Introduction to local (nonparametric) density estimation. methods

Size: px
Start display at page:

Download "Introduction to local (nonparametric) density estimation. methods"

Transcription

1 Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014

2 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest eghbor desty estmato. Local desty estmato s also referred to as o-parametrc desty estmato. To make thgs clear, let s frst look at parametrc desty estmato. I parametrc desty estmato, we ca assume that there exsts a desty fucto whch ca be determed by a set of parameters. The set of parameters are estmated from the sample data ad are later used desgg the classfer. However, some practcal stuatos the assumpto that there exsts a parametrc form of the desty fucto does ot hold true. For example, t s very hard to ft a multmodal probablty dstrbuto wth a smple fucto. I ths case, we eed to estmate the desty fucto the oparametrc way, whch meas that the desty fucto s estmated locally based o a small set of eghborg samples. Because of ths localty, local (oparametrc) desty estmato s less accurate tha parametrc desty estmato. I the followg text the word local s preferred over oparametrc. It s oteworthy that t s very dffcult to obta a accurate local desty estmato, especally whe the dmeso of the feature space s hgh. So why do we bother usg local desty estmato? Ths s because our goal s ot to get a accurate estmato, but rather to use the estmato to desg a well performed classfer. The accuracy of local desty estmato does ot ecessarly lead to a poor decso rule.. Geeral Prcple I local desty estmato the desty fucto p (x) ca be approxmated by k p ( x) (1) v where v s the volume of a small rego R aroud pot x, s the total umber of samples x ( =1,, ) draw accordg to p (x), ad k s the umber of x s whch fall to rego R. The reaso why p (x) ca be calculated ths way s that p (x) does ot vary much wth a relatvely small rego, thus the probablty mass of rego R ca be approxmated by p (x)v, whch equals k /. Some examples of rego R dfferet dmesos: ) le segmet oe-dmeso, ) crcle or rectagle two-dmeso, ) sphere or cube three-dmeso, v) hyper sphere or hypercube d-dmeso (d > 3). Three codtos we eed to pay atteto to whe usg formula (1) are: ) lm v 0. Ths s because f v s fxed, the p (x) oly represets the average probablty desty as grows larger, but what we eed s the pot probablty desty,

3 so we should have v 0 whe. ) lm k. Ths s to make sure that we do ot get zero probablty desty. ) lm k / 0. Ths s to make sure that p (x) does ot dverge. 3. Parze Desty Estmato I Parze desty estmato v s drectly determed by whle k s a radom varable whch deotes the umber of samples that fall to v. Assume that the rego R s a d-dmesoal hypercube wth ts edge legth h, thus v = (h ) d The equvalet codtos whch meet the aforemetoed three codtos are: lm v 0 ad lm v Therefore v ca be chose as v h / or v h / l, where h s a adjustable costat. Now that the relatoshp betwee v ad s defed, the ext step s to determe k. To determe k, we defe a wdow fucto as follows: x x 1 h 0 x x h else 1 where x s ( = 1,,, ) are the gve samples ad x s the pot where the desty s to be estmated. Thus we have k x x 1 h k 1 x x p ( x) v v 1 h The fucto s called a Parze wdow fucto, whch eables us to cout the umber of sample pots the hypercube wth ts edge legth h. Accordg to [], usg hypercube as the wdow fucto may lead to dscotuty the estmato. Ths s due to the supermposto of sharp pulses cetered at the gve sample pots whe h s small. To overcome ths shortcomg, we ca cosder a more geeral form of wdow fucto rather tha the hypercube. Note that f the followg two codtos are met, the estmated p (x) s guarateed to be proper.

4 ( x) 0 ad ( x) dx 1 Therefore a better choce of wdow fucto whch removes dscotuty ca be Gaussa wdow: The estmated desty s gve by x x 1 1 x x exp h h p x x x exp () v 1 h Cosder a oe-dmeso case, assume that v h /, thus h v h /, where h s a adjustable costat. Substtute to formula () we have x x x x p ( x) exp exp v 1 h h 1 h / We ca see that f equals oe, p (x) s just the wdow fucto. If approaches fty, p (x) ca coverge to ay complex form. If s relatvely small, p (x) s very sestve to the value of h. I geeral small h leads to the ose error whle large h leads to the over-smoothg error, whch ca be llustrated by the followg example. I ths expermet samples are 5000 pots o -D plae wth Gaussa dstrbuto. The mea vector s [1 ], ad the covarace matrx s [1 0; 0 1]. Choose rectagle 4 Parze wdow wth h 4/, thus v ( h) 16 /. Fg. 1 shows the sample dstrbuto. Fg. shows the deal probablty desty dstrbuto. Fg. 3 shows the result of Parze desty estmato. Fgure sample pots o -D plae wth Gaussa dstrbuto

