Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 900

Search results for: Bhattacharyya distance.

900 Distances over Incomplete Diabetes and Breast Cancer Data Based on Bhattacharyya Distance

Authors: Loai AbdAllah, Mahmoud Kaiyal

Abstract:

Missing values in real-world datasets are a common problem. Many algorithms were developed to deal with this problem, most of them replace the missing values with a fixed value that was computed based on the observed values. In our work, we used a distance function based on Bhattacharyya distance to measure the distance between objects with missing values. Bhattacharyya distance, which measures the similarity of two probability distributions. The proposed distance distinguishes between known and unknown values. Where the distance between two known values is the Mahalanobis distance. When, on the other hand, one of them is missing the distance is computed based on the distribution of the known values, for the coordinate that contains the missing value. This method was integrated with Wikaya, a digital health company developing a platform that helps to improve prevention of chronic diseases such as diabetes and cancer. In order for Wikaya’s recommendation system to work distance between users need to be measured. Since there are missing values in the collected data, there is a need to develop a distance function distances between incomplete users profiles. To evaluate the accuracy of the proposed distance function in reflecting the actual similarity between different objects, when some of them contain missing values, we integrated it within the framework of k nearest neighbors (kNN) classifier, since its computation is based only on the similarity between objects. To validate this, we ran the algorithm over diabetes and breast cancer datasets, standard benchmark datasets from the UCI repository. Our experiments show that kNN classifier using our proposed distance function outperforms the kNN using other existing methods.

Keywords: Missing values, distance metric, Bhattacharyya distance.

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899 A Distance Function for Data with Missing Values and Its Application

Authors: Loai AbdAllah, Ilan Shimshoni

Abstract:

Missing values in data are common in real world applications. Since the performance of many data mining algorithms depend critically on it being given a good metric over the input space, we decided in this paper to define a distance function for unlabeled datasets with missing values. We use the Bhattacharyya distance, which measures the similarity of two probability distributions, to define our new distance function. According to this distance, the distance between two points without missing attributes values is simply the Mahalanobis distance. When on the other hand there is a missing value of one of the coordinates, the distance is computed according to the distribution of the missing coordinate. Our distance is general and can be used as part of any algorithm that computes the distance between data points. Because its performance depends strongly on the chosen distance measure, we opted for the k nearest neighbor classifier to evaluate its ability to accurately reflect object similarity. We experimented on standard numerical datasets from the UCI repository from different fields. On these datasets we simulated missing values and compared the performance of the kNN classifier using our distance to other three basic methods. Our  experiments show that kNN using our distance function outperforms the kNN using other methods. Moreover, the runtime performance of our method is only slightly higher than the other methods.

Keywords: Missing values, Distance metric, Bhattacharyya distance.

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898 3D Liver Segmentation from CT Images Using a Level Set Method Based on a Shape and Intensity Distribution Prior

Authors: Nuseiba M. Altarawneh, Suhuai Luo, Brian Regan, Guijin Tang

Abstract:

Liver segmentation from medical images poses more challenges than analogous segmentations of other organs. This contribution introduces a liver segmentation method from a series of computer tomography images. Overall, we present a novel method for segmenting liver by coupling density matching with shape priors. Density matching signifies a tracking method which operates via maximizing the Bhattacharyya similarity measure between the photometric distribution from an estimated image region and a model photometric distribution. Density matching controls the direction of the evolution process and slows down the evolving contour in regions with weak edges. The shape prior improves the robustness of density matching and discourages the evolving contour from exceeding liver’s boundaries at regions with weak boundaries. The model is implemented using a modified distance regularized level set (DRLS) model. The experimental results show that the method achieves a satisfactory result. By comparing with the original DRLS model, it is evident that the proposed model herein is more effective in addressing the over segmentation problem. Finally, we gauge our performance of our model against matrices comprising of accuracy, sensitivity, and specificity.

Keywords: Bhattacharyya distance, distance regularized level set (DRLS) model, liver segmentation, level set method.

