John Buffington
email: john.buffington@ltu.se

Mehdi Amini
email: mamini@memphis.edu

Timur Keskinturk

Jack Buffington

Development of a Product Design and Supply Chain Fulfillment System for Discontinuous Innovation

Keywords: AGENT BASED SYSTEMS, FUZZY METHODS, PRODUCT DESIGN, INTEGRATION, PRODUCT generative customization, discontinuous innovation generative customization, discontinuous innovation, product design

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Introduction

Traditionally, a discontinuous innovation has been perceived as a function of a product invention, but for it to occur, dynamic forces from seemingly unrelated processes or industries often must converge upon the environment in order to develop a new consumer market. For instance, an Apple iPod is a discontinuous innovation, less as a function of its ability to store digital music files on a portable hard drive system, and more due to its ability to transform within processes and behaviors via supply chains and consumers, including related industries (such as the music industry). The SmartPhone was able to be adopted within consumer markets due to the size and functionality of the device, but the convergence of previously unrelated industries of mobile telephony, such as computer operating systems and applications was the critical element of the discontinuity of the innovation.

With less than 10% of new product designs being classified as discontinuous or radical innovation [START_REF] Griffin | PDMA Research on New Product Development Practices: Updating Trends and Benchmarking Best Practices[END_REF], the question is whether a conventional NPD solely focused on product design is suitable for creation of products that must extend beyond its own physical functionality.

Typically, a conventional method of product development, such as StageGate TM , is a combination of linear, structured, processes linked as a closed system, which restricts creativity to a single body (such as a product developer within the OEM firm). One alternative to a linear, closed system process of product development that solely focuses on the product is Buffington's (2010) concept of generative customization (Figure 1 represents its high level logical design) that is more of a framework than a process to develop discontinuous innovation. In this model, it is understood that design and fulfillment factors must extend beyond traditional definitions of new product development (NPD). In a study to understand consumer behavior and innovation, Buffington (2010) found generative customization to be a superior alternative to a traditional NPD design approach for discontinuous innovation. The purpose of this study is to build upon this conceptual product design and supply chain fulfillment system of generative customization for discontinuous innovation. This paper will be structured as follows: next, a literature review will be presented discussing generative design and product design (front-end element), agent-based modeling and supply chain network (back-end system), and generative customization as a research concept. Following this literature review, using these concepts a conceptual system design structure will be developed. (2011) expanded upon this point for discontinuous innovations, and found that generativity best serves discontinuous innovation, since it follows a process "on the edge of chaos", which is a necessary condition for the creation of new functionality and markets, beyond the development of a product. [START_REF] Krishnamurti | On learning and knowledge[END_REF] noted that

Literature Review

generative learning requires an individual to focus on intuition and transcendence, which are typically not descriptors used in conventional product design and fulfillment systems; it describes more of a process than an end result. [START_REF] Weick | The Generative Properties of Richness[END_REF] described generativity as an evocative power or aptitude that is opposite of adaptation.

McCormack et al ( 2004) described as generative design as the production of an artifact and experience, which is difficult to define.

Generative design processes enable complex relationships to form rather than undertaking a linear, reductionist approach and simplifying them for the sake of the supply chain system, or within the inter and intra organizational systems. McCormack et al (2004) found generative design to be defined in terms of dynamic processes and their outcomes. In the use of creative evolutionary systems (CES), entirely new designs can be generated from little or no knowledge guided by the process [START_REF] Kicinger | Evolutionary computation and structural design: A survey of the state-of-the-art[END_REF] rather than the designer. Avital and Te'eni (2009) defines these generative schemes as "simple recipes that allow anyone with basic skills to create or put together", suggesting that "simple instructions will lead to infinite variations". In research and practice, there is scant evidence of its ability to achieve creativity in an NPD process beyond a few industries, such as architecture, music and software development. Furthermore, generative design has been utilized solely to develop physical product designs, and to our understanding, has yet to consider definitions of critical processes and relationships to these product designs as well.

The digitization of the design process has become a revolution, and has transformed into a unique field of endeavor, beginning to evolve its own unique methodologies, yet still largely unformulated [START_REF] Oxman | Theory and design in the first digital age[END_REF]. While simple applications of Computer Aided Design (CAD) have been in existence as early as the 1960s (Sutherland's SKETCHPAD at MIT), its purpose has been focused almost entirely upon the digitization of form, rather than an improvement on design and market process possibilities. [START_REF] Szykman | A foundation for interoperability in next-generation product development systems[END_REF] noted that "traditional CAD systems are limited to representations of geometric data and other types of information relating to geometry such as constraints, parametric information, features, and so on". [START_REF] Hanna | Teaching Parametric Design in Code and Construction[END_REF] noted that the focus of CAD is on the representation of the final form described geometrically. [START_REF] Larocca | Development of an ICAD Model for Blended Wing Body Aircraft Design[END_REF] noted the difference between traditional CAD and generative design being that the latter represents the engineering intent behind the geometric design. [START_REF] Liu | Generative Design in an Agent Based Collaborative Design System[END_REF] acknowledged the progress made in CAD tools over the decades, but noted the large limitations associated with it being able to support human cognition and creativity. [START_REF] Pande | A generative process planning system for parts produced by rapid prototyping[END_REF] developed a generative planning process using computer aided process planning (CAPP) to become a link between CAD and computer aided manufacturing (CAM), further extending possibilities of CAD. Generative design is a method that has been implemented in various fields (art, architecture, communication design, product design) where the output is generated by a set of rules or an algorithm, and has a design schema, a means of creating variation and a means of selecting desirable outcomes (Wikipedia, 2010). Therefore, it is different from CAD because it must be based upon modeling knowledge that is created communally on an agreed set of standards [START_REF] Krish | Generative Design[END_REF], as opposed to part of a closed system. In this process, hundreds, or even thousands of design possibilities can be generated through the use of generative schemes and data, which allows a group of designers to change parameters in order to create an innovative design. Therefore, a new design schema is not simply a product of the parameters used to generate it, but rather also includes the conditions used by designers in this process. This is due to the design not being represented in final form, but rather as associative, with changes being enabled through computation [START_REF] Hanna | Teaching Parametric Design in Code and Construction[END_REF]. These communalities of form may also be possible to be extended to communalities of process, to the authors of this paper.

The challenge of generative design is how to frame this human-computer interaction process, for a product-process combination. If generative design adheres to the rules set forth by Simon in his book The Sciences of the Artificial (1996) then the design is contingent upon the goals of the designer and the purposes for which the system is designed. Avital and Te'eni (2008) outlined this human-computer interaction process through the definition of generative capacity (a human's ability to reframe reality and produce something ingenious) and generative fit (an attribute of a system that is • Visualization (to see an object from multiple perspectives)

• Simulation (to the underlying process or the object's behavior, capabilities, or response to stimuli in different context

• Abstraction (the ability to move swiftly between levels of granularity to identify emergent patterns, commonalities, or anomalies

• Integration (ability to merge views of various parallel subsystems or cross-cuts of objects)

• Communication (one's ability to talk and share information with other agents with no regard to institutionally imposed boundaries

The advancement of theory in generative design is beginning to establish characteristics in evolutionary science. Avital and Te'eni (2008) noted the difference between conventional computer assisted design and generative evolutionary design in being the former specified by predetermined parameters while the latter is focused on the generation or discovery of unique and unexpected solutions. [START_REF] Coblani | Generative Design Using the Design DNA[END_REF] developed the concept of Design DNA (DDNA) in the development of generative design.

