Small Number Of Clusters Research Articles

Clustering of heterogeneous networks with directional flows based on “Snake” similarities

Aggregated network level modeling and control of traffic in urban networks have recently gained a lot of interest due to unpredictability of travel behaviors and high complexity of physical modeling in microscopic level. Recent research has shown the existence of well-defined Macroscopic Fundamental Diagrams (MFDs) relating average flow and density in homogeneous networks. The concept of MFD allows to design real-time traffic control schemes specifically hierarchical perimeter control approaches to alleviate or postpone congestion. Considering the fact that congestion is spatially correlated in adjacent roads and it propagates spatiotemporaly with finite speed, describing the main pockets of congestion in a heterogeneous city with small number of clusters is conceivable. In this paper, we propose a three-step clustering algorithm to partition heterogeneous networks into connected homogeneous regions, which makes the application of perimeter control feasible. The advantages of the proposed method compared to the existing ones are the ability of finding directional congestion within a cluster, robustness with respect to parameters calibration, and its good performance for networks with low connectivity and missing data. Firstly, we start to find a connected homogeneous area around each road of the network in an iterative way (i.e. it forms a sequence of roads). Each sequence of roads, defined as ‘snake’, is built by starting from a single road and iteratively adding one adjacent road based on its similarity to join previously added roads in that sequence. Secondly, based on the obtained sequences from the first step, a similarity measure is defined between each pair of the roads in the network. The similarities are computed in a way that put more weight on neighboring roads and facilitate connectivity of the clusters. Finally, Symmetric Non-negative Matrix Factorization (SNMF) framework is utilized to assign roads to proper clusters with high intra-similarity and low inter-similarity. SNMF partitions the data by providing a lower rank approximation of the similarity matrix. The proposed clustering framework is applied in medium and large-size networks based on micro-simulation and empirical data from probe vehicles. In addition, the extension of the algorithm is proposed to deal with the networks with sparse measurements where information of some links is missing. The results show the effectiveness and robustness of the extended algorithm applied to simulated network under different penetration rates (percentage of links with data).

K-means Based Cluster Analysis of Residential Smart Meter Measurements

A clustering module based on the k-means cluster analysis method was developed. Smart meter based residential load profiles were used to validate the clustering module. Several case studies were implemented using daily and segmented load profiles of individual and aggregated smart meters. Simulation results defined in terms of the relationship between the clustering ratio and the segmentation time window reveal that the minimum clustering ratio is obtained for the shortest time window of segmentation. Results also show that a small number of clusters is recommended for highly correlated load profiles.

Generalized estimating equations in cluster randomized trials with a small number of clusters: Review of practice and simulation study.

Generalized estimating equations are a common modeling approach used in cluster randomized trials to account for within-cluster correlation. It is well known that the sandwich variance estimator is biased when the number of clusters is small (≤40), resulting in an inflated type I error rate. Various bias correction methods have been proposed in the statistical literature, but how adequately they are utilized in current practice for cluster randomized trials is not clear. The aim of this study is to evaluate the use of generalized estimating equation bias correction methods in recently published cluster randomized trials and demonstrate the necessity of such methods when the number of clusters is small. Review of cluster randomized trials published between August 2013 and July 2014 and using generalized estimating equations for their primary analyses. Two independent reviewers collected data from each study using a standardized, pre-piloted data extraction template. A two-arm cluster randomized trial was simulated under various scenarios to show the potential effect of a small number of clusters on type I error rate when estimating the treatment effect. The nominal level was set at 0.05 for the simulation study. Of the 51 included trials, 28 (54.9%) analyzed 40 or fewer clusters with a minimum of four total clusters. Of these 28 trials, only one trial used a bias correction method for generalized estimating equations. The simulation study showed that with four clusters, the type I error rate ranged between 0.43 and 0.47. Even though type I error rate moved closer to the nominal level as the number of clusters increases, it still ranged between 0.06 and 0.07 with 40 clusters. Our results showed that statistical issues arising from small number of clusters in generalized estimating equations is currently inadequately handled in cluster randomized trials. Potential for type I error inflation could be very high when the sandwich estimator is used without bias correction.

