Non-parametric Clustering Algorithm Research Articles

UNIC: A fast nonparametric clustering

Clustering is among the tools for exploring, analyzing, and deriving information from data. In the case of large data sets, the real burden to the application of clustering algorithms can be their complexity and demand of control parameters. We present a new fast nonparametric clustering algorithm, UNIC, to address these challenges. To identify clusters, the algorithm evaluates the distances between selected points and other points in the set. While assessing these distances, it employs methods of robust statistics to identify the cluster borders. The performance of the proposed algorithm is assessed in an experimental study and compared with several existing clustering methods over a variety of benchmark data sets.

Semi-supervised learning through hierarchical clustering for interactive aerospace image analysis

A new semi-supervised classification algorithm based on the non-parametric clustering algorithm HCA is proposed. The algorithm obtains hierarchical segmentation result where additional classes that are not represented in the training samples can be found. High performance of the algorithm allows using it in interactive mode. Experimental studies confirm that the proposed algorithm provides aerospace image classification in conditions of limited number of training samples.

Semiparametric Clustering: A Robust Alternative to Parametric Clustering.

Clustering aims at naturally grouping the data according to the underlying data distribution. The data distribution is often estimated using a parametric or nonparametric model, e.g., Gaussian mixture or kernel density estimation. Compared with nonparametric models, parametric models are statistically stable, i.e., a small perturbation of data points leads to a small change in the estimated density. However, parametric models are highly sensitive to outliers because the data distribution is far away from the parametric assumptions in the presence of outliers. Given a parametric clustering algorithm, this paper shows how to turn this algorithm into a robust one. The idea is to modify the original parametric density into a semiparametric one. The high-density data that form the core of each cluster are modeled with the original parametric density. The low-density data are often far away from the cluster cores and may have an arbitrary shape, thus are modeled using a nonparametric density. A combination of parametric and nonparametric clustering algorithms is used to group the data modeled as a semiparametric density. From the robust statistical point of view, the proposed method has good robustness properties. We test the proposed algorithm on several synthetic and 70 UCI data sets. The results indicate that the semiparametric method could significantly improve the clustering performance.

POI Based Urban Commercial Centers Identification and Classification

Recently, urban commercial centers’ layout attracts great attention in research fields of urban planning and economics. This paper proposes an approach, which can identify the boundary of urban commercial centers and classify these centers into a hierarchical classes by using open data source POIs. Two non-parametric algorithm: modified ASCDT and the k-means clustering algorithm are used to measure the agglomeration effect of commercial industries, detect industry centers and commercial centers, and classify industry centers and commercial centers. As a case study, this approach is applied to Harbin, China and find out 374 commercial centers in four levels. To some extent, this result reflects the reality of commercial centers distribution of Harbin. This result provide the information support of layout planning and location decision-making for planners and investors. The non-parametric adaptive clustering algorithm and accessible POIs data are used to make the approach applying to extensive region and more cities.

New Binding Mode of SLURP Protein to a7 Nicotinic Acetylcholine Receptor Revealed by Computer Simulations

SLURP-1 is a member of three-finger toxin-like proteins. Their characteristic feature is a set of three beta strands extruding from hydrophobic core stabilized by disulfide bonds. Each beta-strand carries a flexible loop, which is responsible for recognition. SLURP-1 was recently shown to act as an endogenous growth regulator of keratinocytes and tumor suppressor by reducing cell migration and invasion by antagonizing the pro-malignant effects of nicotine. This effect is achieved through allosteric interaction with alpha7 nicotinic acetylcholine receptors (alpha-7 nAChRs) in an antagonist-like manner. Moreover, this interaction is unaffected by several well-known agents specifically alpha-bungarotoxin. In this work, we carry out the conformational analysis of the SLURP-1 by a microsecond-long full-atom explicit solvent molecular dynamics simulations followed by clustering, to identify representative states. To achieve this timescale we employed a GPU-accelerated version of GROMACS modeling package. To avoid human bias in clustering we used a non-parametric clustering algorithm Affinity Propagation adapted for biomolecules and HPC environments. Then, we applied protein-protein molecular docking of the ten most massive clusters to alpha7-nAChRs in order to test if structural variability can affect binding. Docking simulations revealed the unusual binding mode of one of the minor SLURP-1 conformations.

