L10: Linear discriminants analysis
|
|
- Loraine Cunningham
- 8 years ago
- Views:
Transcription
1 L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU
2 Lnear dscrmnant analyss, two-classes Objectve LDA seeks to reduce dmensonalty whle preservng as much of the class dscrmnatory nformaton as possble Assume we have a set of D-dmensonal samples x (, x (, x (N, N of whch belong to class ω, and N to class ω We seek to obtan a scalar y by projectng the samples x onto a lne y = w T x Of all the possble lnes we would lke to select the one that maxmzes the separablty of the scalars x x x CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU x
3 In order to fnd a good projecton vector, we need to defne a measure of separaton The mean vector of each class n x-space and y-space s μ = N x ω x and μ = N y ω y = N x ω w T x = w T μ We could then choose the dstance between the projected means as our objectve functon J w = μ μ = w T μ μ However, the dstance between projected means s not a good measure snce t does not account for the standard devaton wthn classes x Ths axs yelds better class separablty x Ths axs has a larger dstance between means CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU
4 Fsher s soluton Fsher suggested maxmzng the dfference between the means, normalzed by a measure of the wthn-class scatter For each class we defne the scatter, an equvalent of the varance, as s = y μ y ω where the quantty s + s s called the wthn-class scatter of the projected examples The Fsher lnear dscrmnant s defned as the lnear functon w T x that maxmzes the crteron functon J w = μ μ s +s Therefore, we are lookng for a projecton where examples from the same class are projected very close to each other and, at the same tme, the projected means are as farther apart as possble x x CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU
5 To fnd the optmum w, we must express J(w) as a functon of w Frst, we defne a measure of the scatter n feature space x T S = x ω x μ x μ S + S = S W where S W s called the wthn-class scatter matrx The scatter of the projecton y can then be expressed as a functon of the scatter matrx n feature space x s = y ω y μ = w T x w T x ω μ = = x ω w T x μ x μ T w = w T S w s + s = w T S W w Smlarly, the dfference between the projected means can be expressed n terms of the means n the orgnal feature space μ μ = w T μ w T μ = w T T μ μ μ μ w = w T S B w The matrx S B s called the between-class scatter. Note that, snce S B s the outer product of two vectors, ts rank s at most one We can fnally express the Fsher crteron n terms of S W and S B as J w = wt S B w w T S W w CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU S B
6 To fnd the maxmum of J(w) we derve and equate to zero d dw J w = d dw w T S W w d wt S B w w T S B w w T S W w = 0 w T S dw B w d wt S W w dw w T S W w S B w w T S B w S W w = 0 Dvdng by w T S W w = 0 w T S W w S w T S W w Bw wt S B w S w T S W w Ww = 0 S B w JS W w = 0 S W S B w Jw = 0 Solvng the generalzed egenvalue problem (S W S B w = Jw) yelds w = arg max wt S B w w T S W w = S W μ μ Ths s know as Fsher s lnear dscrmnant (96), although t s not a dscrmnant but rather a specfc choce of drecton for the projecton of the data down to one dmenson CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU 6
7 Example Compute the LDA projecton for the followng D dataset X = {(,), (,), (,), (,6), (,)} X = {(9,0), (6,8), (9,), (8,7), (0,8)} SOLUTION (by hand) The class statstcs are S =.8..6 S = x w LDA μ =.0.6 T ; μ = T The wthn- and between-class scatter are S B = S W =.6..8 The LDA projecton s then obtaned as the soluton of the generalzed egenvalue problem S W S B v = λv S W S B λi = Or drectly by w = S W μ μ v v =.6 v v v v =.9.9 =.9.9 T.89 λ λ x = 0 λ =.6 CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU 7
8 LDA, C classes Fsher s LDA generalzes gracefully for C-class problems Instead of one projecton y, we wll now seek (C ) projectons [y, y, y C ] by means of (C ) projecton vectors w arranged by columns nto a projecton matrx W = [w w w C ]: y = w T x y = W T x Dervaton x The wthn-class scatter generalzes as S W = C = S where S = x μ x μ T x ω and μ = N x ω x And the between-class scatter becomes S B = N μ μ μ μ T C = where μ = N x x = N C = N μ Matrx S T = S B + S W s called the total scatter S W S W S B S B S B S W x CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU 8
