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Nonlinear multichannel singular spectrum analysis of the tropical Pacific climate variability using a.. Hsieh, William W.; Wu, Aiming 2002

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Nonlinear multichannel singular spectrum analysis of the tropicalPacific climate variability using a neural network approachWilliam W. Hsieh and Aiming WuDepartment of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, CanadaReceived 4 May 2001; revised 3 January 2001; accepted 11 January 2001; published 18 July 2002.[1] Singular spectrum analysis (SSA), a linear (univariate and multivariate) time seriestechnique, performs principal component analysis (PCA) on an augmented data setcontaining the original data and time-lagged copies of the data. Neural network theory hasmeanwhile allowed PCA to be generalized to nonlinear PCA (NLPCA). In this paper,NLPCA is further extended to perform nonlinear SSA (NLSSA): First, SSA is applied toreduce the dimension of the data set; the leading principal components (PCs) of the SSAthen become inputs to an NLPCA network (with a circular node at the bottleneck). Thisnetwork performs the NLSSA by nonlinearly combining all the input SSA PCs. TheNLSSA is applied to the tropical Pacific sea surface temperature anomaly (SSTA) fieldand to the sea level pressure anomaly (SLPA) field for the 1950–2000 period. Unlike SSAmodes, which display warm and cool periods of similar duration and intensity, NLSSAmode 1 shows the warm periods to be shorter and more intense than the cool periods, asobserved for the El Nin˜o-Southern Oscillation. Also, in SSTA NLSSA mode 1 the peakwarm event is centered in the eastern equatorial Pacific, while the peak cool event islocated around the central equatorial Pacific, an asymmetry not found in the individualSSA modes. A quasi-triennial wave of about a 39 month period is found in NLSSA mode2 of the SSTA and of the SLPA. INDEX TERMS: 3339 Meteorology and Atmospheric Dynamics:Ocean/atmosphere interactions (0312, 4504); 3220 Mathematical Geophysics: Nonlinear dynamics; 4215Oceanography: General: Climate and interannual variability (3309); 4522 Oceanography: Physical: El Nin˜o;KEYWORDS: El Nin˜o, Southern Oscillation, neural networks, singular spectrum analysis, tropical Pacific1. Introduction[2] Principal component analysis (PCA), also known asempirical orthogonal function (EOF) analysis, is a classicalmultivariate statistical technique used to analyze a set ofvariables {xi}[von Storch and Zwiers, 1999]. It is com-monly used to reduce the dimensionality of the data set {xi}and to extract features (or recognize patterns).[3] By the 1980s, interests in chaos theory and dynam-ical systems led to further extension of PCA to singularspectrum analysis (SSA): The information contained in acontinuous variable and its first to (L C0 1)th derivativescan be approximated by a discrete time series and thesame time series lagged by 1,..., L C0 1 time steps [Elsnerand Tsonis, 1996, chap. 4]. Thus a given time series andits lagged versions can be regarded as a set of variables{xi}(i =1,..., L), an augmented data set, which can beanalyzed by the PCA. This resulting method is the SSAwith window L. In the multivariate case where there ismore than one time series, one can again make laggedcopies of the time series, treat the lagged copies as extravariables, and apply the PCA to this augmented data set,resulting in the multichannel SSA (MSSA) method, alsocalled the space-time PCA (ST-PCA) method, or theextended EOF (EEOF) method (though in typical EEOFapplications, only a small number of lags are used). Forbrevity we will use the term SSA to denote both SSA andMSSA. A recent review of the SSA method is given byGhil et al. [2002].[4] Neural network (NN) models, which first becamepopular in the late 1980s, have significantly advancednonlinear empirical modeling. Each member of thehierachy of classical multivariate methods (multiple linearregression, PCA, and canonical correlation analysis(CCA)) has been nonlinearly generalized by NN models:nonlinear multiple regression [Rumelhart et al., 1986],nonlinear PCA (NLPCA) [Kramer, 1991], and nonlinearCCA (NLCCA) [Hsieh, 2000]. The tropical Pacific climatevariability has been studied by the NLPCA method [Mon-ahan, 2001; Hsieh, 2001a] and by the NLCCA method[Hsieh, 2001b].