Efficient spatial clustering algorithm using binary tree

In this paper we present an efficient k-Means clustering algorithm for two dimensional data. The proposed algorithm re-organizes dataset into a form of nested binary tree*. Data items are compared at each node with only two nearest means with respect to each dimension and assigned to the one that has the closer mean. The main intuition of our research is as follows: We build the nested binary tree. Then we scan the data in raster order by in-order traversal of the tree. Lastly we compare data item at each node to the only two nearest means to assign the value to the intendant cluster. In this way we are able to save the computational cost significantly by reducing the number of comparisons with means and also by the least use to Euclidian distance formula. Our results showed that our method can perform clustering operation much faster than the classical ones. © Springer-Verlag Berlin Heidelberg 2005

Clustering of Pick bodies in the dentate gyrus in Pick's disease

In patients with Pick's disease (PD), high densities of tau positive Pick bodies (PB) have been observed within the granule cell layer of the dentate gyrus. This study investigated the spatial patterns of PB along the granule cell layer in coronal sections of the hippocampus in eight patients with PD. In all patients, there was evidence of clustering of PB within the granule cell layer; however, there was considerable variation in the pattern of clustering. In five patients, the clusters of PB were regularly distributed along the dentate gyms, and in two of these patients, the smaller clusters were aggregated into larger superclusters. In three patients, a single large cluster of PB, more than 1200 μm in diameter, was present. Clustering of PB may reflect a primary degenerative process within the granule cells or the degeneration of pathways that project to the dentate gyrus.

Clustering and spatial correlations of the neuronal cytoplasmic inclusions, astrocytic plaques and ballooned neurons in corticobasal degeneration

This study tested three hypotheses: (1) that there is clustering of the neuronal cytoplasmic inclusions (NCI), astrocytic plaques (AP) and ballooned neurons (BN) in corticobasal degeneration (CBD), (2) that the clusters of NCI and BN are not spatially correlated, and (3) that the lesions are correlated with disease ‘stage’. In 50% of the regions, clusters of lesions were 400–800 µm in diameter and regularly distributed parallel to the tissue boundary. Clusters of NCI and BN were larger in laminae II/III and V/VI, respectively. In a third of regions, the clusters of BN and NCI were negatively spatially correlated. Cluster size of the BN in the parahippocampal gyrus (PHG) was positively correlated with disease ‘stage’. The data suggest the following: (1) degeneration of the cortico-cortical pathways in CBD, (2) clusters of NCI and BN may affect different anatomical pathways and (3) BN may develop after the NCI in the PHG.

Clustering and periodicity of neurofibrillary tangles in the upper and lower cortical laminae in Alzheimer's disease

In Alzheimer's disease (AD), neurofibrillary tangles (NFT) occur within neurons in both the upper and lower cortical laminae. Using a statistical method that estimates the size and spacing of NFT clusters along the cortex parallel to the pia mater, two hypotheses were tested: 1) that the cluster size and distribution of the NFT in gyri of the temporal lobe reflect degeneration of the feedforward (FF) and feedback (FB) cortico-cortical pathways, and 2) that there is a spatial relationship between the clusters of NFT in the upper and lower laminae. In 16 temporal lobe gyri from 10 cases of sporadic AD, NFT were present in both the upper and lower laminae in 11/16 (69%) gyri and in either the upper or lower laminae in 5/16 (31%) gyri. Clustering of the NFT was observed in all gyri. A significant peak-to-peak distance was observed in the upper laminae in 13/15 (87%) gyri and in the lower laminae in 8/ 12 (67%) gyri, suggesting a regularly repeating pattern of NFT clusters along the cortex. The regularly distributed clusters of NFT were between 500 and 800 μm in size, the estimated size of the cells of origin of the FF and FB cortico-cortical projections, in the upper laminae of 6/13 (46%) gyri and in the lower laminae of 2/8 (25%) gyri. Clusters of NFT in the upper laminae were spatially correlated (in phase) with those in the lower laminae in 5/16 (31%) gyri. The clustering patterns of the NFT are consistent with their formation in relation to the FF and FB cortico-cortical pathways. In most gyri, NFT clusters appeared to develop independently in the upper and lower laminae.

