Review: Classification Outline


 Bryce Ball
 1 years ago
 Views:
Transcription
1 Data Miig CS 341, Sprig 2007 Decisio Trees Neural etworks Review: Lecture 6: Classificatio issues, regressio, bayesia classificatio Pretice Hall 2 Data Miig Core Techiques Classificatio Clusterig Associatio Rules Classificatio Outlie Goal: Provide a overview of the classificatio problem ad itroduce some of the basic algorithms Classificatio Problem Overview Classificatio Techiques Regressio Bayesia classificatio Distace Decisio Trees Rules Neural Networks Pretice Hall 3 Pretice Hall 4 Classificatio Outlie Goal: Provide a overview of the classificatio problem ad itroduce some of the basic algorithms Classificatio Problem Overview Classificatio Techiques Regressio Bayesia classificatio Classificatio Problem Give a database D={t 1,t 2,,t } ad a set of classes C={C 1,,C,C m }, the Classificatio Problem is to defie a mappig f:dc where each t i is assiged to oe class. Actually divides D ito equivalece classes. Predictio is similar, but may be viewed as havig ifiite umber of classes. Pretice Hall 5 Pretice Hall 6 1
2 Classificatio Examples Teachers classify studets grades as A, B, C, D, or F. Idetify mushrooms as poisoous or edible. Predict whe a river will flood. Idetify idividuals with credit risks. Speech recogitio Patter recogitio Pretice Hall 7 Classificatio Ex: Gradig If x >= 90 the grade =A. If 80<=x<90 the grade =B. If 70<=x<80 the grade =C. If 60<=x<70 the grade =D. If x<60 the grade =F. <80 <70 x <50 F <90 >=90 Pretice Hall 8 x x x >=70 C >=60 D >=80 B A Classificatio Ex: Letter Recogitio View letters as costructed from 5 compoets: Letter A Letter B Letter C Letter D Letter E Letter F Pretice Hall 9 Classificatio Techiques Approach: 1. Create specific model by evaluatig traiig data (or usig domai experts kowledge). 2. Apply model developed to ew data. Classes must be predefied Most commo techiques use DTs, NNs,, or are based o distaces or statistical methods. Pretice Hall 10 Defiig Classes Issues i Classificatio Partitioig Based Distace Based Missig Data Igore Replace with assumed value Overfittig Large set of traiig data Filter out erroeous or oisy data Measurig Performace Classificatio accuracy o test data Cofusio matrix OC Curve Pretice Hall 11 Pretice Hall 12 2
3 Classificatio Accuracy Classificatio Performace True positive (TP) t i Predicted to be i C j ad is actually i it. False positive (FP) t i Predicted to be i C j but is ot actually i it. True egative (TN) t i ot predicted to be i C j ad is ot actually i it. False egative (FN) t i ot predicted to be i C j but is actually i it. True Positive False Positive False Negative True Negative Pretice Hall 13 Pretice Hall 14 Cofusio Matrix A m x m matrix Etry C i,j idicates the umber of tuples assiged to C j, but where the correct class is C i The best solutio will oly have o zero values o the diagoal. Pretice Hall 15 Height Example Data N am e G e d e r H eig h t O u tp u t1 O u tp u t2 K ristia F 1.6m S ho rt M ed iu m Jim M 2m T all M ed iu m M ag gie F 1.9m M e diu m T all M arth a F 1.88 m M e diu m T all S tep ha ie F 1.7m S ho rt M ed iu m B o b M 1.85 m M e diu m M ed iu m K a th y F 1.6m S ho rt M ed iu m D ave M 1.7m S ho rt M ed iu m W orth M 2.2m T all T all S teve M 2.1m T all T all D eb bie F 1.8m M e diu m M ed iu m T o dd M 1.95 m M e diu m M ed iu m K im F 1.9m M e diu m T all A m y F 1.8m M e diu m M ed iu m W ye tte F 1.75 m M e diu m M ed iu m Pretice Hall 16 Cofusio Matrix Example Operatig Characteristic Curve Usig height data example with Output1 (correct) ad Output2 (actual) assigmet Actual Assigmet Membership Short Medium Tall Short Medium Tall Pretice Hall 17 Pretice Hall 18 3
