Normalisation with respect to pattern
|
|
- Berniece Curtis
- 6 years ago
- Views:
Transcription
1 Normalisatio with respect to patter Iwoa Müller-Fr czek Nicolaus Copericus Uiversity i Toru«, Polad Abstract The article presets a ew ormalisatio method of diagostic variables - ormalisatio with respect to the patter. The ormalisatio preserves some importat descriptive characteristics of variables: skewess, kurtosis ad the Pearso correlatio coeciets. It is particularly useful i dyamical aalysis, whe we work with the whole populatio of objects ot a sample, for example i regioal studies. After proposed trasformatio variables are comparable ot oly betwee themselves but also across time. The we ca use them, for example, to costruct composite variables. keywords: ormalisatio, stadardisatio, composite variable, sythetic measure 1 Itroductio I regioal studies we ofte eed to compare regios objects with respect to aalyzed complex or composite pheomeo. Complex pheomeo is a qualitative pheomeo, that is characterized by some quatitative features, called diagostic variables. Each object is the idetied with a poit of the multidimesioal real space. Oe of the tools of regioal research are composite variable or sythetic measure. Composite variable is created to reect multidimesioal poits objets i the oe-dimesioal space. May advaced methods of costructig sythetic variables have bee developed, however the simplest methods are ofte used i practice. There Author's Address: I. Müller-Fr czek, Faculty of Ecoomic Scieces ad Maagemet, Nicolaus Copericus Uiversity, ul. Gagaria 13 a, Toru«, Polad, muller@umk.pl 1
2 are a lot of such examples see [2], oe of them is very popular Huma Developmet Idex HDI, which raks coutries ito four tiers of socio-ecoomic developmet. Util 2010 HDI was a uiformly weighted sum of three idicators describig: life expectacy, educatio, ad icome per capita. Oe of the step of the costructio of sythetic measure is brigig diagostic variables to comparability, called ormalisatio or stadardisatio. Normalisatio deprives variables their uits ad uies their rages. There are a lot of ormalisatio formulas see [4], [5], [8]. Choosig a proper method is importat because ormalisatio iueces o results of object orderig. The usual stochastic approach ca be used to determie parameters eeded to ormalisatio. The we treat values of variable observatios as a radomly selected sample of the populatio. This approach should ot be used i regioal research, where we work with the whole populatio of objects. I this case we should use a descriptive determiistic approach. Normalisatio formulas are most ofte give for static aalysis, this is for a xed poit i time. A ormalisatio problem appears whe we wat to compare situatios of regios at several time poits. The the variables should also be comparable across time. To achieve this eect i the stochastic approach oe ca use all values of variable both for objects ad for time to determie parameters eeded for ormalisatio. However, this solutio is cotroversial i descriptive approach see [9], i additio, it requires icesat coversio of results whe later observatios occur. I this case we should rather use curret observatios, so after usual ormalisatio variables are ot comparable across time. The we ca ot compare the values of sythetic measures, we ca oly compare rakigs. To solve this problem i the metioed Huma Developmet Idex, the parameters of feature scalig are xed o levels, that are ot related to variable distributio. The levels are justied by substative reasos. For example, the age of 85 was established as the maximum life expectacy at birth. The article proposes a ew method of feature ormalisatio - ormalisatio with respect to the patter or patter ormalisatio for short. This ame was ispired by the Hellwig's paper see [3], [1]. The method is cosistet with the static approach, but it ca be used to compare objects at dieret time poits. The method meets the requiremets of ormalisatio that are suggested i literature see [4], [6]. It preserves skewess ad kurtosis. Moreover, the absolute values of the Pearso correlatio coeciets are ot chaged after ormalisatio. I the rs step of the patter ormalisatio the ature of variable is determied i the cotext of aalyzed complex pheomeo. We distiguish stimulats ad destimulats. Stimulat is a diagostic variable that has a 2
