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This chapter provides an end-to-end introduction to statistics; this highlights how statistics can be used to develop models from data, to quantify the uncertainty of such models, and to make decisions under uncertainty. The chapter also discusses how random variables are the key modeling paradigm that is used in statistics to characterize and quantify uncertainty and risk.
This chapter provides a discussion on multivariate random variables, which are collections of univariate random variables. The chapter discusses how the presence of multiple random variables gives rise to concepts of covariance and correlation, which capture relationships that can arise between variables. The chapter also discussed the multivariate Gaussian model, which is widely used in applications.
This chapter discusses how to apply principles of statistics, optimization, and linear algebra in advanced techniques of data science and machine learning. The chapter shows how to use principal component analysis and singular value decomposition for analyzing complex datasets and discusses advanced estimation techniques such as logistic regression, Gaussian process models, and neural networks.
Using wearable sensors to evaluate workers’ performance is challenging with existing sensor techniques. It requires detecting not only limb motions but also the onset and offset of specific actions. Commonly used inertial measurement units (IMUs) can be combined with surface electromyography (sEMG) to detect muscular activity. However, sEMG requires skin preparation and careful sensor placement, and can be affected by sweat or motion artifacts. To address these limitations, we used a wearable system combining IMUs and force-sensing resistors (FSRs), where IMUs capture joint kinematics and FSRs detect grasping actions. The system included three IMUs (on the trunk, upper arm, and forearm) and two FSR arrays (on the upper and lower arms). The system was first validated in a laboratory setting against an optical motion capture system with 10 healthy young adults performing isolated upper limb movements and mimicking lifting tasks. The results showed high agreement in joint angle estimation (coefficient of multiple correlation = 0.95 $ \pm $ 0.04), with a maximum root mean square error of 8.7 $ \pm $ 2.92°, and a mean absolute timing error for grasp detection of −0.59 seconds. To evaluate its applicability in real-world scenarios, a pilot in-field test was then conducted with two manufacturing workers (using and not using a passive shoulder exoskeleton) during a repetitive panel-packing task. The test shows highly consistent grasping detection, which allowed segmenting the task with a small variability in task duration (maximum coefficient of variation = 5.16$ \% $). These findings demonstrate the feasibility of using the proposed method in industrial environments to analyze upper limb motion and grasping activity.
We study the dynamics of salt fingers in the regime of slow salinity diffusion (small inverse Lewis number) and strong stratification (large density ratio), focusing on regimes relevant to Earth’s oceans. Using three-dimensional direct numerical simulations in periodic domains, we show that salt fingers exhibit rich, multiscale dynamics in this regime, with vertically elongated fingers that are twisted into helical shapes at large scales by mean flows and disrupted at small scales by isotropic eddies. We use a multiscale asymptotic analysis to motivate a reduced set of partial differential equations that filters internal gravity waves and removes inertia from all parts of the momentum equation except for the Reynolds stress that drives the helical mean flow. When simulated numerically, the reduced equations capture the same dynamics and fluxes as the full equations in the appropriate regime. The reduced equations enforce zero helicity in all fluctuations about the mean flow, implying that the symmetry-breaking helical flow is generated spontaneously by strictly non-helical fluctuations.
Cross-shelf transport in the inner continental shelf is governed by wind, wave and tidal interactions, but the role of Langmuir circulation (LC), induced by wave–current interaction and modulated by tides, has remained under-studied in this setting. We develop a Reynolds-averaged Navier–Stokes (RANS) model incorporating the Craik–Leibovich vortex force to resolve LC, coupled with a mass-conserving undertow and oscillating along-shelf tidal currents, and compare results against field data from the Martha’s Vineyard Coastal Observatory (MVCO). Under strong wave forcing (significant wave height $H_{\textit{sig}} = 2.12\,\mathrm{m}$ and significant wave period $T_w = 5.8\,\mathrm{s}$), LC persists throughout the tidal cycle, reducing vertical shear in the tidally averaged cross-shelf velocity profile compared with simulations excluding LC. During peak tidal velocity (reaching $25\,\mathrm{cm\,s^{-1}}$ with period of $ 12.42\,\mathrm{h}$), LC is temporarily suppressed but reforms rapidly as tidal energy declines, sustaining high vertical mixing. Conversely, under weak wave forcing ( $H_{\textit{sig}} = 0.837\,\mathrm{m}$, $T_w = 4.3\,\mathrm{s}$), tidal currents persistently suppress LC, resulting in a cross-shelf undertow profile with greater vertical shear compared with strong-wave conditions. Model–observation comparisons show that only simulations including both the Craik–Leibovich vortex force and tidal forcing reproduce the observed undertow structure at MVCO. These results demonstrate that accurate prediction of cross-shelf transport at tidal and subtidal time scales requires resolving both the generation and disruption of LC by tides.
