Security Analysis of a Single Sign-On Mechanism for Distributed Computer Networks Abstract-Single sign-on (SSO) is a new authentication mechanism that enables a legal user with a single credential to be authenticated by multiple service providers in a distributed computer network. Recently, Chang and Lee proposed a new SSO scheme and claimed its security by providing well-organized security arguments. In this paper, however, we demonstrative that their scheme is actually insecure as it fails to meet credential privacy and soundness of authentication. Specifically, we present two impersonation attacks. The first attack allows a malicious service provider, who has successfully communicated with a legal user twice, to recover the user's credential and then to impersonate the user to access resources and services offered by other service providers. In another attack, an outsider without any credential may be able to enjoy network services freely by impersonating any legal user or a nonexistent user. We identify the flaws in their security arguments to explain why attacks are possible against their SSO scheme. Our attacks also apply to another SSO scheme proposed by Hsu and Chuang, which inspired the design of the Chang-Lee scheme. Moreover, by employing an efficient verifiable encryption of RSA signatures proposed by Ateniese, we propose an improvement for repairing the Chang-Lee scheme. We promote the formal study of the soundness of authentication as one open problem.Index Terms-Authentication, distributed computer networks, information security, security analysis, single sign-on (SSO). In this paper, we demonstrated two effective impersonation attacks on Chang and Lee's single sign-on (SSO) scheme [19] . The first attack shows that their scheme cannot protect the privacy of a user's credential, and thus, a malicious service provider can impersonate a legal user in order to enjoy the resources and services from other service providers. The second attack violates the soundness of authentication by giving an outside attacker without credential the chance to impersonate even a non-existent user and then freely access resources and services provided by service providers. We also discussed why their well-organized security arguments are not strong enough to guarantee the security of their SSO scheme. In addition, we explained why Hsu and Chuang's scheme [12] is also vulnerable to these attacks. Furthermore, by employing an efficient verifiable encryption of RSA signatures introduced by Ateniese [21] , we proposed an improved Chang-Lee scheme to achieve soundness and credential privacy. As future work, it is interesting to formally define authentication soundness and construct efficient and provably secure single sign-on schemes. Based on the draft of this work [36] , a preliminary formal model addressing the soundness of SSO has been proposed in [37] . Further research is necessary to investigate the maturity of this model and study how the security of the improved SSO scheme proposed in this paper can be formally proven. In [19] , Chang and Lee provided a well-organized security analysis to show that their SSO scheme is secure. However, the two impersonation attacks presented in the previous section mean that their SSO scheme is actually not secure. So, why is their analysis not enough to guarantee the security of their scheme? What is the security flaw in their scheme leading to the above attacks? And what could we learn from these attacks to prevent similar situations in the future design of SSO schemes? These are the topics of this section.. In [19] , the security of the Chang-Lee SSO scheme has been analyzed in three different ways: 1) BAN logic [29] was used to show the correctness of the Chang-Lee schemes; 2) informal security arguments were given to demonstrate that their scheme can resist some attacks, including impersonation attacks; and 3) a formal security proof was given to prove that their scheme is a secure authenticated key exchange (AKE) protocol [30] . However, these security analyses and proofs still do not guarantee the full security of the Chang-Lee scheme and there are a number of reasons for this. First, as early as the 1990s, it was known that although BAN logic had been shown useful to identify some attacks, it could approve protocols which are actually unsound in practice because of some technical weaknesses in the logic [31] . Moreover, in [19] the authors did not give details to show how the BAN logic can be used to prove that their scheme guarantees mutual authentication. In fact, at the end of Section V-A of [19] , the authors claimed to be able to: "prove that and are able to authenticate each other using our protocol." but they provided no argument to show why each party could not be impersonated by an attacker. Second, the authors did discuss informally why their scheme could withstand impersonation attacks by considering two scenarios, for example, an attacker re-uses previous nonce to forge message or selects random credential to compute by . However, such informal arguments neither strongly confirm their scheme's security against these two concrete attacks nor exclude the exis-tence of other scenarios of impersonation attacks, such as those presented in previous sections. Finally, their formal proof about AKE only focuses on the session key security, i.e., an attacker with all reasonable resources is not able to know the session key established between the two parties under the computational DH (CDH) assumption (refer to [19, Theorem 1] , and not the security of mutual authentication. According to the definitions given by Bellare and Rogaway [30] , one fundamental requirement of a secure AKE protocol is that there be a secure mutual authentication in the first place.. From the above, we can see that it is the use of credential proof which leads to the above two attacks against the Chang-Lee SSO scheme. More specifically, is a kind of knowledge proof which shows that a prover (usually played by user ) knows credential . However, this is not a secure proof as a malicious verifier (i.e., service provider ) can recover and an outside attacker may be able to get authenticated without a credential. Based on this observation, a natural improvement on the Chang-Lee scheme would be to replace non-interactive proof by a rigorous but interactive ZK proof [20] that shows the prover's knowledge of secret without revealing any additional information about credential . In other words, using the verifiably encrypted signature introduced in [33] , user can encrypt his/her credential under the public key of a trusted party and verifiably convince service provider that the ciphertext does contain w.r.t. 's identity without allowing to get any additional information about credential . Compared with two modular exponentiations used for generating and verifying proof , however, ZK proofs for showing the possession of an RSA signature usually require hundreds of modulo exponentiations [32] , [33] since these proofs rely on inefficient "cut and choose" method, i.e., binary challenges.. From the two attacks presented above, we can learn that both credential privacy and soundness are crucial for SSO schemes. As mentioned in Section III-A, credential privacy has been studied in Yang et al. [10] and Han et al. [17] . To the best of our knowledge, however, there is surprisingly, no existing research which has given a careful treatment of soundness. For example, Han et al. [17] did not investigate soundness, though they did carefully study how to formally define credential forgery and recovery attacks from outsiders, users, service providers and their potential collusion. According to the most traditional form of authentication, a user will be authenticated if he/she can provide a valid pair of user name and password (i.e. credential), and soundness is obviously satisfied because a user is not able to go through authentication without providing a valid credential which is registered and maintained by a server. In complex scenarios, like the Chang-Lee scheme, the situation may be less obvious and, in fact, quite challenging. For this reason, the problem remains an open one for future study. The question of formally defining the soundness of SSO/authentication schemes and rigorously proving them for concrete solutions remains an interesting and important one.. Finally, it must be noted that the analysis above shows only that the Chang-Lee SSO scheme fails to achieve secure authentication, without violating its security for achieving user anonymity and session key privacy.. IV. ATTACKS ON THE HSU-CHUANG SCHEME Here, we briefly highlight the difference between the Chang-Lee scheme [19] and the Hsu-Chuang scheme [12] to see why the above describe impersonation attacks apply to this latter as well. The two schemes have similar structures and use similar notations, but the technical details differ. In summary, the Hsu-Chuang scheme is differs from the Chang-Lee scheme in three ways. First, in the Hsu-Chuang scheme user 's credential is a naive RSA signature signed by the trusted party , i.e., , where is 's identity selected by him/herself. Second, to authenticate itself, service provider sends signature , where is the DH key material generated by , is the current timestamp, and is 's identity. Finally, for user authentication user issues and sends proof to , who validates by checking if . For more detail, see [12] or Section II of [19] .. As pointed out in [19] , the Hsu-Chuang scheme is vulnerable to impersonation attack as an attacker can forge a valid credential w.r.t. identity by simply selecting random and then computing . This attack can be excluded if a specific encoding format is required for identities and the credential is issued by using a secure hash , i.e., , as in the Chang-Lee scheme. According to the discussion in Section III, the Hsu-Chuang scheme is still not secure even with such a countermeasure. The reason is that our two attacks against the Chang-Lee scheme apply to the Hsu-Chuang scheme as well. This means that the Hsu-Chuang scheme also fails to satisfy both credential privacy and soundness of authentication. In addition, there is another flaw in the Hsu-Chuang scheme. Attacker can impersonate service provider to cheat legal users, as service authentication is conducted by using a non-traditional RSA signature, . By communicating with twice attacker can get messages and satisfying and . Once (this holds with probability about 0.6, as we discussed in Section III-A), can run the extended Euclidean algorithm [22, pp. 290-292 ] to find two integers and such that in (without knowing the factors of 's RSA modulus). Hence, can recover by computing . After that, can impersonate to any legal user by using the value of to issue signature , without knowing 's RSA private key .

