Schedule for: 17w5070 - Distributed Data for Dynamics and Manifolds

The second eigenfunction of the Laplace-Beltrami operator (often called Fiedler vector in discrete settings) follows the pattern of the overall shape of an object. This geometric property is well known and used for various applications including mesh processing, feature extraction, manifold learning, spectral embedding and the minimum linear arrangement problem. Surprisingly, this geometric property has not been precisely formulated yet. This problem is directly related to the somewhat obscure hot spots conjecture in differential geometry that postulates the behavior of heat diffusion near boundary. The aim of the talk is to discuss and raise the awareness of the problem. As an application of the hot spots conjecture, we show how the second eigenfunction alone can be used for shape modeling of elongated anatomical structures such as hippocampus and mandible, and determining the diameter of large-scale brain networks. This talk is based on Chung et al. 2015 Medical Image Analysis 22:63076.

Based on two EHR Big Data sets with sample sizes n=10 and 50 million respectively, we derived different types of disease-disease networks using the longitudinal information. We establish both short-term and long-term directed networks as well as the simultaneously-occurring undirected network of 1660 PheWAS disease groups. Among 2,753,940 possible disease pairs, we identified 646,969 for long-term and 10,587 for short-term significant pairs, respectively, which were observed in at least five patients and had relative risk (RR) > 1 with significance at 0.05 level after Bonferroni corrections. Among 1,376,970 possible disease pairs of simultaneous occurrence, we identified 18,137 which were observed in at least five patients and had RR > 1 with significance at 0.05 level after Bonferroni corrections. For the short-term network, the top out-degree diseases are more likely pregnancy and kidney related diseases; while for the long-term network, the top out-degree diseases are more likely chronical diseases. More clinical implications from these findings will be discussed. This project requires multidisciplinary technologies, including medical record databases, ontology, high-performance computing, computational modeling, large-scale optimization, machine learning and statistics. I will also discuss how to form a multidisciplinary team to collaborate on a Big Data project, which has potential to have a high impact in many scientific fields and people’s daily life.

The problem of modeling the dynamical regulation process within a gene network has been of great interest for a long time. We propose to model this dynamical system with a large number of nonlinear ordinary differential equations (ODEs), in which the regulation function is estimated directly from data without any parametric assumption. Most current research assumes the gene regulation network is static, but in reality, the connection and regulation function of the network may change with time or environment. This change is reflected in our dynamical model by allowing the regulation function varying with the gene expression and forcing this regulation function to be zero if no regulation happens. We introduce a statistical method called functional SCAD to estimate a time-varying sparse and directed gene regulation network, and, simultaneously, to provide a smooth estimation of the regulation function and identify the interval in which no regulation effect exists. The finite sample performance of the proposed method is investigated in a Monte Carlo simulation study. Our method is demonstrated by estimating a time-varying directed gene regulation network of 20 genes involved in muscle development during the embryonic stage of Drosophila melanogaster.

Advances in medical imaging acquisition are constantly increasing the complexity of data representing anatomical objects. In particular, some of these imaging modalities allow a richer representation of anatomical manifolds, as a geometric object coupled with a function defined on the geometric object itself, i.e. textured surfaces. We introduce a statistical model which enables a joint study of the variability of the functional objects and the geometric objects, with the latter typically living in a non-euclidean space. This requires the formulation of models whose estimates are constrained to lie in the original non-euclidean space, which is achieved by use of PDE regularized techniques. Moreover, we assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries.

The direction of outlyingness is crucial to describing the centrality of multivariate functional data. Motivated by this idea, we generalize classical depth to directional outlyingness for functional data. We investigate theoretical properties of functional directional outlyingness and find that it naturally decomposes functional outlyingness into two parts: magnitude outlyingness and shape outlyingness which represent the centrality of a curve for magnitude and shape, respectively. Using this decomposition, we provide a visualization tool for the centrality of curves. Furthermore, we design an outlier detection procedure based on functional directional outlyingness. This criterion applies to both univariate and multivariate curves and simulation studies show that it outperforms competing methods. Weather and electrocardiogram data demonstrate the practical application of our proposed framework. We further discuss an outlyingness matrix for multivariate functional data classification as well as plots for multivariate functional data visualization and outlier detection. The talk is based on joint work with Wenlin Dai.

There has been extensive work on data depth-based methods for robust multivariate data analysis. Recent developments have moved to infinite-dimensional objects such as functional data. In this work, we propose a new notion of depth, the total variation depth, for functional data. As a measure of depth, its properties are studied theoretically, and the associated outlier detection performance is investigated through simulations. Compared to magnitude outliers, shape outliers are often masked among the rest of samples and harder to identify. We show that the proposed total variation depth has many desirable features and is well suited for outlier detection. In particular, we propose to decompose the total variation depth into two components that are associated with shape and magnitude outlyingness, respectively. This decomposition allows us to develop an effective procedure for outlier detection and useful visualization tools, while naturally accounting for the correlation in functional data. Finally, the proposed methodology is demonstrated using real datasets of curves, images, and video frames. The talk is based on joint work with Huang Huang.

