Linear Non-Gaussian Structural Equation Models
|
|
- Percival Reed
- 5 years ago
- Views:
Transcription
1 IMPS 8, Durham, NH Linar Non-Gaussian Structural Equation Modls Shohi Shimizu, Patrik Hoyr and Aapo Hyvarinn Osaka Univrsity, Japan Univrsity of Hlsinki, Finland
2 Abstract Linar Structural Equation Modling (linar SEM) Analyzs causal rlations Covarianc-basd SEM Uss covarianc structur alon for modl idntification A numbr of indistinguishabl modls Linar non-gaussian SEM Uss non-gaussian structurs for modl idntification Maks many modls distinguishabl
3 SEM and causal analysis SEM is oftn usd for causal analysis basd on non-primntal data Assumption: th data gnrating procss is rprsntd by a SEM modl If th assumption is rasonabl, SEM provids causal information 3
4 Limitations of covarianc-basd SEM Covarianc-basd SEM cannot distinguish btwn many modls Eampl 4
5 Linar non-gaussian SEM Many obsrvd data ar considrably non- Gaussian (Miccri, 989; Hyvarinn t al. ) Non-Gaussian structurs of data ar usful (Bntlr 983; Mooijaart 985) Non-Gaussianity distinguish btwn th two modls (Shimizu t al. 6) : 5
6 Indpndnt componnt analysis (ICA) Obsrvd random vctor is modld as As s i ar indpndnt and non-gaussian Zro mans and unit variancs A is a constant matri Typically squar, # variabls # indpndnt componnts Idntifiabl up to prmutation of th columns (Mooijaart 985; Comon, 994) 6
7 ICA stimation An altrnativ prssion of ICA (As): ~ whr W A s calld a ~, W rcovring Find such W that maimizs indpndnc of componnts of s ˆ W Many proposals (Hyvarinn t al. ) matri W ~ is stimatd up to prmutation of th rows: W PW ~ 7
8 ICA stimation An altrnativ prssion of ICA (As): ~ whr W A s calld a ~, W rcovring Find such W that maimizs indpndnc of componnts of s ˆ W Many proposals (Hyvarinn t al. ) matri W ~ is stimatd up to prmutation of th rows: W PW ~ 8
9 ICA stimation An altrnativ prssion of ICA (As): ~ whr W A s calld a ~, W rcovring Find such W that maimizs indpndnc of componnts of s ˆ W Many proposals (Hyvarinn t al. ) matri W ~ is stimatd up to prmutation of th rows: W PW ~ 9
10 Discovry of linar non-gaussian acyclic modls Shimizu, Hoyr, Hyvarinn and Krminn (6)
11 Linar non-gaussian acyclic modl (LiNGAM) Dirctd acyclic graphs (DAG) i can b arrangd in a ordr k(i) Assumptions: Linarity Etrnal influncs ar indpndnt and ar non-gaussian i i b j + k ( ij j) < k ( i) i or B +
12 Goal W know Data X is gnratd by W do NOT know Path cofficints: bij Ordrs k(i) Etrnal influncs: i B + What w obsrv is data X only Goal Estimat B and k(i) using data X only!
13 Ky ida First, rlat LiNGAM with ICA as follows: B + ( I B) A - ICA! ~ quivalntly ( I B) W Du to th prmutation indtrminacy, ICA givs: W PW ~ Can find a corrct P Th corrct prmutation is th only on that has no zros in th diagonal 3
14 Ky ida First, rlat LiNGAM with ICA as follows: B + ( I B) A - ICA! ~ quivalntly ( I B) W Du to th prmutation indtrminacy, ICA givs: W PW ~ Can find a corrct P Th corrct prmutation is th only on that has no zros in th diagonal 4
15 Ky ida First, rlat LiNGAM with ICA as follows: B + ( I B) A - ICA! ~ quivalntly ( I B) W Du to th prmutation indtrminacy, ICA givs: W PW ~ Can find a corrct P Th corrct prmutation is th only on that has no zros in th diagonal 5
16 Ky ida First, rlat LiNGAM with ICA as follows: B + ( I B) A - ICA! ~ quivalntly ( I B) W Du to th prmutation indtrminacy, ICA givs: W PW ~ Can find th corrct P Th corrct prmutation is th only on that has no zros in th diagonal 6
17 Illustrativ ampl Considr th modl: B +.6 Goal Estimat th path dirction btwn and obsrving only and 7
18 Prform ICA Rlation of th LiNGAM modl with ICA: ~ W W ~ Du to th prmutation indtrminacy, ICA might giv: ( ~ ) W PW.8 8
19 Prform ICA Rlation of th LiNGAM modl with ICA: ~ W W ~ Du to th prmutation indtrminacy, ICA might giv: W ( ~ ) PW.6 9
20 Find th corrct P Find a prmutation of th rows of W so that it has no zros in th diagonal In th ampl Prmut th rows W W ~
21 Find th corrct P Find a prmutation of th rows of W so that it has no zros in th diagonal In th ampl Prmut th rows W W ~
