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Abstract: We propose a novel non-parametric adaptive anomaly detection algorithm for high dimensional data based on score functions derived from nearest neighbor graphs on $n$-point nominal data. Anomalies are declared whenever the score of a test sample falls below $\alpha$, which is supposed to be the desired false alarm level. The resulting anomaly detector is shown to be asymptotically optimal in that it is uniformly most powerful for the specified false alarm level, $\alpha$, for the case when the anomaly density is a mixture of the nominal and a known density. Our algorithm is computationally efficient, being linear in dimension and quadratic in data size. It does not require choosing complicated tuning parameters or function approximation classes and it can adapt to local structure such as local change in dimensionality. We demonstrate the algorithm on both artificial and real data sets in high dimensional feature spaces.
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Title: Forced Evolution in Silico by Artificial Transposons and their Genetic Operators: The John Muir Ant Problem
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Abstract: Modern evolutionary computation utilizes heuristic optimizations based upon concepts borrowed from the Darwinian theory of natural selection. We believe that a vital direction in this field must be algorithms that model the activity of genomic parasites, such as transposons, in biological evolution. This publication is our first step in the direction of developing a minimal assortment of algorithms that simulate the role of genomic parasites. Specifically, we started in the domain of genetic algorithms (GA) and selected the Artificial Ant Problem as a test case. We define these artificial transposons as a fragment of an ant's code that possesses properties that cause it to stand apart from the rest. We concluded that artificial transposons, analogous to real transposons, are truly capable of acting as intelligent mutators that adapt in response to an evolutionary problem in the course of co-evolution with their hosts.
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Title: On the characterization of the regions of feasible trajectories in the workspace of parallel manipulators
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Abstract: It was shown recently that parallel manipulators with several inverse kinematic solutions have the ability to avoid parallel singularities [Chablat 1998a] and self-collisions [Chablat 1998b] by choosing appropriate joint configurations for the legs. In effect, depending on the joint configurations of the legs, a given configuration of the end-effector may or may not be free of singularity and collision. Characterization of the collision/singularity-free workspace is useful but may be insufficient since two configurations can be accessible without collisions nor singularities but it may not exist a feasible trajectory between them. The goal of this paper is to define the maximal regions of the workspace where it is possible to execute trajectories. Twodifferent families of regions are defined : 1. those regions where the end-effector can move between any set of points, and 2. the regions where any continuous path can be tracked. These regions are characterized from the notion of aspects and free-aspects recently defined for parallel manipulators [Chablat 1998b]. The construction of these regions is achieved by enrichment techniques and using an extension of the octree structures to spaces of dimension greater than three. Illustrative examples show the interest of this study to the optimization of trajectories and the design of parallel manipulators.
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Title: Distinguishing Cause and Effect via Second Order Exponential Models
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Abstract: We propose a method to infer causal structures containing both discrete and continuous variables. The idea is to select causal hypotheses for which the conditional density of every variable, given its causes, becomes smooth. We define a family of smooth densities and conditional densities by second order exponential models, i.e., by maximizing conditional entropy subject to first and second statistical moments. If some of the variables take only values in proper subsets of R^n, these conditionals can induce different families of joint distributions even for Markov-equivalent graphs. We consider the case of one binary and one real-valued variable where the method can distinguish between cause and effect. Using this example, we describe that sometimes a causal hypothesis must be rejected because P(effect|cause) and P(cause) share algorithmic information (which is untypical if they are chosen independently). This way, our method is in the same spirit as faithfulness-based causal inference because it also rejects non-generic mutual adjustments among DAG-parameters.
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Title: Word Sense Disambiguation Using English-Spanish Aligned Phrases over Comparable Corpora
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Abstract: In this paper we describe a WSD experiment based on bilingual English-Spanish comparable corpora in which individual noun phrases have been identified and aligned with their respective counterparts in the other language. The evaluation of the experiment has been carried out against SemCor. We show that, with the alignment algorithm employed, potential precision is high (74.3%), however the coverage of the method is low (2.7%), due to alignments being far less frequent than we expected. Contrary to our intuition, precision does not rise consistently with the number of alignments. The coverage is low due to several factors; there are important domain differences, and English and Spanish are too close languages for this approach to be able to discriminate efficiently between senses, rendering it unsuitable for WSD, although the method may prove more productive in machine translation.
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Title: Which graphical models are difficult to learn?
