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GEOPHYSICAL RESEARCH LETTERS, VOL. 37, L03801, doi:10.1029/2009GL041478, 2010
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Application of partial least squares regression to the diagnosis of year‐to‐year variations in Pacific Northwest snowpack and Atlantic hurricanes Brian V. Smoliak,1 John M. Wallace,1 Mark T. Stoelinga,2 and Todd P. Mitchell3 Received 27 October 2009; revised 21 December 2009; accepted 4 January 2010; published 2 February 2010.
[1] Application of the method of partial least squares (PLS) regression to geophysical data is illustrated with two cases: (1) finding sea level pressure patterns over the North Pacific associated with dynamically‐induced winter‐to‐winter variations in snowpack in the Cascade mountains of western Washington state and (2) finding patterns of sea surface temperature over the tropical oceans that modulate Atlantic hurricane activity on a year‐to‐year basis. In both examples two robust patterns in the “predictor field” are identified that, in combination, account for over half the variance in the target time series. Citation: Smoliak, B. V., J. M. Wallace, M. T. Stoelinga, and T. P. Mitchell (2010), Application of partial least squares regression to the diagnosis of year‐ to‐year variations in Pacific Northwest snowpack and Atlantic hurricanes, Geophys. Res. Lett., 37, L03801, doi:10.1029/ 2009GL041478.
1. Introduction [2] Time series of local climatic variables exhibit large year‐to‐year variability in response to changes in the planetary‐scale atmospheric circulation. This “dynamically‐ induced” variability may be of interest in its own right, or it may be desirable to remove it in order to more clearly reveal secular trends or abrupt jumps in the time series. One approach that is often used to isolate the influence of a specific phenomenon such as El Niño‐Southern Oscillation (ENSO) or the North Atlantic Oscillation on a local climatic variable is to correlate the time series of the variable with an index of the phenomenon. A more general approach is to (1) regress a field such as sea level pressure (SLP) onto the time series of the local climate variable, (2) project the seasonal‐mean field of that variable onto the associated (spatial) pattern, and (3) regress out the time series of the climatic index from the time series of the local climatic variable [e.g., Wallace et al., 1995; Ogi et al., 2008; Thompson et al., 2009]. [3] Here we explore the use of partial least squares (PLS) regression, which under certain conditions, makes it possible to regress out multiple, mutually orthogonal time series.
1 Department of Atmospheric Sciences, University of Washington, Seattle, Washington, USA. 2 3TIER Inc., Seattle, Washington, USA. 3 Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, Washington, USA.
Copyright 2010 by the American Geophysical Union. 0094‐8276/10/2009GL041478$05.00
2. Background on PLS Regression [4] PLS‐regression was first described by Wold [1966], who demonstrated its utility in the field of econometrics. It quickly took hold in the social sciences [Wold, 2001], and has since been developed and applied in a wide variety of fields such as chemometrics [Wold et al., 2001], computational biology [Tan et al., 2004], and neuroimaging [McIntosh and Lobaugh, 2004]. The technique has been applied in the geosciences for the purposes of paleoclimate reconstruction [Kalela‐Brundin, 1999] and statistical prediction [McIntosh et al., 2005]. However, to the best of the authors’ knowledge, it has not been widely used in diagnosing the factors that contribute to the variability of geophysical time series. [5] PLS‐regression embodies the well‐known concept of partial correlation; it seeks to predict one or more dependent variables, Y, with a set of independent variables, X. The predictors Z, which are linear combinations of X, are referred to as latent vectors or PLS components and maximize (1) variance explained in Y and (2) correlation between X and Y. [6] In the examples considered in this paper, the predictors are atmospheric and oceanic fields xij observed at regularly spaced times (i) that match the times of a scalar predictand time series yi, and at an array of grid points (j) in the space domain. The predictors and predictand are standardized prior to carrying out the analysis. The procedure begins by calculating correlation coefficients between the predictand yi and each of the predictor time series xij to obtain a correlation map rj. Then the maps xij for each of the observation times i are projected onto the correlation map rj to obtain the first predictor time series z1. Using conventional least squares fitting procedures, z1 is regressed out of both the predictand time series and each of the predictor time series to obtain a residual predictor field xij and predictand time series yi. The procedure is repeated on the residual matrices to obtain a second predictor time series z2, and so on. The respective predictor time series, z1, z2, … zn, are mutually orthogonal by construction and each successive predictor explains a smaller amount of covariance between X and Y. The predictor time series zk are rearranged in order of decreasing variance of Y and the optimal number to be retained is determined by cross validation. A complete mathematical description of the method is given by Abdi [2010]. [7] When applied to a multivariate field Y, PLS‐regression bears a relationship to several methods involving linear regression that are more widely used in analyzing geophysical fields: maximal covariance analysis (MCA, also known as singular value decomposition (SVD) analysis Bretherton et al. [1992]), canonical correlation analysis (CCA, also