9 Smlarly, we defne the mean vector and scatter matrces for the projected samples as μ = N μ = N C = T y ω y S W = y μ y μ y ω y y S B = C = N μ μ μ μ T From our dervaton for the two-class problem, we can wrte S W = W T S W W S B = W T S B W Recall that we are lookng for a projecton that maxmzes the rato of between-class to wthn-class scatter. Snce the projecton s no longer a scalar (t has C dmensons), we use the determnant of the scatter matrces to obtan a scalar objectve functon J W = S B S W = WT S B W W T S W W And we wll seek the projecton matrx W that maxmzes ths rato CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU 9
10 It can be shown that the optmal projecton matrx W s the one whose columns are the egenvectors correspondng to the largest egenvalues of the followng generalzed egenvalue problem NOTES W = w w w C = arg max WT S B W W T S W W S B λ S W w = 0 S B s the sum of C matrces of rank and the mean vectors are constraned by C C = μ = μ Therefore, S B wll be of rank (C ) or less Ths means that only (C ) of the egenvalues λ wll be non-zero The projectons wth maxmum class separablty nformaton are the egenvectors correspondng to the largest egenvalues of S W S B LDA can be derved as the Maxmum Lkelhood method for the case of normal class-condtonal denstes wth equal covarance matrces CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU 0
11 LDA vs. PCA Ths example llustrates the performance of PCA and LDA on an odor recognton problem Fve types of coffee beans were presented to an array of gas sensors For each coffee type, snffs were performed and the response of the gas sensor array was processed n order to obtan a 60-dmensonal feature vector Results From the D scatter plots t s clear that LDA outperforms PCA n terms of class dscrmnaton Ths s one example where the dscrmnatory nformaton s not algned wth the drecton of maxmum varance PCA Sensor response normalzed data Sulaw esy Kenya Araban Sumatra Colomba LDA axs axs 00 axs axs 0. axs axs CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU
12 Lmtatons of LDA LDA produces at most C feature projectons If the classfcaton error estmates establsh that more features are needed, some other method must be employed to provde those addtonal features LDA s a parametrc method (t assumes unmodal Gaussan lkelhoods) If the dstrbutons are sgnfcantly non-gaussan, the LDA projectons may not preserve complex structure n the data needed for classfcaton = = = = x LDA wll also fal f dscrmnatory nformaton s not n the mean but n the varance of the data P C A L D A x CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU
13 Varants of LDA Non-parametrc LDA (Fukunaga) NPLDA relaxes the unmodal Gaussan assumpton by computng S B usng local nformaton and the knn rule. As a result of ths The matrx S B s full-rank, allowng us to extract more than (C ) features The projectons are able to preserve the structure of the data more closely Orthonormal LDA (Okada and Tomta) OLDA computes projectons that maxmze the Fsher crteron and, at the same tme, are par-wse orthonormal The method used n OLDA combnes the egenvalue soluton of S W S B and the Gram- Schmdt orthonormalzaton procedure OLDA sequentally fnds axes that maxmze the Fsher crteron n the subspace orthogonal to all features already extracted OLDA s also capable of fndng more than (C ) features Generalzed LDA (Lowe) GLDA generalzes the Fsher crteron by ncorporatng a cost functon smlar to the one we used to compute the Bayes Rsk As a result, LDA can produce projects that are based by the cost functon,.e., classes wth a hgher cost C j wll be placed further apart n the low-dmensonal projecton Multlayer perceptrons (Webb and Lowe) It has been shown that the hdden layers of mult-layer perceptrons perform nonlnear dscrmnant analyss by maxmzng Tr[S B S T ], where the scatter matrces are measured at the output of the last hdden layer CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU
14 Other dmensonalty reducton methods Exploratory Projecton Pursut (Fredman and Tukey) EPP seeks an M-dmensonal (M=, typcally) lnear projecton of the data that maxmzes a measure of nterestngness Interestngness s measured as departure from multvarate normalty Ths measure s not the varance and s commonly scale-free. In most mplementatons t s also affne nvarant, so t does not depend on correlatons between features. [Rpley, 996] In other words, EPP seeks projectons that separate clusters as much as possble and keeps these clusters compact, a smlar crteron as Fsher s, but EPP does NOT use class labels Once an nterestng projecton s found, t s mportant to remove the structure t reveals to allow other nterestng vews to be found more easly Unnterestng Interestng x x x CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU x
15 Sammon s non-lnear mappng (Sammon) Ths method seeks a mappng onto an M-dmensonal space that preserves the nter-pont dstances n the orgnal N-dmensonal space Ths s accomplshed by mnmzng the followng objectve functon E d, d = j d P,P j d P,Pj d P,P j The orgnal method dd not obtan an explct mappng but only a lookup table for the elements n the tranng set Newer mplementatons based on neural networks do provde an explct mappng for test data and also consder cost functons (e.g., Neuroscale) Sammon s mappng s closely related to Mult Dmensonal Scalng (MDS), a famly of multvarate statstcal methods commonly used n the socal scences We wll revew MDS technques when we cover manfold learnng x x P j P j d(p, P j )= d(p, P j ) ",j P P x x x CSCE 666 Pattern Analyss Rcardo Guterrez-Osuna CSE@TAMU
Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)
Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton
More informationCS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements
Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems
More informationLecture 5,6 Linear Methods for Classification. Summary
Lecture 5,6 Lnear Methods for Classfcaton Rce ELEC 697 Farnaz Koushanfar Fall 2006 Summary Bayes Classfers Lnear Classfers Lnear regresson of an ndcator matrx Lnear dscrmnant analyss (LDA) Logstc regresson
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationLogistic Regression. Steve Kroon
Logstc Regresson Steve Kroon Course notes sectons: 24.3-24.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationHow To Calculate The Accountng Perod Of Nequalty
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationSingle and multiple stage classifiers implementing logistic discrimination
Sngle and multple stage classfers mplementng logstc dscrmnaton Hélo Radke Bttencourt 1 Dens Alter de Olvera Moraes 2 Vctor Haertel 2 1 Pontfíca Unversdade Católca do Ro Grande do Sul - PUCRS Av. Ipranga,
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationGeorey E. Hinton. University oftoronto. Email: zoubin@cs.toronto.edu. Technical Report CRG-TR-96-1. May 21, 1996 (revised Feb 27, 1997) Abstract
The EM Algorthm for Mxtures of Factor Analyzers Zoubn Ghahraman Georey E. Hnton Department of Computer Scence Unversty oftoronto 6 Kng's College Road Toronto, Canada M5S A4 Emal: zoubn@cs.toronto.edu Techncal
More informationLearning from Multiple Outlooks
Learnng from Multple Outlooks Maayan Harel Department of Electrcal Engneerng, Technon, Hafa, Israel She Mannor Department of Electrcal Engneerng, Technon, Hafa, Israel maayanga@tx.technon.ac.l she@ee.technon.ac.l
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationLoop Parallelization
- - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze
More informationONE of the most crucial problems that every image
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 10, OCTOBER 2014 4413 Maxmum Margn Projecton Subspace Learnng for Vsual Data Analyss Symeon Nktds, Anastasos Tefas, Member, IEEE, and Ioanns Ptas, Fellow,
More informationKernel Methods for General Pattern Analysis
Kernel Methods for General Pattern Analyss Nello Crstann Unversty of Calforna, Davs nello@support-vector.net Overvew Kernel Methods are a new class of pattern analyss algorthms whch can operate on very
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationData Visualization by Pairwise Distortion Minimization
Communcatons n Statstcs, Theory and Methods 34 (6), 005 Data Vsualzaton by Parwse Dstorton Mnmzaton By Marc Sobel, and Longn Jan Lateck* Department of Statstcs and Department of Computer and Informaton