Bayesian exploratory clustering with entropy Chinese restaurant process

Data exploration is essential to data analytics, especially when one is confronted with massive datasets. Clustering is a commonly used technique in data exploration, since it can automatically group data instances into a list of meaningful categories, and capture the natural structure of data. Tra ditional finite mixture model requires the number of clusters to be specified in advance of analyzing the data, and this parameter is crucial to the clustering performance. Chinese restaurant process (CRP) mixture model provides an alternative to this problem, allowing the model complexity to grow as more data instances are observed. Although CRP provides the flexibility to create a new cluster for subsequent data instances, one still has to determine the hyperparameter of the prior and the parameters for the base distribution in the likelihood part. This work proposes a non-parametric clustering algorithm based on CRP with two main differences. First, we propose to create a new cluster based on entropy of the posterior, whereas the CRP uses a hyperparameter to control the probability of creating a new cluster. Second, we propose to dynamically adjust the parameters of the base distribution according to the mean of the observed data owing to Chebyshev’s inequality. Additionally, detailed derivation and update rules are provided to perform posterior inference with the proposed collapsed Gibbs sampling algorithm. The experimental results indicate that the proposed algorithm avoids to specify the number of clusters and works well on several datasets.

Inference of genetic relatedness between viral quasispecies from sequencing data

BackgroundRNA viruses such as HCV and HIV mutate at extremely high rates, and as a result, they exist in infected hosts as populations of genetically related variants. Recent advances in sequencing technologies make possible to identify such populations at great depth. In particular, these technologies provide new opportunities for inference of relatedness between viral samples, identification of transmission clusters and sources of infection, which are crucial tasks for viral outbreaks investigations.ResultsWe present (i) an evolutionary simulation algorithm Viral Outbreak InferenCE (VOICE) inferring genetic relatedness, (ii) an algorithm MinDistB detecting possible transmission using minimal distances between intra-host viral populations and sizes of their relative borders, and (iii) a non-parametric recursive clustering algorithm Relatedness Depth (ReD) analyzing clusters’ structure to infer possible transmissions and their directions. All proposed algorithms were validated using real sequencing data from HCV outbreaks.ConclusionsAll algorithms are applicable to the analysis of outbreaks of highly heterogeneous RNA viruses. Our experimental validation shows that they can successfully identify genetic relatedness between viral populations, as well as infer transmission clusters and outbreak sources.

Semi-Supervised Deep Learning Using Pseudo Labels for Hyperspectral Image Classification.

Deep learning has gained popularity in a variety of computer vision tasks. Recently, it has also been successfully applied for hyperspectral image classification tasks. Training deep neural networks, such as a convolutional neural network for classification requires a large number of labeled samples. However, in remote sensing applications, we usually only have a small amount of labeled data for training because they are expensive to collect, although we still have abundant unlabeled data. In this paper, we propose semi-supervised deep learning for hyperspectral image classification-our approach uses limited labeled data and abundant unlabeled data to train a deep neural network. More specifically, we use deep convolutional recurrent neural networks (CRNN) for hyperspectral image classification by treating each hyperspectral pixel as a spectral sequence. In the proposed semi-supervised learning framework, the abundant unlabeled data are utilized with their pseudo labels (cluster labels). We propose to use all the training data together with their pseudo labels to pre-train a deep CRNN, and then fine-tune using the limited available labeled data. Further, to utilize spatial information in the hyperspectral images, we propose a constrained Dirichlet process mixture model (C-DPMM), a non-parametric Bayesian clustering algorithm, for semi-supervised clustering which includes pairwise must-link and cannot-link constraints-this produces high-quality pseudo-labels, resulting in improved initialization of the deep neural network. We also derived a variational inference model for the C-DPMM for efficient inference. Experimental results with real hyperspectral image data sets demonstrate that the proposed semi-supervised method outperforms state-of-the-art supervised and semi-supervised learning methods for hyperspectral classification.

Visualizing correlated motion with HDBSCAN clustering.

