Analyzing Frequencies
|
|
- Madeline Hardy
- 5 years ago
- Views:
Transcription
1 Frquncy (# ndvduals) Frquncy (# ndvduals) /3/16 H o : No dffrnc n obsrvd sz frquncs and that prdctd by growth modl How would you analyz ths data? 15 Obsrvd Numbr 15 Expctd Numbr from growth modl Abalon Sz (mm) Abalon Sz (mm) Analyzng Frquncs Goodnss of Ft Tsts Basd on Ch Squar Kolmogorov-Smrnov/ Shapro Wlks Tsts 1
2 /3/16 Ch Squar Statstc A fundamntal way to analyz frquncy data Usd n both Goodnss of Ft and Basd on th dvaton btwn obsrvd (o) and xpctd frquncs () c Th tst statstc = n = 1 ( o - ) Ch Squar Statstc c = n = 1 o( - ) Approxmats a Ch Squar dstrbuton f th followng assumptons ar tru Obsrvatons ar classfd nto catgors ndpndntly. No mor than about % of clls hav obsrvatons (o) lss than 5. If ths s volatd thn probablts drvd from th Ch Squar Statstc can b msladng. Mal Fmal Undrgraduat dgr Cll Graduat Dgr 11 33
3 /3/16 Analyzng Frquncs Goodnss of Ft Tsts Basd on Ch Squar Is a sampl of data consstnt wth a gvn probablty dstrbuton??? Exampl: A survy s mad of lngths of abalon. Th goal s to dtrmn f th frquncy dstrbuton of szs s that prdctd by a growth modl. Bcaus masurmnts ar takn undrwatr, oftn n cracks thy ar catgorzd nto 1 mm ntrvals Goodnss of Ft Exampl - Abalon Sz Sz Catgory Numbr obsrvd Sum 557 Count 1. Indvdual ar arrangd nto catgors SIZE Proporton pr Bar 3
4 /3/16 Goodnss of Ft Exampl - Abalon Sz Sz Catgory Numbr obsrvd Expctd proportons 1. Indvdual ar arrangd nto catgors. Entr xpctd proportons Sum Goodnss of Ft Exampl - Abalon Sz Sz Catgory Numbr obsrvd Expctd proportons Expctd frquncs 1. Indvdual ar arrangd nto catgors. Entr xpctd proportons 3. Calculat xpctd frquncs 557 x.3 = Sum
5 /3/16 Goodnss of Ft Exampl - Abalon Sz Sz Catgory Numbr obsrvd Expctd proportons Expctd frquncs Indvdual ar arrangd nto catgors. Entr xpctd proportons 3. Calculat xpctd frquncs (xpctd proporton x sum of obsrvatons) Sum Goodnss of Ft Exampl - Abalon Compar th dstrbutons Thy appar dffrnt Dtrmn th probablty that thy ar ndd dffrnt 5
6 /3/16 Sz Catgory Numbr obsrvd Goodnss of Ft Exampl - Abalon Expctd proportons Expctd frquncs Ch Squar valu 1. Indvdual ar arrangd nto catgors. Entr xpctd proportons 3. Calculat xpctd frquncs (xpctd proporton x sum of obsrvatons) 4. Calculat ndvdual Ch squar valus - ( o ) ( 16 - ) =.3 Goodnss of Ft Exampl - Abalon Sz Catgory Numbr obsrvd Expctd proportons Expctd frquncs Ch Squar valu Indvdual ar arrangd nto catgors. Entr xpctd proportons 3. Calculat xpctd frquncs (xpctd proporton x sum of obsrvatons) 4. Calculat ndvdual Ch squar valus 5. Sum Valus c = n = 1 ( o - ) 6
7 /3/16 Dnsty c = n = 1 Goodnss of Ft Exampl Abalon ( o - ) Ch Squar dstrbuton wth 9 df = Ch Squar Us probablty calculator for dmonstraton df = (1-1) = 9 c Valu, crtcal (p=.5) = 16.9 P(36.95, 9) <.1 1. Indvdual ar arrangd nto catgors. Entr xpctd proportons 3. Calculat xpctd frquncs (xpctd proporton x sum of obsrvatons) 4. Calculat ndvdual Ch squar valus 5. Sum Valus 6. Dtrmn probablty that obsrvd dstrbuton quals th xpctd dstrbuton Us Ch Squar tabls or cumulatv dstrbuton functon wth df = k 1 (usually) If dsrd dtrmn crtcal valu for rjcton Goodnss of Ft Exampl - Abalon Concluson Null hypothss can b rjctd Th obsrvd dstrbuton dffrs from th xpctd dstrbuton at p<.1 7
8 /3/16 Analyzng Frquncs Goodnss of Ft Tsts Basd on Ch Squar Shapro Wlks / Kolmogorov-Smrnov Tsts Kolmogorov-Smrnov Tsts (KS) Th on-sampl Kolmogorov-Smrnov tst s usd to compar th shap and locaton of a sampl dstrbuton to a spcfd dstrbuton. Th Kolmogorov-Smrnov tst and ts gnralzatons ar among th handst of dstrbuton-fr tsts. Th tst statstc s basd on th maxmum dffrnc btwn two cumulatv dstrbuton functons (CDF). In th on-sampl tst, on of th CDF s s contnuous (typcally th xpctd dstrbuton) and th othr s dscrt (typcally th obsrvd). 8
9 /3/16 Goodnss of Ft Exampl (KS) - Abalon 1. Dtrmn th dstrbuton wth whch you want to compar to th obsrvd dstrbuton (rcall w ar comparng cumulatv dstrbutons). Assum w wsh to compar to a normal dstrbuton.5 Dnsty functon 1. Cumulatv functon Proporton Cummulatv Proporton Sz Sz Goodnss of Ft Exampl - Abalon 1. Dtrmn th dstrbuton wth whch you want to compar to th obsrvd dstrbuton (rcall w ar comparng cumulatv dstrbutons). Assum w wsh to compar to a normal dstrbuton Count Obsrvd data Man = 61 Standard Dvaton = Cumulatv Dnsty Expctd functon (undr assumpton of normalty) Cummulatv Proporton Man = 61 Standard Dvaton = Sz Sz 9
