\\\\( \nonumber \newcommand{\bevisslut}{$\blacksquare$} \newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}} \newcommand{\transp}{\hspace{-.6mm}^{\top}} \newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace} \newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}} \newcommand{\eqnl}{} \newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}} \newcommand{\vekind}[2]{{_\mathrm{#1}\mathbf{#2}}} \newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}} \newcommand{\diver}[2]{{\mathrm{div}\mathbf{#1} (#2)}} \newcommand{\rot}[1]{{\mathbf{rot}\mathbf{(#1)}}} \newcommand{\am}{\mathrm{am}} \newcommand{\gm}{\mathrm{gm}} \newcommand{\E}{\mathrm{E}} \newcommand{\Span}{\mathrm{span}} \newcommand{\mU}{\mathbf{U}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mB}{\mathbf{B}} \newcommand{\mC}{\mathbf{C}} \newcommand{\mD}{\mathbf{D}} \newcommand{\mE}{\mathbf{E}} \newcommand{\mF}{\mathbf{F}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mM}{\mathbf{M}} \newcommand{\mN}{\mathbf{N}} \newcommand{\mQ}{\mathbf{Q}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mW}{\mathbf{W}} \newcommand{\mX}{\mathbf{X}} \newcommand{\ma}{\mathbf{a}} \newcommand{\mb}{\mathbf{b}} \newcommand{\mc}{\mathbf{c}} \newcommand{\md}{\mathbf{d}} \newcommand{\me}{\mathbf{e}} \newcommand{\mn}{\mathbf{n}} \newcommand{\mr}{\mathbf{r}} \newcommand{\mv}{\mathbf{v}} \newcommand{\mw}{\mathbf{w}} \newcommand{\mx}{\mathbf{x}} \newcommand{\mxb}{\mathbf{x_{bet}}} \newcommand{\my}{\mathbf{y}} \newcommand{\mz}{\mathbf{z}} \newcommand{\reel}{\mathbb{R}} \newcommand{\mL}{\bm{\Lambda}} \newcommand{\mnul}{\mathbf{0}} \newcommand{\trap}[1]{\mathrm{trap}(#1)} \newcommand{\Det}{\operatorname{Det}} \newcommand{\adj}{\operatorname{adj}} \newcommand{\Ar}{\operatorname{Areal}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Rum}{\operatorname{Rum}} \newcommand{\diag}{\operatorname{\bf{diag}}} \newcommand{\bidiag}{\operatorname{\bf{bidiag}}} \newcommand{\spanVec}[1]{\mathrm{span}{#1}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\Rot}{\operatorname{\mathbf{Rot}}} \newcommand{\Jac}{\operatorname{Jacobi}} \newcommand{\Tan}{\operatorname{Tan}} \newcommand{\Ort}{\operatorname{Ort}} \newcommand{\Flux}{\operatorname{Flux}} \newcommand{\Cmass}{\operatorname{Cm}} \newcommand{\Imom}{\operatorname{Im}} \newcommand{\Pmom}{\operatorname{Pm}} \newcommand{\IS}{\operatorname{I}} \newcommand{\IIS}{\operatorname{II}} \newcommand{\IIIS}{\operatorname{III}} \newcommand{\Le}{\operatorname{L}} \newcommand{\app}{\operatorname{app}} \newcommand{\M}{\operatorname{M}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\compl}{\mathbb{C}} \newcommand{\e}{\mathrm{e}} \\\\)

Reading and exercises in Time Series Analysis - spring 2025

The book is available here as pdf: Time Series Analysis.

You can also buy it in the bookstore.

For each week the chapters or sections to be read are listed below.

Sample solutions are available here: Solutions. They are from the book website, but some been corrected a bit.

Week 1 - Introduction

  • Book
    • Chapter 1: Introduction
    • Chapter 2 (not 2.8): Multivariate random variables
  • Exercises

Week 2 - Regression based methods, 1st part

The General Linear Model (GLM): Ordinary Least Squares (OLS) and Maximum Likelihood (ML) estimation.

Recursive Least Squares (RLS) and global trend models, as well as some principles needed to find a good model, such as model selection.

