Program - Statistics in Complex Systems
Wednesday, April 11
Time | Speaker | Title |
9.00-9.10 | Michael Sørensen | Welcome |
9.10-10.00 | Piotr Zwiernik | Totally positive exponential families, graphical models, and convex optimization |
10.00-10.50 | Martin Jaggi | Learning in a Distributed and Heterogeneous Environment |
10.50-11.10 | Coffee break | |
11.10-12.00 | Caroline Uhler | Maximum likelihood estimation for totally positive log-concave densities |
12.00-16.00 | Lunch and break | |
16.00-16.50 | Kasper Daniel Hansen | Co-expression networks are associated with the role of the epigenetic machinery in neurological dysfunction |
17.10-18.00 | Cecilia Holmgren | Split trees and Galton Watson trees: Two important classes of random trees |
18.00- | Symposium dinner |
Thursday, April 12
Thursday, April 12
Time | Speaker | Title |
9.00-9.50 | Markus Reiss | A nonparametric problem for SPDEs |
10.00-10.50 | Subhash Lele | Covariance models for spatio-temporal processes on a regular grid: Flexible, yet computationally simple |
10.50-11.10 | Coffee break | |
11.10-12.00 | Sonja Greven | Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains |
12.00-13.00 | Lunch | |
13.00- | Louisiana |
Friday, April 13
Friday, April 13
Time | Speaker | Title |
9.00-9.50 | Rajen Shah | Low-priced lunch in conditional independence testing |
10.00-10.25 | Søren Wengel Mogensen | Markov equivalence in graphical models for partially observed stochastic processes |
10.25-10.50 | Jacob Østergaard | Inferring network connectivity using cointegration analysis |
10.50-11.10 | Coffee break | |
11.10-12.00 | Marc Hoffmann | Nonparametric estimation in age-dependent models in a large population limit |
12.00-14.00 | Lunch and break | |
14.00-14.50 | Mathias Drton | Causal Discovery in Linear Non-Gaussian Models |
Abstracts
Abstracts
Piotr Zwiernik, Totally positive exponential families, graphical models, and convex optimization
Martin Jaggi, Learning in a Distributed and Heterogeneous Environment
We discuss recent directions in optimization algorithms used for the training of machine learning systems, such as generalized linear models (regression, classification) and deep learning. For distributed optimization when using many machines, as well as for integrated compute devices with varying compute and memory capacities (such as GPUs paired with regular compute nodes), we present ideas from convex optimization which help accelerating training. In particular, we study importance sampling methods and primal-dual gap techniques for these purposes.
Caroline Uhler, Maximum likelihood estimation for totally positive log-concave densities
We consider the problem of non-parametric density estimation under the shape constraint that the logarithm of the density function is concave and supermodular (i.e., the density is log-concave and totally positive). Given n independent and identically distributed random vectors, we first prove that the MLE under these shape constraints exists with probability one. We then characterize the domain of the MLE and show that it is in general larger than the convex hull of the observations. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a tent function and we provide a conditional gradient method for computing the MLE. In the more general setting where d>2, we provide a conditional gradient method under a more restrictive shape constraint known as translation supermodularity.
Kasper Daniel Hansen, Co-expression networks are associated with the role of the epigenetic machinery in neurological dysfunction
Markus Reiss, A nonparametric problem for SPDEs
In the first part drift estimation for stochastic ordinary and partial differential equations (SODEs/SPDEs) will be reviewed. The basic difference is that simple drift parameters can be identified from continuous SPDE observations, but not from SODE observations. In the second part we consider the specific problem of estimating the space-dependent (nonparametric) diffusivity of a stochastic heat equation from time-continuous observations with space resolution h. The rather counterintuitive result and its efficiency as h -> 0 will be discussed. This is joint work with Randolf Altmeyer.