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


Variational Models for Image Colorization via Chromaticity and Brightness Decomposition Abstract-Colorization refers to an image processing task which recovers color in grayscale images when only small regions with color are given. We propose a couple of variational models using chromaticity color components to colorize black and white images. We first consider total variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Second, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization. I N this paper, we discuss the problem of recovering a color image from a grayscale image with color data given only in small regions. This problem has been considered by Fornasier in [26] , where the author introduced various models for the task and connected with inpainting literature. A specific application is considered in [26] , which consists in recovering frescoes (paintings) by Mantegna in an Italian church which was destroyed during World War II. There are photos of the full frescoes available in black and white, while only few real pieces of frescoes with original colors are remaining. The objective is to reconstruct the original color of the frescoes (image) from few remaining pieces with the original color and full black and white image. In [26] , the author used RGB color model with a nonlinear distortion function for fitting the grayscale data, and presented successful numerical results. Some mathematical analysis of the color restoration model of [26] has been presented by Fornasier and March in [27] .. This color restoration task is closely related to colorization of black and white images in computer graphics. In [33] , the authors notice the neighboring pixels with similar intensities should have similar color, and use nonlocal fitting term to match the colors. In [40] , Sapiro suggests minimizing difference between the gradient of luminance and the gradient of Manuscript received October 17, 2006 ; revised May 21, 2007 . This work was supported by the IMA. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Dimitri Van De Ville.. S. H. Kang is with the Department of Mathematics, University of Kentucky, Lexington, KY 40515 USA (e-mail: skang@ms.uky.edu).. R. March is with the Istituto per le Applicazioni del Calcolo, CNR, 00161, Rome, Italy (e-mail: r.march@iac.cnr.it).. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.. Digital Object Identifier 10.1109/TIP.2007.903257. color. The change in color is constrained by the change in luminance, and the method reduces to solving linear or nonlinear Poisson equations. A fast algorithm using weighted distance image blending technique is studied in [51] . The authors utilize Dijkstra's shortest path algorithm for fast computation. More related literature, some using segmented images can be found in [21] , [31] , and [43] . We propose variational colorization models using chromaticity and brightness color model. Color images can be treated in various ways [30] . There are typical linear models such as red, green and blue (RGB) channels and cyan, magenta, and yellow (CMY); Luminance separated color systems such as YIQ (luminance, hue, and saturation) which is used in standard color TV broadcasting, and YCbCr (Luminance, two color-difference components) widely used for digital video; nonlinear color systems closer to human color perception such as hue, saturation, and value (HSV). In more mathematical settings, color images can also be treated as 3-D vectorial functions [11] as well as tensor products of different color components such as Chromaticity and Brightness (CB). The related literature on different color image models can be found in [16] , [32] , [38] , [41] , [45] , and [46] .. In this paper, we use chromaticity and brightness (CB) color model. If denotes the image domain, from typical RGB color model image is separated into the brightness component , and the chromaticity component , where denotes the Euclidean norm of vectors. In [16] , the authors showed that using CB model gives better color control and detail recovery for color image denoising compared to different color settings. For example, it is better to separate chrominance (color information) from the brightness (or luminance) to better control the details of the images. This is consistent with the fact that the essential geometrical information of images is contained in the gray level, as it has been shown in [13] . Therefore, treating brightness separately from the chromaticity can give more flexibility in detail recovery. On the other hand, it is not ideal to separate the chrominance into more than one components. By separating the color components, it unavoidably introduces undesirable art effects and results in the color which are not close to human vision perception. For example, if chromaticity is separated to hue and saturation, since hue does not have gray as a color, any white (or black) color will have maximum (or minimum) saturation with any random color in hue. Instead, chromaticity is ideal to represent color and it is self contained as one component representing the chrominance.. The brightness component is a grayscale image; however, the chromaticity component stores the color information which takes values on the unit sphere . For this kind of constraints there are various numerical methods introduced to accurately compute nonflat features. Some mathematical analysis on constraints is studied in [28] , and numerical methods are explored in [48] . In this paper, we introduce variational colorization models which use a penalty term to deal with constraint. The penalization method simplifies the computation by avoiding the direct numerical implementation of the constraint. Convergence of solutions of the variational problems with penalty term to solutions of the original problem with constraint will be shown, as the penalty parameter tends to infinity. In practice, large values of the penalty parameter still yield accurate computed results.. We note that depending on how the grayscale images are given, either as brightness, luminance, or value, we can apply our models for each case with simple modification. In some applications (see, for instance, [26] and [27] ) true brightness is not given, and the gray level datum is given by a nonlinear map . In this case, after computation of chromaticity via the variational method, the following nonlinear system has to be solved at each point with respect to the unknown vector (1) In some applications of colorization [40] , the gray level datum is luminance . In this case, the map is linear and it is given by the convex combination , where , , . The nonlinear system (1) can be solved explicitly, and the solution is given by. In the work by Fornasier [26] , [27] about the fresco restoration, the gray level is given by old pictures and the nonlinear map is computed by means of a statistical method. In this case the 3 3 nonlinear system (1) has to be solved numerically at each point . In the following, we assume that the gray level datum is brightness, and we use brightness as a representation of such values, i.e., brightness, luminance, or value.. We propose a couple of variational models for colorization. The total variation (TV) minimizing denoising model by Rudinet al. [39] has been extended to various image processing tasks, one of which is TV inpainting method [19] . The colorization task can also be understood as inpainting the colors (as Sapiro's insight [40] ). We introduce TV minimizing colorization using chromaticity and brightness color model. Second, we further modify our model using weighted harmonic maps for colorization. This model appropriately adds edge information from the Brightness data, while it reconstructs smooth color values for each homogeneous region. In spirit, this has relations to Mumford-Shah functional for segmentation [36] . The proposed weighted harmonic map functional is flexible in modeling, and it can easily be extended to include texture colorization. In Section III, we combined image decomposition techniques to colorize textured regions. This paper is organized as follows. In Section II, we propose TV colorization using CB model. We show a convergence property of the variational problem with penalty term to the TV colorization model with constraint, and we present numerical results for TV colorization. In Section III, we propose a functional based on weighted harmonic maps for colorization. We show an analogous convergence property for a penalized version of the model, and we present numerical results including an application to texture colorization. We explored variational colorization models: TV colorization and colorization via weighted harmonic map. Calculus of variations provides flexibility in modeling, and mathematical analysis yields a sound basis for the models. By using the chromaticity and brightness color model, chrominance is accurately modeled. We consider a penalized version of the variational problems in order to handle the nonconvex constraint . We discuss the consistency of such models by proving existence and convergence of minimizers of the functionals with penalty term to minimizers for the ideal model. Numerical results are shown to validate the model.. Numerical implementation shows effectiveness of TV colorization. Complicated details can be recovered with simple calculation given reasonable chrominance information. The weighted harmonic map model can deal with bigger colorization area. With only a small area of color given, color can be diffused to large region. With this model, color blending is naturally achieved by geodesic direction in chromaticity space, Notice the difference between the values of g(jrB j) and g(jrB j), and its effects in the results. All images are in color [1] .. and it can deal with texture colorization combined with image decomposition methods.

