Integer-valued Vector Research Articles

Security of Neural Network-Based Key Agreement Protocol for Smart Grids

Recent developments in quantum computing pose a significant threat to the asymmetric cryptography currently in use. Neural cryptography offers a potential alternative that is resistant to attacks of known quantum computer algorithms. The considered solution is lightweight and computationally efficient. If a quantum computer algorithm were successfully implemented, it could expose IoT sensors and smart grid components to a wide range of attack vectors. Given the lightweight nature of neural cryptography and the potential risks, neural cryptography could have potential applications in both IoT sensors and smart grid systems. This paper evaluates one of the suggested enhancements: the use of integer-valued input vectors that accelerate the synchronization of the Tree Parity Machine. This enhancement introduces a new parameter M that indicates the minimum and maximum values of input vector elements. This study evaluates the nonbinary version of the mutual learning algorithm in a simulated insecure environment. The results indicate that, while the Nonbinary Tree Parity Machine may involve some trade-offs between security and synchronization time, the speed improvement is more substantial than the decrease in security. The impact of this enhancement is particularly significant for smaller adjustments to parameter M.

An efficient algorithm for isometrically embedding weighted trees into low-dimensional [formula omitted]-normed spaces

We present an efficient algorithm for isometrically embedding weighted trees into low-dimensional ℓ∞-normed spaces. The proposed algorithm takes an n-vertex weighted tree as input and maps each of its vertices onto an integer-valued vector in Rd, where d=Θ(logn). The overall running time of the proposed algorithm is Θ(nlog2n). We also present many applications of this algorithm in the context of competitive programming, including novel tasks for which we do not know any alternative efficient solution. Essentially, the low dimensionality of the resulting vectors allows the efficient answering of various distance-related queries. By means of experiment, we have also compared different implementations of the proposed algorithm with regards to both running time and dimensionality of the resulting embedding for different classes of trees.

Cramér-Rao bound for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model

Performance lower bounds are known to be a fundamental design tool in parametric estimation theory. A plethora of deterministic bounds exist in the literature, ranging from the general Barankin bound to the well-known Cramér-Rao bound (CRB), the latter providing the optimal mean square error performance of locally unbiased estimators. In this contribution, we are interested in the estimation of mixed real- and integer-valued parameter vectors. We propose a closed-form lower bound expression leveraging on the general CRB formulation, being the limiting form of the McAulay-Seidman bound. Such formulation is the key point to take into account integer-valued parameters. As a particular case of the general form, we provide closed-form expressions for the Gaussian observation model. One noteworthy point is the assessment of the asymptotic efficiency of the maximum likelihood estimator for a linear regression model with mixed parameter vectors and known noise covariance matrix, thus complementing the rather rich literature on that topic. A representative carrier-phase based precise positioning example is provided to support the discussion and show the usefulness of the proposed lower bound.

A permutational-based Differential Evolution algorithm for feature subset selection

This paper describes a permutational-based Differential Evolution algorithm implemented in a wrapper scheme to find a feature subset to be applied in the construction of a near-optimal classifier. In this approach, the relevance of a feature chosen to build a better classifier is represented through its relative position in an integer-valued vector, and by using a permutational-based mutation operator, it is possible to create new feasible candidate solutions only. Furthermore, to provide a controlled diversity rate in the population, a straightforward repair-based recombination operator is utilized to evolve a population of candidate solutions. Unlike the other approaches in the existing literature using integer-valued vectors and requiring a predefined subset size, in this approach, this size is determined by an additional element included in the encoding scheme, allowing to find an adequate feature subset size to each specific dataset. Experimental results show that this approach is an effective way to create more accurate classifiers as they are compared with those obtained by other similar approaches.

Adding noise to Markov cohort state-transition model in decision modeling and cost-effectiveness analysis.

