Mixture model clustering

Blashfield, R. K. (1976). Mixture model tests of cluster analysis Accuracy of four agglomerative hierarchical methods. Psychological Bulletin, 83, 377-388. 

Cleland, C., Rothschild, L., Haslam, N. (2000). Detecting latent taxa Monte Carlo comparison of taxometric, mixture model, and clustering procedures. Psychological Reports, 87, 37-47. 

McLachlan, G. J. (1988). Mixture models Inference and applications to clustering. New York Marcel Dekker. 

Examples of nonhierarchical clustering [22] methods include Gaussian mixture models, means, and fuzzy C means. They can be subdivided into hard and soft clustering methods. Hard classification methods such as means assign pixels to membership of only one cluster whereas soft classifications such as fuzzy C means assign degrees of fractional membership in each cluster. 

Mixture Models Broken-Down Ice Structures. Historically, the mixture models have received considerably more attention than the uniformist, average models. Somewhat arbitrarily, we divide these as follows (1) broken-down ice lattice models (i.e., ice-like structural units in equilibrium with monomers) (2) cluster models (clusters in equilibrium with monomers) (3) models based on clathrate-like cages (again in equilibrium with monomers). In each case, it is understood that at least two species of water exist—namely, a bulky species representing some... 

Before proceeding, it is important to recall the significant feature which appears to distinguish the cluster model from the two other prominent mixture models—i.e., the broken-down ice lattice and the clathrate hydrate cage structures. The latter two theories allow for the existence of discrete sites in water, owing to the cavities present either in the ice... 

We have discussed some examples which indicate the existence of thermal anomalies at discrete temperatures in the properties of water and aqueous solutions. From these and earlier studies at least four thermal anomalies seem to occur between the melting and boiling points of water —namely, approximately near 15°, 30°, 45°, and 60°C. Current theories of water structure can be divided into two major groups—namely, the uniformist, average type of structure and the mixture models. Most of the available experimental evidence points to the correctness of the mixture models. Among these the clathrate models and/or the cluster models seem to be the most probable. Most likely, the size of these cages or clusters range from, say 20 to 100 molecules at room tempera-... 

This notion of co-operative build-up of a cluster of hydrogen-bonded water molecules is central to Frank and Wen s mixture model for water (p. 236). 

Clustering can be viewed as a density estimation problem. The basic premise used in such an estimation is that in addition to the observed variables (i.e., descriptors) for each compound there exists an unobserved variable indicating the cluster membership. The observed variables are assumed to arrive from a mixture model, and the mixture labels (cluster identifiers) are hidden. The task is to find parameters associated with the mixture model that maximize the likelihood of the observed variables given the model. The probability distribution specified by each cluster can take any form. Although mixture model methods have found little use in chemical applications to date, they are mentioned here for completeness and because they are obvious candidates for use in the future. 

The experimental RED (Figure 4a) has been compared to various RED calculated from model clusters of different size and shape. Spherical models give always the best agreement, thus Figure 4b is a distribution calculated for a 50 50 mixture of 6-atoms (7 A) octahedral clusters and of 135-atoms (17.4 A) spherical clusters (96 % or the atoms are in the iarge clusters). 

The number of models that describe the structure and properties of liquid water is enormous. They can be subdivided into two groups the uniform continuum models and the cluster or mixture models. The main difference between these two classes of models is their treatment of the H-bond network in liquid water whereas the former assumes that a full network of H-bonds exists in liquid water, in the latter the network is considered broken at melting and that the liquid water is a mixture of various aggregates or clusters. The uniform continuum models stemmed from the classical publications of Bernal and Fowler, Pople, and Bernal.Among the cluster or mixture models, reviewed in refs 2—6 and 12, one should mention the models of Samoilov, Pauling, Frank and Quist, and Nemethy and Scheraga. ... 

The results of our calculations indicated that in all cases (the cubic sample, mono-, bi-, tri-, and more layers) the small amount of 13—20% of broken H-bonds, usually considered enough for melting, is not sufficient to break up the network of H-bonds into separate clusters. The so-called cluster or mixture models are not consistent with the results of the present simulations. From our results one can conclude that liquid water can be considered to consist of a deformed network with some H-bonds ruptured. In the case of bulk ice more than 61% of the H-bonds has to be broken for its complete fragmentation into clusters to occur. The same result was obtained via percolation theory. 

Relation between the Number of Broken Bonds and Structure. Percolation Threshold. Figure 2 presents the fraction of water molecules in small clusters as a function of the fractions of broken H bonds. The calculations show that the small amount of 13—20% of broken H bonds, usually considered to occur in melting, is not sufficient to disintegrate the network of H bonds into separate clusters and that the overwhelming majority of water molecules (>99%) belongs to a new distorted but unbroken network. This result was also obtained by us before when we assumed equal probability of rupture of H bonds and also by others a long time ago. It may be used as a test for any models of the water structure. For instance, the so-called cluster or mixture models are not consistent with the above conclusion. 

We wish to compare the valence band density of states (DOS) of f.c.c. and h.c.p. metals with and without stacking faults. We therefore adopt a mixture of the f.c.c. and h.c.p. structures as a representative of the stacking fault structure of either of these structures. To calculate the DOS we summed up the squares of the coefficients of molecular orbital wave functions and convoluted the summed squares with the Gaussian of full width 0.5 eV at half maximum. For these DOS calculations we chose the metals Mg, Ti, Co, Cu and Zn. The model clusters employed here for both the f.c.c. and the h.c.p. structures were made of 13 atoms i.e., a central atom and 12 equidistant neighbor atoms. These structures are shown in Fig. 1. We reproduced the typical electronic structures in bulk materials by extracting the molecular orbitals localized only on the central atom from all the molecular orbitals which contributed - those localized on ligand atoms as well as on the central atom. To perform calculations we take the symmetry of the cluster as C3, and the number... 

G. J. McLachlan and K. E. Basford, Mixture Models—Inference and Applications to Clustering. Marcel Dekker, New York, 1988. 

McLachlan, G.J., Bean, R.W., and Peel, D. (2002) A mixture model-based approach to the clustering of microarray expression data. Bioinformatics 18, 413 22. 

Ng, S.K., McLachlan, G.J., Wang, K., Ben-Tovim Jones, L., and Ng, S.-W. (2006) A mixture model with random-effects components for clustering correlated gene-expression profiles. Bioinformatics 22, 1745-1752. 

The EM-based clustering is one of the well-known stadstical mixture models, first proposed by Dempster et al. (1977). The results of EM clustering are different... 

Press, W.H. Gaussian Mixture Models and k-Means Clustering. Numerical Recipes The Art of Scientific Computing (3rd.). New York Cambridge University Press. 2007. 

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