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


A general model of parameterized OWA aggregation with given orness level The paper proposes a general optimization model with separable strictly convex objective function to obtain the consistent OWA (ordered weighted averaging) operator family. The consistency means that the aggregation value of the operator monotonically changes with the given orness level. Some properties of the problem are discussed with its analytical solution. The model includes the two most commonly used maximum entropy OWA operator and minimum variance OWA operator determination methods as its special cases. The solution equivalence to the general minimax problem is proved. Then, with the conclusion that the RIM (regular increasing monotone quantifier) can be seen as the continuous case of OWA operator with infinite dimension, the paper further proposes a general RIM quantifier determination model, and analytically solves it with the optimal control technique. Some properties of the optimal solution and the solution equivalence to the minimax problem for RIM quantifier are also proved. Comparing with that of the OWA operator problem, the RIM quantifier solutions are usually more simple, intuitive, dimension free and can be connected to the linguistic terms in natural language. With the solutions of these general problems, we not only can use the OWA operator or RIM quantifier to obtain aggregation value that monotonically changes with the orness level for any aggregated set, but also can obtain the parameterized OWA or RIM quantifier families in some specific function forms, which can incorporate the background knowledge or the required characteristic of the aggregation problems. The ordered weighted averaging (OWA) operator, which was introduced by Yager [45] , has attracted much interest among researchers. It provides a general class of parameterized aggregation operators that include the min, max, average. Many applications in the areas of decision making, expert systems, data mining, approximate reasoning, fuzzy system and control have been proposed [20, 21, 29, 37, 53, 57, 60] .. One of the appealing points of OWA operators is the concept of orness [45] . The orness measure reflects the andlike or orlike aggregation result of an OWA operator, which is very important both in theory and applications [13, 15, [50] [51] [52] . The orness of OWA operator, also called ''attitudinal-character", can be used to represent the preference information in aggregation problems [53, 54] . It is clear that the actual type of aggregation performed by an OWA operator depends upon the form of the weight vector [8, [12] [13] [14] [15] [49] [50] [51] [52] . The weight vector determination is usually a prerequisite step in many OWA related applications, and it has become an active topic in recent years [1, 26, 31, 39, 42] . A number of approaches were suggested for obtaining the required OWA operator, i.e., quantifier guided aggregation [45, 47] , exponential smoothing [13] , sample learning [37, 56] , the weights function method [1] , argument dependent methods [41, 43] and the preference relation method [2] . The most commonly used method is to obtain the desired OWA operator under a given orness level [12] [13] [14] [15] 31, 35, 55] , which is usually formulated as a constrained optimization problem. The objective to be optimized can be the (Shannon) entropy [12, 14, 31, 35] , the variance [15, 26] , the maximum dispersion [4, 39] , the (generalized) Rényi entropy [33] or even the preemptive goal programming [3, 40] . O'Hagan [35] suggested the problem of constraint nonlinear programming with a maximum entropy procedure, the solution is called a MEOWA (Maximum Entropy OWA) operator. Filev and Yager [12] further proposed a method to generate MEOWA weight vector by an immediate parameter. Fullér and Majlender [14] transformed the maximum entropy model into a polynomial equation, which can be solved analytically. Liu and Chen [31] proposed general forms of the MEOWA operator with a parametric geometric approach, and discussed its aggregation properties. Apart from maximum entropy OWA operator, Fullér and Majlender [15] suggested the minimal variability OWA operator problem in quadratic programming, and proposed an analytical method for solving it. Liu [26] gave this OWA operator generating method with the equidifferent OWA operator, which seems being a reformulation of [15] , but actually is an extension with a more simple and intuitive process [28, 34] . A closely related work is that of Wang and Parkan [39] . They proposed a linear programming model with minimax disparity approach to get the OWA operator under the desired orness level. The solution equivalence of the minimum variance problem and the minimax disparity problem was proved by Liu recently [30] . Majlender [33] proposed a maximum Rényi entropy OWA operator problem with exponential objective function, which can include the maximum entropy and minimum variance problem as special cases, and an analytical solution was proposed.. Another important closely related topic is OWA aggregation with Regular Increasing Monotone (RIM) quantifier, which was also proposed by Yager [48] . The linguistic quantifiers were proposed by Zadeh [59] , who also classified them with absolute quantifiers, such as ''much more than 10", and relative quantifiers, such as ''a half". Flexibility can be obtained by introducing fuzzy quantifiers which permit a closer representation in the language of daily life. Yager [46, 48] further distinguished the relative quantifiers into three classes. They are called Regular Increasing Monotone (RIM) quantifier, Regular Decreasing Monotone (RDM) quantifier and Regular UniModal (RUM) quantifier, where the RIM quantifier is the basis of all kinds of relative quantifiers [46, 48] . Some RIM quantifiers in natural language are most, many, at least half, some [6, 7, 11, 16, 21, 19, 38] . This RIM quantifier guided aggregation method with OWA operator in natural language [48] has been applied in many areas such as decision analysis, database querying, and computing with words theory [5, 6, 9, 17, 18, 20, 21, 44] . Based on this method, Liu [24, 29] further analyzed the relationship between the OWA operator and the RIM quantifier with the generating function technique. With the generating function in RIM quantifier playing the role of weight vector in OWA operator, the RIM quantifier can be seen as a dimension free continuous OWA aggregation. The maximum entropy RIM quantifier and minimum variance RIM quantifier were proposed, and some properties of them were discussed [24, 27] . A summarization of the OWA operator and the corresponding RIM quantifier determination methods was given in [32] .. In the present paper, a general optimization model with strictly convex objective function to obtain the OWA operator under given orness level is proposed. This approach includes the maximum entropy and the minimum variance problems as special cases. The problem is also more general than the Rényi entropy objective function case. The solution methods and the properties of maximum entropy and minimum variance problems were studied separately, but they can be included into this general model now. The consistent property that the aggregation value for any aggregated set monotonically increases with the given orness value is still kept, which gives more alternatives to represent the preference information in the aggregation of decision making. Furthermore, the equivalence to the minimax problem is proved, which is the generalization of the equivalence of the minimum variance problem and the minimax disparity problem [30] , but the proof is simplified. With the generating function in the RIM quantifier playing the role of the weight vector in the OWA operator, a general model that can include the maximum entropy and minimum variance RIM problems is proposed. Some properties are discussed and the solution equivalence to the minimax problem for RIM quantifier is proved. The RIM quantifier has the advantages of being dimension free, having a simple solution, and having the ability to be connected with natural language terms. When we face the problem that the number of arguments changes in different cases, the RIM quantifier based aggregation method can provide a uniform formula with its membership function. With the analytical solution of these general models, we can make the OWA operator become the interpolation series of a given monotonic function or make the RIM quantifier function obey some specific function shapes, which gives more possible alternatives for the OWA operator and RIM quantifier determination. We can also incorporate some prerequisite information such as the background or the characteristic requirements of the aggregation problem into the aggregation process.. The remainder of this paper is organized as follows. Section 2 gives some preliminaries of OWA operators, the RIM quantifier guided OWA aggregation method, and the generating function representation method of RIM quantifier. Section 3 proposes a general model to obtain OWA operator under given orness level. Some properties of the optimal solution are discussed. The solution equivalence of the general model and the corresponding minimax problem is proved. Section 4 can be seen as the continuous extension of Section 3 with RIM quantifier. As both OWA operators and RIM quantifiers have some common characteristics in both the solution process and in their applications, the conclusions are organized in parallel for easy comparison. This similarity proposes a general model to obtain the RIM quantifier under given orness level. Some properties of the optimal solution are discussed and the solution equivalence to the corresponding minimax problem is proved. As the general models of Sections 3 and 4 are improvements and extensions of the minimum variance problems and the minimax disparity problems for OWA operators and RIM quantifiers, respectively, Section 5 summarizes the solutions and properties of these two kinds of problems in this general framework, so that the similarity between these two kinds of problems can be connected and some existing results are extended. Section 6 considers the problems' solutions from another viewpoint, which can make the OWA operator or the RIM quantifier generating function have a specific function shape. Some special function forms for the OWA operator and RIM quantifier solutions are provided, which gives more alternatives for their determination. Section 7 summarizes the main results and draws conclusions. The paper proposes a general model to obtain the OWA operator with orness as its control parameter. This general model includes the maximum entropy OWA operator and minimum variance OWA operator as special cases. Some properties of its solution are discussed. The solution equivalence to the minimax problem are proved, which is also a generalization of the solution equivalence for the minimum variance and minimax disparity problems. Then, these results are extended to the RIM quantifier case, which corresponds to the OWA operator in continuous form. A general model to obtain the parameterized RIM quantifiers of given orness level is proposed, with the property discussions and the solution equivalence proof to the corresponding minimax problem. With the analytical optimal solution expression of these two kinds problems, the relationship between the OWA operator vector elements or the shape of the RIM quantifier membership function can be observed intuitively. We can not only use the OWA operator or RIM quantifier to get aggregation results consistent to the preference information (orness level), but also can make the obtained optimal OWA operator or RIM quantifier obey some specific function forms by considering the structure or the background information of the aggregation problem. The parameterized OWA operator and RIM quantifier families of some commonly used function forms are provided for possible applications. Whatever the forms of these optimal solutions, they can always be seen as a parameterized extension of the arithmetic mean between the maximum and minimum. Comparing with the case of the OWA operators, the parameterized RIM quantifier families are dimension free in aggregation and can be connected with natural language interpretation.

