Random Measurement Research Articles

Product formulas for multiple stochastic integrals associated with Lévy processes

AbstractIn the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.

Stochastic differential formula and solution of control problem

Exact solution of finite-dimensional linear stochastic differential system with control is derived. Its uniqueness up to stochastic modification is proved. In order to achieve this, detailed grounding of existence for stochastic integral over a process with orthogonal increments is provided. Particularly, density of step-functions set in the set of all integrable functions is proved. Integration by parts formula for stochastic integral is derived. Analogue of Fubini theorem is proved in the case, when one measure is linear and another one is a random orthogonal measure. Stochastic differential formula for finite-dimensional integral functional is established. Formulation of Ito’s stochastic differential formula and its comparison with the result is presented. The solution formula for controlled linear stochastic differential system was conditionally used in applied sciences. Its rigorous proof provides the necessary background for further research.

Polarization of active galactic nuclei with significant VLBI-Gaia displacements

Context. Numerous studies have reported significant displacements in the coordinates of active galactic nuclei (AGNs) between measurements using the very-long-baseline-interferometry (VLBI) technique and those obtained by the Gaia space observatory. There is consensus that these discrepancies do indeed manifest astrometrically resolved sub-components of AGNs rather than random measurement noise. Among other evidence, it has been reported that AGNs with VLBI-to-Gaia displacements (VGDs) pointing downstream of their parsec-scale radio jets exhibit higher optical polarization compared to sources with the opposite (upstream) VGD orientation. Aims. We aim to verify the previously reported connection between optical polarization and a VGD-jet angle using a larger dataset of polarimetric measurements and updated Gaia DR3 positions. We also seek further evidence supporting the disk-jet dichotomy as an explanation of such a connection by using millimeter-wave polarization and multiband optical polarization measurements. Methods. We performed optical polarimetric observations of 152 AGNs using three telescopes. These data are complemented by other publicly available polarimetric measurements of AGNs. We cross-matched public astrometric data from VLBI and Gaia experiments, obtained corresponding positional displacements, and combined this catalog with the polarimetric and jet direction data. Results. Active galactic nuclei with downstream VGDs are confirmed to have significantly higher optical fractional polarization than the upstream sample. At the same time, the millimeter-wavelength polarization of the two samples shows very similar distributions. Conclusions. Our results support the hypothesis that the VGDs pointing down the radio jet are likely caused by a component in the jet emitting highly polarized synchrotron radiation and dominating in the overall optical emission. The upstream-oriented VGDs are likely to be produced by the low-polarization emission of the central engine’s subcomponents, which dominate in the optical.

The Effect of Measurement Error on Hypothesis Testing in Small Sample Structural Equation Modeling: A Comparison of Various Estimation Approaches

Researchers seeking valid statistical inference in the presence of measurement error often apply approaches that ignore measurement error. This may result in biased estimates, inflated type I error rates, diminished power, and therefore, increases the risk of drawing erroneous conclusions. However, current advice on accounting for random measurement error is limited to large samples and traditional linear models. This article aims to address this gap for small samples and recent estimation approaches in structural equation modeling. Our results show substantial type I error rate inflation for approaches that ignore measurement error when the model contains correlated latent predictors.

Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension

AbstractFor large classes of even‐dimensional Riemannian manifolds , we construct and analyze conformally invariant random fields. These centered Gaussian fields , called co‐polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: . They share a fundamental quasi‐invariance property under conformal transformations. In terms of the co‐polyharmonic Gaussian field , we define the Liouville Quantum Gravity measure, a random measure on , heuristically given as and rigorously obtained as almost sure weak limit of the right‐hand side with replaced by suitable regular approximations . In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's ‐curvature: we give a rigorous meaning to the Polyakov–Liouville measure and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2‐dimensional Riemannian manifolds, all compact non‐negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even‐dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds.

