Time Series, Part 1 Content Literature
|
|
- MargaretMargaret Anthony
- 5 years ago
- Views:
Transcription
1 Time Series, Par Coe - Saioariy, auocorrelaio, parial auocorrelaio, removal of osaioary compoes, idepedece es for ime series - Liear Sochasic Processes: auoregressive (AR), movig average (MA), auoregressive movig average (ARMA) - Fi of models AR, MA ad ARMA o saioary ime series - Liear models for o-saioary ime series - Predicio of ime series - Noliear aalysis of ime series wih sochasic models - Noliear aalysis of ime series ad dyamical sysems Lieraure - The Aalysis of Time Series, A Iroducio, Chafield C., Sixh ediio, Chapma & Hall, 4 - Iroducio o ime series ad forecasig, Brockwell P.J. ad Davis R.A., Secod ediio, Spriger, - No-Liear Time Series, A Dyamical Sysem Approach, Tog H., Oxford Uiversiy Press, Noliear Time Series Aalysis, Kaz H. ad Schreiber T., Cambridge Uiversiy Press, 4
2 physiology Real world ime series uivariae ime series mechaics elecroics oly oe ime series limied legh geophysics ecoomy o-saioariy oise
3 observed quaiy variable Χ Defiiios / oaios The values of he observed quaiy chage wih radomess (sochasiciy) a some small or larger degree radom variable (r.v.) Χ The observaios ake place mos ofe a fixed ime seps samplig ime For each ime poi we cosider he value x of he r.v. Χ The se of he values of x over a ime period (give i uis of he samplig ime) (uivariae) ime series x { x, x,, x } If here are simulaeous observaios of more ha oe variable mulivariae ime series We apply mehods ad echiques o he give uivariae or mulivariae ime series i order o ge isigh for he sysem ha geeraes i ime series aalysis The ime series ca be cosidered as realizaio of a sochasic or deermiisic process (dyamical sysem) X
4 close idex volume close idex close idex Exchage idex ad volume of he Ahes Sock Exchage (ASE) 7 ASE idex, period ASE idex, period ASE idex, period x 5 ASE volume, period mohs Predicio? Wha is he idex value omorrow? The day afer? Dyamical sysem? sochasic process? Wha is he mechaism of he Greek sock marke?
5 Geeral Idex of Comsumer Prices Geeral Idex of Cosumer Prices (GICP) Geeral Idex of Comsumer Prices, period Ja - Aug Tred? Seasoaliy / periodiciy? Auocorrelaio? Auoregressio? Predicio?
6 umber of suspos umber of suspos umber of suspos Aual suspo umbers Aual suspos, period Aual suspos, period Aual suspos, period Wha is he mechaism / sysem / process ha geeraes suspos? Is i a periodic sysem + oise? Is i a sochasic sysem? Is i a chaoic sysem? Give he suspo umber for up o 995, wha is he suspo umber i 996? ad he afer? Wha will be he suspo umber i 3, 4?
7 suspo umber Model compariso Geuie predicio Geuie predicios of suspo daa year
8 Wha is he geeraig sysem of a real ime series? x(i) preical EEG Real ime series ical EEG Cadidae sochasic models sochasic ime i secods ime i secods 4 ime idex i Cadidae deermiisic models periodic + oise low dimesioal chaos high dimesioal chaos ime i secods ime i secods ime i secods
9 Drippig waer fauce (origial experime a UC Saa Cruz). x x x 3 ( x3, x) ( x, x ) The observaio of he drippig fauce shows ha for some flow velociy he drops do o ru a cosa ime iervals. Cruchfield e al, Scieific America, 986 The scaer diagram of he daa showed ha he drop flow is o radom. scaer diagram i i ( x, x ) ( xi, xi, xi ) Héo map s.4s.3s i i i observed variable x s w w i oise i i i chaos
10 AE idex Geeral Idex of Comsumer Prices Geeral Idex of Comsumer Prices, period Ja - Aug 5 5 No-saioariy 5 Tred? Seasoaliy / periodiciy? Auroral Elecroje Idex 8 6 Volailiy? 4 Auocorrelaio? ime [ mi]
11 AE idex AE idex 4 8 y, y,, y Auroral Elecroje Idex Variace sabilizig rasformaio Trasform Χ =T(Υ ) ha sabilizes he variace of Υ? Var[ X ] cos simple soluio: X log( Y)? ime [ mi] x, x,, x ime [ mi] Logarihm rasform of Auroral Elecroje Idex Power rasform (Box-Cox): X Y Assumpio: Var[Υ ] chages as a fucio of he red μ λ Χ Var[y ] Y Y log( Y ) Y Oher rasforms? 4 c 3 c c c?
12 AE idex f Z (z) AE idex f Y (y) AE idex f X (x) 4 Auroral Elecroje Idex 5 x y ormal y, y,, y ime [ mi] Logarihm rasform of Auroral Elecroje Idex x x=log(y) ormal X log( Y) ime [ mi] Gaussia rasform of Auroral Elecroje Idex ime [ mi] y x= - (F Y (y)) ormal -4-4 z X F ( Y ) Y?