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationOut-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering
Out-of-Sample Extensons for LLE, Isomap, MDS, Egenmaps, and Spectral Clusterng Yoshua Bengo, Jean-Franços Paement, Pascal Vncent Olver Delalleau, Ncolas Le Roux and Mare Oumet Département d Informatque
More information+ + + - - This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationEstimating the Number of Clusters in Genetics of Acute Lymphoblastic Leukemia Data
Journal of Al Azhar Unversty-Gaza (Natural Scences), 2011, 13 : 109-118 Estmatng the Number of Clusters n Genetcs of Acute Lymphoblastc Leukema Data Mahmoud K. Okasha, Khaled I.A. Almghar Department of
More informationBypassing Synthesis: PLS for Face Recognition with Pose, Low-Resolution and Sketch
Bypassng Synthess: PLS for Face Recognton wth Pose, Low-Resoluton and Setch Abhshe Sharma Insttute of Advanced Computer Scence Unversty of Maryland, USA bhoaal@umacs.umd.edu Davd W Jacobs Insttute of Advanced
More informationLearning from Large Distributed Data: A Scaling Down Sampling Scheme for Efficient Data Processing
Internatonal Journal of Machne Learnng and Computng, Vol. 4, No. 3, June 04 Learnng from Large Dstrbuted Data: A Scalng Down Samplng Scheme for Effcent Data Processng Che Ngufor and Janusz Wojtusak part
More informationStatistical Methods to Develop Rating Models
Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationGeneral Iteration Algorithm for Classification Ratemaking
General Iteraton Algorthm for Classfcaton Ratemakng by Luyang Fu and Cheng-sheng eter Wu ABSTRACT In ths study, we propose a flexble and comprehensve teraton algorthm called general teraton algorthm (GIA)
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationImperial College London
F. Fang 1, C.C. Pan 1, I.M. Navon 2, M.D. Pggott 1, G.J. Gorman 1, P.A. Allson 1 and A.J.H. Goddard 1 1 Appled Modellng and Computaton Group Department of Earth Scence and Engneerng Imperal College London,
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationPerformance Analysis and Coding Strategy of ECOC SVMs
Internatonal Journal of Grd and Dstrbuted Computng Vol.7, No. (04), pp.67-76 http://dx.do.org/0.457/jgdc.04.7..07 Performance Analyss and Codng Strategy of ECOC SVMs Zhgang Yan, and Yuanxuan Yang, School
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationProject Networks With Mixed-Time Constraints
Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationCompiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing
Complng for Parallelsm & Localty Dependence Testng n General Assgnments Deadlne for proect 4 extended to Dec 1 Last tme Data dependences and loops Today Fnsh data dependence analyss for loops General code
More informationHow To Understand The Results Of The German Meris Cloud And Water Vapour Product
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationPRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.
PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato
More informationAbstract. Clustering ensembles have emerged as a powerful method for improving both the
Clusterng Ensembles: {topchyal, Models jan, of punch}@cse.msu.edu Consensus and Weak Parttons * Alexander Topchy, Anl K. Jan, and Wllam Punch Department of Computer Scence and Engneerng, Mchgan State Unversty
More informationA cooperative connectionist IDS model to identify independent anomalous SNMP situations
A cooperatve connectonst IDS model to dentfy ndependent anomalous SNMP stuatons Álvaro Herrero, Emlo Corchado, José Manuel Sáz Department of Cvl Engneerng, Unversty of Burgos, Span escorchado@ubu.es Abstract
More informationInterpreting Patterns and Analysis of Acute Leukemia Gene Expression Data by Multivariate Statistical Analysis
Interpretng Patterns and Analyss of Acute Leukema Gene Expresson Data by Multvarate Statstcal Analyss ChangKyoo Yoo * and Peter A. Vanrolleghem BIOMATH, Department of Appled Mathematcs, Bometrcs and Process
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationVisualization of high-dimensional data with relational perspective map
(2004) 3, 49 59 & 2004 Palgrave Macmllan Ltd. All rghts reserved 1473-8716 $25.00 www.palgrave-journals.com/vs Vsualzaton of hgh-dmensonal data wth relatonal perspectve map James Xnzh L 1 1 Edgehll Dr.