10 /3/16 Goodnss of Ft Exampl (Shapro- Wlks tst) - Abalon Count Man = 61 Standard Dvaton = Cumulatv Dnsty Sz Analyzng Frquncs Most common form of valuatng catgorcal data n th bologcal scncs Usd for counts of obsrvatons mad n two or mor layrs of catgors (varabls) say - sz of abalon nsd and outsd a rsrv 15 Do ths dstrbutons dffr? Abalon Numbr 1 5 Locaton Sz (mm) Outsd Rsrv Insd Rsrv 1
11 /3/16 1. Gnrally contngncy tabls ar analyzd so that nthr varabl s consdrd as a prdctor or rspons varabl. Thr ar occasons whr varabls can b dstngushd nto prdctor and rspons varabls 3. In practc th analyss of contngncy tabls s not affctd by whthr varabls can b charactrzd as rspons or prdctor th hypothss rman th sam Hypothss tstd s on of Indpndnc H : Th two varabls ar ndpndnt. Null Hypothss H 1 : Th two varabls ar assocatd. Altrnatv Hypothss 11
12 /3/16 W can us a Ch-squard tst (or Gnralzd Lnar Modl) to tst for an assocaton btwn ncdnc of tubrculoss and th ABO blood groups. H : Incdnc of Tubrculoss and ABO blood groups ar ndpndnt. H 1 : Incdnc of Tubrculoss and ABO blood groups ar assocatd. Blood Group Tubrculoss Incdnc Modrat/ Advancd O A AB B Mnmal Not prsnt What s mant by ndpndnc mathmatcally? 1. If varabls ar ndpndnt thn P(A+B) = P(A)*P(B) For xampl what s th probablty of gttng a 4 of damonds n a sngl draw from a dck of cards = P(4)*P(Damond) = (1/13)*(1/4) = 1/5 Thrfor th xpctd lklhood of any cll (undr th hypothss of ndpndnc) s gvn by Probablty of Blood Group * Probablty of Tubrculoss Incdnc [P(BG)*P(TI)] And th xpctd frquncy s N [P(BG)*P(TI)] Blood Group Tubrculoss Incdnc Modrat/ Advancd O A AB B Mnmal Not prsnt
13 /3/16 Calculaton of N [P(BG)*P(TI)] 1. Calculat margnal totals and N Tubrculoss Incdnc Modrat/ Advancd Blood Group O A AB B Total Mnmal Not prsnt Total N Calculaton of N [P(BG)*P(TI)] Calculat margnal totals and N Any cll s xpctd lklhood (f factors ar ndpndnt) s gvn by: [P(BG)*P(TI)] = (Column Margnal Total / N) * (Row Margnal Total/N) For xampl: (89/49)*(8/49) =.4 Blood Group Tubrculoss Incdnc Modrat/ Advancd O A AB B Total Mnmal Not prsnt Total N 13
14 /3/16 Calculaton of N [P(BG)*P(TI)] Calculat margnal totals and N Any cll s xpctd lklhood s gvn by: [P(BG)*P(TI)] = (Column Margnal Total / N) * (Row Margnal Total/N) Multply ach clls xpctd lklhood by N: For xampl.4*49 = 1. Blood Group Tubrculoss Incdnc Modrat/ Advancd O A AB B Total 7 (.4) Mnmal Not prsnt Total N Equvalntly N [P(BG)*P(TI)] = N(Column Margnal Total / N) * (Row Margnal Total/N) = ((Column Margnal Total) * (Row Margnal Total))/N Calculaton of N [P(BG)*P(TI)] Calculat xpctd frquncs for all clls not ths s don automatcally n th Stats program O A AB B Modrat/Advancd 49[(89/49)*(8/49)] 49[(87/49)*(8/49)] 49[(89/49)*(8/49)] 49[(55/49)*(8/49)] Mnmal 49[(89/49)*(85/49)] 49[(87/49)*(85/49)] 49[(89/49)*(8/49)] 49[(55/49)*(85/49)] Not prsnt 49[(89/49)*(136/49)] 49[(87/49)*(136/49)] 49[(89/49)*(8/49)] 49[(55/49)*(136/49)] Expctd Frquncs Tubrculoss Incdnc Modrat/ Advancd O A AB B Mnmal Not prsnt
15 /3/16 1. Rcall that. Hnc: c = n = 1 o( - ) c = (7-1) +..(4 3.1) = Wth dgrs of frdom (df) = (rows 1)(columns-1) = (3-1)(4-1) = 6 Modrat/ Advancd Modrat/ Advancd Obsrvd Frquncs O A AB B Mnmal Not prsnt Expctd Frquncs O A AB B Mnmal Not prsnt H : Incdnc of Tubrculoss and ABO blood groups ar ndpndnt. REJECTED H 1 : Incdnc of Tubrculoss and ABO blood groups ar assocatd. SUPPORTED Dnsty Ch squar dstrbuton (6 df) c calculatd P=.18 = Ch Squar valu Blood Group A AB B O 6 5 MA Mn NP MA Mn NP Tubrculoss Incdnc Tubrculoss Incdnc COUNT COUNT 15
16 /3/16 H : Incdnc of Tubrculoss and ABO blood groups ar ndpndnt. REJECTED H 1 : Incdnc of Tubrculoss and ABO blood groups ar assocatd. SUPPORTED Obsrvd Frquncs Expctd Frquncs (undr assumpton that Tubrculoss Incdnc and Blood Groups ar ndpndnt) Blood Groups A AB B O MA Mn NP MA Mn NP MA Mn NP MA Mn NP Tubrculoss Incdnc Extra slds 16