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Abstract: We consider the problem of learning the structure of Ising models (pairwise binary Markov random fields) from i.i.d. samples. While several methods have been proposed to accomplish this task, their relative merits and limitations remain somewhat obscure. By analyzing a number of concrete examples, we show that low-complexity algorithms systematically fail when the Markov random field develops long-range correlations. More precisely, this phenomenon appears to be related to the Ising model phase transition (although it does not coincide with it).
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Title: Calibration of 3-d.o.f. Translational Parallel Manipulators Using Leg Observations
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Abstract: The paper proposes a novel approach for the geometrical model calibration of quasi-isotropic parallel kinematic mechanisms of the Orthoglide family. It is based on the observations of the manipulator leg parallelism during motions between the specific test postures and employs a low-cost measuring system composed of standard comparator indicators attached to the universal magnetic stands. They are sequentially used for measuring the deviation of the relevant leg location while the manipulator moves the TCP along the Cartesian axes. Using the measured differences, the developed algorithm estimates the joint offsets and the leg lengths that are treated as the most essential parameters. Validity of the proposed calibration technique is confirmed by the experimental results.
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Title: Estimation of safety areas for epidemic spread
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Abstract: In this work we study safety areas in epidemic spred. The aim of this work is, given the evolution of epidemic at time $t$, find a safety set at time $t+h$. This is, a random set $K_t+h$ such that the probability that infection reaches $K_t+h$ at time $t+h$ is small. More precisely, inspired on the study of epidemic spread, we consider a model in which the measure $\mu_n(A)$ is the incidence -density of infectives individuals- in the set $A$, at time $n$ and $$\mu_n+1(A)(\omega)=\int_S\pi_n+1(A;s)(\omega)\mu_n(ds)(\omega), for any Borel set A, $$ with random transition kernels of the form $$\pi_n(.;.)(\omega)=\Pi(.;.)(\xi_n(\omega),Y_n(\omega)),$$ where $\xi$, $Y$ satisfy some ergodic conditions. The support of $\mu_n$ is called $S_n$. We also assume that $S_0$ is compact with regular border and that for any $x,y$ the kernel $\Pi(.;.)(x,y)$ has compact support. A random set $K_n+1$ is a safety area of level $\alpha$ if: [$i$)] $K_n+1$ \rm is a function of $S_0, S_1, ...,S_n.$ [$ii$)] $P(K_n+1 \cap S_n+1 \neq \emptyset)\leq \alpha.$ We present a method to find these safety areas and some related results.
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Title: Metric and Kernel Learning using a Linear Transformation
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Abstract: Metric and kernel learning are important in several machine learning applications. However, most existing metric learning algorithms are limited to learning metrics over low-dimensional data, while existing kernel learning algorithms are often limited to the transductive setting and do not generalize to new data points. In this paper, we study metric learning as a problem of learning a linear transformation of the input data. We show that for high-dimensional data, a particular framework for learning a linear transformation of the data based on the LogDet divergence can be efficiently kernelized to learn a metric (or equivalently, a kernel function) over an arbitrarily high dimensional space. We further demonstrate that a wide class of convex loss functions for learning linear transformations can similarly be kernelized, thereby considerably expanding the potential applications of metric learning. We demonstrate our learning approach by applying it to large-scale real world problems in computer vision and text mining.
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Title: Local likelihood estimation of local parameters for nonstationary random fields
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Abstract: We develop a weighted local likelihood estimate for the parameters that govern the local spatial dependency of a locally stationary random field. The advantage of this local likelihood estimate is that it smoothly downweights the influence of far away observations, works for irregular sampling locations, and when designed appropriately, can trade bias and variance for reducing estimation error. This paper starts with an exposition of our technique on the problem of estimating an unknown positive function when multiplied by a stationary random field. This example gives concrete evidence of the benefits of our local likelihood as compared to na\"ive local likelihoods where the stationary model is assumed throughout a neighborhood. We then discuss the difficult problem of estimating a bandwidth parameter that controls the amount of influence from distant observations. Finally we present a simulation experiment for estimating the local smoothness of a local Mat\'ern random field when observing the field at random sampling locations in $[0,1]^2$.
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Title: Learning Exponential Families in High-Dimensions: Strong Convexity and Sparsity
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Abstract: The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions, such as when there is some sparsity pattern of the optimal parameter. This work characterizes a certain strong convexity property of general exponential families, which allow their generalization ability to be quantified. In particular, we show how this property can be used to analyze generic exponential families under L_1 regularization.