Establishment of a culture condition for strong proliferation and enrichment of chicken spermatogonial stem cells in vitro

Poultry spermatogonial stem cells (SSCs) have the potential to serve as a model for studying the basic biology of SSC and they can also be used for biotechnological purposes. However, the small number of SSCs and the presence of the testicular somatic cells with SSCs have limited their applications. Therefore, this study was undertaken for the first time to investigate the effect of a serum-free medium supplemented with a combination of specific growth factors and B27 on the proliferation and enrichment of newborn chicken SSCs in vitro. Newborn chicken testicular cells were cultured in a serum-free DMEM, supplemented with GDNF, bFGF, LIF, and EGF growth factors and also B27 as an alternative for FBS. Presence and maintenance of the SSCs in the enriched cultures were evaluated by detection of alkaline phosphatase (AP) activity and ASZ1, POU5F1, CVH and GPR125 gene expression. A small number of clusters and colonies were emerged in testicular cell cultures before treatment with the enriched cell culture medium. Enrichment of the DMEM with the above indicated factors strongly promoted the proliferation of the chicken SSCs. Moreover, this culture condition declined attachment and maintenance of the testicular somatic cells and thus they decreased gradually in the cultures. The enriched SSCs were positive for AP activity and with detectable levels of ASZ1, POU5F1, CVH and GPR125 gene expression. This study shows that serum-free medium supplemented with a combination of B27 and the above indicated growth factors induces proliferation and enrichment of chicken SSCs in vitro in a short period of time.

Asymmetric Clustering Algorithm with Gravitational Model (ACAGM)

In wireless sensor networks, the formation of a clustered network topology plays a crucial role in maximizing the network lifetime and effectively improving the network performance. Deploying the scalable wireless sensor networks requires multi-hop operations; in other words, intermediate nodes act as gateways. Most existing works on self-organization in wireless sensor networks do not support the scalability so that a large number of clusters could be generated. Furthermore, producing a smaller number of clusters can tend toward a flat architecture. Thus, the main challenge of these protocols is to find the balance between the size and number of clusters generated in order to have greater energy efficiency. In this paper, we propose a new approach of clustering inspired from the formation of the celestial bodies through universal gravitational model. Thus, each CH candidate node will have a G gravitational parameter expressing the capacity to « attract » sensors around its « orbit ». This parameter is obtained from energy discretization between adjacent clusters. Thanks to gravity, the size and the number of clusters will be neatly and dynamically determined to ensure maximum network lifetime.

Wild Bootstrap Inference for Wildly Different Cluster Sizes

SummaryThe cluster robust variance estimator (CRVE) relies on the number of clusters being sufficiently large. Monte Carlo evidence suggests that the ‘rule of 42’ is not true for unbalanced clusters. Rejection frequencies are higher for datasets with 50 clusters proportional to US state populations than with 50 balanced clusters. Using critical values based on the wild cluster bootstrap performs much better. However, this procedure fails when a small number of clusters is treated. We explain why CRVE t statistics and the wild bootstrap fail in this case, study the ‘effective number’ of clusters and simulate placebo laws with dummy variable regressors. Copyright © 2016 John Wiley & Sons, Ltd.

Bisecting K-Means and 1D Projection Divisive Clustering: A Unified Framework and Experimental Comparison

The paper presents a least squares framework for divisive clustering. Two popular divisive clustering methods, Bisecting K-Means and Principal Direction Division, appear to be versions of the same least squares approach. The PDD recently has been enhanced with a stopping criterion taking into account the minima of the corresponding one-dimensional density function (dePDDP method). We extend this approach to Bisecting K-Means by projecting the data onto random directions and compare thus modified methods. It appears the dePDDP method is superior at datasets with relatively small numbers of clusters, whatever cluster intermix, whereas our version of Bisecting K-Means is superior at greater cluster numbers with noise entities added to the cluster structure.

Speech enhancement using Maximum A-Posteriori and Gaussian Mixture Models for speech and noise Periodogram estimation

In speech enhancement, Gaussian Mixture Models (GMMs) can be used to model the Probability Density Function (PDF) of the Periodograms of speech and different noise types. These GMMs are created by applying the Estimate Maximization (EM) algorithm on large datasets of speech and different noise type Periodograms and hence classify them into a small number of clusters whose centroid Periodograms are the mean vectors of the GMMs. These GMMs are used to realize the Maximum A-Posteriori (MAP) estimation of the speech and noise Periodograms present in a noisy speech observation. To realize the MAP estimation, use of a constrained optimization algorithm is proposed in which relatively good enhancement results with high processing times are attained. Due to the use of constraints in the optimization algorithm, incorrect estimation results may arise due to possible local maxima. A simple analytic MAP algorithm is proposed to attain global maximums in lower calculation times. With the new method the complicated MAP formula is simplified as much as possible to find the maxima, through solving a set of equations and not through conventional numerical methods used in optimization. This method results in excellent speech enhancement with a relatively short processing time.

