Correlation Measures for Solute Transport Model Identification and Evaluation

  • Fred Sonnenwald
  • Virginia Stovin
  • Ian Guymer
Part of the GeoPlanet: Earth and Planetary Sciences book series (GEPS)


Correlation measures are used in a range of applications to quantify the similarity between time-series, often between model output and observed data. A software tool implemented by the authors uses optimisation to identify a system’s Residence Time Distribution (RTD) from noisy solute transport laboratory data. As part of the further development of the tool, an investigation has been undertaken to determine the most suitable correlation measures, both for solute transport model identification as an optimisation constraint and as an objective means of solute transport model evaluation. Correlation measures potentially suitable for use with solute transport data were selected for evaluation. The evaluation was carried out by manipulating synthetic dye traces in ways that reflect common solute transport model discrepancies. The conditions tested include change in number of sample points (discretisation/series length), transformation (scaling, etc.), transformation magnitude, and noise. BLC, \({\chi ^{2}}\), FFCBS, \(\mathrm R ^{2}\), RMSD, \(\text{ R}_\mathrm{t}^{2}\), ISE, and APE show favourable characteristics for use in model identification. Of these, \(\text{ R}^{2}\), \(\text{ R}{}_\mathrm{t}^{2}\) and APE are non-dimensional and so are also suitable for model evaluation.


Sample Point Model Identification Solute Transport Correlation Measure Dynamic Time Warping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Anderson M, Woessner W (1992) Applied groundwater modeling: simulation of flow and advective transport. Academic, LondonGoogle Scholar
  2. Berndt DJ, Clifford J (1994) Using dynamic time warping to find patterns in time series. In: Technical report WS-94-03, The AAAI Press, Melno Park, California, pp 359–370Google Scholar
  3. Boudraa AO, Cexus JC, Groussat M, Brunagel P (2008) An energy-based similarity measure for time series. Eurasip J Adv Signal ProcessGoogle Scholar
  4. Chen L, Ozsu MT, Oria V (2005) Robust and fast similarity search for moving object trajectories. In: Widom J, Ozcan F, Chirkova R (eds) Proceedings of the ACM SIGMOD international conference on management of data, Baltimore, MD, pp 491–502Google Scholar
  5. Cox CS, Boucher AR (1989) Data based models: an automatic method for model structure determination. In: IEE colloquium model validation control system design simulation, London, UK, pp 2, 1–2, 4Google Scholar
  6. Fischer HB (1967) The mechanics of dispersion in natural streams. J Hydraul Div ASCE 93(6):187–215Google Scholar
  7. George SC, Burnham KJ, Mahtani JL, Inst Elect Engineers L (1998) Modelling and simulation of hydraulic components for vehicle applications–a precursor to control system design. In: International conference on simulation ’98, 457. Institute of Electrical Engineers, London, pp 126–132Google Scholar
  8. Ghosh AK (2007) Introduction to linear and digital control systems. Prentice-Hall, IndiaGoogle Scholar
  9. Greenwood P, Nikulin M (1996) A guide to chi-squared testing, vol 280. Wiley-InterscienceGoogle Scholar
  10. Hattersley JG, Evans ND, Hutchison C, Cockwell P, Mead G, Bradwell AR, Chappell MJ (2008) Nonparametric prediction of free-lightchain generation in multiple myelomapatients. In: 17th International federation of automatic control world congress (IFAC), Seoul, Korea, pp 8091–8096Google Scholar
  11. Kashefipour S, Falconer R (2000) An improved model for predicting sediment fluxes in estuarine waters. Proceedings of the fourth international hydroinformatics Conference, Iowa, USA,Google Scholar
  12. Miskiewicz J (2010) Entropy correlation distance method. The euro introduction effect on the consumer price index. Phys A 389(8):1677–1687Google Scholar
  13. Moriasi DN, Arnold JG, Van Liew MW, Bingner RL, Harmel RD, Veith TL (2007) Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans ASABE 50(3):885–900Google Scholar
  14. Movahed MS, Jafari GR, Ghasemi F, Rahvar S, Tabar MRR (2006) Multifractal detrended fluctuation analysis of sunspot time series. Theory Exp J Stat MechGoogle Scholar
  15. Nash JE, Sutcliffe JV (1970) River flow forecasting through conceptual models part i–a discussion of principles. J Hydrol 10(3):282–290CrossRefGoogle Scholar
  16. Rodgers JL, Nicewander WA (1988) Thirteen ways to look at the correlation coefficient. Am Stat 42(1):59–66CrossRefGoogle Scholar
  17. Rutherford JC (1994) River mixing. Wiley, Chichester, EnglandGoogle Scholar
  18. Stovin VR, Guymer I, Chappell MJ, Hattersley JG (2010) The use of deconvolution techniques to identify the fundamental mixing characteristics of urban drainage structures. Water Sci Technol 61(8):2075–2081CrossRefGoogle Scholar
  19. The MathWorks Inc (2011) MATLAB R2011a. Natick, MAGoogle Scholar
  20. Vlachos M, Kollios G, Gunopulos D (2002) Discovering similar multidimensional trajectories. In: Agrawal R (ed) 18th International conference on data engineering. Proceedings, IEEE computer society, pp 673–684Google Scholar
  21. Wang Q, Shen Y, Zhang JQ (2005) A nonlinear correlation measure for multivariable data set. Phys D 200(3–4):287–295CrossRefGoogle Scholar
  22. Ye JC, Tang Y, Peng H, Zheng QL, ieee (2004) Ffcbs: a simple similarity measurement for time series. In: Proceedings of the 2004 international conference on intelligent mechatronics and automation, pp 392–396Google Scholar
  23. Young P, Jakeman A, McMurtrie R (1980) An instrumental variable method for model order identification. Automatica 16(3):281–294CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Civil and Structural EngineeringThe University of SheffieldSheffieldUK
  2. 2.School of EngineeringUniversity of WarwickCoventryUK

Personalised recommendations