Pure Random process Pure Random Process or White Noise Process: is a random process {X t, t 0} which has: { σ 2 if k = 0 0 if k 0
|
|
- Nathan Tucker
- 5 years ago
- Views:
Transcription
1 MODULE 9: STATIONARY PROCESSES 7 Lecture 2 Autoregressive Processes 1 Moving Average Process Pure Random process Pure Random Process or White Noise Process: is a random process X t, t 0} which has: E[X t ] = m(constant), i.e., the mean is independent of t. E[X t X t+k ] = σ 2 if k = 0 0 if k 0 It can be verified that the above process is a covariance stationary process. Moving Average Process A Moving Average Process X(t), t 0} is a stochastic process satisfying certain properties. It is represented as: X t = a 0 e t + a 1 e t a h e t h where, a is are real constants and e k is a pure random process with mean 0 and variance σ 2. This collection X(t), t 0} is known as a moving average process. When the constant a h 0 then the above process is called a moving average( MA in short) of order h. Now let C k = E[X t X t+k ] be the correlation coefficient, then (a 0 a k + a 1 a k a h k a h )σ 2 if k h C k = 0 if k > h
2 MODULE 9: STATIONARY PROCESSES 8 Now we define the covariance coefficient ρ k, ρ k = C k = C 0 a0 a k +a 1 a k a h k a h a 2 0 +a a 2 h, k h 0, k > h. It can be verified that the moving average process is a covariance stationary or wide sense stationary process. First Order Markov Process One very important Moving Average Process is the first order Markov process which is defined as: X t + a t X t 1 = e t, a t < 1 where, e t, t T } is a pure random process with mean 0 and variance 1. Now X t can be written as X t = Σ k=0 ρ ke t k where, ρ k = a k for all k. Thus the Markov process of first order can be related to a Moving Average Process of infinite order. 2 Autoregressive Process Autoregressive process(ar) A stochastic process X t, t T } of the form: X t + b 1 X t 1 + b 2 X t b h X t h = e t where, e t is a pure random process with mean 0 and b is are real constants with b h 0, then the corresponding stochastic process is an AR process of order h. AR process of infinite order: When X t = r=0 b re t r, then the process X t, t T } is an AR process of infinite order. Now we consider a special case of Autoregressive processes i.e. an Autoregressive
3 MODULE 9: STATIONARY PROCESSES 9 process of order 2. An Autoregressive process of order 2 will be of the form: X t + b 1 X t 1 + b 2 X t 2 = e t This process is known as the Yule Process. Autoregressive Moving Average Process A stochastic process is called an Autoregressive Moving Average Process(ARMA) if it is of the form: p r=0 b rx t s = q s=0 a se t s where, e t, t T } is a pure random process with mean 0 and b is are real constants with b 0 = 1. The above process is called an ARMA process of order(p, q). If p and q are then it will be called ARMA process of order infinity. Power Spectrum The autoregressive and moving average process are useful in the study of power spectrum. For a wide sense stationary process we define: Covariance function : C k = E[]X t X t+k E[X t ]E[X t+k ] Correlation function: ρ k = = C k V ar(xt ) V ar(x t+k ) π π e ikw df 1 (w) where, F 1 (w) is cdf called integrated spectrum and whenever it s absolutely continuous then the derivative can be written in terms of the density function. Inverse Representation of ρ k : f 1 (w) = 1 k= ρ ke ikw i.e., knowing the correlation coefficient we can determine the density function.
