STATISTICAL method is one branch of mathematical


 Augustus Beasley
 1 years ago
 Views:
Transcription
1 40 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 Optimizig Forest Samplig by usig Lagrage Multipliers Suhud Wahyudi, Farida Agustii Widjajati ad Dea Oktaviati Abstract To obtai iformatio from a populatio, we use a samplig method Oe of samplig techiques that we ca use is double samplig Double samplig is a samplig techique based o the iformatio of first phase which is used as a additioal iformatio obtaiig estimates for the secod phase I this case, we discuss the model of double samplig with regressio estimator The, to obtai the optimal umber of samples for the first ad secod phases, we use Lagrage multipliers The model aalysis result is a formula to calculate the optimal umber of samples for the first phase ad the secod phase Implemetatio of this method is simulated by usig teak stads data from previous studies at Forest Maagemet Uit FMU Madiu which cosists of Sectio Forest Maagemet Uits FSMU Dagaga ad Dugus The calculatio result of data from FSMU Dagaga, we get optimal umber of plots must be observed i image iterpretatio are 49 plots ad field survey are 4 plots Ad with the data from FSMU Dugus, we get optimal umber of plots to be observed i image iterpretatio are 53 plots ad field survey are 0 plots Idex Terms Double samplig, Lagrage multiplier optimizatios, regressio estimator I INTRODUCTION STATISTICAL method is oe brach of mathematical sciece that focuses o the data collectio techique, processig or aalyzig data, ad deductio based o the data I data processig, we aalyze the relatioship betwee two or more variables, ad decide which oe is the most importat To aalyze the data we ca use regressio ad correlatio, i order to determie which variables are itercoected Oe of the discussio i statistical methods is samplig techique I statistical iferece, if we wat to obtai coclusios about the populatio, although without comprehesive observatio, the compositio of idividuals i the populatio, we ca use samplig techique [] We use samplig techique because of time ad cost efficiecy, large eough populatio, the precisio i the executio of observatio, ad the value of beefits I the process, samplig has may techiques that we ca use i various implemetatios of samplig, oe of them is double samplig Double samplig is a samplig techique based o the iformatio of first phase which is used as a additioal iformatio obtaiig estimates for the secod phase Oe implemetatio of double samplig is i forest ivetory [] However, we have to cosider the cost factor of samplig, so that we eed a optimal allocatio betwee the umber Mauscript received March 6, 07; accepted May 4, 07 The authors are with the Departmet of Mathematics, Istitut Tekologi Sepuluh Nopember, Surabaya 60, Idoesia of samples i the first phase ad secod phase To determie the optimal umber of samples, we ca use optimizatio by miimizig the cost fuctio ad defie the variace estimator fuctio as costrait The result of optimizatio is a optimal umber of samples for the first phase ad the secod phase [] We ca use the Lagrage multipliers method to optimizatio Lagrage multiplier method or Lagrage multipliers are itroduced by Joseph Louis de Lagrage Lagrage multiplier method is a method to maximize or miimize a fuctio of several variables by usig λ as its Lagrage multipliers The extesio of the method to a geeral problem of variables with m costraits has bee discussed i [3] Kitikidou explais that samplig optimizatio by usig Lagrage multipliers are computed by miimizig the cost fuctio ad defiig variace estimator fuctio as costraits [3] I this paper, we discuss a samplig optimizatio usig Lagrage multipliers method ad apply it to the forest observatio, especially the teak forest ivetory II DOUBLE SAMPLING, LINEAR REGRESSION MODEL AND REGRESSION ESTIMATOR IN DOUBLE SAMPLING A Two Phase Samplig Double Samplig Double samplig is oe of samplig techiques with two phases I the first