Linear Combination Research Articles

Unsupervised pattern identification in spatial gene expression atlas reveals mouse brain regions beyond established ontology

The rapid growth of large-scale spatial gene expression data demands efficient and reliable computational tools to extract major trends of gene expression in their native spatial context. Here, we used stability-driven unsupervised learning (i.e., staNMF) to identify principal patterns (PPs) of 3D gene expression profiles and understand spatial gene distribution and anatomical localization at the whole mouse brain level. Our subsequent spatial correlation analysis systematically compared the PPs to known anatomical regions and ontology from the Allen Mouse Brain Atlas using spatial neighborhoods. We demonstrate that our stable and spatially coherent PPs, whose linear combinations accurately approximate the spatial gene data, are highly correlated with combinations of expert-annotated brain regions. These PPs yield a brain ontology based purely on spatial gene expression. Our PP identification approach outperforms principal component analysis and typical clustering algorithms on the same task. Moreover, we show that the stable PPs reveal marked regional imbalance of brainwide genetic architecture, leading to region-specific marker genes and gene coexpression networks. Our findings highlight the advantages of stability-driven machine learning for plausible biological discovery from dense spatial gene expression data, streamlining tasks that are infeasible by conventional manual approaches.

Multivariate Stochastic Volatility with Co-Heteroscedasticity

Abstract A new methodology that decomposes shocks into homoscedastic and heteroscedastic components is developed. This specification implies there exist linear combinations of heteroscedastic variables that eliminate heteroscedasticity; a property known as co-heteroscedasticity. The heteroscedastic part of the model uses a multivariate stochastic volatility inverse Wishart process. The resulting model is invariant to the ordering of the variables, which is shown to be important for volatility estimation. By incorporating testable co-heteroscedasticity restrictions, the specification allows estimation in moderately high-dimensions. The computational strategy uses a novel particle filter algorithm, a reparameterization that substantially improves algorithmic convergence and an alternating-order particle Gibbs that reduces the amount of particles needed for accurate estimation. An empirical application to a large Vector Autoregression (VAR) is provided, finding strong evidence for co-heteroscedasticity and that the new method outperforms some previously proposed methods in terms of forecasting at all horizons. It is also found that the structural monetary shock is 98.8 % homoscedastic, and that investment and the SP 500 index are nearly 100 % determined by fat tail heteroscedastic shocks. A Monte Carlo experiment illustrates that the new method estimates well the characteristics of approximate factor models with heteroscedastic errors.

Distributed Ishikawa algorithms for seeking the fixed points of multi-agent global operators over time-varying communication graphs

In this article, the problem of seeking fixed points for global operators over the time-varying graphs in a real Hilbert space is studied. The global operator is a linear combination of local operators, each local operator being accessed privately by one agent for less resource consumption. All agents form a network and they need to cooperate to solve problems. To this end, on the basis of the centralized Ishikawa iteration, the distributed Ishikawa algorithm (D-I) is first proposed. In the sequel, to predigest the calculational complexity, further considering the situation that only the random part of each operator coordinate is calculated in each iteration, the distributed block coordinate Ishikawa algorithm (D-BI) is also designed. The results indicate that the proposed D-I and D-BI algorithms can weakly converge to a fixed point of the multi-agent global operator. Eventually, we give a few numerical examples to illustrate practical benefits of the proposed algorithms.

Magnus force reduction in a shear-thinning fluid

This study aims to investigate the impact of fluid shear-thinning on the Magnus forces acting on a rotating cylinder or a sphere immersed in an unbounded flow using direct numerical simulation. The Carreau model is employed to represent the shear-thinning fluid, with the considered Reynolds number (Re) ranging from 0.01 to 100, Carreau number (Cu) from 0 to 100, power-law index (n) from 0.1 to 1, and viscosity ratio (β) from 0.001 to 0.5. The rotation rate (α) is fixed at 6. A characteristic Reynolds number, Rec, based on a viscosity evaluated at the characteristic shear rate, γ˙α=αU∞/2a, is introduced. It is found that, at a constant Rec, compared to that in a Newtonian fluid, the Magnus force exerted on the rotating cylinder or sphere in the shear-thinning fluid is reduced. This reduction results from the difference in viscosity distribution between the upper and lower sides of the cylinder or sphere. Furthermore, our analysis demonstrates that the logarithmic reduction in the Magnus force coefficient can be expressed as a linear combination of the logarithm of the strain rate difference and the logarithm of the shear strain rate sensitive function at two limit states, Cu→0 or Cu→∞. This work may be helpful to deepen the understanding of complex rheological behavior encountered in swirling flow hydrodynamics.