17 /3/16 H 1 : Incdnc of Tubrculoss and ABO blood groups ar assocatd. SUPPORTED Now w know th varabls ar not ndpndnt, but w do not know what catgory (s) drv th assocaton 1) Us squntal dlton of catgors If thr ar prdctor and rspons varabls thn work on th prdctor varabl Hr, t s unlkly that Tubrculoss causs Blood Groups but Blood Group mght affct Incdnc of Tubrculoss Slct th Catgory that sms most dffrnt from th rst Hr, th dstrbutons of Tubrculoss Incdnc for Blood Groups B and AB sm most dffrnt Dlt that catgory and rpat analyss Rmmbr that df wll chang Blood Group A AB B O 6 5 MA Mn NP MA Mn NP Tubrculoss Incdnc Tubrculoss Incdnc COUNT COUNT H 1 : Incdnc of Tubrculoss and ABO blood groups ar assocatd. SUPPORTED H a : Incdnc of Tubrculoss and ABO blood groups ar ndpndnt (aftr rmoval of group AB) H 1a : Incdnc of Tubrculoss and ABO blood groups ar assocatd (aftr rmoval of group AB) 1) Rmov Blood Group AB from analyss Calculat Ch Squar Valu and assocatd probablty for (3-1)(3-1) = 4 df ` c calculatd = P =.8 Tubrculoss Incdnc and Blood Groups ar not ndpndnt vn aftr rmoval of group AB Try Group B Blood Group A AB B O 6 5 MA Mn NP MA Mn NP Tubrculoss Incdnc Tubrculoss Incdnc COUNT COUNT 17
18 /3/16 H 1 : Incdnc of Tubrculoss and ABO blood groups ar assocatd. SUPPORTED H b : Incdnc of Tubrculoss and ABO blood groups ar ndpndnt (aftr rmoval of group B) H 1b : Incdnc of Tubrculoss and ABO blood groups ar assocatd (aftr rmoval of group B) 1) Rmov Blood Group B from analyss Calculat Ch Squar Valu and assocatd probablty for (3-1)(3-1) = 4 df c calculatd P =.31 = 4.76 Tubrculoss Incdnc and Blood Groups ar ndpndnt aftr rmoval of group B Hnc th assocaton of Tubrculoss Incdnc and Blood Groups s drvn by Blood Group B Blood Group A AB B O 6 5 MA Mn NP MA Mn NP Tubrculoss Incdnc Tubrculoss Incdnc COUNT COUNT 1) Subsqunt analyss could rsolv how Blood Group B s assocatd wth Tubrculoss Incdnc (g mor Modrat and advancd cass than xpctd) How???? A Blood Group AB COUNT B O MA Mn NP MA Mn NP Tubrculoss Incdnc Tubrculoss Incdnc COUNT 18
19 /3/16 Usng Goodnss of Ft Tsts to dtrmn contrbuton of catgors 1) How s Blood Group B assocatd wth Tubrculoss Incdnc 1) Dtrmn xpctd lklhood's of ncdnc basd on ndpndnt varabls (Blood Group x Tubrculoss Incdnc, lavng out group B) LIKELIHOOD MA Mn NP Tubrculoss Incdnc Usng Goodnss of Ft Tsts to dtrmn contrbuton of catgors 1) How s Blood Group B assocatd wth Tubrculoss Incdnc Dtrmn xpctd lklhood's of ncdnc basd on ndpndnt varabls (Blood Group x Tubrculoss Incdnc, lavng out group B) Dtrmn confdnc ntrvals for obsrvd data (for group B) Tubrculoss Incdnc Numbr obsrvd 95% Confdnc ntrval (lowr) 95% Confdnc ntrval (uppr) Modrat / Advancd 13 (.19) 5.95 Mnmal 18 (.179) 9.84 Not Prsnt 4 (.7) (.393) 1.61 (.49) 6.95 (.599)
20 Proporton /3/16 Usng Goodnss of Ft Tsts to dtrmn contrbuton of catgors 1) How s Blood Group B assocatd wth Tubrculoss Incdnc Dtrmn xpctd lklhood's of ncdnc basd on ndpndnt varabls (Blood Group x Tubrculoss Incdnc, lavng out group B) Dtrmn confdnc ntrvals for obsrvd data (for group B) Compar to xpctd lklhood's Tubrculoss Incdnc (Proporton) Numbr obsrvd 95% Confdnc ntrval (lowr) 95% Confdnc ntrval (uppr) Expctd (lklhood's) frquncs Modrat / Advancd (.36) 13 (.19) 5.95 (.393) 1.61 (.77) 4.3 Obsrvd > Expctd Mnmal (.37) 18 Not Prsnt (.436) 4 (.179) 9.84 (.7) (.49) 6.95 (.599) 3.94 (.345) (.577) Obsrvd ~ Expctd Usng Goodnss of Ft Tsts to dtrmn contrbuton of catgors Rsult: Blood Group B s assocatd wth gratr ncdnc of modrat / advancd Tubrculoss MA Mn NP Tubrculoss Incdnc Lklhood Blood Group B
Chapter 6 Student Lecture Notes 6-1
Chaptr 6 Studnt Lctur Nots 6-1 Chaptr Goals QM353: Busnss Statstcs Chaptr 6 Goodnss-of-Ft Tsts and Contngncy Analyss Aftr compltng ths chaptr, you should b abl to: Us th ch-squar goodnss-of-ft tst to dtrmn
More informationST 524 NCSU - Fall 2008 One way Analysis of variance Variances not homogeneous
ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous On way Analyss of varanc Exampl (Yandll, 997) A plant scntst masurd th concntraton of a partcular vrus n plant sap usng ELISA (nzym-lnkd
