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Partial-order-based process mining: a survey and outlook
The field of process mining focuses on distilling knowledge of the (historical) execution of a process based on the operational event data generated and stored during its execution. Most existing process mining techniques assume that the event data describe activity executions as degenerate time intervals, i.e., intervals of the form [t, t], yielding a strict total order on the observed activity instances. However, for various practical use cases, e.g., the logging of activity executions with a nonzero duration and uncertainty on the correctness of the recorded timestamps of the activity executions, assuming a partial order on the observed activity instances is more appropriate. Using partial orders to represent process executions, i.e., based on recorded event data, allows for new classes of process mining algorithms, i.e., aware of parallelism and robust to uncertainty. Yet, interestingly, only a limited number of studies consider using intermediate data abstractions that explicitly assume a partial order over a collection of observed activity instances. Considering recent developments in process mining, e.g., the prevalence of high-quality event data and techniques for event data abstraction, the need for algorithms designed to handle partially ordered event data is expected to grow in the upcoming years. Therefore, this paper presents a survey of process mining techniques that explicitly use partial orders to represent recorded process behavior. We performed a keyword search, followed by a snowball sampling strategy, yielding 68 relevant articles in the field. We observe a recent uptake in works covering partial-order-based process mining, e.g., due to the current trend of process mining based on uncertain event data. Furthermore, we outline promising novel research directions for the use of partial orders in the context of process mining algorithms.
Journal Article
A Partial-Order-Based Model to Estimate Individual Preferences Using Panel Data
In retail operations, customer choices may be affected by stockout and promotion events. Given panel data with the transaction history of customers, and product availability and promotion data, our goal is to predict future individual purchases. We use a general nonparametric framework in which we represent customers by partial orders of preferences. In each store visit, each customer samples a full preference list of the products consistent with her partial order, forms a consideration set, and then chooses to purchase the most preferred product among the considered ones. Our approach involves: (a) defining behavioral models to build consideration sets as subsets of the products on offer, (b) proposing a clustering algorithm for determining customer segments, and (c) deriving marginal distributions for partial preferences under the multinomial logit model. Numerical experiments on real-world panel data show that our approach allows more accurate, fine-grained predictions for individual purchase behavior compared to state-of-the-art alternative methods.
The online appendix is available at
https://doi.org/10.1287/mnsc.2016.2683
.
This paper was accepted by Vishal Gaur, operations management.
Journal Article
An efficient construction of polar codes based on the general partial order
Traditionally, the construction of polar codes requires intense computations to sort all bit channels. In previous works, two types of partial orders (POs) of polar codes were proposed to decrease the computations in the construction process. In this paper, a procedure is presented to employ POs with complexity O(N2), which is lower than the existing procedures, where N=2n is the code length (n≥1). To further reduce the computation complexity, in this paper, we propose a general partial order (GPO), which works at a lower dimension (nu
Journal Article
Implementing the time-to-event continual reassessment method in the presence of partial orders in a phase I head and neck cancer trial
Background
In this article we describe the methodology of the time-to-event continual reassessment method in the presence of partial orders (PO-TITE-CRM) and the process of implementing this trial design into a phase I trial in head and neck cancer called ADePT-DDR. The ADePT-DDR trial aims to find the maximum tolerated dose of an ATR inhibitor given in conjunction with radiotherapy in patients with head and neck squamous cell carcinoma.
Methods
The PO-TITE-CRM is a phase I trial design that builds upon the time-to-event continual reassessment method (TITE-CRM) to allow for the presence of partial ordering of doses. Partial orders occur in the case where the monotonicity assumption does not hold and the ordering of doses in terms of toxicity is not fully known.
Results
We arrived at a parameterisation of the design which performed well over a range of scenarios. Results from simulations were used iteratively to determine the best parameterisation of the design and we present the final set of simulations. We provide details on the methodology as well as insight into how it is applied to the trial.