13 idex ime series y, y,, y Saioariy - red Tred: slow chage of he successive values of y deermiisic red: a fucio of ime μ = f() Plasic deformaio 6 S&P5 sochasic red: radom slow chage μ
14 Removal of red Y X. Deermiisic red : kow or esimaed fucio of ime μ = f() Example: polyomial of degree p p f () a a a p Plasic deformaio μ : mea value as fucio of (slowly varyig mea level) X Y {X } saioary Fi wih firs degree polyomial Fi wih fifh degree polyomial
15 close idex close idex close idex. Sochasic red Idex of he Ahes Sock Exchage (ASE) ASE idex, period orig orig local liear, breakpois polyomial,p= α. Smoohig wih movig average filer Simple filer: movig average orig MA(3) MA(5) ˆ ASE idex, period ASE idex, period q y q 3 ˆ y y y q j jq "q " 4? More geeral filer: movig weighed average q ˆ ay a j j jq q jq Simple movig average: a j, j q,, q q j
16 close idex close idex b. Tred removal wih differecig Oe lag differece or firs differece Y Y Y ( B) Y B: lag operaor BY Y Secod order lag differece Y ( Y ) ( B)( B) Y ( B B ) Y Y Y Y If he red is locally liear, i is removed by firs differeces: Y Y Y X X aa a a a a ( ) a cosa! If he red is locally polyomial or degree p, i is removed by usig p Y? p Y p! c X [show firs: ] 4 ASE idex: firs differeces, period 7-4 ASE idex: firs differeces, period
17 close idex close idex close idex close idex close idex Which mehod for red removal is bes? orig local liear, breakpois polyomial,p= ASE idex, period 7-5 ASE idex dereded, period 7 - local liear, breakpois polyomial,p= orig MA(3) MA(5) ASE idex, period 7-5 ASE idex dereded, period 7 - MA(3) MA(5) Esimaio of red 5 ASE idex: firs differeces, period
18 differece of logs idex relaive chage firs differece more o differecig rasform y, y,, y - ime series S&P S&P5, firs differeces S&P5, relaive chages S&P5, differece of logs chage of he value x y y relaive chage of he value y y x y chage of he logarihm of he value x l y l y
19 umber of suspos umber of suspos Removal of seasoaliy or periodiciy Y s X s : periodic fucio of wih period d 5 5 Aual suspos, period 7 - Aual suspos Period d ad appropriae fucio s?. kow or esimaed periodic fucio s = f() Aual suspos, period X Y s {X } saioary a. Esimaio of s i i=,,d from he averages for each compoe Period d is kow k / d sˆ k y i i jd k j b. Removal of periodiciy usig lag differeces of order d (d-differecig) Y Y Y ( B d ) Y d d
20 year cycle of GICP residual GICP GICP dereded GICP Removal of red ad periodiciy Y s X Y Y s X. Removal of red X Y s Y s. Removal of periodiciy {Χ }: ime series of residuals Firs remove red ad he periodiciy or vice versa? 5 Geeral Idex of Comsumer GICP: Liear Prices, fiperiod /-8/5 3 GICP: Residual of liear fi year GICP: Year cycle esimae GICP: dereded ad deseasoed year o-saioary y y y saioary,,, year x, x,, x Is here iformaio i he residuals?
21 f Y (y) f X (x) idex firs differece Y : he value of he quaiy y, y,, y ime series Time correlaio Sochasic process Y X 6 S&P5 S&P5, firs differeces chage of he value 4-5 x y y fy ( y) 3.5 x -3 Gaussia pdf superimposed o S&P5 6 5 fx ( x) Gaussia pdf superimposed o S&P5 reurs Saic descripio margial disribuio Dyamic descripio? Time correlaio Y X
22 Disribuio ad momes of a sochasic process A sochasic process ca be fully described i erms of he margial ad joi probabiliy disribuios Z f ( y) f ( y, ) Y Y margial disribuio, Z f ( y, y ) f ( y, y,, ) Y, Y Y joi disribuio of r.v.,, 3 Z f ( y, y, y ) f ( y, y, y,,, ) Y, Y, Y 3 Y joi disribuio of 3 r.v. Y yf y y Firs order mome (mea) (, )d Secod order mome Ceral secod order mome Higher order momes Y Y y y f ( y, y,, )dy d y (, ) Y Y ( Y )( Y ) (, ) (, ) auocovariace The probabiliy disribuio ad momes may chage i ime
23 Sric-sese saioariy Saioariy The disribuios do o chage wih ime (equivalely, all momes are cosa) Z,,, 3 Z Z f ( y) f ( y, ) f ( y) Y Y Y f ( y, y ) f ( y, y ) Y, Y Y, Y f ( y, y, y ) f ( y, y, y ) Y, Y, Y 3 Y, Y, Y 3 3 cosa Z Wide-sese saioariy The firs wo momes are cosa i ime Y YY, (, ) ( ) Y Y (, ) (, ) ( ) cosa Z cosa - mea - variace - auocovariace for τ= Y () cosa variace () Y () Y
24 Auocorrelaio Saioary ime series X Auocovariace X X X X X ( )( ) ( ) () Variace X X Auocorrelaio () () () ( ) ( ) a a lag τ. () Noaio: () () Commes: k k ad k k ad k k Auocovariace marix Time correlaio of variables of Measures he memory of X Auocorrelaio marix X
25 X Basic sochasic processes X idepede ad ideically disribued r.v. (iid) P( X x, X x,, X x ) P( X x ) P( X x) P( X x ) E X whie oise (WN), o-correlaed r.v. E XX i j ij 3 Y radom walk (RW) Y Y X X X X E Y Y, Y,, Y Y X iid E Y E Y E X E X? Variace icreases liearly wih ime!
26 Ucorrelaed (whie oise) ad idepede (iid) observaios Chafield C., The Aalysis of Time Series, A Iroducio, 6 h ediio, p. 38 (Chaper 3): Some auhors prefer o make he weaker assumpio ha he z s are muually ucorrelaed, raher ha idepede. This is adequae for liear, ormal processes, bu he sroger idepedece assumpio is eeded whe cosiderig o-liear models (Chaper ). Noe ha a purely radom process is someimes called whie oise, paricularly by egieers. p. (Chaper ): Whe examiig he properies of o-liear models, i ca be very impora o disiguish bewee idepede ad ucorrelaed radom variables. I Secio 3.4., whie oise (or a purely radom process) was defied o be a sequece of idepede ad ideically disribued (i.i.d.) radom variables. This is someimes called sric whie oise (SWN), ad he phrase ucorrelaed whie oise (UWN) is used whe successive values are merely ucorrelaed, raher ha idepede. Of course if successive values follow a ormal (Gaussia) disribuio, he zero correlaio implies idepedece so ha Gaussia UWN is SWN. However, wih o-liear models, disribuios are geerally o-ormal ad zero correlaio eed o imply idepedece. Wei W.W.C., Time Series Aalysis, Uivariae ad Mulivariae Mehods, p. 5:.4 Whie Noise Processes A process {a } is called a whie oise process if i is a sequece of ucorrelaed radom variables from a fixed disribuio wih cosa mea (usually assumed ), cosa variace ad zero auocovariace for lags differe from.
