Random Map Model Research Articles

Optimization of an Autonomous Mobile Robot Path Planning Based on Improved Genetic Algorithms

Mobile robots are intended to operate in a variety of environments, and they need to be able to navigate and travel around obstacles, such as objects and barriers. In order to guarantee that the robot will not come into contact with any obstacles or other objects during its movement, algorithms for path planning have been demonstrated. The basic goal while constructing a route is to find the fastest and smoothest route between the starting point and the destination. This article describes route planning using the improvised genetic algorithm with the Bezier Curve (GA-BZ). This study carried out two main experiments, each using a 20x20 random grid map model with varying percentages of obstacles (5%, 15%, and 30% in the first experiment, and 25% and 50% in the second). In the initial experiments, the population (PN), generation (GN), and mutation rate (MR) of genetic algorithms (GA) will be altered to the following values: (PN = 100, 125, 150, or 200; GN = 100, 125, 150; and MR = 0.1, 0.3, 0.5, 0.7) respectively. The goal is to evaluate the effectiveness of AMR in terms of travel distance (m), total time (s), and total cost (RM) in comparison to traditional GA and GA-BZ. The second experiment examined robot performance utilising GA, GA-BZ, Simulated Annealing (SA), A-Star (A*), and Dijkstra's Algorithms (DA) for path distance (m), time travel (s), and fare trip (RM). The simulation results are analysed, compared, and explained. In conclusion, the project is summarised.

Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

Recent works have shown that random triangulations decorated by critical (p=1∕2) Bernoulli site percolation converge in the scaling limit to a 8∕3-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE6 in two different ways: The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov–Hausdorff topology. There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE6-decorated 8∕3-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called peanosphere convergence. We prove that one in fact has joint convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to 8∕3-LQG decorated by SLE6 in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into C via the so-called Cardy embedding converge to 8∕3-LQG.

The Dirichlet problem for orthodiagonal maps

We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov [4] for isoradial graphs. We observe that by the double circle packing theorem [3], any finite, simple, 3-connected planar map admits an orthodiagonal representation.Our result improves the work of Skopenkov [41] and Werness [47] by dropping all regularity assumptions required in their work and providing effective bounds. In particular, no bound on the vertex degrees is required. Thus, the result can be applied to models of random planar maps that with high probability admit orthodiagonal representation with mesh size tending to 0. In a companion paper [21], we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield [15,25].

Basic properties of the infinite critical-FK random map

In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each q \in [0, \infty] and integer n \geq 1 , this model chooses a planar map of n edges with a probability proportional to the partition function of critical q -Potts model on that map. Sheeld introduced the hamburger–cheeseburer bijection which maps the cFK random maps to a family of random words, and remarked that one can construct infinite cFK random maps using this bijection. We make this idea precise by a detailed proof of the local convergence. When q = 1 , this provides an alternative construction of the UIPQ. In addition, we show that the limit is almost surely one-ended and recurrent for the simple random walk for any q , and mutually singular in distribution for different values of q .

Calculating Value-at-Risk for high-dimensional time series using a nonlinear random mapping model

In this study, we propose a non-linear random mapping model called GELM. The proposed model is based on a combination of the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and the Extreme Learning Machine (ELM), and can be used to calculate Value-at-Risk (VaR). Alternatively, the GELM model is a non-parametric GARCH-type model. Compared with conventional models, such as the GARCH models, ELM, and Support Vector Machine (SVM), the computational results confirm that the GELM model performs better in volatility forecasting and VaR calculation in terms of efficiency and accuracy. Thus, the GELM model can be an essential tool for risk management and stress testing.

Quantum gravity and inventory accumulation

We begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on $\mathbb{Z}^{2}$. In more interesting versions, a $p$ fraction of customers orders the “freshest available” product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on $p$. We then turn our attention to the critical Fortuin–Kastelyn random planar map model, which gives, for each $q>0$, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the $q$-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on $q$. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at $p=1/2$, $q=4$.