Classifying Plant Communities in the North American Coastal Plain With PRISMA Spaceborne Hyperspectral Imagery and the Spectral Mixture Residual

AbstractThe effort to map terrestrial biodiversity, in recent years limited mostly to the use of broadband multispectral remote sensing at decameter scales, can be greatly enhanced by harnessing hyperspectral imagery. Interpretation of hyperspectral imagery may be aided by the Mixture Residual (MR) spectral preprocessing transformation. MR integrates the benefits of spectral mixture analysis with the absorption peak‐enhancing characteristics of continuum removal. MR characterizes each pixel as a linear combination of generic end‐members estimating the spectral continuum, from which the residual of each wavelength is computed and treated as a source of additional information. Using Hyperspectral Precursor of the Application Mission (PRISMA) imagery, we tested the ability of MR‐transformed reflectance as compared to untransformed surface reflectance (SR) to map plant associations and land cover using ground truthing and random forest classifications across four landscapes within the North American Coastal Plain. We used a forward stepwise selection algorithm to choose bands for each classification and subsequently compared these between SR and MR. Our MR classifications distinguished land cover with 5% greater balanced accuracy on average than the SR‐based classifications across all four landscapes. The MR‐based classification that integrated data from all landscapes into a unified model encompassing all 21 land cover types achieved a 76% average balanced accuracy over three iterations. Generally, MR utilized the near‐infrared region to a greater degree than SR while deemphasizing the green peak. Based on our results, MR improves the accuracy of mapping terrestrial biodiversity, likely extending to other current and planned satellite hyperspectral missions.

Translation-based completeness on compact intervals

Given a compact interval I⊆R, and a function f that is a product of a polynomial with a Gaussian, it will be shown that the translates {f(⋅−λ):λ∈Λ} are complete in C(I) if and only if the series of reciprocals of Λ diverge. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299-308]. An additional characterization is obtained when Λ is an arithmetic progression, and the generator f constitutes a linear combination of translates of a function with sufficiently fast decay.

Designing Digital Resources for Multimodal Composition in the Kindergarten Environment

In the contextual field of environmental composition in kindergarten, transforming it into a space for cognitive, social, emotional, i.e., holistic formation and development of the child, multimodality operates with authenticity, magnetism, multilevelity, multidimensionality, and dynamism to overcome schematicity and linearity of thinking, thus stimulating imagination. Leading the way is the understanding that the environmental composing of multimodality in kindergarten is the foundation of quality for the child’s cognitive, affective, sensory cognition and self-knowledge. This paper outlines the parameters of evaluating multimodal digital resources designed by students preparing for future kindergarten teachers. The starting points for their expertise in the relations of autonomy, coherence, justification, adequacy and authenticity as subjectively significant and objective wholes of multimodal composing are argued. Criteria and indicators for the expert evaluation of digital resources for multimodal environmental composing in kindergarten, as modeling and provoking and enriching the individual sensory experience of the child, as an opportunity to stimulate cognitive and personal activity are derived. It is necessary to conclude that through the design of multimodal digital resources, the existing contradictions in the starting conditions of the pedagogical interaction in the kindergarten will be overcome. The main recommendation points to the need for specific training of future children’s teachers for multimodal environmental composition of educational practices in kindergarten, synergizing different modalities, rather than their linear sequential environmental combination.

On the Optimal Combination of Elliptically Distributed Biomarkers to Improve Diagnostic Accuracy.