27 4 X For each order p: Gaussia (ormal) sochasic process f ( x, x,, x ) X, X,, X p p is p-dimesioal Gaussia disribuio Gaussia disribuio is compleely defied by he firs wo momes sric saioariy weak saioariy Example Sochasic process: Is he process weak saioary? E[ X ] E[ A]E[si( )] X Asi( ) A r.v. E[ A] Var[ A] ~ U[, ] E[ XX ] E A si( )si( ( ) )... cos( ) The firs ad secod order momes do o deped o ime.? θ ad A idepede
28 ime series Sample auocovariace / auocorrelaio Sample mea x, x,, x x x ubiased esimae of he mea μ of he ime series? Sample auocovariace Aoher esimae of auocovariace Biased esimaes: ( ) ( x ) () ( ) c() ( xx x ) c x x E[ c ] ( )Var[ x] E[ c] Var[ x] c x x,,, c( ) Sample auocorrelaio r( ) r() c() r ~ N(,Var[ r ]) For large : Var[ r ] ( m m m m 4 mm ) m Var[ r ] m very large m Noaio c() c bias icreases wih he lag τ Noaio r() r Barle formula
29 Auocorrelaio for whie oise x, x,, x whie oise ime series, r ~ N(, )? Tes for idepedece observed saioary ime series residual ime series afer red or periodiciy removal Η : x, x,, x Hypoheses Is i iid? Η Are here correlaios? x x x is iid Η : is whie oise,,, x, x,, x Saisical Sigificace es for auocorrelaio H : H : r R r z / Rejecio regio: / z Bad of isigifica auocorrelaio: a/ x, x,, x whie oise r N(, ) for sigificace level for =.5 Η
30 r() Numerical Example For a ime series of observaios, he auocorrelaio for τ=,, are: Assume ha he ime series is purely radom (Η :ρ=): Var[ r ].5 for =.5, we expec 95% of auocorrelaios o be i he ierval ρ, ρ και ρ τ για τ=3,4, Example of GICP.3.. GICP residual: auocorrelaio Sigificace es Η : for each idepedely A sigificace level =.5, Η is rejeced for τ= Is here ay correlaio i he GICP ime series?
31 r() The Pormaeau sigificace es Q(k) A es for each lag,,k Oe es for all lags ogeher? H :,,,k Tes saisic Q: ~ k Q k r ( k ) / ( ) Box-Pierce Q r j Q rejecio regio R Q k; a Ljug-Box.3. GICP residual: auocorrelaio k 35 3 GICP residual: Pormaeau (Ljug-Box). -. Q 4.6 k ; a H for τ= is rejeced k
32 x() r() x() r() radom ime series. radom ime series: auocorrelaio logisic ime series logisic ime series: auocorrelaio A appropriae sigificace es?
33 close idex r() Q(k) close idex r() Q(k) close idex r() Q(k) close idex ASE idex, period ASE idex: firs differeces, period 7 - Is here correlaio i he reurs ime series of he ASE idex (ime period 7-)? Wha is he appropriae saioary ime series: firs differeces or reurs? Is here correlaio? firs differeces x y y ASE firs differeces: auocorrelaio oliear? 5 5 X X E ASE firs differeces: Pormaeau (Ljug-Box) sample Q X (k,) ASE idex: reurs, period ASE idex: square reurs, period reurs x l y l y square of reurs x l y l y x ( x) ASE reurs: auocorrelaio ASE square reurs: auocorrelaio k ASE reurs: Pormaeau (Ljug-Box) 5 5 sample Q X (k,) k ASE square reurs: Pormaeau (Ljug-Box) sample Q X (k,) k
10.3 Autocorrelation Function of Ergodic RP 10.4 Power Spectral Density of Ergodic RP 10.5 Normal RP (Gaussian RP)
ENGG450 Probabiliy ad Saisics for Egieers Iroducio 3 Probabiliy 4 Probabiliy disribuios 5 Probabiliy Desiies Orgaizaio ad descripio of daa 6 Samplig disribuios 7 Ifereces cocerig a mea 8 Comparig wo reames
More informationBEST LINEAR FORECASTS VS. BEST POSSIBLE FORECASTS
BEST LINEAR FORECASTS VS. BEST POSSIBLE FORECASTS Opimal ear Forecasig Alhough we have o meioed hem explicily so far i he course, here are geeral saisical priciples for derivig he bes liear forecas, ad
More informationSTK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
More informationStochastic Processes Adopted From p Chapter 9 Probability, Random Variables and Stochastic Processes, 4th Edition A. Papoulis and S.
