Random Set Model Research Articles

A Random Consideration Set Model for Demand Estimation, Assortment Optimization, and Pricing

Random Consideration Set Model We operationalize a microfounded consumer choice model—the random consideration set (RCS) choice model of Manzini and Mariotti [Manzini P, Mariotti M (2014) Stochastic choice and consideration sets. Econometrica 82(3):1153–1176]—that captures the limited attention of consumers, assuming that purchases are based on fixed preference orderings with consideration sets formed from independent attentions. We provide a condition for uniquely identifying model parameters and design an efficient algorithm for model parameters estimation. We offer a greedy-like algorithm for assortment optimization, adaptable for optimal assortment subject to cardinality constraint or discovering efficient sets. We extend the model to consider random product preferences, with a 1/2 performance-guaranteed approximation algorithm. Using data from a major U.S. airline, we find that the RCS model outperforms the mixed multinomial logit model in approximately half of the markets, particularly with smaller, less varied data sets.

Bounds on revenue for the random consideration set choice model

The random consideration set choice model is a recently proposed choice model that can capture the stochastic choice behavior of consumers. One advantage of the random consideration set choice model is that it can accommodate some phenomena that cannot be explained by the multinomial logit model. In this paper, we prove revenue bounds when the attention probabilities and preference order over products change for the random consideration set model. The bounds proposed can be used to control revenue differences when consumers' preference over product changes and can be useful for settings like online assortment optimization with a non-stationary environment.

Normal limiting distributions for systems of linear equations in random sets

We consider the binomial random set model [n]p where each element in {1,…,n} is chosen independently with probability p:=p(n). We show that for essentially all regimes of p and very general conditions for a matrix A and a column vector b, the count of specific integer solutions to the system of linear equations Ax=b with the entries of x in [n]p follows a (conveniently rescaled) normal limiting distribution. This applies among others to the number of solutions with every variable having a different value, as well as to a broader class of so-called non-trivial solutions in homogeneous strictly balanced systems. Our proof relies on the delicate linear algebraic study both of the subjacent matrices and the corresponding ranks of certain submatrices, together with the application of the method of moments in probability theory.

Numerical Study of the Influence of Joint Angle on the Failure Behavior of Randomly and Nonpersistently Jointed Rock Mass

The generation of nonpersistent joints in numerical specimens is not easy to achieve in some geotechnical numerical software. In this paper, the synthetic rock mass modeling method was used to construct a group of numerical specimens based on triangular grain-based model and random joint set (RJS) model in universal discrete element code. The randomly distributed nonpersistent joints have seven dominant inclination angles. Then the unconfined compressive strength (UCS) tests of the seven groups of specimens were carried out numerically to obtain some insight into the influence of dominant inclination angle on strength behaviors, failure modes and fracture development. The inclination angles of the nonpersistent RJS have fluctuant effect on the UCS of the specimens. The maximum UCS appeared when the RJS angle is 90°, and small UCS values appeared when the RJS angles are 45°, 60° and 75°. With decreasing RJS angle, the brittleness gradually changes into ductility, and strain hardening occurs. During the whole uniaxial compression process, shear fractures take the dominant role during the preliminary stage and then reduce gradually, while the tensile fractures have an inverse variation trend during the initial stage and significantly increase when the axial strain exceeds 0.6% and then keeps growing until the compression ending. The nonpersistent RJSs with small and large angles have little contribution to the shear fracture development, and only the RJSs with median inclination angles (45°, 60°) significantly promote the shear fracture propagation and development, while the RJS makes an increasing contribution to the tensile fracture propagation and development with the increase in the RJS inclination angle.

Assouad Spectrum Thresholds for Some Random Constructions

AbstractThe Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and the box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad and the Assouad dimension. For common models of random fractal sets, we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition, and we compute the threshold for the Gromov boundary of Galton–Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton–Watson tree result.

A Multi-Scan Labeled Random Finite Set Model for Multi-Object State Estimation

State-space models in which the system state is a finite set–called the multi-object state–have generated considerable interest in recent years. Smoothing for state-space models provides better estimation performance than filtering. In multi-object state estimation, the multi-object filtering density can be efficiently propagated forward in time using an analytic recursion known as the generalized labeled multi-Bernoulli (GLMB) recursion. In this paper, we introduce a multi-scan version of the GLMB model to accommodate the multi-object posterior recursion, and develop efficient numerical algorithms for computing this so-called multi-scan GLMB posterior.

Fast computation algorithm for the random consideration set model

This paper proposes a fast and reliable computation algorithm for the random consideration set model. A simple and numerically tractable representation of the model is developed to reduce the computational burden. I show that the new algorithm outperforms existing methods in terms of accuracy and speed.