22 Find th corrct P Find a prmutation of th rows of W so that it has no zros in th diagonal In th ampl Prmut th rows W W ~
23 Find th corrct P In practic, Pˆ ma P ( ) P T W ii Havily pnalizs small absolut valus in th diagonal 3
24 Simulations: Estimation of B Both supr- and sub-gaussian trnal influncs tstd 5 datasts cratd for ach scattrplot B randomly gnratd at ach trial Numbr of variabls 5 Estimatd bij Gnrating bij, 3, Numbr of obsrvations 4
25 5 Prun B () In practic, du to stimation rrors, w would gt: Nd to find which path cofficints ar actually zros B
26 Find a prmutation that givs a lowr triangular matri Th LiNGAM modl is acyclic Th matri B can b prmutd to b lowr triangular for som prmutation of variabls (Bolln, 989) First, find a simultanous prmutation of rows and columns of B that givs a lowr-triangular B In practic, find a prmutation matri Q that minimizs th sum of th lmnts in its uppr T Qˆ ma QBQ triangular part: ( ) ij Q i j 6
27 Find a prmutation that givs a lowr triangular matri Th LiNGAM modl is acyclic Th matri B can b prmutd to b lowr triangular for som prmutation of variabls (Bolln, 989) First, find a simultanous prmutation of rows and columns of B that givs a lowr-triangular B In practic, find a prmutation matri Q that minimizs th sum of th lmnts in its uppr T Qˆ ma QBQ triangular part: ( ) ij Q i j 7
28 Find a prmutation that givs a lowr triangular matri Th LiNGAM modl is acyclic Th matri B can b prmutd to b lowr triangular for som prmutation of variabls (Bolln, 989) First, find a simultanous prmutation of rows and columns of B that givs a lowr-triangular B In practic, find a prmutation matri Q that minimizs th sum of th lmnts in its uppr triangular part: ( T Qˆ min QBQ ) ij Q i j 8
29 9 Gt a lowr-triangular B B Applying such a simultanous prmutation of th rows and columns, w gt a prmutd B that is as lowr-triangular as possibl St th uppr-triangular lmnts to b zros T QBQ
30 3 Gt a lowr-triangular B B Applying such a simultanous prmutation of th rows and columns, w gt a prmutd B that is as lowr-triangular as possibl St th uppr-triangular lmnts to b zros T QBQ
31 3 Gt a lowr-triangular B B Applying such a simultanous prmutation of th rows and columns, w gt a prmutd B that is as lowr-triangular as possibl St th uppr-triangular lmnts to b zros T QBQ -.5
32 3 Gt a lowr-triangular B B Applying such a simultanous prmutation of th rows and columns, w gt a prmutd B that is as lowr-triangular as possibl St th uppr-triangular lmnts to b zros T QBQ
33 33 Pruning B () Onc w gt a lowr-triangular B, th modl is idntifiabl using covarianc-basd SEM Many isting mthods can b usd for pruning th rmaining path cofficints Wald tst, Bootstrapping, Modl fit Lasso-typ stimators (Tibshirani 996; Zou, 6) tc. +.65
34 To summariz th procdur. Estimat B ICA + finding th corrct row prmutation. Prun stimatd B. Find a row-and-column prmutation that maks stimatd B lowr triangular. Prun rmaining paths using a covarianc-basd mthod. Estimat B. Prun stimatd B
35 Summary of th rgular LiNGAM A linar acyclic modl is idntifiabl basd on non-gaussianity ICA-basd stimation works wll Confidnc intrvals (Konya t al., in progrss) Bttr pruning mthods might b dvlopd Imposing sparsnss in th ICA stag (Zhang & Chang, 6; Hayashi t al. in progrss) lik Lasso (Tibshirani 996) 35
36 Som tnsions
37 Latnt factors (Shimizu t al., 7) A non-gaussian multipl indicator modl: f Bf + d Gf + Suppos that G is idntifid, thn B is idntifid Could idntify G in a data drivn way using a ttradconstraint-basd mthod (Silva t al., 6) 37
38 Latnt classs (Shimizu & Hyvarinn, 8) LiNGAM modl for ach class q: B + ( I B ) ì + ì + A - ICA! q q q q q q q ICA miturs (L t al., ; Mollah t al., 6) Class :.9 Class :
39 Unobsrvd confoundrs (Hoyr t al., in prss Can idntify and distinguish btwn mor modls u u u 39
40 Tim structurs (Hyvarinn t al., 8) Combining LiNGAM and autorgrssiv modl: k ( t) B ( t ) + ( t) In conomtrics: Structural vctor autorgrssion (Swanson & Grangr, 997) Changs ordinary AR cofficints basd on instantanous ffcts: B ( I B ) for ( : AR matri) M > M 4
41 Som variabls ar Gaussian (Hoyr t al., 8) Considr th modl:.6 Can idntify th path dirction if ithr of or is non-gaussian In gnral, thr ist svral quivalnt modls that ntail th sam distribution if som ar Gaussian 4
