Priebe et al. (2001) introduced the class cover catch digraphs and computed the distribution of the domination number of such digraphs for one dimensional data. In higher dimensions these calculations are extremely difficult due to the geometry of the proximity regions; and only upper-bounds are available. In this article, we introduce a new type of data-random proximity map and the associated (di)graph in R d . We find the asymptotic distribution of the domination number and use it for testing spatial point patterns of segregation and association.

Structural abnormalities in temporal lobe, including the superior temporal gyrus (STG) and planum temporale (PT), have been reported in schizophrenia (SCZ) and bipolar disorder (BPD) patients. While most MRI studies have suggested gray matter volume and surface area reduction in temporal lobe regions, few have explored changes in laminar thickness in PT and STG in SCZ and BPD. ROI subvolumes of the STG from 94 subjects were used to yield gray matter volume, gray/white surface area and laminar thickness for STG and PT cortical regions. Morphometric analysis suggests there may be gender and laterality effects on the size and shape of the PT in BPD (n=36) and SCZ (n=31) with reduced laterality in PT in subjects with SCZ but not in BPD. In addition, PT surface area was seen to be larger in males, and asymmetry in PT surface area was larger in BPD. Subjects with SCZ had reduced thickness and smaller asymmetry in PT volume. Thus, the PT probably plays a more sensitive role than the STG in structural abnormalities seen in SCZ.

High throughput drop-on-demand systems for separation and encapsulation of individual target cells from heterogeneous mixtures of multiple cell types is an emerging method in biotechnology that has broad applications in tissue engineering and regenerative medicine, genomics, and cryobiology. However, cell encapsulation in droplets is a random process that is hard to control. Statistical models can provide an understanding of the underlying processes and estimation of the relevant parameters, and enable reliable and repeatable control over the encapsulation of cells in droplets during the isolation process with high confidence level. We have modeled and experimentally verified a microdroplet-based cell encapsulation process for various combinations of cell loading and target cell concentrations. Here, we explain theoretically and validate experimentally a model to isolate and pattern single target cells from heterogeneous mixtures without using complex peripheral systems.

The spatial interaction between two or more classes (or species) has important consequences in many fields and might cause multivariate clustering patterns such as segregation or association. The spatial pattern of segregation occurs when members of a class tend to be found near members of the same class (i.e., conspecifics), while association occurs when members of a class tend to be found near members of the other class or classes. These patterns can be tested using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor (NN) structure, which may result fromamong other patterns -random labeling (RL) or complete spatial randomness (CSR) of points from two or more classes (which is called CSR independence, henceforth). In this article, we consider Dixon's class-specific tests of segregation and introduce a new class-specific test, which is a new decomposition of Dixon's overall chi-square segregation statistic. We demonstrate that the tests we consider provide information on different aspects of the spatial interaction between the classes and are conditional under the CSR independence pattern, but not under the RL pattern. We analyze the distributional properties and prove the consistency of these tests; compare the empirical significant levels (Type I error rates) and empirical power estimates of the tests using extensive Monte Carlo simulations. We demonstrate that the new class-specific tests also have comparable performance with the currently available tests based on NNCTs in terms of Type I error and power. For illustrative purposes, we use three example data sets. We also provide guidelines for using these tests.

Statistical pattern classification methods based on data-random graphs were introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the data points from various classes. Different random graphs result from different definitions of the proximity region associated with each data point and different graph statistics can be employed for data reduction. The approach used in this article is based on a parameterized family of proximity maps determining an associated family of data-random digraphs. The relative arc density of the digraph is used as the summary statistic, providing an alternative to the domination number employed previously. An important advantage of the relative arc density is that, properly re-scaled, it is a U -statistic, facilitating analytic study of its asymptotic distribution using standard U -statistic central limit theory. The approach is illustrated with an application to the testing of spatial patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman and Hodges-Lehmann asymptotic efficacy, and selection of the proximity map parameter to optimize efficacy. Notice that the approach presented here also has the advantage of validity for data in any dimension.