Given its relative newness as a concept, generative design has significant limitations in research and practice. The discussion of this topic thus far in research has been primarily focused on concepts, or applications in niche industries, such as art and architecture. One concern from literature is that generative design solely addresses the domain of form, texture and color [START_REF] Krish | Generative Design[END_REF], but not functional parameters, or as we have proposed, process parameters as well. In consideration of a discontinuous innovation being defined by the consumer market and peripheral (and perhaps design must be extended in the ex ante definition of future markets as well. This is a critical issue that must be addressed in future research.

Agent-Based Modeling in Supply Chain Fulfillment

Despite business environments changing dramatically over the past two decades [START_REF] Mccarthy | New Product Development as a Complex Adaptive System of Decisions[END_REF], supply chain systems have been surprisingly static, and diagrammed in very unimaginable ways. In contrast, [START_REF] Koh | Competing in the 21st century supply chain through supply chain management and enterprise resource planning integration[END_REF] prescribed that companies in the 21 st century will face enormous complexity, and supply chains will need to become more agile and responsive as a prerequisite. Yet supply chain complexity cannot occur in isolation within a supply chain network; Jain et al ( 2009) found marketing, distribution, planning, manufacturing, and purchasing of organizations along the supply chain operating largely independent of one another, with the organizations having their own objectives and being in conflict with one another. Coordination between multiagent systems (MASs) will stretch the limits of possibilities of how supply chain systems act and evolve, and will require a much greater interaction between the disciplines of supply chain and information technology than ever before.

Therefore, despite the efforts to manage supply chain systems within predictable and linear progressions, supply chains in the 21 st century must present themselves as more dynamic and non-linear, leading to endogenous variability being created as companies seek to simplify in this manner. Beyond exogenous factors that are often perceived to be the greatest cause of uncertainty to supply chain systems, endogenous supply chain systems are gaining in complexity as well. Like biological systems, supply within and from agents in the supply chain impact the entire system. [START_REF] Prater | A framework for understanding the interaction of uncertainty and information systems on supply chains[END_REF] found that while supply chain uncertainty is a major topic in supply chain literature, there is a tendency to lump various forms of supply chain complexity under a broad concept of uncertainty with few areas of differentiation. Kim's (2007) research validated this notion through stating that while supply chain research focused much attention on the 'bullwhip effect' (order variability impacting upstream systems), and its exogenous impact, less attention has focused on how endogenous variability could lead to exogenous variability. Furthermore, as is the case with discontinuous innovation, uncertainty is not always a factor that must be mitigated; instead, it should often be embraced, and developed.

Various methods for attempting to reduce supply chain uncertainty include improvements in modeling and forecasting different components of the supply chain [START_REF] Ewing | Industrial production, volatility, and the supply chain[END_REF]. [START_REF] Moon | Enhancing ERP system's functionality with discrete event simulation[END_REF] noted that simulations can often capture and describe the complex interactions within a particular manufacturing or supply chain system where traditional analytical methods fail.

On the other hand, mathematical techniques can assist in the development of parameters in a modeling effort, how individual agent characteristics affect a system, and identification of agents to develop an agent-based modeling (ABM) effort [START_REF] Surana | Supply-chain networks: a complex adaptive systems perspective[END_REF]. Instead of an either or decision, perhaps the most suitable approach is a combination of traditional mathematical techniques for the establishment of agents and a supply chain model, and Similar to the use of generative design in product design processes, the use of ABM in the dynamic development of supply chain networks is in its infancy stages. avoiding the final form of the product design [START_REF] Garud | Incomplete by Design and Designing for Incompleteness[END_REF], there are inherent problems in the identification of peripheral agents involved in a discontinuous innovation, future markets, and the fulfillment of the product as a result. In an expertbased approach to design that is represented in discontinuous innovation [START_REF] Sanders | An Evolving Map of Design Practice and Design Research[END_REF], a collaborative process including the consumers and suppliers (some who are agents unknown) has not been found to be viable.

Generative Customization

Numerous researchers, such as [START_REF] Danneels | Disruptive Technology Reconsidered: A Critique and Research Agenda[END_REF] have found that consumers are almost never able to conceive a radical (or discontinuous) innovation, and their involvement in such a process is inefficient. Given this limitation, which is so critical in a dynamic supply chain system, the possibility of creating a virtual consumer and supplier, via generative design and agent-based modeling, has been proposed in generative customization as a compromise in the lack of a collaborative environment. For this purpose, the use of agent-based modeling techniques can be proven as useful in the establishment of how these products, processes and agents interface with one another in the creation of a discontinuous innovation.

In the development of a generative customization product design and supply chain fulfillment system, the inability to achieve collaboration as a function of the design of the product and process using generativity (creatively and separate from existing consumers and suppliers) poses an immense challenge for the achievement of discontinuous innovation. [START_REF] Danneels | Disruptive Technology Reconsidered: A Critique and Research Agenda[END_REF] and [START_REF] Govindarajan | The Usefulness of Measuring Disruptiveness of Innovations Ex Post in Making Ex Ante Predictions[END_REF] both noted a firm's reluctance to pursue discontinuous innovation as a function of its inability to more accurately predict its potential for success in a marketplace, ex ante. Outside of the use of structured (not necessarily linear) NPD processes and tools, we question how this research problem can be achieved, which leads to our model for generative customization as a set of NPD processes and tools for discontinuous innovation.

The Use of Generative Design and Agent-Based Modeling in Generative Customization

Through a definition of discontinuous innovation and the literature review of generative design, agent-based modeling, and generative customization, a three step process requirement will be established in this paper for the use of generative customization in the development of discontinuous innovation, as is shown below in Table 1. These steps in the process will be discussed through the product design process for generative customization, as will be discussed in this section. Step Definition Explanaton

Product Variation (variate)

The modification of the digital representation of an existing product, or the creation of an entirely new invention (Generative Design).

Establishing a product design as a process (artifact and experience) rather than an invention in final form.

Non Product Variation (variate)

Starting with the base of a new product artifact, the use of generative design to develop non-product factors (agents and processes) for discontinuous innovation. These may be peripheral products or services.

If the product design is an artifact and experience (product variation), it will be extended in definition beyond the physical product to include other factors and markets.

Market Equilibrium Agent Model (MEAM) (interact)

Agent-based process that simulates the product and market variations in the achievement of new markets and peripheral agents (suppliers of products and services), enabling firms to conduct ex ante predictions.

Fusing generative designed product and non-product features in the development of generative customization. Both product design, and fulfillment parameters are created. [START_REF] Schumpeter | Capitalism, socialism, and democracy[END_REF] found a disruptive or discontinuous innovation process to commence from a market in a state of equilibrium, with a disruption (or multiple) supplier, market and the product (not just to the design of the product). Since incumbents in a current market environment are generally comfortable with stability and equilibrium, the goal of the generative customization system will be to interject an element of disruption into the system to enable radical innovation.

In the first step, the product designer will develop completely new product invention designs, or mutations of products from existing designs using generative design techniques. Through an associative, digital definition of the parameter form, the designer is able to make modifications to product parameters (represented in digital form), developing virtually an unlimited number of design permutations. Through this first step, a product invention has been achieved through the process, but not a product innovation, as these forms have been yet to be found as useful to a future consumer market.

Given that discontinuous product designs (inventions) can often be a convergence of components from different products (such as a Smartphone being a convergence of a telephone, computer, and personal audio player, for example), generative design must transform the function as well as form parameters in design, which is relatively unproven in its history.