How important is randomisation in a stepped wedge trial?

In cluster randomised trials, randomisation increases internal study validity. If enough clusters are randomised, an unadjusted analysis should be unbiased. If a smaller number of clusters are included, stratified or matched randomisation can increase comparability between trial arms. In addition, an adjusted analysis may be required; nevertheless, randomisation removes the possibility for systematically biased allocation and increases transparency. In stepped wedge trials, clusters are randomised to receive an intervention at different start times (‘steps’), and all clusters eventually receive it. In a recent study protocol for a ‘modified stepped wedge trial’, the investigators considered randomisation of the clusters (hospital wards), but decided against it for ethical and logistical reasons, and under the assumption that it would not add much to the rigour of the evaluation. We show that the benefits of randomisation for cluster randomised trials also apply to stepped wedge trials. The biggest additional issue for stepped wedge trials in relation to parallel cluster randomised trials is the need to control for secular trends in the outcome. Analysis of stepped wedge trials can in theory be based on ‘horizontal’ or ‘vertical’ comparisons. Horizontal comparisons are based on measurements taken before and after the intervention is introduced in each cluster, and are unbiased if there are no secular trends. Vertical comparisons are based on outcome measurements from clusters that have switched to the intervention condition and those from clusters that have yet to switch, and are unbiased under randomisation since at any time point, which clusters are in intervention and control conditions will have been determined at random. Secular outcome trends are a possibility in many settings. Many stepped wedge trials are analysed with a mixed model, including a random effect for cluster and fixed effects for time period to account for secular trends, thereby combining both vertical and horizontal comparisons of intervention and control clusters. The importance of randomisation in a stepped wedge trial is that the effects of time can be estimated from the data, and bias from secular trends that would otherwise arise can be controlled for, provided the trends are correctly specified in the model.

Automatic color segmentation of breast infrared images using a Gaussian mixture model

Primary breast infrared images are provided in gray-level scale. For better visualization and more accurate diagnosis, they are usually converted to pseudo-color version by camera's software or external algorithms. In pseudo colored version different colors represent different temperatures. The segmentation of these images is done with the aim of division of the image to the regions with different temperatures. This is the first step in analyzing the images and will help in detecting tumor region. In this paper the color segmentation is performed using Gaussian mixture model (GMM). In this method weighted Gaussian components are fit to pixel values in RGB color space. Model parameters are estimated using the popular iterative EM algorithm. After segmentation, clusters are ordered with respect to their average temperature increasing. In order to determine the optimum number of clusters, two cluster validity indices named Calinski–Harabasz and Davies–Bouldin are used. Results showed that Davies–Bouldin index has bias toward small number of clusters. Sometimes this may lead to improper classifying of pixels, so Calinski–Harabasz index is preferred. According to the results and subjective evaluation of the specialist radiologist, the proposed method has good performance in separating regions with different temperature and can be used in screening applications.

Global Illumination Using Well-Separated Pair Decomposition

Instant radiosity methods rely on using a large number of virtual point lights VPLs to approximate global illumination. Efficiency considerations require grouping the VPLs into a small number of clusters that are treated as individual lights with respect to each point to be shaded. Two examples of clustering algorithms are Lightcutsi¾ź[WFA*05] and LightSlicei¾ź[OP11]. In this work, we use the notion of geometric separatedness of point sets as a basis for a data structure for pre-computing and compactly storing a set of candidate VPL clusterings. Our data structure is created prior to rendering, is view-independent and relies only on geometric and radiometric information. For any point to be shaded, we show that a suitable clustering of the VPLs can be efficiently extracted from this data structure. We develop the above framework into an accurate and efficient clustering algorithm based on well-separated pair decompositions which outperforms earlier work in speed and/or quality for diffuse scenes.

The Topology of Overlapping Portfolio Networks

This paper analyzes the topology of the network of common asset holdings, where nodes represent managed portfolios and edge weights capture the impact of liquidations. We consider the degree centrality as the degree in the subnetwork of weak links, where weak links are those that lead to significant liquidations. We explore the applications of this network representation to clustering and forecasting. To validate the weight attribution and the threshold used to define the weak links, we show that the degree centrality is correlated with excess returns, and is significant after we control for the Carhart four factors. The network of weak links has a scale free structure, similar to financial networks of balance sheet exposures. Moreover, a small number of clusters, densely linked, concentrate a significant proportion of the portfolios.

