Time Series

A Data Analysis Approach Using R

Robert Shumway, David Stoffer, Robert H. Shunway, David S. Stoffer, Robert H. Shumway

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9780367221096
• eBook ISBN: 9781000008395
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

The goals of this text are to develop the skills and an appreciation for the richness and versatility of modern time series analysis as a tool for analyzing dependent data. A useful feature of the presentation is the inclusion of nontrivial data sets illustrating the richness of potential applications to problems in the biological, physical, and social sciences as well as medicine. The text presents a balanced and comprehensive treatment of both time and frequency domain methods with an emphasis on data analysis.

Numerous examples using data illustrate solutions to problems such as discovering natural and anthropogenic climate change, evaluating pain perception experiments using functional magnetic resonance imaging, and the analysis of economic and financial problems. The text can be used for a one semester/quarter introductory time series course where the prerequisites are an understanding of linear regression, basic calculus-based probability skills, and math skills at the high school level. All of the numerical examples use the R statistical package without assuming that the reader has previously used the software.

Chapter 1

Time Series Elements

1.1 Introduction

The analysis of data observed at different time points leads to unique problems that are not covered by classical statistics. The dependence introduced by the sampling data over time restricts the applicability of many conventional statistical methods that require random samples. The analysis of such data is commonly referred to as time series analysis.

To provide a statistical setting for describing the elements of time series data, the data are represented as a collection of random variables indexed according to the order they are obtained in time. For example, if we collect data on daily high temperatures in your city, we may consider the time series as a sequence of random variables, x₁, x₂, x₃,…, where the random variable x₁ denotes the high temperature on day one, the variable x₂ denotes the value for the second day, x₃ denotes the value for the third day, and so on. In general, a collection of random variables, {x_t}, indexed by t is referred to as a stochastic process. In this text, t will typically be discrete and vary over the integers t = 0, ±1, ±2,… or some subset of the integers, or a similar index like months of a year.

Historically, time series methods were applied to problems in the physical and environmental sciences. This fact accounts for the engineering nomenclature that permeates the language of time series analysis. The first step in an investigation of time series data involves careful scrutiny of the recorded data plotted over time. Before looking more closely at the particular statistical methods, we mention that two separate, but not mutually exclusive, approaches to time series analysis exist, commonly identified as the time domain approach (Chapter 4 and 5) and the frequency domain approach (Chapter 6 and 7).

1.2 Time Series Data

The following examples illustrate some of the common kinds of time series data as well as some of the statistical questions that might be asked about such data.

Figure 1.1 Johnson & Johnson quarterly earnings per share, 1960-I to 1980-IV (top). The same data logged (bottom).

Example 1.1. Johnson & Johnson Quarterly Earnings

Figure 1.1 shows quarterly earnings per share (QEPS) for the U.S. company Johnson & Johnson and the data transformed by taking logs. There are 84 quarters (21 years) measured from the first quarter of 1960 to the last quarter of 1980. Modeling such series begins by observing the primary patterns in the time history. In this case, note the increasing underlying trend and variability, and a somewhat regular oscillation superimposed on the trend that seems to repeat over quarters. Methods for analyzing data such as these are explored in Chapter 3 (see Problem 3.1) using regression techniques.

If we consider the data as being generated as a small percentage change each year, say r_t (which can be negative), we might write $x_t=(1+r_t)x_t{{}_{–}}_4$, where x_t is the QEPS for quarter t. If we log the data, then $\log (x_t)=\log (1+r_t)+\log (x_t{{}_{–}}_4)$, implying a linear growth rate; i.e., this quarter’s value is the same as last year plus a small amount, log(1 + r_t). This attribute of the data is displayed by the bottom plot of Figure 1.1.

The R code to plot the data for this example is,¹

library(astsa) # we leave this line off subsequent examples

par(mfrow=2:1)

tsplot(jj, ylab="QEPS", type="o", col=4, main="Johnson & Johnson Quarterly Earnings")

tsplot(log(jj), ylab="log(QEPS)", type="o", col=4)

⋄

Figure 1.2 Yearly average global land surface and ocean surface temperature deviations (1880–2017) in °C.

Example 1.2. Global Warming and Climate Change

Two global temperature records are shown in Figure 1.2. The data are (1) annual temperature anomalies averaged over the Earth’s land area, and (2) sea surface temperature anomalies averaged over the part of the ocean that is free of ice at all times (open ocean). The time period is 1880 to 2017 and the values are deviations (°C) from the 1951–1980 average, updated from Hansen et al. (2006). The upward trend in both series during ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Time Series Elements
• 2 Correlation and Stationary Time Series
• 3 Time Series Regression and EDA
• 4 ARMA Models
• 5 ARIMA Models
• 6 Spectral Analysis and Filtering
• 7 Spectral Estimation
• 8 Additional Topics *
• Appendix A R Supplement
• Appendix B Probability and Statistics Primer
• Appendix C Complex Number Primer
• Appendix D Additional Time Domain Theory
• Hints for Selected Exercises
• References
• Index