Random Natural Gradient

Hybrid quantum-classical algorithms appear to be the most promising approach for near-term quantum applications. An important bottleneck is the classical optimization loop, where the multiple local minima and the emergence of barren plateaux make these approaches less appealing. To improve the optimization the Quantum Natural Gradient (QNG) method \cite{stokes2020quantum} was introduced – a method that uses information about the local geometry of the quantum state-space. While the QNG-based optimization is promising, in each step it requires more quantum resources, since to compute the QNG one requires O(m2) quantum state preparations, where m is the number of parameters in the parameterized circuit. In this work we propose two methods that reduce the resources/state preparations required for QNG, while keeping the advantages and performance of the QNG-based optimization. Specifically, we first introduce the Random Natural Gradient (RNG) that uses random measurements and the classical Fisher information matrix (as opposed to the quantum Fisher information used in QNG). The essential quantum resources reduce to linear O(m) and thus offer a quadratic "speed-up", while in our numerical simulations it matches QNG in terms of accuracy. We give some theoretical arguments for RNG and then benchmark the method with the QNG on both classical and quantum problems. Secondly, inspired by stochastic-coordinate methods, we propose a novel approximation to the QNG which we call Stochastic-Coordinate Quantum Natural Gradient that optimizes only a small (randomly sampled) fraction of the total parameters at each iteration. This method also performs equally well in our benchmarks, while it uses fewer resources than the QNG.

Inverse chance-constrained dynamic data envelopment analysis under natural and managerial disposability: Concerning renewable energy efficiency and potentials

The performance of dynamic systems with specific data has been examined in some data envelopment analysis (DEA) studies. However, in many real-life situations, dealing with dynamic systems becomes challenging because of random factors. So, in this paper, a chance-constrained dynamic DEA approach under managerial and natural disposability is first proposed to analyze the dynamic renewable energy efficacy of processes in the presence of random performance measures. Then, the potentials of some performance metrics are addressed for changes of others using the introduced inverse chance-constrained dynamic DEA models. The chance-constrained dynamic DEA techniques and their inverse dynamic stochastic problems are converted into linear problems. The introduced approaches are applied to probe the renewable energy efficiency and potentials of some OECD countries in a span of time. The results show stochastic dynamic DEA and inverse stochastic dynamic DEA provided are practical tools for making the best choices, when there is uncertainty.

An optimization of the Fourier self-calibration algorithm for improved noise suppression capability

Abstract Self-calibration emerges as a promising approach in ultra-precision field, overcoming the limitations imposed by instrument capabilities in conventional calibration. In the self-calibration research, the algorithm based on the Fourier transform firstly established a rigorous theoretical framework. However, despite advancements in alternative self-calibration methods that have improved noise suppression, the traditional Fourier algorithm still faces challenges, particularly in mitigating random measurement noise in the separation results. In this study, we introduce methods on enhancing the noise suppression capabilities of the Fourier self-calibration algorithm. We investigate the hypothesis in the traditional algorithm and find that the computation of weighted averages had minimal effects. Then, we present our method by augmenting measurement positions. Considering both the increase in the number of measurement positions and the enhancement of noise suppression capabilities, augmenting translation positions in the opposite direction is proved to be the most effective strategy. Our algorithm can effectively suppress measurement noise when one-sided point number of the artifact is below 21. While the additional data processing slightly increases runtime, it does not alter the algorithm’s computational complexity, yet it significantly improves noise suppression. The effectiveness of our method is validated through a Monte Carlo comparison of uncertainties with the original algorithm. Experiments on Fizeau interferometer further prove robustness and practical feasibility of our proposed algorithm, with the measurement repeatability being reduced by 26.34% and 25.15% in the errors separated from the stage and the artifact.

Majorization and randomness measures

Abstract A series of papers by Hickey (1982, 1983, 1984) presents a stochastic ordering based on randomness. This paper extends the results by introducing a novel methodology to derive models that preserve stochastic ordering based on randomness. We achieve this by presenting a new family of pseudometric spaces based on a majorization property. This class of pseudometrics provides a new methodology for deriving the randomness measure of a random variable. Using this, the paper introduces the Gini randomness measure and states its essential properties. We demonstrate that the proposed measure has certain advantages over entropy measures. The measure satisfies the value validity property, provides an adequate extension to continuous random variables, and is often more appropriate (based on sensitivity) than entropy in various scenarios.