On the heuristic of approximating polynomials over finite fields by random mappings

The behavior of iterations of functions is frequently approximated by the Brent–Pollard heuristic, where one treats functions as random mappings. We aim at understanding this heuristic and focus on the expected rho length of a node of the functional graph of a polynomial over a finite field. Since the distribution of preimage sizes of a class of functions appears to play a central role in its average rho length, we survey the known results for polynomials over finite fields giving new proofs and improving one of the cases for quartic polynomials. We discuss the effectiveness of the heuristic for many classes of polynomials by comparing our experimental results with the known estimates for different random mapping models. We prove that the distribution of preimage sizes of general polynomials and mappings have similar asymptotic properties, including the same asymptotic average coalescence. The combination of these results and our experiments suggests that these polynomials behave like random mappings, extending a heuristic that was known only for degree [Formula: see text]. We show numerically that the behavior of Chebyshev polynomials of degree [Formula: see text] over finite fields present a sharp contrast when compared to other polynomials in their respective classes.

Structural Transition in Random Mappings

In this paper we characterise the structural transition in random mappings with in-degree restrictions. Specifically, for integers $0~\!\!\leq~\!\!r\leq~\!\!n$, we consider a random mapping model $\hat{T}_n^r$ from $[n]=\{1,2, \ldots , n\}$ into $[n]$ such that $\hat{G}_n^r$, the directed graph on $n$ labelled vertices which represents the mapping $\hat{T}_n^r$, has $r$ vertices that are constrained to have in-degree at most $1$ and the remaining vertices have in-degree at most 2. When $r=n$, $\hat{T}_n^r$ is a uniform random permutation and when $r<n$, we can view $\hat{T}_n^r$ as a 'corrupted' permutation. We investigate structural transition in $\hat{G}_n^r$ as we vary the integer parameter $r$ relative to the total number of vertices $n$. We obtain exact and asymptotic distributions for the number of cyclic vertices, the number of components, and the size of the typical component in $\hat{G}_n^r$, and we characterise the dependence of the limiting distributions of these variables on the relationship between the parameters $n$ and $r$ as $n\to\infty$. We show that the number of cyclic vertices in $\hat{G}_n^r$ is $\Theta({n\over\sqrt{a}})$ and the number of components is $\Theta(\log({n\over \sqrt{a}}))$ where $a=n-r$. In contrast, provided only that $a=n-r\to\infty$, we show that the asymptotic distribution of the order statistics of the normalised component sizes of $\hat{G}_n^r$ is always the Poisson-Dirichlet(1/2) distribution as in the case of uniform random mappings with no in-degree restrictions.

Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, T n D̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂ 1,D̂ 2,…,D̂ n ). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂ 1,D̂ 2,…,D̂ n . As an application of these results, we consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

A general modelling framework for multivariate disease mapping

This paper deals with multivariate disease mapping. We propose a novel framework that encompasses most of the models already proposed. Our framework starts with a simple identity, reformulating Kronecker products of covariance matrices as simple matrix products. This formula is computationally convenient, and its generalizations reproduce most of the proposals in the disease mapping literature. Use of the identity leads to a flexible, general and computationally convenient modelling framework, making it possible to combine spatial dependence structures and different relationships between diseases with limited effort. Moreover, as the proposed modelling framework covers most of the Gaussian Markov random field-based multivariate disease mapping models in the literature, it allows comparison of all these models in a common context, thus helping us to understand them better.

Random Maps with Parameter-Dependent Probabilities

The random map model is a deterministic dynamical system in a finite phase space with n points. A map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as its image. In this paper, a hierarchy of random maps with parameter-dependent probabilities is introduced. Then some discussion is presented about the properties of the invariant measure corresponding to random maps. Also, the Kolmogorov–Sinai (KS) entropy of the random maps is calculated analytically using the invariant measure. Simulation results indicate that the proposed random dynamical system is more complex than the standard logistic map.