3 positive impact o the aalyzed complex pheomeo, while destimulat egative. I regioal research determiig the ature of variables is atural. Most ofte, before ormalisatio, we tur destimulats ito stimulats usig their iverse values. Ufortuately, the variables after coversio lose their iterpretatio ad their distributios are chaged. I the preseted method, we do ot coverse destimulat before ormalisatio. Destimulats ad stimulats are ormalized i dieret ways. Determiig the ature of variable allows us to choose the most beecial observatio amog all values of the variable, maximum for stimulat ad miimum for destimulat. We call this value a patter. Next we covert all values with respect to this patter. After trasformatio we get comparable variables. All of them are destimulats with clear iterpretatio. Patter ormalisatio ca be used i commo costructiio of composite variables istead of other methods of ormalisatio. A possible applicatio is show i [7]. 2 Deitio of patter ormalisatio Suppose that a complex pheomeo observed for N regios is aalyzed. Assume that we caot measure this pheomeo, whereas we kow a collectio of measurable diagostic variables that characterize it. Assume that diagostic variables meet both substative ad statistical requiremets, for more details see for example [9]. Let us cosider oe such variable x x 1, x 2,..., x R, which is a stimulat the we write x S, S deotes the set of stimulats or a destimulat x D aalogously. I the rst step we choose a patter - the most beecial of all values of the variable x. The patter is uique for all objects ad is described by the formula: max x i if x S, x + i 1 mi x i if x D. i After specifyig the patter x + we ca cosider a ew variable u + istead of the variable x give by: x u + x i x + + x if x S, i j1 x j x + j1 x+ x j x i x + 2 j1 x if x D. j x + 3
4 The formula 2 determies a certai trasformatio of iitial variable x x 1, x 2,..., x ito a ew variable u + u + 1, u + 2,..., u +. We call it a ormalisatio with respect to the patter. After this trasformatio the ew variable describes the same aspect of complex pheomeo as described by x. So u + is a diagostic variable of this pheomeo. The patter ormalisatio 2 is ot just a techical procedure. New variable has a clear iterpretatio, u + i species the share of distace betwee the i-th object ad the patter i the total distace of all objects from the patter. The situatio of the i-th object is better whe the value of u + i is lower. The values of variable u + characterize the positios of objects i the whole system. This is the same as for other forms of ormalisatio, but the system is specied i a dieret way. I the case of the patter ormalisatio the system is represeted by the sum of distaces betwee objects ad patter, while i commo ormalisatios descriptive characteristics of the distributio of x are used for this purpose. 3 Properties of variable after ormalisatio The quatitative descriptio of a immeasurable qualitative pheomeo is obtaied usig sythetic measures. Brigig diagostic variables to comparability is the rst step i the costructio of such measure. The patter ormalisatio ca be used for this purpose. Assume that diagostic variables are trasformed with respect to their patters. The the ew set of variables has advatages, which are expected for creatig sythetic variables. These properties ad some proofs are preseted below. A. Basic properties A1. All variables after patter ormalisatio are uitless, o-egative ad limited to iterval [0, 1]. Because of that, the ew set of diagostic variables cotais comparable elemets. A2. Irrespective of the iitial ature, variable after the patter ormalisatio becomes destimulat. It meas that the situatio of the i-th object is better whe the value u + i is lower. I this sese the patter ormalisatio uies the ature of diagostic variables. A3. Trasformig of variables does ot aect the orderig of objects. B. Extreme values after patter ormalisatio 4
5 B1. The variable u + ca take zero value oly for the patter object. Sice the patter is chose amog values of the variable x, zero value is take. u + i 0 x i x +. u + i 0 x i x + j1 x j x + 0 x i x + 0 x i x + B2. The value u + i equals 1 whe all objects are patters except the i-th object. This situatio is rather urealistic. u + i 1 j i x j x +. u + i 1 x i x + j1 x j x + 1 x i x + x j x + x j x + j i j1 B3. The maximum value of u + depeds o the ature of variable x ad it is expressed by: If x S, the: max i u + i max u + i max ix + x i i j1 x+ x j If x D, the: max i x i mi i x i j1 max i x i x j max i x i mi i x i j1 x j mi i x i x+ mi i x i j1 x+ x j max u + i max ix i x + i j1 x j x + max i x i x + j1 x j x + if x S, if x D. max i x i mi i x i j1 max i x i x j. max i x i mi i x i j1 x j mi i x i. 5
