AR(p) + MA(q) = ARMA(p, q)


 Cleopatra Griffin
 3 years ago
 Views:
Transcription
1 AR(p) + MA(q) = ARMA(p, q)
2 Outline 1 3.4: ARMA(p, q) Model 2 Homework 3a Arthur Berg AR(p) + MA(q) = ARMA(p, q) 2/ 12
3 ARMA(p, q) Model Definition (ARMA(p, q) Model) A time series is ARMA(p, q) if it is stationary and satisfies Z t = α + φ 1 Z t φ p Z t p + θ 1 a t θ q a t q +a t ( ) } {{ } } {{ } AR part MA part Using the AR and MA operators, we can rewrite ( ) as φ(b)ż t = θ(b)a t The textbook defines the ARMA process with a slightly different parametrization given by Z t = α + φ 1 Z t φ p Z t p θ 1 a t 1 θ q a t q + a t ( ) Arthur Berg AR(p) + MA(q) = ARMA(p, q) 3/ 12
4 ARMA(p, q) Model Definition (ARMA(p, q) Model) A time series is ARMA(p, q) if it is stationary and satisfies Z t = α + φ 1 Z t φ p Z t p + θ 1 a t θ q a t q +a t ( ) } {{ } } {{ } AR part MA part Using the AR and MA operators, we can rewrite ( ) as φ(b)ż t = θ(b)a t The textbook defines the ARMA process with a slightly different parametrization given by Z t = α + φ 1 Z t φ p Z t p θ 1 a t 1 θ q a t q + a t ( ) Arthur Berg AR(p) + MA(q) = ARMA(p, q) 3/ 12
5 ARMA(p, q) Model Definition (ARMA(p, q) Model) A time series is ARMA(p, q) if it is stationary and satisfies Z t = α + φ 1 Z t φ p Z t p + θ 1 a t θ q a t q +a t ( ) } {{ } } {{ } AR part MA part Using the AR and MA operators, we can rewrite ( ) as φ(b)ż t = θ(b)a t The textbook defines the ARMA process with a slightly different parametrization given by Z t = α + φ 1 Z t φ p Z t p θ 1 a t 1 θ q a t q + a t ( ) Arthur Berg AR(p) + MA(q) = ARMA(p, q) 3/ 12
6 Causality and Invertibility of ARMA(p, q) Facts: The ARMA(p, q) model given by φ(b)z t = θ(b)a t is causal if and only if φ(z) 0 when z 1 i.e. all roots including complex roots of φ(z) lie outside the unit disk. The ARMA(p, q) model given by φ(b)z t = θ(b)a t is invertible if and only if θ(z) 0 when θ 1 i.e. all roots including complex roots of θ(z) lie outside the unit disk. Arthur Berg AR(p) + MA(q) = ARMA(p, q) 4/ 12
7 Causality and Invertibility of ARMA(p, q) Facts: The ARMA(p, q) model given by φ(b)z t = θ(b)a t is causal if and only if φ(z) 0 when z 1 i.e. all roots including complex roots of φ(z) lie outside the unit disk. The ARMA(p, q) model given by φ(b)z t = θ(b)a t is invertible if and only if θ(z) 0 when θ 1 i.e. all roots including complex roots of θ(z) lie outside the unit disk. Arthur Berg AR(p) + MA(q) = ARMA(p, q) 4/ 12
8 Causality and Invertibility of ARMA(p, q) Facts: The ARMA(p, q) model given by φ(b)z t = θ(b)a t is causal if and only if φ(z) 0 when z 1 i.e. all roots including complex roots of φ(z) lie outside the unit disk. The ARMA(p, q) model given by φ(b)z t = θ(b)a t is invertible if and only if θ(z) 0 when θ 1 i.e. all roots including complex roots of θ(z) lie outside the unit disk. Arthur Berg AR(p) + MA(q) = ARMA(p, q) 4/ 12
9 Parameter Redundancy The process Z t = a t ( ) is equivalent to.5z t 1 =.5a t 1 ( ) which is also equivalent to ( ) ( ), i.e. Z t.5z t 1 = a t.5a t 1 Z t in the last representation is still white noise, but has an ARMA(1,1) representation. We remove redundancies in an ARMA model φ(b)z t = θ(b)a t by canceling the common factors in θ(b) and φ(b). Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
10 Parameter Redundancy The process Z t = a t ( ) is equivalent to.5z t 1 =.5a t 1 ( ) which is also equivalent to ( ) ( ), i.e. Z t.5z t 1 = a t.5a t 1 Z t in the last representation is still white noise, but has an ARMA(1,1) representation. We remove redundancies in an ARMA model φ(b)z t = θ(b)a t by canceling the common factors in θ(b) and φ(b). Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
11 Parameter Redundancy The process Z t = a t ( ) is equivalent to.5z t 1 =.5a t 1 ( ) which is also equivalent to ( ) ( ), i.e. Z t.5z t 1 = a t.5a t 1 Z t in the last representation is still white noise, but has an ARMA(1,1) representation. We remove redundancies in an ARMA model φ(b)z t = θ(b)a t by canceling the common factors in θ(b) and φ(b). Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