Monte Carlo localisation of a mobile robot using a Doppler–Azimuth radar

This paper investigates the moving robot localisation problem using a Doppler–Azimuth radar array. The solution is formulated in the framework of nonlinear/non-Gaussian estimation using a particle filter and a random finite set (RFS) model of measurements. The proposed approach assumes the availability of a feature-based map, radar measurements and robot odometry data. The associations between the measurements and the features of the map (landmarks) are unknown. The RFS model is adopted to deal with false and missed detections and uses Murty’s algorithm to reduce computation when solving the association problem. The proposed particle filter incorporates the Kullback–Leibler Distance (KLD)-Sampling to reduce computational time. Monte-Carlo simulation results demonstrate the efficacy of the proposed algorithm.

A study on the reliability analysis of a deep excavation supported with anchored pile walls

AbstractThis paper presents a study performed to examine the applicability and convenience of the random set theory in combination with the finite element method (RS‐FEM), considering a case of a deep excavation. A case history of a 15 m deep multi‐anchored contiguous pile wall in clay is considered. Existing buildings around the excavation have necessitated a careful evaluation on the deformations and the reliability of the system. However, due to the existence of limited investigations, the soil parameters had to be estimated combining the results of these investigations with previous experience and expertness under similar conditions. Plane strain numerical analyses have been performed in order to predict the wall behavior. The parameters in the random set model have been chosen based on the sensitivity analyses. Most probable boundaries of the wall horizontal deformations have been compared with inclinometer readings. As suggested by previous case histories, wall deformations have been observed to fall within the lower third of the range predicted by RS‐FEM.

Efficient pathway enrichment and network analysis of GWAS summary data using GSA-SNP2.

Pathway-based analysis in genome-wide association study (GWAS) is being widely used to uncover novel multi-genic functional associations. Many of these pathway-based methods have been used to test the enrichment of the associated genes in the pathways, but exhibited low powers and were highly affected by free parameters. We present the novel method and software GSA-SNP2 for pathway enrichment analysis of GWAS P-value data. GSA-SNP2 provides high power, decent type I error control and fast computation by incorporating the random set model and SNP-count adjusted gene score. In a comparative study using simulated and real GWAS data, GSA-SNP2 exhibited high power and best prioritized gold standard positive pathways compared with six existing enrichment-based methods and two self-contained methods (alternative pathway analysis approach). Based on these results, the difference between pathway analysis approaches was investigated and the effects of the gene correlation structures on the pathway enrichment analysis were also discussed. In addition, GSA-SNP2 is able to visualize protein interaction networks within and across the significant pathways so that the user can prioritize the core subnetworks for further studies. GSA-SNP2 is freely available at https://sourceforge.net/projects/gsasnp2.

Prediction of Effective Properties of Porous Carbon Electrodes from a Parametric 3D Random Morphological Model

Pore structures have a major impact on the transport and electrical properties of electrochemical devices, such as batteries and electric double-layer capacitors (EDLCs). In this work we are concerned with the prediction of the electrical conductivity, ion diffusivity and volumetric capacitance of EDLC electrodes, manufactured from hierarchically porous carbons. To investigate the dependence of the effective properties on the pore structures, we use a structurally resolved parametric model of a random medium. Our approach starts from 3D FIB-SEM imaging, combined with automatic segmentation. Then, a random set model is fitted to the segmented structures and the effective transport properties are predicted using full field simulations by iterations of FFT on 3D pore space images and calculations based on the geometric properties of the structure model. A parameter study of the model is used to investigate the sensitivity of the effective conductivity and diffusivity to changes in the model parameters. Finally, we investigate the volumetric capacitance of the EDLC electrodes with a geometric model, make a comparison with experimental measurements and do a parameter study to suggest improved microstructures.

Attention, Consideration then Selection Choice Model

Manzini and Mariotti [Manzini, Paola, Marco Mariotti. 2014. Stochastic choice and consideration sets. Econometrica 82(3) 1153-1176.] propose a consideration set based choice model that postulates a full preference ordering as well as exogenous attention probabilities from which consideration sets are formed. The model assumes that consumers select the highest ranked product in their consideration set and the heterogeneity among choices are due to randomness in the consideration sets formation process. While Manzini and Mariotti focus on rationalizing this choice model, we look at its operations applicability. We show how to recover the full ordering and attention probabilities given accurate estimates of choice probabilities or from empirical data. Empirical testing of the Random Consideration Set model on our airline partner’s data shows that it outperforms the Multinomial Logit choice model in test data over all markets. The Random Consideration Set model also performs better than the Mixture of MNLs model giving a better fit on 67.0% of the markets. We show that an assortment that maximizes expected revenues can be found in $O(n)$ time where $n$ is the number of products. Adding a cardinality constraint increases the complexity to $O(n^2)$. We show that the efficient sets discovery problem can be solved in $O(n^2)$ where the goal is to find an assortment to maximize revenues net of the marginal value of capacity. We extend the Manzini and Mariotti Random Consideration Set model to allow multiple market segments with heterogenous attention probabilities and propose a computationally efficient heuristic with good empirical performance. We also extend the model to allow ties in preferences and show that a revenue-ordered assortment has a 1/2 performance guarantee relative to the optimal assortment. We study the pricing problem where the preference ordering are price aware and show under mild assumptions that optimal profits are such that both the profit contributions and the net value to consumers are aligned with the value gap defined as the difference between the value of the product to consumers and its unit wholesale cost.

