arXiv subjects

Volkmar Liebscher

Publications and source records attributed to Volkmar Liebscher.

16 recordsLinked to original sources

arXiv2025

Tikhonov-Fenichel Reductions and their Application to a Novel Modelling Approach for Mutualism

When formulating a model there is a trade-off between model complexity and (biological) realism. In the present paper we demonstrate how model reduction from a precise mechanistic "super model" to simpler conceptual models using Tikhonov--Fenichel reductions, an algebraic approach to singular perturbation theory, can mitigate this problem. Compared to traditional methods for time scale separations (Tikhonov's theorem, quasi-steady state assumption), Tikhonov--Fenichel reductions have the advantage that we can compute a reduction directly for a separation of rates into slow and fast ones instead of a separation of components of the system. Moreover, we can find all such eductions algorithmically. In this work we use Tikhonov--Fenichel reductions to analyse a mutualism model tailored towards lichens with an explicit description of the interaction. We find: (1) the implicit description of the interaction given in the reductions by interaction terms (functional responses) varies depending on the scenario, (2) there is a tendency for the Mycobiont, an obligate mutualist, to always benefit from the interaction while it can be detrimental for the photobiont, a facultative mutualist, depending on the parameters, (3) our model is capable of describing the shift from mutualism to parasitism, (4) via numerical analyis, that our model experiences bistability with multiple stable fixed points in the interior of the first orthant. To analyse the reductions we formalize and discuss a mathematical criterion that categorizes two-species interactions. Throughout the paper we focus on the relation between the mathematics behind Tikhonov--Fenichel reductions and their biological interpretation.

q-bio.PE↗

arXiv2018

Simultaneous Credible Regions for Multiple Changepoint Locations

Within a Bayesian retrospective framework, we present a way of examining the distribution of \cps through a novel set estimator. For a given level, $α$, we aim at smallest sets that cover all \cps with a probability of at least $1-α$. These so-called smallest simultaneous credible regions, computed for certain values of $α$, provide parsimonious representations of the possible \cp locations. In addition, combining them for a range of different $α$'s enables very informative yet condensed visualisations. Therewith we allow for the evaluation of model choices and the analysis of \cp data to an unprecedented degree. This approach exhibits superior sensitivity, specificity and interpretability in comparison with highest density regions, marginal inclusion probabilities and confidence intervals inferred by \stepR. Whilst their direct construction is usually intractable, asymptotically correct solutions can be derived from posterior samples. This leads to a novel NP-complete problem. Through reformulations into an Integer Linear Program we show empirically that a fast greedy heuristic computes virtually exact solutions.

math.ST↗

arXiv2016

SINOMA - A new approach for estimating linear relationships between noisy serial data streams

Reconstructions of past climates are based on the calibration of available proxy data. This calibration is usually achieved by means of linear regression models. In the recent paleo-climate literature there is an ongoing discussion on the validity of highly resolved climate reconstructions. The reason for this is that the proxy data are noisy, i.e. in addition to the variability that is related to the climate variable of interest, they contain other sources of variability. Inadequate treatment of such noise leads to a biased estimation of regression slopes, resulting in a wrong representation of the real amplitude of past climate variations. Methods to overcome this problem have had a limited success so far. Here, we present a new approach - SINOMA - for noisy serial data streams that are characterized by different spectral characteristics of signal and noise. SINOMA makes use of specific properties of the data streams temporal or spatial structure and by this is able to deliver a precise estimate of the true regression slope and, simultaneously, of the ratio of noise variances present in the predictor and predictand. The paper introduces the underlying mathematics as well as a general description of the presented algorithm. The validity of SINOMA is illustrated with two test data-sets. Finally we address methodological limitations and further potential applications.

stat.AP↗

arXiv2015

On the Balance of Unrooted Trees

We solve a class of optimization problems for (phylogenetic) $X$-trees or their shapes. These problems have recently appeared in different contexts, e.g. in the context of the impact of tree shapes on the size of TBR neighborhoods, but so far these problems have not been characterized and solved in a systematic way. In this work we generalize the concept and also present several applications. Moreover, our results give rise to a nice notion of balance for trees. Unsurprisingly, so-called caterpillars are the most unbalanced tree shapes, but it turns out that balanced tree shapes cannot be described so easily as they need not even be unique.

q-bio.PE↗

arXiv2015

Gromov meets Phylogenetics - new Animals for the Zoo of Biocomputable Metrics on Tree Space

We present a new class of metrics for unrooted phylogenetic $X$-trees derived from the Gromov-Hausdorff distance for (compact) metric spaces. These metrics can be efficiently computed by linear or quadratic programming. They are robust under NNI-operations, too. The local behavior of the metrics shows that they are different from any formerly introduced metrics. The performance of the metrics is briefly analised on random weighted and unweighted trees as well as random caterpillars.

math.MG↗

arXiv2014

The Relation of Spatial and Tensor Product of Arveson Systems --- The Random Set Point of View

We characterise the embedding of the spatial product of two Arveson systems into their tensor product using the random set technique. An important implication is that the spatial tensor product does not depend on the choice of the reference units, i.e. it is an intrinsic construction. There is a continuous range of examples coming from the zero sets of Bessel processes where the two products do not coincide. The lattice of all subsystems of the tensor product is analised in different cases. As a by-product, the Arveson systems coming from Bessel zeros prove to be primitive in the sense of \cite{JMP11a}.

