Seminario de Modelamiento Matemático

The goal of this seminar series is to attract scientists working at the frontier between pure mathematics tools and statistical modeling.
The emphasis of this interdisciplinary seminar is twofold. First we aim to highlight new mathematical approaches that improve the understanding of the properties of complex statistical models. Secondly we aim to explore strengths and limitations of statistical methods from a pure mathematical perspective.

This seminar is held on Thursdays at 13:00 hrs in room 1 of the Faculty of Mathematics

Pablo Marquet. PUC Chile
The recursive structure of nature: Beyond the level of organization paradigm
Sala 1, Departamento de Matematicas UC
Ever since Aristotle, Linnaeus, and Comte, a hierarchical view of nature´s complexity has been the prevalent one to bring order to the structure of nature. This view has been so prominent, that the discipline of biology is structured along this hierarchy of nested levels, from genes to cells, organs, individuals, populations, communities, and ecosystems. However, this view of nature is becoming a straight jacket to make sense of biological complexity. In this talk I will show some mathematical analyses, based on simple birth-death models and diffusion approximations in an  open system context,  that point in a different direction, not to the levels of organization view but to a recursive organization one, where there are no levels but a recursive process that lead to networks of interacting entities where selection acts through differential persistence of the system.
Gonzalo Navarro. Department of Computer Science (Dcc), Faculty of Physical and Mathematical Sciences, University of Chile
Repetitiveness and Indexability
Sala 1, Departamento de Matematicas
Compressed indexes for highly repetitive text collections can reduce the data
size by orders of magnitude while still supporting efficient searches.
Compression of this kind of data requires dictionary-based methods, because
statistical compression fails to capture repetitiveness. Unlike statistical
compression, where the state of the art is mature and indexes reaching entropy
size are already several years old, there is not even a clear concept of
entropy for highly repetitive collections. There is a wealth of measures, some
more ad-hoc and some more principled. Some relations are known between them,
other relations are unknown. It is known that no compressor can reach some
measures, it is known how to reach others, and for some it is unknown whether
this is possible. From the reachable ones, some allow random access to the
compressed text, for others it is unknown how to do it. Finally, some admit
indexed searches, for others we do not know if this is possible. In this talk
I will survey this zoo of measures, show their properties and known relations,
show what is known and unknown about them, and point out several open
questions that relate repetitiveness with indexability.
Rolando Rebolledo. Universidad de Valparaiso
Open-System Population Dynamics: Ecological Examples
Sala 1, Departamento de Matematicas, Primer Piso
Ronny Vallejos. Universidad Tecnica Federico Santa Maria, Valparaiso
Modeling the Reduction of Sample Size for Spatial Datasets
Sala 1, edificio Rolando Chuaqui, PUC
Denis Parra. Puc, Departamento Ciencia de la Computación
Recommendation Systems for Visual Art
Sala 1, Edificio Rolando Chuaqui (dep. Matematicas)
Recommender Systems (RecSys) help people to find relevant items within a large information space by learning user preferences and producing personalized suggestions. Recsys are a well established research area, but most works have focused on recommendation
of movies, music or books. In this talk, I will present a survey of visual recommender systems, which use images
either as a recommendation target or as a signal to model user preferences towards suggesting other types of items. 
Then, I will focus on recommendation of visual art, where I will introduce a deep neural architecture for personalized recommendation
of paintings: CuratorNet. Our experiments indicate that this network performs especially well upon visual one-of-a-kind artworks, i.e., 
items with a single instance which do not allow the use traditional collaborative filtering methods. Finally, I will introduce
and discuss some ideas for future work relating to recent trends in creative deep learning for art, such as style transfer.
Ricardo Borquez. Pontificia Universidad Catolica de Chile
Recurrence properties of martingales
Sala 1, Edificio Rolando Chuaqui, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Macul, Chile.
In this seminar we introduce recent results on recurrence properties of (discrete-parameter) martingales and discuss some of  their implications for statistical inference on martingale laws. The main reference is Bórquez, R., "Recurrence property and pointwise convergence of martingales", Statistics and Probability Letters 135 p. 51-53, 2018.
Felipe Tobar. Universidad de Chile / CMM
On the spectral representation of Gaussian process regression models.
Sala 1, Edificio Rolando Chuaqui, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Macul, Chile.
From astronomy to biomedicine and from audio to climate, spectral estimation gives a meaningful interpretation of the data based on the frequency-wise energy spread of time series. At the same time, these application domains are characterised by noise-corrupted and missing observations, a scenario where Gaussian processes (GP) shine. In this talk, we will review the symbiotic relationship between spectral estimation and GPs: on one hand, we will see how we can design expressive covariance functions for Gaussian process with a specific power spectrum. On the other hand, we will see that using GPs as Bayesian interpolators can be instrumental for spectral estimation under missing and noisy observations. A straightforward result from studying the connection between GPs and SE is a modern probabilistic re-interpretation of classical signal processing concepts such as linear filtering, periodicity detection, band-limitedness, and Shannon-Nyquist sampling. The presented theory will be complemented with pedagogical illustrations and real-world experimental results.
Guido del Pino. PUC Chile
Functional specification of linear models and tensor products
Sala 1, Edificio Rolando Chuaqui, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Macul, Chile.
In this talk we propose to specify a linear model by subspaces generated by a set of linear functions. Truncating these functions leads to the matrix formulation and sheds some light on identifiability concepts.

