Visualising the spectral analysis of time series
|
|
|
- Imogene Nicholson
- 6 years ago
- Views:
Transcription
1 Visualising he specral analysis of ime series Adam M. Sykulski Marie Curie Research Fellow NorhWes Research Associaes (NWRA), Seale, USA & Universiy College London (UCL), UK Slides available online a:
2 Noaion x() : coninuous real-valued saionary process, 2 R x : discree real-valued saionary process, 2 Z : angular frequency, =2 f (f is measured in herz) : ime-lag (posiive or negaive) i = p 1
3 Jus a few equaions Fourier Transform: f x () = R 11 x()e i d, 2 R Inverse Fourier Transform: x() = 1 2 R 1 1 f x()e i d, 2 R Power Specral Densiy: S x () =lim T 1 E 1 2T R T T x()e i d 2 Relaionship wih auocovariance sequence s x ( ) =E[x()x( )]: S x () = R 1 1 s x( )e i d () s x ( ) = 1 2 R 1 1 S x()e i d Percival and Walden, page 65
4 x() =sin(.1) 2 ime series 6 power specral densiy x() S x () (db) db = 1 log 1
5 x() = sin(.1)+.6 sin(.13) 2 ime series 6 power specral densiy x() S x () (db)
6 x() = sin(.1)+.6 sin(.13)+.2 sin(.18) 2 ime series 6 power specral densiy x() S x () (db)
7 x = ", " N (, 1) x ime series S x () (db) power specral densiy
8 x =.9x 1 + ", " N (, 1) x ime series S x () (db) power specral densiy
9 x = ".7" 1, " N (, 1) x ime series S x () (db) power specral densiy
10 Longiude Oceanographic roary specra db " Laiude Complex-valued ime series: z = x + iy 33 32
11 Longiude Oceanographic roary specra db " Laiude Complex-valued ime series: z = x + iy 33 32
12 Solomon Islands 1991 Earhquake X # Y # Y # X # ime (UTC) Complex-valued ime series: z = x + iy
13 ime series ime series 1 1 Y # 1 3 Y # X # X # 1 3 db roary specra : -:/2 :/2 : db roary specra : -:/2 :/2 :
14 Specral Densiy Esimaion Observe: X 1,...,X N from he process x() or x Discree Fourier Transform: J X () = 1 p N P N =1 X e i, 2 R Periodogram: Ŝ X () = J x () 2, 2 R Fourier Frequencies: 2 2 N, 1,...,b N 2 c Percival and Walden, page 196
15 x =.9x 1 + ", " N (, 1) S x () = 2 " cos()+ 2 1 x ime series S x () (db) power specral densiy Periodogram Specral Densiy
16 More lives have been los looking a he raw periodogram han by any oher acion involving ime series John W. Tukey
17 Why? Warning 1: Beware of he Decibels E[1 log 1 {ŜX()}] 1 log 1 S x () = 1 log 1 (e) = (Euler-Mascheroni consan) 2.5 Percival and Walden, page 28
18 x =.9x 1 + ", " N (, 1) S x () = 2 " cos() ime series 3 2 power specral densiy Periodogram Specral Densiy SD (db bias adjused) x 2-2 S x () (db)
19 x =2.767x x x x 4 + ", " N (, 1) S x () = 1 2 " P p k=1 ke i 2 1 ime series 5 4 power specral densiy Periodogram Specral Densiy 5 3 x S x () (db)
20 Wha else? Warning 2: Sampling isn forever Theory: S x () =lim T 1 E 1 2T R T T x()e i d 2 Pracice: Ŝ X () = P N =1 X e i 2 o Convoluion: E nŝx () = R F( )S x ( )d where F( ) ishefejérkernel:f() = 1 2 N sin 2 (N/2) sin 2 (/2) Percival and Walden, page 198
21 x =2.767x x x x 4 + ", " N (, 1) S x () = 1 2 " P p k=1 ke i 2 F() (db) Fejer kernel (N=16) S x () (db) power specral densiy Periodogram Specral Densiy
22 x =2.767x x x x 4 + ", " N (, 1) S x () = 1 2 " P p k=1 ke i 2 F() (db) Fejer kernel (N=1) S x () (db) power specral densiy Periodogram Specral Densiy
23 Poenial Soluion: Tapering o Reminder: E nŝx () = R F( )S x ( )d Propose a daa aper h, such ha we work wih h X New specral esimae: Ŝ (d) X () = P N =1 h X e i 2 Define: H() = P N =1 h e i 2 Then: E nŝ(d) o X () = R H( )S x ( )d Require: R H( )d = P N =1 h2 =1 Percival and Walden, page 26
24 .3 no/recangular aper (N=64) 2 power specral densiy.25 h H() (db)
25 .3 dpss(1) aper (N=64) 2 power specral densiy.25 h H() (db)