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referred to as principal component regression [Barnett and Preisendorfer, 1987]), and procrustes target analysis (PTA, also referred to as redundancy analysis [Richman and Easterling, 1988]). The statistical methods texts of von Storch and Zwiers [1999], Wilks [2005] and van den Dool [2007], which are tailored to the geophysics reading audience, include discussions of MCA, CCA, and PTA, but do not mention PLS‐regression. [8 ] PLS‐regression shares with PTA the property of having clearly delineated predictor (X) and predictand (Y) fields, and both these methods yield patterns whose expansion coefficient time series maximize the fraction of the variance of Y explained by X. In contrast, MCA and CCA make no such distinction between X and Y fields: MCA yields patterns that maximize the squared covariance between the X and Y spatial patterns of the respective modes and CCA yields expansion coefficient time series that maximize the squared correlation between the X and Y time series of the respective modes. PLS‐regression and MCA are performed on the full X and Y matrices regardless of whether the number of spatial elements or gridpoints in the predictor field is smaller than the number of observations, whereas PTA and CCA may require truncation of both X and Y matrices in principal component space, a procedure that can be time consuming to implement if the truncations are to be selected in such a way as to optimize performance.
3. Application to a Time Series of April 1 Snowpack [9] Water stored in the form of snowpack in the Cascade mountain range serves as a source of water and hydroelectric power for the Seattle metropolitan area during the summer and autumn months, when precipitation is light. Year‐ to‐year variations in water storage capacity of the snowpack are often represented in terms of accumulated snow water equivalent (SWE) at the end of the winter season, by convention designated as April 1. The time series of April 1 SWE exhibits large year‐to‐year variability (a coefficient of variation of ∼0.35) in association with variations in the atmospheric circulation over the North Pacific. This dynamically‐induced variability tends to mask any gradual loss in snow pack that might be occurring in response to global warming [Casola et al., 2009]. [10] Stoelinga et al. [2010, hereafter SAM] analyzed a continuous time series of April 1 Cascade snowpack derived from stream flow data for the period of record 1930–2007. They found that when this index is correlated with established indices of winter‐mean circulation patterns in the North Pacific region, such as the November‐March mean North Pacific Index (NPI) defined by Trenberth and Hurrell [1994], substantial annual to interdecadal variability remains in the residual snowpack time series. As an alternative, they made their own definition of the spatial patterns responsible for the dynamically‐induced variability, based on the sea‐ level pressure (SLP) field over the North Pacific sector. For this purpose, they used a method that they referred to as “maximum independent point correlation,” which appears to be identical or at least very closely related to a method described by van den Dool [2007], referred to there as “EOT2.” Klein [1983] employed a similar approach for finding patterns in the 700 hPa height field that contribute to the variability of monthly mean surface air temperature at a
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specified location. The method seeks a limited number of individual spatial points in an independent variable X, (in this case, winter mean sea‐level pressure over the North Pacific region) whose orthogonalized time series together explain the maximum amount of variance in the dependent variable, Y or yi (in this case, April 1 snowpack). [11] SAM found three points in the winter‐mean SLP field that in combination explain 71% of the year‐to‐year variance in the annual time series of April 1 snowpack. Located near the center of the Aleutian low, the first point explains 35% of the variance, roughly comparable to the variance in Cascade snowpack explained by the NPI [Mote et al., 2008]. The second and third points are located near the Alaska panhandle and off the coast of southern California, and explain an additional 24% and 12% of the variance in Cascade snowpack, respectively. SAM defined a “Cascade Snowpack Circulation” (CSC) index based on multiple linear regression of snowpack with respect to the three SLP points, and found that the residual snowpack time series (linearly independent of the CSC index) exhibits a linear downward trend of −2% per decade, statistically significant at the 95% confidence level based on the version of the student t‐test used by Lettenmaier [1976] and Casola et al. [2009]. [12] In this section, PLS‐regression is applied to identify the dynamically‐induced