Diagnostic biomarkers play a critical role in biomedical research, particularly for the diagnosis and prediction of diseases, etc. To enhance diagnostic accuracy, extensive research about combining multiple biomarkers has been developed based on the multivariate normality, which is often not true in practice, as most biomarkers follow distributions that deviate from normality. While the likelihood ratio combination is recognized to be the optimal approach, it is complicated to calculate. To achieve a more accurate and effective combination of biomarkers, especially when these biomarkers deviate from normality, we propose using a receiver operating characteristic (ROC) curve methodology based on the optimal combination of elliptically distributed biomarkers. In this paper, we derive the ROC curve function for the elliptical likelihood ratio combination. Further, proceeding from the derived best combinations of biomarkers, we propose an efficient technique via nonparametric maximum likelihood estimate (NPMLE) to build empirical estimation. Simulation results show that the proposed elliptical combination method consistently provided better performance, demonstrating its robustness in handling various distribution types of biomarkers. We apply the proposed method to two real datasets: Autism/autism spectrum disorder (ASD) and neural tube defects (NTD). In both applications, the elliptical likelihood ratio combination improves the AUC value compared to the multivariate normal likelihood ratio combination and the best linear combination.

Bose-Einstein Condensation of Photons in a Four-Site Quantum Ring.

Thermalization of radiation by contact to matter is a well-known concept, but the application of thermodynamic methods to complex quantum states of light remains a challenge. Here, we observe Bose-Einstein condensation of photons into the hybridized ground state of a coupled four-site ring potential. In our experiment, the periodically closed ring lattice superimposed by a weak harmonic trap for photons is realized inside a spatially structured dye-filled microcavity. Photons thermalize to room temperature, and above a critical photon number macroscopically occupy the symmetric linear combination of the site eigenstates with zero phase winding, which constitutes the ground state of the system. The mutual phase coherence of photons at different lattice sites is verified by optical interferometry.

Bayesian Modeling of Travel Times on the Example of Food Delivery: Part 2—Model Creation and Handling Uncertainty

The e-commerce sector is in a constant state of growth and evolution, particularly within its subdomain of online food delivery. As such, ensuring customer satisfaction is critical for companies working in this field. One way to achieve this is by providing an accurate delivery time estimation. While companies can track couriers via GPS, they often lack real-time data on traffic and road conditions, complicating delivery time predictions. To address this, a range of statistical and machine learning techniques are employed, including neural networks and specialized expert systems, with different degrees of success. One issue with neural networks and machine learning models is their heavy dependence on vast, high-quality data. To mitigate this issue, we propose two Bayesian generalized linear models to predict the time of delivery. Utilizing a linear combination of predictor variables, we generate a practical range of outputs with the Hamiltonian Monte Carlo sampling method. These models offer a balance of generality and adaptability, allowing for tuning with expert knowledge. They were compared with the PSIS-LOO criteria and WAIC. The results show that both models accurately estimated delivery times from the dataset while maintaining numerical stability. A model with more predictor variables proved to be more accurate.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

Equivalent static wind load based on displacement mode and load combination for high‐rise buildings

AbstractUnder the action of the fluctuating wind load, the low frequency part produces background response to the structures, while the high frequency part produces resonance response to the structures. In structural design, equivalent static wind load is usually used to equate the fluctuating wind load. Although there are various methods of evaluating the equivalent static wind load, they did not consider the correlation between modal responses or considered them insufficiently. Therefore, in this paper, the correlation between modal response is considered to evaluate the equivalent static wind load, the displacement response is decomposed by proper orthogonal decomposition (POD) method, the correlation between modal displacement is removed, the equivalent static wind load is expressed in the form of displacement mode, and then the correlation between modal response is fully considered in the extreme value combination. This paper combines the equivalent static wind loads in order to make the wind resistance design more reasonable for high‐rise buildings. First, the formulas of equivalent static wind loads expressed by displacement modes of background and resonant response are deduced based on modal decomposition and POD method. Second, the combination formulas of square‐root‐of‐sum‐square (SRSS) and complete‐quadratic‐combination (CQC) rules for the equivalent static wind loads considering the mean wind loads are proposed. Both the linear combination formula of SRSS for the equivalent static wind load and the weighting factor expressions of background and resonance equivalent static wind load are given. Third, the accuracy and validity of the formulas of equivalent static wind load are verified by a wind tunnel pressure test of a high‐rise building. Finally, the simplified combination coefficient formulas for the equivalent static wind loads are proposed, and the combination of the high‐rise building base's fluctuating equivalent static wind loads of along‐wind direction, across‐wind direction, and torsional direction is analyzed.

