Jekyll2020-10-02T14:13:56+00:00/feed.xmlMario Teixeira Parenteacademic websitePhD defense: Passed2020-09-17T00:00:00+00:002020-09-17T00:00:00+00:00/posts/2020/09/17/phd-defense<p>I am very happy to write that I finally passed my PhD defense.</p>
<p>The defense consisted of a 25 minutes talk on the main outcomes of my research and a subsequent oral examination on contents of the dissertation.</p>
<p>I want to thank the reviewers, examiners, and the chair of the examination board for their participation and interest.</p>I am very happy to write that I finally passed my PhD defense.Submission of my PhD thesis2020-05-25T00:00:00+00:002020-05-25T00:00:00+00:00/posts/2020/05/25/subm-thesis<p>Finally, I made it to submit my PhD thesis “<em>Active Subspaces in Bayesian Inverse Problems</em>”.</p>
<p>The thesis is now going to be reviewed by two reviewers.
If the reviews are then accepted by the department, we can conduct the defense which is the final major step to graduation.</p>
<p>EDIT: A final version of the thesis is available at the <a href="https://mediatum.ub.tum.de/?id=1546065">TUM University library</a>.</p>Finally, I made it to submit my PhD thesis “Active Subspaces in Bayesian Inverse Problems”.Sensitivities in SEIR models: a (very) quick investigation2020-04-18T00:00:00+00:002020-04-18T00:00:00+00:00/posts/2020/04/18/seir-sensit<p>The COVID-19 pandemic recently caused and still causes major problems in several respects.
Also, (mathematical) modelers face difficulties in simulating and predicting the final size of the pandemic.</p>
<p>The dynamics of the pandemic are often simulated by compartmental models.
Although they are known to contain some uncertainties, they can be utilized to reliably demonstrate the effect of intervention strategies.</p>
<h1 id="goal">Goal</h1>
<p>The goal of this post is to show that a particular compartmental model, the <em>SEIR model</em>, is not useful for prediction purposes due to high sensitivities in its model parameters, which is of interest since some of the more sophisticated models are based on SEIR.
For example, the COVID-19 model of the German <a href="https://www.rki.de/"><em>Robert Koch-Institut</em></a> (RKI) is an adjusted SEIR model with more compartments to reflect the complexity of the COVID-19 pandemic; see <a href="https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf">their publication</a>.</p>
<p>We assume that the reader is already familiar with the SEIR model and some statistics.
A short description can be seen on <a href="https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SEIR_model">Wikipedia</a>.</p>
<p><strong>Remark</strong>.
This post is <em>not</em> a scientific statement.
It only/superficially describes the result of a (very) quick investigation of SEIR parameter sensitivities that the author conducted in his spare time.</p>
<h1 id="seir-model">SEIR model</h1>
<p>As a reminder, SEIR (excluding births and deaths) describes the dynamics of an infectious disease with the ODE system
\begin{align}
\frac{dS}{dt} &= -\beta S \frac{I}{N}, \newline
\frac{dE}{dt} &= \beta S \frac{I}{N} - \alpha E, \newline
\frac{dI}{dt} &= \alpha E - \gamma I, \newline
\frac{dR}{dt} &= \gamma I.
\end{align}</p>
<p>It models four compartments (susceptibles – <strong>S</strong>, exposed – <strong>E</strong>, infectious – <strong>I</strong>, removed – <strong>R</strong>) and the transitions between them involving model parameters for transition rates.</p>
<p><br /><center><img src="/assets/images/post-seir-sensit/compartments.png" /></center><br /></p>
<p>The initial conditions for the ODE system above are
\begin{align}
S(0) &= N-I(0), \newline
E(0) &= 0, \newline
I(0) &= I_0, \newline
R(0) &= 0,
\end{align}
where \(N\) is the (fixed) total number of individuals.