The subspaces specify only the deterministics, so that the method applies e.g. to GLM and quantile regression.

In a functorial case, main effecs are defined as linear contrasts of the mean and also in terms of sums of squares.
Reinaldo B. Arellano-Valle. PUC Chile
Scale and Shape Mixtures of Multivariate Skew-Normal Distributions
Sala 1,
We introduce a broad and flexible class of multivariate distributions obtained by both scale and shape mixtures of multivariate skew-normal distributions. We present the probabilistic properties of this family of distributions in detail and lay down the theoretical foundations for subsequent inference with this model. In particular, we study linear transformations, marginal distributions, stochastic representations and hierarchical representations.

We also describe an EM-type algorithm for maximum likelihood estimation of the parameters of the model and demonstrate its implementation on a wind dataset. Our family of multivariate distributions unifies and extends many existing models of the literature that can be seen as submodels of our proposal.
Joint work with: Clécio S. Ferreira1, Department of Statistics, Federal University of Juiz de Fora, Juiz de Fora, Brazil. Marc G. Genton2, CEMSE Division, King Abdullah University of Science and Technology, Thuwal,
Saudi Arabia.
[1] Arellano-Valle, R. B., Ferreira, C. S., and Genton, M. G. (2018) Scale and shape mixtures of multivariate skew-normal distributions, Journal of Multivariate Analysis, 166, 98-110.
Elsa Cazelles . CMM (Center for Mathematical Modelling)
Statistical properties of regularized barycenters in the Wasserstein space and application to the registration of flow cytometry data.
Sala 1, Edificio Rolando Chuaqui, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Macul, Chile.
In this talk, we discuss the study of data that can be described by random probability measures (discrete or absolutely continuous) with support on Rd. The aim is to provide a first order statistical analysis on this space endowed with the Wasserstein distance, which boils down tothe study of the Frechet mean (or barycenter). In particular, we focus on the case of discrete data (or observations) sampled from absolutely continuous probability measures (a.c.) with respect to the Lebesgue measure. We thus introduce an estimator of the barycenter of random measures, penalized by a convex function, making it possible to enforce its a.c.
Another estimator is regularized by adding entropy when computing the Wasserstein distance (which has first been introduced for computational reasons). We are particularly interested in controlling the variance of these estimators.
Thanks to these results, the principle of Goldenshluger and Lepski allows us to obtain an automatic calibration of the regularization parameters. We then apply this work to the registration of multivariate densities, especially for flow cytometry data.
Claire Delplancke. Center of Mathematical Modeling in Santiago, Universidad de Chile
Bayesian Modeling for Inverse Problems? The Example of a Scalable Algorithm for Passive Seismic Tomography in Mining.
Sala 1, Edificio Rolando Chuaqui, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Macul, Chile.