The three plots below demonstrate the flexibility of DHR method to extract trend and seasonal components, seasonally adjust data and assess the effects of exogenous changes (interventions in this case).
The first figure shows the data and model fit, including the standard error band.
The second plot shows seasonally adjusted data with the trend estimate and the accompanying uncertainty band (equivalent to standard error band). This figure illustrates the estimates of instantaneous trend level changes that can be attributed to introduction of specific legislation. It is also easy to pinpoint the data that may possibly be treated as outliers - shown here as being outside 3 standard errors band.
The last figure shows the estimate of the above trend's slope or rate of change together with its standard error band. It is easy to see how the periods of significant change can be distinguished from those where the change is not significant.
For more information see:
Young, P. C. Pedregal D.J., Tych, W. (1999), Dynamic Harmonic Regression, Journal of Forecasting, 18, 369-394.
with some selected application papers showing the versatility of the method:
Tych W., Pedregal D. J., Young P.C., Davies J. (2002) Multi-Rate Forecasting of Telephone Call Demand: A Software Package for Unobserved Components Modelling and Forecasting, International Journal of Forecasting, 18, 673-695
Carling, P.A., Tych, W. and Richardson, K. (2005). The hydraulic scaling of step-pool systems. pp 55-63 in G. Parker and M.H. Garcia (eds) River, Coastal and Estuarine Morphodynamics. Vol 1.Balkema, Taylor and Francis, NY, ISBN: 0 415 39375 2
Boochabun, K., Tych, W., Chappell, N.A., Carling P.A., Lorsirirat K., and Pa-Obsaeng, S. (2004. Statistical modelling of rainfall and river flow in Thailand. Journal of the Geological Society of India, 64, 503-515
Becker, S., Halsall, C. J., Tych, W., Hung, H. H., Attewell, S., Blanchard, P., Li, H., Fellin, P., Stern, G., Billeck, B., Friesen, S. 2006. Resolving the long-term trends of polycyclic aromatic hydrocarbons in the Canadian arctic atmosphere. Environ. Sci. Technol. 40, 3217-3222.
See also the Captain Toolbox pages and its accompanying paper:
C.James Taylor, Diego J. Pedregal, Peter C. Young, Wlodek Tych: Environmental time series analysis and forecasting with the Captain toolbox. Environmental Modelling and Software 22(6): 797-814 (2007)
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Figures from Becker, S., Halsall, C. J., Tych,
W., Su, Y., Hung, H. H., Kallenborn, R. (2008) ©
with annotations by WT
17th March 2009