A study of the effect of term proximity on query expansion Query expansion terms are often used to enhance original query formulations in document retrieval. Such terms are usually selected from the entire documents or from windows or passages surrounding query term occurrences. Arguably, the semantic relatedness between terms weakens with the increase in the distance separating them. In this paper we report a study that was conducted to systematically evaluate different distance functions for selecting query expansion terms. We propose a distance factor that can be effectively combined with the statistical term association measure of mutual information for selecting query expansion terms. Evaluation of the TREC collection shows that distanceweighted mutual information is more effective than mutual information alone in selecting terms for query expansion. Query expansion (QE) is a technique commonly used in information retrieval (IR) [1, 2] to improve retrieval performance by reformulating the original queryeither adding new terms or reweighting the original terms. Query expansion terms can be automatically extracted from documents or taken from knowledge resources, such as thesauri. The advantages of the former techniques are that expansion terms are extracted from the collection, so there is no vocabulary mismatch problem; and secondly, no expensive human-constructed knowledge bases are required. Typically either top-ranked documents in the initially retrieved document set (blind or pseudo-relevance feedback) or documents judged relevant by the user in the retrieved set (relevance feedback) are used to extract query expansion terms. For short and incomplete queries, a substantial improvement can be achieved by using expanded queries [3, 4] . Query expansion terms ideally should have the following characteristics: (a) be semantically related to the original query terms; (b) be good at discriminating between relevant and non-relevant documents. Different approaches to selecting query expansion terms following relevance or pseudo-relevance feedback have been proposed. For instance, in a probabilistic model of IR [4] query expansion terms are selected on the basis of how well they can discriminate between the relevant and non-relevant documents. There have also been attempts to select query expansion terms on the basis of the strength of their association with the original query terms. There exist several statistical association measures to identify significant word associations (also referred to as co-occurrences or collocations) and these measures are commonly used in corpus linguistics to estimate the degree of semantic Term proximity closeness between words. Such measures include the mutual information (MI), t-score, log-likelihood and chi-square tests [5] . These measures have been used to identify significant word collocations in a variety of applications, such as multiword unit extraction [6] , speech recognition [7] , information retrieval [8] , lexicography and lexical analysis [9] , word sense disambiguation [10] , and analysis of aligned corpora [11] . A few studies have been conducted to evaluate the use of term association measures in query expansion to select words that are closely related to query terms [12, 13] . A common approach in these studies is to select query expansion terms from either the entire document or from sections of the document (passages or windows) surrounding the occurrences of query terms. One disadvantage of such approaches is that a decision needs to be made about the span of text surrounding the occurrences of query terms from which query expansion terms are extracted. Span lengths are usually determined empirically, and the optimal span length may vary depending on the collection. Also, within these spans, the association scores of query expansion terms are not affected by how far they are from the original query terms. A study by Beeferman [14] demonstrates, however, that the strength of association between words decays exponentially with the increase in distance. It is, therefore, possible that more accurate estimates of the strength of association between two words can be achieved, if term association measures are combined with a decaying distance factor.. The term 'collocation' has been used differently in the literature. For example, Manning and Schütze understand word collocations as grammatically bound elements occurring in a certain order and having limited compositionality [5] . On the other hand, Hoey [15] understands collocation as long-distance relationships between semantically related words. We recognize the existence of two types of collocation:. • Collocation due to lexical-grammatical or habitual restrictions. These restrictions limit the choice of words that can be used in the same grammatical structures. Collocations of this type occur within short spans, i.e. within the bounds of a syntactic structure, such as a noun phrase (e.g. 'rancid butter', 'white coffee', 'mad cow disease', 'stainless steel').. • Collocation due to a typical occurrence of a word in a certain thematic environment: two words hold a certain lexical-semantic relation, i.e. their meanings are close semantically, therefore they tend to occur in the same topics in texts. This type of collocation may span over longer distances in text than collocations of the previous type. Examples of some of the collocates of this type, identified using the Z-score statistic, are 'nitrogenpollution', 'school-education' [12] . In our approach to the selection of query expansion terms, no explicit distinction is made between these two types of collocates, therefore collocates of both types can potentially be selected for query expansion using the proposed method. However, we assume that the closer a word occurs to the user's query term in a text, the more likely it is related to it semantically; and we propose a method that rewards words co-occurring closer to the instances of the original user's query terms.. Term proximity has been explored extensively in document ranking studies [16] [17] [18] [19] [20] , where several distance factors were proposed. Two common intuitions underlie these approaches: (1) the closer the terms are in a document, the more likely they are topically related; and (2) the closer the query terms are in a document, the more likely the document is relevant to the query. The reasoning behind these intuitions is that a document may contain different topics or talk about different aspects of the same topic, therefore, if terms occur close to each other they are more likely used in related senses and discuss the same subject than is the case if they are located in different parts of the document. For example, Clarke et al. [16] proposed a technique of scoring documents based on term proximity and density. They introduced the notion of 'cover', which is the shortest span of text containing instances of all query terms. Document score is calculated based on two assumptions: (1) the shorter the cover, the more likely the corresponding document is relevant; and (2) the more covers are in a document, the more likely the document is relevant. Evaluation on the Text REtrieval Conference (TREC) data set proved the effectiveness of the method compared to some standard approaches to QE. A similar technique was proposed by Hawking and Thistlewaite [17, 18] , which also demonstrated promising results on the TREC data set. Approaches to document ranking based on term proximity are particularly suitable for distributed information retrieval systems, as they do not rely on collection-wide statistics such as inverse document frequency and average document length.. A related approach that utilizes term proximity implicitly in query expansion is the use of contextindependent or query-biased summaries for the selection of query expansion terms. For example, Lam-Adesina and Jones [21] have evaluated querybiased and context-independent summarization techniques for QE following pseudo-relevance feedback and achieved improvements over standard QE methods using complete documents on the TREC data set.. Gao et al. [22] proposed a combined measure of mutual information with an