Parcellating connectivity in spatial maps.

A common goal in biological sciences is to model a complex web of connections using a small number of interacting units. We present a general approach for dividing up elements in a spatial map based on their connectivity properties, allowing for the discovery of local regions underlying large-scale connectivity matrices. Our method is specifically designed to respect spatial layout and identify locally-connected clusters, corresponding to plausible coherent units such as strings of adjacent DNA base pairs, subregions of the brain, animal communities, or geographic ecosystems. Instead of using approximate greedy clustering, our nonparametric Bayesian model infers a precise parcellation using collapsed Gibbs sampling. We utilize an infinite clustering prior that intrinsically incorporates spatial constraints, allowing the model to search directly in the space of spatially-coherent parcellations. After showing results on synthetic datasets, we apply our method to both functional and structural connectivity data from the human brain. We find that our parcellation is substantially more effective than previous approaches at summarizing the brain’s connectivity structure using a small number of clusters, produces better generalization to individual subject data, and reveals functional parcels related to known retinotopic maps in visual cortex. Additionally, we demonstrate the generality of our method by applying the same model to human migration data within the United States. This analysis reveals that migration behavior is generally influenced by state borders, but also identifies regional communities which cut across state lines. Our parcellation approach has a wide range of potential applications in understanding the spatial structure of complex biological networks.

The Effect of Small Sample Size on Two-Level Model Estimates: A Review and Illustration

Multilevel models are an increasingly popular method to analyze data that originate from a clustered or hierarchical structure. To effectively utilize multilevel models, one must have an adequately large number of clusters; otherwise, some model parameters will be estimated with bias. The goals for this paper are to (1) raise awareness of the problems associated with a small number of clusters, (2) review previous studies on multilevel models with a small number of clusters, (3) to provide an illustrative simulation to demonstrate how a simple model becomes adversely affected by small numbers of clusters, (4) to provide researchers with remedies if they encounter clustered data with a small number of clusters, and (5) to outline methodological topics that have yet to be addressed in the literature.

Finite-sample corrected generalized estimating equation of population average treatment effects in stepped wedge cluster randomized trials.

Stepped wedge designs are increasingly commonplace and advantageous for cluster randomized trials when it is both unethical to assign placebo, and it is logistically difficult to allocate an intervention simultaneously to many clusters. We study marginal mean models fit with generalized estimating equations for assessing treatment effectiveness in stepped wedge cluster randomized trials. This approach has advantages over the more commonly used mixed models that (1) the population-average parameters have an important interpretation for public health applications and (2) they avoid untestable assumptions on latent variable distributions and avoid parametric assumptions about error distributions, therefore, providing more robust evidence on treatment effects. However, cluster randomized trials typically have a small number of clusters, rendering the standard generalized estimating equation sandwich variance estimator biased and highly variable and hence yielding incorrect inferences. We study the usual asymptotic generalized estimating equation inferences (i.e., using sandwich variance estimators and asymptotic normality) and four small-sample corrections to generalized estimating equation for stepped wedge cluster randomized trials and for parallel cluster randomized trials as a comparison. We show by simulation that the small-sample corrections provide improvement, with one correction appearing to provide at least nominal coverage even with only 10 clusters per group. These results demonstrate the viability of the marginal mean approach for both stepped wedge and parallel cluster randomized trials. We also study the comparative performance of the corrected methods for stepped wedge and parallel designs, and describe how the methods can accommodate interval censoring of individual failure times and incorporate semiparametric efficient estimators.

A new biplot procedure with joint classification of objects and variables by fuzzy c-means clustering

Biplot is a technique for obtaining a low-dimensional configuration of the data matrix in which both the objects and the variables of the data matrix are jointly represented as points and vectors, respectively. However, biplots with a large number of objects and variables remain difficult to interpret. Therefore, in this research, we propose a new biplot procedure that allows us to interpret a large data matrix. In particular, the objects and variables are classified into a small number of clusters by using fuzzy $$c$$c-means clustering and the resulting clusters are simultaneously biplotted in lower-dimensional space. This procedure allows us to understand the configurations easily and to grasp the fuzzy memberships of the objects and variables to the clusters. A simulation study and real data example are also provided to demonstrate the effectiveness of the proposed procedure.