Conclusions
Whilst being a very efficient design we highlight some of the difficulties and challenges that come with implementing such a design. As the issue of partial ordering may become more frequent due to the increasing investigations of combination therapies we believe this account will be beneficial to those wishing to implement a design with partial orders.
Trial registration
ADePT-DDR was added to the European Clinical Trials Database (EudraCT number: 2020-001034-35) on 2020-08-07.
Journal Article
Operations on arc diagrams and degenerations for invariant subspaces of linear operators
We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are algebraic groups acting on these varieties, and we give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described.
Journal Article
Using extremal events to characterize noisy time series
Experimental time series provide an informative window into the underlying dynamical system, and the timing of the extrema of a time series (or its derivative) contains information about its structure. However, the time series often contain significant measurement errors. We describe a method for characterizing a time series for any assumed level of measurement error ε by a sequence of intervals, each of which is guaranteed to contain an extremum for any function that ε-approximates the time series. Based on the merge tree of a continuous function, we define a new object called the normalized branch decomposition, which allows us to compute intervals for any level ε. We show that there is a well-defined total order on these intervals for a single time series, and that it is naturally extended to a partial order across a collection of time series comprising a dataset. We use the order of the extracted intervals in two applications. First, the partial order describing a single dataset can be used to pattern match against switching model output (Cummins et al. in SIAM J Appl Dyn Syst 17(2):1589–1616, 2018), which allows the rejection of a network model. Second, the comparison between graph distances of the partial orders of different datasets can be used to quantify similarity between biological replicates.
Journal Article
From Kruskal’s theorem to Friedman’s gap condition
Harvey Friedman’s gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the gap condition can be reconstructed from a small number of well-motivated building blocks: It arises via iterated applications of a uniform Kruskal theorem.
Journal Article
The combinator M and the Mockingbird lattice
We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator${{\\mathbf{M}}}$. This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule${{\\mathbf{M}}} \\mathsf{x}_1 \\to \\mathsf{x}_1 \\mathsf{x}_1$. We prove that the reflexive and transitive closure of this rewrite relation is a partial order on terms on${{\\mathbf{M}}}$and that all connected components of its rewrite graph are Hasse diagram of lattices. This last result is based on the introduction of new lattices on duplicative forests, which are sorts of treelike structures. These lattices are not graded, not self-dual, and not semi-distributive. We present some enumerative properties of these lattices like the enumeration of their elements, of the edges of their Hasse diagrams, and of their intervals. These results are derived from formal power series on terms and on duplicative forests endowed with particular operations.
Journal Article
Optimization with Stochastic Dominance Constraints
We introduce stochastic optimization problems involving stochastic dominance constraints. We develop necessary and sufficient conditions of optimality and duality theory for these models and show that the Lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions. The models and results are illustrated on a portfolio optimization problem.
Journal Article
Equivalence, Partial Order and Lattice of Neighborhood Sequences on the Triangular Grid
In (digital) grids, neighbor relation is a crucial concept; digital distances are based on paths through neighbor points. Digital distances are significant, e.g., in digital image processing for giving an approximation of the Euclidean distance and allowing incremental algorithms on images. Neighborhood sequences (i.e., infinite sequences of the possible types of neighbors) are defining digital distances with a lower rotational dependency than the distances based only on a sole neighborhood. They allow one to change the used neighborhood condition in every step along a path. They are defined in various grids, and they can be periodic. Generalized neighborhood sequences do not need to be periodic. In this paper, the triangular grid is studied. An equivalence and two partial order relations on the set of generalized and periodic neighborhood sequences are shown on this grid. The first partial order, the “faster” relation, is based on distances defined by neighborhood sequences, and it does not provide a lattice but gives a relatively complex relation for neighborhood sequences with a short period. The other partial order, the relation “componentwise dominate”, defines a complete distributive lattice on the set of generalized neighborhood sequences. Finally, a relation of the above-mentioned relations is established. Important differences regarding the cases of the square and triangular grids are also highlighted.
Journal Article