12 Parameter Redundancy The process Z t = a t ( ) is equivalent to.5z t 1 =.5a t 1 ( ) which is also equivalent to ( ) ( ), i.e. Z t.5z t 1 = a t.5a t 1 Z t in the last representation is still white noise, but has an ARMA(1,1) representation. We remove redundancies in an ARMA model φ(b)z t = θ(b)a t by canceling the common factors in θ(b) and φ(b). Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
13 Parameter Redundancy The process Z t = a t ( ) is equivalent to.5z t 1 =.5a t 1 ( ) which is also equivalent to ( ) ( ), i.e. Z t.5z t 1 = a t.5a t 1 Z t in the last representation is still white noise, but has an ARMA(1,1) representation. We remove redundancies in an ARMA model φ(b)z t = θ(b)a t by canceling the common factors in θ(b) and φ(b). Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
14 Example Removing Redundancy Example Consider the process which is equivalent to Z t =.4Z t Z t 2 + a t + a t a t 2 (1.4B.45B 2 ) Z } {{ } t = (1 + B +.25B 2 ) } {{ } φ(b) θ(b) a t Is this process really ARMA(2,2)? Factoring φ(z) and θ(z) gives φ(z) = 1.4z.45z 2 = (1 +.5z)(1.9z) θ(z) = (1 + z +.25z 2 ) = (1 +.5z) 2 Removing the common term (1 +.5z) gives the reduced model Z t =.9Z t 1 +.5a t 1 + a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
15 Example Removing Redundancy Example Consider the process which is equivalent to Z t =.4Z t Z t 2 + a t + a t a t 2 (1.4B.45B 2 ) Z } {{ } t = (1 + B +.25B 2 ) } {{ } φ(b) θ(b) a t Is this process really ARMA(2,2)? Factoring φ(z) and θ(z) gives φ(z) = 1.4z.45z 2 = (1 +.5z)(1.9z) θ(z) = (1 + z +.25z 2 ) = (1 +.5z) 2 Removing the common term (1 +.5z) gives the reduced model Z t =.9Z t 1 +.5a t 1 + a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
16 Example Removing Redundancy Example Consider the process which is equivalent to Z t =.4Z t Z t 2 + a t + a t a t 2 (1.4B.45B 2 ) Z } {{ } t = (1 + B +.25B 2 ) } {{ } φ(b) θ(b) a t Is this process really ARMA(2,2)? Factoring φ(z) and θ(z) gives φ(z) = 1.4z.45z 2 = (1 +.5z)(1.9z) θ(z) = (1 + z +.25z 2 ) = (1 +.5z) 2 Removing the common term (1 +.5z) gives the reduced model Z t =.9Z t 1 +.5a t 1 + a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
17 Example Removing Redundancy Example Consider the process which is equivalent to Z t =.4Z t Z t 2 + a t + a t a t 2 (1.4B.45B 2 ) Z } {{ } t = (1 + B +.25B 2 ) } {{ } φ(b) θ(b) Is this process really ARMA(2,2)? No! Factoring φ(z) and θ(z) gives φ(z) = 1.4z.45z 2 = (1 +.5z)(1.9z) θ(z) = (1 + z +.25z 2 ) = (1 +.5z) 2 Removing the common term (1 +.5z) gives the reduced model Z t =.9Z t 1 +.5a t 1 + a t a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
18 Example Removing Redundancy Example Consider the process which is equivalent to Z t =.4Z t Z t 2 + a t + a t a t 2 (1.4B.45B 2 ) Z } {{ } t = (1 + B +.25B 2 ) } {{ } φ(b) θ(b) Is this process really ARMA(2,2)? No! Factoring φ(z) and θ(z) gives φ(z) = 1.4z.45z 2 = (1 +.5z)(1.9z) θ(z) = (1 + z +.25z 2 ) = (1 +.5z) 2 Removing the common term (1 +.5z) gives the reduced model Z t =.9Z t 1 +.5a t 1 + a t a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
19 Example Removing Redundancy Example Consider the process which is equivalent to Z t =.4Z t Z t 2 + a t + a t a t 2 (1.4B.45B 2 ) Z } {{ } t = (1 + B +.25B 2 ) } {{ } φ(b) θ(b) Is this process really ARMA(2,2)? No! Factoring φ(z) and θ(z) gives φ(z) = 1.4z.45z 2 = (1 +.5z)(1.9z) θ(z) = (1 + z +.25z 2 ) = (1 +.5z) 2 Removing the common term (1 +.5z) gives the reduced model Z t =.9Z t 1 +.5a t 1 + a t a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
20 Example Causality and Invertibility Example (cont.) The reduced model is φ(b)z t = θ(b)a t where φ(z) = 1.9z θ(z) = 1 +.5z There is only one root of φ(z) which is z = 10/9, and 10/9 lies outside the unit circle so this ARMA model is causal. There is only one root of θ(z) which is z = 2, and 2 lies outside the unit circle so this ARMA model is invertible. Arthur Berg AR(p) + MA(q) = ARMA(p, q) 7/ 12