E20 Diffusion Tensor Imaging In Huntington's Disease: Impact Of The Control For Corrupted Data In Comparisons At The Group Level

Background Corrupted gradient directions (GD) in diffusion weighted images may seriously affect reliability of diffusion tensor imaging (DTI)-based comparisons at the group level. In the present study a quality control (QC) algorithm 1 was employed to eliminate corrupted GD from DTI data. Effects of this procedure on comparisons between Huntington disease (HD) subjects and controls were assessed at the group level. Methods Sixty-one HD patients in early stages and forty matched healthy controls were studied in a longitudinal design (baseline and two follow-ups at three time points over 15 months), in a multicenter setting with standardised acquisition protocols on four different MR scanners at four European study sites 2 . QC was used to identify corrupted GD in DTI data sets. Differences in fractional anisotropy (FA) maps at the group level with and without elimination of corrupted GD were analysed. Results The elimination of corrupted GD had an impact on individual FA maps as well as on cross-sectional group comparisons between HD subjects and controls. Following application of the QC algorithm, a lower number of small clusters of FA changes were observed, compared with the analysis without QC-based elimination. However, the grosso modo pattern of regional reductions and increases in FA values was unchanged. Conclusion An impact on the result patterns of the comparison of FA maps between HD subjects and controls was observed depending on whether QC-based elimination of corrupted GD was performed. QC-based elimination of corrupted GD in DTI scans reduces the risk of type I and type II errors in cross-sectional group comparison of FA maps contributing to an increase in reliability and stability of group comparisons. Acknowledgment This work was supported by the European Union under the Seventh Framework programme – PADDINGTON Project, Grant Agreement No. 261358, and the European Huntington’s Disease Network (EHDN), project 070 – PADDINGTON. References Muller HP, et al . PLoS Curr 2011, RRN1232 Muller HP, et al . Neuroimage Clinical 2013,2:161–167

Bayesian estimation with integrated nested Laplace approximation for binary logit mixed models

In multilevel models for binary responses, estimation is computationally challenging due to the need to evaluate intractable integrals. In this paper, we investigate the performance of integrated nested Laplace approximation (INLA), a fast deterministic method for Bayesian inference. In particular, we conduct an extensive simulation study to compare the results obtained with INLA to the results obtained with a traditional stochastic method for Bayesian inference (MCMC Gibbs sampling), and with maximum likelihood through adaptive quadrature. Particular attention is devoted to the case of small number of clusters. The specification of the prior distribution for the cluster variance plays a crucial role and it turns out to be more relevant than the choice of the estimation method. The simulations show that INLA has an excellent performance as it achieves good accuracy (similar to MCMC) with reduced computational times (similar to adaptive quadrature).

Outlier modeling for spectral data reduction.

The spectra in spectral reflectance datasets tend to be quite correlated and therefore they can be represented more compactly using standard techniques such as principal components analysis (PCA) as part of a lossy compression strategy. However, the presence of outlier spectra can often increase the overall error of the reconstructed spectra. This paper introduces a new outlier modeling (OM) method that detects, clusters, and separately models outliers with their own set of basis vectors. Outliers are defined in terms of the robust Mahalanobis distance using the fast minimum covariance determinant algorithm as a robust estimator of the multivariate mean and covariance from which it is computed. After removing the outliers from the main dataset, the performance of PCA on the remaining data improves significantly; however, since outlier spectra are a part of the image, they cannot simply be ignored. The solution is to cluster the outliers into a small number of clusters and then model each cluster separately using its own cluster-specific PCA-derived bases. Tests show that OM leads to lower spectral reconstruction errors of reflectance spectra in terms of both normalized RMS and goodness of fit.

New heuristic for harmonic means clustering

It is well known that some local search heuristics for \(K\)-clustering problems, such as \(k\)-means heuristic for minimum sum-of-squares clustering occasionally stop at a solution with a smaller number of clusters than the desired number \(K\). Such solutions are called degenerate. In this paper, we reveal that the degeneracy also exists in \(K\)-harmonic means (KHM) method, proposed as an alternative to \(K\)-means heuristic, but which is less sensitive to the initial solution. In addition, we discover two types of degenerate solutions and provide examples for both. Based on these findings, we give a simple method to remove degeneracy during the execution of the KHM heuristic; it can be used as a part of any other heuristic for KHM clustering problem. We use KHM heuristic within a recent variant of variable neighborhood search (VNS) based heuristic. Extensive computational analysis, performed on test instances usually used in the literature, shows that significant improvements are obtained if our simple degeneracy correcting method is used within both KHM and VNS. Moreover, our VNS based heuristic suggested here may be considered as a new state-of-the-art heuristic for solving KHM clustering problem.