exponential decaying distance factor, which they evaluated for query translation tasks in CLIR (Cross-language IR). Their approach will be discussed in more detail in Section 2.2. The goal of our study is to systematically evaluate a number of decaying distance factors in combination with the term association measure of mutual information, and their effectiveness in query expansion. We propose a new decaying distance function that shows improved performance over the distance factor proposed by Gao et al. [22] .. The following two hypotheses were formulated and investigated in this study: Hypothesis 1. Query expansion terms ranked by a combination of mutual information with a decaying distance factor lead to a significant performance improvement over the original query terms. Hypothesis 2. Query expansion terms ranked by a combination of mutual information with a decaying distance factor lead to a significant performance improvement over the query expansion terms ranked by mutual information alone.. The rest of the paper is organized as follows: in Section 2 we present the experiments conducted in this study, Section 3 contains the analysis of results, and Section 4 concludes the paper and outlines future work. In this study, motivated by earlier studies on term proximity ranking [16, 17, 20] , experiments on query expansion [1, 3, 4, 12, 28] , and word association models [14, 22, 29] , we systematically investigated the use of term proximity information for the ranking and selection of query term collocates as query expansion terms. Through continuous improvement of the distance factor in the collocate-weighting formula we developed a query expansion term selection method, which shows significant performance gains over the use of original user queries. Combining collocation distance, collocation frequency and mutual information helps select better query expansion terms than the use of MI alone. The difference between the two methods is statistically significant in all but one measure (precision at 5 documents). The evaluation conducted in this study suggests that distanceweighted MI is an effective query expansion technique. The proposed distance-weighted MI function might also be useful for other applications, such as word sense disambiguation and lexicography, which rely on word association measures. However, more task-based evaluations need to be conducted.. Other possible avenues for improvement of the query expansion term selection that could be explored are:. • Use of Part-Of-Speech (POS) information. Jing and Croft [30] studied the effect of different POSs on the selection of word co-occurrences, and found that noun phrases achieved the best performance. Xu and Croft [28] only used noun phrases for query expansion. Evert and Kermes [31] also suggested applying linguistic filters before extracting collocations. A method using noun phrases only in the query expansion may improve retrieval performance.. • Use of the number of distinct relevant documents in query expansion term selection. Collocates located in the proximity of query terms are promising expansion terms as shown in this paper. However, words co-occurring with query terms in several relevant documents are likely to be more useful than words co-occurring with query terms in only one relevant document. Therefore, taking into account the number of relevant documents that a word pair co-occurs in may improve the selection of query terms.. Journal of Information Science, 00 (0) 2006, pp. 1-10 © CILIP, DOI: 10.1177/0165551506065787

O O O O O O B I O B I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O B I O O O O B I O O O O O B I B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Rank-Metric Approach to Error Control in Random Network Coding Abstract-The problem of error control in random network of a basis for V, where each basis vector corresponds to a coding is considered, and a formulation of the problem is given transmitted packet. As V is closed under linear combinations in terms of rank-metric codes. This formulation allows many of of vectors random network coding will (in the absence of the tools developed for rank-metric codes to be applied to random . network coding. A random network code induces a generalized noise) preserve the space, even as it mixes the transmitted decoding problem for rank-metric codes in which the channel vectors themselves. A receiver collects packets, which are may supply partial information about the error in the form of assumed to form a basis for a received space U. Correct erasures (knowledge of an error location not its values) and reception is possible provided that U and V intersection deviations (knowledge of an error value but not its location), in a space of sufficiently large dimension. [5] and addresses it is highly susceptible to errors. The insertion of even a the construction of practical codes. Our main contribution single corrupt packet has the potential, when linearly combined is to show that, for a large class of codes, the subspace with legitimate packets, to affect all packets gathered by an distance metric of [5] and the rank metric (e.g., [6], [7]) information receiver. The problem of error control in random are strongly related, allowing many of the tools from the network coding is therefore of great interest.theory of rank-metric codes to be naturally applied to random In this paper, we focus on end-to-end error control coding. network coding. We note, however, that our approach is not Internal network nodes are assumed to be unaware of the pres-a straightforward application of rank-metric codes. Under ence of an outer code; they simply create outgoing packets as random network coding, two phenomena can occur, called random linear combinations of incoming packets in the usual here erasures and deviations, that depart from the conventional manner of random network coding. Unlike some approaches notion of rank errors. Erasures and deviations are dual to each to error control in network coding (e.g., [2] , [3]) we assume other and correspond to partial information about the error that the source and destination nodes have no knowlege or matrix, akin to the role played by symbol erasures in the at least make no effort to exploit knowledge of the topology Hamming metric. In our context, an erasure corresponds to of the network or of the particular network code used in the the knowledge of an error location but not its value, while a network.deviation correspond to the knowledge of an error value but Two previous "noncoherent" or "channel-blind" approaches not its location. These concepts generalize similar concepts to data transmission in coded networks are closely related to found in the rank-metric literature under the terminology of the work of this paper. Jaggi et al. [4] provide polynomial-"row and column erasures" [8]-[10]. In related work [11], we time rate-optimal network codes that combat Byzantine adver-provide an efficient decoding algorithm for Gabidulin codes saries. Their approach is based on probabilistic assumptions [7] that takes into account erasures and deviations. However, that require both the field size and the packet length to be space limitations make it impossible to describe these results sufficiently large. In contrast, Koetter and Kschischang [5] here. take a more combinatorial approach to the problem. Their keyThe remainder of this paper is organized as follows. In observation is that a random coded network may be regarded Section II, we provide a brief description of the rank metric as a noncoherent multiple-input multiple-output system (over and its properties. In Section III, we provide a formulation a finite field). In this context, an appropriate encoding of of the problem of error control in random network coding. information is the choice by the transmitter of a suitable vector Finally, in Section IV, we demonstrate the strong connection space V, rather than a vector as in classical coding theory. The between certain decoding problems in the rank metric and the choice of the space V is signalled by insertion into the network subspace decoding approach of [5]. Proofs have been omitted 1-4244-1200-5/07/$25.00 ©2007 IEEE.

