# Space-Time Point Processes

## Top PDF Space-Time Point Processes:

### 1 Introduction to Spatial Point Processes

Modern point process theory has a history that can trace its roots back to Poisson in 1837. However, much of the modern theory, which depends heavily on measure theory, was devel- oped in the mid 20th century. Three lectures is not nearly enough time to cover all of the theory and applications. My goal is to give a general introduction to the topics I find instruc- tive and those that I find useful for spatial modeling (at least from the Bayesian framework). Several good texts books are out there for further study. These include, but not limited to, Daley and Vere-Jones (2003), Daley and Vere-Jones (2008), Møller and Waagepetersen (2004) and Illian et al. (2008). Most of my material comes from Møller and Waagepetersen (2004), Illian et al. (2008), Møller, Syversveen, and Waagepetersen (1998) and Møller and Waagepetersen (2007). Most theorems and propositions will be stated without proof. One may think of a spatial point process as a random countable subset of a space S. We will assume that S ⊆ R d . Typically, S will be a d-dimensional box or all of

### Weighted likelihood estimators for point processes

similar magnitude; on the other hand, a large number of earthquakes burst in a short time period in mainshock–aftershock sequences. Models for seismicity should reflect such variations of seismicity structures in space and time in the form of variations of model parameters. A simple and direct way to accomplish this is to fit the spacetime ETAS model or its temporal version to seismicity data from individual regions to obtain model parameters. For example, Utsu et al. (1995) divided the whole Japan region into different subregions and applied the temporal ETAS model to the earthquake data from these subregions. The authors of Utsu et al. (1995) concluded that the ETAS parameters were location dependent. However, such a treatment suffers from unstable estimates when the ETAS model is fitted to a dataset that only contains a small number of earthquakes. Thus, a robust method for estimating variations of model parameters is very important.

### Modern statistics for spatial point processes

Many scientific problems call for new spatial point process methodology for analyzing complex and large data sets (often with marks and possibly in space-time). For example, in the tropical rain forest example, the data for the Beilschmiedia trees are just a very small part of a very large data set containing positions and diameters at breast height for around 300 species recorded over several instances in time, and between-species competition (which is ignored in our analysis) should of course be taken into account (see e.g. Illian, Møller & Waagepetersen (2007)). Another example is the Sloan Digital Sky Survey with millions of galaxies, where it is of interest to model the clustering of galaxies. One question of practical importance is, in which cases might the quick non- likelihood approaches be sufficient and how to choose between them.

### Gaussian processes for state space models and change point detection

Machine learning is the study of algorithms whose performance improves with increased exposure to data. Most computer programs do not become more intel- ligent no matter how much data they process; their entire functionality is specified in advance by the programmer. A machine learning algorithm, by contrast, will become more intelligent as more data is processed. In machine learning, software must learn to match its output to training examples, where the correct output is provided to the algorithm. The classic example is optical character recognizers (OCR): Images of characters, say zero to nine, are provided and the software must output which digit is in the image. As opposed to trying to specify rules about what makes a two a two, example images are provided along with labels (this is known as the training set). A good machine learning algorithm will predict the correct character in a test set when novel images are provided to the algorithm. An OCR is an example of an iid data set, the images are independent of one another and their attributes do not change over time. The canonical examples of machine learning are iid, but we focus on time series data. Examples of time series data include air temperature or stock market returns.

### Approximate Inference for Determinantal Point Processes

estimating the mode, and maximizing likelihood. For DPPs, exactly computing the quantities necessary for the first four of these tasks requires time cubic in the number of items or features of the items. In this thesis, we propose a means of making these four tasks tractable even in the realm where the number of items and the number of features is large. Specifically, we analyze the impact of randomly projecting the features down to a lower-dimensional space and show that the variational distance between the resulting DPP and the original is bounded. In addition to expanding the circumstances in which these first four tasks are tractable, we also tackle the other two tasks, the first of which is known to be NP-hard (with no PTAS) and the second of which is conjectured to be NP-hard. For mode estimation, we build on submodular maximization techniques to develop an algorithm with a multiplicative approximation guarantee. For likelihood maximization, we exploit the generative process associated with DPP sampling to derive an expectation-maximization (EM) algorithm. We experimentally verify the practicality of all the techniques that we develop, testing them on applications such as news and research summarization, political candidate comparison, and product recommendation.