Sochasic Processes Adoped From p Chaper 9 Probabiliy, adom Variables ad Sochasic Processes, 4h Ediio A. Papoulis ad S. Pillai 9. Sochasic Processes Iroducio Le deoe he radom oucome of a experime. To every
More informationth m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)
1 Trasform Techiques h m m m m mome : E[ ] x f ( x) dx h m m m m ceral mome : E[( ) ] ( ) ( x) f ( x) dx A coveie wa of fidig he momes of a radom variable is he mome geeraig fucio (MGF). Oher rasform echiques
More informationComparisons Between RV, ARV and WRV
Comparisos Bewee RV, ARV ad WRV Cao Gag,Guo Migyua School of Maageme ad Ecoomics, Tiaji Uiversiy, Tiaji,30007 Absrac: Realized Volailiy (RV) have bee widely used sice i was pu forward by Aderso ad Bollerslev
More informationMoment Generating Function
1 Mome Geeraig Fucio m h mome m m m E[ ] x f ( x) dx m h ceral mome m m m E[( ) ] ( ) ( x ) f ( x) dx Mome Geeraig Fucio For a real, M () E[ e ] e k x k e p ( x ) discree x k e f ( x) dx coiuous Example
More informationIdeal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory
Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable
More informationThe Central Limit Theorem
The Ceral Limi Theorem The ceral i heorem is oe of he mos impora heorems i probabiliy heory. While here a variey of forms of he ceral i heorem, he mos geeral form saes ha give a sufficiely large umber,
More informationA Note on Prediction with Misspecified Models
ITB J. Sci., Vol. 44 A, No. 3,, 7-9 7 A Noe o Predicio wih Misspecified Models Khresha Syuhada Saisics Research Divisio, Faculy of Mahemaics ad Naural Scieces, Isiu Tekologi Badug, Jala Gaesa Badug, Jawa
More information6/10/2014. Definition. Time series Data. Time series Graph. Components of time series. Time series Seasonal. Time series Trend
6//4 Defiiio Time series Daa A ime series Measures he same pheomeo a equal iervals of ime Time series Graph Compoes of ime series 5 5 5-5 7 Q 7 Q 7 Q 3 7 Q 4 8 Q 8 Q 8 Q 3 8 Q 4 9 Q 9 Q 9 Q 3 9 Q 4 Q Q
More informationO & M Cost O & M Cost
5/5/008 Turbie Reliabiliy, Maieace ad Faul Deecio Zhe Sog, Adrew Kusiak 39 Seamas Ceer Iowa Ciy, Iowa 54-57 adrew-kusiak@uiowa.edu Tel: 39-335-5934 Fax: 39-335-5669 hp://www.icae.uiowa.edu/~akusiak Oulie
More informationInstitute of Actuaries of India
Isiue of cuaries of Idia Subjec CT3-robabiliy ad Mahemaical Saisics May 008 Eamiaio INDICTIVE SOLUTION Iroducio The idicaive soluio has bee wrie by he Eamiers wih he aim of helig cadidaes. The soluios
More informationInference of the Second Order Autoregressive. Model with Unit Roots
Ieraioal Mahemaical Forum Vol. 6 0 o. 5 595-604 Iferece of he Secod Order Auoregressive Model wih Ui Roos Ahmed H. Youssef Professor of Applied Saisics ad Ecoomerics Isiue of Saisical Sudies ad Research
More informationBig O Notation for Time Complexity of Algorithms
BRONX COMMUNITY COLLEGE of he Ciy Uiversiy of New York DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE CSI 33 Secio E01 Hadou 1 Fall 2014 Sepember 3, 2014 Big O Noaio for Time Complexiy of Algorihms Time
More informationOrder Determination for Multivariate Autoregressive Processes Using Resampling Methods
joural of mulivariae aalysis 57, 175190 (1996) aricle o. 0028 Order Deermiaio for Mulivariae Auoregressive Processes Usig Resamplig Mehods Chaghua Che ad Richard A. Davis* Colorado Sae Uiversiy ad Peer
More informationExercise 3 Stochastic Models of Manufacturing Systems 4T400, 6 May
Exercise 3 Sochasic Models of Maufacurig Sysems 4T4, 6 May. Each week a very popular loery i Adorra pris 4 ickes. Each ickes has wo 4-digi umbers o i, oe visible ad he oher covered. The umbers are radomly
More informationSection 8 Convolution and Deconvolution
APPLICATIONS IN SIGNAL PROCESSING Secio 8 Covoluio ad Decovoluio This docume illusraes several echiques for carryig ou covoluio ad decovoluio i Mahcad. There are several operaors available for hese fucios:
More informationF D D D D F. smoothed value of the data including Y t the most recent data.
Module 2 Forecasig 1. Wha is forecasig? Forecasig is defied as esimaig he fuure value ha a parameer will ake. Mos scieific forecasig mehods forecas he fuure value usig pas daa. I Operaios Maageme forecasig
More informationBAYESIAN ESTIMATION METHOD FOR PARAMETER OF EPIDEMIC SIR REED-FROST MODEL. Puji Kurniawan M
BAYESAN ESTMATON METHOD FOR PARAMETER OF EPDEMC SR REED-FROST MODEL Puji Kuriawa M447 ABSTRACT. fecious diseases is a impora healh problem i he mos of couries, belogig o doesia. Some of ifecious diseases
More informationLecture 15 First Properties of the Brownian Motion
Lecure 15: Firs Properies 1 of 8 Course: Theory of Probabiliy II Term: Sprig 2015 Isrucor: Gorda Zikovic Lecure 15 Firs Properies of he Browia Moio This lecure deals wih some of he more immediae properies
More informationPlease, ask questions!
The arrow of ime i ime series Albero Suárez, José Miguel Herádez Lobao, Pablo Morales Mombiela Machie learig group PS, Uiversidad Auóoma de Madrid (Spai) albero.suarez@uam.es Please, ask quesios! The arrow
More information1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4)
7 Differeial equaios Review Solve by he mehod of udeermied coefficies ad by he mehod of variaio of parameers (4) y y = si Soluio; we firs solve he homogeeous equaio (4) y y = 4 The correspodig characerisic
More informationB. Maddah INDE 504 Simulation 09/02/17
B. Maddah INDE 54 Simulaio 9/2/7 Queueig Primer Wha is a queueig sysem? A queueig sysem cosiss of servers (resources) ha provide service o cusomers (eiies). A Cusomer requesig service will sar service
More informationExtremal graph theory II: K t and K t,t
Exremal graph heory II: K ad K, Lecure Graph Theory 06 EPFL Frak de Zeeuw I his lecure, we geeralize he wo mai heorems from he las lecure, from riagles K 3 o complee graphs K, ad from squares K, o complee
More informationProcessamento Digital de Sinal
Deparaeo de Elecróica e Telecouicações da Uiversidade de Aveiro Processaeo Digial de ial Processos Esocásicos uar ado Processes aioar ad ergodic Correlaio auo ad cross Fucio Covariace Fucio Esiaes of he
More informationState and Parameter Estimation of The Lorenz System In Existence of Colored Noise
Sae ad Parameer Esimaio of he Lorez Sysem I Eisece of Colored Noise Mozhga Mombeii a Hamid Khaloozadeh b a Elecrical Corol ad Sysem Egieerig Researcher of Isiue for Research i Fudameal Scieces (IPM ehra
More informationGranger Causality Test: A Useful Descriptive Tool for Time Series Data
Ieraioal OPEN ACCESS Joural Of Moder Egieerig Research (IJMER) Grager Causaliy Tes: A Useful Descripive Tool for Time Series Daa OGUNTADE, E. S 1 ; OLANREWAJU, S. O 2., OJENII, J.A. 3 1, 2 (Deparme of
More informationAdditional Tables of Simulation Results
Saisica Siica: Suppleme REGULARIZING LASSO: A CONSISTENT VARIABLE SELECTION METHOD Quefeg Li ad Ju Shao Uiversiy of Wiscosi, Madiso, Eas Chia Normal Uiversiy ad Uiversiy of Wiscosi, Madiso Supplemeary
More informationMathematical Statistics. 1 Introduction to the materials to be covered in this course
Mahemaical Saisics Iroducio o he maerials o be covered i his course. Uivariae & Mulivariae r.v s 2. Borl-Caelli Lemma Large Deviaios. e.g. X,, X are iid r.v s, P ( X + + X where I(A) is a umber depedig
More informationC(p, ) 13 N. Nuclear reactions generate energy create new isotopes and elements. Notation for stellar rates: p 12
Iroducio o sellar reacio raes Nuclear reacios geerae eergy creae ew isoopes ad elemes Noaio for sellar raes: p C 3 N C(p,) 3 N The heavier arge ucleus (Lab: arge) he ligher icomig projecile (Lab: beam)
More informationLet s express the absorption of radiation by dipoles as a dipole correlation function.