4 Classificatio Outlie Goal: Provide a overview of the classificatio problem ad itroduce some of the basic algorithms Classificatio Problem Overview Classificatio Techiques Regressio Distace Decisio Trees Rules Neural Networks Regressio Assume data fits a predefied fuctio Determie best values for parameters i the model Estimate a output value based o iput values Ca be used for classificatio ad predictio Pretice Hall 19 Pretice Hall 20 Liear Regressio Assume the relatio of the output variable to the iput variables is a liear fuctio of some parameters. Determie best values for regressio coefficiets c 0,c 1,,c,c. Assume a error: y = c 0 +c 1 x 1 + +c x +ε Estimate error usig mea squared error for traiig set: Example: 4.3 Y = C 0 + ε Fid the value for c 0 that best partitio the height values ito classes: short ad medium The traiig data for y i is {1.6, 1.9, 1.88, 1.7, 1.85, 1.6, 1.7, 1.8, 1.95, 1.9, 1.8, 1.75} How? Pretice Hall 21 Pretice Hall 22 Example: 4.4 Liear Regressio Poor Fit Y = c 0 + c 0 x 1 + ε Fid the value for c 0 ad c 1 that best predict the class. Assume 0 for the short class, 1 for the medium class The traiig data for (x i, y i) is {(1.6,0), (1.9,0), (1.88, 0), (1.7, 0), (1.85, 0), (1.6, 0), (1.7,0), (1.8,0), (1.95, 0), (1.9, 0), (1.8, 0), (1.75, 0)} How? Pretice Hall 23 Pretice Hall 24 4
5 Classificatio Usig Regressio Divisio Divisio: Use regressio fuctio to divide area ito regios. Predictio: : Use regressio fuctio to predict a class membership fuctio. Pretice Hall 25 Pretice Hall 26 Predictio Logistic Regressio A geeralized liear model Extesively used i the medical ad social scieces It has the followig form Log e (p /p 1) = c 0 + c 1 x c k x k p is the probability of beig i the class, 1 p is the probability that is ot. The parameters c 0, c 1, c k are usually estimated by maximum likelihood. (maximize the probability of observig the give value.) Pretice Hall 27 Pretice Hall 28 Why Logistic Regressio P is i the rage [0,1] A good model would like to have p value close to 0 or 1 Liear fuctio is ot suitable for p Cosider the odds p/1p. p. As p icreases, the odds (p/1p) p) icreases The odds is i the rage of [0, + ], + asymmetric. The log odds lies i the rage  to +, symmetric. Liear Regressio vs. Logistic Regressio Pretice Hall 29 Pretice Hall 30 5
6 Classificatio Outlie Goal: Provide a overview of the classificatio problem ad itroduce some of the basic algorithms Classificatio Problem Overview Classificatio Techiques Regressio Bayesia classificatio Bayes Theorem Posterior Probability: P(h 1 x i ) Prior Probability: P(h 1 ) Bayes Theorem: Pretice Hall 31 Assig probabilities of hypotheses give a data value. Pretice Hall 32 Naïve Bayes Classificatio Assume that the cotributio by all attributes are idepedet ad that each cotributes equally to the classificatio problem. t i has m idepedet attributes {x i1 P (t( i C j ) P (x( ik C j ) i1,, x im,}. Pretice Hall 33 Example: usig the output1 as classificatio results N a m e G e d e r H e ig h t O u tp u t1 O u tp u t2 K ris ti a F 1.6 m S h o rt M e d iu m J im M 2 m T a ll M e d iu m M a g g ie F 1.9 m M e d iu m T a ll M a rth a F m M e d iu m T a ll S te p h a ie F 1.7 m S h o rt M e d iu m B o b M m M e d iu m M e d iu m K a th y F 1.6 m S h o rt M e d iu m D a v e M 1.7 m S h o rt M e d iu m W o rth M 2.2 m T a ll T a ll S te v e M 2.1 m T a ll T a ll D e b b ie F 1.8 m M e d iu m M e d iu m T o d d M m M e d iu m M e d iu m K im F 1.9 m M e d iu m T a ll A m y F 1.8 m M e d iu m M e d iu m W y e tte F m M e d iu m M e d iu m Pretice Hall 34 Example 4.5 Step1: Calculate the prior probability P (short) = P (medium) = P (tall) = Example 4.5 Step1: Calculate the prior probability P (short) = 4/15 = P (medium) = 8/15 = P (tall) = 3/15 = 0.2 Step 2: Calculate the coditioal probability P(Geder i C j ), Geder i = F or M, C j = short or medium or tall P(Height i C j ) Height i i (0,1.6],(1.6,1.7],(1.7,1.8],(1.8,1.9],(1.9,2.0],(>2.0). Pretice Hall 35 Pretice Hall 36 6