### Testing Separability of Covariances for Space-Time Processes

tivariate process that uses the likelihood ratio test statistic based on estimating the Kronecker product of two unstructured matrices versus estimating a completely un- structured variance-covariance matrix. This test not only applies to spatio-temporal processes, but also to multivariate repeated measures. For large samples the distribu- tion of this statistic can be approximated by a chi-square distribution. We will show that the distribution of the test statistic when the null hypothesis is true will not depend on the type of separable model, and hence the distribution can be approxi- mated for any sample size. This will be especially important for small samples where the Type I error for the chi-square test is very high. In the space-time context, this test will require the number of replicates to be larger than the product of the number of times and locations. In addition to this constraint the test also assumes the data are normally distributed since the normal likelihood is used. However, this test does not require the process to be isotropic or second-order stationary. Furthermore, no particular class of models will need to be specified.

### Space-Time Coding and Space-Time Channel Modelling for Wireless Communications

In this thesis we investigate the effects of the physical constraints such as antenna aperture size, antenna geometry and non-isotropic scattering distribution parame- ters (angle of arrival/departure and angular spread) on the performance of coherent and non-coherent space-time coded wireless communication systems. First, we de- rive analytical expressions for the exact pairwise error probability (PEP) and PEP upper-bound of coherent and non-coherent space-time coded systems operating over spatially correlated fading channels using a moment-generating function-based approach. These analytical expressions account for antenna spacing, antenna ge- ometries and scattering distribution models. Using these new PEP expressions, the degree of the effect of antenna spacing, antenna geometry and angular spread is quantified on the diversity advantage (robustness) given by a space-time code. It is shown that the number of antennas that can be employed in a fixed antenna aperture without diminishing the diversity advantage of a space-time code is de- termined by the size of the antenna aperture, antenna geometry and the richness of the scattering environment.

### A Virtual Space-Time Adaptive Beamforming Method for Space-Time Antijamming

Abstract—Space-time antijamming problem has received signiﬁcant attention recently in the passive radar systems, such as Global Navigation Satellite Systems (GNSS). The space-time beamformer contains two adaptive ﬁlters, i.e., spatial ﬁlter and temporal ﬁlter for canceling interference signals. However, most of the works on space-time antijamming problem presented in the literature require multiple antennas and delay taps. In this paper, a virtual space-time adaptive beamforming method is proposed. The temporal smoothing technique is utilized to add a structure of the received data model for the implementation of the proposed method without delay taps. Compared with the previous works, the presented method oﬀers a number of advantages over other recently proposed algorithms. For example, the space-time weight vector can be obtained by simple algebraic operations with lower computational complexity, since the matrix inversion is avoided. Furthermore, the system overhead can be reduced obviously since the temporal smoothing technology is used instead of multiple delay taps. Simulation results are presented to verify the eﬀectiveness of the proposed method.

### Advances in the Theory of Determinantal Point Processes

Our first mixture representation requires the idea of a k-DPP (Kulesza and Taskar [2011a, 2012]), which is the distribution achieved by sampling from a de- terminantal point process conditional on the event |Y | = k. Given this definition, it is trivial to see that a general determinantal point process can be written as a mixture of k-DPPs. While this may seem too simple to be interesting, there are two reasons to give it some consideration. First, calculating P (|Y | = k) is not a trivial problem. Second, the class of k-DPPs contains distributions which are not determinantal point processes.