MIT Deparme of Chemisry 5.74, Sprig 004: Iroducory Quaum Mechaics II Isrucor: Prof. Adrei Tokmakoff p. 81 Time-Correlaio Fucio Descripio of Absorpio Lieshape Le s express he absorpio of radiaio by dipoles
More informationStationarity and Error Correction
Saioariy ad Error Correcio. Saioariy a. If a ie series of a rado variable Y has a fiie σ Y ad σ Y,Y-s or deeds oly o he lag legh s (s > ), bu o o, he series is saioary, or iegraed of order - I(). The rocess
More informationStationarity and Unit Root tests
Saioari ad Ui Roo ess Saioari ad Ui Roo ess. Saioar ad Nosaioar Series. Sprios Regressio 3. Ui Roo ad Nosaioari 4. Ui Roo ess Dicke-Fller es Agmeed Dicke-Fller es KPSS es Phillips-Perro Tes 5. Resolvig
More informationFresnel Dragging Explained
Fresel Draggig Explaied 07/05/008 Decla Traill Decla@espace.e.au The Fresel Draggig Coefficie required o explai he resul of he Fizeau experime ca be easily explaied by usig he priciples of Eergy Field
More informationComparison between Fourier and Corrected Fourier Series Methods
Malaysia Joural of Mahemaical Scieces 7(): 73-8 (13) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/oural Compariso bewee Fourier ad Correced Fourier Series Mehods 1
More informationOutline. simplest HMM (1) simple HMMs? simplest HMM (2) Parameter estimation for discrete hidden Markov models
Oulie Parameer esimaio for discree idde Markov models Juko Murakami () ad Tomas Taylor (2). Vicoria Uiversiy of Welligo 2. Arizoa Sae Uiversiy Descripio of simple idde Markov models Maximum likeliood esimae
More informationOLS bias for econometric models with errors-in-variables. The Lucas-critique Supplementary note to Lecture 17
OLS bias for ecoomeric models wih errors-i-variables. The Lucas-criique Supplemeary oe o Lecure 7 RNy May 6, 03 Properies of OLS i RE models I Lecure 7 we discussed he followig example of a raioal expecaios
More informationAffine term structure models
/5/07 Affie erm srucure models A. Iro o Gaussia affie erm srucure models B. Esimaio by miimum chi square (Hamilo ad Wu) C. Esimaio by OLS (Adria, Moech, ad Crump) D. Dyamic Nelso-Siegel model (Chrisese,
More informationJuly 24-25, Overview. Why the Reliability Issue is Important? Some Well-known Reliability Measures. Weibull and lognormal Probability Plots
Par I: July 24-25, 204 Overview Why he Reliabiliy Issue is Impora? Reliabiliy Daa Paer Some Well-kow Reliabiliy Measures Weibull ad logormal Probabiliy Plos Maximum Likelihood Esimaor 2 Wha is Reliabiliy?