21 Example Causality and Invertibility Example (cont.) The reduced model is φ(b)z t = θ(b)a t where φ(z) = 1.9z θ(z) = 1 +.5z There is only one root of φ(z) which is z = 10/9, and 10/9 lies outside the unit circle so this ARMA model is causal. There is only one root of θ(z) which is z = 2, and 2 lies outside the unit circle so this ARMA model is invertible. Arthur Berg AR(p) + MA(q) = ARMA(p, q) 7/ 12
22 Example Causality and Invertibility Example (cont.) The reduced model is φ(b)z t = θ(b)a t where φ(z) = 1.9z θ(z) = 1 +.5z There is only one root of φ(z) which is z = 10/9, and 10/9 lies outside the unit circle so this ARMA model is causal. There is only one root of θ(z) which is z = 2, and 2 lies outside the unit circle so this ARMA model is invertible. Arthur Berg AR(p) + MA(q) = ARMA(p, q) 7/ 12
23 ARMA Mathematics Start with the ARMA equation φ(b)z t = θ(b)a t To solve for Z t, we simply divide! That is Z t = θ(b) φ(b) } {{ } ψ(b) Similarly, solving for a t gives a t = φ(b) θ(b) } {{ } π(b) The key is to figuring out the ψ j and π j in ψ(b) = 1 + ψ j B j ψ(b) = 1 + ψ j B j a t Z t MA( ) representation AR( ) representation Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
24 ARMA Mathematics Start with the ARMA equation φ(b)z t = θ(b)a t To solve for Z t, we simply divide! That is Z t = θ(b) φ(b) } {{ } ψ(b) Similarly, solving for a t gives a t = φ(b) θ(b) } {{ } π(b) The key is to figuring out the ψ j and π j in ψ(b) = 1 + ψ j B j ψ(b) = 1 + ψ j B j a t Z t MA( ) representation AR( ) representation Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
25 ARMA Mathematics Start with the ARMA equation φ(b)z t = θ(b)a t To solve for Z t, we simply divide! That is Z t = θ(b) φ(b) } {{ } ψ(b) Similarly, solving for a t gives a t = φ(b) θ(b) } {{ } π(b) The key is to figuring out the ψ j and π j in ψ(b) = 1 + ψ j B j ψ(b) = 1 + ψ j B j a t Z t MA( ) representation AR( ) representation Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
26 ARMA Mathematics Start with the ARMA equation φ(b)z t = θ(b)a t To solve for Z t, we simply divide! That is Z t = θ(b) φ(b) } {{ } ψ(b) Similarly, solving for a t gives a t = φ(b) θ(b) } {{ } π(b) The key is to figuring out the ψ j and π j in ψ(b) = 1 + ψ j B j ψ(b) = 1 + ψ j B j a t Z t MA( ) representation AR( ) representation Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
27 Example MA( ) Representation Example (cont.) Note that ψ(z) = θ(z) φ(z) = 1 +.5z 1.9z = (1 +.5z)(1 +.9z z z 3 + ) for z 1 = 1 + (.5 +.9)z + (.5(.9) )z 2 + (.5(.9 2 ) )z 3 + for z 1 = 1 + (.5 +.9)z + (.5 +.9)(.9)z 2 + (.5 +.9)(.9 2 )z 3 + for z 1 = 1 + (.5 +.9).9 j 1 z j for z 1 } {{ } 1.4 This gives the following MA( ) representation: Z t = θ(b) φ(b) a t = j 1 B j a t = a t j 1 a t j Arthur Berg AR(p) + MA(q) = ARMA(p, q) 9/ 12
28 Example MA( ) Representation Example (cont.) Note that ψ(z) = θ(z) φ(z) = 1 +.5z 1.9z = (1 +.5z)(1 +.9z z z 3 + ) for z 1 = 1 + (.5 +.9)z + (.5(.9) )z 2 + (.5(.9 2 ) )z 3 + for z 1 = 1 + (.5 +.9)z + (.5 +.9)(.9)z 2 + (.5 +.9)(.9 2 )z 3 + for z 1 = 1 + (.5 +.9).9 j 1 z j for z 1 } {{ } 1.4 This gives the following MA( ) representation: Z t = θ(b) φ(b) a t = j 1 B j a t = a t j 1 a t j Arthur Berg AR(p) + MA(q) = ARMA(p, q) 9/ 12
29 Example AR( ) Representation Example (cont.) Similarly, one can show π(z) = φ(z) θ(z). = 1.9z 1 +.5z = (.5) j 1 z j for z 1 This gives the following AR( ) representation: Z t = 1.4 (.5) j 1 a t j + a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 10/ 12
30 Example AR( ) Representation Example (cont.) Similarly, one can show π(z) = φ(z) θ(z). = 1.9z 1 +.5z = (.5) j 1 z j for z 1 This gives the following AR( ) representation: Z t = 1.4 (.5) j 1 a t j + a t Arthur Berg AR(p) + MA(q) = ARMA(p, q) 10/ 12
31 Outline 1 3.4: ARMA(p, q) Model 2 Homework 3a Arthur Berg AR(p) + MA(q) = ARMA(p, q) 11/ 12
