Initial state estimation from limited observations of the heat equation in metric graphs

This paper deals with initial state estimation problems of the heat equation in equilateral metric graphs being admitted to have cycles. Particularly, we are concerned with suitable placements of observation points in order to uniquely determine the initial state from observation data. We give a necessary and sufficient condition for suitable placements of observation points, and such suitable placements are determined from transition matrices of metric graphs. From numerical simulations, we confirm effectiveness of a necessary and sufficient condition.

Computing Different Realizations of Linear Dynamical Systems with Embedding Eigenvalue Assignment

In this paper we investigate realizability of discrete time linear dynamical systems (LDSs) in fixed state space dimension. We examine whether there exist different Θ = (A,B,C,D) state space realizations of a given Markov parameter sequence Y with fixed B, C and D state space realization matrices. Full observation is assumed in terms of the invertibility of output mapping matrix C. We prove that the set of feasible state transition matrices associated to a Markov parameter sequence Y is convex, provided that the state space realization matrices B, C and D are known and fixed. Under the same conditions we also show that the set of feasible Metzler-type state transition matrices forms a convex subset. Regarding the set of Metzler-type state transition matrices we prove the existence of a structurally unique realization having maximal number of non-zero off-diagonal entries. Using an eigenvalue assignment procedure we propose linear programming based algorithms capable of computing different state space realizations. By using the convexity of the feasible set of Metzler-type state transition matrices and results from the theory of non-negative polynomial systems, we provide algorithms to determine structurally different realization. Computational examples are provided to illustrate structural non-uniqueness of network-based LDSs.

Spatiotemporal Road Traffic Anomaly Detection: A Tensor-Based Approach

The increased development of urban areas results in a larger number of vehicles on the road network, leading to traffic congestion, which often leads to potentially dangerous situations that can be described as anomalies. The tensor-based methods emerged only recently in applications related to traffic anomaly detection. They outperform other models regarding simultaneously capturing spatial and temporal components, which are of immense importance in traffic dataset analysis. This paper presents a tensor-based method for extracting the spatiotemporal road traffic patterns represented with the speed transition matrices, with the goal of anomaly detection. A novel anomaly detection approach is presented, which relies on computing the center of mass of the observed traffic patterns. The method was evaluated on a large road traffic dataset and was able to detect the most anomalous parts of the urban road network. By analyzing spatial and temporal components of the most anomalous traffic patterns, sources of anomalies can be identified. Results were validated using the extracted domain knowledge from the Highway Capacity Manual. The anomaly detection model achieved a precision score of 92.88%. Therefore, this method finds its usages for safety experts in detecting potentially dangerous road segments, urban traffic planners, and routing applications.

Mathematical model for forecasting of violations in the field of veterinary medicine using the Markov chain

Activities within the field of veterinary medicine require an efficient governmental administration, including planning, programming, accounting and result analysis, in addition to control and supervision. As a result, it is crucial to have a theoretical understanding of the new forms of interaction between executive authorities at different levels and the economic entities carrying out these activities. Therefore, the improvement of the regulations governing this field is highly required. The solution is also linked to the improvement of the current legislation governing socio-economic processes, using a new reliable model. In this context, the main goal of this study is to develop proposals to improve the current legal regulation of administrative infractions in the field of veterinary medicine. The authors of this paper applied a multi-agent approach to identify groups of law subjects. They also developed a mathematical model using Markov chains to predict infractions in the field of veterinary medicine. The model is formed of “transition matrices” and takes into account the various reaction strategies of regulatory authorities. Such a strategy shows the likelihood of committing infractions and can form a basis for reforming the current legislation ruling the conduct of control and surveillance procedures using a risk-based approach.

On Epistemics in Expected Free Energy for Linear Gaussian State Space Models

Active Inference (AIF) is a framework that can be used both to describe information processing in naturally intelligent systems, such as the human brain, and to design synthetic intelligent systems (agents). In this paper we show that Expected Free Energy (EFE) minimisation, a core feature of the framework, does not lead to purposeful explorative behaviour in linear Gaussian dynamical systems. We provide a simple proof that, due to the specific construction used for the EFE, the terms responsible for the exploratory (epistemic) drive become constant in the case of linear Gaussian systems. This renders AIF equivalent to KL control. From a theoretical point of view this is an interesting result since it is generally assumed that EFE minimisation will always introduce an exploratory drive in AIF agents. While the full EFE objective does not lead to exploration in linear Gaussian dynamical systems, the principles of its construction can still be used to design objectives that include an epistemic drive. We provide an in-depth analysis of the mechanics behind the epistemic drive of AIF agents and show how to design objectives for linear Gaussian dynamical systems that do include an epistemic drive. Concretely, we show that focusing solely on epistemics and dispensing with goal-directed terms leads to a form of maximum entropy exploration that is heavily dependent on the type of control signals driving the system. Additive controls do not permit such exploration. From a practical point of view this is an important result since linear Gaussian dynamical systems with additive controls are an extensively used model class, encompassing for instance Linear Quadratic Gaussian controllers. On the other hand, linear Gaussian dynamical systems driven by multiplicative controls such as switching transition matrices do permit an exploratory drive.

