The study of moving particles (e.g. molecules, virus, vesicles, organelles, or whole cells) is crucial to decipher a plethora of cellular mechanisms within physiological and pathological conditions. Powerful live-imaging approaches enable life scientists to capture particle movements at different scale from cells to single molecules, that are collected in a series of frames. However, although these events can be captured, an accurate quantitative analysis of live-imaging experiments still remains a challenge. Two main approaches are currently used to study particle kinematics: kymographs, which are graphical representation of spatial motion over time, and single particle tracking (SPT) followed by linear linking. Both kymograph and SPT apply a space-time approximation in quantifying particle kinematics, considering the velocity constant either over several frames or between consecutive frames, respectively. Thus, both approaches intrinsically limit the analysis of complex motions with rapid changes in velocity. Therefore, we design, implement and validate a novel reconstruction algorithm aiming at supporting tracking particle trafficking analysis with mathematical foundations. Our method is based on polynomial reconstruction of 4D (3D+time) particle trajectories, enabling to assess particle instantaneous velocity and acceleration, at any time, over the entire trajectory. Here, the new algorithm is compared to state-of-the-art SPT followed by linear linking, demonstrating an increased accuracy in quantifying particle kinematics. Our approach is directly derived from the governing equations of motion, thus it arises from physical principles and, as such, it is a versatile and reliable numerical method for accurate particle kinematics analysis which can be applied to any live-imaging experiment where the space-time coordinates can be retrieved.
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The causal inference literature frequently focuses on estimating the mean of the potential outcome, whereas the quantiles of the potential outcome may carry important additional information. We propose a universal approach, based on the inverse estimating equations, to generalize a wide class of causal inference solutions from estimating the mean of the potential outcome to its quantiles. We assume that an identifying moment function is available to identify the mean of the threshold-transformed potential outcome, based on which a convenient construction of the estimating equation of quantiles of potential outcome is proposed. In addition, we also give a general construction of the efficient influence functions of the mean and quantiles of potential outcomes, and identify their connection. We motivate estimators for the quantile estimands with the efficient influence function, and develop their asymptotic properties when either parametric models or data-adaptive machine learners are used to estimate the nuisance functions. A broad implication of our results is that one can rework the existing result for mean causal estimands to facilitate causal inference on quantiles, rather than starting from scratch. Our results are illustrated by several examples.
It is well-known that mood and pain interact with each other, however individual-level variability in this relationship has been less well quantified than overall associations between low mood and pain. Here, we leverage the possibilities presented by mobile health data, in particular the "Cloudy with a Chance of Pain" study, which collected longitudinal data from the residents of the UK with chronic pain conditions. Participants used an App to record self-reported measures of factors including mood, pain and sleep quality. The richness of these data allows us to perform model-based clustering of the data as a mixture of Markov processes. Through this analysis we discover four endotypes with distinct patterns of co-evolution of mood and pain over time. The differences between endotypes are sufficiently large to play a role in clinical hypothesis generation for personalised treatments of comorbid pain and low mood.
Accounting for exposure measurement errors has been recognized as a crucial problem in environmental epidemiology for over two decades. Bayesian hierarchical models offer a coherent probabilistic framework for evaluating associations between environmental exposures and health effects, which take into account exposure measurement errors introduced by uncertainty in the estimated exposure as well as spatial misalignment between the exposure and health outcome data. While two-stage Bayesian analyses are often regarded as a good alternative to fully Bayesian analyses when joint estimation is not feasible, there has been minimal research on how to properly propagate uncertainty from the first-stage exposure model to the second-stage health model, especially in the case of a large number of participant locations along with spatially correlated exposures. We propose a scalable two-stage Bayesian approach, called a sparse multivariate normal (sparse MVN) prior approach, based on the Vecchia approximation for assessing associations between exposure and health outcomes in environmental epidemiology. We compare its performance with existing approaches through simulation. Our sparse MVN prior approach shows comparable performance with the fully Bayesian approach, which is a gold standard but is impossible to implement in some cases. We investigate the association between source-specific exposures and pollutant (nitrogen dioxide (NO$_2$))-specific exposures and birth outcomes for 2012 in Harris County, Texas, using several approaches, including the newly developed method.