We propose a robust estimator for functional autoregressive models. This estimator, the Depth-based Least Squares (DLS) estimator, down-weights the influence of outliers by using the functional outlyingness as a centrality measure. The DLS estimator consists of two steps: identifying the outliers with a functional boxplot based on a defined depth, then down-weighting the outliers using the functional outlyingness. We prove that the influence function of the DLS estimator is bounded. Through a Monte Carlo study, we show that the DLS estimator performs better than the PCA and robust PCA estimators, which are the most commonly used.

We propose a new perspective on functional regression with a predictor process via the concept of manifold that is intrinsically finite-dimensional and embedded in an infinite-dimensional functional space, where the predictor is contaminated with discrete/noisy measurements. By a method of functional local linear manifold smoothing, we achieve a polynomial rate of convergence that adapts to the intrinsic manifold dimension and the level of noise/sampling contamination with a phase transition phenomenon depending on their interplay. This is in contrast to the logarithmic convergence rate in the literature of functional nonparametric regression. We demonstrate that the proposed method enjoys favourable finite sample performance relative to commonly used methods via simulated and real data examples.

Emboldened by a string of insights gleaned from time series using likelihood-based inference and stochastic dynamical systems models, we undertook to exploit age-specific disease incidence data using an age-structured stochastic transmission model. In this talk, I explain the study's motivation, in questions surrounding the current resurgence of pertussis in countries with high vaccine coverage, and describe the model we formulated to address these questions. I point out the interesting features of the model implementation and the critical aspects of the inference methodology, with special attention to the challenges associated with this high-dimensional context. I highlight the surprises among the scientific conclusions we drew and conclude by speculating on the unreasonable effectiveness of stochastic models in population biology.

Bayesian phylogenetics, which approximates a posterior distribution of phylogenetic trees, has become more and more popular with the development of Monte Carlo methods. Standard Bayesian estimation of phylogenetic trees can handle rich evolutionary models but requires expensive Markov chain Monte Carlo (MCMC) simulations, which may suffer from two difficulties, the curse of dimensionality and the local-trap problem. Our previous work [1] has shown that the combinatorial sequential Monte Carlo (CSMC) method can serve as a good alternative to MCMC in posterior inference over phylogenetic trees. However, the simple proposal distribution used in CSMC is inefficient to combine with MCMC in the framework of the particle Gibbs sampler. Moreover, CSMC is inapplicable to the particle Gibbs with ancestor sampling [2]. In this talk, we will present a more efficient CSMC method, called CSMC-BF, with a more flexible proposal. The proposed CSMC-BF can improve the performance of the particle Gibbs sampler compared with the original CSMC, and can be used in the particle Gibbs with ancestor sampling. We will demonstrate the advantages of the proposed CSMC-BF using simulation studies and real data analysis. References [1] Liangliang Wang, Alexandre Bouchard-Côté, and Arnaud Doucet. Bayesian phylogenetic inference using a combinatorial sequential Monte Carlo method. Journal of the American Statistical Association, 110(512):13621374, 2015. [2] Fredrik Lindsten, Michael I. Jordan, and Thomas B. Schön. Particle Gibbs with ancestor sampling. Journal of Machine Learning Research, 15(1):21452184, 2014.urvival models.

Metabolic systems are among of the oldest applications of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring metabolic models and for formulating reactions has been growing constantly, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. This situation begs the question whether there are representations of metabolic processes that are true over reasonably wide ranges, yet mathematically tractable. A glimpse into such representations is provided by Dynamic Flux Estimation which, under ideal conditions, reveals the actual shapes of functions representing metabolic processes, although not their mathematical formats. While intriguing, DFE is only directly applicable if a pathway system contains as many dependent variables as fluxes. Because most actual systems contain more fluxes than metabolite pools, this requirement is seldom satisfied. Some auxiliary methods have been proposed to alleviate this issue, but they were quite ad hoc. Here I demonstrate a generic strategy that renders DFE applicable to moderately underdetermined pathway systems. A second challenge with DFE is the need to identify explicit functional formats that have shapes as close as possible to those inferred. Clearly, even if this inference is feasible, the result is necessarily biased. As an alternative, I demonstrate that good time series data allow us to circumvent this step and to develop nonlinear dynamic models in an entirely nonparametric fashion. The resulting nonparametric models offer a surprisingly wide range of analytic tools, including stability and sensitivity analyses. I will finish with some comments on dynamical model reduction, using power-law models. Goel, G., I-C. Chou, and E.O. Voit: System estimation from metabolic time series data. Bioinformatics 24, 2505-2511, 2008. Dolatshahi, S., and E.O. Voit: Identification of metabolic pathway systems. Frontiers in Genetics 7:6, 2016. Faraji, M. and E.O. Voit: Nonparametric dynamic modeling. Math. Biosc. 287, 130-146, 2017. Faraji, M. and E.O. Voit: Stepwise inference of likely dynamic flux distributions from metabolic time series data. Bioinformatics, 33 (14): 2165-2172; 2017. Voit, E.O.: The best models of metabolism. WIREs Systems Biology and Medicine. (in press)