[5] In PCA a straight line approximation to the data set issought that accounts for the maximum amount of variancein the data. In NLPCA the straight line is replaced by anopen continuous curve for approximating the data [Kramer,1991]. Kirby and Miranda [1996] introduced a NLPCAwith a circular node at the network bottleneck (henceforthNLPCA.cir), so that the nonlinear principal component(NLPC) as represented by the circular node is an angularvariable q and the NLPCA.cir is capable of approximatingthe data by a closed continuous curve.JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. C7, 3076, 10.1029/2001JC000957, 2002Copyright 2002 by the American Geophysical Union.0148-0227/02/2001JC000957$09.0013 - 1[6] This paper belongs to a series of papers on nonlinearPCA and its extensions. In Hsieh [2001a] both NLPCA andNLPCA.cir were applied to study the tropical Pacific seasurface temperature anomalies (SSTA). W. W. Hsieh andA. Wu (Nonlinear singular spectrum analysis by neuralnetworks, submitted to Neural Networks, 2002, hereinafterreferred to as Hsieh and Wu, submitted manuscript, 2002).developed the nonlinear SSA (NLSSA) method from theNLPCA.cir network, resulting in a new nonlinear timeseries technique. This nonlinear spectral technique allowsthe detection of highly anharmonic oscillations, as wasillustrated by a stretched square wave imbedded in whitenoise, which showed NLSSA to be superior to SSA andclassical Fourier spectral analysis (Hsieh and Wu, submittedmanuscript, 2002). In this present paper the NLSSA methodis used to analyze the tropical Pacific SSTA field, and thesea level pressure anomaly (SLPA) field, for the period from1950 to 2000.2. Theory: From NLPCA to NLSSA[7] Since NLPCA.cir has been explained fully by Hsieh[2001a], we only outline its main properties here. The inputdata are in the form x(t)=[x1,...,xl], where each variablexi(i =1,...,l) is a time series containing n observations. Themodel (Figure 1) is a standard feed forward NN (i.e.,multilayer perceptron), with 3 ‘‘hidden’’ layers of variablesor ‘‘neurons’’ (denoted by circles) sandwiched between theinput layer x on the left and the output layer x0on the right.Next to the input layer (with l neurons) is the encoding layer(with m neurons), followed by the ‘‘bottleneck’’ layer, thenthe decoding layer (with m neurons), and finally, the outputlayer (with l neurons). The jth neuron vj(k)in the kth layerreceives its value from the neurons v(kC01)in the precedinglayer; that is, vj(k)= fk(wj(k)C1 v(kC01)+ bj(k)), where wj(k)is avector of weight parameters, bj(k)is a bias parameter, and thetransfer functions f1and f3are the hyperbolic tangentfunctions, while f2and f4are simply the identity functions.Hence a total of four successive layers of transfer functionsare needed to map from the inputs x to the outputs x0.InNLPCA.cir the bottleneck contains two neurons p and qconfined to lie on a unit circle, i.e., only 1 degree offreedom, as represented by the angle q. Effectively, a non-linear function q = F(x) maps from the higher dimensioninput space to the one-dimensional (1-D) bottleneck space,followed by an inverse transform x0= G(q) mapping fromthe bottleneck space back to the original space, as repre-sented by the outputs. To make the outputs as close to theinputs as possible, the cost function J = hkx C0 x0k2i (i.e., themean square error (MSE)) is minimized (where hC1C1C1i denotesa sample or time mean). Through the optimization thevalues of the weight and bias parameters are solved. Datacompression is achieved by the bottleneck, yielding thenonlinear principal component q.[8] Hsieh [2001a] noted that the NLPCA.cir, with itsability to extract closed curve solutions, is particularly idealfor extracting periodic or wave modes in the data. In SSA itis common to encounter periodic modes, each of which hadto be split into a pair of modes [Elsner and Tsonis, 1996], asthe underlying PCA technique is not capable of modeling aperiodic mode (a closed curve) by a single mode (a straightline). Thus, two SSA modes can easily be combined intoone NLPCA.cir mode. In principle, the original variablesand their lagged versions can be input to the NLPCA.cir toextract the NLSSA solution.[9] The problem is that usually one cannot afford to usemany lags before the large number of input variables to thenetwork results in more model parameters than samples (i.e.,measurements in time). Even in NLPCA or NLPCA.cir, thenumber of input variables can be so large that there aremore model parameters than samples. To avoid this sit-uation, the data are usually first condensed by the classicalPCA method where only the first few leading PCs (i.e., thetime coefficients from the PCA) are retained, resulting infar fewer input variables to the NLPCA or NLPCA.cirnetwork.