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Title: D-optimal designs via a cocktail algorithm
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Abstract: A fast new algorithm is proposed for numerical computation of (approximate) D-optimal designs. This "cocktail algorithm" extends the well-known vertex direction method (VDM; Fedorov 1972) and the multiplicative algorithm (Silvey, Titterington and Torsney, 1978), and shares their simplicity and monotonic convergence properties. Numerical examples show that the cocktail algorithm can lead to dramatically improved speed, sometimes by orders of magnitude, relative to either the multiplicative algorithm or the vertex exchange method (a variant of VDM). Key to the improved speed is a new nearest neighbor exchange strategy, which acts locally and complements the global effect of the multiplicative algorithm. Possible extensions to related problems such as nonparametric maximum likelihood estimation are mentioned.
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Title: Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II
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Abstract: We prove a central limit theorem for a general class of adaptive Markov Chain Monte Carlo algorithms driven by sub-geometrically ergodic Markov kernels. We discuss in detail the special case of stochastic approximation. We use the result to analyze the asymptotic behavior of an adaptive version of the Metropolis Adjusted Langevin algorithm with a heavy tailed target density.
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Title: A Mirroring Theorem and its Application to a New Method of Unsupervised Hierarchical Pattern Classification
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Abstract: In this paper, we prove a crucial theorem called Mirroring Theorem which affirms that given a collection of samples with enough information in it such that it can be classified into classes and subclasses then (i) There exists a mapping which classifies and subclassifies these samples (ii) There exists a hierarchical classifier which can be constructed by using Mirroring Neural Networks (MNNs) in combination with a clustering algorithm that can approximate this mapping. Thus, the proof of the Mirroring theorem provides a theoretical basis for the existence and a practical feasibility of constructing hierarchical classifiers, given the maps. Our proposed Mirroring Theorem can also be considered as an extension to Kolmogrovs theorem in providing a realistic solution for unsupervised classification. The techniques we develop, are general in nature and have led to the construction of learning machines which are (i) tree like in structure, (ii) modular (iii) with each module running on a common algorithm (tandem algorithm) and (iv) selfsupervised. We have actually built the architecture, developed the tandem algorithm of such a hierarchical classifier and demonstrated it on an example problem.
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Title: Particle filtering within adaptive Metropolis Hastings sampling
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Abstract: We show that it is feasible to carry out exact Bayesian inference for non-Gaussian state space models using an adaptive Metropolis Hastings sampling scheme with the likelihood approximated by the particle filter. Furthermore, an adapyive independent Metropolis Hastings sampler based on a mixture of normals proposal is computationally much more efficient than an adaptive random walk proposal because the cost of constructing a good adaptive proposal is negligible compared to the cost of approximating the likelihood. Independent Metropolis Hastings proposals are also attractive because they are easy to run in parallel on multiple processors. We also show that when the particle filter is used, the marginal likelihood of any model is obtained in an efficient and unbiased manner, making model comparison straightforward.
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Title: Causal Inference on Discrete Data using Additive Noise Models
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Abstract: Inferring the causal structure of a set of random variables from a finite sample of the joint distribution is an important problem in science. Recently, methods using additive noise models have been suggested to approach the case of continuous variables. In many situations, however, the variables of interest are discrete or even have only finitely many states. In this work we extend the notion of additive noise models to these cases. We prove that whenever the joint distribution $\prob^(X,Y)$ admits such a model in one direction, e.g. $Y=f(X)+N, N \independent X$, it does not admit the reversed model $X=g(Y)+\tilde N, \tilde N \independent Y$ as long as the model is chosen in a generic way. Based on these deliberations we propose an efficient new algorithm that is able to distinguish between cause and effect for a finite sample of discrete variables. In an extensive experimental study we show that this algorithm works both on synthetic and real data sets.
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Title: Probability matrices, non-negative rank, and parameterizations of mixture models
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Abstract: In this paper we parameterize non-negative matrices of sum one and rank at most two. More precisely, we give a family of parameterizations using the least possible number of parameters. We also show how these parameterizations relate to a class of statistical models, known in Probability and Statistics as mixture models for contingency tables.
