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.
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Notes
- 1.
Experimental design methods, such as \({2^{\rm{k}}}\) factorial designs (see Sect. 6.3.1) also provide a measure of sensitivities and have been in existence for several decades. However, these methods have not been identified as promising since they only provide one-way sensitivity (i.e., the effect on the system response when only one parameter is varied at a time) rather than the multi-response sensitivity sought.
- 2.
There is a possibility of confusion between the bootstrap and Monte Carlo simulation approaches. The tie between them is obvious: both are based on repetitive sampling and then direct examination of the results. A key difference between the methods, however, is that bootstrapping uses the original or initial sample as the population from which to resample, whereas Monte Carlo simulation is based on setting up a sample data generation process for the inputs of the simulation or computational model.
- 3.
The original study suggested an additional phase involving refining the estimates of the strong parameters after the bounded grid search was completed. This could be done by one of several methods such as analytical optimization or genetic algorithms. This step has been intentionally left out in order not to overly burden the reader.
- 4.
There were a large number of parameters identified as strong parameters (about 8–12 out of 20 input parameters). Further, it was not always clear as to which of the three equal-probability intervals to select. Hence, for the two-stage calibration, it was more practical to freeze the weak parameters rather than the strong parameters.
- 5.
Though there is a connotational difference between the words “identification” and “estimation” in the English language, no such difference is usually made in the field of inverse modeling. Estimation is a term widely used in statistical mathematics to denote a similar effect as the term identification which appears in electrical engineering literature.
- 6.
This is a good example of the quote by Einstein expressing the view that ought to be followed by all good analysts: “Everything should be as simple as possible, but not simpler”.
- 7.
Some texts refer to this form as the “state-space” model. Control engineers retain the distinction in both terms, and refer to state space as a specific type of control design technique which is based on the state variable model formulation.
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Problems
Problems
Pr. 11.1
Consider the data in Pr. 10.14 where the Wind Chill (WC) factor is tabulated for different values of ambient temperature and wind velocity. Use this data to evaluate different multi-layer perceptron (MLP) architectures assuming one hidden layer only. You will use 50% of the data for training, 25% as validation and 25% as testing. Compare model goodness of fit and residual behavior of the best identified MLP with the regression models identified in Pr. 10.14.
Pr. 11.2
Consider Fig. 1.5c showing the nth order model thermal network for heat conduction through a wall. The internal node temperatures Ts are the state variables whose values are internal to the system, and have to be expressed in terms of the outdoor temperature (variable u) and indoor air temperatures (variable y) and the individual resistors and capacitors of the system (parameters of the system). You will assume the following two networks, write the model equations for the temperature Ts at each internal node, reframe them in the state variable formulation (Eq. 11.17) and identify the elements of the A and B matrices in terms of the resistances and the capacitors:
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(a)
network with 2 capacitors and 3 resistors (3R2C)
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(b)
network with 3 capacitors and 4 resistors (4R3C).
Pr. 11.3
You will repeat the analysis of the three room compartmental problem presented in Sect. 11.3.5 for the configuration shown in Fig. 11.23.
A building with interconnected and leaky rooms
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(a)
Follow the same procedure to derive the exponential equations for the dynamic response of each of the three rooms assuming the same initial conditions, namely: \({x_1}(0) = 900,\;{x_2}(0) = 0,\;{x_3}(0) = 0\)
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(b)
You will now use these models to generate synthetic data with random noise and perform similar analysis as shown in Table. 11.8. Discuss your results.
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Agami Reddy, T. (2011). Inverse Methods. In: Applied Data Analysis and Modeling for Energy Engineers and Scientists. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9613-8_11
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