phase, we choose uits umber of samples, ad i the secod phase we choose uits that are part of the first phase We use the first phase as a estimator for the secod phase I this case, we use regressio estimator Mea of regressio estimator is [3]: where ŷ = y + bx x y : mea of y from sub sample x : mea of x from sample x : mea of x from sub sample b : estimator of β iace of regressio estimator is [3]: ŷ = Sy r S y : variace of y o subsample r : correlatio coefficiet betwee y ad x : the umber of first sample which is take from N : the umber of subsamples from Optimum allocatio from cost fuctio is [3]: C = C + C
2 WAHYUDI et al: OPTIMIZING FOREST SAMPLING BY USING LAGRANGE MULTIPLIERS 4 C : the total cost of samplig C : the cost of first phase samplig C : the cost of secod phase samplig Let x,x,,x are radom samples from populatio with mea µ da stadard deviatio σ If we use samplig with replacemet ad ulimited populatio the we get []: µ x = µ σ x = σ µ x : mea of mea samplig distributio σx : variace of mea samplig distributio I double samplig, we assume y have ormal distributio, cofidece iterval for mea is [3]: where ŷ the we get ŷ = ε Z α/ B Liear Regressio Model ŷ Z α/ σ < y < ŷ + Z α/ σ = σ, error of estimatio is [3]: ε = Z α/ σ Liear regressio model of populatio is [4]: Y = α + βx + e j where α ad β are costat populatio parameters, ad β is the regressio coefficiet Regressio coefficiet β is [5]: β = N i= x i Xy i Y N i= x i X = σ xy σ x iace of populatio i regressio is [5]: σ = σ y β σ x Correlatio coefficiet of populatio i regressio is [5]: ρ = N i= x i Xy i Y Ni= x i X Ni= y i Y = σ xy σ x σ y Relatio of correlatio coefficiet with regressio coefficiet is [5]: ρ = β σ x σ y From equatio, we ca write: From equatio, we get: σ = σ y ρ σ y = σ ρ Regressio model i sample is [4]: y = a + bx k + e k For k =,,3,, with a ad b are estimators for α ad β, ad e k is error of estimator for kth observatio Regressio coefficiet b is [5]: b = i= x i xy i y i= x i x = S xy S x Correlatio coefficiet of sample i regressio is: r = i= x i xy i y i= x i x i= y i y = S xy S x S y 3 Sy = i= y i i= y i C Regressio Estimator i Double Samplig Aother model of Y is [5]: Y = Y + β If we assume y is a estimator of equatio Y, the we get : y = Y + β + e where e is the error, so E e = 0, the we get: We also get: y = Y + β + e 4 E y = E Y + β + e = Y The above equatio of E y shows that y is a ubiased estimator for Y If we assume i= e ix i x = e w, we get the value of b which i= x i x is a estimator of β as follows: b = β + e w Because of E e = 0, so E e w = 0 The E b = β ad E E b = β E e w = E i= e i x i x i= x i x = D Expectatio ad iace i Multivariate Distributio Geeral variace distributio is defied as [5]: y = Ey E y = E y E y E y = y + E y Expectatio ad variace i multivariate distributio is [5]: E = E E m = E E m m + E m E m + E E m E
3 4 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 E Fisher s Distributio Fisher s variable Fm,m is distributed as [5]: Fm,m = Fm,m where m ad m is degree of freedomi Fisher s distributio If we assume m = ad m =, we get: xi X F, = i= xi X Expectatio of distributio is as follows [5]: where m 3 EF m,m = m m F Optimizatio by usig Lagrage Multipliers Method Optimizatio techique of multivariables with equality costrait have the followig geeral form [6]: miimize f X subject to g j X = 0, for j =,,,m where X = {x,x,,x } T where m If m >, the it caot be solved The first step of this method is the costructio of Lagrage fuctio that is defied as [6]: LX,λ = f X + m j= λ j g j X 5 Theorem [6]: Necessary coditio for a fuctio f X with costrait g j X = 0, where j =,,,m such that it has relative miimum at poit x is first partial derivative of Lagrage fuctio defied as L = L{x,x,,x,λ,λ,,λ } has value zero Theorem [6]: A sufficiet coditio for f X to have relative miimum or maximum at the quadratic, Q, defied by: Q = i= j= L dx i dx j x i x j evaluated at x = x must be positive defiite or egative defiite for all