variability in the same April 1 snowpack time series using North Pacific (20°N – 65°N, 180°W – 110°W) November–March mean SLP as the predictor field. The SLP data are obtained from the NOAA ESRL 20th Century Reanalysis [Compo et al., 2006, also manuscript in preparation, 2009] for the period 1929– 1957, and from the NCEP‐NCAR Reanalysis [Kalnay et al., 1996] for the period 1958–2007. [13] The first two patterns obtained from this analysis are shown in Figure 1. The first (Figure 1a) is marked by positive SLP anomalies over the Gulf of Alaska, indicative of a weakening of the Aleutian low. This pattern is related primarily to fluctuations in temperature over the PNW, consistent with weather forecasters’ association of ridges at ∼140° W with cold weather over the PNW. The second pattern (Figure 1b) is characterized by anomalously low pressure off the coast of British Columbia and enhanced onshore flow at the Washington coast. It is mainly related to precipitation variability over the PNW. [14] The time‐varying index of the first pattern accounts for 48% of the variance in April 1 snowpack, that of the second pattern accounts for another 21%, while the third (not shown) accounts for only 3%. Based on results of leave‐ one‐out cross‐validation, the leading two patterns prove to be statistically significant. Since, by construction, the pattern indices are mutually orthogonal, the explained variances are additive: together, the leading two patterns account for 69% of the variance, almost as much as the combination of the three predictors identified by SAM. [15] The original April 1 snowpack time series exhibits large year‐to‐year variability and has a secular trend of −3% per decade (Figure 1c). Formed by regressing out the time‐varying indices of the spatial patterns derived from PLS‐regression, the residual time series has markedly less variability and a smaller secular trend of −1% per decade (Figure 1d). Neither of these trends is statistically significant at the 95% confidence level. The discrepancy between the residual trend in SAM and the one estimated here is due
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tical trends despite substantial differences in the regression patterns.
4. Application to Power Dissipation Index and SST
Figure 1. (a) First pattern derived from PLS‐regression analysis of November–March mean SLP xij and the time series of April 1 Cascade snowpack yi, represented as the correlation coefficient between local SLP anomalies and the first PLS component z1; the 95% significance level is 0.22 (accounting for autocorrelation in the time series). Contour interval 0.1; the zero contour is bold and the dashed contours are negative. The fraction of variance in X and Y explained by component z1 is shown in the lower right. (b) Second pattern, display as in Figure 1a but for z2. (c) April 1 Cascade snowpack (% of 1961–1990 mean). (d) Residual April 1 Cascade snowpack. to the fact SAM used the Met Office Hadley Centre HadSLP2 dataset, whereas the present study used a combination of ESRL 20th Century Reanalysis and NCEP‐NCAR Reanalysis data. When applied to the same dataset, PLS‐regression and the method used by SAM yield virtually iden-
[16] The destructiveness of tropical cyclones increases with the cube of the wind speed, as reflected by the power dissipation index (PDI) devised by Emanuel [2005]. As in the record analyzed in the previous example, the long‐term record of the annual Atlantic basin PDI (1949–2007, obtained from K. Emanuel) is dominated by large dynamically‐ induced year‐to‐year variability. ENSO is known to modulate hurricane activity through its influence on vertical wind shear in the tropics [Gray, 1984]. Goldenberg et al. [2001] reported that the increase in major hurricane activity between 1995 and 2000 was associated with simultaneous increases in SST and decreases in vertical wind shear over the tropical North Atlantic. They associated the decreases in wind shear with ENSO and the increases in SST with a multidecadal mode of Atlantic SST variability. Vimont and Kossin [2007] showed that a well‐defined dynamical mode of variability intrinsic to the tropics, the Atlantic Meridional Mode (AMM), also plays an important role in regulating hurricane activity by virtue of its influence on local climatic variables such as SST, latent heat fluxes, and vertical wind shear in that sector and Kossin and Vimont [2007] went on to show that SST in the Atlantic sector is more influential than ENSO in regulating Atlantic hurricane activity on a year‐to‐year basis. [17] PLS‐regression is used to identify dynamically‐induced variability in the Atlantic basin PDI using tropical SST (20°S – 20°N, circumpolar) obtained from the NOAA Extended Reconstructed SST dataset, version 3b [Smith et al., 2008]. SST data are July–October means. The time series of global mean SST is subtracted from the time series of SST at each grid point in order to diminish the influence of the general increase in global mean surface temperature associated with global warming. After subjecting the results of this analysis to cross‐ validation, two components emerge as statistically significant. [ 18 ] The pattern associated with the first component (Figure 2a), which explains 31% of the variance of the PDI, appears to be dominated by