More informationUnit - III RANDOM PROCESSES. B. Thilaka Applied Mathematics
Ui - III RANDOM PROCESSES B. Thilaka Applied Mahemaics Radom Processes A family of radom variables {X,s εt, sεs} defied over a give probabiliy space ad idexed by he parameer, where varies over he idex
More informationAuto-correlation of Error Terms
Auo-correlaio of Error Terms Pogsa Porchaiwiseskul Faculy of Ecoomics Chulalogkor Uiversiy (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy Geeral Auo-correlaio () YXβ + ν E(ν)0 V(ν)
More informationLecture 8 April 18, 2018
Sas 300C: Theory of Saisics Sprig 2018 Lecure 8 April 18, 2018 Prof Emmauel Cades Scribe: Emmauel Cades Oulie Ageda: Muliple Tesig Problems 1 Empirical Process Viewpoi of BHq 2 Empirical Process Viewpoi
More informationNotes 03 largely plagiarized by %khc
1 1 Discree-Time Covoluio Noes 03 largely plagiarized by %khc Le s begi our discussio of covoluio i discree-ime, sice life is somewha easier i ha domai. We sar wih a sigal x[] ha will be he ipu io our
More informationLecture 9: Polynomial Approximations
CS 70: Complexiy Theory /6/009 Lecure 9: Polyomial Approximaios Isrucor: Dieer va Melkebeek Scribe: Phil Rydzewski & Piramaayagam Arumuga Naiar Las ime, we proved ha o cosa deph circui ca evaluae he pariy
More informationFOR 496 / 796 Introduction to Dendrochronology. Lab exercise #4: Tree-ring Reconstruction of Precipitation
FOR 496 Iroducio o Dedrochroology Fall 004 FOR 496 / 796 Iroducio o Dedrochroology Lab exercise #4: Tree-rig Recosrucio of Precipiaio Adaped from a exercise developed by M.K. Cleavelad ad David W. Sahle,
More informationMean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs
America Joural of Compuaioal Mahemaics, 04, 4, 80-88 Published Olie Sepember 04 i SciRes. hp://www.scirp.org/joural/ajcm hp://dx.doi.org/0.436/ajcm.04.4404 Mea Square Coverge Fiie Differece Scheme for
More informationDevelopment of Kalman Filter and Analogs Schemes to Improve Numerical Weather Predictions
Developme of Kalma Filer ad Aalogs Schemes o Improve Numerical Weaher Predicios Luca Delle Moache *, Aimé Fourier, Yubao Liu, Gregory Roux, ad Thomas Warer (NCAR) Thomas Nipe, ad Rolad Sull (UBC) Wid Eergy
More informationA Bayesian Approach for Detecting Outliers in ARMA Time Series
WSEAS RASACS o MAEMAICS Guochao Zhag Qigmig Gui A Bayesia Approach for Deecig Ouliers i ARMA ime Series GUOC ZAG Isiue of Sciece Iformaio Egieerig Uiversiy 45 Zhegzhou CIA 94587@qqcom QIGMIG GUI Isiue
More information1 Notes on Little s Law (l = λw)
Copyrigh c 26 by Karl Sigma Noes o Lile s Law (l λw) We cosider here a famous ad very useful law i queueig heory called Lile s Law, also kow as l λw, which assers ha he ime average umber of cusomers i
More informationModeling Time Series of Counts
Modelig ime Series of Cous Richard A. Davis Colorado Sae Uiversiy William Dusmuir Uiversiy of New Souh Wales Sarah Sree Naioal Ceer for Amospheric Research (Oher collaboraors: Richard weedie, Yig Wag)
More informationSpecification of Dynamic Time Series Model with Volatile-Outlier Input Series
America Joural of Applied Scieces 8 (): 49-53, ISSN 546-939 Sciece Publicaios Specificaio of Dyamic ime Series Model wih Volaile-Oulier Ipu Series.A. Lasisi, D.K. Shagodoyi, O.O. Sagodoyi, W.M. hupeg ad
More informationCOS 522: Complexity Theory : Boaz Barak Handout 10: Parallel Repetition Lemma
COS 522: Complexiy Theory : Boaz Barak Hadou 0: Parallel Repeiio Lemma Readig: () A Parallel Repeiio Theorem / Ra Raz (available o his websie) (2) Parallel Repeiio: Simplificaios ad he No-Sigallig Case
More informationThe Moment Approximation of the First Passage Time for the Birth Death Diffusion Process with Immigraton to a Moving Linear Barrier
America Joural of Applied Mahemaics ad Saisics, 015, Vol. 3, No. 5, 184-189 Available olie a hp://pubs.sciepub.com/ajams/3/5/ Sciece ad Educaio Publishig DOI:10.1691/ajams-3-5- The Mome Approximaio of
More informationA Two-Level Quantum Analysis of ERP Data for Mock-Interrogation Trials. Michael Schillaci Jennifer Vendemia Robert Buzan Eric Green
A Two-Level Quaum Aalysis of ERP Daa for Mock-Ierrogaio Trials Michael Schillaci Jeifer Vedemia Rober Buza Eric Gree Oulie Experimeal Paradigm 4 Low Workload; Sigle Sessio; 39 8 High Workload; Muliple
More informationLocalization. MEM456/800 Localization: Bayes Filter. Week 4 Ani Hsieh
Localiaio MEM456/800 Localiaio: Baes Filer Where am I? Week 4 i Hsieh Evirome Sesors cuaors Sofware Ucerai is Everwhere Level of ucerai deeds o he alicaio How do we hadle ucerai? Eamle roblem Esimaig a
More informationActuarial Society of India
Acuarial Sociey of Idia EXAMINAIONS Jue 5 C4 (3) Models oal Marks - 5 Idicaive Soluio Q. (i) a) Le U deoe he process described by 3 ad V deoe he process described by 4. he 5 e 5 PU [ ] PV [ ] ( e ).538!
More informationSupplement for SADAGRAD: Strongly Adaptive Stochastic Gradient Methods"
Suppleme for SADAGRAD: Srogly Adapive Sochasic Gradie Mehods" Zaiyi Che * 1 Yi Xu * Ehog Che 1 iabao Yag 1. Proof of Proposiio 1 Proposiio 1. Le ɛ > 0 be fixed, H 0 γi, γ g, EF (w 1 ) F (w ) ɛ 0 ad ieraio
More informationODEs II, Supplement to Lectures 6 & 7: The Jordan Normal Form: Solving Autonomous, Homogeneous Linear Systems. April 2, 2003
ODEs II, Suppleme o Lecures 6 & 7: The Jorda Normal Form: Solvig Auoomous, Homogeeous Liear Sysems April 2, 23 I his oe, we describe he Jorda ormal form of a marix ad use i o solve a geeral homogeeous
More informationDiscrete-Time Signals and Systems. Introduction to Digital Signal Processing. Independent Variable. What is a Signal? What is a System?