Our goal is to develop an efficient framework for fitting stochastic continuous-time models to experimental data in cognitive psychology. As a simple test problem, we consider data from an eye-tracking study of attention in learning. For each subject, the data for each trial consists of the sequence of stimulus features that the subject fixates on, together with the duration of each fixation. We fit a stochastic differential equation model to this data, using the Approximate Bayesian Computation framework. For an individual subject we infer posterior distributions for the unknown parameters in the model.

High speed fluorescence microscopy has created incredible opportunities to tag nano-/micro-scale biological agents (proteins, organelles, vesicles, etc) and track them dynamically. With opportunities comes challenges. With most of these microscopy methods, other biological entities in the environment (membranes, microtubules, actin networks) cannot be visualized simultaneously. Can we use the dynamic observations from the agents of interest to infer the structure of these other biological entities or at least not corrupt our inference of the dynamics of those original agents? In this talk, I will discuss two such biological challenges along with particular data sets: 1) Molecular motor/cargo complexes (agents) and their interaction with un-visualized microtubules. 2) Tracer particles interacting with membranes (cellular, nuclear, etc ).

We propose a new, more flexible model to tackle the issue of lack of t for conventional functional linear regression. This new model, called the sparse functional additive model, is used to characterize the relationship between a functional predictor and a scalar response of interest. The effect of the functional predictor is represented in a nonparametric additive form, where the arguments are the scaled functional principal component scores. Component selection and smoothing are considered when fitting the model to reduce the variability and enhance the prediction accuracy, while providing an adequate t. To achieve these goals, we propose using the adaptive group LASSO method to select relevant components and smoothing splines to obtain a smoother estimate of those relevant components. Simulation studies show that the proposed estimation method compares favorably with various conventional methods in terms of prediction accuracy and component selection. The advantage of our proposed model and the estimation method is further demonstrated in two real data examples.

We propose a new method for the analysis of functional data defined over spatio-temporal domains. These data can be interpreted as time evolving surfaces or spatially dependent curves. The proposed method is based on regression with differential regularization. We are in particular interested to the case when prior knowledge on the phenomenon under study is available. The prior knowledge is described in terms of a time-dependent Partial Differential Equation (PDE) that jointly models the spatial and temporal variation of the phenomenon. We consider various samplings designs, including geo-statistical and areal data. We show that the corresponding estimation problem are well posed and can be discretized in space by means of the Finite Element method, and in time by means of the Finite Difference method. The model can handle data distributed over spatial domains having complex shapes, such as domains with strong concavities and holes. Moreover, various types of boundary conditions can be considered. The proposed method is compared to existing techniques for the analysis of spatio-temporal models, including space-time kriging and methods based on thin plate splines and soap film smoothing. As a motivating example, we study the blood flow velocity field in the common carotid artery, using data from Echo-Color Doppler.

In the talk, we will present and discuss a general framework for multi-aspect local non-parametric null-hypothesis testing for functional data defined on a common domain (Pini and Vantini, 2017). In detail: “multi-aspect” pertains to the fact the procedure allows the simultaneous investigation of many different data aspects like means, variances, quantiles of functional data and their associated differential and/or integral quantities; “local” pertains instead to the fact the procedure can impute the rejection to aspect-specific regions of the domain; finally, “non-parametric” refers to the fact that the specific implementation of the procedure is permutation-based and thus finite-sample exact and consistent independently on data Gaussianity. For ease of clarity, the focus will be on functional two-population tests and functional one-way ANOVA with an application on the statistical comparison of ultrasound tongue profiles pertaining to different allophones pronounced by the same speaker which can be modelled as functions varying on a spatio-temporal domain (Pini et al. 2017a). Finally, we will quickly show how to extend the approach to deal with more complex testing problems like functional two-way ANOVA and functional-on-scalar linear regression with applications to the analysis of spectral data (Pini et al. 2017b) and human movement data (Pini et al. 2015), respectively. Hébert-Losier, K., Pini, A., Vantini, S., Strandberg, J., Abramowicz, K., Schelin, L., Häger, C. K. (2015): "One-leg hop kinematics 20 years following anterior cruciate ligament rupture: Data revisited using functional data analysis", Clinical Biomechanics, Vol. 30(10), pp. 1153-1161. Pini, A., Vantini, S. (2017): “Interval-Wise Testing for Functional Data”, Journal of Nonparametric Statistics. 29 (2), pp. 407-424. Pini, A., Spreafico, L., Vantini, S., Vietti, A. (2017): Multi-aspect local inference for functional data: analysis of ultrasound tongue profiles. Tech. Rep. MOX 28/2017, Dept. of Mathematics, Politecnico di Milano (https://mox.polimi.it). Pini, A., Vantini, S., Colosimo, B. M., Grasso, M. (2017): “Domain-Selective Functional Analysis of Variance for Supervised Statistical Profile Monitoring of Signal Data”, Journal of the Royal Statistical Society – Series C (to appear).