7 Attribute Example 4.5 (cot d) cout short medium tall Geder M F Height (<1.6] (1.6,1.7] (1.7,1.8] (1.8,1.9] (1.9,2.0] ( >2.0 ) probability p(x i C j ) short medium tall 1/4 2/8 3/3 3/4 6/8 0/3 2/ / / / /8 1/ /3 Example 4.5 (cot d) Give a tuple t ={Adam, M, 1.95m} Step 3: Calculate P(t C j ) P(t short) ) = P(t medium) ) = P(t tall)= Step 4: calculate P(t) P(t) ) = P(t short)p(short)+p(t medium)p(medium)+p(t tall)p(tall) Pretice Hall 37 Pretice Hall 38 Example 4.5 (cot d) Give a tuple t ={Adam, M, 1.95m} Step 3: Calculate P(t C j ) P(t short) ) = ¼ x 0 =0 P(t medium) ) = 2/8 x 1/8 =0.031 P(t tall)= 3/3 x1/3 =0.333 Step 4: calculate P(t) P(t) ) = P(t short)p(short)+p(t medium)p(medium)+p(t tall)p(tall) = Example 4.5 (cot d) Step 5: Calculate P(C j t) usig Bayes Rule P(short t) ) = P(t short)p(short)/p(t) ) = P(medium t) ) = P(tall t)= Last step: classify t based o these probabilities Pretice Hall 39 Pretice Hall 40 Example 4.5 (cot d) Step 5: Calculate P(C j t) usig Bayes Rule P(short t) ) = P(t short)p(short)/p(t) ) = 0 P(medium t) ) = 0.2 P(tall t)= Last step: Classify the ew tuple as tall. A Summary Step 1: Calculate the prior probability of each class. P (C( j ) Step 2: Calculate the coditioal probability for each attribute value, P(Geder i C j ), Step 3: Calculate the coditioal probability P(t C j ) Step 4: calculate the prior probability of a tuple, P(t) Step 5: Calculate the posterior probability for each class give the tuple, P(C j t) usig Bayes Rule Step 6: Classify a tuple based o the P(C j t), the tuple belogs to the class with has the highest posterior probability. Pretice Hall 41 Pretice Hall 42 7
8 Next Lecture: Classificatio: Distacebased algorithms Decisio treebased algorithms HW2 will be aouced! Pretice Hall 43 8
{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationMeasures of Central Tendency
Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationrepresented by 4! different arrangements of boxes, divide by 4! to get ways
Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,
More informationSolving DivideandConquer Recurrences
Solvig DivideadCoquer Recurreces Victor Adamchik A divideadcoquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationNotes on Hypothesis Testing
Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter
More informationMIDTERM EXAM  MATH 563, SPRING 2016
MIDTERM EXAM  MATH 563, SPRING 2016 NAME: SOLUTION Exam rules: There are 5 problems o this exam. You must show all work to receive credit, state ay theorems ad defiitios clearly. The istructor will NOT
More informationApplication and research of fuzzy clustering analysis algorithm under microlecture English teaching mode
SHS Web of Cofereces 25, shscof/20162501018 Applicatio ad research of fuzzy clusterig aalysis algorithm uder microlecture Eglish teachig mode Yig Shi, Wei Dog, Chuyi Lou & Ya Dig Qihuagdao Istitute of
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationCOMP 251 Assignment 2 Solutions
COMP 251 Assigmet 2 Solutios Questio 1 Exercise 8.34 Treat the umbers as 2digit umbers i radix. Each digit rages from 0 to 1. Sort these 2digit umbers ith the RADIXSORT algorithm preseted i Sectio
More informationhp calculators HP 12C Platinum Statistics  correlation coefficient The correlation coefficient HP12C Platinum correlation coefficient
HP 1C Platium Statistics  correlatio coefficiet The correlatio coefficiet HP1C Platium correlatio coefficiet Practice fidig correlatio coefficiets ad forecastig HP 1C Platium Statistics  correlatio coefficiet
More informationCS100: Introduction to Computer Science
Statistics of the Secod Exam CS100: Itroductio to Computer Sciece Lecture 21: Database & Data miig 100 ad above: 8 90 99: 7 80 89: 7 70 79: 5 60 69: 3 Review: Course Objectives Review: Course Objectives
More informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More informationDerivation of the Poisson distribution
Gle Cowa RHUL Physics 1 December, 29 Derivatio of the Poisso distributio I this ote we derive the fuctioal form of the Poisso distributio ad ivestigate some of its properties. Cosider a time t i which
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationClassification Techniques (1)