4 MODULE 9: STATIONARY PROCESSES 10 Representation of C k : C k = eikw df ( w) where, df (w) = C 0 df 1 (w) and whenever F (w) is an absolutely continuous function then the inverse representation of C k can be given: f(w) = 1 k= C ke ikw. Hence, f(w) is called the spectral density function and f 1 (w) is called the normalized spectral density function. The advantage of studying power spectrum is that when we have a stochastic process X(t), t T } having the wide sense stationary property in time domain, the entities C k, ρ k can be found but these are not simple to evaluate. On the other hand, in frequency domain, for the same stochastic process X(t), t T } we have the spectral density and the normalized spectral density function and using the inverse relations we can find C k and ρ k. This is called the spectrum study. EXAMPLE 1. Consider the white noise process, X(t), t T }. Assume E[X t ]=0 and E[X r X s ] = σ 2 if r = s 0 if r s Now we try to find few entities in time domain: C k =0, for k 0 C 0 = σ 2 ρ k =0, for k 0 ρ 0 = C 0 σ 2 = 1 Now we study the same in frequency domain: Spectral density: f(w) = 1 F (w) = k= C ke ikw = σ2, for π w π f(w)dw = σ 2 w, π w π 0, otherwise We can also evaluate the derivative: df (w) = σ2 dw
5 MODULE 9: STATIONARY PROCESSES 11 Now we show that using the inverse relation we can find C k given f(w). C k = e ikw σ 2 if k = 0 df (w) = 0 if k 0
LECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity.
LECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series {X t } is a series of observations taken sequentially over time: x t is an observation
More informationLecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications
Lecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Moving average processes Autoregressive
More informationModule 9: Stationary Processes
Module 9: Stationary Processes Lecture 1 Stationary Processes 1 Introduction A stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space.
More information18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/2013
18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/2013 1. Covariance Stationary AR(2) Processes Suppose the discrete-time stochastic process {X t } follows a secondorder auto-regressive process
More informationMultiresolution Models of Time Series
Multiresolution Models of Time Series Andrea Tamoni (Bocconi University ) 2011 Tamoni Multiresolution Models of Time Series 1/ 16 General Framework Time-scale decomposition General Framework Begin with
More informationTime Series Solutions HT Let fx t g be the ARMA(1, 1) process, where jffij < 1 and j j < 1. Show that the autocorrelation function of
1+ 2 +2ffi ; ρ(h) =ffih 1 ρ(1) for h > 1: Time Series Solutions HT 2008 1. Let fx t g be the ARMA(1, 1) process, X t ffix t 1 = ffl t + ffl t 1 ; fffl t gοwn(0;ff 2 ); where jffij < 1 and j j < 1. Show
More informationChapter 6: Model Specification for Time Series
Chapter 6: Model Specification for Time Series The ARIMA(p, d, q) class of models as a broad class can describe many real time series. Model specification for ARIMA(p, d, q) models involves 1. Choosing
More informationTime Series Solutions HT 2009
Time Series Solutions HT 2009 1. Let {X t } be the ARMA(1, 1) process, X t φx t 1 = ɛ t + θɛ t 1, {ɛ t } WN(0, σ 2 ), where φ < 1 and θ < 1. Show that the autocorrelation function of {X t } is given by
More informationModule 3. Descriptive Time Series Statistics and Introduction to Time Series Models
Module 3 Descriptive Time Series Statistics and Introduction to Time Series Models Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W Q Meeker November 11, 2015
More informationIntroduction to Spectral and Time-Spectral Analysis with some Applications
Introduction to Spectral and Time-Spectral Analysis with some Applications A.INTRODUCTION Consider the following process X t = U cos[(=3)t] + V sin[(=3)t] Where *E[U] = E[V ] = 0 *E[UV ] = 0 *V ar(u) =
More information{ } Stochastic processes. Models for time series. Specification of a process. Specification of a process. , X t3. ,...X tn }
Stochastic processes Time series are an example of a stochastic or random process Models for time series A stochastic process is 'a statistical phenomenon that evolves in time according to probabilistic
More informationESSE Mid-Term Test 2017 Tuesday 17 October :30-09:45
ESSE 4020 3.0 - Mid-Term Test 207 Tuesday 7 October 207. 08:30-09:45 Symbols have their usual meanings. All questions are worth 0 marks, although some are more difficult than others. Answer as many questions
More informationELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random
More informationChapter 4: Models for Stationary Time Series
Chapter 4: Models for Stationary Time Series Now we will introduce some useful parametric models for time series that are stationary processes. We begin by defining the General Linear Process. Let {Y t
More informationENSC327 Communications Systems 19: Random Processes. Jie Liang School of Engineering Science Simon Fraser University