values of dx for which the costraits are satisfied Necessary coditio Q = i= j= L x i x j dx i dx j to be positive or egative defiite for all admissible variatios dx is that each root of the polyomial p i, defied by the followig determiat equatio, be positive or egative L p L L 3 L g g g m L L p L 3 L g g g L L L 3 L m p g m g m g m g g g 3 g = 0 g g g 3 g g m g m g m3 g m where L i j = Lx,λ x i x j ad g i j = g ix x j Observe that equatio 6 is a polyomial of order m, i p III RESULTS AND DISCUSSIONS A Mea of Regressio Estimator i Double Samplig Liear regressio estimator ca be defied as: Y = α + βx Y = α + βx 7 Equatio 7 is liear regressio populatio average If we estimate liear regressio from its sample, where a is a estimator for α, ad b is a estimator for β, so we obtai: y = a + bx 8 y = a + bx 9 Equatio 9 is a liear regressio average equatio i sample From equatio 9, we obtai: a = y bx 0 By substitutig equatio 0 to equatio 8, we obtai estimator regressio equatio as follows: y = y + bx bx = y + bx x If the value of x is ukow, the to compute its estimator, we ca use x = i= x i we obtai: ŷ = y + bx x Equatio is mea estimator equatio of liear regressio i double samplig B iace of Regressio Estimator i Double Samplig After we obtai equatio, substitutig y from equatio 4 ad value of b from equatio, so we obtai aother model of mea estimator equatio of liear regressio i double samplig It is give by: ŷ = Y + β + e + β + e w x x = Y + β x X + e w x X e w + e To obtai the variace of regressio estimator, we use the trivariate distributio theory Mea of trivariate distributio is as follows: E Ŷ = E E 3 ŷ We assume: E 3 ŷ = Y + β x X So we obtai E 3 ŷ, which is a ubiased estimator The we determie E E 3 ŷ as follows: E E 3 ŷ = E Y + β x X = Y + β x X For the ext step, we assume x is ot costat, we obtai: E E 3 ŷ = Y + β x X = Y
4 WAHYUDI et al: OPTIMIZING FOREST SAMPLING BY USING LAGRANGE MULTIPLIERS 43 Equatio is a ubiased estimator of the trivariate distributio The, we get variace of trivariate distributio by the formula : Ŷ = E 3 ŷ E E 3 ŷ + E 3 ŷ + 3 The we determie the secod part of equatio 3, so we obtai: E 3 ŷ = Y + β x X = 0 So that equatio 4 becomes: = E e + x X E + x X E E ee w + x X E ee w x X E 5 The, we solve each part of equatio 5, the first part of equatio 5 is: E e = e E e = σ e = σ Next, we solve the secod part of equatio 5 as follows: x X E = σ E σ F, = 3 The, we solve the third part of equatio 5 as follows: x X E = σ E σ F, = 3 The fourth part of equatio 5 is as follows: E ee w = E 0 = 0 Next, the fifth part of equatio 5 is give by: x X E ee w = x X 0 = 0 From the sixth part of equatio 5, we obtai: x X E = X X E X X E =0 The result from all previous steps is: E 3 ŷ = σ + σ 3 + σ 3 6 If the umber of sample is too big, the 3 ad equatio 6 becomes: = σ + + From aalysis result, equatio 3 becomes: Ŷ = σ β σ x 7 Third part of equatio 3 is: E E 3 ŷ = Y + β x X If ad, the 0 ad 0 I this case, equatio 7 becomes: = β x x = β σ x Ŷ = σ y ρ 8 Next, we determie: Equatio 8 is a variace equatio of regressio estimator E 3 ŷ i populatio If we estimate equatio 8 i sample, the we = E 3 Y + β x X + e w x X ca write: e w + e = E e + x X ŷ = Sy 4 r 9 Equatio 9 is variace equatio of regressio estimator i sample C Samplig Optimizatio by usig Lagrage Multipliers Method First step of optimizatio is determie the objective fuctio ad costrait The objective fuctio is observatio cost fuctio, that ca be writte as: f = C + C The costrait is the variace of regressio estimator i sample as follows: g = Sy r ε Zα/ = 0 The we costruct Lagrage fuctio as i equatio 5: L = C + C + λ Sy r ε Zα/ = C + C + λsy λs yr + λs yr λ ε Optimal coditio for equatio 0 is: 0 L = C λsyr = 0 L = C λsy r = 0 L λ = S y S yr + S yr ε Zα/ = 0 3