More informationThe Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance
JOURNAL OF RESEARCH of the Natonal Bureau of Standards - B. Mathem atca l Scence s Vol. 74B, No.3, July-September 1970 The Dstrbuton of Egenvalues of Covarance Matrces of Resduals n Analyss of Varance
More informationEAR BIOMETRICS IN PERSONAL IDENTIFICATION. M. Sc. Thesis by Bahattin KOCAMAN, B.Sc.
İSANBUL ECHNICAL UNIVERSIY INSIUE OF SCIENCE AND ECHNOLOGY EAR BIOMERICS IN PERSONAL IDENIFICAION M. Sc. hess by Bahattn KOCAMAN, B.Sc. Department : Electroncs and Communcaton Engneerng Programme : Electroncs
More informationAn Empirical Study of Search Engine Advertising Effectiveness
An Emprcal Study of Search Engne Advertsng Effectveness Sanjog Msra, Smon School of Busness Unversty of Rochester Edeal Pnker, Smon School of Busness Unversty of Rochester Alan Rmm-Kaufman, Rmm-Kaufman
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationGRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM
GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM BARRIOT Jean-Perre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jean-perre.barrot@cnes.fr 1/Introducton The
More informationActive Learning for Interactive Visualization
Actve Learnng for Interactve Vsualzaton Tomoharu Iwata Nel Houlsby Zoubn Ghahraman Unversty of Cambrdge Unversty of Cambrdge Unversty of Cambrdge Abstract Many automatc vsualzaton methods have been. However,
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationAn Algorithm for Data-Driven Bandwidth Selection
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO. 2, FEBRUARY 2003 An Algorthm for Data-Drven Bandwdth Selecton Dorn Comancu, Member, IEEE Abstract The analyss of a feature space
More informationNonlinear Time Series Analysis in a Nutshell
9 Nonlnear Tme Seres Analyss n a Nutshell Ralph Gregor Andrzejak Contents 9. Introducton...25 9.2 Dynamcs and Tme Seres... 27 9.2. Nonlnear Determnstc Dynamcs Lorenz Dynamcs... 27 9.2.2 Nosy Determnstc
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationIntra-day Trading of the FTSE-100 Futures Contract Using Neural Networks With Wavelet Encodings
Submtted to European Journal of Fnance Intra-day Tradng of the FTSE-00 Futures Contract Usng eural etworks Wth Wavelet Encodngs D L Toulson S P Toulson Intellgent Fnancal Systems Lmted Sute 4 Greener House
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationPoint cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000-014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
More informationAn Analysis of the relationship between WTI term structure and oil market fundamentals in 2002-2009
MPRA Munch Personal RePEc Archve An Analyss of the relatonshp between WTI term structure and ol market fundamentals n 00-009 Mleno Cavalcante Petrobras S.A., Unversdade de Fortaleza. August 00 Onlne at
More informationThe Development of Web Log Mining Based on Improve-K-Means Clustering Analysis
The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationQuantization Effects in Digital Filters
Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value
More informationSVM Tutorial: Classification, Regression, and Ranking
SVM Tutoral: Classfcaton, Regresson, and Rankng Hwanjo Yu and Sungchul Km 1 Introducton Support Vector Machnes(SVMs) have been extensvely researched n the data mnng and machne learnng communtes for the
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationProbabilistic Linear Classifier: Logistic Regression. CS534-Machine Learning
robablstc Lnear Classfer: Logstc Regresson CS534-Machne Learnng Three Man Approaches to learnng a Classfer Learn a classfer: a functon f, ŷ f Learn a probablstc dscrmnatve model,.e., the condtonal dstrbuton
More informationFixed income risk attribution
5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two
More information1 De nitions and Censoring
De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence
More informationDimensionality Reduction for Data Visualization
Dmensonalty Reducton for Data Vsualzaton Samuel Kask and Jaakko Peltonen Dmensonalty reducton s one of the basc operatons n the toolbox of data-analysts and desgners of machne learnng and pattern recognton
