Exact Decomposition Research Articles

Exact Simulation of Quadratic Intensity Models

We develop efficient algorithms of exact simulation for quadratic stochastic intensity models that have become increasingly popular for modeling events arrivals, especially in economics, finance, and insurance. They have huge potential to be applied to many other areas such as operations management, queueing science, biostatistics, and epidemiology. Our algorithms are developed by the principle of exact distributional decomposition, which lies in a fully analytical expression for the joint Laplace transform of quadratic process and its integral newly derived in this paper. They do not involve any numerical Laplace inversion, have been validated by extensive numerical experiments, and substantially outperform all existing alternatives in the literature. Moreover, our algorithms are extendable to multidimensional point processes and beyond Cox processes to additionally incorporate two-sided random jumps with arbitrarily distributed sizes in the intensity for capturing self-exciting and self-correcting effects in event arrivals. Applications to portfolio loss modeling are provided to demonstrate the applicability and flexibility of our algorithms. History: Accepted by Bruno Tuffin, Area Editor for Simulation. Funding: This work was supported by the Beijing University of Posts and Telecommunications [Grant 2022RC58], the Shanghai University of Finance and Economics [Grant 2020110930], the National Natural Science Foundation of China [Grant 71401147], and Graduate Innovation Fund of Shanghai University of Finance and Economics [CXJJ-2023-387]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0323 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0323 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

On a fair and risk‐averse urban air mobility resource allocation problem under demand and capacity uncertainties

AbstractUrban air mobility (UAM) is an emerging air transportation mode to alleviate the ground traffic burden and achieve zero direct aviation emissions. Due to the potential economic scaling effects, the UAM traffic flow is expected to increase dramatically once implemented, and its market can be substantially large. To be prepared for the era of UAM, we study the fair and risk‐averse urban air mobility resource allocation model (FairUAM) under passenger demand and airspace capacity uncertainties for fair, safe, and efficient aircraft operations. FairUAM is a two‐stage model, where the first stage is the aircraft resource allocation, and the second stage is to fairly and efficiently assign the ground and airspace delays to each aircraft provided the realization of random airspace capacities and passenger demand. We show that FairUAM is NP‐hard even when there is no delay assignment decision or no aircraft allocation decision. Thus, we recast FairUAM as a mixed‐integer linear program (MILP) and explore model properties and strengthen the model formulation by developing multiple families of valid inequalities. The stronger formulation allows us to develop a customized exact decomposition algorithm with both benders and L‐shaped cuts, which significantly outperforms the off‐the‐shelf solvers. Finally, we numerically demonstrate the effectiveness of the proposed method and draw managerial insights when applying FairUAM to a real‐world network.

All-optical quantum computing using cubic phase gates

If suitable quantum optical interactions were available, transforming the field mode operators in a nonlinear fashion, the all-photonics platform could be one of the strongest contenders for realizing a quantum computer. While single-photon qubits may be processed directly, “brighter” logical qubits may be embedded in individual oscillator modes, using so-called bosonic codes, for an in-principle fault-tolerant processing. In this paper, we show how elements of all-optical, universal, and fault-tolerant quantum computation can be implemented using only beam splitters together with single-mode cubic phase gates in reasonable numbers, and possibly off-line squeezed-state or single-photon resources. Our approach is based on a decomposition technique combining exact gate decompositions and approximate Trotterization. This allows for efficient decompositions of certain nonlinear continuous-variable multimode gates into the elementary gates, where the few cubic gates needed may even be weak or all identical, thus facilitating potential experiments. The final gate operations include two-mode controlled phase rotation and three-mode Rabi-type Hamiltonian gates, which are shown to be employable for realizing high-fidelity single-photon two-qubit entangling gates or creating high-quality Gottesman-Kitaev-Preskill states. We expect our method to be of general use with various applications, including those that rely on quartic Kerr-type interactions. Published by the American Physical Society 2024

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations—often referred to as persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