5 44 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 From equatio, we obtai: From equatio, we obtai: λ = C S yr 4 λ = C S y r From equatios 4 ad 5, we obtai: 5 = C r r C 6 = From equatio 3, we get: r C C r Sy S yr + S yr = ε 7 8 The, substitutig equatio 7 to equatio 8, we obtai: S y ε r = S yr + S y C C r r C r C C r + r C r r = 9 ε By solvig the above problem usig Lagrage multipliers method, we get equatio 9 which is a formula for calculatig the umber of optimal plots o first phase We ca write it as follows: C Syr + Sy C r r opt = ε 30 Next, by substitutig equatio 6 to equatio 8, we obtai: S y r + S y C ε C r r = 3 Equatio 3 is the formula for calculatig the umber of optimal plots o secod phase We ca write it as follows: S y r C + Sy C r r opt = ε 3 D Implemetatio of Samplig Optimizatio by usig Lagrage Multipliers Method The method is implemeted by usig simulatio of image iterpretatio data ad field survey data Image iterpretatio data is the data forest picture obtaied from observatios with remote sesig The result of remote sesig is calculated by software util we get the diameter, desity, ad umber of trees per plot, the we ca also calculate tree volume per plot The, we check the result of image iterpretatios i the field To determie the potetial of a forest, it is impossible to observe all objects i forest Thus, we eed to take some samples I previous research of Fathia Amalia R D, she takes 76 plot samples for first phase samplig which is i image iterpretatio ad 38 plot samples for secod phase without kowig whether the umber of samples is optimum or ot I this paper, we eed to calculate the optimal umber of samples i image iterpretatio ad i the field Samples were observed i the form of plots where the plot cosists of several trees Data of previous observatio result ca be used for calculatig the optimal umber of samples which must be observed o image iterpretatio ad field We use data from FMU Perum Perhutai Madiu II, East Java, which icludes data from FSMU Dagaga ad Dugus For calculatig the umber of optimal samples, we use the followig parameters: TABLE I SUM AND AVERAGE OF TREE VOLUME Locatio Parameter m 3 FSMU Dagaga FSMU Dugus /0 ha Sum of V image samples V f ield Sum of V image samples Sum of V f ield Sum of Vimage samples Sum of Vf ield Average of V f ield Average V image Average V image samples Observatio cost cosists of two types: image observatio cost ad field observatio cost Image observatio cost is the total of cost which is used to buy image, image processig cost, ad image map pritig cost Field observatio cost is icluded i trasportatio cost, employee salary ad etc So that, we obtai the cost per hectare: TABLE II OBSERVATION COST Locatio FSMU Dagaga FSMU Dugus Cost Rp/ha Image Iterpretatio Field The to determie the optimal umber of samples i the first phase opt ad secod phase opt, we calculate S y value first by usig formula i equatio 3, ad also calculate r by usig formula i equatio 9 Next, we calculate opt by usig formula i equatio 30 ad calculate opt by usig formula i equatio 3 We calculate all step by usig Matlab, for locatio FSMU Dagaga we obtai optimal umber of samples that must be
6 WAHYUDI et al: OPTIMIZING FOREST SAMPLING BY USING LAGRANGE MULTIPLIERS 45 observed for first phase opt is 49 plots image iterpretatio ad umber of samples that must be observed for the secod phase opt is 4 plots field survey With the same way, for FSMU Dugus the umber of optimal samples that must be observed is 53 plots image iterpretatio ad 0 plots field survey IV CONCLUSIONS Based o aalysis result ad discussio, we obtai the followig coclusios: Result from formula aalysis i samplig ad optimizatio by usig Lagrage multipliers method, we obtai the umber of optimal samples i the formula for first phase ad secod phase is: C Syr + Sy C r r opt = ε S y r C + Sy C r r opt = ε S y : variacey from the secod phase sample r : correlatio coefficiet C : cost of first phase samplig C : cost of secod phase samplig ε : error i estimatio Z α/ : value of radom variables i stadard ormal distributio From the calculatio results, the umber of optimal samples i image iterpretatio ad field survey with FSMU Dagaga data, we