Aila Särkkä

Gibbs point processes for studying the development of spatial-temporal stochastic processes

Although many studies of marked point processes analyse patterns in terms of purely spatial relationships, in real life spatial structure often develops dynamically through time. Here we use a specific space-time stochastic process to generate such patterns, with the aim of determining purely spatial summary measures from which we can infer underlying generating mechanisms of space-time stochastic processes. We use marked Gibbs processes in the estimation procedure, since these are commonly used models for point patterns with interactions, and can also be chosen to ensure that they possess similar interaction structure to the space-time processes under study. We restrict ourselves to Strauss-type pairwise interaction processes, as these are simple both to construct and interpret. Our analysis not only highlights the way in which Gibbs models are able to capture the interaction structure of the generating process, but it also demonstrates that very few statistical indicators are needed to determine the essence of the process. This contrasts markedly with the relatively large number of estimators usually needed to characterise a process in terms of spectral, autocorrelation or K-function representations. We show that the Strauss-type procedure is robust, i.e. it is not crucial to know the exact process-generating mechanism. Moreover, if we do possess additional information about the true mechanism, then the procedure becomes even more effective.

The gamma distribution model for pulsed-field gradient NMR studies of molecular-weight distributions of polymers

Self-diffusion in polymer solutions studied with pulsed-field gradient nuclear magnetic resonance (PFG NMR) is typically based either on a single self-diffusion coefficient, or a log-normal distribution of self-diffusion coefficients, or in some cases mixtures of these. Experimental data on polyethylene glycol (PEG) solutions and simulations were used to compare a model based on a gamma distribution of self-diffusion coefficients to more established models such as the single exponential, the stretched exponential, and the log-normal distribution model with regard to performance and consistency. Even though the gamma distribution is very similar to the log-normal distribution, its NMR signal attenuation can be written in a closed form and therefore opens up for increased computational speed. Estimates of the mean self-diffusion coefficient, the spread, and the polydispersity index that were obtained using the gamma model were in excellent agreement with estimates obtained using the log-normal model. Furthermore, we demonstrate that the gamma distribution is by far superior to the log-normal, and comparable to the two other models, in terms of computational speed. This effect is particularly striking for multi-component signal attenuation. Additionally, the gamma distribution as well as the log-normal distribution incorporates explicitly a physically plausible model for polydispersity and spread, in contrast to the single exponential and the stretched exponential. Therefore, the gamma distribution model should be preferred in many experimental situations.

The analysis of marked point patterns evolving through space and time

A maximum pseudo-likelihood approach has previously been developed for fitting pairwise interaction models to patterns generated by growth-interaction processes that are sampled at fixed time points. This approach is now extended, not only by estimating the parameters of the process through time, but also by employing least squares estimation since likelihood based approaches are much more computationally demanding. First, simple stochastic models are used to demonstrate that least squares methods are as powerful as likelihood-based approaches, as well as being mathematically and computationally simpler. The algorithm generates simulations of the deterministic growth-interaction and stochastic immigration-death process, and through these the parameter estimates are determined. Logistic and linear growth are then combined with (symmetric) disc-interaction and (asymmetric) area-interaction processes, and between them these generate a variety of mark-point spatial structures. A robustness study shows that the procedure works well in that the presence, structure and strength of a growth-interaction process can be determined even when an incorrectly presumed model is employed. Thus, the technique is likely to prove to be very useful in general practical applications where the underlying process generating mechanism is almost certain to be unknown. Finally, the procedure is applied to the analysis of a new Swedish pine forest data set for which tree location and diameter at breast height were recorded in 1985, 1990 and 1996.

Statistical analysis of intramembranous particles using freeze fracture specimens

We studied the point processes of intramembranous particles of mitochondrial membranes from HeLa cells using the freeze fracture technique. Three groups – under normal conditions, after exposition with rotenone, and after exposition with sodium acid – were compared. First, we used several summary statistics in order to study the two‐dimensional point patterns of intramembranous particles within each group. Then, we compared the patterns in different groups by bootstrap tests using the K‐function and the nearest neighbour distance function G(r). Estimation of the G‐function provided significant results but no significant differences between groups were found using the classical K‐function; estimation of G(r) should therefore not be omitted when studying observed planar point patterns.

Interacting neighbour point processes: Some models for clustering

We introduce a class of spatial point processesinteracting neighbour point (INP) processes, where the density of the process can be written by means of local interactions between a point and subsets of its neighbourhood but where the processes may not be Ripley-Kelly Markov processes with respect to this neighbourhood. We show that the processes are iterated Markov processes defined by Hayat and Gubner (1996). Furthermore, we pay special attention to a subclass of interacting neighbour processes, where the density belongs to the exponential family and all neighbours of a point affect it simultaneously. A simulation study is presented to show that some simple processes of this subclass can produce clustered patterns of great variety. Finally, an empirical example is given.

