An investigation of humus disintegration by spatial-temporal regression analysis



We examine the hypothesis of an increase of humus disintegration by analyzing chemical substances measured in the seepage water of a German forest. Problems arise because of a large percentage of missing observations. We use a regression model with spatial and temporal effects constructed in an exploratory data analysis. Spatial dependencies are modeled by random effects and an autoregressive structure for observations in distinct soil depths resulting in a recursive linear mixed model structure. Temporal dependencies are included by an autoregressive structure of the random effects. For parameter estimation an EM algorithm is deduced assuming the errors to be Gaussian. As a result of the data analysis we specify chemical substances which possibly affect the process of humus disintegration. In particular, we find evidence that the presence of aluminum ions is important, but because of the high correlations among the regressors this might be due to confounding with iron.

Key Words

Autoregressive model Maximum likelihood estimation Missing data Mixed effects Recursive linear model Spatial-temporal correlations 


  1. Buonaccorsi, J. P., and Elkinton, J. S. (2002), “Regression Analysis in a Spatial-Temporal Context: Least Squares, Generalized Least Squares, and the Use of the Bootstrap,” Journal of Agricultural, Biological, and Environmental Statistics, 7, 4–20.CrossRefGoogle Scholar
  2. Cressie, N. A. C. (1993), Statistics for Spatial Data, New York: Wiley.Google Scholar
  3. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977), “Maximum Likelihood from Incomplete Data via the EM Algorithm” (with discussion), Journal of the Royal Statistical Society, Series B, 39, 1–38.MATHMathSciNetGoogle Scholar
  4. Eichhorn, J. (1995), “Stickstoffsättigung und ihre Auswirkungen auf das Buchenwaldökosystem der Fallstudie Zierenberg,” Berichte des Forschungszentrums Waldökosysteme, Reihe A, 124, Göttingen, Germany.Google Scholar
  5. Eichhorn, J., and Hüttermann, A. (1999), “Mechanisms of Humus Dynamics and Nitrogen Mineralisation,” in Going Underground—Ecological Studies in Forest Soils, eds. N. Rastin and J. Bauhus, Trivandrum (India): Research Signpost, pp. 239–277.Google Scholar
  6. Fried, R. (1999), “Räumlich-zeitliche Modellierung der Kohlenstoffkonzentration im Waldbodensickerwasser zur Untersuchung der Hypothese der Humusdisintegration,” PhD thesis, Department of Mathematics, Darmstadt University of Technology, D-64289 Darmstadt, Germany.Google Scholar
  7. — (2001), “Autoregressive Linear Mixed Models for Investigating Humus Disintegration,” Biometrical Journal, 43, 757–766.CrossRefMathSciNetGoogle Scholar
  8. Fried, R., Eichhorn, J., and Paar, U. (2001), “Exploratory Analysis and a Stochastic Model for Humus Disintegration,” Environmental Monitoring and Assessment, 68, 273–295.CrossRefGoogle Scholar
  9. Harrison, P. J., and Stevens, C. F. (1976), “Bayesian Forecasting,” (with discussion), Journal of the Royal Statistical Society, Series B, 38, 205–246.MATHMathSciNetGoogle Scholar
  10. Heinz, J., and Spellucci, P. (1994), “A Successful Implementation of the Pantoja-Mayne SQP Method,” Optimization Methods and Software, 4, 1–28.CrossRefGoogle Scholar
  11. Hodges, J. L., and Lehmann, E. L. (1954), “Testing the Approximate Validity of Statistical Hypotheses,” Journal of the Royal Statistical Society, Series B, 16, 259–268.MathSciNetGoogle Scholar
  12. Jones, R.H. (1980), “Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations,” Technometrics, 22, 389–395.MATHCrossRefMathSciNetGoogle Scholar
  13. Kalman, R. E. (1960), “A New Approach to Linear Filtering and Prediction Problems,” Journal of Basic Engineering, 82, 34–45.Google Scholar
  14. Longford, N. T. (1993), Random Coefficient Models, Oxford: Clarendon Press.MATHGoogle Scholar
  15. Schwarz, G. (1978), “Estimating the Dimension of a Model,” The Annals of Statistics, 6, 461–464.MATHCrossRefMathSciNetGoogle Scholar
  16. Shumway, R. H., and Stoffer, D. S. (1982), “An Approach to Time Series Smoothing and Forecasting Using the EM Algorithm,” Journal of Time Series Analysis, 3, 253–264.MATHCrossRefGoogle Scholar
  17. Ulrich, B. (1981), “Theoretische Betrachtung des Ionenkreislaufs in Waldökosystemen,” Zeitschrift für Pflanzenern ährung und Bodenkunde, 144, 647–659.CrossRefGoogle Scholar
  18. Wermuth, N. (1980), “Linear Recursive Equations, Covariance Selection, and Path Analysis,” Journal of the American Statistical Association, 75, 963–972.MATHCrossRefMathSciNetGoogle Scholar
  19. West, M., and Harrison, J. (1989), Bayesian Forecasting and Dynamic Models, New York: Springer.MATHGoogle Scholar
  20. Wu, C. F. J. (1983), “On the Convergence Properties of the EM Algorithm,” The Annals of Statistics, 11, 95–103.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© International Biometric Society 2004

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of DortmundGermany

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