obtai the umber of optimal plots that must be observed i image iterpretatio is 49 plots ad i field survey is 4 plots The other side, with FSMU Dugus data we obtai the umber of optimal plots that must be observed i image iterpretatio is 53 plots ad i field survey is 0 plots So that, if the umber of samples is suitable with that calculatio result, the we obtai a optimal samplig REFERENCES [] R Walpole, Pegatar statistika, edisi ke3 Itroductio to statistics PT Gramedia Pustaka Utama, 990 [] P Malamassam, Modul Mata Kuliah Ivetarisasi Huta Makassar: Hasauddi Uiversity, 009 [3] K Kitikidou, Optimizig forest samplig by usig Lagrage multipliers, America Joural of Operatios Research, vol, o, pp 94 99, 0 [4] S Makridakis, S Wheelwright, ad V McGee, Metode da Aplikasi Peramala Jilid Ir Utug Sus Ardiyato, M Sc & Ir Abdul Basith, M Sc Terjemaha Jakarta: Erlagga, 999 [5] P de Vries, Samplig Theory for Forest Ivetory Wageige: Wageige Agricultural Uiversity, 986 [6] D Lukato, Pegatar Optimasi No Liier Yogyakarta: Gajah Mada Uiversity, 000
Properties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Crosssectioal data. 2. Time series data.
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationLinear Regression Models
Liear Regressio Models Dr. Joh MellorCrummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More informationEstimating Confidence Interval of Mean Using. Classical, Bayesian, and Bootstrap Approaches
Iteratioal Joural of Mathematical Aalysis Vol. 8, 2014, o. 48, 23752383 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ijma.2014.49287 Estimatig Cofidece Iterval of Mea Usig Classical, Bayesia,
More informationSome Exponential RatioProduct Type Estimators using information on Auxiliary Attributes under Second Order Approximation
; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 09770)]; ISSN 0975556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial RatioProduct
More informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 00900 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationSimple Linear Regression
Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio
More informationSampling Error. Chapter 6 Student Lecture Notes 61. Business Statistics: A DecisionMaking Approach, 6e. Chapter Goals
Chapter 6 Studet Lecture Notes 61 Busiess Statistics: A DecisioMakig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 61 Chapter Goals After completig this chapter, you should
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol DiscreteEvent System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol DiscreteEvet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationGoodnessOfFit For The Generalized Exponential Distribution. Abstract
GoodessOfFit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationBootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests
Joural of Moder Applied Statistical Methods Volume 5 Issue Article 5 Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed AlHa Ebrahem
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationModified Ratio Estimators Using Known Median and CoEfficent of Kurtosis
America Joural of Mathematics ad Statistics 01, (4): 95100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad CoEfficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry
More informationAssessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions
Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationVaranasi , India. Corresponding author
A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationImproved exponential estimator for population variance using two auxiliary variables
OCTOGON MATHEMATICAL MAGAZINE Vol. 7, No., October 009, pp 66767 ISSN 5657, ISBN 979735550, www.hetfalu.ro/octogo 667 Improved expoetial estimator for populatio variace usig two auxiliar variables