ENSC327 Communications Systems 19: Random Processes Jie Liang School of Engineering Science Simon Fraser University 1 Outline Random processes Stationary random processes Autocorrelation of random processes
More information1 Linear Difference Equations
ARMA Handout Jialin Yu 1 Linear Difference Equations First order systems Let {ε t } t=1 denote an input sequence and {y t} t=1 sequence generated by denote an output y t = φy t 1 + ε t t = 1, 2,... with
More informationChapter 6. Random Processes
Chapter 6 Random Processes Random Process A random process is a time-varying function that assigns the outcome of a random experiment to each time instant: X(t). For a fixed (sample path): a random process
More informationUniversity of Oxford. Statistical Methods Autocorrelation. Identification and Estimation
University of Oxford Statistical Methods Autocorrelation Identification and Estimation Dr. Órlaith Burke Michaelmas Term, 2011 Department of Statistics, 1 South Parks Road, Oxford OX1 3TG Contents 1 Model
More informationClassic Time Series Analysis
Classic Time Series Analysis Concepts and Definitions Let Y be a random number with PDF f Y t ~f,t Define t =E[Y t ] m(t) is known as the trend Define the autocovariance t, s =COV [Y t,y s ] =E[ Y t t
More informationWe will only present the general ideas on how to obtain. follow closely the AR(1) and AR(2) cases presented before.
ACF and PACF of an AR(p) We will only present the general ideas on how to obtain the ACF and PACF of an AR(p) model since the details follow closely the AR(1) and AR(2) cases presented before. Recall that
More informationTime Series Analysis. Solutions to problems in Chapter 5 IMM
Time Series Analysis Solutions to problems in Chapter 5 IMM Solution 5.1 Question 1. [ ] V [X t ] = V [ǫ t + c(ǫ t 1 + ǫ t + )] = 1 + c 1 σǫ = The variance of {X t } is not limited and therefore {X t }
More informationTime Series Examples Sheet
Lent Term 2001 Richard Weber Time Series Examples Sheet This is the examples sheet for the M. Phil. course in Time Series. A copy can be found at: http://www.statslab.cam.ac.uk/~rrw1/timeseries/ Throughout,
More informationElements of Multivariate Time Series Analysis
Gregory C. Reinsel Elements of Multivariate Time Series Analysis Second Edition With 14 Figures Springer Contents Preface to the Second Edition Preface to the First Edition vii ix 1. Vector Time Series
More informationParametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes (cont d)
Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes (cont d) Electrical & Computer Engineering North Carolina State University Acknowledgment: ECE792-41 slides
More informationLecture 1: Fundamental concepts in Time Series Analysis (part 2)
Lecture 1: Fundamental concepts in Time Series Analysis (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC)
More informationLecture 1: Brief Review on Stochastic Processes
Lecture 1: Brief Review on Stochastic Processes A stochastic process is a collection of random variables {X t (s) : t T, s S}, where T is some index set and S is the common sample space of the random variables.
More informationSome Time-Series Models
Some Time-Series Models Outline 1. Stochastic processes and their properties 2. Stationary processes 3. Some properties of the autocorrelation function 4. Some useful models Purely random processes, random
More informationStat 248 Lab 2: Stationarity, More EDA, Basic TS Models
Stat 248 Lab 2: Stationarity, More EDA, Basic TS Models Tessa L. Childers-Day February 8, 2013 1 Introduction Today s section will deal with topics such as: the mean function, the auto- and cross-covariance
More informationChapter 3 - Temporal processes
STK4150 - Intro 1 Chapter 3 - Temporal processes Odd Kolbjørnsen and Geir Storvik January 23 2017 STK4150 - Intro 2 Temporal processes Data collected over time Past, present, future, change Temporal aspect
More informationStatistics 349(02) Review Questions
Statistics 349(0) Review Questions I. Suppose that for N = 80 observations on the time series { : t T} the following statistics were calculated: _ x = 10.54 C(0) = 4.99 In addition the sample autocorrelation
More information3. ARMA Modeling. Now: Important class of stationary processes
3. ARMA Modeling Now: Important class of stationary processes Definition 3.1: (ARMA(p, q) process) Let {ɛ t } t Z WN(0, σ 2 ) be a white noise process. The process {X t } t Z is called AutoRegressive-Moving-Average
More informationTime Series Analysis
Time Series Analysis Christopher Ting http://mysmu.edu.sg/faculty/christophert/ christopherting@smu.edu.sg Quantitative Finance Singapore Management University March 3, 2017 Christopher Ting Week 9 March
More informationTime Series 2. Robert Almgren. Sept. 21, 2009
Time Series 2 Robert Almgren Sept. 21, 2009 This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models
More informationIV. Covariance Analysis
IV. Covariance Analysis Autocovariance Remember that when a stochastic process has time values that are interdependent, then we can characterize that interdependency by computing the autocovariance function.