Using this first stage (product variation), generative design techniques are used to create non-product parameters to enable the product invention. If the first stage is considered the traditional core product design, the non-product variation can be the development of peripheral products and services to support the core invention.

Continuing to use the Smartphone as an example, a wireless headphone and car MP3 As Schumpeter (1983) noted, there is a difference between a product invention and innovation, with an innovation being a function of the market environment. From [START_REF] Garcia | Co-opetition for the Diffusion of Resistant Innovations: A Case Study in the Global Wine Industry using an Agent-based Model[END_REF], the market environment for discontinuous innovation will be defined in terms of two heterogeneous agents, as follows:

1) Future Consumer Adoption = f (individual preferences, consumer network effects, firm adoption, advertising)

2) Future Firm and Supply Chain Adoption = f (innovativeness, firm network effects, consumer adoption)

The As is shown in Figure 2, the third step will establish a product design and supply chain fulfillment model, created as a market equilibrium agent model (MEAM), to dynamically interface the product and non-product clusters to establish the discontinuous innovation.

The variation processes provides the definitions of the product and non-product parameters to be established via associative definitions, not defined in final form. The ability to variate both the product and non-product components/parameters do not appear to have a precedent in extant research, and needs to be explored in future research. While [START_REF] Krish | Generative Design[END_REF] discussed the variation of a limited range of products through the use of generative design, we believe that there is potential for many products, but also the inclusion of non-product parameters as well.

After the variate stage has occurred, the final stage is the interaction between the product innovation and the existing market equilibrium (or MEAM) is the introduction of the interact phase of generative customization, as is noted in Table 1.

This process is defined through agent based modeling (ABM), based upon Garcia's (2005) conceptual map, is illustrated in Figure 3.

The goal of this agent based model is to simulate an amended diffusion of innovation process, based upon [START_REF] Rogers | Diffusion of innovations[END_REF] model, but existing of a complex adaptive system (holistic) view of diffusion, not just related to the consumer. A fallacious assumption in discontinuous innovation is that only the consumer needs to adopt and change in this process, but in reality, diffusion needs to occur in various, and unstructured ways within Therefore, the product designer must establish numerous simulations of this process, because the difference between an invention and an innovation is the relationship that the product has with its environment, as noted in the following proposition:

Proposition DI : The development of a discontinuous innovation via generative customization occurs through the dynamic interaction between product and nonproduct parameters, which have been developed and interacted through a generative and agent-based modeling simulation process.

From this process, an ex ante prediction process can occur to measure the potential of a discontinuous innovation, a process that is not possible today [START_REF] Danneels | Disruptive Technology Reconsidered: A Critique and Research Agenda[END_REF].

This dynamic environment will model the nature of a discontinuous innovation itself, and provide the firm with an improved structural framework to make new product development decisions.

Conclusions

The cruel irony facing firms when considering the development and implementation of discontinuous innovation is while there appears to be a greater frequency and severity of its occurrences of late (Birkinshaw et al, 2006), the process and risk associated with doing it remains greater than ever. When faced with the consideration of a radical innovation strategy through the use of conventionally linear and finite tools (even including CAD systems), the cliché going to a gunfight with a knife can often come to mind by firms, leading them to play it safe through incremental innovations, or staying safely (at least for the time being) within the comfortable confines of the market equilibrium. The problem posed in this research paper, and in Buffington's (2010) definition of generative customization, is how to utilize the capabilities of computer technology and creative human cognition to improve the capabilities of seeking and achieving discontinuous innovation in product design. After a review of extant literature, a system design has been developed that achieves this purpose, at least in concept. This was accomplished through using the emerging technologies of generative design and agentbased modeling to create discontinuous innovation both in the form of a product design, supply chain process, the identification of future markets, and of new agents involved in the process.

Through the development of a three step generative customization methodological process, as is shown in Table 1, the fusion of generative design and agent based modeling was accomplished in the steps of product variation, non-product variation, and interaction is fashioned for the achievement of generative fit, as noted in Avital and Te'eni (2008). Therefore, the goal of this generative customization methodology is not in the achievement of a discontinuous design of a product, per se, but rather in the development of all parameters necessary for the development of a future consumer market. From this, Proposition DI is introduced that defines discontinuous innovation as a function of a dynamic interfacing between these product and non-product parameters to improve the predictive nature of it. We believe that this is an original and significant finding in the measurement and development discontinuous innovation that is only possible through these emerging tools (and not possible through the use of linear tools, such as differential equations).

  and enhancing that generative capacity in people). According to Avital and Te'eni (2008), a system has high generative fit, and can evoke generative capacity if it has the following attributes:

  ) players as much as by the product design itself, generative

  both endogenous and exogenous forms of variability, and supply chain research hasn't progressed far in understanding how these forms of variability

  -based modeling tool, establishing less abstract modeling capabilities to understand and patterns issues of complexity and uncertainty.[START_REF] Surana | Supply-chain networks: a complex adaptive systems perspective[END_REF] found a useful amalgam of simulation and analytical techniques for the use of understanding and predicting supply chain systems.However, there are limitations in the use of agent-based modeling in supply chain systems. One, simulations are developed offline and separate from enterprise software packages[START_REF] Yoon | Simulation-based real-time decision making for manufacturing automation systems: a review[END_REF], limiting observations to single event analysis.[START_REF] Moon | Enhancing ERP system's functionality with discrete event simulation[END_REF] found that traditional simulation models are usually not built for repetitive and continuous usage partially due to the laborious data collection process. Difficulty in automatic collection of data and continuous update of the data for the built simulation model discourages further usage of the initial model. In their study, Al-Zubaidi and Tyler (2004) noted a major requirement of the simulation model is that it should capture the dynamic and stochastic nature of consumer behavior at the retail store, which is not possible without easily accessible and usable operational data.

  Buffington and McCubbrey (2011) identified a major problem in the development of the generative customization concept being that while the front-end product design process could occur through the use of generative design processes and tools (Proposition P 3 ), there are no new product development (NPD) processes established for the fulfillment of this design in a mass market supply chain system. Given the , and largely unstructured approach of generative design, often

  environment, creating a transformational change to the consumer,

  of peripheral product developments, as is the case of mobile applications, personal GPS, and Internet based delivery sites. The non-product developments of a discontinuous innovation can become more prolific to a consumer market potential than the core invention itself. Through the use of creative generative design principles, these possibilities can be created as a function of the core product development.

  development of a discontinuous innovation requires a combination of a new product invention, a new consumer market, and/or a new firm and supply chain. As an example, the development of a digital audio player is not useful without the development of a compressed audio file format, the development of online sources as a delivery mechanism, and other factors unrelated to the development of a portable audio player with a computer hard drive. As such, the development of the electronic component and these peripherals operate as an open innovation system (Chesbrough,2003), with all consumers and suppliers are modeled in a market equilibrium agent model (MEAM), as opposed discrete and linear consumer and supplier agents solely existing supply chain system. The MEAM process will define the third step of generative customization, to model both the product invention and the development of the market into one discontinuous innovation process.