More informationLogistic Regression I. HRP 261 2/10/ am
Logstc Rgrsson I HRP 26 2/0/03 0- am Outln Introducton/rvw Th smplst logstc rgrsson from a 2x2 tabl llustrats how th math works Stp-by-stp xampls to b contnud nxt tm Dummy varabls Confoundng and ntracton
More informationOutlier-tolerant parameter estimation
Outlr-tolrant paramtr stmaton Baysan thods n physcs statstcs machn larnng and sgnal procssng (SS 003 Frdrch Fraundorfr fraunfr@cg.tu-graz.ac.at Computr Graphcs and Vson Graz Unvrsty of Tchnology Outln
More informationNaresuan University Journal: Science and Technology 2018; (26)1
Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Th Dvlopmnt o a Corrcton Mthod or Ensurng a Contnuty Valu o Th Ch-squar Tst wth a Small Expctd Cll Frquncy Kajta Matchma 1 *, Jumlong Vongprasrt and
More information10/7/14. Mixture Models. Comp 135 Introduction to Machine Learning and Data Mining. Maximum likelihood estimation. Mixture of Normals in 1D
Comp 35 Introducton to Machn Larnng and Data Mnng Fall 204 rofssor: Ron Khardon Mxtur Modls Motvatd by soft k-mans w dvlopd a gnratv modl for clustrng. Assum thr ar k clustrs Clustrs ar not rqurd to hav
More informationte Finance (4th Edition), July 2017.
Appndx Chaptr. Tchncal Background Gnral Mathmatcal and Statstcal Background Fndng a bas: 3 2 = 9 3 = 9 1 /2 x a = b x = b 1/a A powr of 1 / 2 s also quvalnt to th squar root opraton. Fndng an xponnt: 3
More informationGrand Canonical Ensemble
Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls
More informationReview - Probabilistic Classification
Mmoral Unvrsty of wfoundland Pattrn Rcognton Lctur 8 May 5, 6 http://www.ngr.mun.ca/~charlsr Offc Hours: Tusdays Thursdays 8:3-9:3 PM E- (untl furthr notc) Gvn lablld sampls { ɛc,,,..., } {. Estmat Rvw
More informationSoft k-means Clustering. Comp 135 Machine Learning Computer Science Tufts University. Mixture Models. Mixture of Normals in 1D
Comp 35 Machn Larnng Computr Scnc Tufts Unvrsty Fall 207 Ron Khardon Th EM Algorthm Mxtur Modls Sm-Suprvsd Larnng Soft k-mans Clustrng ck k clustr cntrs : Assocat xampls wth cntrs p,j ~~ smlarty b/w cntr
More informationLucas Test is based on Euler s theorem which states that if n is any integer and a is coprime to n, then a φ(n) 1modn.
Modul 10 Addtonal Topcs 10.1 Lctur 1 Prambl: Dtrmnng whthr a gvn ntgr s prm or compost s known as prmalty tstng. Thr ar prmalty tsts whch mrly tll us whthr a gvn ntgr s prm or not, wthout gvng us th factors
More informationUNIT 8 TWO-WAY ANOVA WITH m OBSERVATIONS PER CELL
UNIT 8 TWO-WAY ANOVA WITH OBSERVATIONS PER CELL Two-Way Anova wth Obsrvatons Pr Cll Structur 81 Introducton Obctvs 8 ANOVA Modl for Two-way Classfd Data wth Obsrvatons r Cll 83 Basc Assutons 84 Estaton
More informationToday s logistic regression topics. Lecture 15: Effect modification, and confounding in logistic regression. Variables. Example
Today s stc rgrsson tocs Lctur 15: Effct modfcaton, and confoundng n stc rgrsson Sandy Eckl sckl@jhsh.du 16 May 28 Includng catgorcal rdctor crat dummy/ndcator varabls just lk for lnar rgrsson Comarng
More informationEEC 686/785 Modeling & Performance Evaluation of Computer Systems. Lecture 12
EEC 686/785 Modlng & Prformanc Evaluaton of Computr Systms Lctur Dpartmnt of Elctrcal and Computr Engnrng Clvland Stat Unvrsty wnbng@.org (basd on Dr. Ra Jan s lctur nots) Outln Rvw of lctur k r Factoral
More informationCOMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP
ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng
More informationHeisenberg Model. Sayed Mohammad Mahdi Sadrnezhaad. Supervisor: Prof. Abdollah Langari
snbrg Modl Sad Mohammad Mahd Sadrnhaad Survsor: Prof. bdollah Langar bstract: n ths rsarch w tr to calculat analtcall gnvalus and gnvctors of fnt chan wth ½-sn artcls snbrg modl. W drov gnfuctons for closd
More informationEconomics 600: August, 2007 Dynamic Part: Problem Set 5. Problems on Differential Equations and Continuous Time Optimization
THE UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND Economcs 600: August, 007 Dynamc Part: Problm St 5 Problms on Dffrntal Equatons and Contnuous Tm Optmzaton Quston Solv th followng two dffrntal quatons.
