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Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

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300 12. PCA for Time Series and Other Non-Independent DataWe have already seen a number of examples where the data are timeseries, but where no special account is taken of the dependence betweenobservations. Section 4.3 gave an example of a type that is common inatmospheric science, where the variables are measurements of the samemeteorological variable made at p different geographical locations, and then observations on each variable correspond to different times. Section 12.2largely deals with techniques that have been developed for data of this type.The examples given in Section 4.5 and Section 6.4.2 are also illustrations ofPCA applied to data for which the variables (stock prices and crime rates,respectively) are measured at various points of time. Furthermore, one ofthe earliest published applications of PCA (Stone, 1947) was on (economic)time series data.In time series data, dependence between the x vectors is induced bytheir relative closeness in time, so that x h and x i will often be highlydependent if |h−i| is small, with decreasing dependence as |h−i| increases.This basic pattern may in addition be perturbed by, for example, seasonaldependence in monthly data, where decreasing dependence for increasing|h − i| is interrupted by a higher degree of association for observationsseparated by exactly one year, two years, and so on.Because of the emphasis on time series in this chapter, we need to introducesome of its basic ideas and definitions, although limited space permitsonly a rudimentary introduction to this vast subject (for more informationsee, for example, Brillinger (1981); Brockwell and Davis (1996); or Hamilton(1994)). Suppose, for the moment, that only a single variable is measured atequally spaced points in time. Our time series is then ...x −1 ,x 0 ,x 1 ,x 2 ,....Much of time series analysis is concerned with series that are stationary, andwhich can be described entirely by their first- and second-order moments;these moments areµ = E(x i ), i = ...,−1, 0, 1, 2,...γ k = E[(x i − µ)(x i+k − µ)], i = ...,−1, 0, 1, 2,... (12.1.1)k = ...,−1, 0, 1, 2,...,where µ is the mean of the series and is the same for all x i in stationaryseries, and γ k ,thekth autocovariance, is the covariance between x i andx i+k , which depends on k but not i for stationary series. The informationcontained in the autocovariances can be expressed equivalently in terms ofthe power spectrum of the seriesf(λ) = 12π∞∑k=−∞γ k e −ikλ , (12.1.2)where i = √ −1andλ denotes angular frequency. Roughly speaking, thefunction f(λ) decomposes the series into oscillatory portions with differentfrequencies of oscillation, and f(λ) measures the relative importance ofthese portions as a function of their angular frequency λ. For example, if a

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