1,357 research outputs found
A heuristics approach for computing the largest eigenvalue of a pairwise comparison matrix
Pairwise comparison matrices (PCMs) are widely used to capture subjective human judgements, especially in the context of the Analytic Hierarchy Process (AHP). Consistency of judgements is normally computed in AHP context in the form of consistency ratio (CR), which requires estimation of the largest eigenvalue (Lmax) of PCMs. Since many of these alternative methods do not require calculation of eigenvector, Lmax and hence the CR of a PCM cannot be easily estimated. We propose in this paper a simple heuristics for calculating Lmax without any need to use Eigenvector Method (EM). We illustrated the proposed procedure with larger size matrices. Simulation is used to compare the accuracy of the proposed heuristics procedure with actual Lmax for PCMs of various sizes. It has been found that the proposed heuristics is highly accurate, with errors less than 1%. The proposed procedure would avoid biases and help managers to make better decisions. The advantage of the proposed heuristics is that it can be easily calculated with simple calculations without any need for specialised mathematical procedures or software and is independent of the method used to derive priorities from PCMs
The Use of the AHP in Civil Engineering Projects
Most engineering, economic, social and institutional decisions are made with explicit notions of optimal behavior and implicit human motivations. In such a process, manipulation of both tangible and intangible data and satisfaction of multiple criteria are essential to the success of decision-making. In this paper an approach to multiple-criteria decision making known as the analytic hierarchy process (AHP) is presented. Some mathematical details of the procedure are briefly discussed. The application of the method to a real life civil engineering project for the selection of an appropriate bridge design is also presented.multi-criteria decision making, analytic hierarchy process, bridge design
Sparse integrative clustering of multiple omics data sets
High resolution microarrays and second-generation sequencing platforms are
powerful tools to investigate genome-wide alterations in DNA copy number,
methylation and gene expression associated with a disease. An integrated
genomic profiling approach measures multiple omics data types simultaneously in
the same set of biological samples. Such approach renders an integrated data
resolution that would not be available with any single data type. In this
study, we use penalized latent variable regression methods for joint modeling
of multiple omics data types to identify common latent variables that can be
used to cluster patient samples into biologically and clinically relevant
disease subtypes. We consider lasso [J. Roy. Statist. Soc. Ser. B 58 (1996)
267-288], elastic net [J. R. Stat. Soc. Ser. B Stat. Methodol. 67 (2005)
301-320] and fused lasso [J. R. Stat. Soc. Ser. B Stat. Methodol. 67 (2005)
91-108] methods to induce sparsity in the coefficient vectors, revealing
important genomic features that have significant contributions to the latent
variables. An iterative ridge regression is used to compute the sparse
coefficient vectors. In model selection, a uniform design [Monographs on
Statistics and Applied Probability (1994) Chapman & Hall] is used to seek
"experimental" points that scattered uniformly across the search domain for
efficient sampling of tuning parameter combinations. We compared our method to
sparse singular value decomposition (SVD) and penalized Gaussian mixture model
(GMM) using both real and simulated data sets. The proposed method is applied
to integrate genomic, epigenomic and transcriptomic data for subtype analysis
in breast and lung cancer data sets.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS578 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices
The aim of the paper is to obtain some theoretical and numerical properties of Saatyâs and Koczkodajâs inconsistencies of pairwise comparison matrices (PRM). In the case of 3 Ă 3 PRM, a differentiable one-to-one correspondence is given between Saatyâs inconsistency ratio and Koczkodajâs inconsistency index based on the elements of PRM. In order to make a comparison of Saatyâs and Koczkodajâs inconsistencies for 4 Ă 4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n Ă n PRM is formulated, the elements aij (i < j) of which were randomly chosen from the ratio scale ... ...
with equal probability 1/(2M â 1) and a ji is defined as 1/a ij . By statistical analysis, the empirical distributions of the maximal eigenvalues of the PRM depending on the dimension number are obtained. As the dimension number increases, the shape of distributions gets similar to that of the normal ones. Finally, the inconsistency of asymmetry is dealt with, showing a different type of inconsistency
Information Retrieval Performance Enhancement Using The Average Standard Estimator And The Multi-criteria Decision Weighted Set
Information retrieval is much more challenging than traditional small document collection retrieval. The main difference is the importance of correlations between related concepts in complex data structures. These structures have been studied by several information retrieval systems. This research began by performing a comprehensive review and comparison of several techniques of matrix dimensionality estimation and their respective effects on enhancing retrieval performance using singular value decomposition and latent semantic analysis. Two novel techniques have been introduced in this research to enhance intrinsic dimensionality estimation, the Multi-criteria Decision Weighted model to estimate matrix intrinsic dimensionality for large document collections and the Average Standard Estimator (ASE) for estimating data intrinsic dimensionality based on the singular value decomposition (SVD). ASE estimates the level of significance for singular values resulting from the singular value decomposition. ASE assumes that those variables with deep relations have sufficient correlation and that only those relationships with high singular values are significant and should be maintained. Experimental results over all possible dimensions indicated that ASE improved matrix intrinsic dimensionality estimation by including the effect of both singular values magnitude of decrease and random noise distracters. Analysis based on selected performance measures indicates that for each document collection there is a region of lower dimensionalities associated with improved retrieval performance. However, there was clear disagreement between the various performance measures on the model associated with best performance. The introduction of the multi-weighted model and Analytical Hierarchy Processing (AHP) analysis helped in ranking dimensionality estimation techniques and facilitates satisfying overall model goals by leveraging contradicting constrains and satisfying information retrieval priorities. ASE provided the best estimate for MEDLINE intrinsic dimensionality among all other dimensionality estimation techniques, and further, ASE improved precision and relative relevance by 10.2% and 7.4% respectively. AHP analysis indicates that ASE and the weighted model ranked the best among other methods with 30.3% and 20.3% in satisfying overall model goals in MEDLINE and 22.6% and 25.1% for CRANFIELD. The weighted model improved MEDLINE relative relevance by 4.4%, while the scree plot, weighted model, and ASE provided better estimation of data intrinsic dimensionality for CRANFIELD collection than Kaiser-Guttman and Percentage of variance. ASE dimensionality estimation technique provided a better estimation of CISI intrinsic dimensionality than all other tested methods since all methods except ASE tend to underestimate CISI document collection intrinsic dimensionality. ASE improved CISI average relative relevance and average search length by 28.4% and 22.0% respectively. This research provided evidence supporting a system using a weighted multi-criteria performance evaluation technique resulting in better overall performance than a single criteria ranking model. Thus, the weighted multi-criteria model with dimensionality reduction provides a more efficient implementation for information retrieval than using a full rank model
Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices
The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 Ă 3 and 4 Ă 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants
Consistency of the decision-maker in pair-wise comparisons
Most authors assume that the natural behaviour of the
decision-maker is being inconsistent. This paper investigates the main sources of inconsistency and analyses methods for reducing or eliminating inconsistency. Decision support systems can contain interactive modules for
that purpose. In a system with consistency control, there are three stages. First, consistency should be checked: a consistency measure is needed. Secondly, approval or rejection has to be decided: a threshold value of
inconsistency measure is needed. Finally, if inconsistency is âhighâ, corrections
have to be made: an inconsistency reducing method is needed. This paper reviews the difficulties in all stages. An entirely different approach is to elaborate a decision support system in order to force the decision-maker to give consistent values in each step of answering pair-wise comparison questions. An interactive questioning procedure resulting in consistent (sub) matrices has been demonstrated
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