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Title: Feature-Weighted Linear Stacking
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Abstract: Ensemble methods, such as stacking, are designed to boost predictive accuracy by blending the predictions of multiple machine learning models. Recent work has shown that the use of meta-features, additional inputs describing each example in a dataset, can boost the performance of ensemble methods, but the greatest reported gains have come from nonlinear procedures requiring significant tuning and training time. Here, we present a linear technique, Feature-Weighted Linear Stacking (FWLS), that incorporates meta-features for improved accuracy while retaining the well-known virtues of linear regression regarding speed, stability, and interpretability. FWLS combines model predictions linearly using coefficients that are themselves linear functions of meta-features. This technique was a key facet of the solution of the second place team in the recently concluded Netflix Prize competition. Significant increases in accuracy over standard linear stacking are demonstrated on the Netflix Prize collaborative filtering dataset.
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Title: Strange Bedfellows: Quantum Mechanics and Data Mining
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Abstract: Last year, in 2008, I gave a talk titled \it Quantum Calisthenics. This year I am going to tell you about how the work I described then has spun off into a most unlikely direction. What I am going to talk about is how one maps the problem of finding clusters in a given data set into a problem in quantum mechanics. I will then use the tricks I described to let quantum evolution lets the clusters come together on their own.
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Title: An Optimal Method For Wake Detection In SAR Images Using Radon Transformation Combined With Wavelet Filters
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Abstract: A new fangled method for ship wake detection in synthetic aperture radar (SAR) images is explored here. Most of the detection procedure applies the Radon transform as its properties outfit more than any other transformation for the detection purpose. But still it holds problems when the transform is applied to an image with a high level of noise. Here this paper articulates the combination between the radon transformation and the shrinkage methods which increase the mode of wake detection process. The latter shrinkage method with RT maximize the signal to noise ratio hence it leads to most optimal detection of lines in the SAR images. The originality mainly works on the denoising segment of the proposed algorithm. Experimental work outs are carried over both in simulated and real SAR images. The detection process is more adequate with the proposed method and improves better than the conventional methods.
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Title: Novel Intrusion Detection using Probabilistic Neural Network and Adaptive Boosting
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Abstract: This article applies Machine Learning techniques to solve Intrusion Detection problems within computer networks. Due to complex and dynamic nature of computer networks and hacking techniques, detecting malicious activities remains a challenging task for security experts, that is, currently available defense systems suffer from low detection capability and high number of false alarms. To overcome such performance limitations, we propose a novel Machine Learning algorithm, namely Boosted Subspace Probabilistic Neural Network (BSPNN), which integrates an adaptive boosting technique and a semi parametric neural network to obtain good tradeoff between accuracy and generality. As the result, learning bias and generalization variance can be significantly minimized. Substantial experiments on KDD 99 intrusion benchmark indicate that our model outperforms other state of the art learning algorithms, with significantly improved detection accuracy, minimal false alarms and relatively small computational complexity.
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Title: Breast Cancer Detection Using Multilevel Thresholding
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Abstract: This paper presents an algorithm which aims to assist the radiologist in identifying breast cancer at its earlier stages. It combines several image processing techniques like image negative, thresholding and segmentation techniques for detection of tumor in mammograms. The algorithm is verified by using mammograms from Mammographic Image Analysis Society. The results obtained by applying these techniques are described.
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Title: Slow Learners are Fast
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Abstract: Online learning algorithms have impressive convergence properties when it comes to risk minimization and convex games on very large problems. However, they are inherently sequential in their design which prevents them from taking advantage of modern multi-core architectures. In this paper we prove that online learning with delayed updates converges well, thereby facilitating parallel online learning.
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Title: An Innovative Scheme For Effectual Fingerprint Data Compression Using Bezier Curve Representations
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Abstract: Naturally, with the mounting application of biometric systems, there arises a difficulty in storing and handling those acquired biometric data. Fingerprint recognition has been recognized as one of the most mature and established technique among all the biometrics systems. In recent times, with fingerprint recognition receiving increasingly more attention the amount of fingerprints collected has been constantly creating enormous problems in storage and transmission. Henceforth, the compression of fingerprints has emerged as an indispensable step in automated fingerprint recognition systems. Several researchers have presented approaches for fingerprint image compression. In this paper, we propose a novel and efficient scheme for fingerprint image compression. The presented scheme utilizes the Bezier curve representations for effective compression of fingerprint images. Initially, the ridges present in the fingerprint image are extracted along with their coordinate values using the approach presented. Subsequently, the control points are determined for all the ridges by visualizing each ridge as a Bezier curve. The control points of all the ridges determined are stored and are used to represent the fingerprint image. When needed, the fingerprint image is reconstructed from the stored control points using Bezier curves. The quality of the reconstructed fingerprint is determined by a formal evaluation. The proposed scheme achieves considerable memory reduction in storing the fingerprint.