A random mapping with preferential attachment

AbstractIn this article we investigate the asymptotic structure of a random mapping model with preferential attachment, T, which maps the set {1,2,…,n} into itself. The model T was introduced in a companion article [Hansen and Jaworski, Random Struct Algorithms 33 (2008), 105–126] and the asymptotic structure of the associated directed graph G which represents the action of T on the set {1,2,…,n} was investigated in [Hansen and Jaworski, Random Struct Algorithms 33 (2008), 105–126] and [Hansen and Jaworski, Adv Appl Probab 40 (2008), 183–205] in the case when the attraction parameter ρ > 0 is fixed as n →∞. In this article we consider the asymptotic structure of G when the attraction parameter ρ  ρ(n) is a function of n as n →∞. We show that there are three distinct regimes during the evolution of G: (i) ρn →∞ as n →∞, (ii) ρn → β > 0 as n →∞, and (iii) ρn → 0 as n →∞. It turns out that the asymptotic structure of G is, in some cases, quite different from the asymptotic structure of well‐known models such as the uniform random mapping model and models with an attracting center. In particular, in regime (ii) we obtain some interesting new limiting distributions which are related to the incomplete gamma function. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Local properties of random mappings with exchangeable in-degrees

In this paper we investigate the ‘local’ properties of a random mapping model, TnD̂, which maps the set {1, 2, …, n} into itself. The random mapping TnD̂, which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a collection of exchangeable random variables D̂1, …, D̂n which satisfy In the random digraph, GnD̂, which represents the mapping TnD̂, the in-degree sequence for the vertices is given by the variables D̂1, D̂2, …, D̂n, and, in some sense, GnD̂ can be viewed as an analogue of the general independent degree models from random graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of GnD̂ - for example, the numbers of predecessors and successors of v in GnD̂. We show that the distribution of several variables associated with the local structure of GnD̂ can be expressed in terms of expectations of simple functions of D̂1, D̂2, …, D̂n. We also consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest.

Local properties of random mappings with exchangeable in-degrees

In this paper we investigate the ‘local’ properties of a random mapping model, T n D̂, which maps the set {1, 2, …, n} into itself. The random mapping T n D̂ , which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a collection of exchangeable random variables D̂ 1, …, D̂ n which satisfy In the random digraph, G n D̂ , which represents the mapping T n D̂ , the in-degree sequence for the vertices is given by the variables D̂ 1, D̂ 2, …, D̂ n , and, in some sense, G n D̂ can be viewed as an analogue of the general independent degree models from random graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of G n D̂ - for example, the numbers of predecessors and successors of v in G n D̂ . We show that the distribution of several variables associated with the local structure of G n D̂ can be expressed in terms of expectations of simple functions of D̂ 1, D̂ 2, …, D̂ n . We also consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest.

Random mappings with exchangeable in‐degrees

AbstractIn this article we introduce a new random mapping model, $T_n^{\hat D}$, which maps the set {1,2,…,n} into itself. The random mapping $T_n^{\hat D}$ is constructed using a collection of exchangeable random variables $\hat{D}_1, \ldots.,\hat{D}_n$ which satisfy $\sum_{i=1}^n\hat{D}_i=n$. In the random digraph, $G_n^{\hat D}$, which represents the mapping $T_n^{\hat D}$, the in‐degree sequence for the vertices is given by the variables $\hat{D}_1, \hat{D}_2, \ldots, \hat{D}_n$, and, in some sense, $G_n^{\hat D}$ can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions of $\hat{D}_1, \hat{D}_2, \ldots, \hat{D}_n$. We also consider two special examples of $T_n^{\hat D}$ which correspond to random mappings with preferential and anti‐preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008

A random model for mapping imprinted quantitative trait loci in a structured pedigree: An implication for mapping canine hip dysplasia