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Graph-theoretic formalization of hybridization in DNA sticker complexes Abstract Sticker complexes are a formal graph-based data model for a restricted class of DNA complexes, motivated by potential applications to databases. This data model allows for a purely declarative definition of hybridization. We introduce the notion of terminating hybridization, which intuitively means that only a finite number of different products can be generated. We characterize this notion in purely graphtheoretic terms. Under a finite alphabet, each product is shown to be of polynomial size. Yet, terminating hybridization can still produce results of exponential size, in that there may be exponentially many different (nonisomorphic) finished products. We indicate a class of complexes where hybridization is guaranteed to be polynomially bounded. Since Adleman's experiment (Adleman 1994) , DNA computing has greatly evolved, and many different modes of computation have been invented and investigated (Amos 2005; Benenson et al. 2004; Boneh et al. 1996; Dirks and Pierce 2004; Paun et al. 1998; Qian et al. 2011; Rothemund 1996 , Roweis et al. 1998 , Sakamoto 1999 , Seelig et al. 2006 , Soloveichik et al. 2010 Soloveichik and Winfree 2005; Winfree et al. 1998) . A major goal throughout this evolution has been to achieve autonomy of computation: computation without human or machine mediation, e.g., by self-assembly or strand displacement.. At the same time, DNA computing has also high potential for database applications (Arita et al. 1997; Chen et al. 2005; Reif 2002; Yamamoto 2006) . Indeed, the nanoscale and relative indestructibility of single DNA strands are very promising properties for database storage. Moreover, the highly parallel mode of operation that can be achieved in DNA computing is a nice match with the bulkprocessing nature of database computations.. Autonomy of computation is perhaps less crucial for databases, where indeed traditionally a strict line is drawn between the data, and the query or update operations performed on the data (Garcia-Molina et al. 2009 ). Also, in database theory (Abiteboul et al. 1995) , one expects formal data models defined on the logical level, and declarative definitions of the basic data manipulation operations.. In the present paper, in the context of a formal data model of DNA complexes, we focus on hybridization, one of the cornerstone operations in DNA computing. The data model is that of sticker complexes, a graph-theoretically defined formalization of DNA complexes of a limited format. Sticker complexes have been shown in an earlier paper (Gillis and Van den Bussche 2012) to be adequate for database computations in DNA@. Indeed, while it is relatively straightforward to represent relational databases in DNA, a good data model for database computation must also be able to represent all intermediate data structures needed to support database operations. Specifically, it has been shown that sticker complexes are adequate to support a complete simulation of the operations of the relational algebra, which provides a set of core operations in relational databases (Garcia-Molina et al. 2009 ). The intermediate data structures involved in the simulation of the relational algebra are quite complex as they need to support the creation of circular strands.. The problem addressed in the present paper is to understand the well-definedness and termination of the hybridization operation on sticker complexes. Here we are considering hybridization as a database operation, like the Cartesian product (related to the relational join). When we want to construct the Cartesian product U 9 V of two sets U and V, with U of size m and V of size n, we need in principle n copies of every element of U, and m copies of every element of V, so that we have enough ''material'' to construct the m 9 n-element set {(u, v) 9 u [ U & v [ V}. When more copies are provided of some elements of U or V, some duplicate pairs can be constructed, but no really new information is generated. When hybridization has this behavior, we say it terminates.. The main result of this paper is to provide a purely graphtheoretic characterization of termination of hybridization, which will also imply that termination is decidable for sticker complexes. This result emphasizes the restricted nature of the sticker complex data model, since it is well known that termination is undecidable for Turing-universal computation models (Hopcroft and Ullman 1979) . The investigation of computation models that are not computationally complete, and the corresponding search for the right balance between sufficient expressive power and low complexity, is one of the hallmarks of database theory (Abiteboul et al. 1995) .. We also investigate complexity issues related to DNA hybridization. Even when hybridization in a given DNA complex terminates, depending on the structure of the complex, an exponential amount of material may be required to produce the complete result. This problem was already present in Adleman's solution to the Hamiltonian Path problem (Hartmanis 1995) , and we show it can still occur within the limited context of sticker complexes. Since such exponential behavior is undesirable, and also not needed to support typical database operations, we would like to avoid it.. We will show that the result of hybridization splits up, graph-theoretically, in a number of connected components, and each component is polynomial in size. Hence, the exponentiality is confined to the possible number of distinct components. Furthermore, we identify a broad family of classes of DNA complexes, called c-bounded complexes, within which hybridization is guaranteed to require only a polynomial amount of resources.. An extended abstract of this paper, without proofs, was presented at DNA 17 (Brijder et al. 2011) . A natural extension of our approach would be to account for probabilities or error rates on the results produced (finished or unfinished) during hybrization. Of course, error modeling in DNA computation, and secondary structure prediction, are well-known research problems, e.g., (Chen and Kao 2010; Dimitrov and Zuker 2004) . In previous work (Gillis and Van den Bussche 2012) two of us have defined a database-oriented DNA programming language, called DNAQL, with the goal of understanding the database side of DNA computing. Various open problems remain in connection with this language, including guaranteeing well-definedness through a type system, and understanding the expressive power.. Obviously, we would also like to see the sticker complex data model justified physically (or understand what are the unrealistic aspects), either experimentally or by simulation. In one of the first papers on DNA computing, Reif already defined a formal data structure of DNA complexes (Reif 1999) . Our data structures are simpler in an effort to avoid unrealistic or otherwise complicated and unmanageable secondary structures. (Reif avoids these by invoking an oracle for feasibility.) Our simplification is that single strands are either all-positive or all-negative, and moreover, negative strands have length at most two 1 . The short negative strands can be thought of as stickers; thus the name ''sticker complexes''. Our previous work showed that the restrictions of sticker complexes do not preclude interesting database computations. An important feature of our model, which is lacking in Reif's, is the formal distinction between the structural content of a complex, and the complex as used in reactions, with multiples of each connected component present in surplus quantities.. The use of short stickers in DNA computing originates with Roweis et al. (Roweis et al. 1998) , where stickers were used to turn bits on or off. We use stickers to bind strands together so that possibly complex secondary structures are formed.. The present work also fits in a recent trend of integrating formal methods [such as process calculi in computational systems biology (Cardelli 2005) ] with DNA computing (Cardelli 2009, Majumder and Reif 2009 ). Yet the formalisms we use are different from process calculi and comprise mainly set theory, graph theory, and logic-based query languages. The computational power of hybridization in various models of formal languages has been intensively studied, e.g., (Paun et al. 1998; Winfree et al. 1998 ).. Last but not least, our formal model of hybridization is strikingly comparable in spirit to the model of Jonoska, McColm and Staninska (Jonoska et al. 2011) . That paper introduces the notion of a ''pot type''; for the purpose of hybridization, the model of pot types and our model of sticker complexes are roughly equivalent (Brijder et al. 2011) . It was shown that weak satisfiability is decidable in polynomial time. Using our own terminology as introduced in Sect. 4, a complex is weakly satisfiable if it admits at least one ''finished'' component. In contrast, we show that termination is decidable in polynomial time. Termination is a stronger property than weak satisfiability; a terminating complex is weakly satisfiable, but a complex may be weakly satisfiable without terminating. Jonoska et al. also define the notion of stable complex, which corresponds to our notion of maximal matching as defined in Sect. 4.. In an earlier paper, Jonoska and McColm considered the problem of given a pot type and a bound b, to decide whether there exists a finished component of size b (Jonoska and McColm 2009) . They showed that nondeterministic-time complexity classes can be captured in this manner. In comparison, in Sect. 6 of the present paper, we give complexity results of a very different kind. Specifically, we show that the size of any finished component of a terminating pot type is polynomially bounded. Moreover, for a restricted class of pot types, called c-bounded, we show that also the number of different finished components is polynomially bounded.