  the process, often newly created and involved. In this case, market equilibrium exists in a heterogeneous agent mode (Figure2), that includes not just consumers and suppliers for the existing product, but including suppliers and consumers within the market, not necessarily involved in the current environment, but influencing, as is suggested in complex adaptive systems and open system(Chesbrough, 2003) theory. It is critical in an establishment of the MEAM, as shown in Figure2, to represent the entire possible market, not just the existing supply chain, given that discontinuous innovation is the disruption of an existing market to the creation of new consumers, suppliers and market conditions; given the difficult of ex ante predictions, this is a difficult, unstructured step. As a result of this process, a supply chain system emerges and self-organizes, based upon the definition of the product design. As a result, a radical invention is transformed into a discontinuous innovation through the result of the efficient fulfillment of the product design.The simulation of the variated product into the MEAM is the introduction of an invention into this market equilibrium to understand the nature of the disruption, as shown in the below mathematical problem:DI x = f (PI x , NP x ,Cx, S x )Where, DI x = A definition of discontinuous innovation solutions, as a result of the interaction between the variables (to define an ex ante consumer market and peripheral agents). PI x = An introduction of a discontinuous product invention NP x = An introduction of a discontinuous non-product invention C x = A definition of the consumer response, as noted by Garcia (2007) S x = A definition of supplier's response, as noted by[START_REF] Garcia | Co-opetition for the Diffusion of Resistant Innovations: A Case Study in the Global Wine Industry using an Agent-based Model[END_REF]

  To our knowledge, this is the first study that seeks to integrate the use of generative design and agent based modeling in the understanding, development and measurement of discontinuous innovation. While there has been some research related to agent based modeling and supply chain systems, and scarce, but some in relation to generative design and NPD, there needs to be a further examination of these factors, both separately and in conjunction with one another. Future studies can be individual case studies, surveys within an industry related to NPD practices, and modeling and simulations of this concept.

  Chiva et al(2010) stated that a generative system implies being able to see beyond the situation and questioning operating norms within its environment. Buffington andMcCubbrey

	F o
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	l y
	2.1 Generative Design and Product Design
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Table 1 -

 1 Generative Customization Product Design and Fulfillment Methodology

http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.ukInternational Journal of Production Research

Al-Zubaidi, H., Tyler, D. ( 2004) "A simulation model of quick response replenishment of seasonal clothing", International Journal of Retail & Distribution Management, Vol. 32 Iss: 6, pp.320 -327 Avital, Michel, and Dov Te'eni. (2009). "From generative fit to generative capacity:

exploring an emerging dimension of information systems design and task performance". Information Systems Journal. 19 (4): 345-367. Balakrishnan, P. V., and Jacob, V., (1996). "Genetic Algorithms for Product Design". Management Science. 42 (8): 1105-1117. Birkinshaw, Julian, John Bessant, and Rick Delbridge. 2007. "NETWORKS -Finding, Forming, and Performing: Creating Networks for Discontinuous Innovation". California Management Review. 49 (3): 67. Boden, M. The Creative Mind: myths and mechanisms. London Cardinal, 1992. Bruderer, E., and Singh, J. (1996). Organizational Evolution, Learning and Selection: A Genetic Algorithm Based Model. Academy of Management Journal 39:1322-49 (October). Buffington, J. (2010). Development of Generative Customization for Consumer Mass Markets. , Industrial Management and Data Systems, Volume 111 (1), October. Buffington, J. and McCubbrey, D. (2011). A Conceptual Framework of Generative Customization as a Solution for Product Innovation Conceptualization and Fulfillment, European Journal of Innovation Management, forthcoming. Chesbrough, H. (2003). Open innovation: The new imperative for creating and profiting from technology. Boston, Mass: Harvard Business School Press. Chiva et al (2010). Adaptive and Generative Learning: Implications from Complexity


I W Harry
email: ian.harry@astro.cf.ac.uk

S Fairhurst
email: stephen.fairhurst@astro.cf.ac.uk

A coherent triggered search for single spin compact binary coalescences in gravitational wave data

In this paper we present a method for conducting a coherent search for single spin compact binary coalescences in gravitational wave data and compare this search to the existing coincidence method for single spin searches. We propose a method to characterize the regions of the parameter space where the single spin search, both coincident and coherent, will increase detection efficiency over the existing non-precessing search. We also show example results of the coherent search on a stretch of data from LIGO's fourth science run but note that a set of signal based vetoes will be needed before this search can be run to try to make detections.

Introduction

The Laser Interferometer Gravitational-wave Observatory (LIGO) and Virgo scientific collaborations have performed many searches for compact binary coalescence (CBC) signals in data taken by gravitational wave interferometers [START_REF] Abadie | Search for Gravitational Waves from Compact Binary Coalescence in LIGO and Virgo Data from S5 and VSR1[END_REF][START_REF] Abbott | Search for Gravitational Waves from Low Mass Compact Binary Coalescence in 186 Days of LIGO's fifth Science Run[END_REF][START_REF] Abbott | Search for Gravitational Waves from Low Mass Binary Coalescences in the First Year of LIGO's S5 Data[END_REF]. The majority of these searches have utilized template waveforms where the spins of the individual components are neglected. In some areas of the parameter space spin can have a significant effect on the evolution of the system, and consequently the emitted gravitational waveform [START_REF] Theocharis | Spin induced orbital precession and its modulation of the gravitational wave forms from merging binaries[END_REF][START_REF] Grandclement | Searching for gravitational waves from the inspiral of precessing binary systems. I: Reduction of detection efficiency[END_REF], leading to a poor match with the non-spinning templates. In these regions of parameter space the use of templates incorporating spin will provide an increase in search sensitivity.

Incorporating spin into template waveforms in a gravitational wave search is a complex problem. In a non-spinning search for CBCs with circular orbits, a source is described by nine physical parameters [START_REF] Thorne | Gravitational radiation[END_REF]. The majority of these do not affect the signal morphology, but serve to change the overall amplitude, phase or coalescence time of the signal and are easily maximized over [START_REF] Allen | FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries[END_REF]. Therefore, template placement can be restricted to the two dimensional space of component masses [START_REF] Babak | A template bank to search for gravitational waves from inspiralling compact binaries. I: Physical models[END_REF]. A spinning CBC in a circular orbit, however, is described by 15 physical parameters [START_REF] Thorne | Gravitational radiation[END_REF]. The challenge is to formulate a method to detect any manner of spinning system while limiting the number of templates, such that an analysis can be run in a reasonable amount of time. The problem is simplified if the spins are aligned with the orbital angular momentum. In this case the system will have no precession and is described by just two extra parameters -the spins of both bodies in the direction of the angular momentum. Furthermore, these non-precessing waveforms are well described Confidential: not for distribution. Submitted to IOP Publishing for peer review 25 March 2011 by a single spin parameter [START_REF] Ajith | Inspiral-merger-ringdown waveforms for black-hole binaries with non-precessing spins[END_REF], and it is therefore feasible to search for non-precessing waveforms using a three dimensional template bank.

At the time of writing only one search for CBCs in LIGO and Virgo data using spinning templates with precession has been published [START_REF] Abbott | Search of S3 LIGO data for gravitational wave signals from spinning black hole and neutron star binary inspirals[END_REF]. This search utilized a phenomenological waveform family designed to capture precessional effects [START_REF] Buonanno | Detection template families for precessing binaries of spinning compact binaries: Adiabatic limit[END_REF], but was later abandoned because it was not found to increase efficiency relative to the nonspinning search [START_REF] Abbott | Search for Gravitational Waves from Low Mass Binary Coalescences in the First Year of LIGO's S5 Data[END_REF][START_REF] Van | Template banks to search for compact binaries with spinning components in gravitational wave data[END_REF]. This was due to the ability of the phenomenological templates to match non-stationarities in the data and the lack of an effective signal consistency test to veto them, such as the χ 2 test used in the non spinning search [START_REF] Allen | A χ 2 time-frequency discriminator for gravitational wave detection[END_REF].

