM query. To solve the above problems, the paper proposes two constrained OPSM query methods, which exploit userdefined constraints to search relevant results from two kinds of indices introduced. In this paper, extensive experiments are conducted on real datasets, and experiment results demonstrate that the multi-dimension index (cIndex) and enumerating sequence index (esIndex) based queries have better performance than brute force search.

An apriori-based algorithm for mining semi-order-preserving submatrix

Order-preserving submatrices OPSMs find objects that exhibit a coherent pattern with the same linear ordering in subspace. In general, this problem can be reducible to a special case of the sequential pattern mining problem, where a pattern and its supporting sequences uniquely specify an OPSM. In this paper, we extend the idea of order-preserving submatrix and define a new model semi-order-preserving submatrix or SOPSM that can be generalised to cover most existing bicluster models, and then propose a novel exact algorithm for mining all significant SOPSMs. To reduce the computational costs, we further propose a pruning technique and design an improved data structure for prefix tree to speed up the running time of the algorithm. A set of extensive experiments have been performed which demonstrate the effectiveness and efficiency of our method in mining SOPSMs.

An apriori-based algorithm for mining semi-order-preserving submatrix

Order-preserving submatrices (OPSMs) find objects that exhibit a coherent pattern with the same linear ordering in subspace. In general, this problem can be reducible to a special case of the sequential pattern mining problem, where a pattern and its supporting sequences uniquely specify an OPSM. In this paper, we extend the idea of order-preserving submatrix and define a new model semi-order-preserving submatrix (or SOPSM) that can be generalised to cover most existing bicluster models, and then propose a novel exact algorithm for mining all significant SOPSMs. To reduce the computational costs, we further propose a pruning technique and design an improved data structure for prefix tree to speed up the running time of the algorithm. A set of extensive experiments have been performed which demonstrate the effectiveness and efficiency of our method in mining SOPSMs.

A new approach for the deep order preserving submatrix problem based on sequential pattern mining

As an effective biclustering model, order-preserving submatrix (OPSM) has been widely applied to biological gene expression data mining, which can capture the general tendency of the gene expression under some experimental conditions. Recently, biologists hope to find deep OPSMs with long patterns and comparatively fewer support rows, which are important for the interpretation of gene regulatory networks. However, the traditional exact mining algorithms based on Apriori principle could not deal with the deep OPSM problem, which often take a large minimum support threshold for pattern pruning, and inevitably miss some significant deep OPSMs. In this paper, a new exact algorithm is proposed for mining deep OPSMs. Firstly all the common subsequences shared by every two rows are found out, then the row sets corresponding to the same common subsequences are formed. Finally all the deep OPSMs with support beyond the given threshold will be obtained. Experiments have been done in both real and synthetic data sets, and the results show that this new algorithm is capable of mining all the deep OPSMs over a small support. Under different thresholds of minimum support, this algorithm reveals better performance than the traditional sequential pattern mining algorithms.

Mining order-preserving submatrices from probabilistic matrices

Order-preserving submatrices (OPSMs) capture consensus trends over columns shared by rows in a data matrix. Mining OPSM patterns discovers important and interesting local correlations in many real applications, such as those involving biological data or sensor data. The prevalence of uncertain data in various applications, however, poses new challenges for OPSM mining, since data uncertainty must be incorporated into OPSM modeling and the algorithmic aspects. In this article, we define new probabilistic matrix representations to model uncertain data with continuous distributions. A novel probabilistic order-preserving submatrix (POPSM) model is formalized in order to capture similar local correlations in probabilistic matrices. The POPSM model adopts a new probabilistic support measure that evaluates the extent to which a row belongs to a POPSM pattern. Due to the intrinsic high computational complexity of the POPSM mining problem, we utilize the anti-monotonic property of the probabilistic support measure and propose an efficient Apriori-based mining framework called ProbApri to mine POPSM patterns. The framework consists of two mining methods, UniApri and NormApri , which are developed for mining POPSM patterns, respectively, from two representative types of probabilistic matrices, the UniDist matrix (assuming uniform data distributions) and the NormDist matrix (assuming normal data distributions). We show that the NormApri method is practical enough for mining POPSM patterns from probabilistic matrices that model more general data distributions. We demonstrate the superiority of our approach by two applications. First, we use two biological datasets to illustrate that the POPSM model better captures the characteristics of the expression levels of biologically correlated genes and greatly promotes the discovery of patterns with high biological significance. Our result is significantly better than the counterpart OPSMRM (OPSM with repeated measurement) model which adopts a set-valued matrix representation to capture data uncertainty. Second, we run the experiments on an RFID trace dataset and show that our POPSM model is effective and efficient in capturing the common visiting subroutes among users.

Economic Regionalization based on Order-preserving Submatrix

In the paper, We compared the classical K-means clustering algorithm, the fuzzy clustering with the Order-Preserving Submatrix(OPSM) biclustering algorithm on the dataset of regional fiscal revenue which is collected from the National Statistical Yearbook. The experimental results proves that the OPSM biclustering algorithm could get more interesting results than the classical K-means algorithm and the fuzzy clustering algorithm, which shows more detailed information than the latter either.

