Abstract
The problem of assessing the independence of time series arises in many situations, including evaluating the spatial synchrony of populations in different locations over time. Tests for independence generally have relied on assuming a particular dynamic model for each of the series, and those that do not, require long series. We adapt a test for association between spatial processes to provide a model-free (MF) test for independence between two time series under the assumption that each series is stationary and normally distributed. We evaluate the performance of the test through simulations and compare it with the naive (N) test, which ignores serial correlations, as well as with tests based on residuals from fitting specific dynamic models. We also consider additional tests that involve bootstrap** the MF and N tests. We find that the MF test generally preserves the desired test size, although this is not the case in some extreme settings. The MF test is clearly superior to residual-based tests that arise from fitting an incorrect model. The bootstrap tests are not as robust as the general MF test, but when they are valid, they seem to be more powerful. We also examine the robustness of the procedure to the additional measurement errors present in many applications, explore the extent to which some deficiencies in the MF test are due to estimation of the unknown covariances, and investigate the effect of nonnormality on the MF test. We illustrate the MF test’s performance through an example assessing mouse populations from different locations.
Similar content being viewed by others
References
Adler, G. H. (1994), “Tropical Forest Fragmentation and Isolation Promote Asynchrony Among Populations of a Frugivorous Rodent,” Journal of Animal Ecology, 63 (4), 903–911.
Alpargu, G., and Dutilleul, P. (2003), “To Be or not to Be Valid in Testing the Significance of the Slope in Simple Quantitative Linear Models With Autocorrelated Errors,” Journal of Statistical Computation and Simulation, 73, 165–180.
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994), Time Series Analysis: Forecasting and Control, Englewood Cliffs, NJ: Prentice Hall.
Brockwell, P. J., and Davis, R. A. (1996), Introduction to Time Series and Forecasting, New York: Springer.
Buonaccorsi, J. P., Elkington, J. S., Evans, S. R., and Liebhold, A. M. (2001), “Measuring and Testing for Spatial Synchrony,” Ecology, 82 (6), 1668–1679.
Clifford, P., and Richardson, S. (1985), “Testing the Association Between Two Spatial Processes,” Statistics and Decisions, 2, 155–160.
Clifford, P., Richardson, S., and Hémon, D. (1989), “Assessing the Significance of the Correlation Between Two Spatial Processes,” Biometrics, 45, 123–134.
Davison, A. C., and Hinkley, D. V. (1997), Bootstraps Methods and Their Application, Cambridge: Cambridge University Press.
Dennis, B., and Taper, M. L. (1994), “Density Dependence in Time Series Observations of Natural Populations: Estimation and Testing,” Ecology, 64, 205–224.
Duchesne, P., and Roy, R. (2003), “Robust Tests for Independence of Two Time Series,” Statistica Sinica, 13, 827–852.
Dutilleul, P. (1993), “Modifying the t Test for Assessing the Correlation Between Two Spatial Processes,” Biometrics, 49, 305–314.
El Himdi, K., and Roy, R. (1997), “Tests for Non-Correlation of Two Multivariate ARMA Time Series,” Canadian Journal of Statistics, 25, 233–256.
El Himdi, K., Roy, R., and Duchesne, P. (2003), “Test for Non-Correlation of Two Time Series: A Nonparametric Approach,” in Mathematical Statistics and Applications: Festschrift for Constance van Eden. IMS Lecture Notes Monograph Series, Vol. 42, eds. D. Froda, C. Léger, and M. Moore, Hayward, CA: Institute of Mathematical Statistics, pp. 397–416.
Elkinton, J. S., Healy, W. M., and Buonaccorsi, J. (1996), “Interactions Among Gypsy Moths, White-Footed Mice, and Acorns,” Ecology, 77, 2332–2342.
Fromentin, J.M., Stenseth, N. C., Gjosaeter, J., Johannessen, T., and Planque, B. (1998), “Long-Term Fluctuations in Cod and Pollack Along the Norwegian Skagerrak Coast,” Marine Ecology—Progress Series, 162, 265–278.
Garber, S. D., and Burger, J. (1995), “A 20-yr Study Documenting the Relationship Between Turtle Decline and Human Recreation,” Ecological Applications, 5 (4), 1151–1162.
Graybill, F. A. (1976), Theory and Application of the Linear Model, North Scituate, MA: Duxbury Press.
Hallin, M., Jurečková, J., Picek, J., and Zahaf, T. (1999), “Nonparametric Tests of Independence of Two Autoregressive Time Series Based on Autoregression Rank Scores,” Journal of Statistical Planning and Inference, 75, 319–330.
Haugh, L. D. (1976), “Checking the Independence of Two Covariance-Stationary Time Series: A Univariate Residual Cross-Correlation Approach,” Journal of the American Statistical Association, 71, 378–385.
Hawkins, B. A., and Holyoak, M. (1998), “Transcontinental Crashes of Insect Populations?” American Naturalist, 152 (3), 480–484.
Hong, Y. (1996), “Testing for Independence Between Two Covariance Stationary Series,” Biometrika, 83, 615–625.
Li, H., and Madala, G. S. (1996), “Bootstrap** Time Series Models” (with discussion), Econometric Reviews, 15, 115–195.
Liebhold, A. M., Koenig, W. D., and Bjornstad, O. N. (2004), “Spatial Synchrony in Population Dynamics,” Annual Review of Ecology, Evolution and Systematics, 35, 467–490.
Mackinrogalska, R., and Nabaglo, L. (1990), “Geographical Variation in Cyclic Periodicity and Synchrony in the Common Vole, Microtus-Arvalis,” Oikos, 59 (3), 343–348.
Moran, P. A. P. (1953), “The Statistical Analysis of the Canadian Lynx Cycle. II. Synchronization and Meteorology,” Australian Journal of Zoology, 1, 291–298.
Pham, D. T., Roy, R., and Cédras, L. (2003), “Tests for Non-Correlation of Two Cointegrated ARMA Time Series,” Journal of Time Series Analysis, 24 (5), 553–577.
Pollard, E. (1991), “Synchrony of Population Fluctuations—the Dominant Influence of Widespread Factors on Local Butterfly Populations,” Oikos, 60 (1), 7–10.
Pyper, B. J., and Peterman, R. M. (1998), “Comparison of Methods to Account for Autocorrelation in Correlation Analyses of Fish Data,” Canadian Journal of Fisheries and Aquatic Sciences, 55 (9), 2127–2140.
Royama, T. (1992), Analytical Population Dynamics, New York: Chapman & Hall.
Stuart, A. (1955), “A Paradox in Statistical Estimation,” Biometrika, 42, 527–529.
Williams, D. W., and Liebhold, A. M. (1995), “Influence of Weather Oil the Synchrony of Gypsy-Moth (Lepidoptera, Lymantriidae) Outbreaks in New-England,” Environmental Entomology, 24 (5), 987–995.
Yang, M. C. K., and Rao, P. V. (1993), “Testing Concomitancy Between 2-Physiological Pulse Series,” Statistics in Medicine, 12 (21), 2043–2055.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alpargu, G., Buonaccorsi, J. A model-free test for independence between time series. JABES 14, 115–132 (2009). https://doi.org/10.1198/jabes.2009.0007
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1198/jabes.2009.0007