42 Som othr tnsions Cyclic modls (Lacrda t al., 8) Fwr quivalnt modls than covariancbasd approach Nonlinarity (Zhang & Chan, 7; Sun t al., 7) Modl fit statistics ar undr dvlopmnt Non-Gaussian structurs 4
43 Conclusion Us of non-gaussianity in SEM is usful for modl idntification Many obsrvd data ar considrably non-gaussian Th non-gaussian approach can b a good option 43
44 Most of our paprs and Matlab/Octav cod ar availabl on our wbpags Googl will find us! 44
Data Assimilation 1. Alan O Neill National Centre for Earth Observation UK
Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal
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 informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More informationChapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional
Chaptr 13 GMM for Linar Factor Modls in Discount Factor form GMM on th pricing rrors givs a crosssctional rgrssion h cas of xcss rturns Hors rac sting for charactristic sting for pricd factors: lambdas
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
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 informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
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 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 informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationOutline. Image processing includes. Edge detection. Advanced Multimedia Signal Processing #8:Image Processing 2 processing
Outlin Advancd Multimdia Signal Procssing #8:Imag Procssing procssing Intllignt Elctronic Sstms Group Dpt. of Elctronic Enginring, UEC aaui agai Imag procssing includs Imag procssing fundamntals Edg dtction
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 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 informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationEstimation of odds ratios in Logistic Regression models under different parameterizations and Design matrices
Advancs in Computational Intllignc, Man-Machin Systms and Cybrntics Estimation of odds ratios in Logistic Rgrssion modls undr diffrnt paramtrizations and Dsign matrics SURENDRA PRASAD SINHA*, LUIS NAVA
More informationSelf-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016
Ronald I. Frank 06 Adjoint https://n.wikipdia.org/wiki/adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac https://n.wikipdia.org/wiki/innr_product_spac
More informationThis test is for two independent Populations. The test is sometimes called the Mann-Whitney U test or the Rank Sum Wilcoxon. They are equivalent.
wo indpndnt Sampls his tst is for two indpndnt Populations. h tst is somtims calld th Mann-Whitny U tst or th Rank Sum Wilcoxon. hy ar quivalnt. h main assumption is that th two sampls ar indpndnt and
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
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 informationDiscovery of Linear Acyclic Models Using Independent Component Analysis
Created by S.S. in Jan 2008 Discovery of Linear Acyclic Models Using Independent Component Analysis Shohei Shimizu, Patrik Hoyer, Aapo Hyvarinen and Antti Kerminen LiNGAM homepage: http://www.cs.helsinki.fi/group/neuroinf/lingam/
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationNetwork Congestion Games
Ntwork Congstion Gams Assistant Profssor Tas A&M Univrsity Collg Station, TX TX Dallas Collg Station Austin Houston Bst rout dpnds on othrs Ntwork Congstion Gams Travl tim incrass with congstion Highway
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 informationARIMA Methods of Detecting Outliers in Time Series Periodic Processes
Articl Intrnational Journal of Modrn Mathmatical Scincs 014 11(1): 40-48 Intrnational Journal of Modrn Mathmatical Scincs Journal hompag:www.modrnscintificprss.com/journals/ijmms.aspx ISSN:166-86X Florida
More informationAS 5850 Finite Element Analysis
AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form
More informationStatus of LAr TPC R&D (2) 2014/Dec./23 Neutrino frontier workshop 2014 Ryosuke Sasaki (Iwate U.)
