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A time series ： A set of observations ， each one

being recorded at a specific time t 。

A discrete-time time series ： The set of times at

which observations are made is a discrete set.

e.g. ： observations made at fixed time intervals.

Continuous time series ： Observations are recorded

continuously over some time interval.

e.g. ： when

tX

0T

].1,0[0 T

1.1 Examples of Time Series

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Example 1.1.1 Australian red wine sales：WINE.DAT

The monthly sales（ in kilolitres） of red wine by Australian winemakers form January 1980 through October 1991.The set consists of 142 times（（ Jan. 1980） ,（ Feb. 1980） , … , （ Oct. 1991）） .It is often convenient to rescale the time axis becomes the set of integers { 1, 2,…, 142} .It appears from the graph that the sales have an upward trend and a seasonal pattern with a peak in July and a though in January.

0T

0T

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Figure 1.1 Australian red wine sales：WINE.DAT

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Example 1.1.2 All-star baseball games, 1993-1995

It also has some missing values no game was played in 1945 and two games were scheduled for each of the years 1959-1962.

year t.in won League National theif , 1

year t.in won LeagueAmerican theif , 1 tX

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Figure 1.2 All-star baseball games, 1993-1995

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Example 1.1.3 Accidental deaths, U.S.A., 1973-1978； DEATHS.DAT

The monthly accidental death figures show a strong seasonal pattern, with the maximum for each year occurring in July and the minimum for each year occurring in February.

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Figure 1.3 Accidental deaths, U.S.A., 1973-1978； DEATHS.DAT

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Example 1.1.4 A signal detection problem； SICNAL.DAT

Figure 1.4 shows simulated values of the series

where is a sequence of independent normal random variables, with mean 0 and variance 0.25.Such a series is often referred to as signal plus noise, the signal, being the smooth function, in this case.

tN,0.20,...,2.0,1.0 , cos tNtX tt

tSt cos

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問題： Given only the data , how can we determine the unknown signal component?

tX

建議方法：One simple approach is to smooth the data by expressing as a sum of sine waves of various frequencies and eliminating the high frequency.

tX

結果： If we do this to the values of shown in Figure 1.4 and retain only the lowest 4% of the frequency components, we obtain the estimate of the signal also shown in Figure 1.4. The waveform of the signal is quite close to that of the true signal in this case, although its amplitude is somewhat smaller.

tX

頻率要高，但振幅要小。

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Figure 1.4 A signal detection problem； SICNAL.DAT

tS

tSt cos

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Example 1.1.5 Population of the U.S.A., 1790-1990： USPOP.DAT

The population of the U.S.A., measured at ten-year intervals, is shown in Figure 1.5.The graph the possibility of fitting a quadratic or exponential trend to the data.

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Figure 1.5 Population of the U.S.A., 1790-1990： USPOP.DAT

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correlated Negative (4) [20,80] Range (3)

50 (2) Stationary )1( 有助於預測

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Nonstationary time series

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Nonstationary time series

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( 1 ) Nonstationary time series( 2 ) Seasonality 同月不同年的相依性

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某段期間政府執行新的空氣污染管制法案，以降低車輛污染之排放量 － 造成數列中觀測值在不同區段中的不一致。( 1 ) Interventions 介入事件。( 2 ) Outliers 離群值可以 Dynamic Model 來分析其影響效應。 （存在外部的干擾事件）

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投入產出型態的轉換函數模型（ Transfer function model ） .若討論變數間的複雜回饋關係，則可應用Vector ARMA model.

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Example 1.1.6 Strikes in the U.S.A., 1951-1980； STRIKE.DAT

The annual numbers of strides in the U.S.A. for the years 1951-1980 are shown in Figure 1.6. They appear to fluctuate erratically （不規則的）about a slowly changing level.

每年美國的罷工件數

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Figure 1.6 Strikes in the U.S.A., 1951-1980； STRIKE.DAT

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1.2 Objectives of Time Series Analysis

Fields of engineering, science, sociology, and economics.

時間序列資料來源：

Study techniques for drawing inferences from time series.

本書的目標：

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Set up a hypothetical probability model to represent the data.

( 1 ) Choose a model ( or family of models ).

( 2 ) Estimate parameters.

( 3 ) Check for goodness of fit to the data.

( 4 ) Use the fitted model to enhance our understanding of the mechanism generating the series.

分析的步驟：

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Once a satisfactory model has been develop, it may be used in a variety of ways depending on the particular field of application.

( 1 ) Provide a compact description of the data.e.g. (a) Represent the accidental deaths data as the sum of the specified trend, and seasonal and random terms.

合理模型之應用：

● 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 （移去季節效應的過程 ).

(b) Uemployment figures: interpret economic statistics.

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( 2 ) Separation ( or filtering ) of noise from signals as in Example 1.1.4.

( 3 ) Prediction of future values of a series.e.g. The red wine sales or the population data.

( 4 ) Testing hypotheses.e.g. Global warming using recorded temperature data.

( 5 ) Predicting one series from observations of another.e.g. Predicting future sales using advertising expenditure （廣告的費用） data.

( 6 ) Controlling future values of a series by adjusting parameters.