Correlated motion analysis provides a method for understanding communication between and dynamic similarities of biopolymer residues and domains. The typical equal-time correlation matrices-frequently visualized with pseudo-colorings or heat maps-quickly convey large regions of highly correlated motion but hide more subtle similarities of motion. Here we propose a complementary method for visualizing correlations within proteins (or general biopolymers) that quickly conveys intuition about which residues have a similar dynamic behavior. For grouping residues, we use the recently developed non-parametric clustering algorithm HDBSCAN. Although the method we propose here can be used to group residues using correlation as a similarity matrix-the most straightforward and intuitive method-it can also be used to more generally determine groups of residues which have similar dynamic properties. We term these latter groups "Dynamic Domains", as they are based not on spatial closeness but rather closeness in the column space of a correlation matrix. We provide examples of this method across three human proteins of varying size and function-the Nf-Kappa-Beta essential modulator, the clotting promoter Thrombin and the mismatch repair protein (dimer) complex MutS-alpha. Although the examples presented here are from all-atom molecular dynamics simulations, this visualization technique can also be used on correlations matrices built from any ensembles of conformations from experiment or computation.

Mapping forest fires by nonparametric clustering analysis

Fires have a noteworthy role to play with regards to ecological and environmental losses in Mediterranean forests. In addition to ecological impacts, fire may create economic, social as well as cultural changes. The detection of fire-scars has critical importance to help decrease losses. In the present study, forest fires recorded in Antalya, one of the most important ecological and tourist regions within the Western Mediterranean, were clustered and mapped. Since the dominant factors and devastation records derived from the cases had nominal-scaled properties, a categorical data-based nonparametric clustering algorithm was performed in this evaluation. The proposed tool, k-modes algorithm, uses modes instead of means for clustering. The algorithm may be implemented quickly and does not make distributional assumptions concerning the available data. It uses a frequency-based method to update the modes of the fires. The derived modes from the maps may be useful information for local authorities to manage. In conclusion, the proposed nonparametric clustering procedure may be employed to build a decision-support system to monitor and identify fire activities and to enhance fire management efficiency.

Lineage diversification of fringe-toed lizards (Phrynosomatidae: Uma notata complex) in the Colorado Desert: Delimiting species in the presence of gene flow

Multi-locus nuclear DNA data were used to delimit species of fringe-toed lizards of the Uma notata complex, which are specialized for living in wind-blown sand habitats in the deserts of southwestern North America, and to infer whether Quaternary glacial cycles or Tertiary geological events were important in shaping the historical biogeography of this group. We analyzed ten nuclear loci collected using Sanger sequencing and genome-wide sequence/single-nucleotide polymorphism (SNP) data collected using restriction-associated DNA (RAD) sequencing. A combination of species discovery methods (concatenated phylogenies, parametric and non-parametric clustering algorithms) and species validation approaches (coalescent-based species tree/isolation-with-migration models) were used to delimit species, infer phylogenetic relationships, and to estimate effective population sizes, migration rates, and speciation times. Uma notata, U. inornata, U. cowlesi, and an undescribed species from Mohawk Dunes, Arizona (U. sp.) were supported as distinct in the concatenated analyses and by clustering algorithms, and all operational taxonomic units were decisively supported as distinct species by ranking hierarchical nested speciation models with Bayes factors based on coalescent-based species tree methods. However, significant unidirectional gene flow (2NM>1) from U. cowlesi and U. notata into U. rufopunctata was detected under the isolation-with-migration model. Therefore, we conservatively delimit four species-level lineages within this complex (U. inornata, U. notata, U. cowlesi, and U. sp.), treating U. rufopunctata as a hybrid population (U. notata×cowlesi). Both concatenated and coalescent-based estimates of speciation times support the hypotheses that speciation within the complex occurred during the late Pleistocene, and that the geological evolution of the Colorado River delta during this period was an important process shaping the observed phylogeographic patterns.

Fast Nonparametric Density-Based Clustering of Large Datasets Using a Stochastic Approximation Mean-Shift Algorithm

Mean-shift is an iterative procedure often used as a nonparametric clustering algorithm that defines clusters based on the modal regions of a density function. The algorithm is conceptually appealing and makes assumptions neither about the shape of the clusters nor about their number. However, with a complexity of O(n2) per iteration, it does not scale well to large datasets. We propose a novel algorithm which performs density-based clustering much quicker than mean shift, yet delivering virtually identical results. This algorithm combines subsampling and a stochastic approximation procedure to achieve a potential complexity of O(n) at each step. Its convergence is established. Its performances are evaluated using simulations and applications to image segmentation, where the algorithm was tens or hundreds of times faster than mean shift, yet causing negligible amounts of clustering errors. The algorithm can be combined with existing approaches to further accelerate clustering.