10 10 Overview Classification Techniques (1) Today Classification Problem Classification based on Regression Distancebased Classification (KNN) Net Lecture Decision Trees Classification using Rules Quality
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More information8 The Poisson Distribution
8 The Poisso Distributio Let X biomial, p ). Recall that this meas that X has pmf ) p,p k) p k k p ) k for k 0,,...,. ) Agai, thik of X as the umber of successes i a series of idepedet experimets, each
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationIntroductory Explorations of the Fourier Series by
page Itroductory Exploratios of the Fourier Series by Theresa Julia Zieliski Departmet of Chemistry, Medical Techology, ad Physics Momouth Uiversity West Log Brach, NJ 7764898 tzielis@momouth.edu Copyright
More informationAlgorithms Chapter 7 Quicksort
Algorithms Chapter 7 Quicksort Assistat Professor: Chig Chi Li 林清池助理教授 chigchi.li@gmail.com Departmet of Computer Sciece ad Egieerig Natioal Taiwa Ocea Uiversity Outlie Descriptio of Quicksort Performace
More informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More information428 CHAPTER 12 MULTIPLE LINEAR REGRESSION
48 CHAPTER 1 MULTIPLE LINEAR REGRESSION Table 18 Team Wis Pts GF GA PPG PPcT SHG PPGA PKPcT SHGA Chicago 47 104 338 68 86 7. 4 71 76.6 6 Miesota 40 96 31 90 91 6.4 17 67 80.7 0 Toroto 8 68 3 330 79.3
More informationSimple Linear Regression
Simple Liear Regressio W. Robert Stepheso Departmet of Statistics Iowa State Uiversity Oe of the most widely used statistical techiques is simple liear regressio. This techique is used to relate a measured
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
More information1 StateSpace Canonical Forms
StateSpace Caoical Forms For ay give system, there are essetially a ifiite umber of possible state space models that will give the idetical iput/output dyamics Thus, it is desirable to have certai stadardized
More informationREACHABILITY AND OBSERVABILITY
Updated: Saturday October 4 8 Copyright F.L. Lewis ll rights reserved RECHILIY ND OSERVILIY Cosider a liear statespace system give by x x + u y Cx + Du m with x t R the iteral state u t R the cotrol iput
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationAlgebra Work Sheets. Contents
The work sheets are grouped accordig to math skill. Each skill is the arraged i a sequece of work sheets that build from simple to complex. Choose the work sheets that best fit the studet s eed ad will
More informationStat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals  the general concept
Statistics 104 Lecture 16 (IPS 6.1) Outlie for today Cofidece itervals Cofidece itervals for a mea, µ (kow σ) Cofidece itervals for a proportio, p Margi of error ad sample size Review of mai topics for
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationCluster Validity Measurement Techniques
Cluster Validity Measuremet Techiques Ferec Kovács, Csaba Legáy, Attila Babos Departmet of Automatio ad Applied Iformatics Budapest Uiversity of Techology ad Ecoomics Goldma György tér 3, H Budapest,
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationArithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...
3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationFirewall Modules and Modular Firewalls
Firewall Modules ad Modular Firewalls H. B. Acharya Uiversity of Texas at Austi acharya@cs.utexas.edu Aditya Joshi Uiversity of Texas at Austi adityaj@cs.utexas.edu M. G. Gouda Natioal Sciece Foudatio
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationAlgorithms and Data Structures DV3. Arne Andersson
Algorithms ad Data Structures DV3 Are Adersso Today s lectures Textbook chapters 14, 5 Iformatiostekologi Itroductio Overview of Algorithmic Mathematics Recurreces Itroductio to Recurreces The Substitutio
More information6 Algorithm analysis
6 Algorithm aalysis Geerally, a algorithm has three cases Best case Average case Worse case. To demostrate, let us cosider the a really simple search algorithm which searches for k i the set A{a 1 a...