26 .3 dpss(2) aper (N=64) 2 power specral densiy.25 h H() (db)
27 .3 dpss(4) aper (N=64) 2 power specral densiy.25 h H() (db)
28 .3 dpss(8) aper (N=64) 2 power specral densiy.25 h H() (db)
29 x =2.767x x x x 4 + ", " N (, 1) 2 1 Fejer kernel (N=1) 5 4 power specral densiy Periodogram Specral Densiy F() (db) S x () (db)
30 x =2.767x x x x 4 + ", " N (, 1) 2 1 dpss(1) kernel (N=1) 5 4 power specral densiy dpss(1) esimae Specral Densiy H() (db) S x () (db)
31 x =2.767x x x x 4 + ", " N (, 1) 2 1 dpss(2) kernel (N=1) 5 4 power specral densiy dpss(2) esimae Specral Densiy H() (db) S x () (db)
32 5 4 3 power specral densiy (N=128) Periodogram Specral Densiy S x () (db)
33 5 4 3 power specral densiy (N=128) dpss(1) esimae Specral Densiy S x () (db)
34 5 4 3 power specral densiy (N=128) dpss(2) esimae Specral Densiy S x () (db)
35 5 4 3 power specral densiy (N=128) dpss(4) esimae Specral Densiy S x () (db)
36 5 4 3 power specral densiy (N=128) dpss(8) esimae Specral Densiy S x () (db)
37 X() Maérn(A =1, =.6,h=.1) S x () = A 2 ( 2 + h 2 ) ime series 3 2 power specral densiy periodogram Specral Densiy x 2-2 S x () (db)
38 X() Maérn(A =1, =.6,h=.1) S x () = A 2 ( 2 + h 2 ) ime series 3 2 power specral densiy dpss(2) aper Specral Densiy x 2-2 S x () (db)
39 X() Maérn(A =1, =.6,h=.1) S x () = A 2 ( 2 + h 2 ) ime series 3 2 power specral densiy dpss(8) aper Specral Densiy x 2-2 S x () (db)
40 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x.5 x Percival and Walden, page 97
41 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
42 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
43 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
44 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
45 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
46 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
47 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
48 In 2-D he aliasing problem is also known as he wagon-wheel effec Link o Video
49 X() Maérn(A =1, =.6,h=.1) 1X A 2 S x () = (( +2 k) 2 + h 2 ) k= ime series 3 2 power specral densiy periodogram Specral Densiy Aliased Specral Densiy x 2-2 S x () (db)
50 Summary Specral analysis of ime series can reveal hidden srucure in a ime series Care mus be aken when using periodograms log-scale: adjus for Euler-Mascheroni Consan shor N or high dynamic range: aper coninuous or subsampled: accoun for aliasing
Signal processing. A. Sestieri Dipartimento di Meccanica e Aeronautica University La Sapienza, Rome
Signal processing A. Sesieri Diparimeno di Meccanica e Aeronauica Universiy La Sapienza, Rome Presenaion layou - Fourier series and Fourier ransforms - Leakage - Aliasing - Analog versus digial signals
Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
COMPARISON OF THE DIFFERENCING PARAMETER ESTIMATION FROM ARFIMA MODEL BY SPECTRAL REGRESSION METHODS. By Gumgum Darmawan, Nur Iriawan, Suhartono
COMPARISON OF THE DIFFERENCING PARAMETER ESTIMATION FROM ARFIMA MODEL BY SPECTRAL REGRESSION METHODS By Gumgum Darmawan, Nur Iriawan, Suharono INTRODUCTION (1) TIME SERIES MODELS BASED ON VALUE OF DIFFERENCING
Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
Stochastic Structural Dynamics. Lecture-6
Sochasic Srucural Dynamics Lecure-6 Random processes- Dr C S Manohar Deparmen of Civil Engineering Professor of Srucural Engineering Indian Insiue of Science Bangalore 560 0 India manohar@civil.iisc.erne.in
CHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
Lecture 9: Advanced DFT concepts: The Exchange-correlation functional and time-dependent DFT