Longitudinal Changes of CT-radiomic and Systemic Inflammatory Features Predict Survival in Advanced Non-Small Cell Lung Cancer Patients Treated With Immune Checkpoint Inhibitors.

This study aims to determine whether longitudinal changes in CT radiomic features (RFs) and systemic inflammatory indices outperform single-time-point assessment in predicting survival in advanced non-small cell lung cancer (NSCLC) treated with immune checkpoint inhibitors (ICIs). We retrospectively acquired pretreatment (T0) and first disease assessment (T1) RFs and systemic inflammatory indices from a single-center cohort of stage IV NSCLC patients and computed their delta (Δ) variation as [(T1-T0)/T0]. RFs from the primary tumor were selected for building baseline-radiomic (RAD) and Δ-RAD scores using the linear combination of standardized predictors detected by LASSO Cox regression models. Cox models were generated using clinical features alone or combined with baseline and Δ blood parameters and integrated with baseline-RAD and Δ-RAD. All models were 3-fold cross-validated. A prognostic index (PI) of each model was tested to stratify overall survival (OS) through Kaplan-Meier analysis. We included 90 ICI-treated NSCLC patients (median age 70y [IQR=42 to 85], 63 males). Δ-RAD outperformed baseline-RAD for predicting OS [c-index: 0.632 (95%CI: 0.628 to 0.636) vs. 0.605 (95%CI: 0.601 to 0.608) in the test splits]. Integrating longitudinal changes of systemic inflammatory indices and Δ-RAD with clinical data led to the best model performance [Integrated-Δ model, c-index: 0.750 (95% CI: 0.749 to 0.751) in training and 0.718 (95% CI: 0.715 to 0.721) in testing splits]. PI enabled significant OS stratification within all the models (P-value <0.01), reaching the greatest discriminative ability in Δ models (high-risk group HR up to 7.37, 95% CI: 3.9 to 13.94, P<0.01). Δ-RAD improved OS prediction compared with single-time-point radiomic in advanced ICI-treated NSCLC. Integrating Δ-RAD with a longitudinal assessment of clinical and laboratory data further improved the prognostic performance.

Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an Mth-order diagram at inverse temperature β and spectral width ωmax from O((βωmax)2M−1) for a direct quadrature to O(M(log(βωmax))M+1), with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca2RuO4, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams. Published by the American Physical Society 2024

Inequalities for linear combinations of orthogonal projections and applications

Inequalities for linear combinations of orthogonal projections and applications

Radiomics Based Differentiation of Glioblastoma and Metastatic Disease: Impact of Different T1-Contrast Enhanced Sequences on Radiomic Features and Model Performance.

To evaluate the radiomics-based model performance for differentiation between glioblastoma (GB) and brain metastases (BM) using magnetization prepared rapid gradient echo (MPRAGE) and volumetric interpolated breath-hold examination (VIBE) T1-contrast enhanced sequences. T1-CE MPRAGE and VIBE sequences acquired in 108 patients (31 GBs and 77 BM) during the same MRI session were retrospectively evaluated. Post standardized image pre-processing and segmentation, radiomics features were extracted from necrotic and enhancing tumor components. Pearson correlation analysis of radiomics features from tumor subcomponents was also performed. A total of 90 machine learning (ML) pipelines were evaluated using a five-fold cross validation. Performance was measured by mean AUC-ROC, Log-loss and Brier scores. A feature-wise comparison showed that the radiomic features between sequences were strongly correlated, with the highest correlation for shape-based features. The mean AUC across the top-ten pipelines ranged between 0.851-0.890 with T1-CE MPRAGE and between 0.869-0.907 with T1-CE VIBE sequence. Top performing models for the MPRAGE sequence commonly used support vector machines, while those for VIBE sequence used either support vector machines or random forest. Common feature reduction methods for top-performing models included linear combination filter and least absolute shrinkage and selection operator (LASSO) for both sequences. For the same ML-feature reduction pipeline, model performances were comparable (AUC-ROC difference range: [-0.078, 0.046]). Radiomic features derived from T1-CE MPRAGE and VIBE sequences are strongly correlated and may have similar overall classification performance for differentiating GB from BM. BM: Brain metastases, GB: glioblastoma, T1-CE: T1 contrast enhanced sequence, MPRAGE: magnetization prepared rapid gradient echo, ML: machine learning, RF: random forest, VIBE: volumetric interpolated breath-hold examination.

