arXiv subjects

Michael Falk

Publications and source records attributed to Michael Falk.

At least 37 records · Page 2Linked to original sources

arXiv2014

The Fixpoint-Iteration Algorithm for Parity Games

It is known that the model checking problem for the modal mu-calculus reduces to the problem of solving a parity game and vice-versa. The latter is realised by the Walukiewicz formulas which are satisfied by a node in a parity game iff player 0 wins the game from this node. Thus, they define her winning region, and any model checking algorithm for the modal mu-calculus, suitably specialised to the Walukiewicz formulas, yields an algorithm for solving parity games. In this paper we study the effect of employing the most straight-forward mu-calculus model checking algorithm: fixpoint iteration. This is also one of the few algorithms, if not the only one, that were not originally devised for parity game solving already. While an empirical study quickly shows that this does not yield an algorithm that works well in practice, it is interesting from a theoretical point for two reasons: first, it is exponential on virtually all families of games that were designed as lower bounds for very particular algorithms suggesting that fixpoint iteration is connected to all those. Second, fixpoint iteration does not compute positional winning strategies. Note that the Walukiewicz formulas only define winning regions; some additional work is needed in order to make this algorithm compute winning strategies. We show that these are particular exponential-space strategies which we call eventually-positional, and we show how positional ones can be extracted from them.

cs.LO↗

arXiv2014

On generalized max-linear models and their statistical interpolation

We propose a way how to generate a max-stable process in $C[0,1]$ from a max-stable random vector in $\mathbb R^d$ by generalizing the \emph{max-linear model} established by \citet{wansto11}. It turns out that if the random vector follows some finite dimensional distribution of some initial max-stable process, the approximating processes converge uniformly to the original process and the pointwise mean squared error can be represented in a closed form. The obtained results carry over to the case of generalized Pareto processes. The introduced method enables the reconstruction of the initial process only from a finite set of observation points and, thus, reasonable prediction of max-stable processes in space becomes possible. A possible extension to arbitrary dimension is outlined.

math.PR↗

arXiv2013

Testing for a δ-neighborhood of a generalized Pareto copula

A multivariate distribution function F is in the max-domain of attraction of an extreme value distribution if and only if this is true for the copula corresponding to F and its univariate margins. Aulbach et al. (2012a) have shown that a copula satisfies the extreme value condition if and only if the copula is tail equivalent to a generalized Pareto copula (GPC). In this paper we propose a chi-square goodness-of-fit test in arbitrary dimension for testing whether a copula is in a certain neighborhood of a GPC. The test can be applied to stochastic processes as well to check whether the corresponding copula process is close to a generalized Pareto process. Since the p-value of the proposed test is highly sensitive to a proper selection of a certain threshold, we also present a graphical tool that makes the decision, whether or not to reject the hypothesis, more comfortable.

math.ST↗

arXiv2012

A multivariate piecing-together approach with an application to operational loss data

The univariate piecing-together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. We propose a multivariate extension. First it is shown that an arbitrary copula is in the domain of attraction of a multivariate extreme value distribution if and only if its upper tail can be approximated by the upper tail of a multivariate GPD with uniform margins. The multivariate PT then consists of two steps: The upper tail of a given copula $C$ is cut off and substituted by a multivariate GPD copula in a continuous manner. The result is again a copula. The other step consists of the transformation of each margin of this new copula by a given univariate distribution function. This provides, altogether, a multivariate distribution function with prescribed margins whose copula coincides in its central part with $C$ and in its upper tail with a GPD copula. When applied to data, this approach also enables the evaluation of a wide range of rational scenarios for the upper tail of the underlying distribution function in the multivariate case. We apply this approach to operational loss data in order to evaluate the range of operational risk.

math.ST↗

arXiv2012

Vanishing products of one-forms and critical points of master functions

Let \A be an affine hyperplane arrangement in $\C^\ell$ with complement $U$. Let $f_1, \..., f_n$ be linear polynomials defining the hyperplanes of \A, and $A^\cdot$ the algebra of differential forms generated by the 1-forms $d \log f_1, \..., d \log f_n$. To each $l \in \C^n$ we associate the master function $Φ=Φ_l = \prod_{i=1}^n f_i^{l_i}$ on $U$ and the closed logarithmic 1-form $ω= d \log Φ$. We assume $ω$ is an element of a rational linear subspace $D$ of $A^1$ of dimension $q>1$ such that the multiplication map $\bigwedge^k(D) \to A^k$ is zero for $p<k\leq q$. With this assumption, we prove every component of the critical locus $\crit(Φ)$ of $Φ$ has codimension at most $p$, and $\crit(Φ)$ is a union of intersections of level sets of rational master functions. We give conditions that guarantee $\crit(Φ)$ is nonempty and every component has codimension equal to $p$, in terms of syzygies among polynomial master functions. If \A is $p$-generic, then $D$ is contained in the degree $p$ resonance variety $\R^p(\A)$ -- in this sense the present work complements previous work on resonance and critical loci of master functions. Any arrangement is 1-generic; in case $p=1$ we give a precise description of $\crit(Φ_l)$ in case $l$ lies in an isotropic subspace $D$ of $A^1$, using the multinet structure on \A corresponding to $D\subseteq \R^1(\A)$. This is carried out in detail for the Hessian arrangement. Finally, for arbitrary $p$ and \A, we establish necessary and sufficient conditions for a set of integral one-forms to span such a subspace, in terms of nested sets of \A, using tropical implicitization.