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More information3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average
5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationDetecting Auto Insurance Fraud by Data Mining Techniques
Detectig Auto Isurace Fraud by Data Miig Techiques Rekha Bhowmik Computer Sciece Departmet Uiversity of Texas at Dallas, USA rxb080100@utdallas.edu ABSTRACT The paper presets fraud detectio method to predict
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationOnesample test of proportions
Oesample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:
More informationArithmetic Sequences
. Arithmetic Sequeces Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms,
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationINDR 262 Optimization Models and Mathematical Programming LINEAR PROGRAMMING MODELS
LINEAR PROGRAMMING MODELS Commo termiology for liear programmig:  liear programmig models ivolve. resources deoted by i, there are m resources. activities deoted by, there are acitivities. performace
More informationChapter 6: CPU Scheduling. Previous Lectures. Basic Concepts. Histogram of CPUburst Times. CPU Scheduler. Alternating Sequence of CPU And I/O Bursts
Multithreadig Memory Layout Kerel vs User threads Represetatio i OS Previous Lectures Differece betwee thread ad process Thread schedulig Mappig betwee user ad kerel threads Multithreadig i Java Thread
More informationMining Educational Data to Analyze Students Performance
(IJACSA Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol., No. 6, 0 Miig Educatioal Data to Aalyze Studets Performace Briesh Kumar Baradwa Research Scholor, Sighaiya Uiversity, Raastha,
More informationCorrelation. example 2
Correlatio Iitially developed by Sir Fracis Galto (888) ad Karl Pearso (8) Sir Fracis Galto 8 correlatio is a much abused word/term correlatio is a term which implies that there is a associatio betwee
More informationChapter 9: Correlation and Regression: Solutions
Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationCustomer Satisfaction Control Application in Quality Assurance Departement at Petra Christian University using Fuzzy Aggregation
Customer Satisfactio Cotrol Applicatio i Quality Assurace Departemet at usig Fuzzy Aggregatio Adreas Hadojo Iformatics Departemet Jl. Siwalakerto 23, Surabaya hadojo@petra.ac.id Rolly Ita Iformatics Departemet
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationYour grandmother and her financial counselor
Sectio 10. Arithmetic Sequeces 963 Objectives Sectio 10. Fid the commo differece for a arithmetic sequece. Write s of a arithmetic sequece. Use the formula for the geeral of a arithmetic sequece. Use the
More informationPage 2 of 14 = T(2) + 2 = [ T(3)+1 ] + 2 Substitute T(3)+1 for T(2) = T(3) + 3 = [ T(4)+1 ] + 3 Substitute T(4)+1 for T(3) = T(4) + 4 After i
Page 1 of 14 Search C455 Chapter 4  Recursio Tree Documet last modified: 02/09/2012 18:42:34 Uses: Use recursio tree to determie a good asymptotic boud o the recurrece T() = Sum the costs withi each level
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationCS100: Introduction to Computer Science
Review: History of Computers CS100: Itroductio to Computer Sciece Maiframes Miicomputers Lecture 2: Data Storage  Bits, their storage ad mai memory Persoal Computers & Workstatios Review: The Role of
More informationOverview on SBox Design Principles
Overview o SBox Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA 721302 What is a SBox? SBoxes are Boolea
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationLinear Algebra II. Notes 6 25th November 2010
MTH6140 Liear Algebra II Notes 6 25th November 2010 6 Quadratic forms A lot of applicatios of mathematics ivolve dealig with quadratic forms: you meet them i statistics (aalysis of variace) ad mechaics
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationChapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1.
Use the followig to aswer questios 6: Chapter 7 I the questios below, describe each sequece recursively Iclude iitial coditios ad assume that the sequeces begi with a a = 5 As: a = 5a,a = 5 The Fiboacci
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationTagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper PartA
Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper PartA UitI. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More information13 Fast Fourier Transform (FFT)
13 Fast Fourier Trasform FFT) The fast Fourier trasform FFT) is a algorithm for the efficiet implemetatio of the discrete Fourier trasform. We begi our discussio oce more with the cotiuous Fourier trasform.
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More information