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)
12.1. Introduction 301series is almost a pure oscillation with angular frequency λ 0 , then f(λ) islarge for λ close to λ 0 and near zero elsewhere. This behaviour is signalled inthe autocovariances by a large value of γ k at k = k 0 , where k 0 is the periodof oscillation corresponding to angular frequency λ 0 (that is k 0 =2π/λ 0 ),and small values elsewhere.Because there are two different but equivalent functions (12.1.1) and(12.1.2) expressing the second-order behaviour of a time series, there aretwo different types of analysis of time series, namely in the time domainusing (12.1.1) and in the frequency domain using (12.1.2).Consider now a time series that consists not of a single variable, but pvariables. The definitions (12.1.1), (12.1.2) generalize readily towhereandΓ k = E[(x i − µ)(x i+k − µ) ′ ], (12.1.3)F(λ) = 12πµ = E[x i ]∞∑k=−∞Γ k e −ikλ (12.1.4)The mean µ is now a p-element vector, and Γ k , F(λ) are(p × p) matrices.Principal component analysis operates on a covariance or correlationmatrix, but in time series we can calculate not only covariances betweenvariables measured at the same time (the usual definition of covariance,which is given by the matrix Γ 0 defined in (12.1.3)), but also covariancesbetween variables at different times, as measured by Γ k ,k ≠0.Thisisincontrast to the more usual situation where our observations x 1 , x 2 ,... areindependent, so that any covariances between elements of x i , x j are zerowhen i ≠ j. In addition to the choice of which Γ k to examine, the fact thatthe covariances have an alternative representation in the frequency domainmeans that there are several different ways in which PCA can be appliedto time series data.Before looking at specific techniques, we define the terms ‘white noise’and ‘red noise.’ A white noise series is one whose terms are all identicallydistributed and independent of each other. Its spectrum is flat, like that ofwhite light; hence its name. Red noise is equivalent to a series that followsa positively autocorrelated first-order autoregressive modelx t = φx t−1 + ɛ t , t = ...0, 1, 2 ...,where φ is a constant such that 0
302 12. PCA for Time Series and Other Non-Independent Dataencountered in this area, with observations corresponding to times andvariables to spatial position, they are not necessarily restricted to suchdata.Time series are usually measured at discrete points in time, but sometimesthe series are curves. The analysis of such data is known as functionaldata analysis (functional PCA is the subject of Section 12.3). The final sectionof the chapter collects together a number of largely unconnected ideasand references concerning PCA in the context of time series and othernon-independent data.12.2 PCA-Related Techniques for (Spatio-)Temporal Atmospheric Science DataIt was noted in Section 4.3 that, for a common type of data in atmosphericscience, the use of PCA, more often referred to as empirical orthogonalfunction (EOF) analysis, is widespread. The data concerned consist of measurementsof some variable, for example, sea level pressure, temperature,...,atp spatial locations (usually points on a grid) at n different times. Themeasurements at different spatial locations are treated as variables and thetime points play the rôle of observations. An example of this type was givenin Section 4.3. It is clear that, unless the observations are well-separatedin time, there is likely to be correlation between measurements at adjacenttime points, so that we have non-independence between observations. Severaltechniques have been developed for use in atmospheric science that takeaccount of correlation in both time and space, and these will be describedin this section. First, however, we start with the simpler situation wherethere is a single time series. Here we can use a principal component-liketechnique, called singular spectrum analysis (SSA), to analyse the autocorrelationin the series. SSA is described in Section 12.2.1, as is its extensionto several time series, multichannel singular spectrum analysis (MSSA).Suppose that a set of p series follows a multivariate first-order autoregressivemodel in which the values of the series at time t are linearly relatedto the values at time (t − 1), except for a multivariate white noise term.An estimate of the matrix defining the linear relationship can be subjectedto an eigenanalysis, giving insight into the structure of the series. Such ananalysis is known as principal oscillation pattern (POP) analysis, and isdiscussed in Section 12.2.2.One idea underlying POP analysis is that there may be patterns in themaps comprising our data set, which travel in space as time progresses, andthat POP analysis can help to find such patterns. Complex (Hilbert) empiricalorthogonal functions (EOFs), which are described in Section 12.2.3,are designed to achieve the same objective. Detection of detailed oscillatorybehaviour is also the aim of multitaper frequency-domain singularvalue decomposition, which is the subject of Section 12.2.4.
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12.1. Introduction 301series is almost a pure oscillation with angular frequency λ 0 , then f(λ) islarge for λ close to λ 0 and near zero elsewhere. This behaviour is signalled inthe autocovariances by a large value of γ k at k = k 0 , where k 0 is the periodof oscillation corresponding to angular frequency λ 0 (that is k 0 =2π/λ 0 ),and small values elsewhere.Because there are two different but equivalent functions (12.1.1) and(12.1.2) expressing the second-order behaviour of a time series, there aretwo different types of analysis of time series, namely in the time domainusing (12.1.1) and in the frequency domain using (12.1.2).Consider now a time series that consists not of a single variable, but pvariables. The definitions (12.1.1), (12.1.2) generalize readily towhereandΓ k = E[(x i − µ)(x i+k − µ) ′ ], (12.1.3)F(λ) = 12πµ = E[x i ]∞∑k=−∞Γ k e −ikλ (12.1.4)The mean µ is now a p-element vector, and Γ k , F(λ) are(p × p) matrices.<strong>Principal</strong> component analysis operates on a covariance or correlationmatrix, but in time series we can calculate not only covariances betweenvariables measured at the same time (the usual definition of covariance,which is given by the matrix Γ 0 defined in (12.1.3)), but also covariancesbetween variables at different times, as measured by Γ k ,k ≠0.Thisisincontrast to the more usual situation where our observations x 1 , x 2 ,... areindependent, so that any covariances between elements of x i , x j are zerowhen i ≠ j. In addition to the choice of which Γ k to examine, the fact thatthe covariances have an alternative representation in the frequency domainmeans that there are several different ways in which PCA can be appliedto time series data.Before looking at specific techniques, we define the terms ‘white noise’and ‘red noise.’ A white noise series is one whose terms are all identicallydistributed and independent of each other. Its spectrum is flat, like that ofwhite light; hence its name. Red noise is equivalent to a series that followsa positively autocorrelated first-order autoregressive modelx t = φx t−1 + ɛ t , t = ...0, 1, 2 ...,where φ is a constant such that 0
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