Note that
\begin{equation} S(t)+E(t)+I(t)+R(t)=N\end{equation}
for all \(t\geq0\).</p>
<p><strong>Remark</strong>.
The unit of time \(t\) is <em>weeks</em>.</p>
<p>The model parameters are:</p>
<ul>
<li>\(\beta\) – transmission rate (average number of contacts per person per time),</li>
<li>\(\alpha\) – latency rate, or equivalently, \(\alpha^{-1}\) – mean duration of the latency period,</li>
<li>\(\gamma\) – recovery rate, or equivalently, \(\gamma^{-1}\) – mean duration of the infection,</li>
<li>\(I_0\) – initial number of infections.</li>
</ul>
<p>One important characteristic of an infection is the so-called <a href="https://en.wikipedia.org/wiki/Basic_reproduction_number"><em>basic reproduction number</em></a> denoted by \(\mathcal{R}_0\).
It indicates the expected number of direct infections caused by exactly one case in a population where all other individuals are susceptible.
For the SEIR model, it can be computed to
\begin{equation}\mathcal{R}_0=\frac{\beta}{\gamma}.\end{equation}</p>
<h1 id="rkis-assumptions">RKI’s assumptions</h1>
<p>In <a href="https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf">their publication</a>, RKI makes the following assumptions:</p>
<ul>
<li>\(\mathcal{R}_0=2\),</li>
<li>\(\alpha^{-1}=3/7\),</li>
<li>\(\gamma^{-1}=9/7\),</li>
<li>\(I_0=1000\).</li>
</ul>
<p>It follows that \(\beta = 14/9\).</p>
<h1 id="sensitivity-study">Sensitivity study</h1>
<p>In the following, we additionally assume a total population size of \(N=80 \cdot 10^6 = 80\text{ million}\).</p>
<p>We set
\begin{equation}\theta = (\beta, \alpha, \gamma, I_0)^\top \in \mathbf{R}^4\end{equation}
as the parameter vector and regard it as a <em>random</em> vector following a uniform distribution \(\mu=\mathcal{U}(R)\) on a rectangle \(R\) such that
\begin{equation}\theta_i \in [\theta_i^-,\theta_i^+]\end{equation}
for all \(i=1,\ldots,4\).
The boundaries of the rectangle are determined by a perturbation of \(\pm p\%\) of the RKI parameters above.
For example,
\begin{equation}
\beta^{\pm} = 14/9 \cdot \left(1 \pm \frac{p}{100}\right).
\end{equation}</p>
<p>Let us set the simulation time to \(T=60\text{ [weeks]}\) and define two maps.
The map
\begin{equation}
\mathcal{G}_1(\theta) := (I(t))_{t=0,1,\ldots,T}
\end{equation}
takes a particular parameter \(\theta\) and computes the corresponding number of infectious individuals for times \(t=0,1,\ldots,T\).
Additionally, the map
\begin{equation}
\mathcal{G}_2(\theta) := \mathop{\mathrm{arg\,max}}_{t=0,1,\ldots,T}{I(t)}
\end{equation}
computes the peak time of the infectious compartment.</p>
<p>For a fixed \(p\in[0,100]\), we investigate two corresponding distributions, \(\mu(\mathcal{G}_1^{-1}(\cdot))\) and \(\mu(\mathcal{G}_2^{-1}(\cdot))\) by sampling \(M=1000\) times from \(\mu\) and computing \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) for each sample.</p>
<p>For a perturbation of \(p=5\%\), the (approximate) distributions are plotted in the following figure.</p>
<center><img src="/assets/images/post-seir-sensit/pert_5perc.svg" /></center>
<p>On the left, we see the distribution \(\mu(\mathcal{G}_1^{-1}(\cdot))\) with its mean, median, and a 95% quantile band, i.e. the band between the 2.5% and 97.5% quantile.
The right plot displays the distribution of the peaks.