More information4.1. Lecture 4: Fitting distributions: goodness of fit. Goodness of fit: the underlying principle
Lecture 4: Fttng dstrbutons: goodness of ft Goodness of ft Testng goodness of ft Testng normalty An mportant note on testng normalty! L4.1 Goodness of ft measures the extent to whch some emprcal dstrbuton
More informationApplied Statistics II - Categorical Data Analysis Data analysis using Genstat - Exercise 2 Logistic regression
Applid Statistics II - Catgorical Data Analysis Data analysis using Gnstat - Exrcis 2 Logistic rgrssion Analysis 2. Logistic rgrssion for a 2 x k tabl. Th tabl blow shows th numbr of aphids aliv and dad
More informationA Probabilistic Characterization of Simulation Model Uncertainties
A Proalstc Charactrzaton of Sulaton Modl Uncrtants Vctor Ontvros Mohaad Modarrs Cntr for Rsk and Rlalty Unvrsty of Maryland 1 Introducton Thr s uncrtanty n odl prdctons as wll as uncrtanty n xprnts Th
More informationLecture 14. Relic neutrinos Temperature at neutrino decoupling and today Effective degeneracy factor Neutrino mass limits Saha equation
Lctur Rlc nutrnos mpratur at nutrno dcoupln and today Effctv dnracy factor Nutrno mass lmts Saha quaton Physcal Cosmoloy Lnt 005 Rlc Nutrnos Nutrnos ar wakly ntractn partcls (lptons),,,,,,, typcal ractons
More informationCHAPTER 33: PARTICLE PHYSICS
Collg Physcs Studnt s Manual Chaptr 33 CHAPTER 33: PARTICLE PHYSICS 33. THE FOUR BASIC FORCES 4. (a) Fnd th rato of th strngths of th wak and lctromagntc forcs undr ordnary crcumstancs. (b) What dos that
More informationObserver Bias and Reliability By Xunchi Pu
Obsrvr Bias and Rliability By Xunchi Pu Introduction Clarly all masurmnts or obsrvations nd to b mad as accuratly as possibl and invstigators nd to pay carful attntion to chcking th rliability of thir
More informationReview Statistics review 14: Logistic regression Viv Bewick 1, Liz Cheek 1 and Jonathan Ball 2
Critical Car Fbruary 2005 Vol 9 No 1 Bwick t al. Rviw Statistics rviw 14: Logistic rgrssion Viv Bwick 1, Liz Chk 1 and Jonathan Ball 2 1 Snior Lcturr, School of Computing, Mathmatical and Information Scincs,
More information??? Dynamic Causal Modelling for M/EEG. Electroencephalography (EEG) Dynamic Causal Modelling. M/EEG analysis at sensor level. time.
Elctroncphalography EEG Dynamc Causal Modllng for M/EEG ampltud μv tm ms tral typ 1 tm channls channls tral typ 2 C. Phllps, Cntr d Rchrchs du Cyclotron, ULg, Blgum Basd on slds from: S. Kbl M/EEG analyss
More informationCHAPTER 7d. DIFFERENTIATION AND INTEGRATION
CHAPTER 7d. DIFFERENTIATION AND INTEGRATION A. J. Clark School o Engnrng Dpartmnt o Cvl and Envronmntal Engnrng by Dr. Ibrahm A. Assakka Sprng ENCE - Computaton Mthods n Cvl Engnrng II Dpartmnt o Cvl and
More information2. Grundlegende Verfahren zur Übertragung digitaler Signale (Zusammenfassung) Informationstechnik Universität Ulm
. Grundlgnd Vrfahrn zur Übrtragung dgtalr Sgnal (Zusammnfassung) wt Dc. 5 Transmsson of Dgtal Sourc Sgnals Sourc COD SC COD MOD MOD CC dg RF s rado transmsson mdum Snk DC SC DC CC DM dg DM RF g physcal
More informationA Note on Estimability in Linear Models
Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): 212-216 DOI: 10.5923/j.statstcs.20140404.06 A Not on Estmablty n Lnar Modls S. O. Adymo 1,*, F. N. Nwob 2 1 Dpartmnt of Mathmatcs and Statstcs,
More informationIntroduction to logistic regression
Itroducto to logstc rgrsso Gv: datast D { 2 2... } whr s a k-dmsoal vctor of ral-valud faturs or attrbuts ad s a bar class labl or targt. hus w ca sa that R k ad {0 }. For ampl f k 4 a datast of 3 data
More informationExternal Equivalent. EE 521 Analysis of Power Systems. Chen-Ching Liu, Boeing Distinguished Professor Washington State University
xtrnal quvalnt 5 Analyss of Powr Systms Chn-Chng Lu, ong Dstngushd Profssor Washngton Stat Unvrsty XTRNAL UALNT ach powr systm (ara) s part of an ntrconnctd systm. Montorng dvcs ar nstalld and data ar
More information8-node quadrilateral element. Numerical integration
Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 -nod quadrlatral lmnt. Numrcal ntgraton h tchnqu usd for th formulaton of th lnar trangl can b formall tndd to construct quadrlatral lmnts as wll
More informationThe Hyperelastic material is examined in this section.