32 Homework 3a Read 4.1 and 4.2. Derive the equation for π(z) on slide 10. Do exercise #3.14(a) only for models (i), (ii), and (iv) Do exercise #3.14(b,c) only for model (ii) Arthur Berg AR(p) + MA(q) = ARMA(p, q) 12/ 12
Introduction to Time Series Analysis. Lecture 6.
Introduction to Time Series Analysis. Lecture 6. Peter Bartlett www.stat.berkeley.edu/ bartlett/courses/153fall2010 Last lecture: 1. Causality 2. Invertibility 3. AR(p) models 4. ARMA(p,q) models 1 Introduction
More informationIntroduction to ARMA Models
Statistics 910, #8 1 Overview Introduction to ARMA Models 1. Modeling paradigm 2. Review stationary linear processes 3. ARMA processes 4. Stationarity of ARMA processes 5. Identifiability of ARMA processes
More informationTime Series Analysis
Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos (UC3MUPM)
More informationLecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay
Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay Business cycle plays an important role in economics. In time series analysis, business cycle
More informationITSMR Reference Manual
ITSMR Reference Manual George Weigt June 5, 2015 1 Contents 1 Introduction 3 1.1 Time series analysis in a nutshell............................... 3 1.2 White Noise Variance.....................................
More informationLecture 2: ARMA(p,q) models (part 3)
Lecture 2: ARMA(p,q) models (part 3) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEANice) Sept. 2011  Jan. 2012 Florian Pelgrin (HEC) Univariate time series Sept.
More information4.3 Moving Average Process MA(q)
66 CHAPTER 4. STATIONARY TS MODELS 4.3 Moving Average Process MA(q) Definition 4.5. {X t } is a movingaverage process of order q if X t = Z t + θ 1 Z t 1 +... + θ q Z t q, (4.9) where and θ 1,...,θ q
More informationDiscrete Time Series Analysis with ARMA Models
Discrete Time Series Analysis with ARMA Models Veronica Sitsofe Ahiati (veronica@aims.ac.za) African Institute for Mathematical Sciences (AIMS) Supervised by Tina Marquardt Munich University of Technology,
More informationSome useful concepts in univariate time series analysis
Some useful concepts in univariate time series analysis Autoregressive moving average models Autocorrelation functions Model Estimation Diagnostic measure Model selection Forecasting Assumptions: 1. Nonseasonal
More informationSales forecasting # 2
Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univrennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting
More informationA Multiplicative Seasonal BoxJenkins Model to Nigerian Stock Prices
A Multiplicative Seasonal BoxJenkins Model to Nigerian Stock Prices Ette Harrison Etuk Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Nigeria Email: ettetuk@yahoo.com
More informationFinancial TIme Series Analysis: Part II
Department of Mathematics and Statistics, University of Vaasa, Finland January 29 February 13, 2015 Feb 14, 2015 1 Univariate linear stochastic models: further topics Unobserved component model Signal
More informationTime Series Analysis 1. Lecture 8: Time Series Analysis. Time Series Analysis MIT 18.S096. Dr. Kempthorne. Fall 2013 MIT 18.S096
Lecture 8: Time Series Analysis MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis 1 Outline Time Series Analysis 1 Time Series Analysis MIT 18.S096 Time Series Analysis 2 A stochastic
More informationAnalysis and Computation for Finance Time Series  An Introduction
ECMM703 Analysis and Computation for Finance Time Series  An Introduction Alejandra González Harrison 161 Email: mag208@exeter.ac.uk Time Series  An Introduction A time series is a sequence of observations
More informationLuciano Rispoli Department of Economics, Mathematics and Statistics Birkbeck College (University of London)
Luciano Rispoli Department of Economics, Mathematics and Statistics Birkbeck College (University of London) 1 Forecasting: definition Forecasting is the process of making statements about events whose
More informationUnivariate and Multivariate Methods PEARSON. Addison Wesley
Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston
More informationAnalysis of algorithms of time series analysis for forecasting sales
SAINTPETERSBURG STATE UNIVERSITY Mathematics & Mechanics Faculty Chair of Analytical Information Systems Garipov Emil Analysis of algorithms of time series analysis for forecasting sales Course Work Scientific
More informationUnivariate Time Series Analysis; ARIMA Models
Econometrics 2 Spring 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Outline of the Lecture () Introduction to univariate time series analysis. (2) Stationarity. (3) Characterizing
More informationAnalysis of Financial Time Series with EViews
Analysis of Financial Time Series with EViews Enrico Foscolo Contents 1 Asset Returns 2 1.1 Empirical Properties of Returns................. 2 2 Heteroskedasticity and Autocorrelation 4 2.1 Testing for
More informationGraphical Tools for Exploring and Analyzing Data From ARIMA Time Series Models
Graphical Tools for Exploring and Analyzing Data From ARIMA Time Series Models William Q. Meeker Department of Statistics Iowa State University Ames, IA 50011 January 13, 2001 Abstract Splus is a highly
More informationTime Series  ARIMA Models. Instructor: G. William Schwert
APS 425 Fall 25 Time Series : ARIMA Models Instructor: G. William Schwert 585275247 schwert@schwert.ssb.rochester.edu Topics Typical time series plot Pattern recognition in auto and partial autocorrelations
More informationEconometria dei mercati finanziari c.a. A.A. 20112012. 4. AR, MA and ARMA Time Series Models. Luca Fanelli. University of Bologna
Econometria dei mercati finanziari c.a. A.A. 20112012 4. AR, MA and ARMA Time Series Models Luca Fanelli University of Bologna luca.fanelli@unibo.it For each class of models considered in these slides
More informationTHE SVM APPROACH FOR BOX JENKINS MODELS
REVSTAT Statistical Journal Volume 7, Number 1, April 2009, 23 36 THE SVM APPROACH FOR BOX JENKINS MODELS Authors: Saeid Amiri Dep. of Energy and Technology, Swedish Univ. of Agriculture Sciences, P.O.Box
More informationTime Series Analysis
Time Series Analysis Autoregressive, MA and ARMA processes Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 212 Alonso and GarcíaMartos
More informationUnivariate Time Series Analysis; ARIMA Models
Econometrics 2 Fall 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Univariate Time Series Analysis We consider a single time series, y,y 2,..., y T. We want to construct simple
More informationPromotional Analysis and Forecasting for Demand Planning: A Practical Time Series Approach Michael Leonard, SAS Institute Inc.
Promotional Analysis and Forecasting for Demand Planning: A Practical Time Series Approach Michael Leonard, SAS Institute Inc. Cary, NC, USA Abstract Many businesses use sales promotions to increase the
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationTime Series Analysis
Time Series Analysis Lecture Notes for 475.726 Ross Ihaka Statistics Department University of Auckland April 14, 2005 ii Contents 1 Introduction 1 1.1 Time Series.............................. 1 1.2 Stationarity
More informationTime Series Analysis
Time Series Analysis Andrea Beccarini Center for Quantitative Economics Winter 2013/2014 Andrea Beccarini (CQE) Time Series Analysis Winter 2013/2014 1 / 143 Introduction Objectives Time series are ubiquitous
More informationAlgebraic Concepts Algebraic Concepts Writing
Curriculum Guide: Algebra 2/Trig (AR) 2 nd Quarter 8/7/2013 2 nd Quarter, Grade 912 GRADE 912 Unit of Study: Matrices Resources: Textbook: Algebra 2 (Holt, Rinehart & Winston), Ch. 4 Length of Study:
More information1 8 solve quadratic equations by using the quadratic formula and the discriminate September with 16, 2016 notes.noteb
WARM UP 1. Write 15x 2 + 6x = 14x 2 12 in standard form. ANSWER x 2 + 6x +12 = 0 2. Evaluate b 2 4ac when a = 3, b = 6, and c = 5. ANSWER 24 3. A student is solving an equation by completing the square.