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897 Differences in Students` Satisfaction with Distance Learning Studies

Authors: Ana Horvat, Maja Krsmanovic, Mladen Djuric

Abstract:

Rapid growth of distance learning resulted in importance to conduct research on students- satisfaction with distance learning because differences in students- satisfaction might influence educational opportunities for learning in a relevant Web-based environment. In line with this, this paper deals with satisfaction of students with distance module at Faculty of organizational sciences (FOS) in Serbia as well as some factors affecting differences in their satisfaction . We have conducted a research on a population of 68 first-year students of distance learning studies at FOS. Using statistical techniques, we have found out that there is no significant difference in students- satisfaction with distance learning module between men and women. In the same way, we also concluded that there is a difference in satisfaction with distance learning module regarding to student-s perception of opportunity to gain knowledge as the classic students.

Keywords: distance learning, students' satisfaction

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896 OWA Operators in Generalized Distances

Authors: José M. Merigó, Anna M. Gil-Lafuente

Abstract:

Different types of aggregation operators such as the ordered weighted quasi-arithmetic mean (Quasi-OWA) operator and the normalized Hamming distance are studied. We introduce the use of the OWA operator in generalized distances such as the quasiarithmetic distance. We will call these new distance aggregation the ordered weighted quasi-arithmetic distance (Quasi-OWAD) operator. We develop a general overview of this type of generalization and study some of their main properties such as the distinction between descending and ascending orders. We also consider different families of Quasi-OWAD operators such as the Minkowski ordered weighted averaging distance (MOWAD) operator, the ordered weighted averaging distance (OWAD) operator, the Euclidean ordered weighted averaging distance (EOWAD) operator, the normalized quasi-arithmetic distance, etc.

Keywords: Aggregation operators, Distance measures, Quasi- OWA operator.

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895 Analyzing Methods of the Relation between Concepts based on a Concept Hierarchy

Authors: Ke Lu, Tetsuya Furukawa

Abstract:

Data objects are usually organized hierarchically, and the relations between them are analyzed based on a corresponding concept hierarchy. The relation between data objects, for example how similar they are, are usually analyzed based on the conceptual distance in the hierarchy. If a node is an ancestor of another node, it is enough to analyze how close they are by calculating the distance vertically. However, if there is not such relation between two nodes, the vertical distance cannot express their relation explicitly. This paper tries to fill this gap by improving the analysis method for data objects based on hierarchy. The contributions of this paper include: (1) proposing an improved method to evaluate the vertical distance between concepts; (2) defining the concept horizontal distance and a method to calculate the horizontal distance; and (3) discussing the methods to confine a range by the horizontal distance and the vertical distance, and evaluating the relation between concepts.

Keywords: Concept Hierarchy, Horizontal Distance, Relation Analysis, Vertical Distance

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894 Using the OWA Operator in the Minkowski Distance

Authors: José M. Merigó, Anna M. Gil-Lafuente

Abstract:

We study different types of aggregation operators such as the ordered weighted averaging (OWA) operator and the generalized OWA (GOWA) operator. We analyze the use of OWA operators in the Minkowski distance. We will call these new distance aggregation operator the Minkowski ordered weighted averaging distance (MOWAD) operator. We give a general overview of this type of generalization and study some of their main properties. We also analyze a wide range of particular cases found in this generalization such as the ordered weighted averaging distance (OWAD) operator, the Euclidean ordered weighted averaging distance (EOWAD) operator, the normalized Minkowski distance, etc. Finally, we give an illustrative example of the new approach where we can see the different results obtained by using different aggregation operators.

Keywords: Aggregation operators, Minkowski distance, OWA operators, Selection of strategies.

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893 The Distance between a Point and a Bezier Curveon a Bezier Surface

Authors: Wen-Haw Chen, Sheng-Gwo Chen

Abstract:

The distance between two objects is an important problem in CAGD, CAD and CG etc. It will be presented in this paper that a simple and quick method to estimate the distance between a point and a Bezier curve on a Bezier surface.

Keywords: Geodesic-like curve, distance, projection, Bezier.

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892 Interaction at a Distance – An Approach for Redesigning for Distance Education

Authors: Martin Henkel

Abstract:

Different forms of interaction are an integral part of modern courses. Traditional courses held on-campus might focus on teacher-student interaction, or student-student interaction, or both. However when these traditional on-campus courses are to be held as distance courses there is a risk that these well-designed interactions will be difficult or impossible to uphold. For example, studentstudent interaction in traditional project assignments might not work well if the students are scattered across the world. Thus, even a welldesigned traditional on-site course cannot without modification be turned into a distance course. Traditional on-site courses simply have to be redesigned to become true distance courses. This paper describes a structured approach which facilitates the redesign of a traditional course into a distance course. The approach is based on that the desired forms of course flexibility are identified, and thereafter that the course activities are redesigned to facilitate interaction in a distance course. The approach is making use of known patterns of pedagogic interaction and existing guidelines for distance education design. The approach is illustrated with an example course in the field of information systems design.