4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6):
More informationLecture 3: Phasor notation, Transfer Functions. Context
EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationOptimal Ordering Policy in a Two-Level Supply Chain with Budget Constraint
Optmal Ordrng Polcy n a Two-Lvl Supply Chan wth Budgt Constrant Rasoul aj Alrza aj Babak aj ABSTRACT Ths papr consdrs a two- lvl supply chan whch consst of a vndor and svral rtalrs. Unsatsfd dmands n rtalrs
More informationLecture 6 More on Complete Randomized Block Design (RBD)
Lecture 6 More on Complete Randomzed Block Desgn (RBD) Multple test Multple test The multple comparsons or multple testng problem occurs when one consders a set of statstcal nferences smultaneously. For
More informationA NEW GENERALISATION OF SAM-SOLAI S MULTIVARIATE ADDITIVE GAMMA DISTRIBUTION*
A NEW GENERALISATION OF SAM-SOLAI S MULTIVARIATE ADDITIVE GAMMA DISTRIBUTION* Dr. G.S. Davd Sam Jayakumar, Assstant Profssor, Jamal Insttut of Managmnt, Jamal Mohamd Collg, Truchraall 620 020, South Inda,
More informationEcon107 Applied Econometrics Topic 10: Dummy Dependent Variable (Studenmund, Chapter 13)
Pag- Econ7 Appld Economtrcs Topc : Dummy Dpndnt Varabl (Studnmund, Chaptr 3) I. Th Lnar Probablty Modl Suppos w hav a cross scton of 8-24 yar-olds. W spcfy a smpl 2-varabl rgrsson modl. Th probablty of
More informationBLOCKS REPLICATION EXPERIMENTAL UNITS RANDOM VERSUS FIXED EFFECTS
DESIGN CONCEPTS: BLOCKS REPLICATION EXPERIMENTAL UNITS RANDOM VERSUS FIXED EFFECTS TREATMENT DESIGNS (PLANS) VS. EXPERIMENTAL DESIGNS Outln: Blockd dsgns Random Block Effcts REML analyss Incomplt Blocks
More informationAPPENDIX: STATISTICAL TOOLS
I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.
More informationGPC From PeakSimple Data Acquisition
GPC From PakSmpl Data Acquston Introducton Th follong s an outln of ho PakSmpl data acquston softar/hardar can b usd to acqur and analyz (n conjuncton th an approprat spradsht) gl prmaton chromatography
More informationHANDY REFERENCE SHEET HRP/STATS 261, Discrete Data
Bary prdctor Bary outcom HANDY REFERENCE SHEE HRP/SAS 6, Dscrt Data x Cotgcy abls Dsas (D No Dsas (~D Exposd (E a b Uxposd (~E c d Masurs of Assocato a /( a + b Rs Rato = c /( c + d RR * xp a /( a+ b c
More informationAnswer Homework 5 PHA5127 Fall 1999 Jeff Stark
Answr omwork 5 PA527 Fall 999 Jff Stark A patint is bing tratd with Drug X in a clinical stting. Upon admiion, an IV bolus dos of 000mg was givn which yildd an initial concntration of 5.56 µg/ml. A fw
More informationA primary objective of a phase II trial is to screen for antitumor activity; agents which are found to have substantial antitumor activity and an
SURVIVAL ANALYSIS A prmary objctv of a phas II tral s to scrn for anttumor actvty; agnts whch ar found to hav substantal anttumor actvty and an approprat spctrum of toxcty ar lkly ncorporatd nto combnatons
More informationFolding of Regular CW-Complexes
Ald Mathmatcal Scncs, Vol. 6,, no. 83, 437-446 Foldng of Rgular CW-Comlxs E. M. El-Kholy and S N. Daoud,3. Dartmnt of Mathmatcs, Faculty of Scnc Tanta Unvrsty,Tanta,Egyt. Dartmnt of Mathmatcs, Faculty
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More informationElectrochemistry L E O
Rmmbr from CHM151 A rdox raction in on in which lctrons ar transfrrd lctrochmistry L O Rduction os lctrons xidation G R ain lctrons duction W can dtrmin which lmnt is oxidizd or rducd by assigning oxidation
More informationDealing with quantitative data and problem solving life is a story problem! Attacking Quantitative Problems
Daling with quantitati data and problm soling lif is a story problm! A larg portion of scinc inols quantitati data that has both alu and units. Units can sa your butt! Nd handl on mtric prfixs Dimnsional
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationFakultät III Wirtschaftswissenschaften Univ.-Prof. Dr. Jan Franke-Viebach
Unvrstät Sgn Fakultät III Wrtschaftswssnschaftn Unv.-rof. Dr. Jan Frank-Vbach Exam Intrnatonal Fnancal Markts Summr Smstr 206 (2 nd Exam rod) Avalabl tm: 45 mnuts Soluton For your attnton:. las do not
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationPipe flow friction, small vs. big pipes
Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction
More informationWhat are those βs anyway? Understanding Design Matrix & Odds ratios
Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.
