arXiv subjects

Michal Černý

Publications and source records attributed to Michal Černý.

7 recordsLinked to original sources

arXiv2026

Score-Driven Rating System for Sports

This paper introduces a score-driven rating system, a generalization of the classical Elo rating system that employs the score, i.e. the gradient of the log-likelihood, as the updating mechanism for player and team ratings. The proposed framework extends beyond simple win/loss game outcomes and accommodates a wide range of game results, such as point differences, win/draw/loss outcomes, or complete rankings. Theoretical properties of the score are derived, showing that it has zero expected value, sums to zero across all players, and decreases with increasing value of a player's rating, thereby ensuring internal consistency and fairness. Furthermore, the score-driven rating system exhibits a reversion property, meaning that ratings tend to follow the underlying unobserved true skills over time. The proposed framework provides a theoretical rationale for existing dynamic models of sports performance and offers a systematic approach for constructing new ones.

cs.LG↗

arXiv2024

Clustering Retail Products Based on Customer Behaviour

The categorization of retail products is essential for the business decision-making process. It is a common practice to classify products based on their quantitative and qualitative characteristics. In this paper we use a purely data-driven approach. Our clustering of products is based exclusively on the customer behaviour. We propose a method for clustering retail products using market basket data. Our model is formulated as an optimization problem which is solved by a genetic algorithm. It is demonstrated on simulated data how our method behaves in different settings. The application using real data from a Czech drugstore company shows that our method leads to similar results in comparison with the classification by experts. The number of clusters is a parameter of our algorithm. We demonstrate that if more clusters are allowed than the original number of categories is, the method yields additional information about the structure of the product categorization.

stat.AP↗

arXiv2022

The NP-hard problem of computing the maximal sample variance over interval data is solvable in almost linear time with high probability

We consider the algorithm by Ferson et al. (Reliable computing 11(3), p. 207-233, 2005) designed for solving the NP-hard problem of computing the maximal sample variance over interval data, motivated by robust statistics (in fact, the formulation can be written as a nonconvex quadratic program with a specific structure). First, we propose a new version of the algorithm improving its original time bound $O(n^2 2^ω)$ to $O(n \log n+n\cdot 2^ω)$, where $n$ is number of input data and $ω$ is the clique number in a certain intersection graph. Then we treat input data as random variables as it is usual in statistics) and introduce a natural probabilistic data generating model. We get $2^ω= O(n^{1/\log\log n})$ and $ω= O(\log n / \log\log n)$ on average. This results in average computing time $O(n^{1+ε})$ for $ε> 0$ arbitrarily small, which may be considered as "surprisingly good" average time complexity for solving an NP-hard problem. Moreover, we prove the following tail bound on the distribution of computation time: hard instances, forcing the algorithm to compute in time $2^{Ω(n)}$, occur rarely, with probability tending to zero at the rate $e^{-n\log\log n}$.

math.OC↗

arXiv2021

Bertram's Pairs Trading Strategy with Bounded Risk

Finding Bertram's optimal trading strategy for a pair of cointegrated assets following the Ornstein--Uhlenbeck price difference process can be formulated as an unconstrained convex optimization problem for maximization of expected profit per unit of time. This model is generalized to the form where the riskiness of profit, measured by its per-time-unit volatility, is controlled (e.g. in case of existence of limits on riskiness of trading strategies imposed by regulatory bodies). The resulting optimization problem need not be convex. In spite of this undesirable fact, it is demonstrated that the problem is still efficiently solvable. In addition, the problem that parameters of the price difference process are never known exactly and are imprecisely estimated from an observed finite sample is investigated (recalling that this problem is critical for practice). It is shown how the imprecision affects the optimal trading strategy by quantification of the loss caused by the imprecise estimate compared to a theoretical trader knowing the parameters exactly. The main results focus on the geometric and optimization-theoretic viewpoint of the risk-bounded trading strategy and the imprecision resulting from the statistical estimates.

q-fin.MF↗

arXiv2019

Walks on hyperplane arrangements and optimization of piecewise linear functions

We propose an exact iterative algorithm for minimization of a class of continuous cell-wise linear convex functions on a hyperplane arrangement. Our particular setup is motivated by evaluation of so-called rank estimators used in robust regression, where every cell of the underlying arrangement corresponds to a permutation of residuals (and we also show that the class of function for which the method works is more general). The main obstacle in the construction of the algorithm is how to find an improving direction while standing in a point incident with exponentially many cells of the arrangement. We overcome this difficulty using Birkhoff Theorem which allows us to express the cone of improving directions in the exponential number of cells using a linear system with quadratic number of variables only.

math.OC↗

arXiv2019

A new polynomially solvable class of quadratic optimization problems with box constraints

We consider the quadratic optimization problem $\max_{x \in C}\ x^T Q x + q^T x$, where $C\subseteq\mathbb{R}^n$ is a box and $r := \mathrm{rank}(Q)$ is assumed to be $\mathcal{O}(1)$ (i.e., fixed). We show that this case can be solved in polynomial time for an arbitrary $Q$ and $q$. The idea is based on a reduction of the problem to enumeration of faces of a certain zonotope in dimension $O(r)$. This paper generalizes previous results where $Q$ had been assumed to be positive semidefinite and no linear term was allowed in the objective function. Positive definiteness was a strong restriction and it is now relaxed. Generally, the problem is NP-hard; this paper describes a new polynomially solvable class of instances, larger than those known previously.

math.OC↗

arXiv2016

Interval Linear Algebra and Computational Complexity

This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding solvability of a linear system, computing inverse matrix, eigenvalues, checking positive (semi)definiteness or stability. We discuss these problems and relations between them from the view of computational complexity. Many problems in interval linear algebra are intractable, hence we emphasize subclasses of these problems that are easily solvable or decidable. The aim of this work is to provide a basic insight into this field and to provide materials for further reading and research.

cs.CC↗