# Inverse Methods

## Abstract

A broad description of inverse methods (introduced in Sect. 1.3.3) is that they pertain to the case when the system under study already exists, and one uses measured or observed system behavior to aid in the model building. The three types of inverse problems were classified as: (i) calibration of white-box models (which can be either relatively simple or complex-coupled simulation models) which requires that one selectively manipulate certain parameters of the model to fit observed data. Monte Carlo methods which are powerful random sampling techniques are described along with regional sensitivity analyses as a means of reducing model order of complex simulation models; (ii) model selection and parameter estimation involving positing either black-box or grey-box models, and using regression methods to identify model parameters based on some criterion of error minimization (the least squares regression method and the maximum likelihood method being the most popular); and (iii) control problems where input states and/or boundary conditions are inferred from knowledge of output states and model parameters. This chapter elaborates on these methods and illustrates their approach with case study examples. Local polynomial regression methods are also briefly described as well as the multi-layer perceptron approach, which is a type of neural network modeling method that is widely used in modeling non-linear or complex phenomena. Further, the selection of a grey-box model based more on policy decisions rather than on how well a model fits the data is illustrated in the framework of dose-response models. Finally, variable model formulation and compartmental modeling appropriate for describing dynamic behavior of linear systems are introduced along with a discussion of certain identifiability issues in practice.