math.OA↗

arXiv2013

Complexity $L^0$-penalized M-Estimation: Consistency in More Dimensions

We study the asymptotics in $L^2$ for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains - e.g. images - by piecewise smooth functions. We introduce a fairly general setting which comprises most of the presently popular partitions of signal- or image- domains like interval-, wedgelet- or related partitions, as well as Delaunay triangulations. Then we prove consistency and derive convergence rates. Finally, we illustrate by way of relevant examples that the abstract results are useful for many applications.

math.ST↗

arXiv2011

An algebraic analysis of the two state Markov model on tripod trees

Methods of phylogenetic inference use more and more complex models to generate trees from data. However, even simple models and their implications are not fully understood. Here, we investigate the two-state Markov model on a tripod tree, inferring conditions under which a given set of observations gives rise to such a model. This type of investigation has been undertaken before by several scientists from different fields of research. In contrast to other work we fully analyse the model, presenting conditions under which one can infer a model from the observation or at least get support for the tree-shaped interdependence of the leaves considered. We also present all conditions under which the results can be extended from tripod trees to quartet trees, a step necessary to reconstruct at least a topology. Apart from finding conditions under which such an extension works we discuss example cases for which such an extension does not work.

q-bio.PE↗

arXiv2010

The Spatial Product of Arveson Systems is Intrinsic

We prove that the spatial product of two spatial Arveson systems is independent of the choice of the reference units. This also answers the same question for the minimal dilation the Powers sum of two spatial CP-semigroups: It is independent up to cocycle conjugacy.

math.OA↗

arXiv2009

Subsystems of Fock Need Not Be Fock: Spatial CP-Semigroups

We show that a product subsystem of a time ordered system (that is, a product system of time ordered Fock modules), though type I, need not be isomorphic to a time ordered product system. In that way, we answer an open problem in the classification of CP-semigroups by product systems. We define spatial strongly continuous CP-semigroups on a unital C*-algebra and characterize them as those that have a Christensen-Evans generator.

math.OA↗

arXiv2009

Consistencies and rates of convergence of jump-penalized least squares estimators

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in $L^2([0,1))$ our results cover other metrics like Skorokhod metric on the space of càdlàg functions and uniform metrics on $C([0,1])$. We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) Hölder continuous functions of order $0<α\le1$ and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.

math.ST↗

arXiv2007

A Problem of Powers and the Product of Spatial Product Systems

In the 2002 AMS summer conference on ``Advances in Quantum Dynamics'' in Mount Holyoke Robert Powers proposed a sum operation for spatial E0-semigroups. Still during the conference Skeide showed that the Arveson system of that sum is the product of spatial Arveson systems. This product may but need not coincide with the tensor product of Arveson systems. The Powers sum of two spatial E0-semigroups is, therefore, up to cocycle conjugacy Skeide's product of spatial noises.

math.OA↗

arXiv2007

Scale space consistency of piecewise constant least squares estimators -- another look at the regressogram

We study the asymptotic behavior of piecewise constant least squares regression estimates, when the number of partitions of the estimate is penalized. We show that the estimator is consistent in the relevant metric if the signal is in $L^2([0,1])$, the space of càdlàg functions equipped with the Skorokhod metric or $C([0,1])$ equipped with the supremum metric. Moreover, we consider the family of estimates under a varying smoothing parameter, also called scale space. We prove convergence of the empirical scale space towards its deterministic target.

math.ST↗

arXiv2005

Segmentation of Time Series: Parameter Dependence of Blake-Zisserman and Mumford-Shah Functionals and the Transition from Discrete to Continuous

The paper deals with variational approaches to the segmentation of time series into smooth pieces, but allowing for sharp breaks. In discrete time, the corresponding functionals are of Blake-Zisserman type. Their natural counterpart in continuous time are the Mumford-Shah functionals. Time series which minimise these functionals are proper estimates or representations of the signals behind recorded data. We focus on consistent behaviour of the functionals and the estimates, as parameters vary or as the sampling rate increases. For each time continuous time series $f\in L^2 (\lbrack 0,1\rbrack)$ we take conditional expectations w.r.t. to $σ$-algebras generated by finer and finer partitions of the time domain into intervals, and thereby construct a sequence $(f_n)_{n\in\N}$ of discrete time series. As $n$ increases this amounts to sampling the continuous time series with more and more accuracy. Our main result is consistent behaviour of segmentations w.r.t. to variation of parameters and increasing sampling rate.

math.FA↗

arXiv2005

Constructing Units in Product Systems

We prove a criterion that allows to construct units in product systems of correspondences with prescribed infinitesimal characterizations. This criterion summarizes proofs of known results and new applications. It also frees the hypothesis from the assumption that the units are contained in a product system of time ordered Fock modules.

math.OA↗

arXiv2003

Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

In a series of papers Tsirelson constructed from measure types of random sets and generalised random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups. This paper establishes the converse: Each continuous tensor product systems of Hilbert spaces comes with measure types of distributions of random (closed) sets in [0,1] or $R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system and the range of the invariant is characterized. Moreover, based on a detailed study of this kind of measure types, we construct for each stationary factorizing measure type a continuous tensor product systems of Hilbert spaces such that this measure type arises as the before mentioned invariant. The measure types of the above described kind are connected with representations of the corresponding $L^\infty$-spaces. This leads to direct integral representations of the elements of a given product system which combine well under tensor products. Using this structure in a constructive way, we can relate to any (type III) product system a product system of type $II_0$ preserving isomorphy classes. Thus, the classification of type III product systems reduces to that of type II ones. Further, we show that all consistent measurable structures on an algebraic continuous tensor product systems of Hilbert spaces yield isomorphic product systems. Thus the measurable structure of a continuous tensor product systems of Hilbert spaces is essentially determined by its algebraic one.

math.PR↗