Complete Coverage Path Planning Based on Improved Genetic Algorithm for Unmanned Surface Vehicle

Complete Coverage Path Planning (CCPP) is a key technology for Unmanned Surface Vehicles (USVs) that require complete coverage on the water surface, such as water sample collection, garbage collection, water field patrol, etc. When facing complex and irregular boundaries, the traditional CCPP-based boustrophedon method may encounter many problems and challenges, such as multiple repeated regions, multiple turns, and the easy occurrence of local optima. The traditional genetic algorithm also has some shortcomings. The fixed fitness function, mutation operator and crossover operator are not conducive to the evolution of the population and the production of better offspring. In order to solve the above problems, this paper proposes a CCPP method based on an improved genetic algorithm, including a stretched fitness function, an adaptive mutation operator, and a crossover operator. The algorithm combines the key operators in the fireworks algorithm. Then, the turning and obstacle avoidance during the operation of the Unmanned Surface Vehicle are optimized. Simulation and experiments show that the improved genetic algorithm has higher performance than the exact unit decomposition method and the traditional genetic algorithm, and has more advantages in reducing the coverage path length and repeating the coverage area. This proves that the proposed CCPP method has strong adaptability to the environment and has practical application value in improving the efficiency and quality of USV related operations.

Exact Nonlinear Decomposition of Ideal-MHD Waves Using Eigenenergies

In this paper, we introduce a new method for exact decomposition of propagating, nonlinear magnetohydrodynamic (MHD) disturbances into their component eigenenergies associated with the familiar slow, Alfvén, and fast wave eigenmodes, and the entropy and field-divergence pseudoeigenmodes. First, the mathematical formalism is introduced, where it is illustrated how the ideal-MHD eigensystem can be used to construct a decomposition of the time variation of the total energy density into contributions from the eigenmodes. The decomposition method is then demonstrated by applying it to the output of three separate nonlinear MHD simulations. The analysis of the simulations confirms that the component wave modes of a composite wavefield are uniquely identified by the method. The slow, Alfvén, and fast energy densities are shown to evolve in exactly the way expected from comparison with known linear solutions and nonlinear properties, including processes such as mode conversion. Along the way, some potential pitfalls for the numerical implementation of the decomposition method are identified and discussed. We conclude that the exact, nonlinear decomposition method introduced is a powerful and promising tool for understanding the nature of the decomposition of MHD waves as well as analyzing and interpreting the output of dynamic MHD simulations.

Path-Based Formulations for the Design of On-demand Multimodal Transit Systems with Adoption Awareness

This paper reconsiders the On-Demand Multimodal Transit Systems (ODMTS) Design with Adoptions problem (ODMTS-DA) to capture the latent demand in on-demand multimodal transit systems. The ODMTS-DA is a bilevel optimization problem, for which Basciftci and Van Hentenryck proposed an exact combinatorial Benders decomposition. Unfortunately, their proposed algorithm only finds high-quality solutions for medium-sized cities and is not practical for large metropolitan areas. The main contribution of this paper is to propose a new path-based optimization model, called P-Path, to address these computational difficulties. The key idea underlying P-Path is to enumerate two specific sets of paths which capture the essence of the choice model associated with the adoption behavior of riders. With the help of these path sets, the ODMTS-DA can be formulated as a single-level mixed-integer programming model. In addition, the paper presents preprocessing techniques that can reduce the size of the model significantly. P-Path is evaluated on two comprehensive case studies: the midsize transit system of the Ann Arbor – Ypsilanti region in Michigan (which was studied by Basciftci and Van Hentenryck) and the large-scale transit system for the city of Atlanta. The experimental results show that P-Path solves the Michigan ODMTS-DA instances in a few minutes, bringing more than two orders of magnitude improvements compared with the existing approach. For Atlanta, the results show that P-Path can solve large-scale ODMTS-DA instances (about 17 millions variables and 37 millions constraints) optimally in a few hours or in a few days. These results show the tremendous computational benefits of P-Path which provides a scalable approach to the design of on-demand multimodal transit systems with latent demand. History: Accepted by Andrea Lodi, Design & Analysis of Algorithms—Discrete. Funding: This work was partially supported by National Science Foundation Leap-HI [Grant 1854684] and the Tier 1 University Transportation Center (UTC): Transit - Serving Communities Optimally, Responsively, and Efficiently (T-SCORE) from the U.S. Department of Transportation [69A3552047141]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0014 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0014 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Unified Framework for Choice-Based Facility Location Problem