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Title: Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?
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Abstract: The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk Metropolis algorithm. The proposal distribution has the following time-dependent covariance matrix at step $n+1$ \[ S_n = Cov(X_1,...,X_n) + \epsilon I, \] that is, the sample covariance matrix of the history of the chain plus a (small) constant $\epsilon>0$ multiple of the identity matrix $I$. The lower bound on the eigenvalues of $S_n$ induced by the factor $\epsilon I$ is theoretically convenient, but practically cumbersome, as a good value for the parameter $\epsilon$ may not always be easy to choose. This article considers variants of the AM algorithm that do not explicitly bound the eigenvalues of $S_n$ away from zero. The behaviour of $S_n$ is studied in detail, indicating that the eigenvalues of $S_n$ do not tend to collapse to zero in general.
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Title: Bayes estimators for phylogenetic reconstruction
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Abstract: Tree reconstruction methods are often judged by their accuracy, measured by how close they get to the true tree. Yet most reconstruction methods like ML do not explicitly maximize this accuracy. To address this problem, we propose a Bayesian solution. Given tree samples, we propose finding the tree estimate which is closest on average to the samples. This ``median'' tree is known as the Bayes estimator (BE). The BE literally maximizes posterior expected accuracy, measured in terms of closeness (distance) to the true tree. We discuss a unified framework of BE trees, focusing especially on tree distances which are expressible as squared euclidean distances. Notable examples include Robinson--Foulds distance, quartet distance, and squared path difference. Using simulated data, we show Bayes estimators can be efficiently computed in practice by hill climbing. We also show that Bayes estimators achieve higher accuracy, compared to maximum likelihood and neighbor joining.
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Title: Generalized Discriminant Analysis algorithm for feature reduction in Cyber Attack Detection System
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Abstract: This Generalized Discriminant Analysis (GDA) has provided an extremely powerful approach to extracting non linear features. The network traffic data provided for the design of intrusion detection system always are large with ineffective information, thus we need to remove the worthless information from the original high dimensional database. To improve the generalization ability, we usually generate a small set of features from the original input variables by feature extraction. The conventional Linear Discriminant Analysis (LDA) feature reduction technique has its limitations. It is not suitable for non linear dataset. Thus we propose an efficient algorithm based on the Generalized Discriminant Analysis (GDA) feature reduction technique which is novel approach used in the area of cyber attack detection. This not only reduces the number of the input features but also increases the classification accuracy and reduces the training and testing time of the classifiers by selecting most discriminating features. We use Artificial Neural Network (ANN) and C4.5 classifiers to compare the performance of the proposed technique. The result indicates the superiority of algorithm.
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Title: A New Computational Schema for Euphonic Conjunctions in Sanskrit Processing
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Abstract: Automated language processing is central to the drive to enable facilitated referencing of increasingly available Sanskrit E texts. The first step towards processing Sanskrit text involves the handling of Sanskrit compound words that are an integral part of Sanskrit texts. This firstly necessitates the processing of euphonic conjunctions or sandhis, which are points in words or between words, at which adjacent letters coalesce and transform. The ancient Sanskrit grammarian Panini's codification of the Sanskrit grammar is the accepted authority in the subject. His famed sutras or aphorisms, numbering approximately four thousand, tersely, precisely and comprehensively codify the rules of the grammar, including all the rules pertaining to sandhis. This work presents a fresh new approach to processing sandhis in terms of a computational schema. This new computational model is based on Panini's complex codification of the rules of grammar. The model has simple beginnings and is yet powerful, comprehensive and computationally lean.
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Title: ANN-based Innovative Segmentation Method for Handwritten text in Assamese
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Abstract: Artificial Neural Network (ANN) s has widely been used for recognition of optically scanned character, which partially emulates human thinking in the domain of the Artificial Intelligence. But prior to recognition, it is necessary to segment the character from the text to sentences, words etc. Segmentation of words into individual letters has been one of the major problems in handwriting recognition. Despite several successful works all over the work, development of such tools in specific languages is still an ongoing process especially in the Indian context. This work explores the application of ANN as an aid to segmentation of handwritten characters in Assamese- an important language in the North Eastern part of India. The work explores the performance difference obtained in applying an ANN-based dynamic segmentation algorithm compared to projection- based static segmentation. The algorithm involves, first training of an ANN with individual handwritten characters recorded from different individuals. Handwritten sentences are separated out from text using a static segmentation method. From the segmented line, individual characters are separated out by first over segmenting the entire line. Each of the segments thus obtained, next, is fed to the trained ANN. The point of segmentation at which the ANN recognizes a segment or a combination of several segments to be similar to a handwritten character, a segmentation boundary for the character is assumed to exist and segmentation performed. The segmented character is next compared to the best available match and the segmentation boundary confirmed.