O O O B O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O B B O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


An AIG-Based QBF-solver using SAT for preprocessing In this paper we present a solver for Quantified Boolean Formulas (QBFs) which is based on And-Inverter Graphs (AIGs). We use a new quantifier elimination method for AIGs, which heuristically combines cofactor-based quantifier elimination with quantification using BDDs and thus benefits from the strengths of both data structures. Moreover, we present a novel SAT-based method for preprocessing QBFs that is able to efficiently detect variables with forced truth assignments, allowing for an elimination of these variables from the input formula. We describe the used algorithm which heavily relies on the incremental features of modern SATsolvers. Experimental results demonstrate that our preprocessing method can significantly improve the performance of QBF preprocessing and thus is able to accelerate the overall solving process when used in combination with state-of-the-art QBF-solvers. In particular, we integrated the preprocessing technique as well as the quantifier elimination method into the QBF-solver AIGSolve, allowing it to outperform state-of-the-art solvers. Quantified Boolean Formulas (QBF) are a generalization of propositional formulas by adding existential as well as universal quantifiers. The addition of quantifiers on the one hand allows for the compact representation of many different problems from verification (e.g. [28, 9, 15] ), planning (e.g. [25] ), and other domains, but on the other hand comes at the price of increased complexity: determining the satisfiability of a QBF is a PSPACE-complete problem and thus assumed to be harder to solve than the SAT problem for propositional formulas.. The importance of the problem has given rise to the development of a number of powerful QBF-solvers which are able to tackle QBF This work was partly supported by the German Research Council (DFG) as part of the Transregional Collaborative Research Center "Automatic Verification and Analysis of Complex Systems" (SFB/TR 14 AVACS, http://www.avacs.org/).. problems originating from practical applications. Such solvers include search-based approaches [32, 13] which apply an extension of the DPLL algorithm known from SAT [7] , as well as solvers based on eliminating variables by different methods, such as symbolic skolemization [3] , resolution and expansion [4] , and symbolic quantifier elimination using AIGs [23] .. In this paper we propose a novel method for quantifier elimination which is based on the symbolic quantifier elimination using AIGs [23] , but combines the advantages of AIGs and BDDs [5] by integrating both AIG and BDD techniques. The success of our quantifier elimination method relies on a close interaction between AIG and BDD techniques controlled by observing the sizes of intermediate results during the solution process.. Another improvement presented in this paper is based on preprocessing. Similar to the SAT domain, where preprocessing [10] has proven to be very effective in accelerating the solving process, various preprocessors for QBF instances have been presented (e.g. [27] ), successfully supporting QBF-solvers by substantially simplifying the input formulas.. One technique used by several solvers for SAT formulas is known as failed literal detection [18] or as unit propagation look-ahead. Given a CNF φ and a variable x occurring in φ this method speculatively performs unit propagation for φ ∧ x and φ ∧ x. If a conflict occurs during the first (second) propagation, x (x) is called a failed literal and φ is simplified by permanently assigning x to false (true).. In [29] the authors describe a similar algorithm for the use in a decision procedure for logic programming problems. They also point out that it is not necessary to examine all literals, instead literals implied by non-failed literals may be skipped.. The detection of a failed literal x can also be achieved by recursive learning [17] with depth 0, if the assignment of x to true necessarily implies conflicting assignments. By increasing the recursion depth, recursive learning is able to detect even more 'failed literals' than a pure unit propagation look-ahead. However, for reasons of efficiency the recursion depth of recursive learning is usually limited to small constants.. In this paper we present a novel method for preprocessing QBFs called constant detection, that lifts the idea of failed literal detection to the QBF domain but differs from the former in two substantial aspects:. (1) Since the QBF problem is much harder to solve than the SAT problem, we can afford using a more powerful and more expensive method for constant detection than unit propagation: in fact we apply full SAT-checks for detecting constants.. (2) In order to avoid futile checks for constants, we apply a sophisticated learning scheme, that exploits satisfying assignments returned by the SAT-solver to eliminate candidates for future constants checks.. The paper is structured as follows. In Sect. 2 we briefly review the logic of Quantified Boolean Formulas, in Sect. 3 we describe our algorithm for constant detection and in particular the learning scheme, in Sect. 4 we describe the integration of constant detection and the improved quantifier elimination scheme based on a close interaction between AIGs and BDDs into the QBF-solver AIGSolve [23] . In the last section we report the results for an extensive set of experiments that demonstrate the effectiveness of our approach. In this paper, we presented a novel preprocessing technique for QBF deploying incremental SAT-checks to prove that variables are constant in all satisfying assignments and thus can be eliminated. As shown in the experiments, the key component of this method is the exploitation of satisfying assignments computed by the SATsolver, which is especially effective when the solver's decision heuristics are modified to produce "helpful" assignments. Furthermore, we described an original hybrid quantifier elimination scheme for AIGs, which combines AIG and BDD-based quantification to profit from both representations. Experimental results on the instances from the QBF Evaluation 2008 show that the integration of the presented methods into the QBF solver AIGSolve considerably increases the solver's performance and lets it significantly outperform other state-of-the-art solvers.. For the future we plan to extend the hybrid quantifier elimination algorithm by integrating and adapting other promising techniques such as methods based on functional composition [14] .

O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Multiobjective Process Controllability Analysis A new approach for process controllability analysis by using multiobjective optimisation techniques is proposed. Within the approach, a set of performance specifications, such as minimum control error and input effort with can make the closed-loop system satisfy all performance specifications simultaneously. Therefore, for the process plant, these performance specifications are achievable. Otherwise, they are unachievable. There is a Pareto-optimal set or a trade-off curve in the performance space to separate these two areas.The paper shows that such trade-off curves can be used for process controllability analysis, and therefore, can be applied to control structure selection problems. The issue of input-output controllability analysis has received increasing attention for a few decades. Input-output controllability is the ability of a plant to achieve acceptable control performance. Various tools and techniques have been developed and are available in the literature to quantify the inherent input-output controllability of a plant (Skogestad and Postlethwaite, 1996) . However most of these tools are mainly open-loop controllability indices. Each individual index only addresses one aspect of process controllability, which causes performance limitation, such as input constraints, unstable poles and zeros. It is still an open area to predict performance limitation jointly imposed by these factors. Nonlinear optimisation has been used to predict performance limitation for a plant with input constraints and unstable zeros . The minimum input usage required to stabilize an unstable plant has been derived by Glover (1986) and recently by Havre and Skogestad (2001) . Åström (2000) presented some results for performance limitations in SISO systems. For MIMO systems, the impact of unstable poles and zeros on closed-loop sensitivity and complementary sensitivity functions has been extensively studied by Chen (2000) .. In this paper, performance limitations are presented in trade-off curves, which are used in process controllability analysis. Boyd and Barratt (1991) have revealed that most control system design specifications are affine and convex functions of the controller to be designed. The trade-off curves of these functions have been calculated as Pareto-optimal performance set.. As summarised by Scherer et al. (1997) , many control performance criteria can be represented as a set of linear matrix inequalities (LMI) 1 (Boyd et al., 1994) . These performance criteria include H ∞ and H 2 norms of certain closed-loop transfer functions and pole placement regions. Using LMI, various, even inconsistent performance requirements can be identified as feasible and infeasible in the performance space. These two areas are separated by the Pareto-optimal performance set. The Pareto-optimal performance set gives a clear picture about what is the achievable performance of a process control system and what kind of performance trade-off is necessary for control design.. The paper is organized as follows. Section 2 introduces the multiobjective control design specifications used in this paper. Two types of Paretooptimal performance curves are designated for multiobjective controllability analysis. These curves are solvable by off-the-shelf software with a small modification. Section 3 provides a complete case study to show the usage of these Pareto diagrams in control structure selection. The paper is concluded in Section 4.. 1 A tutorial paper on LMI for chemical process control practitioners is available elsewhere (VanAntwerp and Braatz, 2000) .