More informationLINEAR STOCHASTIC MODELS
LINEAR STOCHASTIC MODELS Let {x τ+1,x τ+2,...,x τ+n } denote n consecutive elements from a stochastic process. If their joint distribution does not depend on τ, regardless of the size of n, then the process
More information3 Theory of stationary random processes
3 Theory of stationary random processes 3.1 Linear filters and the General linear process A filter is a transformation of one random sequence {U t } into another, {Y t }. A linear filter is a transformation
More informationSTAD57 Time Series Analysis. Lecture 23
STAD57 Time Series Analysis Lecture 23 1 Spectral Representation Spectral representation of stationary {X t } is: 12 i2t Xt e du 12 1/2 1/2 for U( ) a stochastic process with independent increments du(ω)=
More informationModule 4. Stationary Time Series Models Part 1 MA Models and Their Properties
Module 4 Stationary Time Series Models Part 1 MA Models and Their Properties Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W. Q. Meeker. February 14, 2016 20h
More informationTime Series: Theory and Methods
Peter J. Brockwell Richard A. Davis Time Series: Theory and Methods Second Edition With 124 Illustrations Springer Contents Preface to the Second Edition Preface to the First Edition vn ix CHAPTER 1 Stationary
More informationAutomatic Autocorrelation and Spectral Analysis
Piet M.T. Broersen Automatic Autocorrelation and Spectral Analysis With 104 Figures Sprin ger 1 Introduction 1 1.1 Time Series Problems 1 2 Basic Concepts 11 2.1 Random Variables 11 2.2 Normal Distribution
More informationCovariance Stationary Time Series. Example: Independent White Noise (IWN(0,σ 2 )) Y t = ε t, ε t iid N(0,σ 2 )
Covariance Stationary Time Series Stochastic Process: sequence of rv s ordered by time {Y t } {...,Y 1,Y 0,Y 1,...} Defn: {Y t } is covariance stationary if E[Y t ]μ for all t cov(y t,y t j )E[(Y t μ)(y
More informationSpectral Analysis - Introduction
Spectral Analysis - Introduction Initial Code Introduction Summer Semester 011/01 Lukas Vacha In previous lectures we studied stationary time series in terms of quantities that are functions of time. The
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Chapter 9 Multivariate time series 2 Transfer function
More informationLecture 2: ARMA(p,q) models (part 2)
Lecture 2: ARMA(p,q) models (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC) Univariate time series Sept.
More informationStochastic Processes: I. consider bowl of worms model for oscilloscope experiment:
Stochastic Processes: I consider bowl of worms model for oscilloscope experiment: SAPAscope 2.0 / 0 1 RESET SAPA2e 22, 23 II 1 stochastic process is: Stochastic Processes: II informally: bowl + drawing
More informationINDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.
INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture Regression on Principal components
More informationEcon 623 Econometrics II Topic 2: Stationary Time Series
1 Introduction Econ 623 Econometrics II Topic 2: Stationary Time Series In the regression model we can model the error term as an autoregression AR(1) process. That is, we can use the past value of the
More informationTime Series Examples Sheet
Lent Term 2001 Richard Weber Time Series Examples Sheet This is the examples sheet for the M. Phil. course in Time Series. A copy can be found at: http://www.statslab.cam.ac.uk/~rrw1/timeseries/ Throughout,
More informationEC402: Serial Correlation. Danny Quah Economics Department, LSE Lent 2015
EC402: Serial Correlation Danny Quah Economics Department, LSE Lent 2015 OUTLINE 1. Stationarity 1.1 Covariance stationarity 1.2 Explicit Models. Special cases: ARMA processes 2. Some complex numbers.
