Oecologia

, Volume 156, Issue 2, pp 287–304 | Cite as

Wavelet analysis of ecological time series

  • Bernard Cazelles
  • Mario Chavez
  • Dominique Berteaux
  • Frédéric Ménard
  • Jon Olav Vik
  • Stéphanie Jenouvrier
  • Nils C. Stenseth
Population Ecology - Original Paper

Abstract

Wavelet analysis is a powerful tool that is already in use throughout science and engineering. The versatility and attractiveness of the wavelet approach lie in its decomposition properties, principally its time-scale localization. It is especially relevant to the analysis of non-stationary systems, i.e., systems with short-lived transient components, like those observed in ecological systems. Here, we review the basic properties of the wavelet approach for time-series analysis from an ecological perspective. Wavelet decomposition offers several advantages that are discussed in this paper and illustrated by appropriate synthetic and ecological examples. Wavelet analysis is notably free from the assumption of stationarity that makes most methods unsuitable for many ecological time series. Wavelet analysis also permits analysis of the relationships between two signals, and it is especially appropriate for following gradual change in forcing by exogenous variables.

Keywords

Ecological time series Transient dynamics Non-stationarity Discontinuities Wavelets Wavelet analysis Wavelet Power Spectrum Wavelet coherency Environmental forcing 