In functional linear regression, one conventional approach is to first perform functional principal component analysis (FPCA) on the functional predictor and then use the first few leading functional principal component (FPC) scores to predict the response variable. The leading FPCs estimated by the conventional FPCA stand for the major source of variation of the functional predictor, but these leading FPCs may not be mostly correlated with the response variable, so the prediction accuracy of the functional linear regression model may not be optimal. In this paper, we propose a supervised version of FPCA by considering the correlation of the functional predictor and response variable. It can automatically estimate leading FPCs, which represent the major source of variation of the functional predictor and are simultaneously correlated with the response variable. Our supervised FPCA method is demonstrated to have a better prediction accuracy than the conventional FPCA method by using one real application on electroencephalography (EEG) data and three carefully-designed simulation studies.

Dimension reduction techniques play a very important role in analyzing a set of functional data that possess temporal or spatial dependence. Of these dimension reduction techniques, functional principal components (FPCs) analysis remains as a popular approach that extracts a set of latent components by maximizing variance in a set of dependent functional data. However, this technique may fail to adequately capture temporal or spatial autocorrelation in a functional data set. Functional maximum autocorrelation factors (FMAFs) are proposed for modelling and forecasting a temporal/spatially dependent functional data. FMAFs find linear combinations of original functional data that have maximum autocorrelation and are decreasingly predictable functions of time. We show that FMAFs can be obtained by searching for the rotated components that have smallest integrated first derivatives. Through a basis function expansion, a set of scores are obtained by multiplying extracted FMAFs with original functional data. Then, these scores are forecast using a vector autoregressive model under stationarity. Conditional on fixed FMAFs and observed functional data, the point forecasts are obtained by multiplying forecast scores with FMAFs. Interval forecasts can also be obtained by forecasting bootstrapped FMAF scores. Through a set of Monte Carlo simulation results, we study the finite-sample properties of the proposed FMAFs. Wherever possible, we compare the performance between the FMAFs and FPCs. Presentor: Han Lin Shang (co-authored with Giles Hooker and Steven Roberts)

In oceanic research, it is challenging to understand the patterns of winds and waves due to the complicated spatio-temporal dynamics. In this work, we propose new spectra-based methods for clustering hourly data of wind speed and wave height observed in the entire Red Sea. By clustering time series observed from different locations together, we identify spatial regions that share similar wind and wave directional spectra. We show that it is necessary to consider directional spectra for winds and waves, and that the clustering results may be very different, ignoring the direction.

Epilepsy is a neurological disorder that negatively affects the visual, audial and motor abilities of a patient. During an epileptic seizure attack, the patient may experience symptoms ranging from visual hallucinations to sense of disassociation. About 1% of American adults suffer from active epilepsy. In the more severe cases, doctors and patients may opt for surgery to remove the origin or focal point of seizures in the brain. The invasive treatment requires clear understanding of the seizure origin. A key brain imaging modality in the study of seizure origin is the electroencephalogram (EEG). Epileptic EEG signals are highly complex time series typically analyzed through their spectral features. Building on elements of computational topology, we propose a framework for analyzing single-trial epileptic EEG signals via their topological features in the time domain. An application to a multichannel dataset showed results of topological indifference before and during a seizure attack in the signals close to the diagnosed seizure origin. This finding was not identified by earlier spectral analyses of the same dataset.

The analysis of complex data distributed over large or highly textured regions poses new challenges for spatial statistics. Available methods usually rely on global assumptions about the stationarity of the field generating the data and are unsuitable for large, textured or convoluted spatial domains, with holes or barriers. We here propose a novel approach for spatial prediction which cope with the data and the domain complexities through iterative random domain decompositions. The method is general and apt to the analysis of different types object data. A case study on the analysis and spatial prediction of density data relevant to the study of dissolved oxygen depletion in the Chesapeake Bay (US) will illustrate the potential of the novel approach. This is a joint work with Alessandra Menafoglio and Giorgia Gaetani, at MOX-Department of Mathematics, Politecnico di Milano.