math.AG↗

arXiv2012

The multivariate Piecing-Together approach revisited

The univariate Piecing-Together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. A multivariate extension was established by Aulbach et al. (2012a): The upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula that coincides in its central part with C. In addition to Aulbach et al. (2012a), we achieve in the present paper an exact representation of the PT-copula's upper tail, giving further insight into the multivariate PT approach. A variant based on the empirical copula is also added. Furthermore our findings enable us to establish a functional PT version as well.

math.PR↗

arXiv2012

Testing for a generalized Pareto process

We investigate two models for the following setup: We consider a stochastic process X \in C[0,1] whose distribution belongs to a parametric family indexed by \vartheta \in Θ \subset R. In case \vartheta = 0, X is a generalized Pareto process. Based on n independent copies X(1),...,X(n) of X, we establish local asymptotic normality (LAN) of the point process of exceedances among X(1),...,X(n) above an increasing threshold line in each model. The corresponding central sequences provide asymptotically optimal sequences of tests for testing H0 : \vartheta = 0 against a sequence of alternatives Hn : \vartheta = \varthetan converging to zero as n increases. In one model, with an underlying exponential family, the central sequence is provided by the number of exceedances only, whereas in the other one the exceedances themselves contribute, too. However it turns out that, in both cases, the test statistics also depend on some additional and usually unknown model parameters. We, therefore, consider an omnibus test statistic sequence as well and compute its asymptotic relative efficiency with respect to the optimal test sequence.

math.ST↗

arXiv2011

Sojourn Times and the Fragility Index

We investigate the sojourn time above a high threshold of a continuous stochastic process Y on [0,1]. It turns out that the limit, as the threshold increases, of the expected sojourn time given that it is positive, exists if the copula process corresponding to Y is in the functional domain of attraction of of an extreme value process. This limit coincides with the limit of the fragility index corresponding to finite (n-)dimensional distributions of Y as n and the threshold increase. If the process is in a certain neighborhood of a generalized Pareto process, then we can replace the constant threshold by a general threshold function and we can compute the asymptotic sojourn time distribution. An extreme value process is a prominent example. Given that there is an exceedance at some t_0 above the threshold, we can also compute the asymptotic distribution of the time cluster length, which the process spends above the threshold function.

math.PR↗

arXiv2011

On Max-Stable Processes and the Functional D-Norm

We introduce a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. The distribution function G of a continuous max-stable process on [0,1] is introduced and it is shown that G can be represented via a norm on functional space, called D-norm. This is in complete accordance with the multivariate case and leads to the definition of functional generalized Pareto distributions (GPD) W. These satisfy W=1+log(G) in their upper tails, again in complete accordance with the uni- or multivariate case. Applying this framework to copula processes we derive characterizations of the domain of attraction condition for copula processes in terms of tail equivalence with a functional GPD. δ-neighborhoods of a functional GPD are introduced and it is shown that these are characterized by a polynomial rate of convergence of functional extremes, which is well-known in the multivariate case.

math.PR↗

arXiv2011

Local asymptotic normality in δ-neighborhoods of standard generalized Pareto processes

De Haan and Pereira (2006) provided models for spatial extremes in the case of stationarity, which depend on just one parameter β > 0 measuring tail dependence, and they proposed different estimators for this parameter. This framework was supplemented in Falk (2011) by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold, yielding in particular asymptotic efficient estimators. The estimators investigated in these papers are based on a finite set of points t1,...,td, at which observations are taken. We generalize this approach in the context of functional extreme value theory (EVT). This more general framework allows estimation over some spatial parameter space, i.e., the finite set of points t1,...,td is replaced by t in [a,b]. In particular, we derive efficient estimators of β based on those processes in a sample of iid processes in C[0,1] which exceed a given threshold function.

math.ST↗

arXiv2011

Asymptotic Conditional Distribution of Exceedance Counts: Fragility Index with Different Margins