References

  1. Barbraud C, Weimerskirch H (2001) Emperor penguins and climate change. Nature 411:183–186PubMedCrossRefGoogle Scholar
  2. Bartlett MS (1954) Problèmes de l’analyse spectrale des séries temporelles stationnaires. Publ Inst Stat Univ Paris 3:119–134Google Scholar
  3. Benton TG, Plaistow SJ, Coulson TN (2006) Complex population dynamics and complex causation: devils, details and demography. Proc R Soc Lond B 273:1173–1181CrossRefGoogle Scholar
  4. Bierman SM, Fairbairn JP, Petty SJ, Elston DA, Tidhar D, Lambin X (2006) Changes over time in the spatiotemporal dynamics of cyclic populations of field voles (Microtus agrestis L.). Am Nat 167:583–590PubMedCrossRefGoogle Scholar
  5. Bjørnstad ON, Grenfell BT (2001) Noisy clock: time series analysis of population fluctuations in animals. Science 293:638–643PubMedCrossRefGoogle Scholar
  6. Bradshaw GA, Spies TA (1992) Characterizing canopy gap structure in forest using wavelet analysis. J Ecol 80:205–215CrossRefGoogle Scholar
  7. Buonaccorsi JP, Elkinton JS, Evans SR, Liebhold AM (2001) Measuring and testing spatial synchrony. Ecology 82:1628–1679CrossRefGoogle Scholar
  8. Cazelles B (2001) Blowout bifurcation with non-normal parameters in population dynamics. Phys Rev E 64:032901CrossRefGoogle Scholar
  9. Cazelles B (2004) Symbolic dynamics for identifying similarity between rhythms of ecological time series. Ecol Lett 7:755–763CrossRefGoogle Scholar
  10. Cazelles B, Chau NP (1997) Using the Kalman filter and dynamic models to assess the changing HIV/AIDS epidemic. Math Biosci 140:131–154PubMedCrossRefGoogle Scholar
  11. Cazelles B, Hales S (2006) Infectious diseases, climate influences and nonstationarity. PLoS Med 3:1212–1213 (e328)CrossRefGoogle Scholar
  12. Cazelles B, Stone L (2003) Detection of imperfect population synchrony in an uncertain world. J Anim Ecol 72:953–968CrossRefGoogle Scholar
  13. Cazelles B, Bottani S, Stone L (2001) Unexpected coherence and conservation. Proc Royal Soc Lond B 268:2595–2602CrossRefGoogle Scholar
  14. Cazelles B, Chavez M, McMichael AJ, Hales S (2005) Nonstationary influence of El Niño on the synchronous dengue epidemics in Thailand. PLoS Med 2:313–318 (e106)CrossRefGoogle Scholar
  15. Chatfield JR (1989) The analysis of time series: an introduction. Chapman & Hall, LondonGoogle Scholar
  16. Cummings DAT,, Irizarry RA, Huang NE, Endy TP, Nisalak A, Ungchusak K, Burke DS (2004) Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand. Nature 427:344–347PubMedCrossRefGoogle Scholar
  17. Cushing JM, Dennis B, Desharnais RA, Costantino RF (1998) Moving toward an unstable equilibrium: saddle nodes in population systems. J Anim Ecol 67:298–306CrossRefGoogle Scholar
  18. Dale MRT, Mah M (1998) The use of wavelets for spatial pattern analysis in ecology. J Veg Sci 9:805–815CrossRefGoogle Scholar
  19. Daubechies I (1992) Ten lectures on wavelets. SIAM monographs, PhiladelphiaGoogle Scholar
  20. Duncan CJ, Duncan SR, Scott S (1996) Whooping cough epidemics in London, 1701–1812: infection dynamics, seasonal forcing and the effects of malnutrition. Proc R Soc Lond B 263:445–450CrossRefGoogle Scholar
  21. Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman & Hall, LondonGoogle Scholar
  22. Elton CS (1924) Fluctuations in the numbers of animals, their causes and effects. Br J Exp Biol 2:119–163Google Scholar
  23. Elton CS, Nicholson M (1942) The ten-year cycle in numbers of the lynx in Canada. J Anim Ecol 11:215–244CrossRefGoogle Scholar
  24. Forchhammer MC, Post E (2004) Using large-scale climate indices in climate change ecology studies. Popul Ecol 46:1–12CrossRefGoogle Scholar
  25. Foster G (1996) Wavelets for period analysis of unevenly sampled time series. Astronom J 112:1709–1729CrossRefGoogle Scholar
  26. Gabor D (1946) Theory of communication. J Inst Electr Eng 93:429–457Google Scholar
  27. Grenfell BT, Bjørnstad ON, Kappey J (2001) Travelling waves and spatial hierarchies in measles epidemics. Nature 414:716–723PubMedCrossRefGoogle Scholar
  28. Hare SR, Mantua NJ (2000) Empirical evidence for North Pacific regime shifts in 1997 and 1989. Prog Oceanogr 47:103–145CrossRefGoogle Scholar
  29. Hastings A (2001) Transient dynamics and persistence of ecological systems. Ecol Lett 4:215–220CrossRefGoogle Scholar
  30. Haydon DT, Greenwood PE, Stenseth NC, Saitoh T (2003) Spatio-temporal dynamics of the grey-sided vole in Hokkaido: identifying coupling using state-based Markov-chain modelling. Proc R Soc Lond B 270:435–445CrossRefGoogle Scholar
  31. Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, Yen NC, Tung CC, Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc Lond A 454:903–995CrossRefGoogle Scholar
  32. Ives AR, Dennis B, Cottingham KL, Carpenter SR (2003) Estimating community stability and ecological interactions from time series data. Ecol Monogr 73:301–330CrossRefGoogle Scholar
  33. Jenouvrier S, Weimerskirch H, Barbraud C, Park YH, Cazelles B. (2005) Evidence of a shift in the cyclicity of Antarctic seabirds dynamics linked to climate. Proc R Soc Lond B 272:887–895CrossRefGoogle Scholar
  34. Johnson DM, Bjørnstad ON, Liebhold AM (2006) Landscape mosaic induces traveling waves of insect outbreaks. Oecologia 148:51–60PubMedCrossRefGoogle Scholar
  35. José MV, Bishop RF (2003) Scaling properties and symmetrical patterns in the epidemiology of rotavirus infection. Phil Trans R Soc Lond B 358:1625–1641CrossRefGoogle Scholar
  36. Keitt TH, Fischer J (2006) Detection of scale-specific community dynamics using wavelets. Ecology 87:2895–2904PubMedCrossRefGoogle Scholar
  37. Keitt TH, Urband DL (2005) Scale-specific inference using wavelets. Ecology 86:2497–2504CrossRefGoogle Scholar