The choice-based facility location (CBFL) problem arises in various industrial and business contexts. The problem stands on a decentralized perspective: Companies set up chains of facilities, and customers determine from which chain or facility to seek service according to their own preferences. Essentially, customer preferences or choices play a key role in characterizing various CBFL problems, which differ mainly in the models or rules used to characterize the choice. Consequently, a large number of formulations appear and are often solved by dedicatedly designed approaches in the literature. Such a situation significantly complicates practitioners’ decision-making process when they are facing practical problems but are unsure which ad hoc model is suitable for their cases. In this article, we address this dilemma by providing a unified modeling framework based on the concept of preference dominance. Specifically, we conceptualize the choice behavior as a sequential two-step procedure: Given a set of open facilities, each customer first forms a nondominated consideration set and then splits the buying power within the set. Such an interpretation renders practitioners high modeling flexibility as they can tailor how preference dominance is constructed according to their specific contexts. In particular, we show that our model can represent several streams of CBFL problems. To support our model’s applicability, we design an efficient exact decomposition algorithm. Extensive computational studies reveal that although the algorithm is designed for a general purpose, it outperforms most approaches that are tailored for ad hoc problems by a large margin, which justifies both the effectiveness and the efficiency of the unified framework. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0366 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0366 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Code and Data Repository for Unified Framework for Choice-Based Facility Location Problem

The choice-based facility location (CBFL) problem arises in various industrial and business contexts. The problem is based on a decentralized perspective: Companies set up chains of facilities, and customers determine from which chain or facility to seek service according to their own preferences. Essentially, customer preferences or choices play a key role in characterizing various CBFL problems, which differ mainly in the models or rules employed to characterize the choice. Consequently, a large number of formulations appear and are often solved by dedicatedly designed approaches in the literature. Such a situation significantly complicates practitioners' decision-making process when they are facing practical problems but are unsure which ad hoc model is suitable for their cases. In this article, we address this dilemma by providing a unified modeling framework based on the concept of preference dominance. Specifically, we conceptualize the choice behavior as a sequential two-step procedure: Given a set of open facilities, each customer first forms a nondominated consideration set and then splits the buying power within the set. Such an interpretation renders practitioners high modeling flexibility, as they can tailor how preference dominance is constructed according to their specific contexts. In particular, we show that our model can represent several streams of CBFL problems. To support the applicability of our model, we design an efficient exact decomposition algorithm. Extensive computational studies reveal that although the algorithm is designed for a general purpose, it outperforms most approaches that are tailored for ad hoc problems by a large margin, which justifies both the effectiveness and the efficiency of the unified framework. This repository provides data for the problem and code for the algorthm. The main folders are 'data', 'src', and 'results'. 'data': four datasets generator (and instances) used in the paper. 'src': the source code for the decomposition algorithm. 'results': high-resoultion figures in our paper All experimental results can be found in the paper (manuscript and electronic companion).