### Keywords

Dioxide Transportation Radar Stratification Posit### References

- ASHRAE 14-2002, Guideline 14-2002: Measurement of Energy and Demand Savings, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta.Google Scholar
- Bard, Y., 1974. Nonlinear Parameter Estimation, Academic Press, New York.MATHGoogle Scholar
- Claridge, D.E. and M. Liu, 2001. HVAC System Commissioning, Chap. 7.1 Handbook of Heating, Ventilation and Air Conditioning, J.F. Kreider (editor), CRC Press, Boca Raton, FL.Google Scholar
- Clarke, J.A., 1993. Assessing building performance by simulation,
*Building and Environment*, 28(4), pp.419-427.CrossRefGoogle Scholar - Crump, K.S., 1984. An improved procedure for low-dose carcinogenic risk assessment from animal data,
*Journal of Environmental Pathology, Toxicology and Oncology*, (5) 4/5:, pp 339-349.Google Scholar - Devore J., and N. Farnum, 2005. Applied Statistics for Engineers and Scientists, 2
^{nd}Ed., Thomson Brooks/Cole, Australia.Google Scholar - Edwards, C.H. and D.E. Penney, 1996. Differential Equations and Boundary Value Problems, Prentice Hall, Englewood Cliffs, NJ.MATHGoogle Scholar
- Esteban, J., A. Starr, R. Willetts, P. Hannah and P. Bryanston-Cross, 2005. A review of data fusion models and architectures: Towards engineering guidelines,
*Neural Computing and Applications*, (14), pp. 273-281.CrossRefGoogle Scholar - Evans, W. C., 1996. Linear Systems, Compartmental Modeling, and Estimability Issues in IAQ Studies, in Tichenor, B.,
*Characterizing Sources of Indoor Air Pollution and Related Sink Effects*, ASTM STP 1287, pp. 239-262.Google Scholar - Fausett, L., 1993. Fundamentals of Neural Network: Architectures, Algorithms, and Applications, Prentice Hall, Englewood Cliffs, NJ.Google Scholar
- Franklin, G.F., J.D. Powell and A. Emami-Naeini,1994. Feedback Control of Dynamic Systems, 3
^{rd}Ed., Addison-Wesley, Reading, MA.Google Scholar - Godfrey, K., 1983. Compartmental Models and Their Application, Academic Press, New York.Google Scholar
- Haykin, S., 1999. Neural Networks, 2nd ed., Prentice Hall, NJ.MATHGoogle Scholar
- Hammersley, J.M. and D.C. Handscomb, 1964. Monte Carlo Methods, Methuen and Co., London.CrossRefMATHGoogle Scholar
- Helton, J.C. and F.J. Davis, 2003. “Latin hypercube sampling and the propagation of uncertainty of complex systems”,
*Reliability Engineering and System Safety*, vol. 81, pp. 23-69.CrossRefGoogle Scholar - Hofer, E., 1999. “Sensitivity analysis in the context of uncertainty analysis for computationally intensive models”,
*Computer Physics Communication*, vol. 117, pp. 21-34.CrossRefGoogle Scholar - Hornberger, G.M. and R.C. Spear, 1981. “An approach to the preliminary analysis of environmental systems”,
*Journal of Environmental Management*, vol.12, pp. 7-18.Google Scholar - Iman, R.L. and J.C. Helton, 1985. “A Comparison of Uncertainty and Sensitivity Analysis Techniques for Computer Models”, Sandia National Laboratories report NUREG/CR-3904, SAND 84-1461.Google Scholar
- IPCC, 1996. Climate Change 1995: The Science of Climate Change, Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, UK.Google Scholar
- Kammen, D.M. and Hassenzahl, D.M., 1999.
*Should We Risk It*, Princeton University Press, Princeton, NJGoogle Scholar - Kawashima, M., C.E. Dorgan and J.W. Mitchell, 1998, Hourly thermal load prediction for the next 24 hours by ARIMA, EWMA, LR, and an artificial neural network,
*ASHRAE Trans. 98(2),*Atlanta, GA.Google Scholar - Lam, J.C. and S.C.M. Hui, 1996. “Sensitivity analysis of energy performance of office buildings”,
*Building and Environment*, vol.31, no.1, pp 27-39.CrossRefGoogle Scholar - Mayer, T., F. Sebold, A. Fields, R. Ramirez, B. Souza and M. Ciminelli, 2003. “DrCEUS: Energy and demand usage from commercial on-site survey data”,
*Proc. of the International Energy program Evaluation Conference*, Aug. 19-22, Seattle, WA.Google Scholar - Masters, G.M. and W.P. Ela, 2008. Introduction to Environmental Engineering and Science, 3
^{rd}Ed. Prentice Hall, Englewood Cliffs, NJ.Google Scholar - Miller, R.C. and J.E. Seem, 1991. Comparison of artificial neural networks with traditional methods of predicting return time from night or weekend setback,
*ASHRAE Trans., 91(2)*, Atlanta, GA*.,*Google Scholar - Pindyck, R.S. and D.L. Rubinfeld, 1981. Econometric Models and Economic Forecasts, 2nd Edition, McGraw-Hill, New York, NY.Google Scholar
- Reddy, T.A., 2006. Literature review on calibration of building energy simulation programs: uses, problems, procedures, uncertainty and tools,
*ASHRAE Transactions*, vol.12, no.1, pp.177-196.Google Scholar - Reddy, T.A., I. Maor and C. Ponjapornpon, 2007a, “Calibrating detailed building energy simulation programs with measured data- Part I: General methodology”, HVAC&R Research Journal, vol. 13(2), March.Google Scholar
- Reddy, T.A., I. Maor and C. Ponjapornpon, 2007b, “Calibrating detailed building energy simulation programs with measured data- Part II: Application to three case study office buildings”, HVAC&R Research Journal, vol. 13(2), March.Google Scholar
- Saltelli, A., K. Chan and E.M. Scott (eds.) 2000. Sensitivity Analysis, John Wiley and Sons, Chichester.MATHGoogle Scholar
- Saltelli, A., 2002. “Sensitivity analysis for importance assessment”,
*Risk Analysis*, vol. 22, no.2, pp. 579-590.CrossRefGoogle Scholar - Sinha, N.K. and B. Kuszta, 1983. Modeling and Identification of Dynamic Systems, Van Nostrand Reinhold Co., New York.Google Scholar
- Sonderegger, R., J. Avina, J. Kennedy and P. Bailey, 2001.Deriving loadshapes from utility bills through scaled simulation,
*ASHRAE seminar presentation*, Kansas City, MI.Google Scholar - Spears, R., T. Grieb and N. Shiang, 1994. “Parameter uncertainty and interaction in complex environmental models”,
*Water Resources Research*, vol. 30 (11), pp 3159-3169.CrossRefGoogle Scholar - SPSS, 1997.Nural Connection- Applications Guide, SPSS Inc and Recognition Systems Inc., Chicago, IL.Google Scholar
- Sun, J. and T.A. Reddy, 2006. Calibration of building energy simulation programs using the analytic optimization approach,
*HVAC&R Research Journal*, vol.12, no.1, pp.177-196.CrossRefGoogle Scholar - Wasserman, P.D., 1989. Neural Computing: Theory and Practice, Van Nostrand Reinhold, New York.Google Scholar
- Winkelmann, F.C., B.E. Birdsall, W.F. Buhl, K.L. Ellington, A.E. Erdem, J.J. Hirsch, and S. Gates, 1993. DOE-2 Supplement, Version 2.1E, Lawrence Berkeley National Laboratory, November.Google Scholar