An incremental nonparametric Bayesian clustering-based traversable region detection method

Navigation capability in complex and unknown outdoor environments is one of the major requirements for an autonomous vehicle and a robot that perform tasks such as a military mission or planetary exploration. Robust traversability estimation in unknown environments would allow the vehicle or the robot to devise control and planning strategies to maximize their effectiveness. In this study, we present a self-supervised on-line learning architecture to estimate the traversability in complex and unknown outdoor environments. The proposed approach builds a model by clustering appearance data using the newly proposed incremental nonparametric Bayesian clustering algorithm. The clusters are then classified as being either traversable or non-traversable. Because our approach effectively groups unknown regions with similar properties, while the vehicle is in motion without human intervention, the vehicle can be deployed to new environments by automatically adapting to changing environmental conditions. We demonstrate the performance of the proposed clustering algorithm through intensive experiments using synthetic and real data and evaluate the viability of the traversability estimation using real data sets collected in outdoor environment.

Bayesian Nonparametric Clustering for Positive Definite Matrices.

Symmetric Positive Definite (SPD) matrices emerge as data descriptors in several applications of computer vision such as object tracking, texture recognition, and diffusion tensor imaging. Clustering these data matrices forms an integral part of these applications, for which soft-clustering algorithms (K-Means, expectation maximization, etc.) are generally used. As is well-known, these algorithms need the number of clusters to be specified, which is difficult when the dataset scales. To address this issue, we resort to the classical nonparametric Bayesian framework by modeling the data as a mixture model using the Dirichlet process (DP) prior. Since these matrices do not conform to the Euclidean geometry, rather belongs to a curved Riemannian manifold,existing DP models cannot be directly applied. Thus, in this paper, we propose a novel DP mixture model framework for SPD matrices. Using the log-determinant divergence as the underlying dissimilarity measure to compare these matrices, and further using the connection between this measure and the Wishart distribution, we derive a novel DPM model based on the Wishart-Inverse-Wishart conjugate pair. We apply this model to several applications in computer vision. Our experiments demonstrate that our model is scalable to the dataset size and at the same time achieves superior accuracy compared to several state-of-the-art parametric and nonparametric clustering algorithms.

Abstract B2-59: PhyloSpan: Using multi-mutation reads to resolve subclonal architectures from heterogeneous tumor samples

Abstract We have developed a new method that uses high-throughput reads that span multiple somatic point mutations to reconstruct multiple, genetically diverse subclonal populations from one or more heterogeneous tumor samples. Tumors often contain multiple, genetically diverse subclonal populations, as predicted by the clonal theory of cancer. These subclonal populations develop through successive waves of expansion and selection and have differing abilities to metastasize and resist treatment. Identifying these sub-populations and their evolutionary relationships can help identify driver mutations associated with cancer development and progression. Subclonal reconstruction algorithms attempt to infer the prevalence and genotype of multiple, genetically-related subclonal populations using the variant allele frequency (VAF) of somatic variants. To date, these algorithms exclusively use data on individual somatic mutations. This restriction greatly reduces their ability to fully resolve phylogenic ambiguities. In some cases, it is possible to determine the mutation status of >1 mutation in a single cell, for example, when single reads cover multiple single nucleotide variants (SNVs). This type of information can add considerable power to the phylogenetic reconstruction of the tumor subclonal population. We have developed the PhyloSpan algorithm which attempts to infer the states of multiple SNVs in single cells, and then exploits that information in subclonal reconstruction. Our algorithm starts with phasing somatic SNVs by looking for reads / read-pairs that cover both a somatic mutation and germline heterozygous single nucleotide polymorphism (SNP). These germline SNPs are often available through profiling of normal tissue. PhyloSpan then identifies SNVs that are on the same chromosome and close enough to be covered by a single read or paired reads. These pairs of mutations provide more phylogenetic certainty than can be found by looking at mutations independently. For example, if those SNVs are found in the same evolutionary branch, then we expect to see some reads containing both mutations. If however, the SNVs are an separate branches then no reads should show both SNVs. PhyloSpan integrates this phylogenetic information, along with information about the VAF of each somatic SNV in order to perform subclonal reconstruction. Incorporating these various types of information, especially given the substantial uncertainty in phasing and NGS read content, requires a rigorous statistical approach and so we have developed a Bayesian non-parametric tree-based clustering algorithm, based on our existing PhyloWGS method. This algorithm not only infers the number of subclonal populations and their genotype but also provides a measure of uncertainty about this inference, enabling users to determine which parts of the subclonal reconstruction are certain and which parts remain ambiguous. While the number of SNVs a short-read length distance away from another SNV is small, a handful of such pairs are all that is needed to eliminate a substantial amount of ambiguity in subclonal reconstruction. Furthermore, long (>10k) read technologies, such as PacBio, can be used to supplement short read sequence. Our approach generalizes to permit the integration of single-cell sequencing with bulk tumor sequencing. Furthermore, we can also use our framework to identify a small number of SNVs for which low throughput assays would be most useful to resolve subclonal reconstruction ambiguity. We will present results applying our algorithm to whole genome sequencing data showing the added value of considering multiple SNVs compared to independent SNVs. Citation Format: Amit G. Deshwar, Levi Boyles, Jeff Wintersinger, Paul C. Boutros, Yee Whye Teh, Quaid Morris, Quaid Morris. PhyloSpan: Using multi-mutation reads to resolve subclonal architectures from heterogeneous tumor samples. [abstract]. In: Proceedings of the AACR Special Conference on Computational and Systems Biology of Cancer; Feb 8-11 2015; San Francisco, CA. Philadelphia (PA): AACR; Cancer Res 2015;75(22 Suppl 2):Abstract nr B2-59.