Lecure 9: Advanced DFT conceps: The Exchange-correlaion funcional and ime-dependen DFT Marie Curie Tuorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dep. of Chemisry and Couran Insiue
2.160 System Identification, Estimation, and Learning. Lecture Notes No. 8. March 6, 2006
2.160 Sysem Idenificaion, Esimaion, and Learning Lecure Noes No. 8 March 6, 2006 4.9 Eended Kalman Filer In many pracical problems, he process dynamics are nonlinear. w Process Dynamics v y u Model (Linearized)
6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
Solutions to the Exam Digital Communications I given on the 11th of June = 111 and g 2. c 2
Soluions o he Exam Digial Communicaions I given on he 11h of June 2007 Quesion 1 (14p) a) (2p) If X and Y are independen Gaussian variables, hen E [ XY ]=0 always. (Answer wih RUE or FALSE) ANSWER: False.
Reliability of Technical Systems
eliabiliy of Technical Sysems Main Topics Inroducion, Key erms, framing he problem eliabiliy parameers: Failure ae, Failure Probabiliy, Availabiliy, ec. Some imporan reliabiliy disribuions Componen reliabiliy
Figure 18: Top row: example of a purely continuous spectrum (left) and one realization
1..5 S(). -.2 -.5 -.25..25.5 64 128 64 128 16 32 requency time time Lag 1..5 S(). -.5-1. -.5 -.1.1.5 64 128 64 128 16 32 requency time time Lag Figure 18: Top row: example o a purely continuous spectrum
Anno accademico 2006/2007. Davide Migliore
Roboica Anno accademico 2006/2007 Davide Migliore migliore@ele.polimi.i Today Eercise session: An Off-side roblem Robo Vision Task Measuring NBA layers erformance robabilisic Roboics Inroducion The Bayesian
( ) = b n ( t) n " (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2.
Andrei Tokmakoff, MIT Deparmen of Chemisry, 3/14/007-6.4 PERTURBATION THEORY Given a Hamilonian H = H 0 + V where we know he eigenkes for H 0 : H 0 n = E n n, we can calculae he evoluion of he wavefuncion
Second quantization and gauge invariance.
1 Second quanizaion and gauge invariance. Dan Solomon Rauland-Borg Corporaion Moun Prospec, IL Email: dsolom@uic.edu June, 1. Absrac. I is well known ha he single paricle Dirac equaion is gauge invarian.
Sensors, Signals and Noise
Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1 Noise Descripion Noise Waveforms and Samples Saisics of Noise
GMM - Generalized Method of Moments
GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................
Kalman filtering for maximum likelihood estimation given corrupted observations.
alman filering maimum likelihood esimaion given corruped observaions... Holmes Naional Marine isheries Service Inroducion he alman filer is used o eend likelihood esimaion o cases wih hidden saes such
IB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
Phys1112: DC and RC circuits
Name: Group Members: Dae: TA s Name: Phys1112: DC and RC circuis Objecives: 1. To undersand curren and volage characerisics of a DC RC discharging circui. 2. To undersand he effec of he RC ime consan.
Direct Current Circuits. February 19, 2014 Physics for Scientists & Engineers 2, Chapter 26 1
Direc Curren Circuis February 19, 2014 Physics for Scieniss & Engineers 2, Chaper 26 1 Ammeers and Volmeers! A device used o measure curren is called an ammeer! A device used o measure poenial difference
1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
KEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
6.003: Signal Processing
6.003: Signal Processing Coninuous-Time Fourier Transform Definiion Examples Properies Relaion o Fourier Series Sepember 5, 08 Quiz Thursday, Ocober 4, from 3pm o 5pm. No lecure on Ocober 4. The exam is
Differential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
Dynamic Effects of Feedback Control!