Supplier selection under disruption risk with hybrid procurement

In today’s global landscape, supply chains face a plethora of risks due to the increasingly dynamic and turbulent business environment. In this study, we introduce a supplier selection model with hybrid procurement under disruption risk. Specifically, we consider the supplier selection problem of a firm with multiple units (or business segments) demanding several products, raw materials, and services. The firm has a centralized supplier selection strategy at the corporate level. However, instead of allocating orders to selected suppliers, the corporate management allows each unit to use its most preferred suppliers from the selected ones in a decentralized manner. To mitigate the impact of disruptions, less preferred suppliers of each unit serve as a backup. We present a mathematical model for this problem and study the structural properties of our formulation. We develop an exact Benders decomposition algorithm, and also present a method to generate upper and lower bounds. We compare the performance of the proposed solution algorithms on large-scale instances. Finally, we present a numerical example to demonstrate the impact of decentralized procurement with different objectives. Our results show that the expected procurement cost can increase significantly if suppliers are selected without considering the decentralized decisions of units.

Optimal guidance track generation for precision agriculture: A review of coverage path planning techniques

AbstractThe Complete Coverage Path Planning (CCPP) problem is a subfield of industrial motion planning that has applications in various domains, ranging from mobile robotics to treatment applications. Especially in precision agriculture with a high level of automation, the use of CCPP techniques is essential for efficient resource utilization, reduced soil compaction, and increased yields. This paper reviews the CCPP problem in the context of machines operating in agricultural fields and proposes a methodological approach consisting of three steps: Generating the Guidance Tracks (i.e., the track system along which the path should be oriented), determining the traversing sequence through these tracks, and planning a smooth and drivable path. This paper provides an in‐depth review of optimization‐based approaches that deal with the first step, the generation of the guidance track system. Thereby, a comprehensive and pedagogical approach for the generation of guidance tracks for arbitrarily shaped two‐dimensional regions of interest is provided, along with an overview and detailed elaboration on different exact cellular decomposition techniques found in literature. Furthermore, cost functions are outlined for the different approaches presented in this work, which are utilized to generate optimal guidance tracks. Finally, this survey serves as an introductory guide for research and practitioners to solve the CCPP problem effectively and efficiently.

Faster Bootstrapping via Modulus Raising and Composite NTT

FHEW-like schemes utilize exact gadget decomposition to reduce error growth and ensure that the bootstrapping incurs only polynomial error growth. However, the exact gadget decomposition method requires higher computation complexity and larger memory storage. In this paper, we improve the efficiency of the FHEWlike schemes by utilizing the composite NTT that performs the Number Theoretic Transform (NTT) with a composite modulus. Specifically, based on the composite NTT, we integrate modulus raising and gadget decomposition in the external product, which reduces the number of NTTs required in the blind rotation from 2(dg + 1)n to 2(⌈dg⌉/2 + 1)n. Furthermore, we develop a modulus packing technique that uses the Chinese Remainder Theorem (CRT) and composite NTT to bootstrap multiple LWE ciphertexts through one blind rotation process.We implement the bootstrapping algorithms and evaluate the performance on various benchmark computations using both binary and ternary secret keys. Our results of the single bootstrapping process indicate that the proposed approach achieves speedups of up to 1.7 x, and reduces the size of the blind rotation key by 50% under specific parameters. Finally, we instantiate two ciphertexts in the packing procedure, and the experimental results show that our technique is around 1.5 x faster than the two bootstrapping processes under the 127-bit security level.

An exact decomposition technique for the deadline-constrained discrete time/cost trade-off problem with discounted cash flows

This paper addresses the deadline-constrained discrete time/cost trade-off problem with discounted cash flows in project scheduling optimization. Our objective is to optimize the net present value (NPV) of cash flows by selecting execution modes for activities while considering project deadlines. To address this problem, we propose an adapted model that combines the activity-on-node representation of the project network with the milestone activity payment system commonly used in real-world scenarios. This enhanced model accurately depicts project dynamics, enabling improved decision-making in NPV optimization. To solve the problem, we employ the generalized benders decomposition technique, which provides an exact solution approach to handle the computational complexity of the problem. Our approach introduces three types of feasibility cuts. The first type eliminates infeasible solutions by analyzing subproblem infeasibility. The second type enhances the generation of high-quality optimality cuts by iteratively introducing constraints to the master problem, utilizing dominated paths in the project network graph. Lastly, the third type targets the elimination of inferior infeasible solutions. Additionally, we generate a comprehensive dataset encompassing various scenarios and parameter ranges from existing literature. This dataset serves as a valuable resource for further research, with online accessibility and the latest solutions available for approximately 32,400 generated instances. Through our research, we demonstrate the superiority of our proposed solution procedure compared to existing methods in the field. By considering the specific requirements of project scheduling with NPV optimization and project deadlines, our approach contributes to more effective project planning and decision-making in real-world contexts.