More informationIntroduction to Time Series Analysis. Lecture 22.
Introduction to Time Series Analysis. Lecture 22. 1. Review: The smoothed periodogram. 2. Examples. 3. Parametric spectral density estimation. 1 Review: Periodogram The periodogram is defined as I(ν) =
More informationTime Series Graphs. Model ACF PACF. White Noise All zeros All zeros. AR(p) Exponential Decay P significant lags before dropping to zero
Time Series Graphs Model ACF PACF White Noise All zeros All zeros AR(p) Exponential Decay P significant lags before dropping to zero MA(q) q significant lags before dropping to zero Exponential Decay ARMA(p,q)
More informationTime Series Analysis
Time Series Analysis Identifying possible ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos
More informationDetermine If An Equation Represents a Function
Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The
More informationMaximum Likelihood Estimation of an ARMA(p,q) Model
Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates
More information(Refer Slide Time: 1:42)
Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture  10 Curves So today we are going to have a new topic. So far
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
More informationGates, Circuits and Boolean Functions
Lecture 2 Gates, Circuits and Boolean Functions DOC 112: Hardware Lecture 2 Slide 1 In this lecture we will: Introduce an electronic representation of Boolean operators called digital gates. Define a schematic
More informationFactoring A Quadratic Polynomial
Factoring A Quadratic Polynomial If we multiply two binomials together, the result is a quadratic polynomial: This multiplication is pretty straightforward, using the distributive property of multiplication
More informationA FULLY INTEGRATED ENVIRONMENT FOR TIMEDEPENDENT DATA ANALYSIS
A FULLY INTEGRATED ENVIRONMENT FOR TIMEDEPENDENT DATA ANALYSIS Version 1.4 July 2007 First edition Intended for use with Mathematica 6 or higher Software and manual: Yu He, John Novak, Darren Glosemeyer
More informationDo Electricity Prices Reflect Economic Fundamentals?: Evidence from the California ISO
Do Electricity Prices Reflect Economic Fundamentals?: Evidence from the California ISO Kevin F. Forbes and Ernest M. Zampelli Department of Business and Economics The Center for the Study of Energy and
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationTraffic Safety Facts. Research Note. Time Series Analysis and Forecast of Crash Fatalities during Six Holiday Periods Cejun Liu* and ChouLin Chen
Traffic Safety Facts Research Note March 2004 DOT HS 809 718 Time Series Analysis and Forecast of Crash Fatalities during Six Holiday Periods Cejun Liu* and ChouLin Chen Summary This research note uses
More informationLecture 1: Univariate Time Series B41910: Autumn Quarter, 2008, by Mr. Ruey S. Tsay
Lecture 1: Univariate Time Series B41910: Autumn Quarter, 2008, by Mr. Ruey S. Tsay 1 Some Basic Concepts 1. Time Series: A sequence of random variables measuring certain quantity of interest over time.
More informationLecture 20: Transmission (ABCD) Matrix.
Whites, EE 48/58 Lecture 0 Page of 7 Lecture 0: Transmission (ABC) Matrix. Concerning the equivalent port representations of networks we ve seen in this course:. Z parameters are useful for series connected
More informationTons of Free Math Worksheets at:
Topic : Equations of Circles  Worksheet 1 form which has a center at (6, 4) and a radius of 5. x 28x+y 28y12=0 circle with a center at (4,4) and passes through the point (7, 3). center located at
More informationTime Series Analysis in Economics. Klaus Neusser
Time Series Analysis in Economics Klaus Neusser May 26, 2015 Contents I Univariate Time Series Analysis 3 1 Introduction 1 1.1 Some examples.......................... 2 1.2 Formal definitions.........................
More informationSoftware Review: ITSM 2000 Professional Version 6.0.