### Insertion and deletion tolerance of point processes

Proof of Lemma 5.1. Let Π be a translation-invariant ergodic insertion-tolerant point process. Let the occupied region be given by a union of balls of radius R > 0 . By ergodicity, if K(Π) is the number of unbounded clusters, then K(Π) is a fixed constant a.s. Assume that K(Π) < ∞ . It suffices to show that P (K(Π) ≤ 1) > 0 . Since K(Π) < ∞ , there exists N > 0 so that every unbounded cluster intersects B(0, N) with positive probability. Consider the finite set S := (R/4) Z d ∩ B(0, N) . For each x ∈ S , let U

### Survival models for censored point processes

Hougaard 1987 gives a good overview of the analysis of multivariate survival data, and also discusses some aspects of recurrent event data in the form of counts, and Poisson mixture mode[r]

### A product space with the fixed point property

It has been known from the outset of the study of this property (around the early sixties of last century) that it depends strongly on ¨nice¨ geometrical properties of the space. A seminal work for this theory, due to Kirk [1] established that those Banach spaces with normal structure (NS), have the (WFPP). In particular uniformly convex Banach spaces have normal structure. A long time open major question in metric fixed point theory is: Does every reflexive Banach space have (FPP)? (See [2] for more about this problem). A special case of this question is: Does every superreflexive Banach space have (FPP)? Although superreflexive spaces have the fixed point property for isometries [3] the question for general nonexpansive mappings remains still unsolved.

### Politics and Space/Time

The aim here is not to disagree in total with these formulations, but to indicate what they imply. What they both point to is a contrast between temporal movement on the one hand, and on the other a notion of space as instantaneous connections between things at one moment. For Jameson, the latter type of (inadequate) history-telling has replaced the former. And if this is true then it is indeed inade- quate. But while the contrast—the shift in balance—to which both authors are drawing attention is a valid one, in the end the notion of space as only systems of simultaneous relations, the flashing of a pin- ball machine, is inadequate. For, of course, the temporal movement is also spatial; the moving elements have spatial relations to each other. And the ‘spatial’ interconnections which flash across can only be constituted temporally as well. Instead of linear process counterposed to flat surface (which anyway reduces space from three to two dimen- sions), it is necessary to insist on the irrefutable four-dimensionality (indeed, n-dimensionality) of things. Space is not static, nor time spaceless. Of course spatiality and temporality are different from each other, but neither can be conceptualized as the absence of the other. The full implications of this will be elaborated below, but for the moment the point is to try to think in terms of all the dimensions of space-time. It is a lot more difficult than at first it might seem. Second, we need to conceptualize space as constructed out of interre- lations, as the simultaneous coexistence of social interrelations and interactions at all spatial scales, from the most local level to the most global. Earlier it was reported how, in human geography, the recogni- tion that the spatial is socially constituted was followed by the perhaps even more powerful (in the sense of the breadth of its implications) recognition that the social is necessarily spatially constituted too. Both points (though perhaps in reverse order) need to be grasped at this moment. On the one hand, all social (and indeed physical) phenom- ena/activities/relations have a spatial form and a relative spatial loca- tion. The relations which bind communities, whether they be ‘local’ societies or worldwide organizations; the relations within an indus- trial corporation; the debt relations between the South and the North; the relations which result in the current popularity in European cities of music from Mali. The spatial spread of social relations can be intimately local or expansively global, or anything in between. Their spatial extent and form also changes over time (and there is consider- able debate about what is happening to the spatial form of social relations at the moment). But, whichever way it is, there is no getting away from the fact that the social is inexorably also spatial.

### On the Metric of Space Time

Such results would have another consequence. Tomil- chik [5] and Wulfman [7,8] have both argued that the magnitudes of the “Pioneer Anomalies” roughly corre- spond to the value of the Hubble constant because the anomalies are primarily due to the assumption that the space-time is Minkowskian. However, Turyshev, et al. [19] argue that the anomalies can be explained by a re- coil effect which had not been properly accounted for. The proposed experiments would determine the extent to which each explanation is correct.