Inter-functional analysis of high-throughput phenotype data by non-parametric clustering and its application to photosynthesis.

Phenomics is the study of the properties and behaviors of organisms (i.e. their phenotypes) on a high-throughput scale. New computational tools are needed to analyze complex phenomics data, which consists of multiple traits/behaviors that interact with each other and are dependent on external factors, such as genotype and environmental conditions, in a way that has not been well studied. We deployed an efficient framework for partitioning complex and high dimensional phenotype data into distinct functional groups. To achieve this, we represented measured phenotype data from each genotype as a cloud-of-points, and developed a novel non-parametric clustering algorithm to cluster all the genotypes. When compared with conventional clustering approaches, the new method is advantageous in that it makes no assumption about the parametric form of the underlying data distribution and is thus particularly suitable for phenotype data analysis. We demonstrated the utility of the new clustering technique by distinguishing novel phenotypic patterns in both synthetic data and a high-throughput plant photosynthetic phenotype dataset. We biologically verified the clustering results using four Arabidopsis chloroplast mutant lines. Software is available at www.msu.edu/~jinchen/NPM. Supplementary data are available at Bioinformatics online. jinchen@msu.edu, kramerd8@cns.msu.edu or rongjin@cse.msu.edu.

Abstract 4865: PhyloSpan: using multi-mutation reads to resolve subclonal architectures from heterogeneous tumor samples

Abstract We have developed a new method that uses high-throughput reads that span multiple somatic point mutations to reconstruct multiple, genetically diverse subclonal populations from one or more heterogeneous tumor samples. Subclonal reconstruction algorithms attempt to infer the prevalence and genotype of multiple, genetically-related subclonal populations using the variant allele frequency (VAF) of somatic variants. To date, these algorithms exclusively use data on individual somatic mutations. This restriction greatly reduces their ability to fully resolve phylogenic ambiguities. In some cases, it is possible to determine the mutation status of >1 mutation in a single cell, for example, when single reads cover multiple single nucleotide variants (SNVs). This type of information can add considerable power to the phylogenetic reconstruction of the tumor subclonal population. We have developed the PhyloSpan algorithm which attempts to infer the states of multiple SNVs in single cells, and then exploits that information in subclonal reconstruction. Our algorithm starts with phasing somatic SNVs by looking for reads / read-pairs that cover both a somatic mutation and germline heterozygous single nucleotide polymorphism (SNP). These germline SNPs are often available through profiling of normal tissue. PhyloSpan then identifies SNVs that are on the same chromosome and close enough to be covered by a single read or paired reads. These pairs of mutations provide more phylogenetic certainty than can be found by looking at mutations independently. For example, if those SNVs are found in the same evolutionary branch, then we expect to see some reads containing both mutations. If however, the SNVs are an separate branches then no reads should show both SNVs. PhyloSpan integrates this phylogenetic information, along with information about the VAF of each somatic SNV in order to perform subclonal reconstruction. Incorporating these various types of information requires a rigorous statistical approach, and so we have developed a Bayesian non-parametric tree-based clustering algorithm. This algorithm not only infers the number of subclonal populations and their genotype but also provides a measure of uncertainty about this inference, enabling users to determine which parts of the subclonal reconstruction are certain and which parts remain ambiguous. While the number of SNVs a short-read length distance away from another SNV is small, a handful of such pairs are all that is needed to eliminate a substantial amount of ambiguity in subclonal reconstruction. Furthermore, long read technologies, such as PacBio, can be used to supplement short reads. Our approach generalizes to permit the integration of single-cell sequencing with bulk tumor sequencing. We will present results applying our algorithm to whole genome sequencing data showing the added value of considering multiple SNVs compared to independent SNVs. Citation Format: Amit G. Deshwar, Levi Boyles, Jeff Wintersinger, Paul C. Boutros, Yee Whye Teh, Quaid Morris. PhyloSpan: using multi-mutation reads to resolve subclonal architectures from heterogeneous tumor samples. [abstract]. In: Proceedings of the 106th Annual Meeting of the American Association for Cancer Research; 2015 Apr 18-22; Philadelphia, PA. Philadelphia (PA): AACR; Cancer Res 2015;75(15 Suppl):Abstract nr 4865. doi:10.1158/1538-7445.AM2015-4865