Managing Information Uncertainty in Wave Height Modeling for the Offshore Structural Analysis through Random Set

This chapter presents a reliability study for an offshore jacket structure with emphasis on the features of nonconventional modeling. Firstly, a random set model is formulated for modeling the random waves in an ocean site. Then, a jacket structure is investigated in a pushover analysis to identify the critical wave direction and key structural elements. This is based on the ultimate base shear strength. The selected probabilistic models are adopted for the important structural members and the wave direction is specified in the weakest direction of the structure for a conservative safety analysis. The wave height model is processed in a P-box format when it is used in the numerical analysis. The models are applied to find the bounds of the failure probabilities for the jacket structure. The propagation of this wave model to the uncertainty in results is investigated in both an interval analysis and Monte Carlo simulation. The results are compared in context of information content and numerical accuracy. Further, the failure probability bounds are compared with the conventional probabilistic approach.

Book review: Review of Particle Filters for Random Set Models

If the reader is able to mentally strip out the unnecessary and sometimes burdensome random set framework used in many places, he will find a very fine book filled with interesting examples illustrating the power of particle filters. If fact, he will find a book much like a sequel to Beyond the Kalman Filter.

Book review: Review of Particle Filters for Random Set Models

Branko Ristic has given us a superb new book that combines two hot topics (particle filters and random sets) into a delicious feast of algorithms, Monte Carlo simulations, important applications, and mathematics that is new for normal engineers, all served up with clarity and thoroughness. The seminal research on random sets applied to tracking and data fusion problems is due to Ron Mahler [1]. It is clearly a good idea to apply random sets to such problems; in fact, it is hard to avoid using them either implicitly or explicitly. For example, the set of hypotheses for the well-known multiple hypothesis tracking (MHT) algorithms are random sets. The set of hypotheses for MHT is obviously random, because it depends on the measurements (which are random owing to noise errors and random target states). This is exactly the viewpoint adopted in [5], in which one is forced to compute equivalence classes of random sets in order to obtain a set of hypotheses that is mutually exclusive and collectively exhaustive, as required for any rational probability model of the situation (Bayesian or otherwise). It turns out that “fuzzy sets” correspond to equivalence classes of random sets, as explained by Goodman [6]. The set of particles in particle filters is also random, because it is generated from random numbers, but almost no papers explicitly use non-trivial random set theory for particle filter design. In fact, there are very few papers that explicitly talk about random sets and particle filters together, with the conspicuous exception of [20]. Nevertheless, random sets are a mature and well documented mathematical tool, as shown in [1], [2], [3], and [4]. You should not be afraid of random set mathematics; Ron Mahler has written a series of highly accessible tutorials on using random finite sets which avoid the pathologies and complexity of random unfettered sets (e.g., [26] and [28]).

Random fields of bounded variation and computation of their variation intensity

AbstractThe main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called thevariation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

Iterated Boolean random varieties and application to fracture statistics models

Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in R2 and R3 and on random planes in R3. The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set K and the Choquet capacity T (K) are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical results on the probability of fracture of convex specimens in the framework of the weakest link assumption are derived, and used to compare geometrical effects on the sensitivity of materials to fracture.

An overview of particle methods for random finite set models

This overview paper describes the particle methods developed for the implementation of the class of Bayes filters formulated using the random finite set formalism. It is primarily intended for the readership already familiar with the particle methods in the context of the standard Bayes filter. The focus in on the Bernoulli particle filter, the probability hypothesis density (PHD) particle filter and the generalised labelled multi-Bernoulli (GLMB) particle filter. The performance of the described filters is demonstrated in the context of bearings-only target tracking application.

Four random permutations conjugated by an adversary generate Sn with high probability

AbstractWe prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group Sn generate a transitive subgroup with probability for some independent of n, even when an adversary is allowed to conjugate each of the four by a possibly different element of . In other words, the cycle types already guarantee generation of a transitive subgroup; by a well known argument, this implies generation of An or except for probability as . The analysis is closely related to the following random set model. A random set is generated by including each independently with probability . The sumset is formed. Then at most four independent copies of are needed before their mutual intersection is no longer infinite. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 409–428, 2016

Comments on "Joint Detection and Estimation of Multiple Objects From Image Observations

The above article [1] introduced an algorithm for multitarget track-before-detect based on a multi-Bernoulli random finite set model (MB-TBD). This new algorithm was compared with the Histogram Probabilistic Multi-Hypothesis Tracker (H-PMHT) on simulated data examples containing multiple targets with non-linear dynamics. The authors reported poor performance from H-PMHT and described several deficiencies of the algorithm. This note highlights unnecessary assumptions made in the assessment of H-PMHT and repeats two of the simulation examples after relaxing them. We demonstrate a substantial improvement in performance compared with the originally published results. The simulation example is also shown to be a relatively high signal to noise problem and good performance is obtained from a conventional detect-then-track algorithm.