[10] An analogous approach to reduce greatly the numberof input variables to the network can be used for theNLSSA, except that instead of PCA, SSA is used toprefilter the data. First, the original data are analyzed bythe SSA with window L. Only the first few leading SSAmodes are retained, and their PCs, [x1,...,xl], are thenserved as input variables to the NLPCA.cir network. TheNLPCA.cir finds a continuous curve solution by nonli-nearly relating the PCs, thereby giving NLSSA mode 1.Hsieh [2001a] pointed out that the general configuration ofthe NLPCA.cir can model not only closed curve solutionsbut also open curve solutions like the Kramer [1991]NLPCA. Because the NLPCA.cir is more general than theNLPCA (with one bottleneck neuron), it is used here toperform the NLSSA. More details of the NLSSA theory aregiven by Hsieh and Wu (submitted manuscript, 2002).[11] A small amount of weight penalty was added to thecost function as by Hsieh [2001a] and Hsieh and Wu(submitted manuscript, 2002), with the penalty parameterP = 0.02. Because of local minima in the cost function, thereis no guarantee that the optimization algorithm reaches theglobal minimum. Hence an ensemble of 30 NNs withrandom initial weights and bias parameters was run. Also,20% of the data were randomly selected as validation dataand withheld from the training of the NNs. Runs where theMSE was larger for the validation data set than for theFigure 1. A schematic diagram of the NN model forcalculating the NLPCA with a circular node at thebottleneck (NLPCA.cir).13 - 2 HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSISFigure 2. The SSA (i.e., ST-PCA or EEOF) modes (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6 for thetropical Pacific SSTA. The contour plots display the space-time eigenvectors (loading patterns), showingthe SSTA along the equator as a function of the lag. Solid contours indicate positive anomalies, anddashed contours indicate negative anomalies, with the zero contour indicated by the thick solid curve. In aseparate panel beneath each contour plot, the PC of each SSA mode is also plotted as a time series (whereeach tick mark on the abscissa indicates the start of a year). The time of the PC is synchronized to the lagtime of 0 month in the space-time eigenvector.HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS 13 - 3training data set were rejected to avoid overfitted solu-tions. Then the NN with the smallest overall MSE (i.e.,the MSE computed over both training and validation data)was selected as the solution. Separate runs were madeusing m =2,...,6. The overall MSE dropped withincreasing m but eventually levelled off. Following theprinciple of parsimony, we chose the smallest m when thelevelling occurred as the solution. For the SSTA NLSSAmodes, m = 4, while for SLPA modes, m =6.3. NLSSA of the Tropical Pacific SSTA[12] The Smith et al. [1996] monthly SST for the domain21C176S–21C176N, 123C176E–69C176W during January 1950 to Decem-ber 2000 were used. The original 2C176 C2 2C176 data wereaveraged to 2C176 latitude C2 4C176 longitude resolution (to reducethe memory requirement in the subsequent SSA calcula-tion), then the climatological seasonal cycle was subtractedand the linear trend was removed from the SST to yield theSSTA data.[13] As we want to resolve the El Nin˜o-SouthernOscillation (ENSO) variability, a window of 73 monthswas chosen. With a lag interval of 3 months, the originalplus 24 lagged copies of the SSTA data formed theaugmented SSTA data set. The first eight SSA modesexplain 12.4, 11.7, 7.1, 6.7, 5.4, 4.4, 3.5, and 2.8%,respectively, of the total variance of the augmented dataset (with the first six modes shown in Figure 2). The firsttwo modes have space-time eigenvectors (i.e., loadingpatterns) showing an oscillatory timescale of about 48months, comparable to the ENSO timescale, with themode 1 anomaly pattern occurring about 12 months beforea very similar mode 2 pattern; that is, the two patterns arein quadrature. The PC time series also show similartimescales for modes 1 and 2. Modes 3 and 5 showlonger timescale fluctuations, while modes 4 and 6 showshorter timescale fluctuations, around the 30 month time-scale.