Subhash Lele, Covariance models for spatio-temporal processes on a regular grid: Flexible, yet computationally simple
Mountain pine beetles (MPB) is a major forest pest in North America. Due to the climate change and other anthropogenic factors, the MPBs are spreading across large regions of Canada, substantially affecting forestry and forestry related economy in British Columbia and Alberta. Understanding the spread of MPB is important to devise biological control systems. The data on MPB are available on a regular grid across space and time. Two important impediments to modelling large spatial or spatio-temporal data are (a) Specification of the spatial covariance structure, (b) Conducting likelihood inference that involves computation of determinant and inversion of large matrices. We utilize a flexible class of covariance models, called separable covariance models, to model the dependence structure much more flexibly than the commonly used isotropic models. These models allow us to reduce the computational complexity by several orders of magnitude and at the same time, to increase the model flexibility by allowing geometric and other kinds of anisotropies. We will discuss the statistical and computational implications of separable covariance models for analyzing large amounts of spatio-temporal Gaussian and non-Gaussian data. This is joint work with Dean Koch and Mark Lewis.
Sonja Greven, Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains
We consider principal component analysis for multivariate functional data on different domains that may differ in dimension. This allows us to investigate the main joint modes of variation and joint dimension reduction for several curves and images. The theoretical basis for the approach is given in terms of a Karhunen-Loève Theorem. For the practically relevant case of a finite Karhunen-Loève representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers an estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. For the resulting estimators, asymptotic results are derived. The approach can be extended to finite univariate expansions in general, not necessarily orthonormal bases. It is also applicable for sparse functional data or data with measurement error, as naturally occurs in longitudinal studies. The new method is shown to be competitive to existing approaches for the special case of densely observed functional data on a common one-dimensional domain. The motivating application is a neuroimaging study, where the goal is to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline. A flexible R implementation in the packages MFPCA and funData is available on CRAN. This is joint work with Clara Happ.
Rajen Shah, Low-priced lunch in conditional independence testing
It is a common saying that testing for conditional independence, i.e., testing whether X is independent of Y, given Z, is a hard statistical problem if Z is a continuous random variable. We provide a formalisation of this result and show that a test with correct size does not have power against any alternative. Given the non-existence of a uniformly valid conditional independence test, we argue that tests must be designed so their suitability for a particular problem setting may be judged easily. To address this need, we propose to nonlinearly regress X on Z, and Y on Z and then compute a test statistic based on the sample covariance between the residuals, which we call the generalised covariance measure (GCM). We prove that validity of this form of test relies almost entirely on the weak requirement that the regression procedures are able to estimate the conditional means X given Z, and Y given Z, at a slow rate. While our general procedure can be tailored to the setting at hand by combining it with any regression technique, we develop the theoretical guarantees for kernel ridge regression. A simulation study shows that the test based on GCM is competitive with state of the art conditional independence tests.
Søren Wengel Mogensen, Markov equivalence in graphical models for partially observed stochastic processes
When studying multivariate stochastic processes, one can ask if a coordinate process is independent of another process. However, this ignores the temporal component of the model. Instead, one can ask if the past of a coordinate process is predictive of or influences the present of another process. This can be formalized using the notion of (conditional) local independence. This notion of independence resembles conditional independence of random variables, but in contrast to conditional independence, local independence treats past and present differently which means that B can be locally independent of A given C without A being locally independent of B given C. To make a graphical representation of local independence models, earlier work has used directed graphs in which each vertex corresponds to a coordinate process and introduced the notion of delta-separation in these graphs. We extend this work and introduce a larger class of graphs along with mu-separation to represent local independence models in partially observed systems of multivariate stochastic processes, i.e., settings where unmeasured and unknown processes may act as "confounders" of measured processes. We discuss properties of the Markov equivalence classes of these graphs, in particular a central maximality property which gives a characterization of Markov equivalence. This characterization is constructive in the sense that it straightforwardly outlines a learning algorithm in the oracle setting assuming that the graph and the local independence model are Markov and faithful. The maximality property also allows us to define another graphical object that concisely represents the equivalence class.
Jacob Østergaard, Inferring network connectivity using cointegration analysis
Marc Hoffmann, Nonparametric estimation in age-dependent models in a large population limit
Mathias Drton, Causal Discovery in Linear Non-Gaussian Models