6 C. Descriptive characteristics of ormalised variables C1. The mea value of u + depeds oly o the umber of objects ad is iversely proportioal to this umber. It is expressed by: u + 1 u + def 1 u + i 1. x i x + j1 x j x + 1 x i x + j1 x j x + 1 C2. The variace of u + is described by: S 2 u + 1 S 2 u + def 1 u + i u + 2 x + x i j1 x+ x j 1 If x S, the: 2 S 2 u + 1 x + x j1 x+ x j x + 2 x i 3 x + x S 2 x 2 x + x 2 The proof is similar whe x D. S 2 x 2 x + x 2. x + x x + 1 j1 x 1 j 2 1 x x i 2 2 x + x 2 x xi x + x C3. The stadard deviatio of u + depeds o the ature of variable x ad it is expressed by: Su + def Sx if x S, S 2 u + x + x Sx if x D. x x + 6 2
7 C4. The coeciet of variatio of u + is give by: CV u + def Su+ u + C5. The 3-rd cetral momet of u + is give by: µ 3 u + def 1 Sx if x S, x + x Sx if x D. x x + u + i u + 3 µ 3 x 3 x + x 3. µ 3 u + 1 x + x i j1 x+ x j 1 3 If x S, the: µ 3 u + 1 x + x j1 x+ x j x + x 4 x + 1 j1 x j 1 x + 3 x i 4 x + x xi x 4 x x + µ 3 x 3 x x The proof is similar whe x D. C6. The absolute value of the coeciet of skewess does ot chage after the patter ormalisatio: { Au + def µ 3u + S 3 u + Ax if x S, Ax if x D. C7. The 4-th cetral momet of u + is give by: µ 4 u + def 1 u + i u + 4 µ 4 x 4 x + x 4. 7
8 µ 4 u + 1 x + x j1 x+ x j 1 4 If x S, the: µ 4 u + 1 x + x j1 x+ x j xi x 1 5 x x + 5 µ 4 x 3 x x + 4 x xi x + x 1 x + x i x + 1 j1 x j The proof is similar whe x D. C8. The kurtosis of u + does ot chage after the patter ormalisatio: Ku + def µ 4u + S 4 u + Kx. D. Liear relatio betwee variables after ormalisatio Assume that two diagostics variables x 1, x 2 are trasformed with respect to their patters. Deote by u + 1 ad u + 2 variables after ormalisatio. D1. The covariace betwee u + 1 ad u + 2 equals: covu 2 1, u + 2 def 1 u + i1 u+ 1 u + i2 u+ 2 covx 1, x 2 2 x + 1 x 1 x + 2 x 2 covx 1, x 2 2 x + 1 x 1 x + 2 x 2 if x 1, x 2 S or x 1, x 2 D, otherwise. covu 2 1, u x i1 x + 1 j1 x j1 x x i2 x + 2 j1 x j2 x
9 Assume that x 1 ad x 2 are stimulats. The proof i other cases is similar. covu 2 1, u x + 1 x 1 j1 x j1 x x + 2 x 2 j1 x j2 x x + 1 x i1 x + 2 x 2 3 x j1 x 1 j1 x j1 x 1 j2 1 x + 1 x i1 x + 3 x x i2 1 x 1 x x 2 1 x1 x i1 3 x + x2 1 x i2 1 x 1 x + x i1 x 1 x i2 x 2 2 x 2 2 x + 1 x 1 x + 2 x 2 covx 1, x 2 2 x + 1 x 1 x + 2 x 2 D2. The absolute value of the Pearso correlatio coeciet of diagostic variables is preserved after the ormalisatio: { corru + 1, u + 2 def covu2 1, u + 2 Su + 1 Su + 2 corrx 1, x 2 if x 1, x 2 S or x 1, x 2 D, corrx 1, x 2 otherwise. E. Dyamic approach Assume that the diagostic variable x is observed i two periods of time the we write x 1 ad x 2 respectively. For each period we choose a patter ad trasform x 1 ad x 2 ito u 1+ ad u 2+ accordig to the formula 2. E1. The values of variables u 1+ ad u 2+ are comparable. Substatiatio. The system is characterized by the sum of distaces betwee objects ad the patter. It chages over time. For give object, if the value of the trasformed variable icreases over time, this meas that the share of distace from this object to the patter i the sum of all distaces icreases, so the situatio of this object becomes worse i compariso with the situatios of other objects. 9
10 4 Summary The ormalisatio of diagostic variables described by formula 2 plays a double role i the costructio of sythetic measure. First, it uies the ature of variables A2. Secodly, it brigs variables to comparability A1. So, after patter ormalisatio diagostic variables become comparable destimulats. The ormalisatio with respect to the patter preserves two importat characteristics of the distributio of diagostic variables - skewess C6 ad kurtosis C8. Moreover, this coversio does ot disrupt liear relatio betwee variables - the absolute value of the Pearso correlatio coeciet is ot chaged D2. This advatages are expected for ormalisatios used for brigig variables to comparability. Ulike other methods the patter ormalisatio is ot just a techical procedure, it has clear iterpretatio. However, the major advatage of the patter ormalisatio over other ormalisatio methods appears i dyamic approach. Although