[14] With the eight PCs as input x1,...,x8to theNLPCA.cir network the resulting NLSSA mode 1 is aclosed curve in the 8-D PC space and is plotted in thex1-x2-x3space in Figure 3. NLSSA mode 1 is basically alargeloopalignedparalleltothex1-x2plane,therebycombin-ing the first two SSA modes. The solution also shows somemodest variations in the x3direction. This NLSSA mode 1explains 24.0% of the total variance of the augmented dataset, essentially that explained by the first two SSA modestogether. The linear PCA solution is shown as a straight linein Figure 3, which is, of course, simply SSA mode 1. Ofinterest is r, the ratio of the MSE of the nonlinear solution tothe MSE of the linear solution. Here r = 0.71.[15] The NLSSA mode 1 space-time loading pattern for agiven value of q can be obtained by mapping from q to theoutputs x0, which are the eight PC values corresponding tothe given q. Multiplying each PC value by its correspondingSSA eigenvector and summing over the eight modes, weobtain the NLSSA mode 1 pattern corresponding to thegiven q.[16] The NLSSA mode 1 loading patterns for various qvalues are shown in Figure 4. Comparing the patterns inFigure 4 with the patterns from the first two SSA modes inFigure 2, we find three notable differences: (1) Thepresence of warm anomalies for 24 months followed bycool anomalies for 24 months in the first two SSA modesis replaced by warm anomalies for 18 months followed bycool anomalies for about 33 months in NLSSA mode 1.Although the cool anomalies can be quite mild for longperiods, they can develop into full La Nin˜a cool events(Figure 4c). (2) The El Nin˜o warm events are strongestnear the eastern boundary, while the La Nin˜a events arestrongest near the central equatorial Pacific in NLSSAmode 1, an asymmetry not found in the individual SSAmodes. (3) The magnitude of the peak positive anomaliesis significantly larger than that of the peak negativeanomalies in NLSSA mode 1 (Figure 4c), again anasymmetry not found in the individual SSA modes. Allthree differences indicate that NLSSA mode 1 is muchcloser to the observed ENSO properties than the first twoSSA modes are.[17] The NLPC q of NLSSA mode 1, plotted as a timeseries (cyclically bounded between C0p and p radians) inFigure 5, reveals q generally increasing with time. Thus thepatterns in Figure 4 generally evolve from Figures 4a–4fwith time.[18] Next, we try to reconstruct the SSTA field fromNLSSA mode 1 for the whole record. The PC at a giventime gives not only the SSTA field at that time but atotal of 25 SSTA fields (spread 3 months apart) coveringthe 73 months in the lag window. Thus (except for thefinal 72 months of the record) the SSTA field at any timeis taken to be the average of the 25 SSTA fieldsproduced by the PC of the current month and theFigure 3. NLSSA mode 1 for the tropical Pacific SSTA.The PCs of SSA modes 1–8 were used as inputs x1,..., x8tothe NLPCA.cir network, with the resulting NLSSA mode 1shown as (densely overlapping) crosses in the x1–x2–x33-Dspace. The projections of this mode onto the x1–x2, x1–x3,and x2–x3planes are denoted by the (densely overlapping)small circles, and the projected data are denoted by dots.For comparison, linear SSA mode 1 is shown by thedashed line in the 3-D space and by the projected solidlines on the 2-D planes.13 - 4 HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSISpreceding 72 months. The SSTA field reconstructed fromNLSSA mode 1 (called the reconstructed component(NLRC1)) during 1971–2000 is shown in Figure 6,which compares well with the observed field, exceptfor the weaker amplitude (Figure 7).[19] To extract NLSSA mode 2, we removed the NLSSAmode 1 x0from the data x and input the residuals (x C0 x0)into the same NLPCA.cir network. Since this NLSSA mode2 mainly combines SSA modes 3 and 4, it is plotted in thex2–x3–x4space (Figure 8), where the most prominentfeature is the large loop in the x3–x4plane. There are alsononlinear relations manifested in the x2–x3and x2–x4planes, as well as in the x1–x3and x1–x4planes (notshown). NLSSA mode 2 explains 12.6% of the totalvariance of the residuals. In contrast, if the residuals wereinput into a PCA model, the leading mode (shown by thestraight lines in Figure 8) would only explain 7.3% of thevariance. For mode 2, r = 0.77.