Mathias Fuchs Analytics services

Mathias is a Mathematician (PhD) by training, with experience in bioinformatics, medical statistics, mathematical statistics, architecture, geometry, and consulting.
Currently, he works for Zaha Hadid Architects, London, as a Senior Researcher.

Mathematical consulting

Some past and current projects

A spatial process interpretation of pedestrian indoor tracks With spatstat & Rhino. We learn simulated and observed pedestrian flows as spatial processes, and extrapolate them in real-time to new geometries, as a Rhino plugin that obtains its data live from a web-based mongo database. This is a demonstration of the basic functionality as a Rhino plugin.

spatial density process visualisation in 2.5 dimensions: tracks and spatial usage from Mathias Fuchs on Vimeo.

Paper
Inferential statistics of stress states in rigid body continuum mechanics Series of four finite element calculations of a 2d rectangular element under shear stress, subject to random perturbations in each iteration, with first and second principal stress lines in blue and orange (deformations not to scale). In this project, we develop a framework for statistical analysis of stress fluctuations as in this series. In each subfigure, the point at the centre is associated with a particular two-by-two Cauchy stress tensor. A component-wise averaging procedure, resulting in the two-by-two mean Cauchy stress is not an adequate averaging procedure because it disrespects the magnitudes of its eigenvalues, the principal stresses. However, we propose a methodology for averaging and comparing large numbers of perturbed stress samples, circumventing this problem. We thereby allow the researcher to compute measures for the degree of certainty of difference in means between two sets of samples. An important application is quality control of finite element simulations. GitHub
Predicting pedestrian density flow

Real-time occupation modelling in Rhino from Mathias Fuchs on Vimeo.

Academic publications

These are my academical publications, see here Here, I'd just like to say a few informal words about how I would describe each of their topics.

Journal Paperps

My first architectural paper. We described outputs of a finite element software in terms of geometrical features, by a linear regression. This was fun and interesting, but the most important part was finding out that the surface normal's z component was an important explanatory variable in that linear regression. We found out about that by investigating the Airy approach to structural mechanics more closely.

  • A.-L.Boulesteix, R.De Bin, X.Jiang, M. Fuchs: IPF-LASSO: Integrative L1-Penalized Regression with Penalty Factors for Prediction Based on Multi-Omics Data. Computational and Mathematical Methods in Medicine (2017), https://doi.org/10.1155/2017/7691937.

We show how penalized regression generalises naturally to multiple data sources, by penalising each modality differently. This sounds easy and straightforward, but it's the details which make it hard. Took a long time to publish - not catchy enough.

Yet another topic: theoretical statistics. This is my second favorite paper on this side. We identified and computed certain covariances between evaluations of error estimators. Furthermore, we identified U-statistics to show how these covariances can be estimated.

  • M. Fuchs: Equivariant K-homology of Bianchi groups in the case of non-trivial class group. Münster Journal of Mathematics 7 (2014), 589–607 http://doi.org/10.17879/58269758846

Back to pure Mathematics. This paper is in spirit not too far away from that of my PhD thesis. I think this paper is the one I am most proud of.

  • M. Fuchs, T.Beissbarth, E.Wingender, K.Jung: Connecting high-dimensional mRNA and miRNA expression data for binary medical classification problems, Computer Methods and Programs in Biomedicine (2013), http://dx.doi.org/10.1016/j.cmpb.2013.05.013

Yet another topic: a general overview of data combination strategies. In retrospect, there are better ones out there, though, than the ones we looked at.

  • V.Halacheva, M. Fuchs, J.Dönitz, T.Reupke, B.Püschel, C.Viebahn: Complex Planar Cell Movement and Oriented Cell Division Build the Early Primitive Streak in the Mammalian Embryo, Developmental Dynamics 240, 1905-1916 (2011), https://dx.doi.org/10.1002/dvdy.22687

Once again, a completely different topic: biology, and in particular developmental biology. The primitive streak is the first morphogenic feature in the nascent mammalian embryo. We applied circular statistics to show that cell divisions prefer certain directions.

  • M.Ante, E.Wingender, M. Fuchs: Integration of gene expression data with prior knowledge for network analysis and validation, BMC Research Notes 4, 520 (2011), https://doi.org/10.1186/1756-0500-4-520

My first non-mathematical paper: Some work on biological databases, of a rather algebraic/non-quantitative flavor. Biological databases ... nothing I really worked on before or after anytime again ....

The title says it all: we compute the group homology of a certain class of groups acting on hyperbolic three-space. Was (mostly) fun. I learned a lot about homological algebra, and about fundamental sets of actions.

Thesis

PhD thesis: M. Fuchs, "Cohomologie cyclique des produits croises associes aux groupes de Lie". Written at the Institut de Mathematiques de Luminy under the direction of Michael Puschnigg at the Universite de la Mediterranee Aix-Marseile, http://arxiv.org/abs/math/0612023

I give a completely new proof of the fact that the group ring of torsion-free discrete co-compact subgroups of SL(n, C) satisfies the Kadison-Kaplansky conjecture: it is free of non-trivial idempotents. Unfortunately, I never came back to that topic later.

Technical Reports
  • M. Fuchs, X. Jiang, A.-L. Boulesteix, 2016. The computationally optimal test set size in simulation studies on supervised learning. [https://epub.ub.uni-muenchen.de/26870/](Technical Report 189), Department of Statistics, Ludwig Maximilian University of Munich

A simulation study on supervised learning is when the following process is carried out repeatedly, with independent repetitions: One draws a learning sample from a distribution. The size of the learning sample, i.e., the number of observations it is comprised of, is to be kept the same throughout. Each observation consists of predictors or (covariables), and an outcome or value of response variable, for instance, a binary prediction target. Furthermore, one evaluates the performance of the predictive model thus achieved, by evaluating it on a test set. Averageing out across all independent iterations is guaranteed to converge towards the true expectated value of the error estimator - the true error. It might seem a little unintuitive that the latter statement holds true regardless of the size of the test set. The reason is that the mentioned expected value is unaffected by the size of the test set. Therefore, it is just a matter of computational convenience to choose it carefully. It is the goal of this paper to perform that choice in an educated way, as a function of the machine computation execution speed.

  • A.-L. Boulesteix, R. De Bin, X. Jiang, M. Fuchs, 2015. IPF-LASSO: integrative L1-penalized regression with penalty factors for prediction based on multi-omics data. [https://epub.ub.uni-muenchen.de/26551/](Technical Report 187), Department of Statistics, Ludwig Maximilian University of Munich

Here, we took a look at the problem of integrating different group of covariables, the prototypical situation being when mRNA and miRNA are to be integrated. We pursue a fairly immediate line of thought: to apply different lasso penalizations to each of them. The details, however, are quite nitty-gritty.

  • M. Fuchs, R. Hornung, R. De Bin, A.-L. Boulesteix, 2013. A U-statistic estimator for the variance of resampling-based error estimators. [http://epub.ub.uni-muenchen.de/17654/](Technical Report 148), Department of Statistics, Ludwig Maximilian University of Munich

We show a bunch of things about the errror estimator of a machine learning algorithm, on a real data sample of fxed size (typically denoted by n in statistics). Among several statements, we show that the studentized error estimator is asymptotically normally distributed. This is a new contribution.