Discree-Time Sigals ad Sysems Iroducio o Digial Sigal Processig Professor Deepa Kudur Uiversiy of Toroo Referece: Secios. -.4 of Joh G. Proakis ad Dimiris G. Maolakis, Digial Sigal Processig: Priciples,
More informationEstimation of the Mean and the ACVF
Chapter 5 Estimatio of the Mea ad the ACVF A statioary process {X t } is characterized by its mea ad its autocovariace fuctio γ ), ad so by the autocorrelatio fuctio ρ ) I this chapter we preset the estimators
More informationConvergence theorems. Chapter Sampling
Chaper Covergece heorems We ve already discussed he difficuly i defiig he probabiliy measure i erms of a experimeal frequecy measureme. The hear of he problem lies i he defiiio of he limi, ad his was se
More informationF.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mathematics
F.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mahemaics Prelim Quesio Paper Soluio Q. Aemp ay FIVE of he followig : [0] Q.(a) Defie Eve ad odd fucios. [] As.: A fucio f() is said o be eve fucio if
More informationThe Eigen Function of Linear Systems
1/25/211 The Eige Fucio of Liear Sysems.doc 1/7 The Eige Fucio of Liear Sysems Recall ha ha we ca express (expad) a ime-limied sigal wih a weighed summaio of basis fucios: v ( ) a ψ ( ) = where v ( ) =
More informationLECTURE DEFINITION
LECTURE 8 Radom Processes 8. DEFINITION A radom process (or sochasic process) is a ifiie idexed collecio of radom variables {X() : T}, defied over a commo probabiliy space. The idex parameer is ypically
More informationResearch Design - - Topic 2 Inferential Statistics: The t-test 2010 R.C. Gardner, Ph.D. Independent t-test
Research Desig - - Topic Ifereial aisics: The -es 00 R.C. Garer, Ph.D. Geeral Raioale Uerlyig he -es (Garer & Tremblay, 007, Ch. ) The Iepee -es The Correlae (paire) -es Effec ize a Power (Kirk, 995, pp
More informationCalculus BC 2015 Scoring Guidelines
AP Calculus BC 5 Scorig Guidelies 5 The College Board. College Board, Advaced Placeme Program, AP, AP Ceral, ad he acor logo are regisered rademarks of he College Board. AP Ceral is he official olie home
More informationELEG5693 Wireless Communications Propagation and Noise Part II
Deparme of Elecrical Egieerig Uiversiy of Arkasas ELEG5693 Wireless Commuicaios Propagaio ad Noise Par II Dr. Jigxia Wu wuj@uark.edu OUTLINE Wireless chael Pah loss Shadowig Small scale fadig Simulaio
More informationA note on deviation inequalities on {0, 1} n. by Julio Bernués*
A oe o deviaio iequaliies o {0, 1}. by Julio Berués* Deparameo de Maemáicas. Faculad de Ciecias Uiversidad de Zaragoza 50009-Zaragoza (Spai) I. Iroducio. Le f: (Ω, Σ, ) IR be a radom variable. Roughly
More informationUsing GLS to generate forecasts in regression models with auto-correlated disturbances with simulation and Palestinian market index data
America Joural of Theoreical ad Applied Saisics 04; 3(: 6-7 Published olie December 30, 03 (hp://www.sciecepublishiggroup.com//aas doi: 0.648/.aas.04030. Usig o geerae forecass i regressio models wih auo-correlaed
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 4 9/16/2013. Applications of the large deviation technique
MASSACHUSETTS ISTITUTE OF TECHOLOGY 6.265/5.070J Fall 203 Lecure 4 9/6/203 Applicaios of he large deviaio echique Coe.. Isurace problem 2. Queueig problem 3. Buffer overflow probabiliy Safey capial for
More informationA Note on Random k-sat for Moderately Growing k
A Noe o Radom k-sat for Moderaely Growig k Ju Liu LMIB ad School of Mahemaics ad Sysems Sciece, Beihag Uiversiy, Beijig, 100191, P.R. Chia juliu@smss.buaa.edu.c Zogsheg Gao LMIB ad School of Mahemaics
More informationHadamard matrices from the Multiplication Table of the Finite Fields
adamard marice from he Muliplicaio Table of he Fiie Field 신민호 송홍엽 노종선 * Iroducio adamard mari biary m-equece New Corucio Coe Theorem. Corucio wih caoical bai Theorem. Corucio wih ay bai Remark adamard
More informationSamuel Sindayigaya 1, Nyongesa L. Kennedy 2, Adu A.M. Wasike 3
Ieraioal Joural of Saisics ad Aalysis. ISSN 48-9959 Volume 6, Number (6, pp. -8 Research Idia Publicaios hp://www.ripublicaio.com The Populaio Mea ad is Variace i he Presece of Geocide for a Simple Birh-Deah-
More informationMaximum Likelihood Estimation for Allpass Time Series Models
Maximum Likelihood Esimaio or Allass Time Series Models Richard A. Davis Dearme o Saisics Colorado Sae Uiversiy h://www.sa.colosae.edu/~rdavis/lecures/magdeburg.d Joi work wih Jay Breid, Colorado Sae Uiversiy
More informationHYPOTHESIS TESTING. four steps
Irodcio o Saisics i Psychology PSY 20 Professor Greg Fracis Lecre 24 Correlaios ad proporios Ca yo read my mid? Par II HYPOTHESIS TESTING for seps. Sae he hypohesis. 2. Se he crierio for rejecig H 0. 3.