The physical template family (PTF) waveforms, which were proposed in [START_REF] Buonanno | Detection template families for precessing binaries of spinning compact binaries: Adiabatic limit[END_REF] and further explored in [START_REF] Pan | A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries[END_REF][START_REF] Buonanno | Quasiphysical family of gravity-wave templates for precessing binaries of spinning compact objects: Application to double-spin precessing binaries[END_REF][START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF] give a different method for searching for spinning binaries with precession. This method uses single-spin precessing waveforms as templates. Making clever use of maximization, it was shown in [START_REF] Pan | A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries[END_REF] that a PTF search could be performed with a four dimensional template bank: the two masses, the magnitude of the spin and the angle between the spin and the orbital plane. This method is especially useful for detecting neutron star-black hole binary (NSBH) systems, where the spin of the neutron star would have a negligible effect on the dynamics of the system [START_REF] Pan | A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries[END_REF]. A coincidence search utilizing the PTF waveforms has been developed [START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF]. Data from each instrument are analysed separately and only events observed with consistent time of arrival, mass and spin parameters in more than one detector are retained. While coincidence requirements for non-spinning searches are well known [START_REF] Robinson | A geometric algorithm for efficient coincident detection of gravitational waves[END_REF], it is less clear how to define coincidence when the additional spin parameters are present.

In a coherent search [START_REF] Abbott | Einstein@Home search for periodic gravitational waves in early S5 LIGO data[END_REF][START_REF] Pai | A data-analysis strategy for detecting gravitational-wave signals from inspiraling compact binaries with a network of laserinterferometric detectors[END_REF][START_REF] Bose | Detection of gravitational waves from inspiraling compact binaries using a network of interferometric detectors[END_REF][START_REF] Bose | Detection of gravitational waves using a network of detectors[END_REF][START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF] the data from all active detectors are combined together before searching for interesting events in the combined data. This circumvents the need for a coincidence test between events in different detectors. Furthermore, a coherent search offers an increased detection efficiency over the coincident technique when more than two detectors are active [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF]. The coherent technique is especially useful when the sky position is known, such as when searching for gravitational waves in coincidence with an electromagnetic transient, such as a gamma-ray burst (GRB) [START_REF] Abbott | Search for gravitational-wave bursts associated with gamma-ray bursts using data from LIGO Science Run 5 and Virgo Science Run 1[END_REF]. Since NSBH and binary neutron star (BNS) mergers are the preferred progenitor model [START_REF] Nakar | Short-hard gamma-ray bursts[END_REF] for short GRB, a coherent single-spin search is ideally suited to this source.

In this paper we describe the implementation of a coherent search for single spin binaries with known sky location, using the PTF waveforms. We briefly review the PTF formalism before deriving the coherent PTF signal-to-noise ratio (SNR). Due to the increased complexity of the spinning waveforms, the coherent SNR has a different distribution than its non-spinning counterpart. In particular, there is a greater chance of obtaining a large value of the spinning SNR, even in Gaussian noise. Thus, there is a trade-off between the improved spinning signal model and the increased false alarm rate at a fixed SNR. We explore the single-spin CBC parameter space to identify regions where spin (and precession) effects are significant enough to make the spinning search worthwhile. We will also briefly discuss some possibilities for vetoing background non-Gaussian transients in the data when using the PTF search and present results of this search run on a short stretch of LIGO's fourth science run (S4) data

The layout of this paper is as follows: In section 2 we briefly review the single detector PTF search and investigate the distribution of the spinning SNR in Gaussian noise. In section 3 we introduce the coherent PTF search and investigate the distribution of the coherent spinning SNR. In section 4 we identify regions of the parameter space where the PTF search offers increased sensitivity over the nonprecessing search. Section 5 briefly describes our search pipeline and the results of these methods applied to a stretch of data from S4.

Spinning Search Using Physical Template Family Waveforms

In this section we give a brief recap of the PTF search and its implementation. We also explore the expected distribution of the spinning SNR in Gaussian noise and compare this to that of the non-precessing search. For a more detailed description of the PTF search and terminology we refer the reader to [START_REF] Pan | A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries[END_REF][START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF]. We will follow the conventions of these earlier publications as much as possible. We will also assume that the reader is familiar with matched-filtering techniques and its application to gravitational wave data analysis, if not we refer the reader to [START_REF] Wainstein | Extraction of Signals from Noise[END_REF][START_REF] Allen | FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries[END_REF].

Single detector analysis

The likelihood ratio of there being a signal h present in the data s for a single detector is given by

Λ(h) = P (s|h) P (s|0) . (1)
Assuming the noise is Gaussian, the log likelihood can be written as

log Λ = (h|s) - 1 2 (h|h). (2)
Where we have defined the single detector inner product

(a|b) = 4 Re ∞ 0 ã(f )[ b(f )] S h (f ) , (3)
and S h (f ) is the noise power spectral density (PSD) of the detector. From this starting point, h must be re-expressed in such a way that it is possible to maximize over the majority of the parameters, leaving us with only a small number of dimensions over which to carry out a templated search. The dominant harmonic of the gravitational waveform can be expanded in terms of the five l = 2 spin-weighted spherical harmonics. The amplitude of each of these terms will depend upon the distance to the source, D; the sky location of the source, (θ, φ); the initial orientation of the source, which is described by three angles: the inclination, ι, and polarization, ψ, of the binary and the orientation of the spin in the orbital plane, ϕ. The waveform for each of these harmonics depends on the two masses, (M 1 , M 2 ); the amplitude of the spin, χ; the angle between the spin and the orbital plane, κ; the initial orbital phase relative to the spin direction, Φ 0 , and the time of coalescence, t c . Consequently the gravitational waveform for a single spin binary can be expressed as

h(t) = 5 I=1 P I (D, θ, φ, ψ, ι, ϕ)Q I (M 1 , M 2 , χ, κ, Φ 0 , t c ). ( 4
)
where P I are five amplitudes and Q I describe five waveform components.

To obtain the PTF detection statistic, a free maximization is carried out over the five P I components as well as the initial orbital phase of the system. The SNR will then depend on 10 components: the 0 and π 2 phases of the five Q I waveforms. Specifically, the inner product is calculated between each component Q I of the waveform and the data

A I = (s|Q I 0 ) and B I = (s|Q I π 2 ), (5)
as well as between the different Q I themselves ‡

M IJ = (Q I 0 |Q J 0 ) = (Q I π 2 |Q J π 2 ) . (6)
The maximized PTF detection statistic is given by [START_REF] Pan | A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries[END_REF][START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF]]

ρ 2 2 = log Λ| max = 1 2 A T M -1 A + B T M -1 B + 1 2 (A T M -1 A -B T M -1 B) 2 + (2A T M -1 B) 2 .( 7
)
The expression for the SNR, ρ, can be simplified by performing a transformation such that both

Q I 0 and Q I π 2
are orthonormal. First, perform a rotation on the Q I 0 to make M IJ diagonal, then normalize the basis vectors. We denote the orthonormal basis Q I 0 . This transformation will also orthonormalize

Q I π 2
and render M IJ the identity matrix. After this transformation, the SNR can be written as

ρ 2 = A • A + B • B + A • A -B • B 2 + 2 A • B 2 . (8)
A free maximization over the five P I amplitudes has been performed. In principle, these depend upon six physical parameters. However, these parameters only enter in four different combinations as

• an amplitude parameter, dependent on (D, θ, φ, ψ),

• the relative sensitivity of the instrument to the + and × polarizations, dependent on (θ, φ, ψ) • the inclination angle, ι • the spin orientation, ϕ.

