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2009 | Buch

Introductory Time Series with R

verfasst von: Andrew V. Metcalfe, Paul S.P. Cowpertwait

Verlag: Springer New York

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SUCHEN

Inhaltsverzeichnis

Frontmatter
1. Time Series Data
Time series are analysed to understand the past and to predict the future, enabling managers or policy makers to make properly informed decisions. A time series analysis quantifies the main features in data and the random variation. These reasons, combined with improved computing power, have made time series methods widely applicable in government, industry, and commerce.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
2. Correlation
Once we have identified any trend and seasonal effects, we can deseasonalise the time series and remove the trend. If we use the additive decomposition method of §1.5, we first calculate the seasonally adjusted time series and then remove the trend by subtraction. This leaves the random component, but the random component is not necessarily well modelled by independent random variables. In many cases, consecutive variables will be correlated. If we identify such correlations, we can improve our forecasts, quite dramatically if the correlations are high. We also need to estimate correlations if we are to generate realistic time series for simulations. The correlation structure of a time series model is defined by the correlation function, and we estimate this from the observed time series.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
3. Forecasting Strategies
Businesses rely on forecasts of sales to plan production, justify marketing decisions, and guide research. A very efficient method of forecasting one variable is to find a related variable that leads it by one or more time intervals. The closer the relationship and the longer the lead time, the better this strategy becomes. The trick is to find a suitable lead variable. An Australian example is the Building Approvals time series published by the Australian Bureau of Statistics. This provides valuable information on the likely demand over the next few months for all sectors of the building industry. A variation on the strategy of seeking a leading variable is to find a variable that is associated with the variable we need to forecast and easier to predict.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
4. Basic Stochastic Models
So far, we have considered two approaches for modelling time series. The first is based on an assumption that there is a fixed seasonal pattern about a trend.We can estimate the trend by local averaging of the deseasonalised data, and this is implemented by the R function decompose. The second approach allows the seasonal variation and trend, described in terms of a level and slope, to change over time and estimates these features by exponentially weighted averages. We used the HoltWinters function to demonstrate this method.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
5. Regression
Trends in time series can be classified as stochastic or deterministic. We may consider a trend to be stochastic when it shows inexplicable changes in direction, and we attribute apparent transient trends to high serial correlation with random error. Trends of this type, which are common in financial series, can be simulated in R using models such as the random walk or autoregressive process (Chapter 4). In contrast, when we have some plausible physical explanation for a trend we will usually wish to model it in some deterministic manner. For example, a deterministic increasing trend in the data may be related to an increasing population, or a regular cycle may be related to a known seasonal frequency. Deterministic trends and seasonal variation can be modelled using regression.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
6. Stationary Models
As seen in the previous chapters, a time series will often have well-defined components, such as a trend and a seasonal pattern. A well-chosen linear regression may account for these non-stationary components, in which case the residuals from the fitted model should not contain noticeable trend or seasonal patterns. However, the residuals will usually be correlated in time, as this is not accounted for in the fitted regression model.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
7. Non-stationary Models
As we have discovered in the previous chapters, many time series are nonstationary because of seasonal effects or trends. In particular, random walks, which characterise many types of series, are non-stationary but can be transformed to a stationary series by first-order differencing (§4.4). In this chapter we first extend the random walk model to include autoregressive and moving average terms. As the differenced series needs to be aggregated (or ‘integrated’) to recover the original series, the underlying stochastic process is called autoregressive integrated moving average, which is abbreviated to ARIMA.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
8. Long-Memory Processes
Some time series exhibit marked correlations at high lags, and they are referred to as long-memory processes. Long-memory is a feature of many geophysical time series. Flows in the Nile River have correlations at high lags, and Hurst (1951) demonstrated that this affected the optimal design capacity of a dam. Mudelsee (2007) shows that long-memory is a hydrological property that can lead to prolonged drought or temporal clustering of extreme floods. At a rather different scale, Leland et al. (1993) found that Ethernet local area network (LAN) traffic appears to be statistically self-similar and a long-memory process. They showed that the nature of congestion produced by self-similar traffic differs drastically from that predicted by the traffic models used at that time.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
9. Spectral Analysis
Although it follows from the definition of stationarity that a stationary time series model cannot have components at specific frequencies, it can nevertheless be described in terms of an average frequency composition. Spectral analysis distributes the variance of a time series over frequency, and there are many applications. It can be used to characterise wind and wave forces, which appear random but have a frequency range over which most of the power is concentrated. The British Standard BS6841, “Measurement and evaluation of human exposure to whole-body vibration”, uses spectral analysis to quantify exposure of personnel to vibration and repeated shocks. Many of the early applications of spectral analysis were of economic time series, and there has been recent interest in using spectral methods for economic dynamics analysis (Iacobucci and Noullez, 2005).
Paul S.P. Cowpertwait, Andrew V. Metcalfe
10. System Identification
Vibration is defined as an oscillatory movement of some entity about an equilibrium state. It is the means of producing sound in musical instruments, it is the principle underlying the design of loudspeakers, and it describes the response of buildings to earthquakes. The squealing of disc brakes on a car is caused by vibration. The up and down motion of a ship at sea is a lowfrequency vibration. Spectral analysis provides the means for understanding and controlling vibration.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
11. Multivariate Models
Data are often collected on more than one variable. For example, in economics, daily exchange rates are available for a large range of currencies, or, in hydrological studies, both rainfall and river flow measurements may be taken at a site of interest. In Chapter 10, we considered a frequency domain approach where variables are classified as inputs or outputs to some system. In this chapter, we consider time domain models that are suitable when measurements have been made on more than one time series variable. We extend the basic autoregressive model to the vector autoregressive model, which has more than one dependent time series variable, and look at methods in R for fitting such models.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
12. State Space Models
The state space formulation for time series models is quite general and encompasses most of the models we have considered so far. However, it is usually simpler to use the specific time series models we have already introduced when they are appropriate for the physical situation. Here, we shall focus on applications for which we require parameters to adapt over time, and to do so more quickly than in a Holt-Winters model. The recent turmoil on the world’s stock exchanges is a dramatic reminder that time series are subject to sudden changes. Another desirable feature of state space models is that they can incorporate time series of predictor variables in a straightforward manner.
Paul S.P. Cowpertwait, Andrew V. Metcalfe
Backmatter
Metadaten
Titel
Introductory Time Series with R
verfasst von
Andrew V. Metcalfe
Paul S.P. Cowpertwait
Copyright-Jahr
2009
Verlag
Springer New York
Electronic ISBN
978-0-387-88698-5
Print ISBN
978-0-387-88697-8
DOI
https://doi.org/10.1007/978-0-387-88698-5