More informationSolution of Assignment #2
olution of Assignmnt #2 Instructor: Alirza imchi Qustion #: For simplicity, assum that th distribution function of T is continuous. Th distribution function of R is: F R ( r = P( R r = P( log ( T r = P(log
More informationph People Grade Level: basic Duration: minutes Setting: classroom or field site
ph Popl Adaptd from: Whr Ar th Frogs? in Projct WET: Curriculum & Activity Guid. Bozman: Th Watrcours and th Council for Environmntal Education, 1995. ph Grad Lvl: basic Duration: 10 15 minuts Stting:
More informationNonparametric Methods: Goodness-of-Fit Tests
Nonparamtric Mthods: Goodnss-o-Fit Tsts Chaptr 15 McGraw-Hill/Irwin Copyright 013 by Th McGraw-Hill Companis, Inc. All rights rsrvd. LEARNING OBJECTIVES LO 15-1 Conduct a tst o hypothsis comparing an obsrvd
More informationJones vector & matrices
Jons vctor & matrcs PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs Matr tratmnt of polarzaton Consdr a lght ra wth an nstantanous -vctor as shown k, t ˆ k, t ˆ k t, o o
More informationECE 2210 / 00 Phasor Examples
EE 0 / 00 Phasor Exampls. Add th sinusoidal voltags v ( t ) 4.5. cos( t 30. and v ( t ) 3.. cos( t 5. v ( t) using phasor notation, draw a phasor diagram of th thr phasors, thn convrt back to tim domain
More informationUnit 7 Introduction to Analysis of Variance
PubHlth 640 Sprng 04 7. Introducton to Analyss of Varanc Pag of 8 Unt 7 Introducton to Analyss of Varanc Always graph rsults of an analyss of varanc - Grald van Bll. Analyss of varanc s a spcal cas of
More informationThe Fourier Transform
/9/ Th ourr Transform Jan Baptst Josph ourr 768-83 Effcnt Data Rprsntaton Data can b rprsntd n many ways. Advantag usng an approprat rprsntaton. Eampls: osy ponts along a ln Color spac rd/grn/blu v.s.
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationPhysics of Very High Frequency (VHF) Capacitively Coupled Plasma Discharges
Physcs of Vry Hgh Frquncy (VHF) Capactvly Coupld Plasma Dschargs Shahd Rauf, Kallol Bra, Stv Shannon, and Kn Collns Appld Matrals, Inc., Sunnyval, CA AVS 54 th Intrnatonal Symposum Sattl, WA Octobr 15-19,
More informationLecture 23 APPLICATIONS OF FINITE ELEMENT METHOD TO SCALAR TRANSPORT PROBLEMS
COMPUTTION FUID DYNMICS: FVM: pplcatons to Scalar Transport Prolms ctur 3 PPICTIONS OF FINITE EEMENT METHOD TO SCR TRNSPORT PROBEMS 3. PPICTION OF FEM TO -D DIFFUSION PROBEM Consdr th stady stat dffuson
More informationStat 642, Lecture notes for 01/27/ d i = 1 t. n i t nj. n j
Stat 642, Lecture notes for 01/27/05 18 Rate Standardzaton Contnued: Note that f T n t where T s the cumulatve follow-up tme and n s the number of subjects at rsk at the mdpont or nterval, and d s the
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationSCITECH Volume 5, Issue 1 RESEARCH ORGANISATION November 17, 2015
Journal of Informaton Scncs and Computng Tchnologs(JISCT) ISSN: 394-966 SCITECH Volum 5, Issu RESEARCH ORGANISATION Novmbr 7, 5 Journal of Informaton Scncs and Computng Tchnologs www.sctcrsarch.com/journals
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationPhysics 256: Lecture 2. Physics
Physcs 56: Lctur Intro to Quantum Physcs Agnda for Today Complx Numbrs Intrfrnc of lght Intrfrnc Two slt ntrfrnc Dffracton Sngl slt dffracton Physcs 01: Lctur 1, Pg 1 Constructv Intrfrnc Ths wll occur
More information:2;$-$(01*%<*=,-./-*=0;"%/;"-*
!"#$%'()%"*#%*+,-./-*+01.2(.*3+456789*!"#$%"'()'*+,-."/0.%+1'23"45'46'7.89:89'/' ;8-,"$4351415,8:+#9' Dr. Ptr T. Gallaghr Astrphyscs Rsarch Grup Trnty Cllg Dubln :2;$-$(01*%
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More information15. Stress-Strain behavior of soils
15. Strss-Strain bhavior of soils Sand bhavior Usually shard undr draind conditions (rlativly high prmability mans xcss por prssurs ar not gnratd). Paramtrs govrning sand bhaviour is: Rlativ dnsity Effctiv
More informationExercises for lectures 7 Steady state, tracking and disturbance rejection
Exrc for lctur 7 Stady tat, tracng and dturbanc rjcton Martn Hromčí Automatc control 06-3-7 Frquncy rpon drvaton Automatcé řízní - Kybrnta a robota W lad a nuodal nput gnal to th nput of th ytm, gvn by
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationAdvanced Macroeconomics
Advancd Macroconomcs Chaptr 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY Thms of th chaptr Nomnal rgdts, xpctatonal rrors and mploymnt fluctuatons. Th short-run trad-off btwn nflaton and unmploymnt.