Let $\bm X=(X_1,...,X_d)$ be a random vector, whose components are not necessarily independent nor are they required to have identical distribution functions $F_1,...,F_d$. Denote by $N_s$ the number of exceedances among $X_1,...,X_d$ above a high threshold $s$. The fragility index, defined by $FI=\lim_{s\nearrow}E(N_s\mid N_s>0)$ if this limit exists, measures the asymptotic stability of the stochastic system $\bm X$ as the threshold increases. The system is called stable if $FI=1$ and fragile otherwise. In this paper we show that the asymptotic conditional distribution of exceedance counts (ACDEC) $p_k=\lim_{s\nearrow}P(N_s=k\mid N_s>0)$, $1\le k\le d$, exists, if the copula of $\bm X$ is in the domain of attraction of a multivariate extreme value distribution, and if $\lim_{s\nearrow}(1-F_i(s))/(1-F_κ(s))=γ_i\in[0,\infty)$ exists for $1\le i\le d$ and some $κ\in{1,...,d}$. This enables the computation of the FI corresponding to $\bm X$ and of the extended FI as well as of the asymptotic distribution of the exceedance cluster length also in that case, where the components of $\bm X$ are not identically distributed.

math.ST↗

arXiv2011

Pure braid groups are not residually free

We show that the Artin pure braid group on at least four strands is not residually free. Our results also show that the pure braid group on at least three strands has corank two.

math.GR↗

arXiv2006

Multinets, resonance varieties, and pencils of plane curves

We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a "multinet," a multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement. We derive numerical conditions which impose restrictions on the number of classes, and the line and point multiplicities that can appear in multinets, and allow us to detect whether the associated pencils yield nonlinear fiberings of the complement.

math.AG↗

arXiv2006

Resonance varieties over fields of positive characteristic

This is a revision and update of the part of the preprint math.CO/0405210 concerning field coefficients, line complexes, and the Hessian arrangement. The material from that paper concerning coefficients in arbitrary commutative rings and the deleted B_3 arrangement will appear separately.

math.CO↗

arXiv2005

Complex hyperplane arrangements

This is a slightly revised version (with references added in) of a survey article which appeared in the Spring 2005 edition of the MSRI newsletter, the Emissary. The article describes some of the themes from the Fall 2004 MSRI program on Hyperplane Arrangements and Applications.

math.AG↗

arXiv2004

The line geometry of resonance varieties

Let R^1(A,R) be the degree-one resonance variety over a field R of a hyperplane arrangement A. We give a geometric description of R^1(A,R) in terms of projective line complexes. The projective image of R^1(A,R) is a union of ruled varieties, parametrized by neighborly partitions of subarrangements of A. The underlying line complexes are intersections of special Schubert varieties, easily described in terms of the corresponding partition. We generalize the definition and decomposition of R^1(A,R) to arbitrary commutative rings, and point out the anomalies that arise. In general the decomposition is parametrized by neighborly graphs, which need not induce neighborly partitions of subarrangements of A. We use this approach to show that the resonance variety of the Hessian arrangement over a field of characteristic three has a nonlinear component, a cubic threefold with interesting line structure. This answers a question of A. Suciu. We show that Suciu's deleted B_3 arrangement has resonance components over Z_2 that intersect nontrivially. We also exhibit resonant weights over Z_4 supported on the deleted B_3, which has no neighborly partitions. The modular resonant weights on the deleted B_3 exponentiate to points on the complex torus which lie on, and determine, the translated 1-torus in the first characteristic variety.

math.CO↗

arXiv2001

Line-closed matroids, quadratic algebras, and formal arrangements

Let $G$ be a matroid on ground set \A. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra \E on \A by the ideal \I generated by circuit boundaries. The quadratic closure $\bar{A}(G)$ of $A(G)$ is the quotient of \E by the ideal generated by the degree-two component of \I. We introduce the notion of \nbb set in $G$, determined by a linear order on \A, and show that the corresponding monomials are linearly independent in the quadratic closure $\bar{A}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S.~Yuzvinsky proves the converse false. These results generalize to the degree $r$ closure of $\A(G)$. The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for \A to be free and for the complement $M$ of \A to be a $K(π,1)$ space. Formality of \A is also necessary for $A(G)$ to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of $G$ is not necessary or sufficient for $M$ to be a $K(π,1)$, or for \A to be free.

math.CO↗

arXiv2000

Combinatorial and algebraic structure in Orlik-Solomon algebras

The Orlik-Solomon algebra ${\cal A}(G)$ of a matroid $G$ is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing $G$. On the other hand, some features of the matroid $G$ are reflected in the algebraic structure of ${\cal A}(G)$. In this mostly expository article, we describe recent developments in the construction of algebraic invariants of ${\cal A}(G)$. We develop a categorical framework for the statement and proof of recently discovered isomorphism theorems which suggests a possible setting for classification theorems. Several specific open problems are formulated.

math.CO↗