  • Book
    • 3.1: Introduction
    • 3.2: The GLM, including OLS-, and ML-estimates (but we skip Section 3.2.1.4 on WLS))
    • 3.3: Prediction in the GLM.
    • 11.1 (with introduction): RLS.
  • Additional material
  • Exercises
    • 3.1 (maybe try solve it by hand as in the book and then check result with a computer)
    • 3.4
    • You can start working on Assignment 1: Section 1 to 3 is covered.

Week 3 - Regression based methods, 2nd part

  • Book
    • 3.2.1.4 on WLS
    • 11.1.1 RLS with forgetting
  • Additional material
    • More on Local trend models (more to come)
  • Exercises
    • You can work on Assignment 1: Section 4 and 5 is about local trend models using WLS and RLS.

Week 4 - Introduction to Stochastic Processes, Operators and Linear Systems

  • 4.5: Shift operators (for understanding 5.3)
  • 5.1 and 5.2: Stochastic processes in general
  • 5.3 (only slightly touch 5.3.2): Linear processes
  • Exercises
    • 5.1 (for c != 0) (Q2: see page 117 for MA(1) process)
    • 5.4
    • 5.7

Week 5 - AR, MA and ARMA processes

  • 5.5 (disregard ‘spectra’ like (5.67), (5.72), (5.85), (5.86), (5.112)): MA, AR, and ARMA-processes
  • 5.3.2: Cursory material
  • 5.6: Non-stationary models
  • 5.7: Optimal Prediction
  • 6.4: Estimation of parameters in ARMA models
  • Exercises
    • Identification game
    • 5.5
    • 5.6 (Assume that the process is stationary and invertible.)
    • 5.10 (if time allows. In Question 2 you are expected to find a recursion for gamma(k) for k>2. Skip Question 3.),

Week 6 - ACF and PACF with a focus on model order selection

Identification of univariate time series models, 1st part:

  • 6.1 (with intro): Introduction
  • 6.2.1 (and the introduction to 6.2 (Sec. 6.2.1 (a))): Estimation of auto-covariance and -correlation
  • 6.3 (not 6.3.3): Using the SACF and SPACF for model order selection
  • 6.5: Model order selection
  • 6.6: Model validation
  • Exercises

Week 7 - Linear systems

  • 4.1 (with intro): Linear Systems
  • 4.4: You should disregard Theorem 4.10 and the following example. Furthermore, we shall not discuss Theorem 4.12 until we start to look at the multivariate time series.
  • 6.2.2: Cross-correlation functions
  • Chapter 8: Linear systems and stochastic processes
  • Exercises

Week 8 - Multivariate time series

  • Chapter 9 (read it coarsely, the bi-variate ARMA is good to think about, and then the rest: think like it’s the same as for uni-variate, just extended to be multi-variable in VARMAX models, as in VectorARMAX).
  • MARIMA: Spliids method for parameter estimation in multi-variable ARMAX models.
  • Have a look at the original paper (don’t go too much into details, it’s too cumbersome): Marima_paper

  • Have a look at the marima R package vignette (again, don’t go into details): Marima_vignette

  • Exercises
    • Bivariate-output ARMAX exercise: exercise_bivariate_waterheating
    • If time allows, take a look at 9.1 and its solutions, just to get some points about the theoretical ACF in a bivariate model.

Week 9 - TBD

TBD

Exercises TBD

Week 10 - State space models 1st part

  • 10.1: The Linear Stochastic State Space Model, Sec. 10.1
  • 10.3: The Kalman filter
  • Exercises

Week 11 - State space models 2nd part

  • 10.4 (not 10.4.1): ARMA-models on state space form
  • 10.6: ML-estimates of state space models
  • Exercises

Week 12 - Recursive and adaptive estimation

  • Chapter 11: Recursive and adaptive estimation

Execises:

  • 10.4
  • Get help on Assignment4

Week 13 - Final lecture

During the final lecture we will consider a number of real world problems and discuss what tools from the course can be used to solve them.

  • Exercises
    • Get help on Assignment4