### On Universal Space and Time

In earlier papers [1]-[4], it was shown that the consistency of the concept of time with motion re- quires time and distance to be of the same dimension, and thus measured by the same unit. The arising reduced system of units revealed that mass and energy were only different facets of one entity, and resulted in the well-known mass-energy equivalence formula as a natural consequence. The physical space can be identified with any inertial frame, but when it comes to comparing the results of measurements in two frames, or more, only one frame, say S, can be taken stationary and identified with the physical space, whereas all other inertial frames are moving relative to S. The equivalence of inertial frames as sites of one physical world implies that an intrinsic units system of length, time, mass and charge should be defined in terms of basic constituent physical blocks that have the same identity in all inertial frames. A basic feature of the universal space and time theory (UST) is that the same one time prevails in all inertial frames. The scaling transforma- tions (STs) that relate the geometric distances in two frames, S (s) when chosen the stationary frame, are derived, and applied to explain the Doppler’s effect. The time distance between a mov- ing object in S and an observer depends on its state of motion; and the Euclidean form of the STs is employed to explain arrival of some meta-stable at the earth’s surface despite its short lifetime. The quantitative predicted Doppler’s effect, which is in a striking agreement with the Ives-Stilwell experimental results, coincides with the relativistic prediction for longitudinal motion, but yet predicts a complete absence of a transverse effect at a right angle. In coming parts of this work it will be shown that the UST explains elaborately the drag effect, stellar aberration, and produces naturally the relativistic mechanics. The UST will also be completed through deriving the scaling transformations of the second type, by which the null results of Michelson and Morley experiment, Michelson and Gale experiment, and the Sagnac effect are explained. The current work and our intended future works in UST are new versions containing basic conceptions and visions that didn’t appear in earlier versions [1]-[6].

### Space-time configuration for visualisation in information space

We demonstrate the possibility of exploring the history of the events that have taken place on documents in a project folder, as various members make changes to its content, through explicit spatial syntactic relationships. Further more we provide a tool for managing and inspecting the folders contents: the DocuDrama Timetunnel. Here we present preliminary findings showing how spatialised time-history visualisation may lead to a better understanding of the project related events history. First we outline the motivation and strategy for this approach, followed by a description of the approach and the three-dimensional model with the representation of the various DocuDrama elements. The section on implementation specifies a range of interfaces available in the DocuDrama architecture. Finally, we give an account of example configurations for different three-dimensional DocuDrama models generated, using data from the TOWER 1 application partners. This specification has been implemented as a full prototype, which forms one of the main components in the TOWER environment.

### Is the Space Time a Superconductor?

At the fundamental level, the 4-dimensional space-time of our direct experience might not be a continuum and discrete quantum entities might “collectively” rule its dynamics. Henceforth, it seems natural to think that in the “low-energy” regime some of its distinctive quantum attributes could, in principle, manifest themselves even at macroscopically large scales. Indeed, when confronted with Nature, classical gravitational dynamics of spinning astrophysical bodies is known to lead to paradoxes: to untangle them, dark matter or modifications to the classical law of gravity are openly consid- ered. In this article, the hypothesis of a fluctuating space-time acquiring “at large distances” the properties of a Bose-Einstein condensate is pushed forward: firstly, it is shown that a natural outcome of this picture is the production of monopoles, dyons, and vortex lines of “quantized” gravitomagnetic—or gyrogravitational—flux along the transition phase; the minimal supported “charge” (and multiples of it) being directly linked with a nonzero (minimal) vacuum energy. Thus, a world of vibrating, spinning, interacting strings whose only elements in their construction are our topo- logical concepts of space and time is envisioned, and they are proposed as tracers of the superfluid features of the space-time: the archetypal embodiment of these physical processes being set by the “gravitational roton”, an analogue of Landau’s classic higher-energy excitation used to explain the superfluid properties of helium II. The far and the near field asymptotics of string line solutions are presented and used to deduce their pair-interaction energy. Remarkably, it is found that two stationary, axis-aligned, quantum space-time vortices with the same sense of spin not only exhibit zones of repulsion but also of attraction, depending on their relative geodetic distance.

### MINIMIZATION IN GENERATING SPACE AND FIXED POINT

An important area of fixed point theory is the generating space of quasi 2-metric family, because of its involvement and application to fuzzy and probabilistic 2-metric space and a minimization theorem [1], [3] is to obtain fixed point theorem. In 2008 V. B. Dhagat and V. S. Thakur [2] proved non convex minimization theorem for generating space of quasi 2-metric family. In this paper we prove a minimization theorem for sequence of mappings 𝑇 𝑎 for 𝑎 ∈ 𝑁 and further we prove fixed point theorem as an application of minimization theorem with non commuting condition known as weak compatible.