the curret data are the sole data used to covert variables, the trasformed variables are comparable i time E1. The ormalisatio with respect to the patter seems to be a useful tool i multidimesioal comparative aalysis. It ca be applied wheever variables eed to be comparable, for example i the sythetic aalysis of complex pheomeo. The proposed costructio ca have various modicatios, for example we ca chage the measure of distace or the method of choosig patter. Refereces [1] FANCHETTE, S "Sychroic ad diachroic approaches i the Uesco project o huma resources idicators - Wroclaw taxoomy ad bivariate diachroic aalysis", UNESCO documet, SHS/WS/209, Paris. [2] FREUDENBERG, M. 2003, "Composite Idicators of Coutry Performace: A Critical Assessmet", OECD Sciece, Techology ad Idustry Workig Papers, No. 2003/16, OECD Publishig, Paris. [3] HELLWIG, Z. 1968, "Procedure of Evaluatig High-Level Mapower Data Ad Typology of Coutries by Meas of the Taxoomic Method", upublished UNESCO workig paper, COM/WS/91, Paris. [4] JAJUGA, K., WALESIAK, M. 2000, "Stamdardisatio of Data Set Uder Dieret Measuremet Scales", i Classicatio ad Iformatio Processig at the Tur of the Milleium. Studies i Classicatio, 10
11 Data Aalysis, ad Kowledge Orgaizatio, eds. Decker R., Gaul W., Spriger-Verlag, Berli, Heidelberg, [5] MILLIGAN, G.W., COOPER, M.C. 1988, "A Study of Stadardizatio of Variables i Cluster Aalysis", Joural of Classicatio 5, [6] MŠODAK, A. 2006, "Multirateral Normalisatios of Diagostic Features", Statistics I Trasitio 75, [7] MÜLLER-FR CZEK, I. 2017, "Propozycja miary sytetyczej" [Propositio of Sythetic Measure], Przegl d Statystyczy, 644, [8] STEINLEY, D. 2004, "Stadardizig Variables i K -meas Clusterig" i Classicatio, Clusterig, ad Data Miig Applicatios. Studies i Classicatio, Data Aalysis, ad Kowledge Orgaisatio, eds. Baks D., McMorris F.R., Arabie P., Gaul W., Spriger, Berli, Heidelberg. [9] ZELIA A. 2002, "Some Notes of the Selectio of Normalisatio of Diagostic Variables", Statistics I Trasitio 55,
Chapter 12 - Quality Cotrol Example: The process of llig 12 ouce cas of Dr. Pepper is beig moitored. The compay does ot wat to uderll the cas. Hece, a target llig rate of 12.1-12.5 ouces was established.
More informationµ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion
Poit Estimatio Poit estimatio is the rather simplistic (ad obvious) process of usig the kow value of a sample statistic as a approximatio to the ukow value of a populatio parameter. So we could for example
More information1 Lesson 6: Measure of Variation
1 Lesso 6: Measure of Variatio 1.1 The rage As we have see, there are several viable coteders for the best measure of the cetral tedecy of data. The mea, the mode ad the media each have certai advatages
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationContinuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised
Questio 1. (Topics 1-3) A populatio cosists of all the members of a group about which you wat to draw a coclusio (Greek letters (μ, σ, Ν) are used) A sample is the portio of the populatio selected for
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
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 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 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More information577. Estimation of surface roughness using high frequency vibrations
577. Estimatio of surface roughess usig high frequecy vibratios V. Augutis, M. Sauoris, Kauas Uiversity of Techology Electroics ad Measuremets Systems Departmet Studetu str. 5-443, LT-5368 Kauas, Lithuaia
More informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
More informationChapter 2 Descriptive Statistics
Chapter 2 Descriptive Statistics Statistics Most commoly, statistics refers to umerical data. Statistics may also refer to the process of collectig, orgaizig, presetig, aalyzig ad iterpretig umerical data
More informationEconomics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls
Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls 1 20 10 20 13 23 3 7 18 7 4 5 15 7 29 10 18 10 10 23 4 12 8 6 (1)
More informationResponse Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable
Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated
More information9. Simple linear regression G2.1) Show that the vector of residuals e = Y Ŷ has the covariance matrix (I X(X T X) 1 X T )σ 2.