A novel family of Q1-finite volume element schemes on quadrilateral meshes

A novel family of isoparametric bilinear finite volume element schemes are constructed and analyzed to solve the anisotropic diffusion problems on general convex quadrilateral meshes. These new schemes are obtained by employing a special quadrature rule to approximate the line integrals in classical Q1-finite volume element method. The new quadrature rule is a linear combination of trapezoidal and midpoint rules, and the weights depend on a parameter ωK. The novelty of this work is that, for any fully anisotropic diffusion tensor, we provide some specific ωK to ensure the coercivity result of the proposed schemes on arbitrary parallelogram, quasi-parallelogram, trapezoidal and some general convex quadrilateral meshes. More interesting is that, the parameter ωK can only involves the anisotropic diffusion tensor and the geometry of quadrilateral cell. An optimal H1 error estimate is also proved on quasi-parallelogram meshes. Finally, the theoretical findings are validated by several numerical examples.

Minimax rates of convergence for sliced inverse regression with differential privacy

Sliced inverse regression (SIR) is a highly efficient paradigm used for the purpose of dimension reduction by replacing high-dimensional covariates with a limited number of linear combinations. This paper focuses on the implementation of the classical SIR approach integrated with a Gaussian differential privacy mechanism to estimate the central space while preserving privacy. We illustrate the tradeoff between statistical accuracy and privacy in sufficient dimension reduction problems under both the classical low- dimensional and modern high-dimensional settings. Additionally, we achieve the minimax rate of the proposed estimator with Gaussian differential privacy constraint and illustrate that this rate is also optimal for multiple index models with bounded dimension of the central space. Extensive numerical studies on synthetic data sets are conducted to assess the effectiveness of the proposed technique in finite sample scenarios, and a real data analysis is presented to showcase its practical application.

RF-GymCare: Introducing Respiratory Prior for RF Sensing in Gym Environments

Exercise-induced sudden death has become an increasingly serious health issue in recent years. Fortunately, respiratory monitoring can serve as an effective early prevention solution. In particular, radio frequency (RF) sensing has emerged as a promising solution due to its contactless nature and privacy preservation. Existing RF-based respiratory monitoring solutions for homes, offices, or vehicles would suffer from significant performance degradation when deployed in typical gym workouts, where respiratory and physical movements are mixed in the RF signals in an unpredictable manner, leading to a blind source separation (BSS) problem. What is worse, the prior assumptions of source independence in existing BSS algorithms no longer hold, leaving respiratory monitoring in gym scenarios as an open issue. In this paper, we propose RF-GymCare, which introduces respiratory priors into the BSS problem and utilizes knowledge distillation to transfer these priors, aiming to learn the demixing weight from mixed signals. With the demixing weight, the respiratory signal can be obtained through a linear combination of the mixed signals. Our experiments with 13 hours of data across nine types of exercises demonstrate its superiority over existing methods, achieving a median similarity of 0.75 and a median rate error of 0.5 respiration per minute (RPM), respectively.

Multi–level method of fundamental solutions for solving polyharmonic problems

We consider a multi–level method of fundamental solutions for solving polyharmonic problems governed by ΔNu=0,N∈N∖{1} in both two and three dimensions. Instead of approximating the solution with linear combinations of N fundamental solutions, we show that, with appropriate deployments of the source points, it is possible to employ an approximation involving only the fundamental solution of the operator ΔN. To determine the optimal position of the source points, we apply the recently developed effective condition number method. In addition, we show that when the proposed technique is applied to boundary value problems in circular or axisymmetric domains, with appropriate distributions of boundary and source points, it lends itself to the application of matrix decomposition algorithms. The results of several numerical tests are presented and analysed.