O B I O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Reciprocity and the Emergence of Power Laws in Social Networks Research in network science has shown that many naturally occurring and technologically constructed networks are scale free, that means a power law degree distribution emerges from a growth model in which each new node attaches to the existing network with a probability proportional to its number of links (=degree). Little is known about whether the same principles of local attachment and global properties apply to societies as well. Empirical evidence from six ethnographic case studies shows that complex social networks have significantly lower scaling exponents γ ∼ 1 than have been assumed in the past. Apparently humans do not only look for the most prominent players to play with. Moreover cooperation in humans is characterized through reciprocity, the tendency to give to those from whom one has received in the past. Both variables -reciprocity and the scaling exponent -are negatively correlated (r = −0.767, sig = 0.075). If we include this effect in simulations of growing networks, degree distributions emerge that are much closer to those empirically observed. While the proportion of nodes with small degrees decreases drastically as we introduce reciprocity, the scaling exponent is more robust and changes only when a relatively large proportion of attachment decisions follow this rule. If social networks are less scale free than previously assumed this has far reaching implications for policy makers, public health programs and marketing alike. Networks are a general model to describe complex forms of organization. A network N = (V, E) is defined as a set of vertices V and a set E of unordered pairs of distinct elements of V called links of V . Only two networks with V vertices have no structure. One is the network of isolates, where no link is realized. The other is a completely connected graph in which all possible links are present. Research in network science has shown that some structures are much more likely to occur in the technological and biological world than others [1, 2, 3] . Most importantly, many known complex networks share one fundamental property: they are scale free [4] . Each node in a network has k links; k is also called the degree of a node. Networks are scale free if the non-cumulative degree distribution P (k) follows a power law: P (k) ∼ 1/k γ with an exponent γ usually between 2 and 3. Scale free networks are dominated by a few hubs, nodes with a very high degree. In contrast the vast majority of nodes have little links. This structure proves very efficient in connecting a random pair of nodes with few links. Moreover scale free networks are robust to random failure [5, 6] . In contrast attacks targeted at its hubs can relatively easily disconnect them. To explain the evolution of the scale free typology Barabási and Albert have proposed a model that became known as preferential attachment. In a network that grows over time, each new node links to the existing structure with a probability that is proportional to the number of links a node already has [4] .. Both preferential attachment and scale free typologies are well established models in the natural sciences [2, 7, 8, 9] . However, comparatively little is known about whether the same principles -local exchanges and the emergent overall typologies -apply to the social world as well. This is especially astonishing since sociologists, anthropologists, and other social scientists have studied social organization as social networks for almost a century. Consequently no network domain is better documented than the social world. Hundreds of empirical studies describe the social fabric in small groups: fraternities, clubs, organizations, villages and small scale societies [10, 11, 12, 13, 14, 15] . Additional survey data shed light on personal networks in some complex societies such as the United States and most European countries. Despite this wealth of empirical information we do not know whether, when, and why the social world follows similar laws. This knowledge would be of enormous value for public health institutions, politics, consumer research and marketing alike. If societies are scale free this would make them comparatively easy to control, to influence and to manipulate alike. A characteristic that would play into the hands of those in power.. Besides computer connections, the four most often cited examples of power law distribution in social relationships are citation networks [16] , co-authorships [17] , co-acting relations [4] , and sexual relationships [18] . It is difficult to treat any of these relations as an adequate proxy for the larger construct of social relations in a complex social world. Co-acting, citing and even co-authoring and sexual intercourse are only relatively selected aspects of the spectrum of social interactions humans engage in. While combinations of these networks would make up a better indicator of the latent variable we do not know whether they are correlated or not. Often they will not be. The best scientist is unlikely to be the most popular actor at the same time. Not to speak of lovers. Social activities are not only restricted by talent but also by time. Table 1 : Network statistics of 6 social networks, mostly small-scale societies from Africa bound of social networks size that is related to neocortex size [19] . However, if different social relations are only loosely correlated, or not at all the overall pattern of the multi-relational network would override its effects and make the aggregate significantly less scale free than any of its parts.. From the perspective of a social scientist a better proxy for the social structure may be sets of interactions that link across different societal domains. Friendship could be an indicator of these relationships. For a more in-depth understanding the typology of the networks emerging from more encompassing, time consuming, and emotionally laden relationships needs to be analyzed. Drawing on common sense and empirical observations in different societies we question the fact that 80 percent of all friends really belong to a happy few.. But this is what a power law in friendship ties translates to.

B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Logic of subtyping We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutually-recursive types defined using those type constructors.Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ⊗ and ⊕. An interpretation of S is a function that maps standard connectives into set-theoretical operations (intersection, union, and complement) and modalities into Cartesian product and disjoint union type constructors. This allows S to capture many subtyping properties of the above type constructors. We also consider logics S ρ and S ω ρ that incorporate into S mutually-recursive types over arbitrary and well-founded universes correspondingly.The main results are completeness of the above three logics with respect to appropriate type universes. In addition, we prove Cut elimination theorem for S and establish decidability of S and S ω ρ . We have developed a sequential logical calculus S ω that describes subtyping properties of mutually-recursive types built using Cartesian product and disjoint union. From logical prospective, this calculus is an extension of the classical propositional logic by just two new binary modalities: ⊗ and ⊕. We have proved its completeness with respect to all algebras of terms. In case when the set of fixed point equations is empty, this calculus becomes basic logic of subtyping S. We proved Cut elimination theorem and gave Hilbertstyle axiomatic for S. We also investigated the case of well-founded universes. It turns out that in this case the basic logic of subtyping (without recursive types) stays the same, but subtyping logic of recursive types needs an extra well-foundness inference rule for completeness. Subtyping logic of mutuallyrecursive types over well-founded universe S ω ρ is proven to be decidable. The same question about logic S ρ of all mutually-recursive types remains open.. The most logical next step in this work is the description of subtyping logics of inductive and co-inductive types. Those are logics describing subtyping properties of the least and the greatest solutions of the appropriate system of type equations. Another direction of the future research is the extension of subtyping logic S by modalities corresponding to the other type constructors: constructive and non-constructive function, quotient, dependent product, etc.. The author is thankful to Alexei Kopylov for fruitful discussions at the early stages of this work and to anonymous reviewer for many insightful comments constructive suggestion.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Solve least absolute value regression problems using modified goal programming techniques Scope and PurposeÐLeast absolute value (LAV) regression methods have been widely applied in estimating regression equations. However, most of the current LAV methods are based on the original goal program developed over four decades. On the basis of a modi®ed goal program, this study reformulates the LAV problem using a markedly lower number of deviational variables than used in the current LAV methods. Numerical results indicate that for the regression problems with hundreds of observations, this novel method can save more than 1/3 of the CPU time compared to current LAV methods. Since Charnes et al. [1] formulated the least absolute value (LAV) regression problems as linear programs, numerous algorithms have been developed to solve LAV regression problems. Dodge [2] and Dielman [3] thoroughly review these algorithms.. A LAV regression problem can be stated as follows: consider a linear regression model with n observations Most LAV regression algorithms [1, 4±6] restate Problem 1 as the following linear programming based on the conventional goal programming techniques [1, 7] : Problem 2: x ij , j 1, 2, F F F , m,. Armstrong and Kung [8] provided a dual algorithm to solve the LAV regression problem. Their algorithm was slower than algorithms using the primal problem (i.e. Problem 2) unless à`g ood'' dual feasible solution was available.. Many of the timing comparisons of the LAV algorithms are summarized in Refs. [3, 9] . These investigators confer that the primal LP approach proposed by Armstrong et al. [4] , which is formulated as Problem 2, appears to be best among the LAV regression algorithms.. However, two computational diculties are encountered in solving Problem 2: (i) Problem 2, which contains equality constraints, can only be solved by the big-M method or the two-phase method [10] . Both the big-M method and the two-phase method take a markedly longer time than the simplex method in terms of solving linear programming problems [11] .. (ii) Problem 2 involves too many deviation variables (i.e. An alternative LAV model is proposed by Gonin and Money [12] , which formulates Problem 1 as the following linear programming problem:. Problem 3:. a i r0 and b j P FX Compared with Problem 2, Problem 3 uses half of the deviation variables to reformulate the same Problem. However, the number of constraints in Problem 3 is twice that in Problem 2. Problem 3 is therefore not superior to Problem 2 from the perspective of computational eciency.. In light of the above developments, this paper reformulates a LAV regression problem as a linear program based on modi®ed goal programming techniques. Advantages of the proposed formulation over the current formulation as expressed in Problem 2 are as follows: ®rst, the proposed formulation can be directly solved by the simplex method instead of by the big-M method or the two-phase method. Second, in terms of solving the regression problem involving n observations, the proposed formulation only requires adding n deviation variables instead of adding 2n deviation variables. The proposed method is therefore more convenient to ®nd a good start basis leading to signi®cant eciencies. Numerical results presented herein demonstrate that the proposed formulation is more computationally ecient than the current formulation. This study reformulates an LAV regression problem as a linear program using lower number of variables. The linear program proposed herein can attain a feasible basic solution. Numerical experiments con®rm that the reformulated form is more computationally ecient than the conventional form of the LAV regression problems.