ENSO‐related variability over the tropical Pacific Ocean. However, if the time series of the first predictor is decomposed into a component linearly proportional to a conventional ENSO index (the cold tongue index, defined as the average SST anomaly over the region 6°N–6°S, 180–90°W minus the global mean SST [Deser and Wallace, 1990]) and a linearly independent residual time series, we find that the ENSO index accounts for only 14% of the variance of the PDI. Hence, in agreement with results of Kossin and Vimont [2007], SST in the Atlantic sector appears to exert a stronger influence on Atlantic hurricane activity than the ENSO cycle. The residual time series accounts for 22% of the variance of the PDI and is associated with a pattern (not shown) that is dominated by variability over the tropical Atlantic sector, including the positive center of action adjacent to the northern coast of South America in Figure 2a. The second predictor pattern derived from PLS‐regression (Figure 2b) has prominent centers of action in the tropical Atlantic sector. The stronger center of action over the tropical North Atlantic was emphasized by Enfield [1996] and the pattern as a whole, with its equatorially‐symmetric dipole, closely
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tween the residual time series, the residual trend determined from the HadISST data was smaller by about a factor of 3 and not statistically significant at the 95% confidence level.
5. Discussion
Figure 2. As in Figure 1, but for annually accumulated values of Emanuel’s [2005] power dissipation index (PDI) for Atlantic basin hurricanes, in units of 1011 m3s−2. The 95% significance level for the correlation coefficients is 0.28. resembles the AMM defined by Vimont and Kossin [2007]. The second PLS component explains 24% of the variance in the PDI. Together, the leading two PLS components account for 55% of the variance of the PDI. [19] The original PDI time series (Figure 2c) exhibits a very weak upward trend that is dwarfed by the year‐to‐year and decade‐to‐decade variability. The residual time series (Figure 2d) exhibits a positive trend in the PDI that is statistically significant at the 95% confidence level. Although the residual PDI time series exhibits less year‐to‐year variability, large interannual variability is still evident, particularly near the beginning of the record. [20] PLS‐regression was applied to the Hadley Centre HadISST dataset [Rayner et al., 2003] and the original Atlantic basin PDI time series to test the robustness of the aforementioned patterns and trends. The patterns (not shown) were found to be nearly identical to those shown in Figure 2, and the residual time series was determined to be strongly correlated (r = 0.92) with the residual time series shown in Figure 2d. Despite the close correspondence be-
[21] In the two examples considered in this paper, the use of PLS‐regression made it possible to explain a much larger fraction of the variance in a geophysical time series than would have been possible using a single index as a predictor, but this is not always the case. For example, we have found that incorporating a second predictor defined on the basis of PLS substantially improves the prediction of April 1 Cascades snowpack, but not the prediction of wintertime temperature or precipitation over the Cascades. In the case of Atlantic basin PDI, PLS‐regression using global tropical SST as the predictor field yields two useful predictor time series, whereas PLS performed using only Atlantic SST yields a single predictor time series with predictive skill (not shown). These examples suggest that PLS‐regression may perform better than simple linear regression in situations in which the target time series yi is influenced by the predictor field X through more than one pathway. [22] In both of the examples considered here, the residual trends derived from the analysis proved to be highly sensitive to the choice of dataset used in the analysis. The true uncertainty inherent in the trends in these noisy time series is evidently much larger than indicated by formal estimates based on the student t‐test, as applied to that series. It is larger because it includes additional contributions from the observational error in the dataset and from the sampling errors inherent in the determination of the regression coefficients upon which the residual time series is based. This caveat applies not only to PLS‐regression analysis but to any method of analysis that is used to extract trends from noisy time series. [23] Despite the fact that the trends in the residual time series derived in this study were not replicated when PLS was applied to other datasets, PLS has yielded satisfactory results in terms of (1) the large fraction of the variance explained by the predictor patterns, (2) the interpretability of the predictor patterns and (3) the robustness of the predictor patterns with respect to sampling variability and the choice of dataset. Its good performance and its ease of use (e.g., as compared to PTA) render PLS‐regression attractive for many geophysical applications. [24] Acknowledgments. This work was supported by the Climate Dynamics Program Office of the National Science Foundation under grant 0812802.
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