Lee, J. & Strazicich, M.C. (2002). Software Review: ITSM 2000 Professional Version 6.0. International Journal of Forecasting, 18(3): 455459 (June 2002). Published by Elsevier (ISSN: 01692070). http://0
More informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
More informationTime Series Analysis of Aviation Data
Time Series Analysis of Aviation Data Dr. Richard Xie February, 2012 What is a Time Series A time series is a sequence of observations in chorological order, such as Daily closing price of stock MSFT in
More informationIntroduction. Logic. Most Difficult Reading Topics. Basic Logic Gates Truth Tables Logical Functions. COMP370 Introduction to Computer Architecture
Introduction LOGIC GATES COMP370 Introduction to Computer Architecture Basic Logic Gates Truth Tables Logical Functions Truth Tables Logical Expression Graphical l Form Most Difficult Reading Topics Logic
More informationLines and Planes in R 3
.3 Lines and Planes in R 3 P. Daniger Lines in R 3 We wish to represent lines in R 3. Note that a line may be described in two different ways: By specifying two points on the line. By specifying one point
More informationTime Series Analysis III
Lecture 12: Time Series Analysis III MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis III 1 Outline Time Series Analysis III 1 Time Series Analysis III MIT 18.S096 Time Series Analysis
More informationVariable. 1.1 Order of Operations. August 17, evaluating expressions ink.notebook. Standards. letter or symbol used to represent a number
1.1 evaluating expressions ink.notebook page 8 Unit 1 Basic Equations and Inequalities 1.1 Order of Operations page 9 Square Cube Variable Variable Expression Exponent page 10 page 11 1 Lesson Objectives
More informationTIME SERIES. Syllabus... Keywords...
TIME SERIES Contents Syllabus.................................. Books................................... Keywords................................. iii iii iv 1 Models for time series 1 1.1 Time series
More informationDIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION
1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: dsorton1@gmail.com Abstract: There are many longstanding
More informationVisualizing Differential Equations Slope Fields. by Lin McMullin
Visualizing Differential Equations Slope Fields by Lin McMullin The topic of slope fields is new to the AP Calculus AB Course Description for the 2004 exam. Where do slope fields come from? How should
More informationPhysics 505 Fall 2007 Homework Assignment #2 Solutions. Textbook problems: Ch. 2: 2.2, 2.8, 2.10, 2.11
Physics 55 Fall 27 Homework Assignment #2 Solutions Textbook problems: Ch. 2: 2.2, 2.8, 2., 2. 2.2 Using the method of images, discuss the problem of a point charge q inside a hollow, grounded, conducting
More informationthe points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
More informationPreliminary Mathematics
Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and
More informationStatistics and Probability Letters
Statistics and Probability Letters 79 (2009) 1884 1889 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Discretevalued ARMA
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationBell Ringer. Solve each equation. Show you work. Check the solution. 8 = 7 + m = m 15 = m = 7 + m 8 = = 8
Bell Ringer Solve each equation. Show you work. the solution. 1. 8 = 7 + m 8 = 7 + m 8 + 7 = 7 + 7 + m 15 = m 8 = 7 + m 8 = 7 + 15 8 = 8 Answers to Homework Worksheet 21 Today s Objectives Solving
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationLecture Notes on Univariate Time Series Analysis and Box Jenkins Forecasting John Frain Economic Analysis, Research and Publications April 1992 (reprinted with revisions) January 1999 Abstract These are
More informationStudy & Development of Short Term Load Forecasting Models Using Stochastic Time Series Analysis
International Journal of Engineering Research and Development eissn: 2278067X, pissn: 2278800X, www.ijerd.com Volume 9, Issue 11 (February 2014), PP. 3136 Study & Development of Short Term Load Forecasting
More informationTime Series Analysis and Forecasting
Time Series Analysis and Forecasting Math 667 Al Nosedal Department of Mathematics Indiana University of Pennsylvania Time Series Analysis and Forecasting p. 1/11 Introduction Many decisionmaking applications
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
More informationŽ Ž Ň Ň Č ú Ď Ř Á Ó ž ž Ú ž ž Ň Á Š ž É Á Ň Ň Ň ú Ú Š Ó ž Ř ú Á ž Ď ú ú Ú Ú Ň Ž Á Ž Ž Á Ž Č É Ó Á Ž Ž Ř Ž Ř Ž Ř Ž Ř Ž Ž Ř Ž Ž Ž Ř Ž Ž Ž Ž Ř ú Ž Ž Ř Ý Š Ď ž Ý ž Ý Ď Ď Ž Á ú Š Á ž Ď ŽŽ Š Á Ň Ý ž Ď ú Č Ú
More informationWorking Paper Pattern recognition in intensive care online monitoring
econstor www.econstor.eu Der OpenAccessPublikationsserver der ZBW LeibnizInformationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Fried,
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More information7. Cauchy s integral theorem and Cauchy s integral formula
7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose
More informationForecasting model of electricity demand in the Nordic countries. Tone Pedersen
Forecasting model of electricity demand in the Nordic countries Tone Pedersen 3/19/2014 Abstract A model implemented in order to describe the electricity demand on hourly basis for the Nordic countries.