5 Fgure. The deal probablty desty dstrbuto Fgure 3. The result of Parze desty estmato Next we chage the value of h ad see how t affects the estmato. Fg. 4 shows the result of Parze desty estmato whe h s twce ts tal value. Fg. 5 shows the result of Parze desty estmato whe h s ts tal value dvded by two. We ca see that the results agree wth the aforesad property of h.

6 Fgure 4. The result of Parze desty estmato whe h s twce ts tal value Fgure 5. The result of Parze desty estmato whe h s ts tal value dvded by two To desg a classfer usg Parze wdow method [3], we estmate the destes for each class ad classfy the test pot by the label correspodg to the maxmum posteror. Below lsts some advatages ad dsadvatages of Parze desty estmato: Advatages: ) p (x) ca coverge to ay complex form whe approaches fty; ) applcable to data wth ay dstrbuto. Dsadvatages: ) eed a large umber of samples to obta a accurate estmato; ) computatoally expesve, ot sutable for feature space wth very hgh dmesos;

7 ) the adjustable costat h has a relatvely heavy fluece o the decso boudares whe s small, ad s ot easy to choose practce. 4. K-Nearest Neghbor Desty Estmato I k-earest eghbor desty estmato (use acroym k-nn the followg text) k s drectly determed by whle v s a radom varable whch deotes the volume that ecompasses just k sample pots sde v ad o ts boudary. If v s a sphere, t ca be gve by h ( hk ) vk ( x) m h ( 1) ( 1) where h s the radus of the sphere wth ceter x. h k equals x lk - x where x lk s the k th closest sample pot to x. The the probablty desty at x s approxmated by px ( ) k v ( x) 1 (3) where k 1 s umber of sample pots o the boudary of v k (x). Most of the tme formula (3) ca be rewrtte as It ca be proved that E[ p( x)] p( x). k 1 p( x) ( k ) v ( x) k I Parze desty estmato v oly depeds o ad s the same for all the test pots, whle k-nn v s smaller at hgh desty area ad s larger at low desty area. Ths strategy seems more reasoable tha the strategy to determe v Parze desty estmato sce ow v s adaptve to the local desty. I practce, whe we wat to classfy data usg k-nn estmato, t turs out that we ca get the posteror p(w x) drectly wthout worryg about p(x). If we have k samples fall to volume v aroud pot x, ad amog the k samples there are k samples belogg to class w, the we have p w x k, k v The posteror p(w x) s gve by p w x,, p w x p w x k m P( x) k p w, x j1 j (4) where m s the umber of classes. Formula (4) tells us oe smple decso rule: the

8 class of a test pot x s the same as the most frequet oe amog the earest k pots of x. Smple ad tutve, s t t? Havg sad that, choosg k k-nn s stll a otrval problem as choosg h Parze desty estmato. Small k leads to osy decso boudares whle large k leads to over-smoothed boudares, whch s llustrated by the followg example. I ths expermet samples are 00 pre-labeled (red or blue) pots. The task s to fd the classfcato boudares uder dfferet k values. Fg. 6-9 show the results. Fgure 6. k-nn decso boudares expermet (k=) Fgure 7. k-nn decso boudares expermet (k=3)

9 Fgure 8. k-nn decso boudares expermet (k=5) Fgure 9. k-nn decso boudares expermet (k=8) I practce we ca use cross-valdato to choose the best k. Below lsts some advatages ad dsadvatages of k-nn: Advatages: ) decso performace s good f s large eough; ) applcable to data wth ay dstrbuto; ) smple ad tutve. Dsadvatages: ) eed a large umber of samples to obta a accurate estmato, whch s evtable local desty estmato; ) computatoally expesve, low effcecy for feature space wth very hgh dmesos; ) choosg the best k s otrval.