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. BetaBinomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 12631277 HIKARI Ltd, www.mhikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture  9 Multicolliearity Dr Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Multicolliearity diagostics A importat questio that
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationSampling, Sampling Distribution and Normality
4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationEstimation of Gumbel Parameters under Ranked Set Sampling
Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 112014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com
More informationSimple Regression. Acknowledgement. These slides are based on presentations created and copyrighted by Prof. Daniel Menasce (GMU) CS 700
Simple Regressio CS 7 Ackowledgemet These slides are based o presetatios created ad copyrighted by Prof. Daiel Measce (GMU) Basics Purpose of regressio aalysis: predict the value of a depedet or respose
More informationCTL.SC0x Supply Chain Analytics
CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationV. Nollau Institute of Mathematical Stochastics, Technical University of Dresden, Germany
PROBABILITY AND STATISTICS Vol. III  Correlatio Aalysis  V. Nollau CORRELATION ANALYSIS V. Nollau Istitute of Mathematical Stochastics, Techical Uiversity of Dresde, Germay Keywords: Radom vector, multivariate
More informationSession 5. (1) Principal component analysis and KarhunenLoève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad KarhueLoève trasformatio Topic 2 of this course explais the image
More informationNYU Center for Data Science: DSGA 1003 Machine Learning and Computational Statistics (Spring 2018)
NYU Ceter for Data Sciece: DSGA 003 Machie Learig ad Computatioal Statistics (Sprig 208) Brett Berstei, David Roseberg, Be Jakubowski Jauary 20, 208 Istructios: Followig most lab ad lecture sectios, we
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationWorksheet 23 ( ) Introduction to Simple Linear Regression (continued)
Worksheet 3 ( 11.511.8) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationOn an Application of Bayesian Estimation
O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, HigashiOsaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma
More informationUnbiased Estimation. February 712, 2008
Ubiased Estimatio February 72, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationStudy on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm
Joural of ad Eergy Egieerig, 05, 3, 43437 Published Olie April 05 i SciRes. http://www.scirp.org/joural/jpee http://dx.doi.org/0.436/jpee.05.34058 Study o Coal Cosumptio Curve Fittig of the Thermal Based
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hardcopy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationCorrelation and Covariance
Correlatio ad Covariace Tom Ilveto FREC 9 What is Next? Correlatio ad Regressio Regressio We specify a depedet variable as a liear fuctio of oe or more idepedet variables, based o covariace Regressio
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 656 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationEstimation of the Population Mean in Presence of NonResponse
Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of NoRespose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationOutput Analysis and RunLength Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad RuLegth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationStatistical Intervals for a Single Sample
3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8 Cofidece Iterval o the Mea
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationLainiotis filter implementation. via Chandrasekhar type algorithm
Joural of Computatios & Modellig, vol.1, o.1, 2011, 115130 ISSN: 17927625 prit, 17928850 olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais
More information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationRegression. Correlation vs. regression. The parameters of linear regression. Regression assumes... Random sample. Y = α + β X.