More informationNotes on Random Processes
otes on Random Processes Brian Borchers and Rick Aster October 27, 2008 A Brief Review of Probability In this section of the course, we will work with random variables which are denoted by capital letters,
More informationLECTURE 10 LINEAR PROCESSES II: SPECTRAL DENSITY, LAG OPERATOR, ARMA. In this lecture, we continue to discuss covariance stationary processes.
MAY, 0 LECTURE 0 LINEAR PROCESSES II: SPECTRAL DENSITY, LAG OPERATOR, ARMA In this lecture, we continue to discuss covariance stationary processes. Spectral density Gourieroux and Monfort 990), Ch. 5;
More informationLecture 3-4: Probability models for time series
Lecture 3-4, page 1 Lecture 3-4: Probability models for time series Outline of lesson 3-4 (chapter 3) The most heavy (and theoretical) part (and lesson) AR(p) processes MA(q) processes ARMA(p,q) processes
More informationIdentifiability, Invertibility
Identifiability, Invertibility Defn: If {ǫ t } is a white noise series and µ and b 0,..., b p are constants then X t = µ + b 0 ǫ t + b ǫ t + + b p ǫ t p is a moving average of order p; write MA(p). Q:
More informationChapter 9: Forecasting
Chapter 9: Forecasting One of the critical goals of time series analysis is to forecast (predict) the values of the time series at times in the future. When forecasting, we ideally should evaluate the
More informationE 4101/5101 Lecture 6: Spectral analysis
E 4101/5101 Lecture 6: Spectral analysis Ragnar Nymoen 3 March 2011 References to this lecture Hamilton Ch 6 Lecture note (on web page) For stationary variables/processes there is a close correspondence
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More informationMAT3379 (Winter 2016)
MAT3379 (Winter 2016) Assignment 4 - SOLUTIONS The following questions will be marked: 1a), 2, 4, 6, 7a Total number of points for Assignment 4: 20 Q1. (Theoretical Question, 2 points). Yule-Walker estimation
More informationStochastic Processes- IV
!! Module 2! Lecture 7 :Random Vibrations & Failure Analysis Stochastic Processes- IV!! Sayan Gupta Department of Applied Mechanics Indian Institute of Technology Madras Properties of Power Spectral Density
More informationStochastic Processes
Elements of Lecture II Hamid R. Rabiee with thanks to Ali Jalali Overview Reading Assignment Chapter 9 of textbook Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic
More informationDiscrete time processes
Discrete time processes Predictions are difficult. Especially about the future Mark Twain. Florian Herzog 2013 Modeling observed data When we model observed (realized) data, we encounter usually the following
More informationChapter 2. Some basic tools. 2.1 Time series: Theory Stochastic processes
Chapter 2 Some basic tools 2.1 Time series: Theory 2.1.1 Stochastic processes A stochastic process is a sequence of random variables..., x 0, x 1, x 2,.... In this class, the subscript always means time.
More informationCh 5. Models for Nonstationary Time Series. Time Series Analysis
We have studied some deterministic and some stationary trend models. However, many time series data cannot be modeled in either way. Ex. The data set oil.price displays an increasing variation from the
More informationReview Session: Econometrics - CLEFIN (20192)
Review Session: Econometrics - CLEFIN (20192) Part II: Univariate time series analysis Daniele Bianchi March 20, 2013 Fundamentals Stationarity A time series is a sequence of random variables x t, t =
More informationE X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl.