Keywords: Distance education, interaction in education, course design.

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891 The Traits That Facilitate Successful Student Performance in Distance Education: The Case of the Distance Education Unit at European University Cyprus

Authors: D. Vlachopoulos, G. Tsokkas

Abstract:

Although it is not intended to identify distance education students as a homogeneous group, recent research has demonstrated that there are some demographic and personality common traits among most of them that provide the basis for the description of a typical distance learning student. The purpose of this paper is to describe these common traits and to facilitate their learning journey within a distance education program. The described research is an initiative of the Distance Education Unit at the European University Cyprus (Laureate International Universities) in the context of its action for the improvement of the students’ performance.

Keywords: Distance education students, successful student performance, European University Cyprus, common traits.

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890 Design and Development of an MPH Program for Distance Education Delivery

Authors: Steven R. Hawks

Abstract:

The Master-s of Public Health (MPH) degree is growing in popularity among a number of higher education institutions throughout the world as a distance education graduate program. This paper offers an overview of program design and development strategies that promote successful distance delivery of MPH programs. Design and development challenges are discussed in terms of type of distance delivery, accreditation, student demand, faculty development, user needs, course content, and marketing strategies. The ongoing development of a distance education MPH program at Utah State University will be used to highlight and consider various aspects of this important but challenging process.

Keywords: Public health, course content, distance education, higher education, graduate students.

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889 Distance Estimation for Radar Systems Using DS-UWB Signals

Authors: Youngpo Lee, Seokho Yoon

Abstract:

In this paper, we propose a distance estimation scheme for radar systems using direct sequence ultra wideband (DS-UWB) signals. The proposed distance estimation scheme averages out the noise by accumulating the correlator outputs of the radar, and thus, helps the radar to employ a short-length DS-UWB signal reducing the correlation processing time. Numerical results confirm that the proposed distance estimation scheme provides a better estimation performance and a reduced correlation processing time compared with those of the conventional DS-UWB radars.

Keywords: Radar, DS-UWB, distance estimation, correlation accumulation.

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888 Using Mean-Shift Tracking Algorithms for Real-Time Tracking of Moving Images on an Autonomous Vehicle Testbed Platform

Authors: Benjamin Gorry, Zezhi Chen, Kevin Hammond, Andy Wallace, Greg Michaelson

Abstract:

This paper describes new computer vision algorithms that have been developed to track moving objects as part of a long-term study into the design of (semi-)autonomous vehicles. We present the results of a study to exploit variable kernels for tracking in video sequences. The basis of our work is the mean shift object-tracking algorithm; for a moving target, it is usual to define a rectangular target window in an initial frame, and then process the data within that window to separate the tracked object from the background by the mean shift segmentation algorithm. Rather than use the standard, Epanechnikov kernel, we have used a kernel weighted by the Chamfer distance transform to improve the accuracy of target representation and localization, minimising the distance between the two distributions in RGB color space using the Bhattacharyya coefficient. Experimental results show the improved tracking capability and versatility of the algorithm in comparison with results using the standard kernel. These algorithms are incorporated as part of a robot test-bed architecture which has been used to demonstrate their effectiveness.

Keywords: Hume, functional programming, autonomous vehicle, pioneer robot, vision.

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887 Fiber Lens Structure for Large Distance Measurement

Authors: Jaemyoung Lee

Abstract:

We propose a new fiber lens structure for large distance measurement in which a polymer layer is added to a conventional fiber lens. The proposed fiber lens can adjust the working distance by properly choosing the refractive index and thickness of the polymer layer. In our numerical analysis for the fiber lens radius of 120 μm, the working distance of the proposed fiber lens is about 10 mm which is about 30 times larger than conventional fiber lens.

Keywords: fiber lens, distance measurement, collimation.