More informationLearning to Classify Ordinal Data: The Data Replication Method
Journal of Machne Learnng Research 8 (7) 393-49 Submtted /6; Revsed 9/6; Publshed 7/7 Learnng to Classfy Ordnal Data: The Data Replcaton Method Jame S. Cardoso INESC Porto, Faculdade de Engenhara, Unversdade
More informationMulticlass sparse logistic regression for classification of multiple cancer types using gene expression data
Computatonal Statstcs & Data Analyss 51 (26) 1643 1655 www.elsever.com/locate/csda Multclass sparse logstc regresson for classfcaton of multple cancer types usng gene expresson data Yongda Km a,, Sunghoon
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationFast Fuzzy Clustering of Web Page Collections
Fast Fuzzy Clusterng of Web Page Collectons Chrstan Borgelt and Andreas Nürnberger Dept. of Knowledge Processng and Language Engneerng Otto-von-Guercke-Unversty of Magdeburg Unverstätsplatz, D-396 Magdeburg,
More informationBERNSTEIN POLYNOMIALS
On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationA Fast Incremental Spectral Clustering for Large Data Sets
2011 12th Internatonal Conference on Parallel and Dstrbuted Computng, Applcatons and Technologes A Fast Incremental Spectral Clusterng for Large Data Sets Tengteng Kong 1,YeTan 1, Hong Shen 1,2 1 School
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationQuestion 2: What is the variance and standard deviation of a dataset?
Queston 2: What s the varance and standard devaton of a dataset? The varance of the data uses all of the data to compute a measure of the spread n the data. The varance may be computed for a sample of
More informationDiscussion Papers. Support Vector Machines (SVM) as a Technique for Solvency Analysis. Laura Auria Rouslan A. Moro. Berlin, August 2008
Deutsches Insttut für Wrtschaftsforschung www.dw.de Dscusson Papers 8 Laura Aura Rouslan A. Moro Support Vector Machnes (SVM) as a Technque for Solvency Analyss Berln, August 2008 Opnons expressed n ths
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationTo Fill or not to Fill: The Gas Station Problem
To Fll or not to Fll: The Gas Staton Problem Samr Khuller Azarakhsh Malekan Julán Mestre Abstract In ths paper we study several routng problems that generalze shortest paths and the Travelng Salesman Problem.
More informationFault tolerance in cloud technologies presented as a service
Internatonal Scentfc Conference Computer Scence 2015 Pavel Dzhunev, PhD student Fault tolerance n cloud technologes presented as a servce INTRODUCTION Improvements n technques for vrtualzaton and performance
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationApplication of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance
Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom
More informationStochastic Protocol Modeling for Anomaly Based Network Intrusion Detection
Stochastc Protocol Modelng for Anomaly Based Network Intruson Detecton Juan M. Estevez-Tapador, Pedro Garca-Teodoro, and Jesus E. Daz-Verdejo Department of Electroncs and Computer Technology Unversty of
More informationNPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationBayesian Network Based Causal Relationship Identification and Funding Success Prediction in P2P Lending
Proceedngs of 2012 4th Internatonal Conference on Machne Learnng and Computng IPCSIT vol. 25 (2012) (2012) IACSIT Press, Sngapore Bayesan Network Based Causal Relatonshp Identfcaton and Fundng Success
More informationBayesian Cluster Ensembles
Bayesan Cluster Ensembles Hongjun Wang 1, Hanhua Shan 2 and Arndam Banerjee 2 1 Informaton Research Insttute, Southwest Jaotong Unversty, Chengdu, Schuan, 610031, Chna 2 Department of Computer Scence &
More informationAn Interest-Oriented Network Evolution Mechanism for Online Communities
An Interest-Orented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationRobust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School
Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management
More informationRegression Models for a Binary Response Using EXCEL and JMP
SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STAT-TECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationAnalysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
More information