Clustering of neuronal inclusions in "dementia with neurofilament inclusions"

Dementia with neurofilament inclusions (DNI) is a new disorder characterized clinically by early-onset dementia and histologically by the presence of intraneural inclusions immunopositive for neurofilament antigens but lacking tau and α-synuclein reactivity. We studied the clustering patterns of the neurofilament inclusions (NI) in regions of the temporal lobe in three cases of DNI to determine whether they have the same spatial patterns as inclusions in the tauopathies and α-synucleinopathies. The NI exhibited a clustered distribution (mean size of clusters 400 μm, range 50-800 μm, SD 687.8) in 24/28 of the areas studied. In 22 of these areas, the clusters exhibited a regular distribution along the tissue parallel to the pia mater or alveus. In 3 cortical areas, there was evidence of a more complex pattern in which the NI clusters were aggregated into larger superclusters. In 6 cortical areas, the size of the clusters approximated to those of the cells of origin of the cortico-cortical pathways but in the remaining areas cluster size was smaller than 400 μm. Despite the unique molecular profile of the NI, their spatial patterns are similar to those shown by filamentous neuronal inclusions in the tauopathies and α-synucleinopathies.

Correlations between the clustering patterns of the pathological changes in sporadic Creutzfeldt-Jakob disease

Correlations between the clustering patterns of the vacuolation ('spongiform change'), prion protein (PrP) deposits, and surviving neurons were studied in the cerebral cortex, hippocampus, and cerebellum in 11 cases of sporadic Creutzfeldt-Jakob disease (sCJD). Differences in the sizes of the clusters of vacuoles were observed between brain regions and in the cerebral cortex, between the upper and lower laminae. With the exception of the parietal cortex, mean cluster size of the vacuoles was similar to that of the PrP deposits in each brain area. Clusters of the vacuoles were spatially correlated with the density of surviving neurons and with the clusters of PrP deposits in 47% and 53% of cortical areas analysed respectively but there were few spatial correlation between the PrP deposits and the density of surviving neurons. The data suggest that the pathology of sCJD may spread through the brain via specific anatomical pathways. Development of the clusters of vacuoles is spatially related to surviving neurons while the appearance of clusters of PrP deposits is related to the development of the vacuolation.

Robust automatic mapping algorithms in a network monitoring scenario

Automatically generating maps of a measured variable of interest can be problematic. In this work we focus on the monitoring network context where observations are collected and reported by a network of sensors, and are then transformed into interpolated maps for use in decision making. Using traditional geostatistical methods, estimating the covariance structure of data collected in an emergency situation can be difficult. Variogram determination, whether by method-of-moment estimators or by maximum likelihood, is very sensitive to extreme values. Even when a monitoring network is in a routine mode of operation, sensors can sporadically malfunction and report extreme values. If this extreme data destabilises the model, causing the covariance structure of the observed data to be incorrectly estimated, the generated maps will be of little value, and the uncertainty estimates in particular will be misleading. Marchant and Lark [2007] propose a REML estimator for the covariance, which is shown to work on small data sets with a manual selection of the damping parameter in the robust likelihood. We show how this can be extended to allow treatment of large data sets together with an automated approach to all parameter estimation. The projected process kriging framework of Ingram et al. [2007] is extended to allow the use of robust likelihood functions, including the two component Gaussian and the Huber function. We show how our algorithm is further refined to reduce the computational complexity while at the same time minimising any loss of information. To show the benefits of this method, we use data collected from radiation monitoring networks across Europe. We compare our results to those obtained from traditional kriging methodologies and include comparisons with Box-Cox transformations of the data. We discuss the issue of whether to treat or ignore extreme values, making the distinction between the robust methods which ignore outliers and transformation methods which treat them as part of the (transformed) process. Using a case study, based on an extreme radiological events over a large area, we show how radiation data collected from monitoring networks can be analysed automatically and then used to generate reliable maps to inform decision making. We show the limitations of the methods and discuss potential extensions to remedy these.