More informationIn this paper we study how the timeseries structure of the demand process affects the value of information
MANAGEMENT SCIENCE Vol. 51, No. 6, June 25, pp. 961 969 issn 25199 eissn 1526551 5 516 961 informs doi 1.1287/mnsc.15.385 25 INFORMS Information Sharing in a Supply Chain Under ARMA Demand Vishal Gaur
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More information3.1 Stationary Processes and Mean Reversion
3. Univariate Time Series Models 3.1 Stationary Processes and Mean Reversion Definition 3.1: A time series y t, t = 1,..., T is called (covariance) stationary if (1) E[y t ] = µ, for all t Cov[y t, y t
More informationCorrelational Research
Correlational Research Chapter Fifteen Correlational Research Chapter Fifteen Bring folder of readings The Nature of Correlational Research Correlational Research is also known as Associational Research.
More informationGENERALIZED PYTHAGOREAN TRIPLES AND PYTHAGOREAN TRIPLE PRESERVING MATRICES. Mohan Tikoo and Haohao Wang
VOLUME 1, NUMBER 1, 009 3 GENERALIZED PYTHAGOREAN TRIPLES AND PYTHAGOREAN TRIPLE PRESERVING MATRICES Mohan Tikoo and Haohao Wang Abstract. Traditionally, Pythagorean triples (PT) consist of three positive
More informationWelcome! 8 th Grade Honors Algebra Informational Meeting
Welcome! 8 th Grade Honors Algebra Informational Meeting Agenda Why Honors Algebra In Middle School? Placement Process Specifics of the Class Student Responsibilities Parent Responsibilities Math Program
More informationNotes 5: More on regression and residuals ECO 231W  Undergraduate Econometrics
Notes 5: More on regression and residuals ECO 231W  Undergraduate Econometrics Prof. Carolina Caetano 1 Regression Method Let s review the method to calculate the regression line: 1. Find the point of
More informationMatrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.
2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true
More informationAdvanced Linear Modeling
Ronald Christensen Advanced Linear Modeling Multivariate, Time Series, and Spatial Data; Nonparametric Regression and Response Surface Maximization Second Edition Springer Preface to the Second Edition
More information4.4 Transforming Circles
Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby 1 Outline of the lecture Identification of univariate time series models, cont.:
More informationEULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.
EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If
More informationChapter 9: Univariate Time Series Analysis
Chapter 9: Univariate Time Series Analysis In the last chapter we discussed models with only lags of explanatory variables. These can be misleading if: 1. The dependent variable Y t depends on lags of
More information15.1 Factoring Polynomials
LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE
More informationHigher Mathematics Homework A
Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(3,1) and B(5,3) 2. Find the equation of the tangent to the circle
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationMatrices: 2.3 The Inverse of Matrices
September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations.
More information5.1 The Unit Circle. Copyright Cengage Learning. All rights reserved.
5.1 The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives The Unit Circle Terminal Points on the Unit Circle The Reference Number 2 The Unit Circle In this section we explore some
More informationEconomics 457: Financial Econometrics
Lutz Kilian Winter 2017 Economics 457: Financial Econometrics Lecture: Monday/Wednesday 4:00PM5:30PM in Lorch 173. Office hours: Monday after class. First Day of Class: Wednesday, January 4. Last Day
More informationSurvey of Algebra Teachers Topic: Teaching Algebra in a Block Schedule
Survey of Algebra Teachers Topic: Teaching Algebra in a Block Schedule Consider the following statements involving teaching algebra in an alternate day extended block schedule. Indicate whether you: 1
More informationSOLVING EQUATIONS WITH EXCEL
SOLVING EQUATIONS WITH EXCEL Excel and Lotus software are equipped with functions that allow the user to identify the root of an equation. By root, we mean the values of x such that a given equation cancels
More informationTIME SERIES ANALYSIS: AN OVERVIEW
Outline TIME SERIES ANALYSIS: AN OVERVIEW Hernando Ombao University of California at Irvine November 26, 2012 Outline OUTLINE OF TALK 1 TIME SERIES DATA 2 OVERVIEW OF TIME DOMAIN ANALYSIS 3 OVERVIEW SPECTRAL
More informationSolving Compound Interest Problems
Solving Compound Interest Problems What is Compound Interest? If you walk into a bank and open up a savings account you will earn interest on the money you deposit in the bank. If the interest is calculated
More informationAn Explicit Expression of Average Run Length of Exponentially Weighted Moving Average Control Chart with ARIMA (p,d,q)(p, D, Q)L Models
Preprints (www.preprints.org) NOT PEERREVIEWED Posted: 5 August 26 doi:.2944/preprints268.46.v Article An Explicit Expression of Average Run Length of Exponentially Weighted Moving Average Control Chart
More information