Stochastic representation for solutions of a system of coupled HJB-Isaacs equations with integral–differential operators

In this paper, we focus on the stochastic representation of a system of coupled Hamilton–Jacobi–Bellman–Isaacs (HJB–Isaacs (HJBI), for short) equations which is in fact a system of coupled Isaacs’ type integral-partial differential equation. For this, we introduce an associated zero-sum stochastic differential game, where the state process is described by a classical stochastic differential equation (SDE, for short) with jumps, and the cost functional of recursive type is defined by a new type of backward stochastic differential equation (BSDE, for short) with two Poisson random measures, whose wellposedness and a prior estimate as well as the comparison theorem are investigated for the first time. One of the Poisson random measures μ appearing in the SDE and the BSDE stems from the integral term of the HJBI equations; the other random measure in BSDE is introduced to link the coupling factor of the HJBI equations. We show through an extension of the dynamic programming principle that the lower value function of this game problem is the viscosity solution of the system of our coupled HJBI equations. The uniqueness of the viscosity solution is also obtained in a space of continuous functions satisfying certain growth condition. In addition, also the upper value function of the game is shown to be the solution of the associated system of coupled Isaacs’ type of integral-partial differential equations. As a byproduct, we obtain the existence of the value for the game problem under the well-known Isaacs’ condition.

Glucose inhibitor tubes in Croatian laboratories: are we doing well?

Reliable and accurate measurement of blood glucose concentration is of crucial importance for making clinical decisions in diagnosis diabetes, gestational diabetes and impaired fasting glucose tolerance. Survey was performed in form of questionnaire. Questionnaire was sent to all Croatian laboratories (N = 204) in electronic form using SurveyMonkey cloud-based software (SurveyMonkey, Inc., San Mateo, USA) as an extra-analytical module of the Croatian EQA (External Quality Assessment) provider Croatian center for external quality assessment (CROQALM) in June 2023. In total 148 (73%) of laboratories responded to the survey. Large proportion of laboratories never use glucose inhibitor tubes for random glucose measurement (more than half) or for glucose function tests (one quarter). Only three laboratories use recommended glycolysis inhibitor citrate. Many other inhibitors are also used, even if some of them are not recommended for plasma glucose measurement. Glucose is almost never (93%) sampled on ice when glucose inhibitor tube is not available. Laboratories in Croatia do not follow the recommended procedures regarding glycolysis inhibitors for glucose determination.

Plaquette models, cellular automata, and measurement-induced criticality

Abstract We present a class of two-dimensional randomized plaquette models, where the multi-spin interaction term, referred to as the plaquette term, is replaced by a single-site spin term with a probability of $1-p$. By varying $p$, we observe a ground state phase transition, or equivalently, a phase transition of the symmetry operator. We find that as we vary $p$, the symmetry operator changes from being extensive to being localized in space. These models can be equivalently understood as 1+1D randomized cellular automaton dynamics, allowing the 2D transition to be interpreted as a 1+1D dynamical absorbing phase transition. In this paper, our primary focus is on the plaquette term with three or five-body interactions, where we explore the universality classes of the transitions. Specifically, for the model with five-body interaction, we demonstrate that it belongs to the same universality class as the measurement-induced entanglement phase transition observed in 1+1D Clifford dynamics, as well as the boundary entanglement transition of the 2D cluster state induced by random bulk Pauli measurements. This work establishes a connection between transitions in classical spin models, cellular automata, and hybrid random circuits.


Gaze transition entropy as a measure of attention allocation in a dynamic workspace involving automation

Real-world work environments require operators to perform multiple tasks with continual support from an automated system. Eye movement is often used as a surrogate measure of operator attention, yet conventional summary measures such as percent dwell time do not capture dynamic transitions of attention in complex visual workspace. This study analyzed eye movement data collected in a controlled a MATB-II task environment using gaze transition entropy analysis. In the study, human subjects performed a compensatory tracking task, a system monitoring task, and a communication task concurrently. The results indicate that both gaze transition entropy and stationary gaze entropy, measures of randomness in eye movements, decrease when the compensatory tracking task required more continuous monitoring. The findings imply that gaze transition entropy reflects attention allocation of operators performing dynamic operational tasks consistently.