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationModeling and Simulation NETW 707
Modelng and Smulaton NETW 707 Lecture 5 Tests for Random Numbers Course Instructor: Dr.-Ing. Magge Mashaly magge.ezzat@guc.edu.eg C3.220 1 Propertes of Random Numbers Random Number Generators (RNGs) must
More informationChapter 13 Aggregate Supply
Chaptr 13 Aggrgat Supply 0 1 Larning Objctivs thr modls of aggrgat supply in which output dpnds positivly on th pric lvl in th short run th short-run tradoff btwn inflation and unmploymnt known as th Phillips
More informationFEFF and Related Codes
FEFF and Rlatd Cods Anatoly Frnl Profssor Physcs Dpartmnt, Yshva Unvrsty, w Yor, USA Synchrotron Catalyss Consortum, Broohavn atonal Laboratory, USA www.yu.du/faculty/afrnl Anatoly.Frnl@yu.du FEFF: John
More informationIntroduction to Condensed Matter Physics
Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan Ovrviw Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl h Hat Capacity Hat capacity h hat capacity of a systm hld at
More informationSeptember 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline
Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationThe Variance-Covariance Matrix
Th Varanc-Covaranc Marx Our bggs a so-ar has bn ng a lnar uncon o a s o daa by mnmzng h las squars drncs rom h o h daa wh mnsarch. Whn analyzng non-lnar daa you hav o us a program l Malab as many yps o
More informationAuthentication Transmission Overhead Between Entities in Mobile Networks
0 IJCSS Intrnatonal Journal of Computr Scnc and twork Scurty, VO.6 o.b, March 2006 Authntcaton Transmsson Ovrhad Btwn Entts n Mobl tworks Ja afr A-Sararh and Sufan Yousf Faculty of Scnc and Tchnology,
More informationElectrochemical Equilibrium Electromotive Force. Relation between chemical and electric driving forces
C465/865, 26-3, Lctur 7, 2 th Sp., 26 lctrochmcal qulbrum lctromotv Forc Rlaton btwn chmcal and lctrc drvng forcs lctrochmcal systm at constant T and p: consdr G Consdr lctrochmcal racton (nvolvng transfr
More informationLECTURE 6 TRANSFORMATION OF RANDOM VARIABLES
LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES TRANSFORMATION OF FUNCTION OF A RANDOM VARIABLE UNIVARIATE TRANSFORMATIONS TRANSFORMATION OF RANDOM VARIABLES If s a rv wth cdf F th Y=g s also a rv. If w wrt
More information4.2 Design of Sections for Flexure
4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt
More informationDiagnostics in Poisson Regression. Models - Residual Analysis
Dagnostcs n Posson Regresson Models - Resdual Analyss 1 Outlne Dagnostcs n Posson Regresson Models - Resdual Analyss Example 3: Recall of Stressful Events contnued 2 Resdual Analyss Resduals represent
More informationEstimation of apparent fraction defective: A mathematical approach
Availabl onlin at www.plagiarsarchlibrary.com Plagia Rsarch Library Advancs in Applid Scinc Rsarch, 011, (): 84-89 ISSN: 0976-8610 CODEN (USA): AASRFC Estimation of apparnt fraction dfctiv: A mathmatical
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More informationPractical: Phenotypic Factor Analysis
Practical: Phnotypic Factor Analysis Big 5 dimnsions Nuroticism & Extravrsion in 361 fmal UvA studnts - Exploratory Factor Analysis (EFA) using R (factanal) with Varimax and Promax rotation - Confirmatory
More informationChapter 14 Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment
Chaptr 14 Aggrgat Supply and th Short-run Tradoff Btwn Inflation and Unmploymnt Modifid by Yun Wang Eco 3203 Intrmdiat Macroconomics Florida Intrnational Univrsity Summr 2017 2016 Worth Publishrs, all
More informationΕρωτήσεις και ασκησεις Κεφ. 10 (για μόρια) ΠΑΡΑΔΟΣΗ 29/11/2016. (d)
Ερωτήσεις και ασκησεις Κεφ 0 (για μόρια ΠΑΡΑΔΟΣΗ 9//06 Th coffcnt A of th van r Waals ntracton s: (a A r r / ( r r ( (c a a a a A r r / ( r r ( a a a a A r r / ( r r a a a a A r r / ( r r 4 a a a a 0 Th
More informationON THE COMPLEXITY OF K-STEP AND K-HOP DOMINATING SETS IN GRAPHS
MATEMATICA MONTISNIRI Vol XL (2017) MATEMATICS ON TE COMPLEXITY OF K-STEP AN K-OP OMINATIN SETS IN RAPS M FARAI JALALVAN AN N JAFARI RA partmnt of Mathmatcs Shahrood Unrsty of Tchnology Shahrood Iran Emals:
More information