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886 The Net as a Living Experience of Distance Motherhood within Italian Culture

Authors: C. Papapicco

Abstract:

Motherhood is an existential human relationship that lasts for the whole life and is always interwoven with subjectivity and culture. As a result of the brain drain, the motherhood becomes motherhood at distance. Starting from the hypothesis that re-signification of the mother at distance practices is culturally relevant; the research aims to understand the experience of mother at a distance in order to extrapolate the strategies of management of the empty nest. Specifically, the research aims to evaluate the experience of a brain drain’s mother, who created a blog that intends to take care of other parents at a distance. Actually, the blog is the only artifact symbol of the Italian culture of motherhood at distance. In the research, a Netnographic Analysis of the blog mammedicervelliinfuga.com is offered with the aim of understanding if the online world becomes an opportunity to manage the role of mother at a distance. A narrative interview with the blog creator was conducted and then the texts were analyzed by means of a Diatextual Analysis approach. It emerged that the migration projects of talented children take on different meanings and representations for parents. Thus, it is shown that the blog becomes a new form of understanding and practicing motherhood at a distance.

Keywords: Brain drain, diatextual analysis, distance motherhood blog, online and offline narrations.

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885 A New Edit Distance Method for Finding Similarity in Dna Sequence

Authors: Patsaraporn Somboonsak, Mud-Armeen Munlin

Abstract:

The P-Bigram method is a string comparison methods base on an internal two characters-based similarity measure. The edit distance between two strings is the minimal number of elementary editing operations required to transform one string into the other. The elementary editing operations include deletion, insertion, substitution two characters. In this paper, we address the P-Bigram method to sole the similarity problem in DNA sequence. This method provided an efficient algorithm that locates all minimum operation in a string. We have been implemented algorithm and found that our program calculated that smaller distance than one string. We develop PBigram edit distance and show that edit distance or the similarity and implementation using dynamic programming. The performance of the proposed approach is evaluated using number edit and percentage similarity measures.

Keywords: Edit distance, String Matching, String Similarity

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884 The Survey and the Comparison of Maximum Likelihood, Mahalanobis Distance and Minimum Distance Methods in Preparing Landuse Map in the Western Part of Isfahan Province

Authors: Ali Gholami, M.Esfadiari, M.H.Masihabadi

Abstract:

In this research three methods of Maximum Likelihood, Mahalanobis Distance and Minimum Distance were analyzed in the Western part of Isfahan province in the Iran country. For this purpose, the IRS satellite images and various land preparation uses in region including rangelands, irrigation farming, dry farming, gardens and urban areas were separated and identified. In these methods, matrix error and Kappa index were calculated and accuracy of each method, based on percentages: 53.13, 56.64 and 48.44, were obtained respectively. Considering the low accuracy of these methods to separate land uses due to spread of the land uses, it-s suggested the visual interpretation of the map, to preparing the land use map in this region. The map prepared by visual interpretation is in high accuracy if it will be accompany with the visit of the region.

Keywords: Aghche Region, land use map, MaximumLikelihood, Mahalanobis Distance and Minimum Distance.

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883 Intrusion Detection based on Distance Combination

Authors: Joffroy Beauquier, Yongjie Hu

Abstract:

The intrusion detection problem has been frequently studied, but intrusion detection methods are often based on a single point of view, which always limits the results. In this paper, we introduce a new intrusion detection model based on the combination of different current methods. First we use a notion of distance to unify the different methods. Second we combine these methods using the Pearson correlation coefficients, which measure the relationship between two methods, and we obtain a combined distance. If the combined distance is greater than a predetermined threshold, an intrusion is detected. We have implemented and tested the combination model with two different public data sets: the data set of masquerade detection collected by Schonlau & al., and the data set of program behaviors from the University of New Mexico. The results of the experiments prove that the combination model has better performances.

Keywords: Intrusion detection, combination, distance, Pearson correlation coefficients.

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882 A Study on Inference from Distance Variables in Hedonic Regression

Authors: Yan Wang, Yasushi Asami, Yukio Sadahiro

Abstract:

In urban area, several landmarks may affect housing price and rents, and hedonic analysis should employ distance variables corresponding to each landmarks. Unfortunately, the effects of distances to landmarks on housing prices are generally not consistent with the true price. These distance variables may cause magnitude error in regression, pointing a problem of spatial multicollinearity. In this paper, we provided some approaches for getting the samples with less bias and method on locating the specific sampling area to avoid the multicollinerity problem in two specific landmarks case.