Efficient random walks for generating random fuzzy measures in Möbius representation in large universe

Random generation of fuzzy measures plays a pivotal role in large-scale decision-making and optimization. Random walks ensure uniform generation and adequate coverage. The Möbius representation of set functions is a valuable tool for establishing the sparse structure of fuzzy measures, with its non-negativity closely linked to monotonicity and convexity/supermodularity checking. We propose three efficient methods for monotonicity verification and convexity/supermodularity verification applicable to random walks in Möbius representations, specifically tailored for universal sets larger than ten inputs and k-order fuzzy measures. We first present the baseline methods by directly inspecting monotonicity and convexity constraints. Building on the observation that the majority of initially generated values exhibit non-negativity, we intentionally track negative Möbius values to enhance the computational performance of these baseline approaches. Further, we introduce the more agile methods that employ insertion and merge sorting techniques for both monotonicity and convexity checks in random walks that involve small perturbations of fuzzy measures. To address sparsity in large-scale scenarios, we focus on two major types of measures: k-additive and k-interactive measures, demonstrating their effectiveness through theoretical analysis and experimental results.

Ambivalent effects of mass cultivation of biodiesel producible green alga Tetraselmis striata on a microbial ecosystem: Evidence from mesocosm experiments

A 280-ton plant for the mass cultivation of Tetraselmis striata was operated from 2012 to 2019 to produce biodiesel fuel in western Korean coastal waters (Incheon) as a pilot project. This was the first instance globally where a microalgal mass cultivation plant for biodiesel production was implemented in coastal waters, and no prior studies had investigated the impact of microalgal mass cultivation plant on surrounding microecosystems. In this study, bioreactors (1× and 10×) mimicking a large-scale T. striata cultivation plant were installed in semi-permeable a mesocosms (5 tons) to assess their impact on the microbial ecosystem. The results showed that the release of large amounts of dissolve organic carbon (DOC) from the T. striata bioreactors. The 10× pond had a DOC concentration of 21.3 mg/L compared to the control pond of 2.1 mg/L. For the Water Quality Index (WQI), the 1× and 10× bioreactor installed mesocosms improved from Class II (Good) at the beginning of the experiment to Class I (Excellent) via decreasing nutrient levels and increasing of DO levels. However, from a biodiversity perspective, the microbial ecosystem deteriorated, with reductions in the diversity of zooplankton, ciliates, and phytoplankton. The correlation analysis and random forest variable importance measures indicated that the primary factor driving these changes was the alteration of the bacterial community due to elevated DOC levels. These findings indicate that while the mass cultivation of T. striata may improve physicochemical water quality, it has adverse effects on biological environments. Therefore, it is crucial to monitor physical, chemical, and biological factors comprehensively when cultivating microalgae on a large scale in marine environments.

Central limit theorem of overlap for the mean field Ghatak–Sherrington model

The Ghatak–Sherrington spin glass model is a random probability measure defined on the configuration space {0,±1,±2,…,±S}N with system size N and S⩾1 finite. This generalizes the classical Sherrington–Kirkpatrick (SK) model on the boolean cube {−1, +1}N to capture more complex behaviors, including the spontaneous inverse freezing phenomenon. We give a quantitative joint central limit theorem for the overlap and self-overlap array at sufficiently high temperature under arbitrary crystal and external fields. Our proof uses the moment method combined with the cavity approach. Compared to the SK model, the main challenge comes from the non-trivial self-overlap terms that correlate with the standard overlap terms.

Deep learning for solving initial path optimization of mean-field systems with memory

We consider the problem of finding the optimal initial investment strategy for a system modelled by a linear McKean–Vlasov (mean-field) stochastic differential equation with delay, driven by Brownian motion and a pure jump Poisson random measure. The goal is to determine the optimal initial values for the system in the period [ − δ , 0 ] , where δ > 0 is a delay constant, before the system starts at t = 0. Due to the delay in the dynamics, the system will, after startup, be influenced by these initial investment values. It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By utilizing this equivalence, we can find implicit expressions for the optimal investment. Moreover, we propose a deep neural network-based algorithm to solve the stochastic control problem with delay. Specifically, we employ a multi-layer feed-forward neural network for control modelling in the interval [ − δ , 0 ] , and use back-propagation to train the feed-forward neural network. The gradient of the loss function is computed using stochastic gradient descent (SGD) with respect to the weights of the network.