Following its introduction over 30 years ago, the Markov cohort state-transition model has been used extensively to model population trajectories over time in health decision modeling and cost-effectiveness analysis studies. We recently showed that a cohort model represents the average of a continuous-time stochastic process on a multidimensional integer lattice governed by a master equation, which represents the time-evolution of the probability function of an integer-valued random vector. By leveraging this theoretical connection, this study introduces an alternative modeling method using a stochastic differential equation (SDE) approach, which captures not only the mean behavior but also the variance of the population process. We show the derivation of an SDE model from first principles, describe an algorithm to construct an SDE and solve the SDE via simulation for use in practice, and demonstrate the two applications of an SDE in detail. The first example demonstrates that the population trajectories, and their mean and variance, from the SDE and other commonly used methods in decision modeling match. The second example shows that users can readily apply the SDE method in their existing works without the need for additional inputs beyond those required for constructing a conventional cohort model. In addition, the second example demonstrates that the SDE model is superior to a microsimulation model in terms of computational speed. In summary, an SDE model provides an alternative modeling framework which includes information on variance, can accommodate for time-varying parameters, and is computationally less expensive than a microsimulation for a typical cohort modeling problem.

The Maker--Breaker Rado Game on a Random Set of Integers

Given an integer-valued matrix $A$ of dimension $\ell \times k$ and an integer-valued vector $b$ of dimension $\ell$, the Maker--Breaker $(A,b)$-game on a set of integers $X$ is the game where Maker and Breaker take turns claiming previously unclaimed integers from $X$, and Maker's aim is to obtain a solution to the system $Ax=b$, whereas Breaker's aim is to prevent this. When $X$ is a random subset of $\{1,\dots,n\}$ where each number is included with probability $p$ independently of all others, we determine the threshold probability $p_0$ for when the game is Maker's or Breaker's win, for a large class of matrices and vectors. This class includes but is not limited to all pairs $(A,b)$ for which $Ax=b$ corresponds to a single linear equation. The Maker's win statement also extends to a much wider class of matrices which include those which satisfy Rado's partition theorem.

Integer Factorization of a Positive-Definite Matrix

This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension of the matrix.

Performance Analysis of an Orienteering Problem-based Trip Scheduler for Electric Vehicles

This paper designs a tour recommendation scheme for electric vehicles and analyzes its performance. Aiming at reducing the time waste brought by long charging time, the recommender not just decides a visiting and charging schedule for the given set of userselected tour spots but also integrate additional places, which have chargers and provide tour activities. Genetic operators, such as reproduction, selection, and mutation, are tailored to create a combinatory schedule of essential selected and optional recommended spots based on a hybrid orienteering model. Its encoding scheme represents a visiting order by a fixedlength integer-valued vector, allowing the omission of recommendable spots. In addition, the fitness function estimates time waste by tracing the battery charge state based on the interdestination distance and stay time along a given route. The performance measurement result obtained from a prototype implementation discovers that our recommendation service can reduce the time waste by up to 12 % for the practical range of parameter setting, compared with the legacy tour scheduling method.

Power Load Distribution for Wireless Sensor and Actuator Networks in Smart Grid Buildings

This paper presents a design and analyzes the performance of an actuator operation scheduler for wireless sensor and actuator networks, aiming at efficiently managing power consumption and distributing peak load in smart grid buildings. To create a schedule within an acceptable response time, a genetic algorithm is designed, and the scheduler places the operations of activated tasks to appropriate time slots in the allocation table. For genetic operations, each schedule is encoded to an integer-valued vector, where each element represents either start time or binary allocation map of the associated task according to the task type. The fitness function evaluates the schedule quality by estimating the load of the peaking slot. Out-task model defines P-Penalty and N-Penalty to account for the extrapower load brought by the delayed start of task operation. The performance measurement results obtained from a prototype implementation reveal that our genetic scheduler reduces the peak load by up to 35.2% for the given parameter set compared with the Earliest scheduling scheme, intelligently compromising two conflicting requirements of even load distribution and small initiation delay.

Reconstructing 2-complexes by a nonnegative integer-valued vector

Over the last decade, hypergraphs have seen a lot of use in electrotechnical problems, network design, multiindex transportation problems, computer modeling of complex dynamical systems, representations of complex industrial control systems, and many other applied problems. For some of these problems (e.g., multiindex transportation problems), a better representation is given by 2-complexes, a special class of hypergraphs in which each edge is incident to three vertices. It is known that each hypergraph corresponds to the vector of degrees of its vertices, but the converse does not hold. In this work, we consider the problem of reconstructing (implementing, realizing) a 2-complex from an arbitrary vector.