O B I I I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Cocreating IT Value: New Capabilities and Metrics for Multifirm Environments After more than two decades of business value of information technology research, organizations continue to seek ways to create greater value from IT investments. However, over the past decade, particularly with internetworking technologies, there is a fundamental transformation taking place in the creation of business value. Now, multiple organizations collectively leverage IT, thus raising important new issues that cannot easily be addressed by frameworks proposed in current IT value research.Most research on IT value has examined relationships between IT investments and organizational outcomes. More recently, the value thesis has expanded to examine complementary resources, capabilities, and other mediating factors in value creation. However, given the centrality of the IT value question to our field, it is important to expand the agenda to include how best to cocreate value from IT in multiorganizational forms (Dhar and Sundararajan 2007). These forms raise new issues of value creation, risk allocation, complementary investments, capability building, adoption, absorption, and incentives among collaborating organizations. This special issue focuses on understanding how IT value emanates in multifirm environments.The strategic information systems that garnered attention in the 1980s focused on how a single company could gain a monopolistic position through IT with respect to its customer (e.g., by imposing high switching costs). These positions were often framed by researchers through Porter's (1980) industry structure view. Contemporary environments with open systems and hypercompetition make it difficult to attain such a position and even more difficult to sustain it (D'Aveni After more than two decades of business value of information technology research, organizations continue to seek ways to create greater value from IT investments. However, over the past decade, particularly with internetworking technologies, there is a fundamental transformation taking place in the creation of business value. Now, multiple organizations collectively leverage IT, thus raising important new issues that cannot easily be addressed by frameworks proposed in current IT value research.. Most research on IT value has examined relationships between IT investments and organizational outcomes. More recently, the value thesis has expanded to examine complementary resources, capabilities, and other mediating factors in value creation. However, given the centrality of the IT value question to our field, it is important to expand the agenda to include how best to cocreate value from IT in multiorganizational forms (Dhar and Sundararajan 2007) . These forms raise new issues of value creation, risk allocation, complementary investments, capability building, adoption, absorption, and incentives among collaborating organizations. This special issue focuses on understanding how IT value emanates in multifirm environments. We have highlighted a contemporary issue of cocreating IT value that is of increasing importance for firms that seek to be agile and innovative. In this paper, we have framed the cocreation of value through IT as emerging from four layers: relationship-specific assets, knowledge-sharing routines, complementary resources and capabilities, and effective governance. We described each layer with examples of firms that cocreated value by exploiting IT. As firms continue to collaborate, we see several new areas of cocreation opportunities by taking an integrative view of the four layers such as interdependencies among the IT value creation layers, the process through which cocreation of IT value occurs, IT platform-based value, and new sources of value.. Through this special issue, our goal was to seek exemplary studies of IT-based cocreation of value. Each of the four studies examines IT-enabled value cocreation from different vantage points. As we surmise above, it is apparent that while the papers here deal with important aspects of value creation, there is a vast domain that we believe will become an instrumental part of the IS value research stream as well as the broader IS research agenda. As the dynamics of competition and cooperation among firms continue to evolve, and ITbased infrastructures, devices, and software tools create opportunities for value cocreation, we look forward to a high incidence of theoretical and empirical work in this area.