Therefore, performing a free maximization over the five P I components means that the maximized P I values may not correspond to a physical set of parameters. This is discussed in [START_REF] Pan | A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries[END_REF] and various methods for projecting onto the physical sub-space have been proposed. For the case of an externally triggered search, where the sky location is known, the situation is unchanged as the P I are still described by the same four unknown parameters.

When the orbital plane of the system does not precess, there is gravitational wave emission in only two of the harmonics, Q 1 and Q 2 . The other components vanish identically. Furthermore, these two harmonics are related by a phase shift: Q 1 = iQ 2 . Thus, the matrix M is degenerate and the PTF maximization breaks down. It is, however, straightforward to maximize over the two remaining amplitudes, and obtain the SNR as

ρ 2 = (s|Q 1 0 ) 2 + (s|Q 1 π 2 ) 2 (Q 1 0 |Q 1 0 ) . ( 9
)
This is identical to the well known SNR for the non-spinning search [START_REF] Allen | FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries[END_REF], and the two phases of Q 1 correspond to the 0 and π 2 phases of the non-precessing template. ‡ We have made the standard assumption that

Q I 0 = iQ I π 2 .
The PTF search allows one to perform a search using single spin waveforms in a reasonable amount of time on a single detector [START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF]. Any event with an SNR above some preset threshold constitutes a single detector "trigger", and candidate events would be required to be observed in more than one detector. However, it remains a challenge to derive a metric on the four dimensional mass and spin space that could be used in generating a template bank and in defining coincidence requirements. Furthermore, a strategy for vetoing non-transient glitches has been suggested [START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF], such a strategy would be needed to make a coincident PTF search viable.

SNR distribution in Gaussian noise

The PTF template waveform will provide a better match than a non-spinning template to a gravitational wave signal from a spinning binary. However, we pay a price since we must filter the data against more waveform components, Q I , thereby increasing the chance of a spurious match with the noise. Additionally, the spinning SNR takes a more complex form (8) than the simple quadratic expression (9) when there is no precession. Here, we will investigate the SNR distributions in Gaussian noise for these two cases. In section 4 we use this to identify regions of parameter space with sufficient spin effects to warrant the use of the PTF search.

Ten filters are used in the calculation of the PTF detection statistic:

( Q I 0 , s) and ( Q I π 2 , s). As both Q I 0 and Q I π 2
are orthonormal, the only remaining freedom is the relation between the

Q I 0 and Q I π 2
terms,

N IJ = ( Q I 0 , Q J π 2 ). (10)
This N IJ is a will be a 5 × 5 antisymmetric matrix which can have 4 non-zero eigenvalues: ±λ 1 , ±λ 2 . The values of these eigenvalues determine the distribution of the PTF detection statistic.

For every NSBH waveform we have tested using the initial LIGO sensitivity curve, the magnitudes of λ 1 and λ 2 have been very close to 1. Thus, although there are ten different waveform components, we find that, in effect, only six of these are independent -the others are linear combinations of these six. There are then only six independent filters and it is not difficult to show that the spinning SNR (8) collapses to a quadratic form that is χ 2 distributed with six degrees of freedom, in Gaussian noise. This is the "best" case for the detection statistic. The "worst" case occurs when both λ 1 and λ 2 are zero and all ten of the filters are independent. In this case, the SNR expression cannot be simplified and its distribution does not correspond to a χ 2 distribution with 10 degrees of freedom as might be expected; the real distribution is somewhat more complex. Both best and worst cases are illustrated in Figure 1.

The SNR (9) for a non-precessing template follows a χ 2 distribution with two degrees of freedom in Gaussian noise. This is also plotted on Figure 1. By comparing the distributions of the PTF and non-precessing SNRs, it is clear that the background triggers produced by the PTF search will have, on average, considerably larger SNR than those produced by the non-precessing search. We explore the affect that this has on a search further in section 4.

Coherent Spinning PTF search

In this section we introduce a multi-detector, coherent formulation of the PTF search. As in [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF], we will restrict attention to the case where the sky location is known. This simplifies a coherent search as the sensitivity of the detectors to the two GW polarizations and the relative time delays between detectors are known. Astrophysically, this is of interest when searching for gravitational waves associated to electromagnetic transients such as GRBs [START_REF] Nakar | Short-hard gamma-ray bursts[END_REF][START_REF] Shibata | Merger of black hole and neutron star in general relativity: Tidal disruption, torus mass, and gravitational waves[END_REF].

The data from various detectors are combined together coherently to form two coherent data streams, with one stream containing the + polarization of any gravitational wave signal present in the network and the second containing the × polarization. The coherent method will offer an improvement in sensitivity over the coincidence method when more than two detectors are used, as only two data streams are searched. For networks with greater than two detectors, it is also possible to construct null streams that will contain no gravitational wave signal. These can be used to construct a consistency test [START_REF] Guersel | Near optimal solution to the inverse problem for gravitational wave bursts[END_REF][START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF].

We begin by formulating the coherent SNR for the spinning PTF search and go on to explore how this will be distributed in Gaussian noise.

Coherent PTF Search Method

To formulate a coherent detection statistic for the PTF templates we draw on many of the methods and techniques that were used in deriving the single detector statistic described in section 2.1 and derived in detail in [START_REF] Pan | A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries[END_REF][START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF]. We follow the conventions of [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF] in extending this to a coherent, multi-detector search. Assuming that the noise in different detectors is independent, the multi-detector likelihood is given by

ln Λ = (h|s) - 1 2 (h|h), (11)
where we have defined a multiple detector matched filter

(a|b) = X (a X |b X ); (12)
the index X runs over the detectors in the network.

As before, we want to maximize over as many of the parameters as possible to minimize the dimension of the required template bank. We start by maximizing this over the distance, D, and initial orbital phase, Φ 0 , to obtain § ln Λ| max(D,Φ0) = 1 2

X,I P X I A X I 2 + X,I P X I B X I 2 Y,J,K P Y J P Y K (Q J 0 |Q K 0 ) Y , (13)
where P I are the amplitudes of the various waveform components Q I , and A I X , B I X are defined as in [START_REF] Grandclement | Searching for gravitational waves from the inspiral of precessing binary systems. I: Reduction of detection efficiency[END_REF]. Although the P I depend upon D the maximized likelihood is independent of it as scaling the distance has an identical effect on both the numerator and denominator of eq. ( 13).

As in the single detector case, we would like to maximize over the P I to eliminate them. However, in the multi-detector case, they are detector dependent since the sensitivity of the detectors to the + and × gravitational wave polarizations will differ. These sensitivities are encoded in the detector response functions, F + and F × , which depend on the sky location of the source in the detector frame. As we are focusing on an externally triggered search, where the sky location is known, these values will be known for each detector. We can then factor the detector dependent terms out of the P X I as

P X I = F X + (θ, φ)S I (D, ι, ψ, ϕ 0 ) + F X × (θ, φ)T I (D, ι, ψ, ϕ 0 ) ( 1 4 )
where S I and T I denote the amplitude of the + and × components respectively of the 5 Q I in the radiation frame. They depend on the distance, D and the angles (ι, ψ, ψ 0 ) that describe the rotations necessary to transform from the source frame to the radiation frame. We can re-cast the log-likelihood into a form which more closely resembles the single detector case by introducing ten-dimensional analogues of the P I and Q I by defining

P α := [S 1 , S 2 , S 3 , S 4 , S 5 , T 1 , T 2 , T 3 , T 4 , T 5 ] Q α 0, π 2 := F + Q 1 0, π 2 ; ...; F + Q 5 0, π 2 ; F × Q 1 0, π 2 ; ...; F × Q 5 0, π 2 . ( 15
)
The change to ten dimensions naturally arises because a multiple detector coherent network is sensitive to both the + and × components of a gravitational wave, whereas a single detector network is only sensitive to one polarization. We also define the multi-detector inner products between signal and waveform components

A α = (s|Q α 0 ) and B α = (s|Q α π 2 ) M αβ = (Q α 0 |Q β 0 ) = (Q α π 2 |Q β π 2
).

