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Lagrangian data assimilation exploits the trajectories of moving tracers as observations to recover the underlying flow field. One major challenge in Lagrangian data assimilation is the intrinsic nonlinearity that impedes using exact Bayesian formulae for the state estimation of high-dimensional systems. In this paper, an analytically tractable mathematical framework for continuous-in-time Lagrangian data assimilation is developed. It preserves the nonlinearity in the observational processes while approximating the forecast model of the underlying flow field using linear stochastic models (LSMs). A critical feature of the framework is that closed analytic formulae are available for solving the posterior distribution, which facilitates mathematical analysis and numerical simulations. First, an efficient iterative algorithm is developed in light of the analytically tractable statistics. It accurately estimates the parameters in the LSMs using only a small number of the observed tracer trajectories. Next, the framework facilitates the development of several computationally efficient approximate filters and the quantification of the associated uncertainties. A cheap approximate filter with a diagonal posterior covariance derived from the asymptotic analysis of the posterior estimate is shown to be skillful in recovering incompressible flows. It is also demonstrated that randomly selecting a small number of tracers at each time step as observations can reduce the computational cost while retaining the data assimilation accuracy. Finally, based on a prototype model in geophysics, the framework with LSMs is shown to be skillful in filtering nonlinear turbulent flow fields with strong non-Gaussian features. • Linear stochastic models are proposed to accelerate Lagrangian data assimilation. • An iterative algorithm is developed to estimate the parameters in the models. • The models are skillful in filtering nonlinear turbulent flow fields. • A novel random tracer selection strategy for data assimilation is proposed. [ABSTRACT FROM AUTHOR]

Knee pain is the most common and disabling symptom in Osteoarthritis (OA). Joint pain is a late manifestation of the OA. In earlier stages of the disease changes in joint structures are shown. Also, formation of bony osteophytes, cartilage degradation, and joint space reduction which are some of the most common, among others. The main goal of this study is to associate radiological features with the joint pain symptom. Univariate and multivariate studies were performed using Bioinformatics tools to determine the relationship of future pain with early radiological evidence of the disease. All data was retrieved from the Osteoarthritis Initiative repository (OAI). A case-control study was done using available data from participants in OAI database. Radiological data was assessed with different OAI radiology groups. We have used quantitative and semi-quantitative scores to measure two different relations between radiological data in three different time points. The goal was to track the appearance and prevalence of pain as a symptom. All predictive models were statistically significant (P ≤ 0,05), obtaining the receiving operating characteristic (ROC) curves with their respective area under the curves (AUC) of 0.6516, 0.6174, and 0.6737 for T-0, T-1 and T-2 in quantitative analysis. For semi-quantitative an AUC of 0.6865, 0.6486, and 0.6406 for T-0, T-1 and T-2. The models obtained in the Bioinformatics study suggest that early joint structure changes can be associated with future joint pain. An image based biomarker that could predict future pain, measured in early OA stages, could become a useful tool to improve the quality of life of people dealing OA. [ABSTRACT FROM AUTHOR]