Genetic imprinting may have played a more notable role in shaping embryonic development of plants, animals, and humans than previously appreciated. Quantitative trait loci that are imprinted ( iQTL) exert monoallelic effects, depending on the parent of origin, which is an exception to the laws of Mendelian genetics. In this article, we present a modified random effect-based mapping model to use in a genome-wide scan for the distribution of iQTL that contribute to genetic variance for a complex trait in a structured pedigree. This model, implemented with the maximum likelihood method, capitalizes on a network of relatedness for maternally and paternally derived alleles through identical-by-descent sharing, thus allowing for the discrimination of the genetic variances due to alleles derived from maternal and paternal parents. The model was employed to map iQTL responsible for canine hip dysplasia in a multihierarchical canine pedigree, founded with seven greyhounds and six Labrador retrievers. Of eight significant QTL detected, three, located on CFA1, CFA8, and CF28, were found to trigger significant parent-of-origin effects on the age of femoral capital ossification measured at the left and right hips of a canine. The detected iQTL provide important candidate regions for fine-mapping of imprinted genes and for studying their structure and function in the control of complex traits.

An analytical calculation of neighbourhood order probabilities for high dimensional Poissonian processes and mean field models

Consider that the coordinates of N points are randomly generated along the edges of a d-dimensional hypercube (random point problem). The probability P(d,N)m,n that an arbitrary point is the mth nearest neighbour to its own nth nearest neighbour (Cox probabilities) plays an important role in spatial statistics. Also, it has been useful in the description of physical processes in disordered media. Here we propose a simpler derivation of Cox probabilities, where we stress the role played by the system dimensionality d. In the limit d → ∞, the distances between pair of points become independent (random link model) and closed analytical forms for the neighbourhood probabilities are obtained both for the thermodynamic limit and finite-size system. Breaking the distance symmetry constraint drives us to the random map model, for which the Cox probabilities are obtained for two cases: whether a point is its own nearest neighbour or not.

Analytical results for the statistical distribution related to a memoryless deterministic walk: Dimensionality effect and mean-field models

Consider a medium characterized by N points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding mu steps (deterministic tourist walk). Each trajectory generated by this dynamics has an initial nonperiodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parametrized by the normalized incomplete beta function Id= I1/4 [1/2, (d+1) /2] . The joint distribution S(N) (mu,d) (t,p) is relevant, and the marginal distributions previously studied are particular cases. We show that, for the memory-less deterministic tourist walk in the euclidean space, this distribution is Sinfinity(1,d) (t,p) = [Gamma (1+ I(-1)(d)) (t+ I(-1)(d) ) /Gamma(t+p+ I(-1)(d)) ] delta(p,2), where t=0, 1,2, ... infinity, Gamma(z) is the gamma function and delta(i,j) is the Kronecker delta. The mean-field models are the random link models, which correspond to d-->infinity, and the random map model which, even for mu=0 , presents nontrivial cycle distribution [ S(N)(0,rm) (p) proportional to p(-1) ] : S(N)(0,rm) (t,p) =Gamma(N)/ {Gamma[N+1- (t+p) ] N( t+p)}. The fundamental quantities are the number of explored points n(e)=t+p and Id. Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.

COARSE-GRAINED OBSERVATION OF DISCRETIZED MAPS

We investigate why discretized versions fN of one-dimensional ergodic maps f : I → I behave in many ways similarly to their continuous counterparts. We propose to register observations of the N × N discretization fN on a coarse M × M grid, with N = cM, c being an integer. We prove that rounding errors behave like uniformly distributed random variables, and by assuming their independence, the M × M incidence matrix AM associated with the continuous map (indicating which of the M equal subintervals is mapped onto which) can be expected to be identical to the incidence matrix BN,M associated with the aforementioned coarse grid, if [Formula: see text], where deg (f) denotes the degree of f. We show how coarse-grained registration can be used as a "digital" definition of an unstable orbit and how this can be applied in real computations. Combination of these results with ideas from the random map model suggests an intuitive explanation for the statistical similarity between f and fN. Our approach is not a rigorous one, however, we hope that the results will be useful for the computational community and may facilitate a rigourous mathematical description.