E X A M Course code: Course name: Number of pages incl. front page: 6 MA430-G Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours Resources allowed: Notes: Pocket calculator,
More informationIf we want to analyze experimental or simulated data we might encounter the following tasks:
Chapter 1 Introduction If we want to analyze experimental or simulated data we might encounter the following tasks: Characterization of the source of the signal and diagnosis Studying dependencies Prediction
More information9) Time series econometrics
30C00200 Econometrics 9) Time series econometrics Timo Kuosmanen Professor Management Science http://nomepre.net/index.php/timokuosmanen 1 Macroeconomic data: GDP Inflation rate Examples of time series
More informationSwitching Regime Estimation
Switching Regime Estimation Series de Tiempo BIrkbeck March 2013 Martin Sola (FE) Markov Switching models 01/13 1 / 52 The economy (the time series) often behaves very different in periods such as booms
More informationStochastic Processes. A stochastic process is a function of two variables:
Stochastic Processes Stochastic: from Greek stochastikos, proceeding by guesswork, literally, skillful in aiming. A stochastic process is simply a collection of random variables labelled by some parameter:
More informationIntroduction to Signal Processing
to Signal Processing Davide Bacciu Dipartimento di Informatica Università di Pisa bacciu@di.unipi.it Intelligent Systems for Pattern Recognition Signals = Time series Definitions Motivations A sequence
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 II. Basic definitions A time series is a set of observations X t, each
More informationEconomics Department LSE. Econometrics: Timeseries EXERCISE 1: SERIAL CORRELATION (ANALYTICAL)
Economics Department LSE EC402 Lent 2015 Danny Quah TW1.10.01A x7535 : Timeseries EXERCISE 1: SERIAL CORRELATION (ANALYTICAL) 1. Suppose ɛ is w.n. (0, σ 2 ), ρ < 1, and W t = ρw t 1 + ɛ t, for t = 1, 2,....
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 III. Stationary models 1 Purely random process 2 Random walk (non-stationary)
More informationCONTENTS NOTATIONAL CONVENTIONS GLOSSARY OF KEY SYMBOLS 1 INTRODUCTION 1
DIGITAL SPECTRAL ANALYSIS WITH APPLICATIONS S.LAWRENCE MARPLE, JR. SUMMARY This new book provides a broad perspective of spectral estimation techniques and their implementation. It concerned with spectral
More informationSOME BASICS OF TIME-SERIES ANALYSIS
SOME BASICS OF TIME-SERIES ANALYSIS John E. Floyd University of Toronto December 8, 26 An excellent place to learn about time series analysis is from Walter Enders textbook. For a basic understanding of
More informationLecture 25: Review. Statistics 104. April 23, Colin Rundel
Lecture 25: Review Statistics 104 Colin Rundel April 23, 2012 Joint CDF F (x, y) = P [X x, Y y] = P [(X, Y ) lies south-west of the point (x, y)] Y (x,y) X Statistics 104 (Colin Rundel) Lecture 25 April
More informationLecture Notes 7 Stationary Random Processes. Strict-Sense and Wide-Sense Stationarity. Autocorrelation Function of a Stationary Process
Lecture Notes 7 Stationary Random Processes Strict-Sense and Wide-Sense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Continuity and Integration of Random Processes
More informationEE538 Final Exam Fall :20 pm -5:20 pm PHYS 223 Dec. 17, Cover Sheet
EE538 Final Exam Fall 005 3:0 pm -5:0 pm PHYS 3 Dec. 17, 005 Cover Sheet Test Duration: 10 minutes. Open Book but Closed Notes. Calculators ARE allowed!! This test contains five problems. Each of the five
More informationNational Sun Yat-Sen University CSE Course: Information Theory. Maximum Entropy and Spectral Estimation
Maximum Entropy and Spectral Estimation 1 Introduction What is the distribution of velocities in the gas at a given temperature? It is the Maxwell-Boltzmann distribution. The maximum entropy distribution
More information1 Simulating normal (Gaussian) rvs with applications to simulating Brownian motion and geometric Brownian motion in one and two dimensions
Copyright c 2007 by Karl Sigman 1 Simulating normal Gaussian rvs with applications to simulating Brownian motion and geometric Brownian motion in one and two dimensions Fundamental to many applications
More informationLecture 2: Univariate Time Series
Lecture 2: Univariate Time Series Analysis: Conditional and Unconditional Densities, Stationarity, ARMA Processes Prof. Massimo Guidolin 20192 Financial Econometrics Spring/Winter 2017 Overview Motivation:
More informationSTAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 2 - Probability Models Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 34 Agenda 1 Introduction 2 Stochastic Process Definition 1 Stochastic Definition
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationCointegration, Stationarity and Error Correction Models.