The corresponding 95% quantile interval here is \([20, 24]\).</p>
<p>The same quantities are plotted for \(p=10\%\) in the following figure.</p>
<center><img src="/assets/images/post-seir-sensit/pert_10perc.svg" /></center>
<p>The 95% quantile interval for the infectious peaks on the right is \([19, 27]\).</p>
<p><strong>Remark</strong>.
This is not a serious sensitivity analysis.
There might be parameters that are more sensitive than others which is not visible by the investigated quantities and plots.</p>
<h1 id="conclusion">Conclusion</h1>
<p>The predictions of SEIR models are subject to uncertainties caused by sensitivities in its parameters.</p>
<p>For example, a perturbation of \(p=10\%\), which is likely to occur in practice, causes an uncertainty in the peak time of infectious individuals of about 8 weeks in the sense of a 95% quantile interval.</p>
<h1 id="source-code">Source code</h1>
<p>The source code to reproduce the above figures is put in a repository at <a href="https://bitbucket.org/m-parente/uq-tools/src/master/examples/epidemiology/">bitbucket</a>.</p>
<p>Since the samples for the plotted distributions are independent, we computed the corresponding ODE solutions in parallel using a program called <a href="https://github.com/TACC/launcher">launcher</a>.</p>The COVID-19 pandemic recently caused and still causes major problems in several respects. Also, (mathematical) modelers face difficulties in simulating and predicting the final size of the pandemic.Published: Generalized bounds for active subspaces2020-02-18T00:00:00+00:002020-02-18T00:00:00+00:00/posts/2020/02/18/asm-poincare-pub<p>I am very proud to annouce to our article <em>Generalized bounds for active subspaces</em> from <em>Jonas Wallin</em>, <em>Barbara Wohlmuth</em>, and me was accepted and published in <a href="https://projecteuclid.org/euclid.ejs"><em>Electronic Journal of Statistics</em></a> which is open access.</p>
<p>I explained <a href="/posts/2019/10/06/asm-poincare-prepr">here</a> (<em>v1</em>) and <a href="% link _posts/2020-02-03-asm-poincare-rev.md %}">here</a> (<em>v2</em>, revised) what the article is about.</p>
<p><strong>Journal link:</strong> <a href="https://doi.org/10.1214/20-EJS1684">doi:10.1214/20-EJS1684</a><br />
<strong>arXiv link:</strong> <a href="https://arxiv.org/abs/1910.01399">arXiv:1910.01399</a></p>I am very proud to annouce to our article Generalized bounds for active subspaces from Jonas Wallin, Barbara Wohlmuth, and me was accepted and published in Electronic Journal of Statistics which is open access.Revised preprint: Generalized bounds for active subspaces2020-02-03T00:00:00+00:002020-02-03T00:00:00+00:00/posts/2020/02/03/asm-poincare-rev<p><em>Jonas Wallin</em>, <em>Barbara Wohlmuth</em>, and I put a revised version of our article <em>Generalized bounds for active subspaces</em> in the <a href="https://arxiv.org/abs/1910.01399">arXiv</a>.
The main changes consist of a formalization of our results to a theorem/proof style, the consideration of a particular supremum (more below), and a revision of the section on future work with MGH distributions (multivariate generalized hyperbolics).</p>
<p>In the former version, our counterexample to existing theoretical results considered an <em>arbitrary</em> orthogonal transformation of input variables that, however, was used before as a particular defined transformation.
Since related quantities appear in error bounds, we now consider the supremum of the related quantities over the set of all orthogonal matrices which makes it valid for us to keep regard arbitrary transformations.