The role of pylon-induced vortex structures on flame stabilisation within a supersonic pylon-cavity flameholder is numerically investigated. The study examines how the fuel jet interacts with the vortices produced by three distinct pylon-cavity flameholder geometries labelled as P0, P1 and P2. P0 represents the pyramidal-shaped baseline pylon configuration, whereas P1 and P2 consist of parallel and slanted grooves on the pylon slant surfaces with respect to the supersonic crossflow for the generation of instream vortices. The selection criteria for P1 and P2 are based on reduced effective blockage area compared with P0. The inlet flow Mach number used for the investigation is 2.2. A sonic ${{\rm{H}}_2}$ fuel injection at 2.5 bar and 250 K is used for all the test cases. Steady RANS reactive flow simulations are used for the assessments. An 18-step Jachimowski chemical kinetic scheme is used to model ${{\rm{H}}_2}$-air reaction mechanism. The flowfield structures within the pylon-cavity flameholder are categorised into two, (i) pylon-cavity geometry-induced vortex structures (II, III, and IV) and (ii) fuel jet vortex pairs (FJVP and SFJVP). The study shows that the interaction between these two decides the reactant mixture formation within the flameholder and the flame stabilisation. This study identifies four different flame-holding locations – L1, L2, L3, and L4 – and their strength depends on the pylon configuration. Overall the P2 configuration is found to perform better than the others in terms of high heat release magnitude and flame spread within the combustor.
This chapter provides an overview of different theoretical random variable models that can be used to model random phenomena encountered in applications. The chapter discusses the types of behavior that different models capture and provides some preliminary discussion on how to determine model parameters from data.
A Lagrangian description of bubble swarms has largely eluded both experimental and numerical efforts. Now, in a tour de force of deep-learning-enabled optical tracking measurements, Huang et al. (2025 J. Fluid. Mech.1014, R1) have managed to follow the three-dimensional trajectories of $10^5$ deforming and overlapping bubbles within a swarm, perhaps for long enough to witness their approach to the diffusive limit. Their results reveal that bubble swarms exhibit a dispersion law strikingly reminiscent of classical Taylor dispersion in isotropic turbulence, but with an earlier, undulatory transition from the ballistic-to-diffusive regime. Huang et al. (2025 J. Fluid Mech.1014, R1), have helped close the loop on our understanding of Lagrangian bubble dispersion – from self-stirring swarms to bubbles in isotropic turbulence.
This chapter discusses techniques that help us estimate parameters and summarizing statistics for random variables from data. The chapter discusses techniques such as the method of moments, least-squares, and maximum likelihood. The chapter also touches on concepts of Monte Carlo simulation, which is a technique that can be used to approximate the summarizing statistics of random variables from random samples or from data. The chapter also highlights how one can characterize the quality of such approximations using the central limit theorem and the law of large numbers.
This chapter discusses techniques to measure uncertainty/risk and to make decisions that explicitly take risk into consideration. The chapter also discusses how to use principles of statistics and optimization in advanced decision-making techniques such as stochastic programming, flexibility analysis, and Bayesian optimization.
Despite their widespread use, purely data-driven methods often suffer from overfitting, lack of physical consistency, and high data dependency, particularly when physical constraints are not incorporated. This study introduces a novel data assimilation approach that integrates Graph Neural Networks (GNNs) with optimization techniques to enhance the accuracy of mean flow reconstruction, using Reynolds-averaged Navier–Stokes (RANS) equations as a baseline. The method leverages the adjoint approach, incorporating RANS-derived gradients as optimization terms during GNN training, ensuring that the learned model adheres to physical laws and maintains consistency. Additionally, the GNN framework is well-suited for handling unstructured data, which is common in the complex geometries encountered in computational fluid dynamics. The GNN is interfaced with the finite element method for numerical simulations, enabling accurate modeling in unstructured domains. We consider the reconstruction of mean flow past bluff bodies at low Reynolds numbers as a test case, addressing tasks such as sparse data recovery, denoising, and inpainting of missing flow data. The key strengths of the approach lie in its integration of physical constraints into the GNN training process, leading to accurate predictions with limited data, making it particularly valuable when data are scarce or corrupted. Results demonstrate significant improvements in the accuracy of mean flow reconstructions, even with limited training data, compared to analogous purely data-driven models.