LINKÖPINGS UNIVERSITET Matematiska Istitutioe Matematisk Statistik HT1-2015 TAMS24 9. Simple liear regressio G2.1) Show that the vector of residuals e = Y Ŷ has the covariace matrix (I X(X T X) 1 X T )σ
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 information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationChapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008
Chapter 6 Part 5 Cofidece Itervals t distributio chi square distributio October 23, 2008 The will be o help sessio o Moday, October 27. Goal: To clearly uderstad the lik betwee probability ad cofidece
More informationProperties 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. Cross-sectioal data. 2. Time series data.
More informationLecture 24 Floods and flood frequency
Lecture 4 Floods ad flood frequecy Oe of the thigs we wat to kow most about rivers is what s the probability that a flood of size will happe this year? I 100 years? There are two ways to do this empirically,
More informationt distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference
EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The
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 informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationChapter 23: Inferences About Means
Chapter 23: Ifereces About Meas Eough Proportios! We ve spet the last two uits workig with proportios (or qualitative variables, at least) ow it s time to tur our attetios to quatitative variables. For
More informationEconomics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator
Ecoomics 24B Relatio to Method of Momets ad Maximum Likelihood OLSE as a Maximum Likelihood Estimator Uder Assumptio 5 we have speci ed the distributio of the error, so we ca estimate the model parameters
More informationAnna Janicka Mathematical Statistics 2018/2019 Lecture 1, Parts 1 & 2
Aa Jaicka Mathematical Statistics 18/19 Lecture 1, Parts 1 & 1. Descriptive Statistics By the term descriptive statistics we will mea the tools used for quatitative descriptio of the properties of a sample
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 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 be doe
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More informationNCSS Statistical Software. Tolerance Intervals
Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided
More informationGeometry of LS. LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT
OCTOBER 7, 2016 LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT Geometry of LS We ca thik of y ad the colums of X as members of the -dimesioal Euclidea space R Oe ca
More informationCorrelation Regression
Correlatio Regressio While correlatio methods measure the stregth of a liear relatioship betwee two variables, we might wish to go a little further: How much does oe variable chage for a give chage i aother
More informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationA Decomposition of the Herfindahl Index of Concentration
MPRA Muich Persoal RePEc Archive A Decompositio of the Herfidahl Idex of Cocetratio Giacomo de Gioia 26 November 207 Olie at https://mpra.ub.ui-mueche.de/82944/ MPRA Paper No. 82944, posted 27 November
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More information6 Sample Size Calculations
6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig
More informationChapter 12 EM algorithms The Expectation-Maximization (EM) algorithm is a maximum likelihood method for models that have hidden variables eg. Gaussian
Chapter 2 EM algorithms The Expectatio-Maximizatio (EM) algorithm is a maximum likelihood method for models that have hidde variables eg. Gaussia Mixture Models (GMMs), Liear Dyamic Systems (LDSs) ad Hidde
More informationChi-Squared Tests Math 6070, Spring 2006
Chi-Squared Tests Math 6070, Sprig 2006 Davar Khoshevisa Uiversity of Utah February XXX, 2006 Cotets MLE for Goodess-of Fit 2 2 The Multiomial Distributio 3 3 Applicatio to Goodess-of-Fit 6 3 Testig for
More informationA New Conception of Measurement Uncertainty Calculation
Vol. 14 (013) ACTA PHYSICA POLONICA A No. 3 Optical ad Acoustical Methods i Sciece ad Techology 013 A New Coceptio of Measuremet Ucertaity Calculatio J. Jakubiec Silesia Uiversity of Techology, Faculty
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationLecture 7: Density Estimation: k-nearest Neighbor and Basis Approach
STAT 425: Itroductio to Noparametric Statistics Witer 28 Lecture 7: Desity Estimatio: k-nearest Neighbor ad Basis Approach Istructor: Ye-Chi Che Referece: Sectio 8.4 of All of Noparametric Statistics.
More informationCURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:
CURRICULUM INSPIRATIONS: wwwmaaorg/ci MATH FOR AMERICA_DC: wwwmathforamericaorg/dc INNOVATIVE CURRICULUM ONLINE EXPERIENCES: wwwgdaymathcom TANTON TIDBITS: wwwjamestatocom TANTON S TAKE ON MEAN ad VARIATION
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter 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 informationSample Size Determination (Two or More Samples)
Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie
More informationKolmogorov-Smirnov type Tests for Local Gaussianity in High-Frequency Data
Proceedigs 59th ISI World Statistics Cogress, 5-30 August 013, Hog Kog (Sessio STS046) p.09 Kolmogorov-Smirov type Tests for Local Gaussiaity i High-Frequecy Data George Tauche, Duke Uiversity Viktor Todorov,
More informationSampling Distributions, Z-Tests, Power
Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationSimple Linear Regression
Chapter 2 Simple Liear Regressio 2.1 Simple liear model The simple liear regressio model shows how oe kow depedet variable is determied by a sigle explaatory variable (regressor). Is is writte as: Y i
More informationEstimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya
More informationEconomics Spring 2015
1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationChi-squared tests Math 6070, Spring 2014
Chi-squared tests Math 6070, Sprig 204 Davar Khoshevisa Uiversity of Utah March, 204 Cotets MLE for goodess-of fit 2 2 The Multivariate ormal distributio 3 3 Cetral limit theorems 5 4 Applicatio to goodess-of-fit
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationStudy on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm
Joural of ad Eergy Egieerig, 05, 3, 43-437 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 informationInvestigating the Significance of a Correlation Coefficient using Jackknife Estimates
Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN 2307-4531 (Prit & Olie) http://gssrr.org/idex.php?joural=jouralofbasicadapplied ---------------------------------------------------------------------------------------------------------------------------
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationSoo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7:
0 Multivariate Cotrol Chart 3 Multivariate Normal Distributio 5 Estimatio of the Mea ad Covariace Matrix 6 Hotellig s Cotrol Chart 6 Hotellig s Square 8 Average Value of k Subgroups 0 Example 3 3 Value
More informationIf, for instance, we were required to test whether the population mean μ could be equal to a certain value μ
STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially
More informationOBJECTIVES. Chapter 1 INTRODUCTION TO INSTRUMENTATION FUNCTION AND ADVANTAGES INTRODUCTION. At the end of this chapter, students should be able to:
OBJECTIVES Chapter 1 INTRODUCTION TO INSTRUMENTATION At the ed of this chapter, studets should be able to: 1. Explai the static ad dyamic characteristics of a istrumet. 2. Calculate ad aalyze the measuremet
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
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 an Application of Bayesian Estimation
O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, Higashi-Osaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma
More informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9
Hypothesis testig PSYCHOLOGICAL RESEARCH (PYC 34-C Lecture 9 Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationChapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers
Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio:
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More information11 Hidden Markov Models
Hidde Markov Models Hidde Markov Models are a popular machie learig approach i bioiformatics. Machie learig algorithms are preseted with traiig data, which are used to derive importat isights about the
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationTopic 18: Composite Hypotheses
Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationA Cobb - Douglas Function Based Index. for Human Development in Egypt
It. J. Cotemp. Math. Scieces, Vol. 7, 202, o. 2, 59-598 A Cobb - Douglas Fuctio Based Idex for Huma Developmet i Egypt E. Khater Istitute of Statistical Studies ad Research Dept. of Biostatistics ad Demography
More informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:
PSet ----- Stats, Cocepts I Statistics 7.3. Cofidece Iterval for a Mea i Oe Sample [MATH] The Cetral Limit Theorem. Let...,,, be idepedet, idetically distributed (i.i.d.) radom variables havig mea µ ad
More informationMA238 Assignment 4 Solutions (part a)
(i) Sigle sample tests. Questio. MA38 Assigmet 4 Solutios (part a) (a) (b) (c) H 0 : = 50 sq. ft H A : < 50 sq. ft H 0 : = 3 mpg H A : > 3 mpg H 0 : = 5 mm H A : 5mm Questio. (i) What are the ull ad alterative
More informationBivariate Sample Statistics Geog 210C Introduction to Spatial Data Analysis. Chris Funk. Lecture 7
Bivariate Sample Statistics Geog 210C Itroductio to Spatial Data Aalysis Chris Fuk Lecture 7 Overview Real statistical applicatio: Remote moitorig of east Africa log rais Lead up to Lab 5-6 Review of bivariate/multivariate
More informationA RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (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 XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationCov(aX, cy ) Var(X) Var(Y ) It is completely invariant to affine transformations: for any a, b, c, d R, ρ(ax + b, cy + d) = a.s. X i. as n.