A Study on Lao Phuan Vernacular House in Ngan Area, Khoun District, Xieng Khuang Province

The study on vernacular houses of the Lao Phuan ethnic group aims to determine the area that still retains most of its original form, as defined by the fieldwork of Ngan area in Khoun District, Xieng Khouang Province as a case study. The selected area is the only one that still has a large number of traditional houses left behind. Through the study of relevant documents, physical surveys with maps and observations, random measurements of house components, and in-depth interviews with village chiefs, the elderly, and residents in the village: These Lao Phuan vernacular houses are designed and built to meet the needs of their occupations and to suit the nature or environment of the area in which they reside. Including their traditions, culture, and way of living. This can be seen from using materials that are easy to find, durable in use, and abundant locally, such as using Longleng wood as a roofing material. In addition, to being a natural sense, it is also an extra sun and rain-resistant material, making it usable for decades. At the same time, the style of the house has long eaves with small windows, which is due to the fact that the area is located in an area with high rainfall. In addition, the stilt house form will be used as a storage area for daily production equipment, firewood, fishing tools, and weaving equipment, as well as for survival and safety conditions. The use of space inside the house, in addition to allocating the area to serve a basic living, also allocated a large hall area to accommodate guests or relatives who visit regularly. Especially when there are traditional events held in the village or gift ceremonies inside the house. Another important point is that the area has to adapt to the dry season (winter). Therefore, two fireplaces have been set up; the first is in the cooking area. The second point is in the living area to use the fireplace to keep warm, the process of building a house still follows the strict traditions and beliefs that have been practiced for a long time.

Multimodal Deep Representation Learning for Quantum Cross-Platform Verification.

Cross-platform verification, a critical undertaking in the realm of early-stage quantum computing, endeavors to characterize the similarity of two imperfect quantum devices executing identical algorithms, utilizing minimal measurements. While the random measurement approach has been instrumental in this context, the quasiexponential computational demand with increasing qubit count hurdles its feasibility in large-qubit scenarios. To bridge this knowledge gap, here we introduce an innovative multimodal learning approach, recognizing that the formalism of data in this task embodies two distinct modalities: measurement outcomes and classical description of compiled circuits on explored quantum devices, both containing unique information about the quantum devices. Building upon this insight, we devise a multimodal neural network to independently extract knowledge from these modalities, followed by a fusion operation to create a comprehensive data representation. The learned representation can effectively characterize the similarity between the explored quantum devices when executing new quantum algorithms not present in the training data. We evaluate our proposal on platforms featuring diverse noise models, encompassing system sizes up to 50 qubits. The achieved results demonstrate an improvement of 3 orders of magnitude in prediction accuracy compared to the random measurements and offer compelling evidence of the complementary roles played by each modality in cross-platform verification. These findings pave the way for harnessing the power of multimodal learning to overcome challenges in wider quantum system learning tasks.

Blocked Gibbs Sampler for Hierarchical Dirichlet Processes

Posterior computation in hierarchical Dirichlet process (HDP) mixture models is an active area of research in nonparametric Bayes inference of grouped data. Existing literature almost exclusively focuses on the Chinese restaurant franchise (CRF) analogy of the marginal distribution of the parameters, which can mix poorly and has a quadratic complexity with the sample size. A recently developed slice sampler allows for efficient blocked updates of the parameters, but is shown to be statistically unstable in our article. We develop a blocked Gibbs sampler that employs a truncated approximation of the underlying random measures to sample from the posterior distribution of HDP, which produces statistically stable results, is highly scalable with respect to sample size, and is shown to have good mixing. The heart of the construction is to endow the shared concentration parameter with an appropriately chosen gamma prior that allows us to break the dependence of the shared mixing proportions and permits independent updates of certain log-concave random variables in a block. En route, we develop an efficient rejection sampler for these random variables leveraging piece-wise tangent-line approximations. Supplementary materials, which include substantive additional details and code, are available online.