The log likelihood equation can then be written as ln Λ| max(D,Φ0) = 1 2

P α P β A α A β + B α B β P α P β M αβ . ( 16
)
We proceed, as before, by transforming to an orthonormal basis

Q α 0 , Q α π 2
for the waveform components. Then, maximizing freely over P α yields the coherent PTF SNR

ρ 2 coh = A • A + B • B + A • A -B • B 2 + 2 A • B 2 , ( 17
)
§ The Y subscript in the inner product in the denominator denotes the fact that the PSD of detector Y is used in evaluating the inner product. We do not require the noise PSDs of the different detectors to be the same where, as before, the tilde denotes that we are in the orthonormal basis. This is very similar in form to the single detector statistic in equation [START_REF] Allen | FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries[END_REF]. When the network is only sensitive to one polarization, the matrix M αβ becomes degenerate and the maximization procedure must be re-visited. Here it is natural to remove all terms corresponding to the second polarization and reduce to 5 dimensions, as in the single detector search. Additionally, in section 2.1 we noted that when the template has no precession the single detector PTF SNR collapses to the familiar SNR formalism used in the non-spinning search. Similarly, in the coherent PTF search, when the template has no precession, the coherent SNR will collapse to the non-spinning coherent SNR given in [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF].

The coherent SNR of equation ( 17) can be used as a detection statistic in performing a coherent search using PTF templates, as we explore in section 5. In the single detector search, we maximized freely over five P I which were dependent upon four physical parameters. In the coherent search, the P α still depend on only 4 parameters but we are now maximizing over ten amplitudes. This clearly introduces a lot of unnecessary, and unphysical, freedom. However, we should note that a naive coincidence search, in which the P I are maximized independently for each detector, would allow for a similar freedom as. Consequently, for a network with three or more detectors, the coherent search provides a sensitivity improvement. We are currently investigating alternative methods of constructing the coherent SNR which might eliminate these un-physical degrees of freedom. Once the filters A and B have been calculated, it might be possible, using a combination of analytical and numerical techniques, to marginalize over physically realizable values of the P α . The use of marginalization would also make it possible to down-weight "unlikely" sets of physical parameters, for example binaries at small distances with poor alignment.

SNR distribution in Gaussian noise

In section 2.2 we explored how the single detector PTF statistic is distributed in Gaussian noise. For the coherent PTF search we can use a similar strategy to investigate the distribution of the coherent SNR. In the coherent case there are twenty filters A α and B α and we have constructed the detection statistic such that Q α 0 and Q α π 2 are orthonormal. As before, the only freedom is the relationship between the 0 and π 2 terms encoded in

N αβ = ( Q α 0 , Q β π 2 ). ( 18
)
This is a 10 × 10 antisymmetric matrix comprised of four 5 × 5 blocks, each of which is antisymmetric. Therefore this matrix can have 8 non-zero eigenvalues: ±λ 1,2,3,4 . These eigenvalues determine the distribution of coherent SNR in Gaussian noise -for smaller eigenvalues, the large SNR tail of the distribution becomes more significant. In the tests that we have performed, using the initial LIGO sensitivity curve and NSBH precessing templates, all four eigenvalues give values close to unity; the "best" case in which there are 12 independent waveform components. However, the distribution does not collapse to a χ 2 . In Figure 2, we demonstrate that this gives a distribution similar to a χ 2 distribution with 12 degrees of freedom. In the "worst" case, where all of the eigenvalues are equal to 0, there are 20 independent waveform components and this distribution is also shown in Figure 2.

Identifying where the PTF search is most beneficial

In sections 2 and 3 we have derived the spinning SNR that can be used to perform a gravitational wave search using single spin inspiral waveforms as templates. We have demonstrated that, on the average, background triggers will have larger values of SNR in the PTF search than in the non-precessing search. At the same time, precessing PTF waveforms will be a better match to any spinning, precessing signals in the data. This begs the question as to whether it is preferable to use a search with nonprecessing waveforms or single spin PTF waveforms to detect precessing systems. The PTF triggers will match the waveform better but this comes at the cost of searching a larger parameter space.

To quantify this, in Table 1 we give the SNR that corresponds to a false alarm probability (FAP) of 10 -10 in Gaussian noise for the various searches. We chose this value because it roughly corresponds to the loudest background events we observe when running the search on 2000 seconds of Gaussian noise, as is appropriate for a GRB search. The figures in the table show that the PTF search must obtain 26% more signal power (SNR squared) to be more efficient in the single detector case at this FAP and 38% more signal power for the coherent case.

There are large areas of the parameter space where precession will not significantly affect the evolution of the binary and thus a non-precessing template will pick up the majority of the power in a precessing signal. In these areas it would be better to search for the spinning signal with a non-precessing template, achieving a lower FAP than for the PTF search using an exactly matching template. Equivalently, when a system has little precession, the majority of power is contained in the Q 1 and Q 2 components of the PTF waveform and these two components are very similar, up to an overall phase shift. We can then consider performing a "restricted PTF" search, where we filter only these two components of the waveform against the data. This serves to reduce the FAP at a fixed SNR while losing only a small amount of the power in the signal.

To do this, we test every template waveform, before filtering, to determine whether the template would be more likely to detect a matching signal below a false alarm probability of 10 -10 using the restricted or full PTF search. This can be calculated by simulating a large number of gravitational wave signals, with masses and spin matching those of the template, uniformly distributed in volume and orientation. Then, simply count the number of simulated signals expected to give an SNR greater than the value corresponding to a FAP 10 -10 (given in Table 1) for both methods. Whichever of the PTF or restricted methods is expected to perform better is then used when filtering the data with that template. Using this method, we are able search the full parameter space of NSBH binaries in a single search, including non-spinning, nonprecessing, marginally precessing and fully precessing configurations. This method works equally well for the single detector or the coherent search.

In Figure 3 we illustrate the fraction of templates analysed by the full PTF statistic, as a function of the masses, for the coherent search. The splitting of the templates into full and restricted does not require filtering against the data, but it does make use of the PSDs of the detectors. For this study, we use data from the three LIGO detectors during the S4 run.

A template bank was generated by taking a standard non-spinning template bank [START_REF] Cokelaer | Gravitational waves from inspiralling compact binaries: hexagonal template placement and its efficiency in detecting physical signals[END_REF] in the mass space and, for each value of the masses, creating 15 templates with identical masses but spin parameters gridded over the two dimensional spin space, as described in [START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF]. The precessing single spin templates are most needed in the high mass ratio region of the parameter space. For this template bank, there are 35395 templates to be analysed with the restricted method and 14660 templates to be analysed with the full PTF method.

Search method and example results

In section 3 we derived a detection statistic appropriate for a coherent search using the PTF waveforms as templates. In section 4 we described a method with which one can identify where the PTF search is most needed and to split a template bank into those templates that should be analysed with the full PTF statistic and those that should be analysed with the restricted PTF statistic. We have combined these two methods together to create a search pipeline that can be used to coherently analyse gravitational wave data to search for precessing NSBH signals associated to short GRBs. We will briefly describe the analysis procedure before presenting an example result.