Quantification of behavior is critical in applications ranging from neuroscience, veterinary medicine and animal conservation efforts. A common key step for behavioral analysis is first extracting relevant keypoints on animals, known as pose estimation. However, reliable inference of poses currently requires domain knowledge and manual labeling effort to build supervised models. We present a series of technical innovations that enable a new method, collectively called SuperAnimal, to develop unified foundation models that can be used on over 45 species, without additional human labels. Concretely, we introduce a method to unify the keypoint space across differently labeled datasets (via our generalized data converter) and for training these diverse datasets in a manner such that they don't catastrophically forget keypoints given the unbalanced inputs (via our keypoint gradient masking and memory replay approaches). These models show excellent performance across six pose benchmarks. Then, to ensure maximal usability for end-users, we demonstrate how to fine-tune the models on differently labeled data and provide tooling for unsupervised video adaptation to boost performance and decrease jitter across frames. If the models are fine-tuned, we show SuperAnimal models are 10-100$\times$ more data efficient than prior transfer-learning-based approaches. We illustrate the utility of our models in behavioral classification in mice and gait analysis in horses. Collectively, this presents a data-efficient solution for animal pose estimation.
Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree $n$ converge at the root-exponential rate $O(\exp(-2\rho\sqrt{n}))$ with $\rho>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-\rho^2$. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.
A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.
Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to investigating the effective dynamics for slow-fast stochastic dynamical systems. Given observation data on a short-term period satisfying some unknown slow-fast stochastic systems, we propose a novel algorithm including a neural network called Auto-SDE to learn invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also validated to be accurate, stable and effective through numerical experiments under various evaluation metrics.
Detecting differences in gene expression is an important part of single-cell RNA sequencing experiments, and many statistical methods have been developed for this aim. Most differential expression analyses focus on comparing expression between two groups (e.g., treatment vs. control). But there is increasing interest in multi-condition differential expression analyses in which expression is measured in many conditions, and the aim is to accurately detect and estimate expression differences in all conditions. We show that directly modeling single-cell RNA-seq counts in all conditions simultaneously, while also inferring how expression differences are shared across conditions, leads to greatly improved performance for detecting and estimating expression differences compared to existing methods. We illustrate the potential of this new approach by analyzing data from a single-cell experiment studying the effects of cytokine stimulation on gene expression. We call our new method "Poisson multivariate adaptive shrinkage", and it is implemented in an R package available online at //github.com/stephenslab/poisson.mash.alpha.
We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of recovering the solution operator of hyperbolic PDEs is the presence of characteristics, along which the associated Green's function is discontinuous. Therefore, a central component of our algorithm is a rank detection scheme that identifies the approximate location of the characteristics. By combining the randomized singular value decomposition with an adaptive hierarchical partition of the domain, we construct an approximant to the solution operator using $O(\Psi_\epsilon^{-1}\epsilon^{-7}\log(\Xi_\epsilon^{-1}\epsilon^{-1}))$ input-output pairs with relative error $O(\Xi_\epsilon^{-1}\epsilon)$ in the operator norm as $\epsilon\to0$, with high probability. Here, $\Psi_\epsilon$ represents the existence of degenerate singular values of the solution operator, and $\Xi_\epsilon$ measures the quality of the training data. Our assumptions on the regularity of the coefficients of the hyperbolic PDE are relatively weak given that hyperbolic PDEs do not have the ``instantaneous smoothing effect'' of elliptic and parabolic PDEs, and our recovery rate improves as the regularity of the coefficients increases.
This work focuses on the task of property targeting: that is, generating molecules conditioned on target chemical properties to expedite candidate screening for novel drug and materials development. DiGress is a recent diffusion model for molecular graphs whose distinctive feature is allowing property targeting through classifier-based (CB) guidance. While CB guidance may work to generate molecular-like graphs, we hint at the fact that its assumptions apply poorly to the chemical domain. Based on this insight we propose a classifier-free DiGress (FreeGress), which works by directly injecting the conditioning information into the training process. CF guidance is convenient given its less stringent assumptions and since it does not require to train an auxiliary property regressor, thus halving the number of trainable parameters in the model. We empirically show that our model yields up to 79% improvement in Mean Absolute Error with respect to DiGress on property targeting tasks on QM9 and ZINC-250k benchmarks. As an additional contribution, we propose a simple yet powerful approach to improve chemical validity of generated samples, based on the observation that certain chemical properties such as molecular weight correlate with the number of atoms in molecules.
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