10 5. Referece [1] Mrelle Bout, ECE66: Statstcal Patter Recogto ad Decso Makg Processes, Purdue Uversty, Sprg 014 [] [3]

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall

More information

6. Nonparametric techniques

6. Nonparametric techniques 6. Noparametrc techques Motvato Problem: how to decde o a sutable model (e.g. whch type of Gaussa) Idea: just use the orgal data (lazy learg) 2 Idea 1: each data pot represets a pece of probablty P(x)

More information

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,

More information

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution: Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed

More information

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg

More information

Nonparametric Density Estimation Intro

Nonparametric Density Estimation Intro Noarametrc Desty Estmato Itro Parze Wdows No-Parametrc Methods Nether robablty dstrbuto or dscrmat fucto s kow Haes qute ofte All we have s labeled data a lot s kow easer salmo bass salmo salmo Estmate

More information

Bayes (Naïve or not) Classifiers: Generative Approach

Bayes (Naïve or not) Classifiers: Generative Approach Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg

More information

CHAPTER VI Statistical Analysis of Experimental Data

CHAPTER VI Statistical Analysis of Experimental Data Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

X ε ) = 0, or equivalently, lim

X ε ) = 0, or equivalently, lim Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece

More information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample

More information

The Mathematical Appendix

The Mathematical Appendix The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.

More information

Point Estimation: definition of estimators

Point Estimation: definition of estimators Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.

More information

Lecture 7: Linear and quadratic classifiers

Lecture 7: Linear and quadratic classifiers Lecture 7: Lear ad quadratc classfers Bayes classfers for ormally dstrbuted classes Case : Σ σ I Case : Σ Σ (Σ daoal Case : Σ Σ (Σ o-daoal Case 4: Σ σ I Case 5: Σ Σ j eeral case Lear ad quadratc classfers:

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted

More information

Special Instructions / Useful Data

Special Instructions / Useful Data JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth

More information

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur

More information

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The

More information

Unsupervised Learning and Other Neural Networks

Unsupervised Learning and Other Neural Networks CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all

More information

An Introduction to. Support Vector Machine

An Introduction to. Support Vector Machine A Itroducto to Support Vector Mache Support Vector Mache (SVM) A classfer derved from statstcal learg theory by Vapk, et al. 99 SVM became famous whe, usg mages as put, t gave accuracy comparable to eural-etwork

More information

L5 Polynomial / Spline Curves

L5 Polynomial / Spline Curves L5 Polyomal / Sple Curves Cotets Coc sectos Polyomal Curves Hermte Curves Bezer Curves B-Sples No-Uform Ratoal B-Sples (NURBS) Mapulato ad Represetato of Curves Types of Curve Equatos Implct: Descrbe a

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:

More information

Dimensionality Reduction and Learning

Dimensionality Reduction and Learning CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that

More information

PTAS for Bin-Packing

PTAS for Bin-Packing CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,

More information

CHAPTER 3 POSTERIOR DISTRIBUTIONS

CHAPTER 3 POSTERIOR DISTRIBUTIONS CHAPTER 3 POSTERIOR DISTRIBUTIONS If scece caot measure the degree of probablt volved, so much the worse for scece. The practcal ma wll stck to hs apprecatve methods utl t does, or wll accept the results

More information

KLT Tracker. Alignment. 1. Detect Harris corners in the first frame. 2. For each Harris corner compute motion between consecutive frames

KLT Tracker. Alignment. 1. Detect Harris corners in the first frame. 2. For each Harris corner compute motion between consecutive frames KLT Tracker Tracker. Detect Harrs corers the frst frame 2. For each Harrs corer compute moto betwee cosecutve frames (Algmet). 3. Lk moto vectors successve frames to get a track 4. Itroduce ew Harrs pots

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ  1 STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ

More information

To use adaptive cluster sampling we must first make some definitions of the sampling universe:

To use adaptive cluster sampling we must first make some definitions of the sampling universe: 8.3 ADAPTIVE SAMPLING Most of the methods dscussed samplg theory are lmted to samplg desgs hch the selecto of the samples ca be doe before the survey, so that oe of the decsos about samplg deped ay ay

More information

Solving Constrained Flow-Shop Scheduling. Problems with Three Machines

Solving Constrained Flow-Shop Scheduling. Problems with Three Machines It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632

More information

Simulation Output Analysis

Simulation Output Analysis Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5

More information

Class 13,14 June 17, 19, 2015

Class 13,14 June 17, 19, 2015 Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral

More information

Lecture 3. Sampling, sampling distributions, and parameter estimation

Lecture 3. Sampling, sampling distributions, and parameter estimation Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called