More informationChapter 9 Autocorrelation
Chaper 9 Aocorrelaio Oe of he basic assmpios i liear regressio model is ha he radom error compoes or disrbaces are ideically ad idepedely disribed So i he model y = Xβ +, i is assmed ha σ if s = E (, s)
More informationChapter 11 Autocorrelation
Chaper Aocorrelaio Oe of he basic assmpio i liear regressio model is ha he radom error compoes or disrbaces are ideically ad idepedely disribed So i he model y = Xβ +, i is assmed ha σ if s = E (, s) =
More informationOnline Supplement to Reactive Tabu Search in a Team-Learning Problem
Olie Suppleme o Reacive abu Search i a eam-learig Problem Yueli She School of Ieraioal Busiess Admiisraio, Shaghai Uiversiy of Fiace ad Ecoomics, Shaghai 00433, People s Republic of Chia, she.yueli@mail.shufe.edu.c
More informationChapter Chapter 10 Two-Sample Tests X 1 X 2. Difference Between Two Means: Different data sources Unrelated. Learning Objectives
Chaper 0 0- Learig Objecives I his chaper, you lear how o use hypohesis esig for comparig he differece bewee: Chaper 0 Two-ample Tess The meas of wo idepede populaios The meas of wo relaed populaios The
More informationPrinciples of Communications Lecture 1: Signals and Systems. Chih-Wei Liu 劉志尉 National Chiao Tung University
Priciples of Commuicaios Lecure : Sigals ad Sysems Chih-Wei Liu 劉志尉 Naioal Chiao ug Uiversiy cwliu@wis.ee.cu.edu.w Oulies Sigal Models & Classificaios Sigal Space & Orhogoal Basis Fourier Series &rasform
More informationApplying the Moment Generating Functions to the Study of Probability Distributions
3 Iformaica Ecoomică, r (4)/007 Applyi he Mome Geerai Fucios o he Sudy of Probabiliy Disribuios Silvia SPĂTARU Academy of Ecoomic Sudies, Buchares I his paper, we describe a ool o aid i provi heorems abou
More informationN! AND THE GAMMA FUNCTION
N! AND THE GAMMA FUNCTION Cosider he produc of he firs posiive iegers- 3 4 5 6 (-) =! Oe calls his produc he facorial ad has ha produc of he firs five iegers equals 5!=0. Direcly relaed o he discree! fucio
More informationOBJECTIVES OF TIME SERIES ANALYSIS
OBJECTIVES OF TIME SERIES ANALYSIS Undersanding he dynamic or imedependen srucure of he observaions of a single series (univariae analysis) Forecasing of fuure observaions Asceraining he leading, lagging
More informationSemiparametric and Nonparametric Methods in Political Science Lecture 1: Semiparametric Estimation
Semiparameric ad Noparameric Mehods i Poliical Sciece Lecure : Semiparameric Esimaio Michael Peress, Uiversiy of Rocheser ad Yale Uiversiy Lecure : Semiparameric Mehods Page 2 Overview of Semi ad Noparameric
More informationStatistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons
Statistical Aalysis o Ucertaity for Autocorrelated Measuremets ad its Applicatios to Key Comparisos Nie Fa Zhag Natioal Istitute of Stadards ad Techology Gaithersburg, MD 0899, USA Outlies. Itroductio.
More informationEstimation for Parameter-Driven State-Space Space Models:
Esimaio for Parameer-Drive Sae-Sace Sace Models: Richard A. Davis ad Gabriel Rodriguez-Yam Colorado Sae Uiversiy h://www.sa.colosae.edu/~rdavis/lecures Joi work wih: William Dusmuir Uiversiy of New Souh
More informationDavid Randall. ( )e ikx. k = u x,t. u( x,t)e ikx dx L. x L /2. Recall that the proof of (1) and (2) involves use of the orthogonality condition.
! Revised April 21, 2010 1:27 P! 1 Fourier Series David Radall Assume ha u( x,) is real ad iegrable If he domai is periodic, wih period L, we ca express u( x,) exacly by a Fourier series expasio: ( ) =
More informationA TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY
U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 2, 206 ISSN 223-7027 A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY İbrahim Çaak I his paper we obai a Tauberia codiio i erms of he weighed classical
More information2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)
Mah PracTes Be sure o review Lab (ad all labs) There are los of good quesios o i a) Sae he Mea Value Theorem ad draw a graph ha illusraes b) Name a impora heorem where he Mea Value Theorem was used i he
More informationEGR 544 Communication Theory
EGR 544 Commuicaio heory 7. Represeaio of Digially Modulaed Sigals II Z. Aliyazicioglu Elecrical ad Compuer Egieerig Deparme Cal Poly Pomoa Represeaio of Digial Modulaio wih Memory Liear Digial Modulaio
More informationExtended Laguerre Polynomials
I J Coemp Mah Scieces, Vol 7, 1, o, 189 194 Exeded Laguerre Polyomials Ada Kha Naioal College of Busiess Admiisraio ad Ecoomics Gulberg-III, Lahore, Pakisa adakhaariq@gmailcom G M Habibullah Naioal College
More informationMath 6710, Fall 2016 Final Exam Solutions
Mah 67, Fall 6 Fial Exam Soluios. Firs, a sude poied ou a suble hig: if P (X i p >, he X + + X (X + + X / ( evaluaes o / wih probabiliy p >. This is roublesome because a radom variable is supposed o be
More information6.003: Signals and Systems
6.003: Sigals ad Sysems Lecure 8 March 2, 2010 6.003: Sigals ad Sysems Mid-erm Examiaio #1 Tomorrow, Wedesday, March 3, 7:30-9:30pm. No reciaios omorrow. Coverage: Represeaios of CT ad DT Sysems Lecures
More informationThe Splice Bootstrap
1 The Splice Boosrap Gerard Keogh Triiy College Dubli. Absrac: This paper proposes a ew boosrap mehod o compue predicive iervals for oliear auoregressive ime series model forecas. This mehod we call he
More informationCSE 241 Algorithms and Data Structures 10/14/2015. Skip Lists
CSE 41 Algorihms ad Daa Srucures 10/14/015 Skip Liss This hadou gives he skip lis mehods ha we discussed i class. A skip lis is a ordered, doublyliked lis wih some exra poiers ha allow us o jump over muliple
More informationK3 p K2 p Kp 0 p 2 p 3 p
Mah 80-00 Mo Ar 0 Chaer 9 Fourier Series ad alicaios o differeial equaios (ad arial differeial equaios) 9.-9. Fourier series defiiio ad covergece. The idea of Fourier series is relaed o he liear algebra
More informationt = s D Overview of Tests Two-Sample t-test: Independent Samples Independent Samples t-test Difference between Means in a Two-sample Experiment
Overview of Te Two-Sample -Te: Idepede Sample Chaper 4 z-te Oe Sample -Te Relaed Sample -Te Idepede Sample -Te Compare oe ample o a populaio Compare wo ample Differece bewee Mea i a Two-ample Experime
More informationStatistical Estimation
Learig Objecives Cofidece Levels, Iervals ad T-es Kow he differece bewee poi ad ierval esimaio. Esimae a populaio mea from a sample mea f large sample sizes. Esimae a populaio mea from a sample mea f small
More informationApplication of Intelligent Systems and Econometric Models for Exchange Rate Prediction
0 Ieraioal Coferece o Iovaio, Maageme ad Service IPEDR vol.4(0) (0) IACSIT Press, Sigapore Applicaio of Iellige Sysems ad Ecoomeric Models for Exchage Rae Predicio Abu Hassa Shaari Md Nor, Behrooz Gharleghi
More information