A reconstruction algorithm for the essential graph

A standard graphical representative of a Bayesian network structure is a special chain graph, known as an essential graph. An alternative algebraic approach to the mathematical description of this statistical model uses instead a certain integer-valued vector, known as a standard imset. We give a direct formula for the translation of any chain graph describing a Bayesian network structure into the standard imset. Moreover, we present a two-stage algorithm which makes it possible to reconstruct the essential graph on the basis of the standard imset. The core of this paper is the proof of the correctness of the algorithm.

On the evolution of stock vectors in a deterministic integer-valued Markov system

For multistate systems, studies on the evolution of the stock vector have already been done under various hypothesis on the model. For example for a regular Markov chain it is known that the system evolves towards a unique limiting vector, that is a fixed point of the transition matrix and that is independent on the initial stock vector. In this paper, there are examined, under deterministic assumptions, constant size systems in which the integer-valued stock vectors are generated by a regular Markov chain. For these models, properties of the evolution of the integer-valued state sizes are proved and the limiting stock vector is examined. Sufficient conditions on the transition matrix are formulated to have, for two initial integer-valued stock vectors with lp-distance equal to 1, at each step of the trajectory of evolution integer-valued vectors of which the corresponding coordinates differ at most with one.

Properties of calculated predictions of grade sizes and the associated integer valued vectors

In a system modelled by a time-discrete deterministic model, predictions of the distribution of the members over the different classes do not result automatically in an integer valued vector. In this paper, for a constant size system, we discuss how to associate with the calculated vector an integer valued vector. Furthermore we examine whether the evolution of the calculated vectors on the one hand, and the evolution of the associated integer valued vectors on the other hand, have the same properties.

Properties of calculated predictions of grade sizes and the associated integer valued vectors

In a system modelled by a time-discrete deterministic model, predictions of the distribution of the members over the different classes do not result automatically in an integer valued vector. In this paper, for a constant size system, we discuss how to associate with the calculated vector an integer valued vector. Furthermore we examine whether the evolution of the calculated vectors on the one hand, and the evolution of the associated integer valued vectors on the other hand, have the same properties.

Optimal mean-square N-observation digital morphological filters : II. Optimal gray-scale filters

The second part of this two-part study interprets digital gray-scale morphological operations as numerical functionals on integer-valued vectors. The result is a gray-scale morphological statistical estimation theory based on N observation random variables and a consequent theory of mean-square optimization. Whereas in the binary setting the search for optimal structuring elements is clearly restricted by the requirement of binary N-vector structuring elements, gray-scale structuring elements are N-vectors of integers (not confined to the range of the images). Thus, it is necessary to find a set of structuring elements from which both singleand multiple-erosion filters can be optimized, the latter being characterized within the framework of the gray-scale Matheron representation. Consequently, a significant part of the paper is concerned with the derivation of the fundamental set, this set being a minimal family of structuring elements from which it is always possible to select the erosions comprising an optimal filter. As in the case of binary optimization, we treat constrained optimality; however, here we apply it to construction of optimal linear filters. Finally, we demonstrate that erosion optimization is equivalent to dilation optimization.

Finding the vertices nearest to a point in a hypercube

An algorithm is presented and analyzed for the following problem: given a point in the unit hypercube, iteratively output its nearest vertex, its second nearest vertex, and so forth. A more general version of this problem is also addressed, namely: given a d-long vector w of nonnegative reals, iteratively output nonnegative integer-valued vectors v 1, v 2,… where v n gives the n th smallest value of w T· v.

A Measure of “smoothness” of Multidimensional Distributions of Integer-Valued Random Vectors

A Measure of “smoothness” of Multidimensional Distributions of Integer-Valued Random Vectors

An arithmetic method of proving local theorems for arrays of independent integer-valued random vectors

An arithmetic method of proving local theorems for arrays of independent integer-valued random vectors