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


Did federal regulation discourage facilities-based entry into US local telecommunications markets? In the United States from 2001 to 2006, federal regulations allowed entrants to lease from incumbents at relatively low cost all of the network infrastructure necessary to provide local phone service. These platform entrants could then provide phone service without installing any of their own equipment. Advocates of this policy claimed that it was needed to provide an economically feasible means by which entrants could serve residential customers. Critics contended that the policy substantially deterred loop entry whereby entrants installed their own switching equipment and leased only the wires that connect a customer's premises to the telephone network. An analysis of panel data for each state over this period indicates that the policy's critics may have been correct. The crossprice elasticity of loop entry with respect to platform price was roughly 1.0. A back of the envelope calculation suggests that loop entry may have decreased by roughly 20% due to platform entry price reductions. This paper addresses whether platform entry affected loop entry in U.S. local telecommunications markets. The results suggest that platform entry may have limited loop entry. As such, some support is provided to those who believe that platform entry was a hindrance to introducing competition to local telephone markets. Further, the estimate of 750,000 forgone lines of loop may be a lower bound, as the analysis above only measures the incremental effect of platform price decreases on loop entry. Loop entry could have also been discouraged by the relatively low prices set for platform entry when the option was introduced. Also, there is no measurement of the potential effect of platform entry on other forms of phone service, such as those provided by cable companies and voice-over-internet-protocol providers.. Nevertheless, platform entry was widely implemented by entrants and arguably provided some customers with choices that they otherwise would not have had. Over 16 million phone lines were provided by entrants utilizing platform entry as of 2004.. For those policymakers who viewed platform entry as a relatively close substitute to loop entry, platform entry may be considered as a regulatory success.. The future of platform entry is muddled. Soon after it became clear that platform entry at regulated low prices would be eliminated, the two largest platform entry players, AT&T and MCI, announced large asset write downs and were eventually acquired by other companies. The incumbents claim that they are open to continuing to offer platform entry and some entrants have entered into agreements to continue providing service via platform entry. However, time will tell whether platform entry at privately negotiated prices will emerge as a viable entry model. * p < 0.10, ** p < 0.05, *** p < 0.01. Panel-corrected standard errors are reported in parentheses.. The Prais-Winsten estimates are based on errors that follow an AR(1) process common to each state and heteroskedasticity within states and across states in a given period.. State fixed effects omitted for brevity.

