Abstract
Here we describe a practical, step-by-step primer to scale-dependent correlation (SDC) analysis. The analysis of transitory processes is an important but often neglected topic in ecological studies because only a few statistical techniques appear to detect temporary features accurately enough. We introduce here the SDC analysis, a statistical and graphical method to study transitory processes at any temporal or spatial scale. SDC analysis, thanks to the combination of conventional procedures and simple well-known statistical techniques, becomes an improved time-domain analogue of wavelet analysis. We use several simple synthetic series to describe the method, a more complex example, full of transitory features, to compare SDC and wavelet analysis, and finally we analyze some selected ecological series to illustrate the methodology. The SDC analysis of time series of copepod abundances in the North Sea indicates that ENSO primarily is the main climatic driver of short-term changes in population dynamics. SDC also uncovers some long-term, unexpected features in the population. Similarly, the SDC analysis of Nicholson’s blowflies data locates where the proposed models fail and provides new insights about the mechanism that drives the apparent vanishing of the population cycle during the second half of the series.
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References
Bradshaw GA, McIntosh BA (1994) Detecting climate-induced patterns using wavelet analysis. Environ Pollut 83:135–142
Bradshaw GA, Spies TA (1992) Characterizing canopy gap structure in forests using wavelet analysis. J Ecol 80:205–215
Broomhead DS, King GP (1986) Extracting qualitative dynamics from experimental data. Physica D 20:217–236
Brosofske KD, Chen J, Crow TR, Saunders SC (1999) Vegetation responses to landscape structure at multiple scales across a northern Wisconsin, USA, pine barrens landscape. Plant Ecol 143:203–218
Butler L (1953) The nature of cycles in populations of Canadian mammals. Can J Zool 31:242–262
Casdagli MC (1997) Recurrence plots revisited. Physica D 108:12–44
Chatfield C (1989) The analysis of time series: an introduction. 4th edn. Chapman and Hall, London
Chen X, Cohen JE (2001) Transient dynamics and food-web complexity in the Lotka-Volterra cascade model. Proc R Soc Lond B 268:869–877
Childers DL, Sklar FH, Hutchinson SE (1994) Statistical treatment and comparative analysis of scale-dependent aquatic transect data in estuarine landscapes. Landscape Ecol 9:127–141
Chiu CK, Montefusco L, Puccio L (eds) (1994) Wavelets: theory, algorithms and applications. Academic Press, San Diego
Compagnucci RH, Blanco SA, Figliola MA, Jacovkis PM (2000) Variability in subtropical Andean Argentinean Atuel river; a wavelet approach. Environmetrics 11:251–269
Dale MRT, Mah M (1998) The use of wavelets for spatial pattern analysis in ecology. J Veg Sci 9:805–814
Daubechies I (ed) (1993) Different perspectives on wavelets. American Mathematical Society, Providence, Rhode Island
Dohan K, Whitfield PH (1997) Identification and characterization of water quality transients using wavelet analysis. I. Wavelet analysis methodology. In: Beck MB, Lessard P (eds) Watermatex-97, systems analysis and computing in water quality management: towards a new agenda. Elsevier, Amsterdam
Fortin MJ (1994) Edge detection algorithms for two-dimensional ecological data. Ecology 75:956–965
Fromentin JM, Planqué B (1996) Calanus and environment in the eastern North Atlantic. II. Influence of the North Atlantic Oscillation on C. finmarchicus and C. helgolandicus. Mar Ecol Prog Ser 134:111–118
Gershinov A, Barnett TP (1998) Interdecadal modullation of ENSO teleconnections. Bull Am Meteorol Soc 79:2715–2726
Green DG (1995) Time and spatial analysis. In: Maddy D, Brew JS (eds) Statistical modeling of quaternary science data. Quaternary Research Association, Cambridge
Grenfell BT, Bjørnstad ON, Kappey J (2001) Travelling waves and spatial hierarchies in measles epidemics. Nature 414:6865 716–723
Gurney WSC, Blythe SP, Nisbet RM (1980) Nicholson’s blowflies revisited. Nature 287:17–21
Harper KA, Macdonald SE (2001) Structure and composition of riparian boreal forest: New methods for analyzing edge influence. Ecology 82:649–659