In fact, we should justify why it is enough in our case to consider rotations, a subset of orthogonal transformations, only.</p>
<p>Finally, I want to thank both of my co-authors for their feedback and assistance in revising this manuscript.</p>Jonas Wallin, Barbara Wohlmuth, and I put a revised version of our article Generalized bounds for active subspaces in the arXiv. The main changes consist of a formalization of our results to a theorem/proof style, the consideration of a particular supremum (more below), and a revision of the section on future work with MGH distributions (multivariate generalized hyperbolics).Accepted: Identifying relevant hydrological and catchment properties in active subspaces2019-11-25T00:00:00+00:002019-11-25T00:00:00+00:00/posts/2019/11/25/unmix-karst-daniel-accpt<p>I am very glad that our article <em>Identifying relevant hydrological and catchment properties in active subspaces: An inference study of a lumped karst aquifer model</em> was accepted for publication in <em>Advances in Water Resources</em> (Elsevier).</p>
<p>As <a href="/posts/2019/07/16/unmix-karst-mario-accpt">before</a>, this article comes out of a collaboration between the teams at TUM within the <a href="/posts/2018/03/08/kickoff-unmix"><em>UNMIX project</em></a>.
<em>Daniel Bittner</em> and <em>Gabriele Chiogna</em> from the TUM Chair of Hydrology and River Basin Management and <em>Steven Mattis</em>, <em>Barbara Wohlmuth</em>, and I from the TUM Chair for Numerical Mathematics were interested in hydrological inferences that can be drawn out of looking at subspaces that are dominant for the behavior of karst systems.
In particular, we studied a karst aquifer model that was applied to the Kerschbaum spring in Waidhofen a.d. Ybbs in Austria and used the <em>active subspace method</em> as a technique to study parameter sensitivities and correlations.</p>
<p>Congratulations to Daniel!
I also want to thank everyone who contributed to this article and made this a valuable scientific contribution!</p>
<p><strong>Journal link:</strong> <a href="https://doi.org/10.1016/j.advwatres.2019.103472">doi:10.1016/j.advwatres.2019.103472</a></p>I am very glad that our article Identifying relevant hydrological and catchment properties in active subspaces: An inference study of a lumped karst aquifer model was accepted for publication in Advances in Water Resources (Elsevier).Preprint: Generalized bounds for active subspaces2019-10-06T00:00:00+00:002019-10-06T00:00:00+00:00/posts/2019/10/06/asm-poincare-prepr<p>I am very happy to announce that Jonas Wallin (Department of Statistics, Lund University), Barbara Wohlmtuh (Chair for Numerical Mathematics, TUM), and I put a new article on theory for active subspaces (ASM) on <a href="https://arxiv.org/abs/1910.01399">arXiv</a>.
ASM is already known to be applicable to settings involving probability distributions with compact support or of Gaussian-type.
We investigate ASM in the context of distributions with exponential tails and were able to show that existing bounds are not valid anymore due to arbitrarily large Poincaré constants.
Also, we propose a way for getting weaker, or generalized, bounds that, however, result in a lower order of the error bound.
Indeed, we show how to balance the size the Poincaré constant and the order of the error.
At the end, we suggest an open problem to the community which aims at generalizing the applicability of ASM to a larger class of distributions, i.e., multivariate generalized hyperbolics.
I want to thank both of my co-authors for their feedback and assistance in developing the structure of this manuscript and for the valuable discussions we had.</p>I am very happy to announce that Jonas Wallin (Department of Statistics, Lund University), Barbara Wohlmtuh (Chair for Numerical Mathematics, TUM), and I put a new article on theory for active subspaces (ASM) on arXiv. ASM is already known to be applicable to settings involving probability distributions with compact support or of Gaussian-type. We investigate ASM in the context of distributions with exponential tails and were able to show that existing bounds are not valid anymore due to arbitrarily large Poincaré constants. Also, we propose a way for getting weaker, or generalized, bounds that, however, result in a lower order of the error bound. Indeed, we show how to balance the size the Poincaré constant and the order of the error. At the end, we suggest an open problem to the community which aims at generalizing the applicability of ASM to a larger class of distributions, i.e., multivariate generalized hyperbolics. I want to thank both of my co-authors for their feedback and assistance in developing the structure of this manuscript and for the valuable discussions we had.Another research stay at Lund University2019-08-18T00:00:00+00:002019-08-18T00:00:00+00:00/posts/2019/08/18/lund<p><img src="/assets/images/lund-botgard.jpg" class="img-left no-margin-top" /></p>
<p>From August 12th to August 28th, I am visiting the Department of Statistics at Lund University once again.