Approximate option pricing under a two-factor Heston–Kou stochastic volatility model

Under a two-factor stochastic volatility jump (2FSVJ) model we obtain an exact decomposition formula for a plain vanilla option price and a second-order approximation of this formula, using Itô calculus techniques. The 2FSVJ model is a generalization of several models described in the literature such as Heston (Rev Financ Stud 6(2):327–343, 1993); Bates (Rev Financ Stud 9(1):69–107, 1996); Kou (Manag Sci 48(8):1086–1101, 2002); Christoffersen et al. (Manag Sci 55(12):1914–1932, 2009) models. Thus, the aim of this study is to extend some approximate pricing formulas described in the literature, like formulas in Alòs (Finance Stoch 16(3):403–422, 2012); Merino et al. (Int J Theor Appl Finance 21(08):1850052, 2018); Gulisashvili et al. (J Comput Finance 24(1), 2020), to pricing under the more general 2FSVJ model. Moreover, we provide numerical illustrations of our pricing method and its accuracy and computational advantage under double exponential and log-normal jumps. Numerically, our pricing method performs very well compared to the Fourier integral method. The performance is ideal for out-of-the-money options as well as for short maturities.

Cost minimizing decisions on equipment and charging schedule for electric buses in a single depot

Combined decisions on equipment and charging schedule for electric buses (e-buses) are studied in this paper. The most representative day is considered, in which each e-bus fulfills a transfer task composed of a series of passenger transportation trips. Each trip starts and terminates at given times in the same single depot where e-buses re-charge their batteries. The problem is to determine battery variants for the e-buses, number of chargers of different types and daily cyclic charging schedule of e-buses such that a linear combination of (a) the total daily cost of the charging equipment, batteries wear and consumed energy, and (b) the total number of e-bus charging interruptions (switchings) is minimized. The constraints include an upper bound on the supplied electric power, restoration of the initial battery energy state before the next day transfer task for each e-bus and an upper bound on the maximal number of switchings for each e-bus in each depot visit over all e-buses and visits. A bi-level decomposition approach is proposed, in which decisions and constraints related to the detailed charging schedule and the number of switchings are moved to the lower-level problem. The lower level problem is formulated as Mixed Integer Linear Programming (MILP) problem. The upper-level problem is reduced to a MILP problem using linearization techniques. An exact decomposition scheme is developed for the upper-level MILP problem. A case study and comprehensive computer experiments are provided.

A New Method of Kronecker Product Decomposition

Kronecker product decomposition is often applied in various fields such as particle physics, signal processing, image processing, semidefinite programming, quantum computing, and matrix time series analysis. In the paper, a new method of Kronecker product decomposition is proposed. Theoretical results ensure that the new method is convergent and stable. The simulation results show that the new method is far faster than the known method. In fact, the new method is very applicable for exact decomposition, fast decomposition, big matrix decomposition, and online decomposition of Kronecker products. At last, the extension direction of the new method is discussed.

Multi-wavelength unidirectional forward scattering properties of the arrow-shaped galliumphosphide nanoantenna.