Hastings A, Higgins K (1994) Persistence of transients in spatially structured ecological models. Science 263:1133–1136
Heath MR, Backhaus JO, Richardson K, Mckenzie E, Slagstad D, Beare D, Dunn J, Fraser JG, Gallego A, Hainbucher D, Hay S, Jónasdóttir S, Madden H, Mardaljevic J, Schacht A (1999) Climate fluctuations and the spring invasion of the North Sea by Calanus finmarchicus. Fish Oceanogr 8 [Suppl 1]:163–176
Hoerling MP, Kumar A, Xu T (2001) Robustness of the nonlinear climate response to ENSO’s extreme phases. J Clim 14:1277–1293
Hu Z, Chen Y, Islam S (1998) Multiscaling properties of soil moisture images and decomposition of large- and small-scale features using wavelet transforms. Int J Remote Sens 19:2451–2467
Hurrell JW (1995) Decadal trends in the North Atlantic oscillation: regional temperatures and precipitation. Science 269:676–679
Kendall BE, Briggs CJ, Murdoch WW, Turchin P, Ellner SP, McCauley E, Nisbet RM, Wood SN (1999) Why do populations cycle? A synthesis of statistical and mechanistic modelling approaches. Ecology 80:1789–1805
Labat D, Ababou R, Mangin A (1999) Wavelet analysis in karstic hydrology. II: Rainfall-runoff cross-wavelet analysis. C R Acad Sci II A 329:881–887
Lark RM, Webster R (2001) Changes in variance and correlation of soil properties with scale and location: analysis using an adapted maximal overlap discrete wavelet transform. Eur J Soil Sci 52:547–562
Lau KM, Weng H (1995) Climate signal detection using wavelet transform: how to make a time series sing. Bull Am Meteorol Soc 76:2391–2402
Law R, Edley MT (1990) Transient dynamics of population with age- and size-dependent vital rates. Ecology 71:1863–1870
Legendre P, Fortin JH (1989) Spatial pattern and ecological analysis. Vegetatio 80:107–138
Legendre P, Legendre L (1998) Numerical ecology, 2nd English edn. Developments in ecological modeling, vol 20. Elsevier, Amsterdam
Legendre P, Dallot S, Legendre L (1985) Succession of species within a community: chronological clustering, with applications to marine and freshwater zooplankton. Am Nat 125:257–288
Levin SA (1992) The problem of pattern and scale in ecology. Ecology 73:1943–1967
Li Z, Kafatos M (2000) Interannual variability of vegetation in the United States and its relation to El Nino/Southern Oscillation. Remote Sens Environ 71:239–247
Machu E, Ferret G, Garcon V (1999) Phytoplankton pigment distribution from SeaWiFS data in the subtropical convergence zone south of Africa: a wavelet analysis. Geophys Res Lett 26:1469–1472
Mallat SG (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11:674–693
Manuca R, Savit R (1996) Stationarity and nonstationarity in time series analysis. Physica D 99:134–161
Margalef R (1991) Teoría de los sistemas ecológicos. Publicacions Universitat de Barcelona, Barcelona
Mariotti A, Zeng N, Lau KM (2002) Euro-Mediterranean rainfall and ENSO –a seasonally varying relationship. Geophys Res Lett 29:10
Martínez ND, Lawton JH (1995) Scale and food-web structure-from local to global. Oikos 73:148–154
May RM (1973) Stability and complexity in model ecosystems. Princeton University Press, New Jersey
Neubert MG, Caswell H (1997) Alternatives to resilience for measuring the responses of ecological systems to perturbations. Ecology 78:653–665
Nicholson AJ (1957) The self-adjustment of populations to change. Cold Spring Harbor Symp Quant Biol 22:153–173
Nils S, Chan KS, Tong H, Boonstra R, Boutin S, Krebs CJ, Post E, O’Donoghue M, Yoccoz NG, Forchhammer MC, Hurrell JW (1999) Common dynamic structure of Canada lynx populations within three climatic regions. Science 285:1071–1073
Ottersen G, Planqué B, Belgrano A, Post E, Reid PC, Stenseth NC (2001) Ecological effects of the North Atlantic oscillation. Oecologia 128:1–14
Pascual M, Rodó X, Ellner S, Colwell R, Bouma M (2000) Cholera dynamics and El Niño-southern oscillation. Science 289:1766–1769
Planqué B, Fromentin JM (1996) Calanus and environment in the eastern North Atlantic. I. Spatial and temporal patterns of C. finmarchicus and C. helgolandicus. Mar Ecol Prog Ser 134:101–109
Planqué B, Taylor AH (1998) Long-term changes in zooplankton and the climate of the North Atlantic. ICES J Mar Sci 55:644–654