Dynamic Effecs of Feedback Conrol! Rober Sengel! Roboics and Inelligen Sysems MAE 345, Princeon Universiy, 2017 Inner, Middle, and Ouer Feedback Conrol Loops Sep Response of Linear, Time- Invarian (LTI)
Lecture 15. Dummy variables, continued
Lecure 15. Dummy variables, coninued Seasonal effecs in ime series Consider relaion beween elecriciy consumpion Y and elecriciy price X. The daa are quarerly ime series. Firs model ln α 1 + α2 Y = ln X
How to Deal with Structural Breaks in Practical Cointegration Analysis
How o Deal wih Srucural Breaks in Pracical Coinegraion Analysis Roselyne Joyeux * School of Economic and Financial Sudies Macquarie Universiy December 00 ABSTRACT In his noe we consider he reamen of srucural
Ordinary Differential Equations
Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described
Tracking. Announcements
Tracking Tuesday, Nov 24 Krisen Grauman UT Ausin Announcemens Pse 5 ou onigh, due 12/4 Shorer assignmen Auo exension il 12/8 I will no hold office hours omorrow 5 6 pm due o Thanksgiving 1 Las ime: Moion
III-A. Fourier Series Expansion
Summer 28 Signals & Sysems S.F. Hsieh III-A. Fourier Series Expansion Inroducion. Divide and conquer Signals can be decomposed as linear combinaions of: (a) shifed impulses: (sifing propery) Why? x() x()δ(
Q1) [20 points] answer for the following questions (ON THIS SHEET):
![Q1) [20 points] answer for the following questions (ON THIS SHEET):](/thumbs/87/96063197.jpg "Q1) [20 points] answer for the following questions (ON THIS SHEET):")
Dr. Anas Al Tarabsheh The Hashemie Universiy Elecrical and Compuer Engineering Deparmen (Makeup Exam) Signals and Sysems Firs Semeser 011/01 Final Exam Dae: 1/06/01 Exam Duraion: hours Noe: means convoluion
On a Discrete-In-Time Order Level Inventory Model for Items with Random Deterioration
Journal of Agriculure and Life Sciences Vol., No. ; June 4 On a Discree-In-Time Order Level Invenory Model for Iems wih Random Deerioraion Dr Biswaranjan Mandal Associae Professor of Mahemaics Acharya
di Bernardo, M. (1995). A purely adaptive controller to synchronize and control chaotic systems.
di ernardo, M. (995). A purely adapive conroller o synchronize and conrol chaoic sysems. hps://doi.org/.6/375-96(96)8-x Early version, also known as pre-prin Link o published version (if available):.6/375-96(96)8-x
Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.
Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since
Chapter 4 The Fourier Series and Fourier Transform
Represenaion of Signals in Terms of Frequency Componens Chaper 4 The Fourier Series and Fourier Transform Consider he CT signal defined by x () = Acos( ω + θ ), = The frequencies `presen in he signal are
OBJECTIVES 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
Mobile Ion Effects on SiC MOS Bias- Temperature Instability Measurements
14-15 Aug 2014 1 U.S. Army Research, Developmen and Engineering Command Mobile Ion Effecs on SiC MOS Bias- Temperaure Insabiliy Measuremens Daniel B. Habersa Neil Goldsman (UMD), and Aivars Lelis 14-15
Description of the MS-Regress R package (Rmetrics)
Descriion of he MS-Regress R ackage (Rmerics) Auhor: Marcelo Perlin PhD Suden / ICMA Reading Universiy Email: marceloerlin@gmail.com / m.erlin@icmacenre.ac.uk The urose of his documen is o show he general
Unit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
EECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are Coninuous-Time Signals??? Reading Assignmen: Secion. of Kamen and Heck /22 Course Flow Diagram The arrows here show concepual flow beween ideas.