The search uses much of the same architecture as that described in [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF] and [START_REF] Abadie | Search for gravitational-wave inspiral signals associated with short Gamma-Ray Bursts during LIGO's fifth and Virgo's first science run[END_REF]. Namely we search for gravitational wave signals in the "on-source" time, defined to be [-5,+1) seconds around the reported time of the GRB. Background is estimated from performing 324, trials of 6s duration around the GRB time, but at least 48s away from the on-source time. The coherent PTF search makes use of the same infrastructure as a coherent non-spinning search described in [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF]. In particular, the data handling, PSD estimation and matched filtering routines are the same. Of course, the coherent PTF search makes use of spinning, precessing waveforms in the filtering and computes the SNR given in equation [START_REF] Robinson | A geometric algorithm for efficient coincident detection of gravitational waves[END_REF].

To demonstrate the performance of the coherent PTF, we ran it over a stretch of data from LIGOs S4 run. The data was chosen randomly, subject to the condition that all three of the LIGO detectors were operational at the time. This is the same data as was used to illustrate the template bank splitting in section 4, and the same bank with 15,000 full PTF and 35,000 restricted PTF templates was used.

In Figure 4 we show the distribution of the SNR of the triggers produced using both the full and restricted statistic. This is shown for the stretch of real S4 data and for a stretch of simulated Gaussian data. As expected, the SNRs of triggers in Gaussian noise are larger for the precessing templates than the restricted ones, even though significantly fewer templates were analysed with the full statistic. The results from real data are badly affected by non-Gaussianities in the data. A number of loud transients are clearly visible as short duration peaks of large SNR, while there are an even greater number of quieter peaks throughout the analyzed time. This has a similar effect on both the full and restricted waveforms.

In [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF], we described and developed a number of tools which can be used to effectively remove the majority of the non-Gaussian features from a non-spinning, coherent analysis. These include null stream consistency [START_REF] Guersel | Near optimal solution to the inverse problem for gravitational wave bursts[END_REF], amplitude consistency and χ 2 signal consistency tests [START_REF] Allen | A χ 2 time-frequency discriminator for gravitational wave detection[END_REF][START_REF] Hanna | Searching for gravitational waves from binary systems in non-stationary data[END_REF]. All of these can be applied without modification to the restricted PTF search, and it seems reasonable to expect they would be similarly effective in reducing the effect of non-Gaussianities in the data. However, we currently have no such tools which can be used for the full PTF waveforms. Before using this search on real data we will need to implement a set of tests that can discriminate glitches from real signals for the full statistic. It should be relatively straightforward to implement the null stream consistency test. Unfortunately, as discussed in [START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF], the null stream for the LIGO S4 detectors is constructed only from the two instruments in Hanford. In this stretch of data the loudest background triggers are caused by non-stationarities in the Livingston detector and thus the null stream is ineffective. Alternatively a χ 2 test such as the ones described in [START_REF] Allen | A χ 2 time-frequency discriminator for gravitational wave detection[END_REF][START_REF] Hanna | Searching for gravitational waves from binary systems in non-stationary data[END_REF] could be adapted to this search, [START_REF] Fazi | Development of a physical-template search for gravitational waves from spinning compact-object binaries with LIGO[END_REF] presents a possible way of doing this for single detectors. We are working on developing an alternative version of this χ 2 test, which would test the consistency of the six independent components of a single detector PTF waveform, and then extending this to the coherent analysis. The left panels show the distribution of triggers from templates that were analysed using the "restricted" coherent PTF search, the right panels show the distribution from the templates that were analysed using the "full" coherent PTF search. The top panels were created from analysing Gaussian noise. The bottom panels were created from analysing a stretch of real data from S4. All these plots have been rescaled to use the same y-axis. For the two cases using real data the non-Gaussian spikes extend much higher than is shown, the loudest trigger has an SNR of 39 in the restricted case and 45 in the full case.

Discussion

In this paper we have presented a method for performing a coherent search for precessing, single spin black hole-neutron star coalescences using PTF template waveforms. We have compared the performance to searches using non-precessing waveforms and have identified regions of the parameter space where the PTF search offers increased sensitivity. We have presented a method by which these areas could be identified and demonstrated these techniques on a short stretch of S4 data.

This method should allow for the detection of highly precessing NSBH systems with greater efficiency than the current non-spinning searches. However, more work is required before this search is ready to be used. The main need is for the development of effective methods of separating glitches from real events in the full PTF search, whether performing a coincident or a coherent search. As discussed in section 5, it should be possible to adapt a lot of the methods that have proven effective in nonspinning searches [START_REF] Allen | FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries[END_REF][START_REF] Harry | A targeted coherent search for gravitational waves from compact binary coalescences[END_REF] but this is a non-trivial task.

We also demonstrated that the coincidence and coherent PTF statistics allow for a large degree of unphysical freedom when maximizing over the extrinsic parameters. The efficiency of a PTF search could be increased if this unphysical freedom is reduced or removed. We are looking into methods for achieving this, including marginalizing the likelihood over the physical parameters of the system. In this paper, we have focused on the PTF precessing waveforms. However, many of the techniques we have discussed would be equally applicable to the dominant harmonic of any family of precessing waveforms. In particular, the method of maximizing over freely over the amplitudes of the five components of the l = 2 spin weighted-spherical harmonic is directly applicable to other waveform families. As the catalogue of numerical simulations of precessing binaries grows, these methods may well find applications in searches using such as numerical relativity inspired inspiralmerger-ringdown waveforms.

Figure 1 .

 1 Figure1. The "best" and "worst" possible distributions of the single detector PTF SNR squared, this is compared with the non precessing SNR squared.

Figure 2 .

 2 Figure 2. The "best" and "worst" possible distributions of the coherent PTF SNR squared as well as the distribution of the non spinning SNR squared.

Figure 3 .

 3 Figure 3. The fraction of templates analysed by the full PTF statistic as a function of the masses in the NSBH region of the parameter space.

Figure 4 .

 4 Figure 4. The distribution of triggers found by the coherent PTF search in the off-source data.The left panels show the distribution of triggers from templates that were analysed using the "restricted" coherent PTF search, the right panels show the distribution from the templates that were analysed using the "full" coherent PTF search. The top panels were created from analysing Gaussian noise. The bottom panels were created from analysing a stretch of real data from S4. All these plots have been rescaled to use the same y-axis. For the two cases using real data the non-Gaussian spikes extend much higher than is shown, the loudest trigger has an SNR of 39 in the restricted case and 45 in the full case.

Table 1 .

 1 The SNR corresponding to a FAP of 10 -10 for the non precessing and the PTF search, for both coherent and single detector cases. Here, for the PTF case the single detector and coherent detection statistics are assumed to be χ 2 distributions with 6 and 12 degrees of freedom respectively.

		Non Precessing PTF
	Single detector	6.79	7.63
	Coherent	7.26	8.53

Acknowledgements

We would like to thank Duncan Brown, Sukanta Bose, Diego Fazi, Mark Hannam, Drew Keppel, Andrew Lundgren, Ilya Mandel and Bangalore Sathyaprakash for useful discussions and comments on this manuscript. We would also like to thank the anonymous referees for carefully reading the manuscript and providing useful feedback and comments. IWH was funded by the Science and Technology Facilities Council, UK, studentship ST/F005954/1, SF would like to acknowledge the support of the Royal Society.