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Title: Imputation Estimators Partially Correct for Model Misspecification
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Abstract: Inference problems with incomplete observations often aim at estimating population properties of unobserved quantities. One simple way to accomplish this estimation is to impute the unobserved quantities of interest at the individual level and then take an empirical average of the imputed values. We show that this simple imputation estimator can provide partial protection against model misspecification. We illustrate imputation estimators' robustness to model specification on three examples: mixture model-based clustering, estimation of genotype frequencies in population genetics, and estimation of Markovian evolutionary distances. In the final example, using a representative model misspecification, we demonstrate that in non-degenerate cases, the imputation estimator dominates the plug-in estimate asymptotically. We conclude by outlining a Bayesian implementation of the imputation-based estimation.
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Title: Comments on "Particle Markov chain Monte Carlo" by C. Andrieu, A. Doucet, and R. Hollenstein
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Abstract: This is the compilation of our comments submitted to the Journal of the Royal Statistical Society, Series B, to be published within the discussion of the Read Paper of Andrieu, Doucet and Hollenstein.
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Title: Examples as Interaction: On Humans Teaching a Computer to Play a Game
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Abstract: This paper reviews an experiment in human-computer interaction, where interaction takes place when humans attempt to teach a computer to play a strategy board game. We show that while individually learned models can be shown to improve the playing performance of the computer, their straightforward composition results in diluting what was earlier learned. This observation suggests that interaction cannot be easily distributed when one hopes to harness multiple human experts to develop a quality computer player. This is related to similar approaches in robot task learning and to classic approaches to human learning and reinforces the need to develop tools that facilitate the mix of human-based tuition and computer self-learning.
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Title: Irregular sets and Central Limit Theorems for dependent triangular arrays
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Abstract: In previous papers, we studied the asymptotic behaviour of $S_N(A,X)=(2N+1)^-d/2\sum_n \in A_N X_n,$ where $X$ is a centered, stationary and weakly dependent random field, and $A_N=A \cap [-N,N]^d$, $A \subset ^d$. This leads to the definition of asymptotically measurable sets, which enjoy the property that $S_N(A;X)$ has a Gaussian weak limit for any $X$ belonging to a certain class. Here we extend this type of results to the case of weakly dependent triangular arrays and present an application of this technique to regression models. Indeed, we prove that CLT and related results hold for $X_n^N=\varphi(\xi_n^N,Y_n^N), n \in ^d$, where $\varphi$ satisfies certain regularity conditions, $\xi$ and $Y$ are independent random fields, $\xi$ is weakly dependent and $Y$ satisfies some Strong Law of Large Numbers.
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Title: Identification and quantification of Granger causality between gene sets
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Abstract: Wiener and Granger have introduced an intuitive concept of causality between two variables which is based on the idea that an effect never occurs before its cause. Later, Geweke has generalized this concept to a multivariate Granger causality, i.e., n variables Granger-cause another variable. Although Granger causality is not "effective causality", this concept is useful to infer directionality and information flow in observational data. Granger causality is usually identified by using VAR models due to their simplicity. In the last few years, several VAR-based models were presented in order to model gene regulatory networks. Here, we generalize the multivariate Granger causality concept in order to identify Granger causalities between sets of gene expressions, i.e., whether a set of n genes Granger-causes another set of m genes, aiming at identifying and quantifying the flow of information between gene networks (or pathways). The concept of Granger causality for sets of variables is presented. Moreover, a method for its identification with a bootstrap test is proposed. This method is applied in simulated and also in actual biological gene expression data in order to model regulatory networks. This concept may be useful to understand the complete information flow from one network or pathway to the other, mainly in regulatory networks. Linking this concept to graph theory, sink and source can be generalized to node sets. Moreover, hub and centrality for sets of genes can be defined based on total information flow. Another application is in annotation, when the functionality of a set of genes is unknown, but this set is Granger caused by another set of genes which is well studied. Therefore, this information may be useful to infer or construct some hypothesis about the unknown set of genes.
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