[20] The space-time loading patterns associated with thisNLSSA mode 2 (Figure 9) reveal oscillations at around thetriennial period. The corresponding NLPC q is also plottedin Figure 5. NLRC2, the SSTA field reconstructed fromFigure 4. The SSTA NLSSA mode 1 space-time loading patterns for (a) q =0C176,(b)q =60C176,(c)q = 120C176,(d) q = 180C176,(e)q = 240C176, and (f) q = 300C176. The contour plots display the SSTA anomalies along theequator as a function of the lag time. The contour interval is 0.2C176C.HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS 13 - 5NLSSA mode 2 during 1971–2000 (Figure 10), manifestsoscillations at around the 39 month period.[21] What is striking about this NLSSA mode 2 whencompared with NLSSA mode 1 is its great regularity. Herethe warm and cool anomaly patterns are approximatelymirror images of each other, unlike the large differences(in duration and intensity) between warm and cool anomalypatterns found in NLSSA mode 1 (Figure 6). The oscilla-tions at around the 39 month period, except for a few yearswhen the oscillations faded away, are far more regular(Figure 10) than the fluctuations generated by NLSSA mode1 (Figure 6). The great regularity strongly suggests that thisquasi-triennial oscillation of NLSSA mode 2 is a muchmore linear phenomenon than NLSSA mode 1.[22] We next plot the spatial anomaly patterns duringpeak warming and during peak cooling in the Nino 3 region(150C176–90C176W, 5C176S–5C176N) in the eastern equatorial Pacific.Searching over all lags and q from 0C176 to 360C176, we found thelag and q when the maximum value of the loading in theNino 3 region occurred, as well as for the minimum value.The spatial anomaly patterns for NLSSA modes 1 and 2during maximum and minimum SSTA in the Nino 3 regionare shown in Figure 11. For NLSSA mode 1 the coolpattern (Figure 11b) is centered much farther offshore fromPeru than the warm pattern (Figure 11a), while for NLSSAmode 2 such a difference is not found between the warmpattern (Figure 11c) and the cool pattern (Figure 11d). Theanomalies are more closely confined to the equator formode 1 than those for mode 2.[23] For the period 1971–2000, comparing the recon-structed SSTA from NLSSA mode 1 (Figure 6), that fromNLSSA mode 2 (Figure 10), and the observed SSTA(Figure 7), we note that during the major El Nin˜o events(e.g., 1972, 1982–1983, and 1997) both NLSSA mode 1and mode 2 developed warm anomalies, while during LaNin˜a events (e.g., 1973–1974 and 1988) both NLSSAmode 1 and mode 2 developed cool anomalies. Intrigu-ingly, in Figures 6, 7, and 10 the positive and negativeanomalies in Figures 6a, 7a, and 10a for 1971–1985match quite well with the anomalies in Figures 6b, 7b,and 10b for 1986–2000, as though history nearly repeateditself after 15 years. The sensitivity of these results to thechoice of the window L = 73 months was checked byrepeating the calculations with L = 97 months, whichyielded essentially the same results.4. NLSSA of the Tropical Pacific SLPA[24] The tropical Pacific monthly SLP data from theComprehensive Ocean-Atmosphere Data Set [Woodruffet al., 1987] for the domain 21C176S–21C176N, 121C176E–71C176Wduring January 1950 to December 2000 were used. The2C176 C2 2C176 resolution SLP data were interpolated for missingvalues and averaged into 2C176 latitude C2 4C176 longitudegridded data. The seasonal cycle and the linear trendwere then removed from the SLP to give the SLPA data.[25] The first eight SSA modes of the SLPA accountedfor 7.9, 7.1, 5.0, 4.9, 4.0, 3.1, 2.5, and 1.9%, respectively,of the total variance of the augmented data. The first twomodes displayed the Southern Oscillation (SO), the east-west seesaw of SLPA at around the 50 month period,while the higher modes displayed fluctuations at aroundthe quasi-biennial oscillation (QBO) [Hamilton, 1998]average period of 28 months (not shown).[26] The eight leading PCs of the SSA were then used asinputs, x1,...,x8, to the NLPCA.cir network, yieldingNLSSA mode 1 for the SLPA. This mode accounts for17.1% of the variance of the augmented data, significantlymore than the variance explained by the first two SSA modes(15.0%). This is not surprising as the NLSSA mode did morethan just combine SSA modes 1 and 2; it also nonlinearlyconnects SSA mode 3 to SSA modes 1 and 2 (Figure 12). InFigure 5. The NLPC q time series for SSTA NLSSA mode 1 (bottom curve) and mode 2 (second curvefrom bottom) and for the SLPA NLSSA mode 1 (second curve from top) and mode 2 (top curve). Thetime series have been vertically displaced by multiples of 2p radians for better visualization.13 - 6 HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSISthe x1-x3plane the bowl-shaped projected solution impliesthat PC3 tends to be positive when PC1 takes on either largepositive or large negative values. Similarly, in the x2-x3planethe hill-shaped projected solution indicates that PC3 tends tobe negative when PC2 takes on large positive or negativevalues. These curves reveal nonlinear interactions betweenthe longer timescale SSA modes 1 and 2, and the shortertimescale SSA mode 3. Here r = 0.68, smaller than thatFigure 6. The reconstructed SSTA of NLSSA mode 1 from January 1971 to December 2000. Thecontour plots display the SSTA along the equator as a function of time, with a contour interval of 0.2C176C.HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS 13 - 7found for SSTA mode 1, indicating somewhat strongernonlinearity in the SLPA than in the SSTA.