Power of Random Processes 1/40
Power of Random Processes 40 Power of a Random Process Recall : For deerminisic signals insananeous power is For a random signal, is a random variable for each ime. hus here is no single # o associae wih
WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
Γ(h)=0 h 0. Γ(h)=cov(X 0,X 0-h ). A stationary process is called white noise if its autocovariance
A family, Z,of random vecors : Ω R k defined on a probabiliy space Ω, A,P) is called a saionary process if he mean vecors E E =E M = M k E and he auocovariance marices are independen of. k cov, -h )=E
Traveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
Review - Quiz # 1. 1 g(y) dy = f(x) dx. y x. = u, so that y = xu and dy. dx (Sometimes you may want to use the substitution x y
Review - Quiz # 1 (1) Solving Special Tpes of Firs Order Equaions I. Separable Equaions (SE). d = f() g() Mehod of Soluion : 1 g() d = f() (The soluions ma be given implicil b he above formula. Remember,
Random Processes 1/24
Random Processes 1/24 Random Process Oher Names : Random Signal Sochasic Process A Random Process is an exension of he concep of a Random variable (RV) Simples View : A Random Process is a RV ha is a Funcion
Properties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
Estimation of Poses with Particle Filters
Esimaion of Poses wih Paricle Filers Dr.-Ing. Bernd Ludwig Chair for Arificial Inelligence Deparmen of Compuer Science Friedrich-Alexander-Universiä Erlangen-Nürnberg 12/05/2008 Dr.-Ing. Bernd Ludwig (FAU
Two Popular Bayesian Estimators: Particle and Kalman Filters. McGill COMP 765 Sept 14 th, 2017
Two Popular Bayesian Esimaors: Paricle and Kalman Filers McGill COMP 765 Sep 14 h, 2017 1 1 1, dx x Bel x u x P x z P Recall: Bayes Filers,,,,,,, 1 1 1 1 u z u x P u z u x z P Bayes z = observaion u =
Chapter 5. Heterocedastic Models. Introduction to time series (2008) 1
Chaper 5 Heerocedasic Models Inroducion o ime series (2008) 1 Chaper 5. Conens. 5.1. The ARCH model. 5.2. The GARCH model. 5.3. The exponenial GARCH model. 5.4. The CHARMA model. 5.5. Random coefficien
12: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME. Σ j =
1: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME Moving Averages Recall ha a whie noise process is a series { } = having variance σ. The whie noise process has specral densiy f (λ) = of
23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé
Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070
ES 250 Practice Final Exam
ES 50 Pracice Final Exam. Given ha v 8 V, a Deermine he values of v o : 0 Ω, v o. V 0 Firs, v o 8. V 0 + 0 Nex, 8 40 40 0 40 0 400 400 ib i 0 40 + 40 + 40 40 40 + + ( ) 480 + 5 + 40 + 8 400 400( 0) 000
STAD57 Time Series Analysis. Lecture 5
STAD57 Time Series Analysis Lecure 5 1 Exploraory Daa Analysis Check if given TS is saionary: µ is consan σ 2 is consan γ(s,) is funcion of h= s If no, ry o make i saionary using some of he mehods below:
KINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
Math 10C: Relations and Functions PRACTICE EXAM
Mah C: Relaions and Funcions PRACTICE EXAM. Cailin rides her bike o school every day. The able of values shows her disance from home as ime passes. An equaion ha describes he daa is: ime (minues) disance
13.3 Term structure models
13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)
Y 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
Institute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler
MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion
Linear Time-invariant systems, Convolution, and Cross-correlation
Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An
Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis
Inernaional Economeric Review (IER) Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis Muhammad Irfan Malik and Aiq-ur-Rehman Inernaional Islamic Universiy Islamabad and Inernaional
Vector autoregression VAR. Case 1
Vecor auoregression VAR So far we have focused mosl on models where deends onl on as. More generall we migh wan o consider oin models ha involve more han one variable. There are wo reasons: Firs, we migh
System of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
An Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
Laplace Transforms. Examples. Is this equation differential? y 2 2y + 1 = 0, y 2 2y + 1 = 0, (y ) 2 2y + 1 = cos x,
Laplace Transforms Definiion. An ordinary differenial equaion is an equaion ha conains one or several derivaives of an unknown funcion which we call y and which we wan o deermine from he equaion. The equaion
Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
Guest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
A Dynamic Model of Economic Fluctuations
CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model
EXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
Chapter 3: Signal Transmission and Filtering. A. Bruce Carlson Paul B. Crilly 2010 The McGraw-Hill Companies
Communicaion Sysems, 5e Chaper 3: Signal Transmission and Filering A. Bruce Carlson Paul B. Crilly 00 The McGraw-Hill Companies Chaper 3: Signal Transmission and Filering Response of LTI sysems Signal
(10) (a) Derive and plot the spectrum of y. Discuss how the seasonality in the process is evident in spectrum.