Polymath: Low-Latency MPC via Secure Polynomial Evaluations and Its Applications

While the practicality of secure multi-party computation (MPC) has been extensively analyzed and improved over the past decade, we are hitting the limits of efficiency with the traditional approaches of representing the computed functionalities as generic arithmetic or Boolean circuits. This work follows the design principle of identifying and constructing fast and provably-secure MPC protocols to evaluate useful high-level algebraic abstractions; thus, improving the efficiency of all applications relying on them. We present Polymath, a constant-round secure computation protocol suite for the secure evaluation of (multi-variate) polynomials of scalars and matrices, functionalities essential to numerous data-processing applications. Using precise natural precomputation and high-degree of parallelism prevalent in the modern computing environments, Polymath can make latency of secure polynomial evaluations of scalars and matrices independent of polynomial degree and matrix dimensions. We implement our protocols over the HoneyBadgerMPC library and apply it to two prominent secure computation tasks: privacy-preserving evaluation of decision trees and privacy-preserving evaluation of Markov processes. For the decision tree evaluation problem, we demonstrate the feasibility of evaluating high-depth decision tree models in a general n-party setting. For the Markov process application, we demonstrate that Poly-math can compute large powers of transition matrices with better online time and less communication.

The Predictive Power of Transition Matrices

When working with Markov chains, especially if they are of order greater than one, it is often necessary to evaluate the respective contribution of each lag of the variable under study on the present. This is particularly true when using the Mixture Transition Distribution model to approximate the true fully parameterized Markov chain. Even if it is possible to evaluate each transition matrix using a standard association measure, these measures do not allow taking into account all the available information. Therefore, in this paper, we introduce a new class of so-called "predictive power" measures for transition matrices. These measures address the shortcomings of traditional association measures, so as to allow better estimation of high-order models.

A global random walk on grid algorithm for second order elliptic equations

In this paper we develop stochastic simulation methods for solving large systems of linear equations, and focus on two issues: (1) construction of global random walk algorithms (GRW), in particular, for solving systems of elliptic equations on a grid, and (2) development of local stochastic algorithms based on transforms to balanced transition matrix. The GRW method calculates the solution in any desired family of prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula. The use in local random walk methods of balanced transition matrices considerably decreases the variance of the random estimators and hence decreases the computational cost in comparison with the conventional random walk on grids algorithms.

The parallel convergence of income and educational achievements: joint distribution dynamics

Human capital is an important factor of economic growth. Previous studies show that convergence patterns of income and education differ on a regional level. The purpose of this article is to verify whether there is statistical relationship between the patterns of educational achievements convergence and income convergence processes in Poland on the regional and local level. The paper describes and applies an innovative way for the formal verification of similarities in convergence patterns on the basis of transition matrices and kernel density estimation. The analysis is performed on Polish data for educational achievements (lagged exam results of 15-year old pupils) and income per capita worker on the regional and local level for the period 2003–2015. Despite the occurrence of (weak) convergence for both phenomena, each has a different course. Therefore, the processes of the income convergence and educational achievements convergence in Poland were not parallel.

Land Use Modeling Predicts Divergent Patterns of Change Between Upper and Lower Elevations in a Subalpine Watershed of the Alps

The synergic influence of land use and climate change on future forest dynamics is hard to disentangle, especially in human-dominated forest ecosystems. Forest gain in mountain ecosystems often creates different spatial–temporal patterns between upper and lower elevation belts. We analyzed land cover dynamics over the past 50 years and predicted Business as Usual future changes on an inner subalpine watershed by using land cover maps, derived from five aerial images, and several topographic, ecological, and anthropogenic predictors. We analyzed historical landscape patterns through transition matrices and landscape metrics and predicted future forest ecosystem change by integrating multi-layer perceptron and Markov chain models for short-term (2050) and long-term (2100) timespans. Below the maximum timberline elevation of the year 1965, the dominant forest dynamic was a gap-filling process through secondary succession at the expense of open areas leading to an increase of landscape homogeneity. At upper elevations, the main observed dynamic was the colonization of unvegetated soil through primary succession and timberline upward shift, with an increasing speed over the last years. Future predictions suggest a saturation of open areas in the lower part of the watershed and stronger forest gain at upper elevations. Our research suggests an increasing role of climate change over the last years and on future forest dynamics at a landscape scale.