[27] The NLPC q (Figure 5) shows q generally increasingwith time. The space-time loading patterns for NLSSAmode 1 at various q reveal that the negative phase of theSO is much shorter and more intense than the positive phase(not shown), in agreement with observations and in contrastto SSA modes 1 and 2, where the negative and positiveFigure 7. The observed SSTA along the equator from January 1971 to December 2000. The data havebeen low-pass filtered, with periods of 12 months or less removed. Note, the contour interval is 0.5C176Cinstead of the 0.2C176C in Figure 6.13 - 8 HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSISphases of the SO are about equal in duration and magnitude.The SLPA field reconstructed from NLSSA mode 1(NLRC1) during 1971–2000 is shown in Figure 13, whichalso reveals the negative phase of the SO to be more intensebut of shorter duration than the positive phase.[28] After NLSSA mode 1 had been removed from thedata, the residual data were then input into the NLPCA.cirnetwork to extract NLSSA mode 2. This mode is not shownin the PC space as it resembles Figure 8 for SSTA NLSSAmode 2. SLPA NLSSA mode 2 accounts for 8.2% of thevariance, versus 4.8% accounted for by the leading PCAmode of the residual data. Here r = 0.77, the same as that forSSTA mode 2.[29] The space-time loading patterns of NLSSA mode 2reveal east-west seesaw oscillations in the SLPA, but ataround the 39 month period (not shown). The SLPA fieldreconstructed from NLSSA mode 2 (NLRC2) during 1971–2000 (Figure 14) also manifests oscillations at around the 39month period.[30] Again, comparing SLPA NLSSA mode 2 (Figure 14)with SLPA NLSSA mode 1 (Figure 13), we note that thepositive and negative anomaly patterns of mode 2 areapproximately mirror images of each other, in contrast tothe large differences (in duration and intensity) between thepositive and negative anomaly patterns found in NLSSAmode 1. The oscillations at around the 39 month period,except for a few years when the oscillations faded, are farmore regular (Figure 14) than the fluctuations generated byNLSSA mode 1 (Figure 13). The great regularity stronglysuggests that this quasi-triennial oscillation of NLSSA mode2 is a much more linear phenomenon than NLSSA mode 1.The quasi-triennial oscillations in the SLPA (Figure 14)coincide very well with those in the SSTA (Figure 10).[31] The spatial SLPA patterns for NLSSA modes 1 and 2during maximum and minimum SLPA in the Nino 3 regionwere also calculated. Again, the strongest SLPA in NLSSAmode 2 tend to occur farther away from the equator than thestrongest SLPA in NLSSA mode 1, analogous to what wasfound for the SSTA (Figure 11).[32] For the period 1971–2000, comparing NLRC1(Figure 13) and NLRC2 (Figure 14), we note that duringthe major El Nin˜o events (e.g., 1972, 1982–1983, and1997) both NLSSA mode 1 and mode 2 developednegative SLPA in the eastern equatorial Pacific (andpositive anomalies in the west), while during La Nin˜aevents (e.g., 1973–1974 and 1988) both NLSSA mode 1and mode 2 developed positive SLPA in the eastern equa-torial Pacific. Again, the positive and negative anomalies inFigures 13a and 14a for 1971–1985 match quite well withthe anomalies in Figure 13b and 14b for 1986–2000, asthough history nearly repeated itself after 15 years.[33] To explore relations between the NLRC1 for theSSTA and that for the SLPA, the NLRC1 of the SSTAaveraged over the Nino 3 region was lagged correlated withthat of the SLPA over Nino 3. The peak correlation of 0.56occurred when SLPA leads SSTA by 1 month. In the Nino3.4 region (170C176–120C176W, 5C176S–5C176N) the peak correlationwas 0.54 with SLPA leading SSTA by 2 months.