Numerical simulation of hydraulic fracturing and associated microseismicity using finite-discrete element method

Hydraulic fracturing (HF) technique has been extensively used for the exploitation of unconventional oil and gas reservoirs. HF enhances the connectivity of less permeable oil and gas-bearing rock formations by fluid injection, which creates an interconnected fracture network and increases the hydrocarbon production. Meanwhile, microseismic (MS) monitoring is one of the most effective approaches to evaluate such stimulation process. In this paper, the combined finite-discrete element method (FDEM) is adopted to numerically simulate HF and associated MS. Several post-processing tools, including frequency-magnitude distribution (b-value), fractal dimension (D-value), and seismic events clustering, are utilized to interpret numerical results. A non-parametric clustering algorithm designed specifically for FDEM is used to reduce the mesh dependency and extract more realistic seismic information. Simulation results indicated that at the local scale, the HF process tends to propagate following the rock mass discontinuities; while at the reservoir scale, it tends to develop in the direction parallel to the maximum in-situ stress.

A nonparametric clustering algorithm with a quantile-based likelihood estimator.

Clustering is a representative of unsupervised learning and one of the important approaches in exploratory data analysis. By its very nature, clustering without strong assumption on data distribution is desirable. Information-theoretic clustering is a class of clustering methods that optimize information-theoretic quantities such as entropy and mutual information. These quantities can be estimated in a nonparametric manner, and information-theoretic clustering algorithms are capable of capturing various intrinsic data structures. It is also possible to estimate information-theoretic quantities using a data set with sampling weight for each datum. Assuming the data set is sampled from a certain cluster and assigning different sampling weights depending on the clusters, the cluster-conditional information-theoretic quantities are estimated. In this letter, a simple iterative clustering algorithm is proposed based on a nonparametric estimator of the log likelihood for weighted data sets. The clustering algorithm is also derived from the principle of conditional entropy minimization with maximum entropy regularization. The proposed algorithm does not contain a tuning parameter. The algorithm is experimentally shown to be comparable to or outperform conventional nonparametric clustering methods.

Sensing-Transmission Edifice Using Bayesian Nonparametric Traffic Clustering in Cognitive Radio Networks

In cognitive radio networks, the main objective of spectrum sensing is to exploit spectrum holes left by the primary users (PUs). Different PUs' traffic patterns might provide different opportunities for second user (SU) spectrum access. In this paper, we identify the PUs' traffic patterns and then maximize SU transmission accordingly. First a theoretical framework is developed to cluster PU traffic patterns based on a Bayesian nonparametric inference model, in which the number of traffic types is unknown. Second, in order to exploit the spectrum holes, we study a sensing-transmission structure to optimize the SU transmission strategy. Specifically, we exploit the short and long transmission opportunities based on the PU traffic pattern and channel idle time distribution. Finally, we propose a threshold-based sensing-transmission method that optimizes the SU utility, while protecting PU transmissions. Both sensing and transmission errors are considered for perfect sensing with/without acknowledgement-based transmission and imperfect sensing, respectively. From the simulation results, we show that the proposed technique outperforms the nonparametric mean shift clustering algorithm. Furthermore, we utilize these clustering results to optimize the SU's transmission strategy with perfect and imperfect sensing. We compare our proposed technique with the probabilistic sensing-transmission structure and show the performance gain in terms of throughput.