O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Strategies for shape matching using skeletons Skeletons are often used as a framework for part-based shape description and matching. This paper describes some useful strategies that can be employed to improve the performance of such shape matching algorithms. Firstly, it is important that ligaturesensitive information be incorporated into the part decomposition and shape matching processes. Secondly, part decomposition should be treated as a dynamic process in which the selection of the final decomposition of a shape is deferred until the shape matching stage. Thirdly, both local and global measures must be employed when computing shape dissimilarity. Finally, skeletal segments must be weighted by appropriate visual saliency measures during the part matching process. These saliency measures include curvature and ligature based measures. Experimental results show that the incorporation of these strategies significantly improves shape database retrieval accuracy. Medial or skeleton-based approaches to shape analysis have been widely used and proven to be very effective in representing and classifying 2D shapes [6] , [9] , [24] , [26] , [27] , [29] . This is because skeletons are very amenable to part-based (local) description of shapes and it has been acknowledged that part-based representation is necessary in order to handle common shape matching problems such as occlusion, limb growth and articulation, part addition and deletion [16] , [26] . Additionally, Kimia [17] provides a comprehensive discussion of the possible role of medial geometry in the human visual system. Neurophysiological experiments by Lee et al. [19] , [20] observed peak neural activity in the V1 cells at localities of medial axis and psychophysical experiments such as those by Kovacs [18] suggest that medial points may be used in encoding shape information.. Unfortunately, a common criticism of medial-based representation of shapes is that skeletonization algorithms are sensitive to small boundary deformation and ligatures [1] .. Ligatures are problematic because they can appear and disappear with non-visually salient changes in the shape (see Fig. 4a ). If not properly handled, such unstable skeletal features can alter skeletal topology and skew the matching cost between visually similar shapes [32] . Some like van Eede et al. [34] addressed this problem with the use of canonical skeletons that eliminates unstable medial branches while retaining only salient part structure of shape. It is not necessary that ligatures have to be removed before shape matching, but it is important that ligature-sensitive information be strategically incorporated into the shape matching algorithm as is discussed in this paper.. The use of a part-based approach to handle problems such as occlusion requires a shape to be decomposed into meaningful parts. This approach gives rise to another problem, that is, how should a shape be decomposed into parts? Assuming that a shape has now been decomposed into it component parts, another question arises. How should the parts of one shape be matched to another? Is local part-to-part matching sufficient or should shapes also be compared based on a more global geometric descriptor? This paper addresses these questions with some useful strategies that can be incorporated into partbased shape matching algorithms that employ skeletons. In order to test their efficacy, the strategies presented are implemented within the multiresolution gradient vector field (MGVF) skeleton framework of [11] , [13] , which is briefly introduced in section 2. An important feature of the MGVF framework that is relevant to this work is the availability of a ligature-sensitive measure in the form of a vector field directional disparity map.. Sections 3, 4, 5 and 6 discuss these strategies and their respective implementation. Section 7 presents experimental results, which supports the performance improvement resulting from the adoption of these strategies. Fig. 1 provides the reader with an overview of the entire part-based shape matching process. It also serves as a guide which describes how the various aspects of the proposed strategies presented in this paper come together at various stages of the shape analysis process. Several strategies were proposed to improve the performance of skeleton-based shape matching algorithms. Firstly, the unavoidable presence of ligatures in medial-based shape representation must be explicitly accounted for. It was shown how this can be implemented in both the part matching and part decomposition processes. In the former, as a form of visual saliency measure that can be used to weigh the computed part matching cost. In the case of part decomposition, ligature-sensitive information was incorporated into the merging process of basic skeletal components such that the numerous combinations of these components can be suitably reduced in a visually coherent manner based on the notion of visual continuity along the skeleton. Additionally, the selection of the shape's part decomposition option is deferred until the shape matching stage. This is done in acknowledgement of the fact that the ideal decomposition of a shape is dependent on the shape it is being compared with. The use of curvature saliency along a skeleton was also suggested as a means to moderate the effect of curvature error when comparing skeletal parts. Additionally, experimental results suggest that shape matching can be made more robust if both local and global inter-part measures are used to compute shape dissimilarity. However, it was observed that the emphasis between local and global distances is some what dependent on the nature of the shape matching application and data set. These numerous strategies were implemented and tested within the MGVF skeleton framework. Experimental results showed that incorporation of these suggested strategies do produced significant improvement in shape retrieval accuracy. Application of these strategies on other skeleton-based shape matching algorithms should yield similar benefits, albeit their implementations may be realized differently.

O O B I O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O


Time Series Forecasting: Obtaining Long Term Trends with Self-Organizing Maps Kohonen self-organisation maps are a well know classification tool, commonly used in a wide variety of problems, but with limited applications in time series forecasting context. In this paper, we propose a forecasting method specifically designed for multi-dimensional long-term trends prediction, with a double application of the Kohonen algorithm. Practical applications of the method are also presented. Time series forecasting is a problem encountered in many fields of applications, as finance (returns, stock markets), hydrology (river floods), engineering (electrical consumption), etc. Many methods designed for time series forecasting perform well (depending on the complexity of the problem) on a rather short-term horizon but are rather poor on a longer-term one. This is due to the fact that these methods are usually designed to optimize the performance at short term, their use at longer term being not optimized. Furthermore, they generally carry out the prediction of a single value while the real problem sometimes requires predicting a vector of future values in one step. For example, in the case of some a priori known periodicity, it could be interesting to predict all values for a period as a whole. But forecasting a vector requires either more complex * G. Simon is funded by the Belgian F.R.I.A.. † M. Verleysen is Senior Research Associate of the Belgian F.N.R.S. models (with potential loss of performance for some of the vector components) or many distinct single value predicting models (with potential loss of the correlation information between the various values). Methods able to forecast a whole vector with the same precision for each of its components are thus of great interest. While enlarging the prediction horizon is of course of primary interest for practitioners, there is of course some limit to the accuracy that can be expected for a long-term forecast. The limitation is due to the availability of the information itself, and not to possible limitations of the forecasting methods. Indeed, there is no doubt that, whatever forecasting method is used, predicting at long term (i.e. many time steps in advance) is more difficult that predicting at short term, because of the missing information in the unknown future time steps (those between the last known value and the one to predict). At some term, all prediction methods will thus fail. The purpose of the method presented in this paper is not to enlarge the time horizon for which accurate predictions could be expected, but rather to enlarge the horizon for which we can have insights about the future evolution of the series. By insights, we mean some information of interest to the practitioner, even if it does not mean accurate predictions. For example, are there bounds on the future values ? What can we expect in average ? Are confidence intervals on future values large or narrow ?. Predicting many steps in advance could be realized in a straightforward way, by subsampling the known sequence, then using any short-term prediction method. However, in this case, the loss of information (used for the forecast) is obviously even higher, due to the lower resolution of the known sequence. Furthermore, such solution does not allow in a general way to introduce a stochastic aspect to the method, which is a key issue in the proposed method. Indeed, to get insights about the future evolution of a series through some statistics (expected mean, variance, confidence intervals, quartiles, etc.), several predictions should be made in order to extract such statistics. The predictions should differ; a stochastic prediction method is able to generate several forecasts by repeated Monte-Carlo runs. In the method presented in this paper, the stochastic character of the method results from the use of random draws on a probability law.. Another attractive aspect of the method presented in this paper is that it can be used to predict scalar values or vectors, with the same expected precision for each component in the case of vector prediction. Having at disposal a time series of values x(t) with 1 ≤ t ≤ n, the prediction of a vector can be defined as follows :. where d is the size of the vector to be