More information

ECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity

ECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data

More information

MIMA Group. Chapter 4 Non-Parameter Estimation. School of Computer Science and Technology, Shandong University. Xin-Shun SDU

MIMA Group. Chapter 4 Non-Parameter Estimation. School of Computer Science and Technology, Shandong University. Xin-Shun SDU Grou M D L M Chater 4 No-Parameter Estmato X-Shu Xu @ SDU School of Comuter Scece ad Techology, Shadog Uversty Cotets Itroducto Parze Wdows K-Nearest-Neghbor Estmato Classfcato Techques The Nearest-Neghbor

More information

Chapter 3 Sampling For Proportions and Percentages

Chapter 3 Sampling For Proportions and Percentages Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys

More information

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators

More information

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen.

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen. .5 x 54.5 a. x 7. 786 7 b. The raked observatos are: 7.4, 7.5, 7.7, 7.8, 7.9, 8.0, 8.. Sce the sample sze 7 s odd, the meda s the (+)/ 4 th raked observato, or meda 7.8 c. The cosumer would more lkely

More information

A Combination of Adaptive and Line Intercept Sampling Applicable in Agricultural and Environmental Studies

A Combination of Adaptive and Line Intercept Sampling Applicable in Agricultural and Environmental Studies ISSN 1684-8403 Joural of Statstcs Volume 15, 008, pp. 44-53 Abstract A Combato of Adaptve ad Le Itercept Samplg Applcable Agrcultural ad Evrometal Studes Azmer Kha 1 A adaptve procedure s descrbed for

More information

Department of Agricultural Economics. PhD Qualifier Examination. August 2011

Department of Agricultural Economics. PhD Qualifier Examination. August 2011 Departmet of Agrcultural Ecoomcs PhD Qualfer Examato August 0 Istructos: The exam cossts of sx questos You must aswer all questos If you eed a assumpto to complete a questo, state the assumpto clearly

More information

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR Pot Patter Aalyss Part I Outle Revst IRP/CSR, frst- ad secod order effects What s pot patter aalyss (PPA)? Desty-based pot patter measures Dstace-based pot patter measures Revst IRP/CSR Equal probablty:

More information

Ideal multigrades with trigonometric coefficients

Ideal multigrades with trigonometric coefficients Ideal multgrades wth trgoometrc coeffcets Zarathustra Brady December 13, 010 1 The problem A (, k) multgrade s defed as a par of dstct sets of tegers such that (a 1,..., a ; b 1,..., b ) a j = =1 for all

More information

LINEAR REGRESSION ANALYSIS

LINEAR REGRESSION ANALYSIS LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for

More information

Chapter 11 Systematic Sampling

Chapter 11 Systematic Sampling Chapter stematc amplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of

More information

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose

More information

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture) CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.

More information

Lecture 3 Probability review (cont d)

Lecture 3 Probability review (cont d) STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto

More information

ENGI 3423 Simple Linear Regression Page 12-01

ENGI 3423 Simple Linear Regression Page 12-01 ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable

More information

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch

More information

D. VQ WITH 1ST-ORDER LOSSLESS CODING

D. VQ WITH 1ST-ORDER LOSSLESS CODING VARIABLE-RATE VQ (AKA VQ WITH ENTROPY CODING) Varable-Rate VQ = Quatzato + Lossless Varable-Legth Bary Codg A rage of optos -- from smple to complex A. Uform scalar quatzato wth varable-legth codg, oe

More information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst

More information

Chapter 14 Logistic Regression Models

Chapter 14 Logistic Regression Models Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as

More information

Applications of Multiple Biological Signals

Applications of Multiple Biological Signals Applcatos of Multple Bologcal Sgals I the Hosptal of Natoal Tawa Uversty, curatve gastrectomy could be performed o patets of gastrc cacers who are udergoe the curatve resecto to acqure sgal resposes from

More information

1 Onto functions and bijections Applications to Counting

1 Onto functions and bijections Applications to Counting 1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of

More information

Block-Based Compact Thermal Modeling of Semiconductor Integrated Circuits

Block-Based Compact Thermal Modeling of Semiconductor Integrated Circuits Block-Based Compact hermal Modelg of Semcoductor Itegrated Crcuts Master s hess Defese Caddate: Jg Ba Commttee Members: Dr. Mg-Cheg Cheg Dr. Daqg Hou Dr. Robert Schllg July 27, 2009 Outle Itroducto Backgroud