Keywords: Landmarks, hedonic regression, distance variables, collinearity, multicollinerity.

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881 Accelerating GLA with an M-Tree

Authors: Olli Luoma, Johannes Tuikkala, Olli Nevalainen

Abstract:

In this paper, we propose a novel improvement for the generalized Lloyd Algorithm (GLA). Our algorithm makes use of an M-tree index built on the codebook which makes it possible to reduce the number of distance computations when the nearest code words are searched. Our method does not impose the use of any specific distance function, but works with any metric distance, making it more general than many other fast GLA variants. Finally, we present the positive results of our performance experiments.

Keywords: Clustering, GLA, M-Tree, Vector Quantization .

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880 Accuracy of Divergence Measures for Detection of Abrupt Changes

Authors: P. Bergl

Abstract:

Numerous divergence measures (spectral distance, cepstral distance, difference of the cepstral coefficients, Kullback-Leibler divergence, distance given by the General Likelihood Ratio, distance defined by the Recursive Bayesian Changepoint Detector and the Mahalanobis measure) are compared in this study. The measures are used for detection of abrupt spectral changes in synthetic AR signals via the sliding window algorithm. Two experiments are performed; the first is focused on detection of single boundary while the second concentrates on detection of a couple of boundaries. Accuracy of detection is judged for each method; the measures are compared according to results of both experiments.

Keywords: Abrupt changes detection, autoregressive model, divergence measure.

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879 A Research on Inference from Multiple Distance Variables in Hedonic Regression – Focus on Three Variables

Authors: Yan Wang, Yasushi Asami, Yukio Sadahiro

Abstract:

In urban context, urban nodes such as amenity or hazard will certainly affect house price, while classic hedonic analysis will employ distance variables measured from each urban nodes. However, effects from distances to facilities on house prices generally do not represent the true price of the property. Distance variables measured on the same surface are suffering a problem called multicollinearity, which is usually presented as magnitude variance and mean value in regression, errors caused by instability. In this paper, we provided a theoretical framework to identify and gather the data with less bias, and also provided specific sampling method on locating the sample region to avoid the spatial multicollinerity problem in three distance variable’s case.

Keywords: Hedonic regression, urban node, distance variables, multicollinerity, collinearity.

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878 The Prospects and Challenges of Open Learning and Distance Education in Malawi

Authors: Andrew Chimpololo

Abstract:

Open and distance learning is a fairly new concept in Malawi. The major public provider, the Malawi College of Distance Education, rolled out its activities only about 40 years ago. Over the years, the demand for distance education has tremendously increased. The present government has displayed positive political will to uplift ODL as outlined in the Malawi Growth and Development Strategy as well as the National Education Sector Plan. A growing national interest in education coupled with political stability and a booming ICT industry also raise hope for success. However, a fragile economy with a GNI per capita of -US$ 200 over the last decade, poor public funding, erratic power supply and lack of expertise put strain on efforts towards the promotion of ODL initiatives. Despite the challenges, the nation appears determined to go flat out and explore all possible avenues that could revolutionise education access and equity through ODL.

Keywords: challenges, distance education, Malawi, openlearning, prospects.

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877 Similarity Measure Functions for Strategy-Based Biometrics

Authors: Roman V. Yampolskiy, Venu Govindaraju

Abstract:

Functioning of a biometric system in large part depends on the performance of the similarity measure function. Frequently a generalized similarity distance measure function such as Euclidian distance or Mahalanobis distance is applied to the task of matching biometric feature vectors. However, often accuracy of a biometric system can be greatly improved by designing a customized matching algorithm optimized for a particular biometric application. In this paper we propose a tailored similarity measure function for behavioral biometric systems based on the expert knowledge of the feature level data in the domain. We compare performance of a proposed matching algorithm to that of other well known similarity distance functions and demonstrate its superiority with respect to the chosen domain.

Keywords: Behavioral Biometrics, Euclidian Distance, Matching, Similarity Measure.