An arrow-shaped gallium phosphide nanoantenna exhibits both near-field electric field enhancement and far-field unidirectional scattering, and the interference conditions involve electric and magnetic quadrupoles as well as toroidal dipoles. By using long-wavelength approximation and exact multipole decomposition, the interference conditions required for far-field unidirectional transverse light scattering and backward near-zero scattering at multiple wavelengths are determined. The near-field properties are excellent, as exemplified by large Purcell factors of 4.5×109 for electric dipole source excitation, 464.68 for magnetic dipole source excitation, and 700V/m for the field enhancement factor. The degree of enhancement of unidirectional scattering is affected by structural parameters such as the angle and thickness of the nanoantenna. The arrow-shaped nanoantenna is an efficient platform to enhance the electric field and achieve high directionality of light scattering. Moreover, the nanostructure enables flexible manipulation of light waves and materials, giving rise to superior near-field and far-field performances, which are of great importance pertaining to the practicability and application potential of optical antennas in applications such as spectroscopy, sensing, displays, and optoelectronic devices.

Coupled-mode separation of multiply scattered wavefield components in two-dimensional waveguides.

For a waveguide that is invariant in one of the horizontal directions, this paper presents a mathematically exact partial-wave decomposition of the wavefield for assessment of multiple scattering by horizontally displaced medium anomalies. The decomposition is based on discrete coupled-mode theory and combination of reflection/transmission matrices. In particular, there is no high-frequency ray approximation. Full details are presented for the scalar case with plane-wave incidence from below. An application of interest concerns ground motion induced by seismic waves, which may be severely amplified by local medium anomalies such as alluvial valleys. Global optimization techniques are used to design an artificial medium termination at depth for a normal-mode representation of the field. A derivation of a horizontal source array to produce an incident plane wave gives, as a by-product, an extension of a previous Fourier-transform relation involving Bessel functions. Like purely numerical methods, such as finite differences and finite elements, the method can handle all kinds of (two-dimensional) anomaly shapes.

Multi-period distribution networks with purchase commitment contracts

Retailers which deliver products directly to their customer locations often rely on Logistics Service Intermediaries (LSI) for order management, warehousing, transportation and distribution services. Usually, the LSI acts as a shipper and subcontracts the transportation to carriers for long-haul and last-mile delivery services. All agents interact and are connected through cross-docking facilities. As the demand from customers may vary significantly over time, the shipper’s requirements for transportation evolve accordingly at the tactical level. This creates opportunities for the shipper to take advantage of medium-term contracts with the carriers at prices lower than those offered by the spot market. The study focuses on the tactical design, through dynamic contracts, of a suitable network of cross-docking facilities and related transportation capacities (belonging to different carriers) to reduce the shipper’s operational costs. In this article, we propose an MILP formulation for the multi-period planning problem with minimum purchase commitment contracts faced by the shipper. We propose exact and heuristic decomposition methods for the the model, respectively, based on combinatorial Benders cuts and on relax-and-repair approaches. The performance of these algorithms is experimentally compared to that of commercial solvers (branch-and-cut and classical Benders). The numerical results show that our methods perform comparatively well for the solution of large size instances and brings economic benefits to the shipper.

Information Ratio = Selection × Breadth + Sizing

Factor-based performance attribution is an established practice among quantitative and fundamental strategies alike. Although popular, this attribution technique is not of great help to investors interested in understanding the source of their idiosyncratic performance. Of special interest is the ability to separate stock selection skill and sizing skill. Methods aimed at measuring these skills are inspired by analogies to sports. In this article, the author presents an exact decomposition of the information ratio. The IR is the sum of a breadth-adjusted stock selection skill and a sizing skill. The definition of breadth relies on an intuitive measure of concentration—the Herfindahl Index—rather than on the “square root of <italic>n</italic>” measure. These quantities can be compared so that the portfolio manager can determine the percentage of their realized performance originating from each term. In addition, the decomposition empowers the manager to improve their risk-adjusted performance by choosing optimal position sizing, determined by the observed performance of their portfolio. Finally, the decomposition can be further extended to long–short analysis, and provides a natural explanation of an empirical phenomenon: The idiosyncratic performance of the long side of a strategy is often larger than that of the short side.