Regressio Correlatio vs. regressio Predicts Y from X Liear regressio assumes that the relatioship betwee X ad Y ca be described by a lie Regressio assumes... Radom sample Y is ormally distributed with
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationA NOTE ON THE TOTAL LEAST SQUARES FIT TO COPLANAR POINTS
A NOTE ON THE TOTAL LEAST SQUARES FIT TO COPLANAR POINTS STEVEN L. LEE Abstract. The Total Least Squares (TLS) fit to the poits (x,y ), =1,,, miimizes the sum of the squares of the perpedicular distaces
More informationSolutions to Odd Numbered End of Chapter Exercises: Chapter 4
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd Numbered Ed of Chapter Exercises: Chapter 4 (This versio July 2, 24) Stock/Watso  Itroductio to Ecoometrics
More informationPaired Data and Linear Correlation
Paired Data ad Liear Correlatio Example. A group of calculus studets has take two quizzes. These are their scores: Studet st Quiz Score ( data) d Quiz Score ( data) 7 5 5 0 3 0 3 4 0 5 5 5 5 6 0 8 7 0
More informationo <Xln <X2n <... <X n < o (1.1)
Metrika, Volume 28, 1981, page 257262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3337 HIKARI Ltd, www.mhikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationQuestion 1: Exercise 8.2
Questio 1: Exercise 8. (a) Accordig to the regressio results i colum (1), the house price is expected to icrease by 1% ( 100% 0.0004 500 ) with a additioal 500 square feet ad other factors held costat.
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationFirst Year Quantitative Comp Exam Spring, Part I  203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I1 Part I  203A A radom variable X is distributed with the margial desity: >
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Twoparameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationSIMPLE LINEAR REGRESSION AND CORRELATION ANALYSIS
SIMPLE LINEAR REGRESSION AND CORRELATION ANALSIS INTRODUCTION There are lot of statistical ivestigatio to kow whether there is a relatioship amog variables Two aalyses: (1) regressio aalysis; () correlatio
More information5.1 A mutual information bound based on metric entropy
Chapter 5 Global Fao Method I this chapter, we exted the techiques of Chapter 2.4 o Fao s method the local Fao method) to a more global costructio. I particular, we show that, rather tha costructig a local
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 5766 ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More informationIntroduction to Econometrics (3 rd Updated Edition) Solutions to Odd Numbered End of Chapter Exercises: Chapter 4
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd Numbered Ed of Chapter Exercises: Chapter 4 (This versio August 7, 204) 205 Pearso Educatio, Ic. Stock/Watso
More informationSample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for
Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they
More informationNew Ratio Estimators Using Correlation Coefficient
New atio Estimators Usig Correlatio Coefficiet Cem Kadilar ad Hula Cigi Hacettepe Uiversit, Departmet of tatistics, Betepe, 06800, Akara, Turke. emails : kadilar@hacettepe.edu.tr ; hcigi@hacettepe.edu.tr
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationMOMENTMETHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMETMETHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationA LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!
A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited
More information6.867 Machine learning, lecture 7 (Jaakkola) 1
6.867 Machie learig, lecture 7 (Jaakkola) 1 Lecture topics: Kerel form of liear regressio Kerels, examples, costructio, properties Liear regressio ad kerels Cosider a slightly simpler model where we omit
More informationTopic 15: Maximum Likelihood Estimation
Topic 5: Maximum Likelihood Estimatio November ad 3, 20 Itroductio The priciple of maximum likelihood is relatively straightforward. As before, we begi with a sample X (X,..., X of radom variables chose
More informationChapter 1 (Definitions)
FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple
More informationUCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences
UCLA STAT 13 Itroductio to Statistical Methods for the Life ad Health Scieces Istructor: Ivo Diov, Asst. Prof. of Statistics ad Neurolog Sample Size Calculatios & Cofidece Itervals for Proportios Teachig
More informationMatrix Representation of Data in Experiment
Matrix Represetatio of Data i Experimet Cosider a very simple model for resposes y ij : y ij i ij, i 1,; j 1,,..., (ote that for simplicity we are assumig the two () groups are of equal sample size ) Y
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 20052006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationA Risk Comparison of Ordinary Least Squares vs Ridge Regression
Joural of Machie Learig Research 14 (2013) 15051511 Submitted 5/12; Revised 3/13; Published 6/13 A Risk Compariso of Ordiary Least Squares vs Ridge Regressio Paramveer S. Dhillo Departmet of Computer
More informationII. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation
II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso productmomet correlatio
More information