January 01 Final Exam Quesions: Mark W. Wason (Poins/Minues are given in Parenheses) (15) 1. Suppose ha y follows he saionary AR(1) process y = y 1 +, where = 0.5 and ~ iid(0,1). Le x = (y + y 1 )/. (11)
Solutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
Licenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A
Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In
Reading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
Radical Expressions. Terminology: A radical will have the following; a radical sign, a radicand, and an index.
Radical Epressions Wha are Radical Epressions? A radical epression is an algebraic epression ha conains a radical. The following are eamples of radical epressions + a Terminology: A radical will have he
THE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant).
THE WAVE EQUATION 43. (S) Le u(x, ) be a soluion of he wave equaion u u xx = 0. Show ha Q43(a) (c) is a. Any ranslaion v(x, ) = u(x + x 0, + 0 ) of u(x, ) is also a soluion (where x 0, 0 are consans).
Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims
Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,
( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
Exponential Smoothing
Exponenial moohing Inroducion A simple mehod for forecasing. Does no require long series. Enables o decompose he series ino a rend and seasonal effecs. Paricularly useful mehod when here is a need o forecas
Chapter Q1. We need to understand Classical wave first. 3/28/2004 H133 Spring
Chaper Q1 Inroducion o Quanum Mechanics End of 19 h Cenury only a few loose ends o wrap up. Led o Relaiviy which you learned abou las quarer Led o Quanum Mechanics (1920 s-30 s and beyond) Behavior of
On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature
On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check
Schrödinger problem and W 2 -geodesics
Schrödinger problem and W 2 -geodesics join work wih Nicola Gigli Luca Tamanini Universié Paris Nanerre SISSA En l honneur de P. Caiaux e C. Léonard Toulouse, le 7 juin 2017 L. Tamanini (Paris Nanerre
Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number
5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
Receivers, Antennas, and Signals. Professor David H. Staelin Fall 2001 Slide 1
Receivers, Anennas, and Signals Professor David H. Saelin 6.66 Fall 00 Slide A Subjec Conen A Human Processor Transducer Radio Opical, Infrared Acousic, oher Elecromagneic Environmen B C Human Processor
THE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI
THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach
Stability. Coefficients may change over time. Evolution of the economy Policy changes
Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable,
Signals and Systems Linear Time-Invariant (LTI) Systems
Signals and Sysems Linear Time-Invarian (LTI) Sysems Chang-Su Kim Discree-Time LTI Sysems Represening Signals in Terms of Impulses Sifing propery 0 x[ n] x[ k] [ n k] k x[ 2] [ n 2] x[ 1] [ n1] x[0] [
Solutions: Wednesday, November 14
Amhers College Deparmen of Economics Economics 360 Fall 2012 Soluions: Wednesday, November 14 Judicial Daa: Cross secion daa of judicial and economic saisics for he fify saes in 2000. JudExp CrimesAll
5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
Wavelet Methods for Time Series Analysis. What is a Wavelet? Part I: Introduction to Wavelets and Wavelet Transforms. sines & cosines are big waves
Wavele Mehods for Time Series Analysis Par I: Inroducion o Waveles and Wavele Transforms waveles are analysis ools for ime series and images as a subjec, waveles are relaively new (983 o presen) a synhesis
Ordinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
Cash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
Sample Autocorrelations for Financial Time Series Models. Richard A. Davis Colorado State University Thomas Mikosch University of Copenhagen
Sample Auocorrelaions for Financial Time Series Models Richard A. Davis Colorado Sae Universiy Thomas Mikosch Universiy of Copenhagen Ouline Characerisics of some financial ime series IBM reurns NZ-USA
CH Sean Han QF, NTHU, Taiwan BFS2010. (Joint work with T.-Y. Chen and W.-H. Liu)
CH Sean Han QF, NTHU, Taiwan BFS2010 (Join work wih T.-Y. Chen and W.-H. Liu) Risk Managemen in Pracice: Value a Risk (VaR) / Condiional Value a Risk (CVaR) Volailiy Esimaion: Correced Fourier Transform