Status of LAr TPC R&D (2) 214/Dc./23 Nutrino frontir workshop 214 Ryosuk Sasaki (Iwat U.) Tabl of Contnts Dvlopmnt of gnrating lctric fild in LAr TPC Introduction - Gnrating strong lctric fild is on of
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More informationEstimation of linear non-gaussian acyclic models for latent factors
Estimation of linear non-gaussian acyclic models for latent factors Shohei Shimizu a Patrik O. Hoyer b Aapo Hyvärinen b,c a The Institute of Scientific and Industrial Research, Osaka University Mihogaoka
More informationCS 361 Meeting 12 10/3/18
CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm
More informationRecursive Estimation of Dynamic Time-Varying Demand Models
Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 Rcursiv Estimation of Dynamic im-varying Dmand Modls Alxandr Efrmov Abstract: h papr prsnts an implmntation of a st of rcursiv algorithms
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationDifference -Analytical Method of The One-Dimensional Convection-Diffusion Equation
Diffrnc -Analytical Mthod of Th On-Dimnsional Convction-Diffusion Equation Dalabav Umurdin Dpartmnt mathmatic modlling, Univrsity of orld Economy and Diplomacy, Uzbistan Abstract. An analytical diffrncing
More informationMCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems
MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a
More informationECE 650 1/8. Homework Set 4 - Solutions
ECE 65 /8 Homwork St - Solutions. (Stark & Woods #.) X: zro-man, C X Find G such that Y = GX will b lt. whit. (Will us: G = -/ E T ) Finding -valus for CX: dt = (-) (-) = Finding corrsponding -vctors for
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
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 informationWhere k is either given or determined from the data and c is an arbitrary constant.
Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationText: WMM, Chapter 5. Sections , ,
Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl
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 informationWhat is the product of an integer multiplied by zero? and divided by zero?
IMP007 Introductory Math Cours 3. ARITHMETICS AND FUNCTIONS 3.. BASIC ARITHMETICS REVIEW (from GRE) Which numbrs form th st of th Intgrs? What is th product of an intgr multiplid by zro? and dividd by
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationThus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.
Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn
More informationTransitional Probability Model for a Serial Phases in Production
Jurnal Karya Asli Lorkan Ahli Matmatik Vol. 3 No. 2 (2010) pag 49-54. Jurnal Karya Asli Lorkan Ahli Matmatik Transitional Probability Modl for a Srial Phass in Production Adam Baharum School of Mathmatical
More informationSara Godoy del Olmo Calculation of contaminated soil volumes : Geostatistics applied to a hydrocarbons spill Lac Megantic Case
wwwnvisol-canadaca Sara Godoy dl Olmo Calculation of contaminatd soil volums : Gostatistics applid to a hydrocarbons spill Lac Mgantic Cas Gostatistics: study of a PH contamination CONTEXT OF THE STUDY
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 information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationInference Methods for Stochastic Volatility Models
Intrnational Mathmatical Forum, Vol 8, 03, no 8, 369-375 Infrnc Mthods for Stochastic Volatility Modls Maddalna Cavicchioli Cá Foscari Univrsity of Vnic Advancd School of Economics Cannargio 3, Vnic, Italy
More informationSquare of Hamilton cycle in a random graph
Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationMultiple-Choice Test Introduction to Partial Differential Equations COMPLETE SOLUTION SET
Mltipl-Choic Tst Introdction to Partial Diffrntial Eqations COMPLETE SOLUTION SET 1. A partial diffrntial qation has (A on indpndnt variabl (B two or mor indpndnt variabls (C mor than on dpndnt variabl
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationPanel Data Analysis Introduction
Panl Data Analysis Introduction Modl Rprsntation N-first or T-first rprsntation Poold Modl Fixd Effcts Modl Random Effcts Modl Asymptotic Thory N, or T N, T Panl-Robust Infrnc Panl Data Analysis Introduction
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 information3-2-1 ANN Architecture
ARTIFICIAL NEURAL NETWORKS (ANNs) Profssor Tom Fomby Dpartmnt of Economics Soutrn Mtodist Univrsity Marc 008 Artificial Nural Ntworks (raftr ANNs) can b usd for itr prdiction or classification problms.
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 informationRational Approximation for the one-dimensional Bratu Equation
Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 5 Rational Approximation for th on-dimnsional Bratu Equation Moustafa Aly Soliman Chmical Enginring Dpartmnt, Th British Univrsity in
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More information2. Laser physics - basics
. Lasr physics - basics Spontanous and stimulatd procsss Einstin A and B cofficints Rat quation analysis Gain saturation What is a lasr? LASER: Light Amplification by Stimulatd Emission of Radiation "light"
More informationNumbering Systems Basic Building Blocks Scaling and Round-off Noise. Number Representation. Floating vs. Fixed point. DSP Design.
