《时间序列与预测》英文原著导论

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Introduction

1.1 Examples of Time Series

1.2 Objectives of Time Series Analysis

1.3 Some Simple Time Series Models

1.4 Stationary Models and the Autocorrelation Function

1.5 Estimation and Elimination of Trend and Seasonal Components

1.6 Testing the Estimated Noise Sequence

In this chapter we introduce some basic ideas of time series analysis and stochastic

processes. Of particular importance are the concepts of stationarity and the autocovari-

ance and sample autocovariance functions. Some standard techniques are described

for the estimation and removal of trend and seasonality (of known period) from

an observed time series. These are illustrated with reference to the data sets in

Section 1.1. The calculations in all the examples can be carried out using the time

series package ITSM, the professional version of which is av ailable at http://extras.

springer.com. The data sets are contained in files with names ending in .TSM. For

example, the Australian red wine sales are filed as WINE.TSM. Most of the topics

cov ered in this chapter will be developed more fully in later sections of the book. The

reader who is not already familiar with random variables and random vectors should

first read Appendix A, where a concise account of the required background is given.

1.1 Examples of Time Series

A time series is a set of observations x

, each one being recorded at a specific time t.

A discrete-time time series (the type to which this book is primarily devoted) is one

in which the set T

of times at which observations are made is a discrete set, as is the

case, for example, when observations are made at fixed time intervals. Continuous-

time time series are obtained when observations are recorded continuously over some

time interval, e.g., when T

=[0, 1].

P.J. Brockwell, R.A. Davis, Introduction to Time Series and Forecasting,

Springer Te xts in Statistics, DOI 10.1007/978-3-319-29854-2_1

1.2 Objectives of Time Series Analysis 5

Figure 1-6

Strikes in the

U.S.A., 1951–1980

(thousands)

1950 1960 1970 1980

3.5 4.0 4.5 5.0 5.5 6.0

1.2 Objectives of Time Series Analysis

The examples considered in Section 1.1 are an extremely small sample from the

multitude of time series encountered in the fields of engineering, science, sociology,

and economics. Our purpose in this book is to study techniques for drawing inferences

from such series. Before we can do this, however, it is necessary to set up a hypothetical

probability model to represent the data. After an appropriate family of models has

been chosen, it is then possible to estimate parameters, check for goodness of fit to

the data, and possibly to use the fitted model to enhance our understanding of the

mechanism generating the series. Once a satisfactory model has been developed, it

may be used in a variety of ways depending on the particular field of application.

The model may be used simply to provide a compact description of the data. We

may, for example, be able to represent the accidental deaths data of Example 1.1.3 as

the sum of a specified trend, and seasonal and random terms. For the interpretation

of economic statistics such as unemployment figures, it is important to recognize

the presence of seasonal components and to remove them so as not to confuse

them with long-term trends. This process is known as seasonal adjustment.Other

applications of time series models include separation (or filtering) of noise from signals

as in Example 1.1.4, prediction of future v alues of a series such as the red wine

sales in Example 1.1.1 or the population data in Example 1.1.5, testing hypotheses

such as global warming using recorded temperature data, predicting one series from

observations of another, e.g., predicting future sales using advertising expenditure data,

and controlling future values of a series by adjusting parameters. Time series models

are also useful in simulation studies. For e xample, the performance of a reservoir

depends heavily on the random daily inputs of water to the system. If these are modeled

as a time series, then we can use the fitted model to simulate a large number of

independent sequences of daily inputs. Knowing the size and mode of operation

of the reservoir, we can determine the fraction of the simulated input sequences that

cause the reservoir to run out of water in a given time period. This fraction will then be

an estimate of the probability of emptiness of the reservoir at some time in the given

period.

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