Platt T, Denman KL (1975) Spectral analysis in ecology. Annu Rev Ecol Syst 6:189–210
Powell TM, Steele JH (eds) (1995) Ecological time series. Chapman & Hall, New York
Readshaw JL, Cuff WR (1980) A model of Nicholson’s blowfly cycles and its relevance to predation theory. J Anim Ecol 49:1005–1010
Reid PC, Borges MF, Svendsen E (2001) A regime shift in the North Sea circa 1988 linked to changes in the North Sea horse mackerel fishery. Fish Res 50:163–171
Rodó X (2001) Reversal of three global atmospheric fields linking changes in SST anomalies in the Pacific, Atlantic and Indian oceans at tropical latitudes and midlatitudes. Clim Dyn 18:203–217
Rodó X, Baert E, Comín FA (1997) Variations in seasonal rainfall in southern Europe during the present century: relationship with the North Atlantic Oscillation and the El Nino-Southern Oscillation. Clim Dyn 13:275–284
Rodó X, Pascual M, Fuchs G, Faruque ASG (2002) ENSO and cholera: a non-stationary link related to climate change. Proc Natl Acad Sci USA 99:12901–12906
Saunders SC, Chen J, Crow TR, Brosofske KD (1998) Hierarchical relationships between landscape structure and temperature in a managed forest landscape. Landscape Ecol 13:381–395
Steele JH (ed) (1978) Spatial pattern in plankton communities. Plenum, New York
Stephen JA, Jordan MB, Taylor AH, Proctor R (1998) The effects of fluctuations in North Sea flows on zooplankton abundance. J Plankton Res 5:943–956
Stepnowski A, Moszynski M (2000) Inverse problem solution techniques as applied to indirect in situ estimation of fish target strength. J Acoust Soc Am 107:2554–2562
Stokes TK, Gurney WS, Nisbet RM, Blythe SP (1988) Parameter evolution in a laboratory insect population. Theor Popul Biol 34:248–265
Taylor AH (1995) North-south shifts of the Gulf Stream and their climatic connection with the abundance of zooplankton in the UK and surrounding seas. ICES J Mar Sci 52:711–721
Taylor AH, Jordan MB, Stephens JA (1998) Gulf Stream shifts following ENSO events. Nature 393:638
Thomas DM, Wasserstein-Robbins F (1999) Analysis of a non-autonomous Nicholson Blowfly model. Physica A 273:198–211
Torrence C, Compo GP (1998) A practical guide to wavelet analysis. Bull Am Meteorol Soc 79:61–78
Wang B, Wang Y (1996) Temporal structure of the Southern Oscillation as revealed by waveform and wavelet analysis. J Clim 9:1586–1598
Whitfield PH, Dohan K (1997) Identification and characterization of water quality transients using wavelet analysis. II. Application to electronic water quality data. In: Beck MB, Lessard P (eds) Watermatex-97, systems analysis and computing in water quality management: towards a new agenda. Elsevier, Amsterdam
Acknowledgements
We thank S. Pueyo, J.J. Camarero, J. Piera, F. Bartomeus, and X. de Pedro at the Department of Ecology (Universitat de Barcelona, UB) for their fruitful discussions during the development of the SDC method. We gratefully acknowledge M. Pascual, R. Solow, E. Ranta, C. Osenberg, A. Steward-Oaten, and two anonymous reviewers for their useful comments on preliminary versions of this manuscript. We also especially thank Parc Científic de Barcelona (UB), Servei Meteorològic de Catalunya, CIRIT (ACES98 29/7), Moelco (Research branch), and BATMARE (Marie Curie Training Site at the Institutt for fiskeri- og marinbiologi, Universiteten i Bergen, EVK3-CT-2000-57129) for support, funding, and computing facilities. NAO and ENSO indices were kindly provided by the Climate Analysis Section, NCAR, Boulder, Colo., USA (Hurrel 1995). Nicholson’s blowflies and North Sea zooplankton series can be found, respectively, at the URLs of the Monash University and the Sir Alister Hardy Foundation for Ocean Sciences (SAHFOS). Wavelet software (http://paos.colorado.edu/research/wavelets/) was provided by C. Torrence and G.P Compo.
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Rodríguez-Arias, M.À., Rodó, X. A primer on the study of transitory dynamics in ecological series using the scale-dependent correlation analysis. Oecologia 138, 485–504 (2004). https://doi.org/10.1007/s00442-003-1464-4
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DOI: https://doi.org/10.1007/s00442-003-1464-4