Numbring Systms Basic Building Blocks Scaling and Round-off Nois Numbr Rprsntation Viktor Öwall viktor.owall@it.lth.s Floating vs. Fixd point In floating point a valu is rprsntd by mantissa dtrmining th
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More informationSUMMER 17 EXAMINATION
(ISO/IEC - 7-5 Crtifid) SUMMER 7 EXAMINATION Modl wr jct Cod: Important Instructions to aminrs: ) Th answrs should b amind by ky words and not as word-to-word as givn in th modl answr schm. ) Th modl answr
More informationTwo Products Manufacturer s Production Decisions with Carbon Constraint
Managmnt Scinc and Enginring Vol 7 No 3 pp 3-34 DOI:3968/jms9335X374 ISSN 93-34 [Print] ISSN 93-35X [Onlin] wwwcscanadant wwwcscanadaorg Two Products Manufacturr s Production Dcisions with Carbon Constraint
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More informationON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park
Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction
More informationME311 Machine Design
ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform
More informationdx equation it is called a second order differential equation.
TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld
More informationGeneralized Estimating Equations for Zero-Inflated Spatial Count Data
Availabl onlin at www.scincdirct.com Procdia Environmntal Scincs 77 (0 8 86 st Confrnc on Spatial Statistics 0: Mapping Global Chang Gnralizd Estimating Equations for Zro-Inflatd Spatial Count Data Antha
More informationperm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l
h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
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 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 informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationFinite Element Analysis
Finit Elmnt Analysis L4 D Shap Functions, an Gauss Quaratur FEA Formulation Dr. Wiong Wu EGR 54 Finit Elmnt Analysis Roamap for Dvlopmnt of FE Strong form: govrning DE an BCs EGR 54 Finit Elmnt Analysis
More informationL 1 = L G 1 F-matrix: too many F ij s even at quadratic-only level
5.76 Lctur #6 //94 Pag of 8 pag Lctur #6: Polyatomic Vibration III: -Vctor and H O Lat tim: I got tuck on L G L mut b L L L G F-matrix: too many F ij vn at quadratic-only lvl It obviou! Intrnal coordinat:
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationMCB137: Physical Biology of the Cell Spring 2017 Homework 6: Ligand binding and the MWC model of allostery (Due 3/23/17)
MCB37: Physical Biology of th Cll Spring 207 Homwork 6: Ligand binding and th MWC modl of allostry (Du 3/23/7) Hrnan G. Garcia March 2, 207 Simpl rprssion In class, w drivd a mathmatical modl of how simpl
More informationCS 6353 Compiler Construction, Homework #1. 1. Write regular expressions for the following informally described languages:
CS 6353 Compilr Construction, Homwork #1 1. Writ rgular xprssions for th following informally dscribd languags: a. All strings of 0 s and 1 s with th substring 01*1. Answr: (0 1)*01*1(0 1)* b. All strings
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More informationMATHEMATICS (B) 2 log (D) ( 1) = where z =
MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J +
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More information4037 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Lvl MARK SCHEME for th Octobr/Novmbr 0 sris 40 ADDITIONAL MATHEMATICS 40/ Papr, maimum raw mark 80 This mark schm is publishd as an aid to tachrs and candidats,
More informationPair (and Triplet) Production Effect:
Pair (and riplt Production Effct: In both Pair and riplt production, a positron (anti-lctron and an lctron (or ngatron ar producd spontanously as a photon intracts with a strong lctric fild from ithr a
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationINC 693, 481 Dynamics System and Modelling: The Language of Bound Graphs Dr.-Ing. Sudchai Boonto Assistant Professor
INC 693, 48 Dynamics Systm and Modlling: Th Languag o Bound Graphs Dr.-Ing. Sudchai Boonto Assistant Prossor Dpartmnt o Control Systm and Instrumntation Enginring King Mongkut s Unnivrsity o Tchnology
More informationCE 530 Molecular Simulation
CE 53 Molcular Simulation Lctur 8 Fr-nrgy calculations David A. Kofk Dpartmnt of Chmical Enginring SUNY Buffalo kofk@ng.buffalo.du 2 Fr-Enrgy Calculations Uss of fr nrgy Phas quilibria Raction quilibria
More information