[34] Between the NLRC2 of the SSTA and the NLRC2 ofthe SLPA the peak correlation in Nino 3 was 0.96 withSLPA leading SSTA by 1 month, while in Nino 3.4 the peakcorrelation was 0.92 with SLPA leading by 2 months. Thepeak correlations here are much higher than those involvingthe NLRC1, suggesting that the coupling between NLSSAmode 2 of the SSTA and that of the SLPA may be strongerthan the NLSSA mode 1 coupling.5. Summary and Discussion[35] As NN modeling has generalized PCA to NLPCA, itis natural to similarly extend SSA to NLSSA. SSA is simplyPCA applied to an augmented data set (containing theoriginal data set and copies of the data set lagged by arange of time steps). In most NLPCA applications the dataset is first condensed by the PCA method, and the first fewPCs are used as inputs to the NLPCA. Similarly, withNLSSA the data set is first condensed by the SSA, andthe first few PCs from the SSA are chosen as inputs to theNLPCA.cir network (the NLPCAwith a circular node at thebottleneck), which extracts the NLSSA mode by nonli-nearly combining the various SSA modes.[36] In general, NLSSA has several advantages over SSA:(1) The PCs from different SSA modes are linearly uncorre-lated; however, they may have nonlinear relationships thatcan be detected by the NLSSA. (2) Although the SSA modesare not restricted to sinusoidal oscillations in time like theFourier spectral components, in practice, they are inefficientin modeling strongly anharmonic signals (e.g., the stretchedsquare wave by Hsieh and Wu (submitted manuscript,2002)), scattering the signal energy into many SSA modes.The NLSSA recombines the SSA modes to extract theanharmonic signal. (3) As different SSA modes are associ-ated with different timescales (e.g., timescales of the ENSO,QBO, and decadal oscillations), the nonlinear relationsfound by the NLSSA reveal the timescales among whichFigure 8. NLSSA mode 2 for the tropical Pacific SSTAshown in the x2–x3–x43-D PC space. After removingNLSSA mode 1 from the data we input the residuals into thesame NLPCA.cir network to extract NLSSA mode 2. Thedots indicate the residuals projected onto the 2-D planes,and the straight lines indicate the PCA approximation to theresiduals.HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS 13 - 9Figure 9. The SSTA NLSSA mode 2 space-time loading patterns for (a) q =0C176,(b)q =90C176,(c)q = 180C176,and (d) q = 270C176. The contour plots display the SSTA along the equator as a function of the lag time. Thecontour interval is 0.1C176C.13 - 10 HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSISthere are nonlinear relations, thereby disclosing nonlinearrelations between seemingly separate phenomena.[37] For the tropical Pacific SSTA the NLSSA mode 1combines mainly the SSA modes 1 and 2, characterizingthe ENSO phenomenon. However, in the SSA modes thewarm and cool periods are of similar duration and inten-sity, whereas in the NLSSA mode the warm period is ofshorter duration and greater intensity. Furthermore, in theFigure 10. The NLSSA mode 2 reconstructed SSTA along the equator from January 1971 to December2000. The contour interval is 0.1C176C.HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS 13 - 11NLSSA mode the peak warm event is centered in theeastern equatorial Pacific, while the peak cool event iscentered much farther west, a contrast not found in SSAmodes 1 and 2. In short, the NLSSA mode gives a muchmore accurate picture of the ENSO phenomenon than theleading pair of SSA modes.[38] For the tropical Pacific SLPA, NLSSA mode 1 notonly combines SSA modes 1 and 2 with ENSO timescalesbut also reveals nonlinear interactions between these twoand mode 3, which has a QBO timescale. Again, thesymmetry between the positive and negative phases of theSO found in SSA modes 1 and 2 disappeared in NLSSAmode 1, in agreement with observations.[39] An interesting comparison can be made betweenNLSSA mode 1 of the SSTA and that of the SLPA, whichshows that the SLPA is more nonlinear than the SSTA atthe ENSO timescale. This is in contrast to the NLPCAstudy by Monahan [2001] and the NLCCA study by Hsieh[2001b], where the SSTA was found to be more nonlinearthan the SLPA. The difference is that in the two earlierstudies, time lags were not incorporated, so only non-linearity in space was being measured. In fact, NLPCAcan be regarded as a limiting case of NLSSA, with thewindow L = 1 month. By incorporating time lags, NLSSAmeasures the nonlinearity in both space and time andreveals SLPA to be the more nonlinear field because ofthe stronger nonlinear interactions between ENSO andQBO timescales.