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876 Investigating Performance of Numerical Distance Relay with Higher Order Antialiasing Filter

Authors: Venkatesh C., K. Shanti Swarup

Abstract:

This paper investigates the impact on operating time delay and relay maloperation when 1st,2nd and 3rd order analog antialiasing filters are used in numerical distance protection. RC filter with cut-off frequency 90 Hz is used. Simulations are carried out for different SIR (Source to line Impedance Ratio), load, fault type and fault conditions using SIMULINK, where the voltage and current signals are fed online to the developed numerical distance relay model. Matlab is used for plotting the impedance trajectory. Investigation results shows that, about 75 % of the simulated cases, numerical distance relay operating time is not increased even-though there is a time delay when higher order filters are used. Relay maloperation (selectivity) also reduces (increases) when higher order filters are used in numerical distance protection.

Keywords: Antialiasing, capacitive voltage transformers, delay estimation, discrete Fourier transform (DFT), distance measurement, low-pass filters, source to line impedance ratio (SIR), protective relaying.

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875 Maximum Distance Separable b-Symbol Repeated-Root γ-Constacylic Codes over a Finite Chain Ring of Length 2

Authors: Jamal Laaouine, Mohammed Elhassani Charkani

Abstract:

Let p be a prime and let b be an integer. MDS b-symbol codes are a direct generalization of MDS codes. The γ-constacyclic codes of length pˢ over the finite commutative chain ring Fₚm [u]/ < u² > had been classified into four distinct types, where is a nonzero element of the field Fₚm. Let C₃ be a code of Type 3. In this paper, we obtain the b-symbol distance db(C₃) of the code C₃. Using this result, necessary and sufficient conditions under which C₃ is an MDS b-symbol code are given.

Keywords: constacyclic code, repeated-root code, maximum distance separable, MDS codes, b-symbol distance, finite chain rings

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874 Graphs with Metric Dimension Two-A Characterization

Authors: Sudhakara G, Hemanth Kumar A.R

Abstract:

In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. β (G) = 2 ) is characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever β (G) = 2 . We also find a bound for cardinality of any distance partite set with reference to a given vertex, when ever β (G) = 2 . Also, in a graph G with β (G) = 2 , a bound for cardinality of any distance partite set as well as a bound for number of vertices in any sub graph H of G is obtained in terms of diam H .

Keywords: Metric basis, Distance partition, Metric dimension.

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873 Web Application for Evaluating Tests in Distance Learning Systems

Authors: Bogdan Walek, Vladimir Bradac, Radim Farana

Abstract:

Distance learning systems offer useful methods of learning and usually contain a final course test or another form of test. The paper proposes a web application for evaluating tests using an expert system in distance learning systems. The proposed web application is appropriate for didactic tests or tests with results for subsequent studying follow-up courses. The web application works with test questions and uses an expert system and LFLC tool for test evaluation. After test evaluation, the results are visualized and shown to the student.

Keywords: Distance learning, test, uncertainty, fuzzy, expert system, student.

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872 Terminal Wiener Index for Graph Structures

Authors: J. Baskar Babujee, J. Senbagamalar,

Abstract:

The topological distance between a pair of vertices i and j, which is denoted by d(vi, vj), is the number of edges of the shortest path joining i and j. The Wiener index W(G) is the sum of distances between all pairs of vertices of a graph G. W(G) = i

Keywords: Graph, Degree, Distance, Pendent vertex, Wiener index, Tree.

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871 The Mutated Distance between Two Mixture Trees

Authors: Wan Chian Li, Justie Su-Tzu Juan, Yi-Chun Wang, Shu-Chuan Chen

Abstract:

The evolutionary tree is an important topic in bioinformation. In 2006, Chen and Lindsay proposed a new method to build the mixture tree from DNA sequences. Mixture tree is a new type evolutionary tree, and it has two additional information besides the information of ordinary evolutionary tree. One of the information is time parameter, and the other is the set of mutated sites. In 2008, Lin and Juan proposed an algorithm to compute the distance between two mixture trees. Their algorithm computes the distance with only considering the time parameter between two mixture trees. In this paper, we proposes a method to measure the similarity of two mixture trees with considering the set of mutated sites and develops two algorithm to compute the distance between two mixture trees. The time complexity of these two proposed algorithms are O(n2 × max{h(T1), h(T2)}) and O(n2), respectively

Keywords: evolutionary tree, mixture tree, mutated site, distance.

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