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


Estimation of parameters for trend-renewal processes Abstract Methods of estimating unknown parameters of a trend function for trend-renewal processes are investigated in the case when the renewal distribution function is unknown. If the renewal distribution is unknown, then the likelihood function of the trend-renewal process is unknown and consequently the maximum likelihood method cannot be used. In such a situation we propose three other methods of estimating the trend parameters. The methods proposed can also be used to predict future occurrence times. The performance of the estimators based on these methods is illustrated numerically for some trend-renewal processes for which the statistical inference is analytically intractable. We consider a class of estimation problems for the stochastic model determined by the trend-renewal process (TRP) which is defined to be a time-transformed renewal process (time-transformed RP), where the time transformation is given by a trend function λ(·). It contains the nonhomogeneous Poisson process (NHPP) and RP as special cases and serves as a useful reliability model for repairable systems.. The TRP was introduced and investigated first by Lindqvist (1993) and by Lindqvist et al. (1994) (see also Lindqvist and Doksum (2003) , and Lindqvist (2006) ). Parametric inference on the parameters of the TRP was considered in the paper of Lindqvist et al. (2003) , where the authors also proposed corresponding models, called heterogeneous trend-renewal processes, that extend the TRP to cases involving unobserved heterogeneity. Nonparametric maximum likelihood (ML) estimation of the trend function of a TRP under the often natural condition that λ(·) is monotone was considered by Heggland and Lindqvist (2007) . Peña and Hollander (2004) presented a general class of models that allows the researcher to incorporate the effect of interventions performed on a unit after each event occurrence, the impact of accumulating events on a unit, the effect of unobservable random effects of frailties, and the effect of covariates that could be time-dependent. The ML estimators of this general models parameters were presented, and their finite and asymptotic properties were ascertained by Stocker and Peña (2007) .. In the present paper we take up the problem of estimating unknown trend parameters of a TRP in the case when its renewal distribution is unknown. If the renewal distribution is unknown, then the likelihood function of the TRP is unknown and consequently the ML method cannot be used. In this case we propose three possible methods of estimating the trend parameters. The problem of estimating trend parameters of the TRP with unknown renewal distribution may be of interest in the situation when we observe several systems, of the same kind, working in different environments and we are interesting in examining and comparing their trend functions, whatsoever their renewal distribution is.. The article is organized as follows. In Sect. 2 we recall the definition of the TRP and the form of its likelihood function. A special case of the TRP with a Weibull renewal distribution and power-law type trend function is considered in Sect. 3. The likelihood function and the likelihood equations for estimating the parameters of that process are given. The likelihood equations are also presented in the form which is used in simulation study to obtain the ML estimators of the TRP parameters. In Sect. 4 we propose three methods of estimating the trend parameters in the case when the ML methods can not be used. The estimation problem of the trend parameters in some special case of the TRP is considered in Sect. 5. In Sect. 6 the estimators proposed are examined and compared with the ML estimators (obtained under the additional assumption that the renewal distribution has a known parametric form) through a computer simulation study. Some real data are examined in Sect. 6.3. Section 7 contains conclusions and some prospects. We observe a good performance of the estimators α CLS and β CLS obtained by the CLS method. This method leads to satisfactory accuracy of these estimators in the TRP(F, λ(·)) model considered with unspecified F in comparison to the ML estimators α ML and β ML for this model with specified F . The CLS method leads in average to more accurate estimators than the LS and M methods.. The LS method considerably underestimates the parameter α. In most cases considered we have RMSE( α CLS ) < RMSE( α LS ), and in all cases, RMSE( α CLS ) < RMSE( α M ).. The RMSE( α M ) is about two, or even more, times greater than the RMSE( α CLS ). A similar remark concerns the RMSE( β M ) and RMSE( β CLS ). In some cases the RMSE( α CLS ) is even less than the RMSE( α ML ), and the RMSE( β CLS ) is less than the RMSE( β ML ).. For a given number n of failures, the RMSE's of all the estimators in the WPLP(α, β, γ ) become significantly smaller as the parameter γ increases. Remark that, according to formula (27), the variance s of the renewal distribution F decreases evidently as γ increases. For example, s = 1 for γ = 1, s = 0.2732 for γ = 2, s = 0.0787 for γ = 4. A smaller value of γ (a larger value of the variance s) causes larger variability of the estimators (recall that the RMSE determines the mean squared deviation of the estimate from the true value of the parameter-the risk). In the WPLP(α, β, 1), i.e. in the PLP(α, β), the variance of F is equal to 1 and constitutes a great value in reference to the same value of the expectation of the renewal distribution as well as in reference to the assumed value 1 of the sample mean of the transformed working times