Cointegration, Stationarity and Error Correction Models. STATIONARITY Wold s decomposition theorem states that a stationary time series process with no deterministic components has an infinite moving average
More informationMAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS)
MAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS) 15 April 2016 (180 minutes) Professor: R. Kulik Student Number: Name: This is closed book exam. You are allowed to use one double-sided A4 sheet of notes.
More informationTime Series Analysis Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Introduction 1 14.384 Time
More informationSpatial Statistics with Image Analysis. Lecture L02. Computer exercise 0 Daily Temperature. Lecture 2. Johan Lindström.
C Stochastic fields Covariance Spatial Statistics with Image Analysis Lecture 2 Johan Lindström November 4, 26 Lecture L2 Johan Lindström - johanl@maths.lth.se FMSN2/MASM2 L /2 C Stochastic fields Covariance
More informationAutoregressive Moving Average (ARMA) Models and their Practical Applications
Autoregressive Moving Average (ARMA) Models and their Practical Applications Massimo Guidolin February 2018 1 Essential Concepts in Time Series Analysis 1.1 Time Series and Their Properties Time series:
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture State space models, 1st part: Model: Sec. 10.1 The
More informationStochastic Processes
Stochastic Processes Stochastic Process Non Formal Definition: Non formal: A stochastic process (random process) is the opposite of a deterministic process such as one defined by a differential equation.
More informationMAT 3379 (Winter 2016) FINAL EXAM (PRACTICE)
MAT 3379 (Winter 2016) FINAL EXAM (PRACTICE) 15 April 2016 (180 minutes) Professor: R. Kulik Student Number: Name: This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
More informationMarcel Dettling. Applied Time Series Analysis SS 2013 Week 05. ETH Zürich, March 18, Institute for Data Analysis and Process Design
Marcel Dettling Institute for Data Analysis and Process Design Zurich University of Applied Sciences marcel.dettling@zhaw.ch http://stat.ethz.ch/~dettling ETH Zürich, March 18, 2013 1 Basics of Modeling
More informationat least 50 and preferably 100 observations should be available to build a proper model
III Box-Jenkins Methods 1. Pros and Cons of ARIMA Forecasting a) need for data at least 50 and preferably 100 observations should be available to build a proper model used most frequently for hourly or
More informationLecture 4 - Spectral Estimation
Lecture 4 - Spectral Estimation The Discrete Fourier Transform The Discrete Fourier Transform (DFT) is the equivalent of the continuous Fourier Transform for signals known only at N instants separated
More informationTIME SERIES ANALYSIS. Forecasting and Control. Wiley. Fifth Edition GWILYM M. JENKINS GEORGE E. P. BOX GREGORY C. REINSEL GRETA M.
TIME SERIES ANALYSIS Forecasting and Control Fifth Edition GEORGE E. P. BOX GWILYM M. JENKINS GREGORY C. REINSEL GRETA M. LJUNG Wiley CONTENTS PREFACE TO THE FIFTH EDITION PREFACE TO THE FOURTH EDITION
More information5 Transfer function modelling
MSc Further Time Series Analysis 5 Transfer function modelling 5.1 The model Consider the construction of a model for a time series (Y t ) whose values are influenced by the earlier values of a series
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Input-Output systems The z-transform important issues
More informationAdvanced Econometrics
Advanced Econometrics Marco Sunder Nov 04 2010 Marco Sunder Advanced Econometrics 1/ 25 Contents 1 2 3 Marco Sunder Advanced Econometrics 2/ 25 Music Marco Sunder Advanced Econometrics 3/ 25 Music Marco
More informationSome functional (Hölderian) limit theorems and their applications (II)
Some functional (Hölderian) limit theorems and their applications (II) Alfredas Račkauskas Vilnius University Outils Statistiques et Probabilistes pour la Finance Université de Rouen June 1 5, Rouen (Rouen
More informationLecture 4a: ARMA Model
Lecture 4a: ARMA Model 1 2 Big Picture Most often our goal is to find a statistical model to describe real time series (estimation), and then predict the future (forecasting) One particularly popular model
More information