Approximation Algorithms for a Minimization Variant of the Order-Preserving Submatrices and for Biclustering Problems

Finding a largest Order-Preserving SubMatrix, OPSM, is an important problem arising in the discovery of patterns in gene expression. Ben-Dor et al. formulated the problem in Ben-Dor et al. [2003]. They further showed that the problem is NP-complete and provided a greedy heuristic for the problem. The complement of the OPSM problem, called MinOPSM, is to delete the least number of entries in the matrix so that the remaining submatrix is order preserving. We devise a 5-approximation algorithm for the MinOPSM based on a formulation of the problem as a quadratic, nonseparable set cover problem. An alternative formulation combined with a primal-dual algorithm improves the approximation factor to 3. The complexity of both algorithms for a matrix of size m × n is O ( m 2 n ). We further comment on the related biclustering problem.

On the Deep Order-Preserving Submatrix Problem: A Best Effort Approach

Order-preserving submatrix (OPSM) has been widely accepted as a biologically meaningful cluster model, capturing the general tendency of gene expression across a subset of experiments. In an OPSM, the expression levels of all genes induce the same linear ordering of the experiments. The OPSM problem is to discover those statistically significant OPSMs from a given data matrix. The problem is reducible to a special case of the sequential pattern mining problem, where a pattern and its supporting sequences uniquely specify an OPSM. Unfortunately, existing methods do not scale well to massive data sets containing thousands of experiments and hundreds of thousands of genes, which are common in today's gene expression analysis. In particular, deep OPSMs, corresponding to long patterns with few supporting sequences, incur explosive computational costs in their discovery and are completely pruned off by existing methods. However, it is of particular interest of biologists to determine small groups of genes that are tightly coregulated across many experiments, and some pathways or processes may require as few as two genes to act in concert. In this paper, we study the discovery of deep OPSMs from massive data sets. We propose a novel best effort mining framework Kiwi that exploits two parameters k and w to bound the available computational resources and search a selected search space, and does what it can to find as many as possible deep OPSMs. Extensive biological and computational evaluations on real data sets demonstrate the validity and importance of the deep OPSM problem, and the efficiency and effectiveness of the Kiwi mining framework.

Solving the Order-Preserving Submatrix Problem via Integer Programming

In this paper we consider the order-preserving submatrix (OPSM) problem. This problem is known to be NP-hard. Although in recent years some heuristic methods have been presented to find OPSMs, they lack the guarantee of optimality. We present exact solution approaches based on linear mixed 0–1 programming formulations and develop algorithmic enhancements to aid in solvability. Encouraging computational results are reported both for synthetic and real biological data. In addition, we discuss theoretical computational complexity issues related to finding fixed patterns in matrices.

Algorithmic and Complexity Issues of Three Clustering Methods in Microarray Data Analysis

The complexity, approximation and algorithmic issues of several clustering problems are studied. These non-traditional clustering problems arise from recent studies in microarray data analysis. We prove the following results. (1) Two variants of the Order-Preserving Submatrix problem are NP-hard. There are polynomial-time algorithms for the Order-Preserving Submatrix problem when the condition or gene sets are fixed. (2) Three variants of the Smooth Clustering problem are NP-hard. The Smooth Subset problem is approximable with ratio 0.5, but it cannot be approximable with ratio 0.5 + δ for any δ > 0 unless NP = P. (3) The inferring plaid model problem is NP-hard.

On mining micro-array data by Order-Preserving Submatrix

We study the problem of pattern-based subspace clustering which is clustering by pattern similarity finds objects that exhibit a coherent pattern of rises and falls in subspaces. Applications of pattern-based subspace clustering include DNA micro-array data analysis. Our goal is to devise pattern-based clustering methods that are capable of: discovering useful patterns of various shapes, and discovering all significant patterns. Our approach is to extend the idea of Order-Preserving Submatrix (OPSM). We devise a novel algorithm for mining OPSM, show that OPSM can be generalised to cover most existing pattern-based clustering models and propose a number of extensions to the original OPSM model.

Discovering local structure in gene expression data: the order-preserving submatrix problem.

This paper concerns the discovery of patterns in gene expression matrices, in which each element gives the expression level of a given gene in a given experiment. Most existing methods for pattern discovery in such matrices are based on clustering genes by comparing their expression levels in all experiments, or clustering experiments by comparing their expression levels for all genes. Our work goes beyond such global approaches by looking for local patterns that manifest themselves when we focus simultaneously on a subset G of the genes and a subset T of the experiments. Specifically, we look for order-preserving submatrices (OPSMs), in which the expression levels of all genes induce the same linear ordering of the experiments (we show that the OPSM search problem is NP-hard in the worst case). Such a pattern might arise, for example, if the experiments in T represent distinct stages in the progress of a disease or in a cellular process and the expression levels of all genes in G vary across the stages in the same way. We define a probabilistic model in which an OPSM is hidden within an otherwise random matrix. Guided by this model, we develop an efficient algorithm for finding the hidden OPSM in the random matrix. In data generated according to the model, the algorithm recovers the hidden OPSM with a very high success rate. Application of the methods to breast cancer data seem to reveal significant local patterns.