O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Measuring similarity to detect qualified links The early success of link-based ranking algorithms was predicated on the assumption that links imply merit of the target pages. However, today many links exist for purposes other than to confer authority. Such links bring noise into link analysis and harm the quality of retrieval. In order to provide high quality search results, it is important to detect them and reduce their influence. In this paper, a method is proposed to detect such links by considering multiple similarity measures over the source pages and target pages. With the help of a classifier, these noisy links are detected and dropped. After that, link analysis algorithms are performed on the reduced link graph. The usefulness of a number of features are also tested. Experiments across 53 query-specific datasets show our approach almost doubles the performance of Kleinberg's HITS and boosts Bharat and Henzinger's imp algorithm by close to 9% in terms of precision. It also outperforms a previous approach focusing on link farm detection. In modern web search engines, link-based ranking algorithms play an important role. Typical link analysis algorithms are based on the assumption that links confer authority. However, this assumption is often broken on the real web. As a result, the retrieval performance based on such naive link analysis is often disappointing. According to our experiments on more than fifty query-specific datasets, on average only four out of the top ten results generated by the HITS algorithm [15] are considered relevant to the query (details in Section 5) .. The prevalence of links that do not (or should not) confer authority is an important reason that makes link analysis less effective. Examples of such links are links that are created for the purpose of advertising or navigation like those in Figure 1 . These types of links are common on the Web. From a person's view, these links do carry some information that the authors of the web pages want to promote. However, from the perspective of link analysis algorithms, these links are noisy information because they do not show the authors' recommendation of the target pages. Traditional link analysis algorithms do not distinguish such noise from useful information. As a consequence, the target pages of these links could get unmerited higher ranking. Therefore, in order to provide better retrieval quality, the influence of such links needs to be reduced.. For years, researchers have been working on improving the quality of link analysis ranking, most often through improvements to the traditional PageRank [21] and HITS [15] algorithms. Some other work focuses on detecting and demoting web pages that do not deserve the ranking generated by traditional link analysis. However, little work has asked which links should be used in web link analysis.. In this paper, we introduce the notion of "qualified links"-links that are qualified to make a recommendation regarding the target page. We propose to detect qualified links using a classifier which, based on a number of similarity measures of the source page and target page of a link, makes the decision that whether the link is "qualified". After this, the "unqualified links" are filtered out, which leaves only the "qualified links". Link analysis algorithms are then performed on the reduced web graph and generate the resulting authority ranking. We also studied a number of features In this paper, we presented the approach of identifying qualified links by computing a number of similarity measures of their source and target pages. Through experiments on 53 query-specific datasets, we showed that our approach improved precision by 9% compared to the Bharat and Henzinger imp variation of HITS.. This paper is merely a preliminary study, demonstrating the potential of our approach. The following limitations can be addressed in future work.. • The classifier and similarity measures being used are quite simple. It is expected that the use of a better classification algorithm and an advanced set of similarity measures would produce a better result. For example, examining the similarity of text in and around a link to its target (as in [5] ) might fare better, especially for multi-topic hubs.. • The punishment of removing "unqualified links" might be too stringent. A manual examination of the experimental results revealed that some authoritative pages are removed in addition to poor quality pages. Weighting the links by their quality could be a better alternative than the current binary weighting.. • The computational complexity of "qualified link analysis" is an issue that requires careful consideration. Although the index of the corpus could be made available before hand, computing the similarity scores is still expensive considering the size of the web. Potential solutions include using fewer features, using features that are easy to compute, and utilizing simple classification algorithms. We tested one possible extension, which builds a thresholding classifier based on anchor text similarity. The classifier simply categorizes the links within the first eight buckets as "qualified links", and the rest as "unqualified". This approach gives a precision of Figure 12 : Retrieval performance of Q-PR on WebBase dataset negative class ("unqualified links") at 97.05% on the labeled training set. This classifier is then applied to the 53 queryspecific datasets. The retrieval performance is between that of imp and Q-HITS (precision@10 being 73.02%, score@10 being 0.93).. • As mentioned in Section 5.1, the data collection process used an existing search engine, which could introduce certain bias into the dataset. Building a dataset from a large web crawl can solve such a problem.. • The experiment testing Qualified PageRank is still rudimentary. In order to fully examine its usefulness, more experimental work on global datasets is needed.. • This approach is not a panacea for "unqualified links". We did not differentiate the different types of "unqualified links". Some types of links are perhaps more difficult to identify than others. Finer-grained discrimination might further boost retrieval quality. As a preliminary investigation in this direction, we trained a multi-class classifier to distinguish each individual type of "unqualified link". The result showed that the classifier was effective in finding spam links, while not very helpful in finding other types of "unqualified links". This remains a topic for future study. In this paper we are concerned with the classification of hyperlinks regarding their usefulness in web link analysis. Thus, prior work in this area is quite relevant. Davison [8] first proposed the automatic recognition of nepotistic links-links that are present for reason other than merit. While that work considered seventyfive features, only a few dealt with content, looking at page titles and meta descriptions, and overlapping outgoing link sets. Rather than binary features, our current work focuses on a few similarity measures, and additionally considers the full content of the pages. Benczur et al. [1] proposed to detect nepotistic links using language models. In this method, a link is down-weighted if its source and target page are not related based on their language models. This approach is based on the assumption that pages that are connected by non-nepotistic links must be sufficiently similar, which is not required in our model. Chakrabarti et al. [5] extend HITS by increasing the weights of links whose anchor text (or surrounding text) incorporates terms from the query. Our approach does not examine text specifically in or around the anchor, and more importantly, is not query-specific.. While we focus on content, more work has considered the analysis of link structure to eliminate or down-weight links (e.g., to combat web spam). Here we consider a representative set of such work. Lempel and Moran [16] defined a tightly-knit community (TKC) as a small but highly connected set of sites. Even though such a community is not quite relevant to the query, it may still be ranked highly by link-based ranking algorithms. The authors proposed SALSA, a stochastic approach for link structure analysis, and demonstrated that it is less vulnerable to the TKC effect than HITS. Li et al. [17] pointed out the small-in, large-out link problem with HITS, in which a community is associated with a root with few in-links but many out-links. Such communities may dominate HITS results even if they are not very relevant. The authors addressed this problem by assigning appropriate weights to the inlinks of root. Wu and Davison [25] proposed a two-step algorithm to identify link farms. The first step generates a seed set based on the intersection of in-link and out-links of web pages. The second step expands the seed set to include pages pointing to many pages within the seed set. The links between these identified spam pages are then re-weighted and a ranking algorithm is applied to the modified link graph. Carvalho et al. [7] also proposed algorithms to detect noisy links at site level by examining the link structure among web sites.. Finally, we note that many other approaches to web spam detection have been explored. This includes Drost and Scheffer's work on spam identification [9] , the work by Fetterly et al. [10] and Ntoulas et al. [19] on spam detection using statistical analysis (of links and content), and Gyöngyi et al.'s work on using trust to demote spam [11] . These approaches typically focus on the identification of specific pages that should be labeled as spam rather than the links between them. The early success of link-based ranking algorithms was predicated on the assumption that links imply merit of the target pages. However, in many instances this assumption is no longer valid. An evident example is spam links-links that are created for the sole purpose of manipulating the ranking algorithm of a search engine. The presence of link spam makes link analysis less effective. Another example is navigational links, where links are created for easy access to other pages regardless of relevance. Links between different regional web sites of the same company (http://foobar.com/ and http://foobar.co.uk/), and links to the "terms of use" of a web site can be considered as examples of navigational links. Although navigational links are not created for the purpose of spamming, they should also be considered less valuable for link analysis since they hardly imply authority of their target pages.. Based on this motivation, we introduce the notion of "a qualified link". A qualified link is a link on a page that is qualified to make a recommendation regarding the target page, and is in some sense the opposite of a nepotistic link [8] .. Besides spam links and navigational links, other types of "unqualified links" include advertising links and irrelevant links. Advertising links are links created for the purpose of advertising. Irrelevant links can be considered as the collection of other "unqualified links", such as links pointing to a required document viewer or to a particular web browser for the desired display of the web page.. To determine whether a link is qualified or not, we propose to build a binary classifier based on the characteristics of links. Then based on the decision of that classifier, a filtering process can be performed on the web graph which retains only the "qualified" links. Finally, link analysis algorithms can run on the reduced graph and generate rankings.. Alternatively, if the classifier is able to generate a reasonable probability of a link being "qualified", link analysis may be performed on a weighted graph where each edge in the web graph is weighted by its relative qualification.. There are a variety of metrics one might use to measure the qualification of a link. The measures we use in this work are the similarity scores of the source page and the target page, such as content similarity and URL similarity. The similarity measures are detailed in Section 4. The classifier considers these similarity measures, and makes a decision (of a link being "qualified" or not) or predict a probability (of how likely a link being "qualified"). The process of this approach is visualized in Figure 2 . Kleinberg [15] proposed that web documents had two important properties, called hubness and authority, as well as a mechanism to calculate them. In his Hyperlink-Induced Topic Search (HITS) approach to broad topic information discovery, the score of a hub (authority) depended on the sum of the scores of the connected authorities (hubs):. Kleinberg calculated these scores on the subset of the web that included top-ranked pages for a given query, plus those pages that pointed to or were referenced by that set. Bharat and Henzinger [2] proposed a number of improvements to HITS. One of the changes is an algorithm called imp, which re-weights links involved in mutually reinforcing relationships and drops links within the same host. They found that imp made a significant improvement over the original HITS.. Page and Brin [21, 3] proposed an alternative model of page importance, called the random surfer model. In that model, a surfer on a given page i, with probability (1 − d) chooses to select uniformly one of its outlinks O(i), and with probability d to jump to a random page from the entire web W . The PageRank score for node i is defined as the stationary probability of finding the random surfer at node i. One formulation of PageRank is. PageRank is a topic-independent measure of the importance of a web page, and must be combined with one or more measures of query relevance for ranking the results of a search.