[40] Perhaps the most intriguing achievement of theNLSSA is the isolation of a quasi-triennial oscillation ofabout 39 month period in both SSTA NLSSA mode 2(Figures 9 and 10) and SLPA NLSSA mode 2 (Figure 14).In fact, the correlation between the SSTA and the SLPA inthis 39 month oscillation is much higher than that found inthe main ENSO signal (NLSSA mode 1) at the 4–5 yearperiod. While a 3 year oscillation has been known to existFigure 11. The NLSSA mode 1 SSTA when the eastern equatorial Pacific (Nino 3 region) is (a)warmest and (b) coolest. For comparison, the NLSSA mode 2 reconstructed SSTA when the Nino 3region is (c) warmest and (d) coolest. The contour interval is 0.3C176C in Figures 11a and 11b and 0.15C176CinFigures 11c and 11d.Figure 12. The NLSSA mode 1 for the tropical PacificSLPA plotted in the x1–x2–x33-D PC space and its 2-Dplanes. The straight lines indicate the linear SSA mode 1.13 - 12 HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSISin the SO for a long time and, in fact, has been known todominate over the 5 year oscillation during 1882–1926[Troup, 1965], its close proximity to the main ENSO signalat the 4–5 year period renders a clean extraction difficult.NLSSA reveals this 39 month signal to be, in fact,considerably more regular, without the east-west shift ofthe SSTA between warm and cold events as found inNLSSA mode 1, and hence likely to be more linear thanFigure 13. The reconstructed SLPA of NLSSA mode 1 along the equator from January 1971 toDecember 2000. The contour interval is 0.2 mbar.HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS 13 - 13the main signal (NLSSA mode 1). The 39 month period isalso longer than most of the QBO periods reported byprevious studies using linear techniques [Ghil et al., 2002].Here the longer 39 month period appears to arise fromnonlinearly combining the SSA modes of QBO periodsand the longer-period SSA modes 1 and 2. This 39 monthsignal also appears less closely confined to the equatorthan the main signal.Figure 14. The NLSSA mode 2 reconstructed SLPA along the equator from January 1971 to December2000. The contour interval is 0.1 mbar.13 - 14 HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS[41] Acknowledgments. Support through strategic and researchgrants from the Natural Sciences and Engineering Research Council ofCanada is gratefully acknowledged.ReferencesElsner, J. B., and A. A. Tsonis, Singular Spectrum Analysis, Plenum, NewYork, 1996.Ghil, M., et al., Advanced spectral methods for climatic time series, Rev.Geophys., 10.1029/2000RG000092, in press, 2002.Hamilton, K., Dynamics of the tropical middle atmosphere: A tutorialreview, Atmos.-Ocean, 36, 319–354, 1998.Hsieh, W. W., Nonlinear canonical correlation analysis by neural networks,Neural Networks, 13, 1095–1105, 2000.Hsieh, W. W., Nonlinear principal component analysis by neural networks,Tellus, Ser. A, 53, 599–615, 2001a.Hsieh, W. W., Nonlinear canonical correlation analysis of the tropicalPacific climate variability using a neural network approach, J. Clim.,14, 2528–2539, 2001b.Kirby, M. J., and R. Miranda, Circular nodes in neural networks, NeuralComp., 8, 390–402, 1996.Kramer, M. A., Nonlinear principal component analysis using autoassocia-tive neural networks, AIChE J., 37, 233–243, 1991.Monahan, A. H., Nonlinear principal component analysis: Tropical Indo-Pacific sea surface temperature and sea level pressure, J. Clim., 14,219–233, 2001.Rumelhart, D. E., G. E. Hinton, and R. J. Williams, Learning internalrepresentations by error propagation, in Parallel Distributed Processing,vol. 1, edited by D. E. Rumelhart, J. L. McClelland, and P. R. Group,pp. 318–362, MIT Press, Cambridge, Mass., 1986.Smith, T. M., R. W. Reynolds, R. E. Livezey, and D. C. Stokes, Recon-struction of historical sea surface temperatures using empirical orthogonalfunctions, J. Clim., 9, 1403–1420, 1996.Troup, A. J., The ‘‘Southern Oscillation,’’ Q. J. R. Meteorol. Soc., 91, 490–506, 1965.von Storch, H., and F. W. Zwiers, Statistical Analysis in Climate Research,Cambridge Univ. Press, New York, 1999.Woodruff, S. D., R. J. Slutz, R. L. Jenne, and P. M. Steurer, A compre-hensive ocean-atmosphere data set, Bull. Am. Meteorol. Soc., 68, 1239–1250, 1987.C0C0C0C0C0C0C0C0C0C0C0W. W. Hsieh and A. Wu, Department of Earth and Ocean Sciences,University of British Columbia, Vancouver, B.C., Canada V6T 1Z4.(whsieh@eos.ubc.ca)HSIEH AND WU: NONLINEAR MULTICHANNEL SINGULAR SPECTRUM ANALYSIS 13 - 15

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