Modelling interactions between trees by means of field observations

The Gibbs model and the pseudo-likelihood estimation method are applied to study interactions between trees of pine and spruce forests in the southern half of Finland. The coordinates of trees have been measured on small sample plots in the original data. Only some trees from the plots are needed in parameter estimation through field measurements.

Some edge correction methods for marked spatio-temporal point process models

Three edge correction methods for (marked) spatio-temporal point processes are proposed. They are all based on the idea of placing an approximated expected behaviour of the process at hand (simulated realisations) outside the study region which interacts with the data during the estimation. These methods are applied to the so-called growth-interaction model. The specific choices of growth function and interaction function made are purely motivated by the forestry applications considered. The parameters of the growth and interaction functions, i.e. the parameters related to the development of the marks, are estimated using the least-squares approach together with the proposed edge corrections. Finally, the edge corrected estimation methods are applied to a data set of Swedish Scots pine.

Characterization of pore structure of polymer blended films used for controlled drug release

The characterization of the pore structure in pharmaceutical coatings is crucial for understanding and controlling mass transport properties and function in controlled drug release. Since the drug release rate can be associated with the film permeability, the effect of the pore structure on the permeability is important to study. In this paper, a new approach for characterizing the pore structure in polymer blended films was developed based on an image processing procedure for given two-dimensional scanning electron microscopy images of film cross-sections. The focus was on different measures for characterizing the complexity of the shape of a pore. The pore characterization developed was applied to ethyl cellulose (EC) and hydroxypropyl cellulose (HPC) blended films, often used as pharmaceutical coatings, where HPC acts as the pore former. It was studied how two different HPC viscosity grades influence the pore structure and, hence, mass transport through the respective films. The film with higher HPC viscosity grade had been observed to be more permeable than the other in a previous study; however, experiments had failed to show a difference between their pore structures. By instead characterizing the pore structures using tools from image analysis, statistically significant differences in pore area fraction and pore shape were identified. More specifically, it was found that the more permeable film with higher HPC viscosity grade seemed to have more extended and complex pore shapes than the film with lower HPC viscosity grade. This result indicates a greater degree of connectivity in the film with higher permeability and statistically confirms hypotheses on permeability from related experimental studies.

Analysis of spatial structure of epidermal nerve entry point patterns based on replicated data

Epidermal nerve fiber (ENF) density and morphology are used to diagnose small fiber involvement in diabetic, HIV, chemotherapy induced, and other neuropathies. ENF density and summed length of ENFs per epidermal surface area are reduced, and ENFs may appear clustered within the epidermis in subjects with small fiber neuropathy compared to healthy subjects. Therefore, it is important to understand the spatial behaviour of ENFs in healthy and diseased subjects. This work investigates the spatial structure of ENF entry points, which are locations where the nerves enter the epidermis (the outmost living layer of the skin). The study is based on suction skin blister specimens from two body locations of 25 healthy subjects. The ENF entry points are regarded as a realization of a spatial point process and a second‐order characteristic, namely Ripley’s K function, is used to investigate the effect of covariates (e.g. gender) on the degree of clustering of ENF entry points. First, the effects of covariates are evaluated by means of pooled K functions for groups and, secondly, the statistical significance of the effects and individual variation are characterized by a mixed model approach. Based on our results the spatial pattern of ENFs in samples taken from calf is affected by the covariates but not in samples taken from foot.

Identifying directional persistence in intracellular particle motion using Hidden Markov Models

Particle tracking is a widely used and promising technique for elucidating complex dynamics of the living cell. The cytoplasm is an active material, in which the kinetics of intracellular structures are highly heterogeneous. Tracer particles typically undergo a combination of random motion and various types of directed motion caused by the activity of molecular motors and other non-equilibrium processes. Random switching between more and less directional persistence of motion generally occurs. We present a method for identifying states of motion with different directional persistence in individual particle trajectories. Our analysis is based on a multi-scale turning angle model to characterize motion locally, together with a Hidden Markov Model with two states representing different directional persistence. We define one of the states by the motion of particles in a reference data set where some active processes have been inhibited. We illustrate the usefulness of the method by studying transport of vesicles along microtubules and transport of nanospheres activated by myosin. We study the results using mean square displacements, durations, and particle speeds within each state. We conclude that the method provides accurate identification of states of motion with different directional persistence, with very good agreement in terms of mean-squared displacement between the reference data set and one of the states in the two-state model.