O O O O O O O O O O O O O O O O O O O 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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


Intrusion detection using autonomous agents AAFID is a distributed intrusion detection architecture and system, developed in CERIAS at Purdue University. AAFID was the ®rst architecture that proposed the use of autonomous agents for doing intrusion detection. With its prototype implementation, it constitutes a useful framework for the research and testing of intrusion detection algorithms and mechanisms. We describe the AAFID architecture and the existing prototype, as well as some design and implementation experiences and future research issues. Ó The intrusion detection ®eld has grown considerably in the last few years, and a large number of intrusion detection systems have been developed to address dierent needs. Intrusion detection is clearly necessary with the growing number of computer systems being connected to networks. We describe an architecture for intrusion detection and system monitoring based on autonomous agents that serves as a research framework for intrusion detection techniques and algorithms.. We start by de®ning some common terms and the motivation for using autonomous agents in an intrusion detection system. A simple example of an intrusion detection system that adheres to the AAFID architecture is shown in Fig. 1(a) . This ®gure shows the four components of the architecture: agents, ®lters, transceivers and monitors. We refer to each one of these components as AAFID entities or simply entities, and to the whole intrusion detection system constituted by them as an AAFID system.. An AAFID system can be distributed over any number of hosts in a network. Each host can contain any number of agents that monitor for interesting events occurring in the host. Agents may use ®lters to obtain data in a system-independent manner. All the agents in a host report their ®ndings to a single transceiver. Transceivers are per-host entities that oversee the operation of all the agents running in their host. They have the ability to start, stop and send con®guration commands to agents. They may also perform data reduction on the data received from the agents. The transceivers report their results to one or more monitors. Each monitor oversees the operation of several transceivers. Monitors have access to network-wide data, therefore they are able to perform higher-level correlation and detect intrusions that involve several hosts. Monitors can be organized in a hierarchical fashion such that a monitor may in turn report to a higher-level monitor. Also, a transceiver may report to more than one monitor to provide redundancy and resistance to the failure of one of the monitors. Ultimately, a monitor is responsible for providing information and getting control commands from a user interface. Fig. 1(b) shows the logical organization corresponding to the physical distribution depicted in Fig. 1(a) .. We now describe each component in greater detail.

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 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 B I 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 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 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 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


Managing space for finite-state verification Finite-state verification (FSV) techniques attempt to prove properties about a model of a system by examining all possible behaviors associated with that model. This approach suffers from the state-explosion problem, where the size of the model or the analysis costs may be exponentially large with respect to the size of the system. Using symbolic data structures to represent subsets of the state space has been shown to usually be an effective optimization approach for hardware verification. The value for software verification, however, is still unclear. In this paper, we investigate applying two symbolic data structures, Binary Decision Diagrams (BDDs) and Zero-suppressed Binary Decision Diagrams (ZDDs), in two FSV tools, LTSA and FLAVERS. We describe an experiment showing that these two symbolic approaches can improve the performance of both FSV tools and are more efficient than two other algorithms that store the state space explicitly. Moreover, the ZDD-based approach often runs faster and can handle larger systems than the BDD-based approach. Finite-state verification (FSV) approaches attempt to prove properties about a model of a system. These approaches are not as general as theorem-proving, but they are usually easier to use. They do suffer, however, from the so-called state-explosion problem, where the size of the model or the analysis costs may be exponentially large with respect to the size of the system being analyzed.. There are a number of different techniques for dealing with this state-explosion problem. Using symbolic data structures such as Binary Decision Diagrams (BDDs) to represent subsets of the state space [8, 15] has been shown to usually be an effective optimization approach for hardware verification. The value for software verification, however, is still unclear [2, 11] .. In this paper we report on our exploration of two symbolic data structures, BDDs and Zero-suppressed Binary Decision Diagrams (ZDDs), applied to two different FSV tools, LTSA [14] and FLAVERS [12] . We describe how the systems to be verified are encoded as Boolean functions and how the verification problem is solved via the Boolean operations. Moreover, we present a BDD-based algorithm and a novel ZDD-based algorithm for implementing a key Boolean operation used in the verification. We demonstrate that these two symbolic approaches can improve the performance of both FSV tools and are more efficient than the tools' native algorithms that store the state space explicitly. In addition, we find that on average, the ZDD-based approach uses only 60% as much time as the BDD-based approach in LTSA and 71% in FLAVERS, and can often handle larger systems than the BDD-based approach. Based on these experimental results, we believe that it would be worthwhile to explore applying this ZDD approach to other FSV techniques.. In the next two sections of this paper, we provide an overview of LTSA and FLAVERS so that the reader can understand how the verification problem is encoded using Boolean functions and solved by Boolean operations. Sections 4 and 5 describe how BDDs and ZDDs, respectively, are applied in these two FSV tools. Section 6 discusses our experimental methodology and Section 7 presents and analyzes the experimental results. Section 8 describes related work, and we conclude in Section 9 with a summary of the results and a discussion of future work. FSV techniques can check whether a property holds for a given system. Once the system model and the property are specified, this check is done automatically by exhaustively exploring all possible states of the system. FSV techniques are especially useful for concurrent systems, where nondeterministic behavior usually makes testing problematic. FSV methods, however, are limited by the state-explosion problem. Much research has been done to ameliorate this problem but few comparative studies have been done using the same approach with different symbolic algorithms.. In this paper, we investigate the use of two symbolic data structures, BDDs and ZDDs, in two FSV tools, LTSA and FLAVERS. To do this, we developed an efficient algorithm to compute transition functions with ZDDs. This algorithm treats missing Boolean variables in a ZDD representation in different ways depending on whether the variable is from the domain set of the Boolean function being represented or not. In our experiments, both symbolic algorithms perform much better than the native, non-symbolic algorithms used by LTSA and FLAVERS, in terms of both runtime and the size of the system that the algorithm can handle under the given resource limits. Moreover, the ZDD-based algorithm is almost always better than the BDD-based one. Our results suggest that the ZDD-based approaches should be more widely investigated. We intend to explore several additional directions. There are several additional variations of the BDD and ZDD based algorithms that appear promising. One variation, for exam-ple, is to change the algorithm so that each composite state visited during the check is used in the computation for the transition functions only once. To implement this, besides the function f visited , another function fnew , which represents the composite states at the current exploration frontier only, is needed. Then only states represented by fnew would be used in the computation for the next frontier. A preliminary experiment with this variation found that it sometimes improves performance but sometimes degrades it.. Another variation we want to pursue is to change the encoding scheme for the ZDD representations. The encoding strategy described in this paper uses log 2 n Boolean variables to encode n FSA states. We call this encoding a binary encoding. An alternative, unary encoding would use one variable for each FSA state. A benefit of the unary encoding is that it enables us to use an algorithm for computing transition functions without encoding all transitions in Boolean functions. We also did a preliminary experiment with this approach and found that it also sometimes helps improve performance but sometimes degrades performance. We intend to do more experiments to explore the system features that make one approach more effective than the other.. Besides these variations, we also want to investigate how the BDD and ZDD based algorithms work when the property does not hold (at least for the system model). In that situation, FSV tools can generate a counterexample, a trace through the model showing how the property is violated, that can help the analyst find out what went wrong. For the LTSA search algorithm, it is easy to generate counterexamples once a violation or deadlock is found. For the state propagation algorithm and the symbolic algorithms, however, more work needs to be done on efficiently generating counterexamples.. Finally, as discussed above, we intend to explore the distribution of 1-assignments for a Boolean function. We believe that this will not only help understand why the ZDD representations of the functions we encounter are more compact than the BDD representations, but also may suggest further optimizations. There are many BDD-based FSV tools, such as [4, 10] . The main difference between our work and these FSV tools is the use of ZDDs.. ZDDs have been used in many domains, such as SAT solvers [9] , and arithmetic polynomial manipulation [16] . ZDDs have also been used in verification [17] . In that work, both BDD and ZDD representations are used and compared in model checking CTL properties of Petri nets. Several symbolic approaches using BDDs and ZDDs to manipulate Petri nets are proposed in that paper, and the experimental results suggest that ZDDs are more suitable.. A number of other investigators have compared the performance of different FSV tools, including those that use symbolic algorithms and non-symbolic algorithms [2, 11] . Since different tools use different modeling languages to describe the system, there is no guarantee that the models used with different tools are really equivalent. This factor may sig- nificantly affect the comparison of different algorithms. In our work, since the algorithms compared are implemented in the same FSV tool, the systems used for comparison are modeled in exactly the same way.