CS 189 Itroductio to Machie Learig Sprig 218 Note 11 1 Caoical Correlatio Aalysis The Pearso Correlatio Coefficiet ρ(x, Y ) is a way to measure how liearly related (i other words, how well a liear model
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 informationSince X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain
Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationECON 3150/4150, Spring term Lecture 3
Itroductio Fidig the best fit by regressio Residuals ad R-sq Regressio ad causality Summary ad ext step ECON 3150/4150, Sprig term 2014. Lecture 3 Ragar Nymoe Uiversity of Oslo 21 Jauary 2014 1 / 30 Itroductio
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More information4 Multidimensional quantitative data
Chapter 4 Multidimesioal quatitative data 4 Multidimesioal statistics Basic statistics are ow part of the curriculum of most ecologists However, statistical techiques based o such simple distributios as
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationEquivalence Between An Approximate Version Of Brouwer s Fixed Point Theorem And Sperner s Lemma: A Constructive Analysis
Applied Mathematics E-Notes, 11(2011), 238 243 c ISSN 1607-2510 Available free at mirror sites of http://www.math.thu.edu.tw/ame/ Equivalece Betwee A Approximate Versio Of Brouwer s Fixed Poit Theorem
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 informationModeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy
Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral
More informationOutput Analysis and Run-Length Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More information4.1 SIGMA NOTATION AND RIEMANN SUMS
.1 Sigma Notatio ad Riema Sums Cotemporary Calculus 1.1 SIGMA NOTATION AND RIEMANN SUMS Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each
More informationRAINFALL PREDICTION BY WAVELET DECOMPOSITION
RAIFALL PREDICTIO BY WAVELET DECOMPOSITIO A. W. JAYAWARDEA Departmet of Civil Egieerig, The Uiversit of Hog Kog, Hog Kog, Chia P. C. XU Academ of Mathematics ad Sstem Scieces, Chiese Academ of Scieces,
More informationGroupe de Recherche en Économie et Développement International. Cahier de Recherche / Working Paper 10-18
Groupe de Recherche e Écoomie et Développemet Iteratioal Cahier de Recherche / Workig Paper 0-8 Quadratic Pe's Parade ad the Computatio of the Gii idex Stéphae Mussard, Jules Sadefo Kamdem Fraçoise Seyte
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationThe Poisson Process *
OpeStax-CNX module: m11255 1 The Poisso Process * Do Johso This work is produced by OpeStax-CNX ad licesed uder the Creative Commos Attributio Licese 1.0 Some sigals have o waveform. Cosider the measuremet
More informationOverview of Estimation
Topic Iferece is the problem of turig data ito kowledge, where kowledge ofte is expressed i terms of etities that are ot preset i the data per se but are preset i models that oe uses to iterpret the data.
More informationGini Index and Polynomial Pen s Parade
Gii Idex ad Polyomial Pe s Parade Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Gii Idex ad Polyomial Pe s Parade. 2011. HAL Id: hal-00582625 https://hal.archives-ouvertes.fr/hal-00582625
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationCoefficient of variation and Power Pen s parade computation
Coefficiet of variatio ad Power Pe s parade computatio Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Coefficiet of variatio ad Power Pe s parade computatio. 20. HAL Id: hal-0058658
More information