  38. Klvana I, Berteaux D, Cazelles B (2004) Porcupine feeding scars and climatic data show ecosystem effects of the solar cycle. Am Nat 164:283–297PubMedCrossRefGoogle Scholar
  39. Koelle K, Pascual M (2004) Disentangling extrinsic from intrinsic factors in disease dynamics: a nonlinear time series approach with an application to cholera. Am Nat 163:901–913PubMedCrossRefGoogle Scholar
  40. Lau KM, Weng H (1995) Climatic signal detection using wavelet transform: how to make a time series sing. Bull Am Meteorol Soc 76:2391–2402CrossRefGoogle Scholar
  41. Liebhold AM, Koening WD, Bjørnstad ON (2004) Spatial synchrony in population dynamics. Annu Rev Ecol Evol Syst 35:467–490CrossRefGoogle Scholar
  42. Liu PC (1994) Wavelet spectrum analysis and ocean wind waves. In: Foufoula-Georgiou E, Kumar P (eds) Wavelets in geophysics. Academic Press, New York, pp 151–166Google Scholar
  43. Mackenzie JMD (1952) Fluctuations in the number of British tetraonids. J Anim Ecol 21:128–153Google Scholar
  44. Mallat S (1998) A wavelet tour of signal processing. Academic Press, San DiegoGoogle Scholar
  45. Ménard F, Marsac F, Bellier B, Cazelles B (2007) Climatic oscillations and tuna catch rates in the Indian Ocean: a wavelet approach to time series analysis. Fish Oceanogr 16:95–104CrossRefGoogle Scholar
  46. Meyers SD, Kelly BG, O’Brien JJ (1993) An introduction to wavelet analysis in oceanography and meteorology with application to the dispersion of Yanai waves. Mon Weather Rev 121:2858–2866CrossRefGoogle Scholar
  47. Mi X, Ren H, Ouyang Z, Wei W, Ma K (2005) The use of the Mexican hat and the Morlet wavelets for detection of ecological patterns. Plant Ecol 179:1–19CrossRefGoogle Scholar
  48. Murdoch WW, Kendall BE, Nisbet RM, Briggs CJ, McCauley E, Bolser R (2002) Single-species models for many-species food webs. Nature 423:541–543CrossRefGoogle Scholar
  49. Nezlin NP, Li BL (2003) Time-series analysis of remoted-sensed chlorophyll and environmental factors in the Santa Monica-San Pedro basin off Southern California. J Mar Syst 39:185–202CrossRefGoogle Scholar
  50. Pikovsky AS, Rosenblum MG, Kurths J (2001) Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, CambridgeGoogle Scholar
  51. Platt T, Denman KL (1975) Spectral analysis in ecology. Annu Rev Ecol Syst 6:189–210CrossRefGoogle Scholar
  52. Reid PC, Borges MF, Svendsen E (2001) A regime shift in the North Sea circa in 1988 linked to changes in the North Sea horse mackerel fishery. Fish Res 50:163–171CrossRefGoogle Scholar
  53. Rodó X., Pascual M, Fuchs G, Faruque SG (2002) ENSO and cholera: a nonstationary link related to climate change? Proc Natl Acad Sci USA 99:12901–12906PubMedCrossRefGoogle Scholar
  54. Rodriguez-Arias MA, Rodó X (2004) A primer on the study of transitory dynamics in ecological series using the scaledependent correlation analysis. Oecologia 138:485–504PubMedCrossRefGoogle Scholar
  55. Rohani P, Earn DJD, Grenfell BT (1999) Opposite patterns of synchrony in sympatric disease metapopulations. Science 286:968–971PubMedCrossRefGoogle Scholar
  56. Rohani P, Green CJ, Mantilla-Beniers NB, Grenfell BT (2003) Ecological interference between fatal diseases. Nature 422:885–888PubMedCrossRefGoogle Scholar
  57. Rosenberg M (2004) Wavelet analysis for detecting anisotropy in point patterns. J Veg Sci 15:277–284CrossRefGoogle Scholar
  58. Rouyer T, Fromentin JM, Stenseth NC, Cazelles B (2008) Analysing multiple time series and extending significance testing in wavelet. Mar Ecol Progr Ser (in press)Google Scholar
  59. Royama T (1992) Analytical population dynamics. Chapman & Hall, LondonGoogle Scholar
  60. Saitoh T, Cazelles B, Vik JO, Viljugrein H, Stenseth NC (2006) Effects of the regime shift on population dynamics of the grey-sided vole in Hokkaido, Japan. Clim Res 32:109–118CrossRefGoogle Scholar
  61. Sinclar ARE, Gosline JM, Holdsworth G, Krebs CJ, Boutin S, Smith JNM, Boonstra R, Dale M (1993) Can the solar cycle and climate synchronize the snowshoe hare cycle in Canada? Evidence from the tree rings and ice cores. Am Nat 141:173–198CrossRefGoogle Scholar
  62. Stenseth NC, Mysterud A, Ottersen G, Hurrel JW, Chan KS, Lima M (2002) Ecological effects of climate fluctuations. Science 297:1292–1296PubMedCrossRefGoogle Scholar
  63. Sweldens W (1998) The lifting scheme: a construction of second generation wavelets. SIAM J Math Anal 29:511–546CrossRefGoogle Scholar
  64. Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD (1992) Testing for nonlinearity in time series: the method of surrogate data. Physica D 58:77–94CrossRefGoogle Scholar
  65. Torrence C, Compo GP (1998) A practical guide to wavelet analysis. Bull Am Meteorol Soc 79:61–78CrossRefGoogle Scholar
  66. Vautard R, Yiou P, Ghil M (1992) Singular spectrum analysis: a toolkit for short, noisy chaotic signals. Physica D 58:95–126CrossRefGoogle Scholar
  67. Xia Y, Bjørnstad ON, Grenfell BT (2004) Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics. Am Nat 164:267–281PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Bernard Cazelles
    • 1
    • 2
  • Mario Chavez
    • 3
  • Dominique Berteaux
    • 4
  • Frédéric Ménard
    • 5
  • Jon Olav Vik
    • 6
  • Stéphanie Jenouvrier
    • 7
  • Nils C. Stenseth
    • 6
  1. 1.Ecole Normale SupérieureCNRS UMR 7625ParisFrance
  2. 2.IRD UR GEODESBondyFrance
  3. 3.CHU Pitié-SalpêtrièreLENA – CNRS UPR 640ParisFrance
  4. 4.Canada Research Chair in Conservation of Northern EcosystemsUniversité du Québec à RimouskiRimouskiCanada
  5. 5.Centre de Recherche Halieutique Méditerrannéen et TropicalIRD, UR 109 THETISSèteFrance
  6. 6.Centre for Ecological and Evolutionary Synthesis (CEES), Department of BiologyUniversity of OsloOsloNorway
  7. 7.Centre d’Etudes Biologiques de ChizéCNRSVilliers en BoisFrance

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