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


Second-Generation Wavelet Collocation Method for the Solution of Partial Differential Equations An adaptive numerical method for solving partial differential equations is developed. The method is based on the whole new class of second-generation wavelets. Wavelet decomposition is used for grid adaptation and interpolation, while a new O(N ) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm. In this paper we demonstrate the algorithm for one particular choice of second-generation wavelets, namely lifted interpolating wavelets on an interval with uniform (regular) sampling. The main advantage of using second-generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling. Thus, the strength of the new method is that it can be easily extended to the whole class of second-generation wavelets, leaving the freedom and flexibility to choose the wavelet basis depending on the application. Many interesting physical systems are characterized by the presence of a wide range of spatial and temporal scales. In particular we are interested in solving problems with localized structures or sharp transitions, which might occur intermittently anywhere in the computational domain or change their locations and scales in space and time. The numerical solution of such problems on uniform grids is impractical, since high-resolution computations are required only in regions where sharp transitions occur. In order to solve these problems in a computationally efficient way, the computational grid should adapt dynamically in time to reflect local changes in the solution.. Several adaptive gridding techniques exist, and this paper will concentrate on one such class of methods, namely wavelet methods. Wavelet methods take advantage of the fact that functions with localized regions of sharp transition are well compressed using wavelet decomposition. The basic idea behind the wavelet decomposition is to represent a function in terms of basis functions, called wavelets, which are localized in both physical and wavenumber spaces [1] [2] [3] [4] . The currently existing wavelet-based numerical algorithms can be roughly classified as either wavelet-Galerkin [5] [6] [7] [8] or wavelet-collocation [9] [10] [11] [12] [13] [14] [15] [16] type. The major difference between these approaches is that wavelet-Galerkin algorithms solve problems in wavelet coefficient space and, in general, can be considered gridless methods, while wavelet-collocation methods solve problem in physical space on a dynamically adaptive computational grid. In wavelet-collocation methods every wavelet is uniquely associated with a collocation point, and thus grid adaptation is simply based on the analysis of wavelet coefficients; i.e., at any given time the computational grid consists of points corresponding to wavelets whose coefficients are greater than a given threshold (a parameter that controls the accuracy of the solution). With this adaptation strategy a solution is obtained on a near-optimal grid for a given accuracy; i.e., the compression of the solution is performed dynamically as opposed to a posteriori as done in data analysis. The major advantage of wavelet-collocation methods is the ease of treating the nonlinear terms. Derivatives in wavelet-collocation methods can be computed in many ways, including application of matrix derivative operators [11, 17] , projection back and forth between wavelet and physical space at every time step [13, 14] , and use of finite difference operators [9, 10, 15, 16, 18, 19] .. Although the wavelet transform with its space/scale localization is an attractive technique to apply to the solution of problems with localized structures, traditional, biorthogonal wavelet transforms have difficulties dealing with boundaries. Traditionally, wavelets ψ j k are defined as translates and dilates of one mother wavelet ψ; i.e., ψ j k (x) = ψ(2 j x − k). Orthogonal and biorthogonal wavelet transforms have been extended to the interval [20 -22] , but a better solution is to abandon the translation/dilation relationship. This leads to what are referred to as second-generation wavelets in the literature [23] . The main advantage of second-generation wavelets is that wavelets are constructed in the spatial domain and can be custom designed for complex domains and irregular sampling.. Second-generation wavelets supply the necessary freedom to deal with boundary conditions, but with a cost. With the loss of translation invariance goes also the Fourier transform, the primary tool used in the creation of most of the first-generation wavelet bases. There are few first-generation wavelets that can be constructed without the use of Fourier techniques developed in [24] . Interpolating wavelets, independently discovered by Donoho [25] and Harten [9] , are an example of such a family. Interpolating wavelets are based on the interpolating subdivision scheme of Deslauriers and Dubuc [26] and are well suited to numerical analysis [9, 10, 14, 16, 27] . Interpolating wavelets, however, do have their shortcomings, which are discussed in detail in Section 2.1. It is desirable to have a larger class of secondgeneration wavelets to build on. Fortunately there is a general method available for the construction of second-generation wavelets, known as the lifting scheme [23, 28] .. The main objective of this paper is to establish a general framework for constructing numerical methods for solving partial differential equations, which are based on second-generation wavelets. The beauty of second-generation wavelets is that the algorithm developed for one particular choice of wavelet basis can be easily extended to the whole class of second-generation wavelets, leaving the freedom and flexibility to choose wavelets depending on applications. In this paper we will demonstrate the method by solving a number of one-dimensional nonlinear test problems on an interval using lifted interpolating wavelets. Extensions of the algorithm to higher dimensions, complex geometries, and irregular sampling will be the subject of further investigation.. The rest of the paper is organized as follows. Section 2 gives a brief introduction of the second-generation wavelets. Two major tools for constructing second-generation wavelets, namely interpolating wavelet transform and lifting, are discussed in detail in Sections 2.1 and 2.2. The efficient implementation of the lifted interpolating wavelet transform algorithm is described in Section 2.3. The numerical algorithm based on the lifted interpolating wavelet transform is introduced in Section 3. Finally, Section 4 contains numerical examples of applications of the new method to the solution of one-dimensional Burgers and modified Burgers equations and the one-dimensional diffusion flame problem. A general framework for constructing adaptive numerical methods for solving partial differential equations, which are based on second-generation wavelets, is developed. Wavelet decomposition is used for grid adaptation and interpolation, while a new O(N ) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. In this paper the method is demonstrated by solving the one-dimensional Burgers and the modified Burgers equations with small viscosities and the laminar diffusion flame problem. The results indicate that the computational grid and associated wavelets can very efficiently adapt to the local irregularities of the solution in order to resolve regions of large gradients. Furthermore, a solution is obtained on a near optimal grid for a given accuracy; i.e., the compression of the solution is performed dynamically as opposed to a posteriori as done in data analysis. Additional strengths of the algorithm are:. 2. The method can easily be extended to the whole class of second-generation wavelets, leaving the freedom and flexibility to choose wavelet basis depending on applications.. 3. The method can handle general boundary conditions and nonlinearities.. Future areas of research include the implementation of the algorithm in higher dimensions, complex geometries, and irregular sampling. This work is currently underway.

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