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

332 12. PCA for Time Series and Other Non-Independent Data12.4.3 Climate Change—Fingerprint TechniquesIn climate change detection, the objective is not only to discover whetherchange is taking place, but also to explain any changes found. If certaincauses are suspected, for example increases in greenhouse gases, variationsin solar output, or volcanic activity, then changes in these quantities can bebuilt into complex atmospheric models, and the models are run to discoverwhat happens to parameters of interest such as the global pattern of temperature.Usually changes in one (or more) of the potential causal variableswill manifest themselves, according to the model, in different ways in differentgeographical regions and for different climatic variables. The predictedpatterns of change are sometimes known as ‘fingerprints’ associated withchanges in the causal variable(s). In the detection of climate change it isusually more productive to attempt to detect changes that resemble suchfingerprints than to search broadly over a wide range of possible changes.The paper by Hasselmann (1979) is usually cited as the start of interestin this type of climate change detection, and much research has been donesubsequently. Very similar techniques have been derived via a number ofdifferent approaches. Zwiers (1999) gives a good, though not exhaustive,summary of several of these techniques, together with a number of applications.North and Wu (2001) describe a number of recent developments,again with applications.The basic idea is that the observed data, which are often values of someclimatic variable x tj , where t indexes time and j indexes spatial position,can be writtenx tj = s tj + e tj .Here s tj is the deterministic response to changes in the potential causalvariables (the signal), and e tj represents the stochastic noise associatedwith ‘normal’ climate variation.Suppose that an optimal detection variable A t at time t is constructed asa linear combination of the observed data x sj for s = t, (t−1),...,(t−l+1)and j =1, 2,...,m, where m is the number of spatial locations for whichdata are available. The variable A t can be written as A t = w ′ x, where x isan ml-vector of observed measurements at times t, (t − 1),...,(t − l + 1),and all m spatial locations, and w is a vector of weights. Then w is chosento maximize the signal to noise ratio[E(A t )] 2var(A t ) = [w′ s t ] 2w ′ Σ e w ,where s t is an ml-vector of known signal at time t and Σ e is the spatiotemporalcovariance matrix of the noise term. It is straightforward to showthat the optimal detector, sometimes known as the optimal fingerprint, hasthe form ŵ = Σ −1e s t . The question then arises: Where does PCA fit intothis methodology?