<em>Prof. Krzysztof Podgórski</em> and <em>Jonas Wallin</em> are supporting me and my research on the theory of <em>active subspaces</em>.</p>
<p>Although the weather is quite rainy, I was able to get a feeling for the city and its inhabitants. Many thanks to Krzysztof, Jonas, and all university affiliates I had contact to for having me here in Lund and for making the environment to work in very smoothly!</p>
<p>P.S. The picture shows a path at the Botanical Garden in Lund.
It was really nice to walk there after work!</p>Accepted: Bayesian calibration and sensitivity analysis for a karst aquifer model using active subspaces2019-07-16T00:00:00+00:002019-07-16T00:00:00+00:00/posts/2019/07/16/unmix-karst-mario-accpt<p>I am very happy that our article <em>Bayesian calibration and sensitivity analysis for a karst aquifer model using active subspaces</em> was accepted in <em>Water Resources Research</em> (Wiley).</p>
<p>The article is a product of two collaborating teams at TUM within the <a href="/posts/2018/03/08/kickoff-unmix"><em>UNMIX project</em></a>.
<em>Daniel Bittner</em> and <em>Gabriele Chiogna</em> from the Chair of Hydrology and River Basin Management and <em>Steven Mattis</em>, <em>Barbara Wohlmuth</em>, and I worked a lot on the revision and improved the quality of the manuscript due to very detailed reviews.</p>
<p>The paper is a description of our parameter calibration approach for a model of the Kerschbaum spring in Waidhofen a.d. Ybbs in Austria.
We used <em>active subspaces</em> as a dimension reduction technique to infer parameters in lower dimensions while saving computational cost and to study parameter sensitivities in a Bayesian inverse problem.</p>
<p>I want to thank everyone who contributed to the article and made this a scientifically valuable contribution!</p>
<p><strong>Journal link:</strong> <a href="https://doi.org/10.1029/2019WR024739">doi:10.1029/2019WR024739</a><br />
<strong>arXiv link:</strong> <a href="https://arxiv.org/abs/1901.03283">arXiv:1901.03283</a></p>I am very happy that our article Bayesian calibration and sensitivity analysis for a karst aquifer model using active subspaces was accepted in Water Resources Research (Wiley).Research week at Lund University2019-05-31T00:00:00+00:002019-05-31T00:00:00+00:00/posts/2019/05/31/lund<p><img src="/assets/images/lund.jpg" class="img-left no-margin-top" /></p>
<p>Recently, I was invited to give a talk for a group in the Department of Statistics of Lund University.
Lund is a small but pretty town in South Sweden near Copenhagen, Denmark.
Since I spent a whole week there, it was possible to have nice in-depth discussions with scientists there about statistical perspectives on problems of my interest.
My talk was about <em>active subspaces in Bayesian inverse problems</em> and I basically reported about my recent experience with it, theoretically and for a hydrological application described in <a href="/posts/2019/01/13/unmix-karst-mario-prepr">this post</a>.</p>
<p>It took me nearly 14.5 hours to come to Lund by train.
However, the journey was very comfortable, calm, and I managed to get something done productively (indeed, I can recommend to ride the train if you want to work focusedly).
Also, comparing landscapes and architectures of different countries (Germany, Denmark, and Sweden) was quite enjoyable.</p>
<p>I really enjoyed my time in Lund and hope to come back for some more research discussions and Swedish culture this summer.</p>
<p>P.S. The image shows a small alley my accommodation was located at.
Alleys of this kind are very typical for Lund.</p>