More information

Lecture 02: Bounding tail distributions of a random variable

Lecture 02: Bounding tail distributions of a random variable CSCI-B609: A Theorst s Toolkt, Fall 206 Aug 25 Lecture 02: Boudg tal dstrbutos of a radom varable Lecturer: Yua Zhou Scrbe: Yua Xe & Yua Zhou Let us cosder the ubased co flps aga. I.e. let the outcome

More information

Lecture 9: Tolerant Testing

Lecture 9: Tolerant Testing Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have

More information

1. BLAST (Karlin Altschul) Statistics

1. BLAST (Karlin Altschul) Statistics Parwse seuece algmet global ad local Multple seuece algmet Substtuto matrces Database searchg global local BLAST Seuece statstcs Evolutoary tree recostructo Gee Fdg Prote structure predcto RNA structure

More information

Random Variables and Probability Distributions

Random Variables and Probability Distributions Radom Varables ad Probablty Dstrbutos * If X : S R s a dscrete radom varable wth rage {x, x, x 3,. } the r = P (X = xr ) = * Let X : S R be a dscrete radom varable wth rage {x, x, x 3,.}.If x r P(X = x

More information

Module 7: Probability and Statistics

Module 7: Probability and Statistics Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to

More information

Cubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem

Cubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem Joural of Amerca Scece ;6( Cubc Nopolyomal Sple Approach to the Soluto of a Secod Order Two-Pot Boudary Value Problem W.K. Zahra, F.A. Abd El-Salam, A.A. El-Sabbagh ad Z.A. ZAk * Departmet of Egeerg athematcs

More information

CS286.2 Lecture 4: Dinur s Proof of the PCP Theorem

CS286.2 Lecture 4: Dinur s Proof of the PCP Theorem CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Aalyss of Varace ad Desg of Exermets-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr Shalabh Deartmet of Mathematcs ad Statstcs Ida Isttute of Techology Kaur Tukey s rocedure

More information

Chapter 9 Jordan Block Matrices

Chapter 9 Jordan Block Matrices Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.

More information

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA THE ROYAL STATISTICAL SOCIETY EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER II STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for

More information

A practical threshold estimation for jump processes

A practical threshold estimation for jump processes A practcal threshold estmato for jump processes Yasutaka Shmzu (Osaka Uversty, Japa) WORKSHOP o Face ad Related Mathematcal ad Statstcal Issues @ Kyoto, JAPAN, 3 6 Sept., 2008. Itroducto O (Ω, F,P; {F

More information

22 Nonparametric Methods.

22 Nonparametric Methods. 22 oparametrc Methods. I parametrc models oe assumes apror that the dstrbutos have a specfc form wth oe or more ukow parameters ad oe tres to fd the best or atleast reasoably effcet procedures that aswer

More information

6.867 Machine Learning

6.867 Machine Learning 6.867 Mache Learg Problem set Due Frday, September 9, rectato Please address all questos ad commets about ths problem set to 6.867-staff@a.mt.edu. You do ot eed to use MATLAB for ths problem set though

More information

Generative classification models

Generative classification models CS 75 Mache Learg Lecture Geeratve classfcato models Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Data: D { d, d,.., d} d, Classfcato represets a dscrete class value Goal: lear f : X Y Bar classfcato

More information

Bayesian Classification. CS690L Data Mining: Classification(2) Bayesian Theorem: Basics. Bayesian Theorem. Training dataset. Naïve Bayes Classifier

Bayesian Classification. CS690L Data Mining: Classification(2) Bayesian Theorem: Basics. Bayesian Theorem. Training dataset. Naïve Bayes Classifier Baa Classfcato CS6L Data Mg: Classfcato() Referece: J. Ha ad M. Kamber, Data Mg: Cocepts ad Techques robablstc learg: Calculate explct probabltes for hypothess, amog the most practcal approaches to certa

More information

A New Family of Transformations for Lifetime Data

A New Family of Transformations for Lifetime Data Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several

More information

A tighter lower bound on the circuit size of the hardest Boolean functions

A tighter lower bound on the circuit size of the hardest Boolean functions Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the

More information

Unit 9. The Tangent Bundle

Unit 9. The Tangent Bundle Ut 9. The Taget Budle ========================================================================================== ---------- The taget sace of a submafold of R, detfcato of taget vectors wth dervatos at

More information

9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d

9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d 9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,

More information

Lecture 07: Poles and Zeros

Lecture 07: Poles and Zeros Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto

More information

STK4011 and STK9011 Autumn 2016

STK4011 and STK9011 Autumn 2016 STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto

More information

Homework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015

Homework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015 Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bg-o ) otato. I ths problem, you wll prove some basc facts

More information

Chapter 8. Inferences about More Than Two Population Central Values

Chapter 8. Inferences about More Than Two Population Central Values Chapter 8. Ifereces about More Tha Two Populato Cetral Values Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha

More information

Median as a Weighted Arithmetic Mean of All Sample Observations

Median as a Weighted Arithmetic Mean of All Sample Observations Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of

More information

Nonparametric Techniques

Nonparametric Techniques Noparametrc Techques Noparametrc Techques w/o assumg ay partcular dstrbuto the uderlyg fucto may ot be kow e.g. mult-modal destes too may parameters Estmatg desty dstrbuto drectly Trasform to a lower-dmesoal

More information

The E vs k diagrams are in general a function of the k -space direction in a crystal

The E vs k diagrams are in general a function of the k -space direction in a crystal vs dagram p m m he parameter s called the crystal mometum ad s a parameter that results from applyg Schrödger wave equato to a sgle-crystal lattce. lectros travelg dfferet drectos ecouter dfferet potetal

More information

Multivariate Transformation of Variables and Maximum Likelihood Estimation

Multivariate Transformation of Variables and Maximum Likelihood Estimation Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty

More information

5 Short Proofs of Simplified Stirling s Approximation

5 Short Proofs of Simplified Stirling s Approximation 5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:

More information

TESTS BASED ON MAXIMUM LIKELIHOOD

TESTS BASED ON MAXIMUM LIKELIHOOD ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal

More information

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?

More information

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:

More information

Naïve Bayes MIT Course Notes Cynthia Rudin

Naïve Bayes MIT Course Notes Cynthia Rudin Thaks to Şeyda Ertek Credt: Ng, Mtchell Naïve Bayes MIT 5.097 Course Notes Cytha Rud The Naïve Bayes algorthm comes from a geeratve model. There s a mportat dstcto betwee geeratve ad dscrmatve models.

More information

18.413: Error Correcting Codes Lab March 2, Lecture 8

18.413: Error Correcting Codes Lab March 2, Lecture 8 18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse

More information

Chapter 8: Statistical Analysis of Simulated Data

Chapter 8: Statistical Analysis of Simulated Data Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample

More information

MATH 247/Winter Notes on the adjoint and on normal operators.

MATH 247/Winter Notes on the adjoint and on normal operators. MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say

More information

Pseudo-random Functions

Pseudo-random Functions Pseudo-radom Fuctos Debdeep Mukhopadhyay IIT Kharagpur We have see the costructo of PRG (pseudo-radom geerators) beg costructed from ay oe-way fuctos. Now we shall cosder a related cocept: Pseudo-radom

More information

Module 7. Lecture 7: Statistical parameter estimation

Module 7. Lecture 7: Statistical parameter estimation Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato

More information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package

More information

Arithmetic Mean and Geometric Mean

Arithmetic Mean and Geometric Mean Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,

More information

1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i.

1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i. CS 94- Desty Matrces, vo Neuma Etropy 3/7/07 Sprg 007 Lecture 3 I ths lecture, we wll dscuss the bascs of quatum formato theory I partcular, we wll dscuss mxed quatum states, desty matrces, vo Neuma etropy

More information

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss

More information

Lecture 2 - What are component and system reliability and how it can be improved?

Lecture 2 - What are component and system reliability and how it can be improved? Lecture 2 - What are compoet ad system relablty ad how t ca be mproved? Relablty s a measure of the qualty of the product over the log ru. The cocept of relablty s a exteded tme perod over whch the expected

More information

Lecture 12: Multilayer perceptrons II

Lecture 12: Multilayer perceptrons II Lecture : Multlayer perceptros II Bayes dscrmats ad MLPs he role of hdde uts A eample Itroducto to Patter Recoto Rcardo Guterrez-Osua Wrht State Uversty Bayes dscrmats ad MLPs ( As we have see throuhout

More information

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67. Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please

More information

9.1 Introduction to the probit and logit models

9.1 Introduction to the probit and logit models EC3000 Ecoometrcs Lecture 9 Probt & Logt Aalss 9. Itroducto to the probt ad logt models 9. The logt model 9.3 The probt model Appedx 9. Itroducto to the probt ad logt models These models are used regressos

More information