predicted, f is the data generating process, p is the number of past values that influence the future values and ε t is a centred noise vector. The past values are gathered in a p-dimensional vector called regressor.. The knowledge of n values of the time series (with n >> p and n >> d) means that relation (1) is known for many (n− p− d+ 1) time steps in the past. The modeling problem then becomes to estimate a function f that models correctly the time series for the whole set of past regressors. The idea of the method is to segment the space of p-dimensional regressors. This segmentation can be seen as a way to make possible a local modeling in each segment. This part of the method is achieved using the Self-Organizing Map (SOM) [1] . The prototypes obtained for each class model locally the regressors of the corresponding class. Furthermore, in order to take into account temporal dependences in the series, deformation regressors are built. Those vectors are constructed as the differences between two consecutive regressors. The set of regressor deformations can also be segmented using the SOM. Once those two spaces are segmented and their dependences characterized, simulations can be performed. Using a kind of Monte-Carlo procedure to repeat the simulations, it is then possible to estimate the distribution of these simulations and to forecast global trends of the time series at long term.. Though we could have chosen some other classical vector quantization (VQ) method as only the clustering property is of interest here, the choice of the SOM tool to perform the segmentation of the two spaces is justified by the fact that SOM are efficient and fast compared to other VQ methods with a limited complexity [2] and that they provide an intuitive and helpful graphical representation.. In the following of this paper, we first recall some basic concepts about the SOM classification tool. Then we introduce the proposed forecasting method, the double vector quantization, for scalar time series and then for vector ones. Next we present some experimental results for both scalar and vector forecastings. A proof of the method stability is given in appendix. In this paper, we have presented a time series forecasting method based on a double classification of the regressors and of their deformations using the SOM algorithm. The use of SOMs Figure 10 . Impact of the variation of n 1 and n 2 on the model generalization ability for the Polish electrical consumption problem.. makes it possible to apply the method both on scalar and vector time series, as discussed in section 3 and illustrated in section 4. A proof of the method stability is given in appendix.. The proposed method is not designed to obtain an accurate forecast of the next values of a series, but rather aims to determine long-term trends. Indeed, its stochastic nature allows repeating simulations by a Monte-Carlo procedure, allowing to compute statistics (variance, confidence intervals, etc.) on the predictions. Such a method could also be used for example in the financial context, for the estimation of volatilities.. Intuitively, the stability property of the method is not surprising. Indeed, the model is designed such that it will mostly produce predictions that are in the range of the observed data. By construction, deformations are chosen randomly according to an empirical probability law and the obtained predictions should stay in the same range. If, for some reason, the prediction is about to exceed this range during one of the simulations, the next deformations will then tend to drive it back inside this range, at least with high probability. Furthermore, as simulations are repeated with the Monte-Carlo procedure, the influence of such unexpected cases will be reduced when the mean is taken to obtain the final predictions. The following of this section is intended to prove this intuitive result.. The proof consists in two steps: it is first shown that the series generated by the model is a Markov chain; secondly, it is demonstrated that this particular type of Markov chain is stable. In order to improve the readability of the proof, lighter notations will be used. For a fixed d and a fixed p, notation X t will represent the vector x t t−p+1 . The last known regressor will be denoted X 0 . The prototype of a cluster C ′ j of deformations will be noted Y j . Finally, hats will be omitted for simplicity as all regressors X t are estimations, except for t = 0.. To prove that the series is a Markov chain, we consider the starting vector of the simulation at time 0. The corresponding initial regressor of the series is denoted X 0 , and C 0 is the corresponding SOM cluster in the regressor space. The deformation that is applied to X 0 at this stage is Y 0 . Then the next values of the series are given by. . . drawn randomly from the transition matrix for clusters C 0 , C 1 , . . . respectively. The series X t is therefore a Markov chain, homogeneous in time (the transition distribution are not time dependant), irreducible and defined over a numerable set (the initial X t are in finite number, and so are the deformations).. To show the stability of this Markov chain and thus the existence of a stationary distribution, Foster's criterion [14] is applied. Note that this criterion is a stronger result which proves the ergodicity of the chain, which in turns implies the stability. Foster's criterion is the following:. A necessary and sufficient condition for an irreducible chain to be ergodic is that there exists a positive function g(.), a positive ε and a finite set A such that:. Since the Markov chain is homogenous, it is sufficient to observe transition Y 0 from X 0 to X 1 . The same development can be deduced for any other transition.. The demonstration is done for two-dimensional regressors but can be generalized easily to other dimensions. Note that in the following, we use g(.) = .. 2 in (14). Before going in further details, let us remark that for a SOM with at least 3 classes in general position, class C 0 covers less than a half plane. Furthermore, we have to distinguish two cases for each cluster. First, the cluster may be included in a finite compact from IR 2 . The second case is the case of an infinite cluster i.e. of a cluster which may does have any neighbour in some direction; this happens to ckusters on the border of the map.. The first case is easely proved. Since X 0 < R 0 , where R 0 can be any constant, then we have by triangular inequality:. As the deformations Y j are in finite number, the maximum of their norm is finite. This proves the first inequality of (14) in an obvious way for the first case (i.e. bounded cluster case). The other case thus appens when X 0 → +∞. This happens in unbounded clusters. The unbounded cluster case is much more technical to prove.. Looking at figure 11 , we see that each unbounded cluster is included in a cone with vertex A and delimited by the normalized vectors a 1 and a 2 . There are two possibilities: either a 1 and a 2 form an acute angle, either an obtuse one, as shown in figure 11 (a) and figure 11(b) respectively.. Before going on and applying Foster's criterion, note that the three following geometrical properties can be proven:. Denoting. we have δ 1 and δ 2 both positive in the acute angle case, while either δ 1 or δ 2 is positive for an obtuse angle. Indeed, using the origin O, we define:. We thus have:. which can be bounded by a strictly positive constant as In the method description, the numbers n 1 and n 2 of prototypes have not been fixed. Indeed, the problem is that different values of n 1 (n 2 ) result in different segmentations in the original (deformation) space and in different conditional distribution in the transition matrix. The model may thus slightly vary.. Selecting the best values for n 1 and n 2 is an important question too. Traditionally, such hyperparameters are estimated by model selection procedures such as AIC, BIC or computationallycostly resampling techniques (Leave-One-Out, kfold cross validation, bootstrap). As it will be shown further in this paper, exact values of n 1 and n 2 are not necessary, as the sensitivity of the method around the optimums is low. A simple validation is then used to choose adequate values for n 1 and n 2 . For that purpose the available data are divided into three subsets: the learning, the validation and the test set. The learning set is used to fix the values of the model parameters, such as the weights of the prototypes in the SOM and the transition matrix. The validation set is used to fix meta-parameters, such as the numbers n 1 and n 2 of prototypes in the SOM maps. The validation set is thus used for model selection. The test set aims to see how the model behaves on unused data that mimic real conditions.. The selection of n 1 and n 2 is done with regards to an error criterion, in our case a sum of squared error criterion, computed over the validation set V S:. Once n 1 and n 2 have been chosen, a new learning is done with a new learning set obtained from the reassembled learning and validation sets. This new learning is only performed once with optimal values for n 1 and n 2 .. Note that, hopefully, the sensitivity of the method to specific values of n 1 and n 2 is not high. This has been experimentally verified in all our simulations, and will be illustrated on the first example (Santa Fe A) in section 4.2.. Another crucial question is the sensitivity of the method to various runs of the SOM algorithm (with the same n 1 and n 2 values). Indeed it is well known that initial conditions largely influence the exact final result of the SOM algorithm (by final result it is meant the prototype locations, and their neighborhood relations) [13] . Nevertheless, as mentioned above, the neighborhood relations of the SOM are used for visualization purposes only; they do not influence the results of the forecast. Moreover, the location of the centroids are used to quantize the space (therefore allowing the estimation of the empirical conditional frequencies of the clusters); small variations in the centroid location have thus a low influence on each prediction generated by the method, and an even lower one on the statistics (mean, confidence intervals, etc.) estimated from the predictions. This last result has been confirmed experimentally in all our simulations, for which no significant difference was observed after different runs of the two SOM algorithms.

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