Input-output models

The general input-output model for discrete data could be written ... 

Number of data points, number of stages or effects Number of inputs/outputs, model horizon Pressure... 

An optimization criterion for determining the output parameters and basis functions is to minimize the output prediction error and is common to all input-output modeling methods. The activation or basis functions used in data analysis methods may be broadly divided into the following two categories ... 

The nature of the input transformation, type of basis functions, and optimization criteria discussed in this section provide a common framework for comparing the wide variety of techniques for input transformation and input-output modeling. This comparison framework is useful for understanding the similarities and differences between various methods it may be used to select the best method for a given task and to identify the challenges for combining the properties of various techniques (Bakshi and Utojo, 1999). 

Methods based on linear projection transform input data by projection on a linear hyperplane. Even though the projection is linear, these methods may result in either a linear or a nonlinear model depending on the nature of the basis functions. With reference to Eq. (6), the input-output model for this class of methods is represented as... 

OLS is also called multiple linear regression (MLR) and is a commonly used method to obtain a linear input-output model for a given data set. The model obtained by OLS for a single output is given by... 

Work on dimension reduction methods for both input and input-output modeling and for interpretation has produced considerable practical interest, development, and application, so that this family of nonlocal methods is becoming a mainstream set of technologies. This section focuses on dimension reduction as a family of interpretation methods by relating to the descriptions in the input and input-output sections and then showing how these methods are extended to interpretation. 

The most serious problem with input analysis methods such as PCA that are designed for dimension reduction is the fact that they focus only on pattern representation rather than on discrimination. Good generalization from a pattern recognition standpoint requires the ability to identify characteristics that both define and discriminate between pattern classes. Methods that do one or the other are insufficient. Consequently, methods such as PLS that simultaneously attempt to reduce the input and output dimensionality while finding the best input-output model may perform better than methods such as PCA that ignore the input-output relationship, or OLS that does not emphasize input dimensionality reduction. 

After Laplace transform, a differential equation of deviation variables can be thought of as an input-output model with transfer functions. The causal relationship of changes can be represented by block diagrams. 

Expressing this quantity system in terms of matrix algebra, an input-output model of Table 2.4b, closed with respect to worker consumption, therefore takes the form... 

It should be noted, however, that our application of the new interpretation does not imply that the traditional labour embodied definition of value should be completely abandoned. Foley (2000 30) is open to the possibility that there may be a role for both the new and traditional interpretations of the value of labour power. As Appendix 4 shows, the labour embodied definition of the value of labour power is nested in the input-output model of the circulation of money between departments of production, regardless of how prices are defined. The deviation of prices from values does not modify the constituent role of the labour embodied measure in the interindustry monetary circuit. It is only when a macroeconomic aggregation is developed under price-value deviations, and in the derivation of the scalar Keynesian multiplier, that a switch to the value-form definition is required. 

This type of scalar multiplier can also be derived from the two-sector Kaleckian schema, as shown by Nell (1988b 112), although this latter multiplier was not applied specifically to the circulation of money. A possible advantage of equation (4.23), since it is derived from an input-output model, is that it could be easily generalized to an n sector framework. 

The following three sections present different model applications to analyse the impacts of hydrogen to the economies using the scenarios described in Section 18.3. In Section 18.4 employment effects for ten European countries will be exemplarily analysed with an input-output model. In Section 18.5, GDP effects for different European countries will be analysed with a general equilibrium model. Section 18.6 presents a system dynamic model, which deals with GDP and employment effects. Section 18.7 summarises the different model approaches, presents and discusses the results, and draws overall economic conclusions. 

Input-output model ISIS 18.4.1 General modelling approach... 

The main elements of the used input output model ISIS are described in the following. At the core of ISIS is a statistical input output model (IO model) used to examine the structural impacts of the various strategies. Other modules for employment effects, qualification structure and job conditions, regional effects and environmental effects were developed or added to analyse other dimensions of sustainability. The results of the scenario calculations from the IO model, i.e., production changes as a result of the different strategies, serve as inputs for the other modules. 

Diagnostic observers consist in the definition of a set of observers from which it is possible to define residuals specific of only one failure [8]. Parity relations are relations derived from an input-output model or a state-space model [11] checking the consistency of process outputs and known process inputs. 

The models used can be either fixed or adaptive and parametric or non-parametric models. These methods have different performances depending on the kind of fault to be treated i.e., additive or multiplicative faults). Analytical model-based approaches require knowledge to be expressed in terms of input-output models or first principles quantitative models based on mass and energy balance equations. These methodologies give a consistent base to perform fault detection and isolation. The cost of these advantages relies on the modeling and computational efforts and on the restriction that one places on the class of acceptable models. 

Annaswamy, A. M., M. Fleifil, J.W. Rumsey, J. P. Hathout, and A.F. Ghoniem. 1997. An input-output model of thermoacoustic instability and active control design. MIT Report No. 9705. Cambridge, MA. 

For continuous process systems, empirical models are used most often for control system development and implementation. Model predictive control strategies often make use of linear input-output models, developed through empirical identification steps conducted on the actual plant. Linear input-output models are obtained from a fit to input-output data from this plant. For batch processes such as autoclave curing, however, the time-dependent nature of these processes—and the extreme state variations that occur during them—prevent use of these models. Hence, one must use a nonlinear process model, obtained through a nonlinear regression technique for fitting data from many batch runs. 

The nonlinear nature of these mixed-integer optimization problems may arise from (i) nonlinear relations in the integer domain exclusively (e.g., products of binary variables in the quadratic assignment model), (ii) nonlinear relations in the continuous domain only (e.g., complex nonlinear input-output model in a distillation column or reactor unit), (iii) nonlinear relations in the joint integer-continuous domain (e.g., products of continuous and binary variables in the schedul-ing/planning of batch processes, and retrofit of heat recovery systems). In this chapter, we will focus on nonlinearities due to relations (ii) and (iii). An excellent book that studies mixed-integer linear optimization, and nonlinear integer relationships in combinatorial optimization is the one by Nemhauser and Wolsey (1988). 

Some factors or covariates may cause deviations from the population typical value generated from system models so that each individual patient may have different PK/PD/disease progression profiles. The relevant covariate effects on drug/disease model parameters are identified in the model development process. Clinical trial simulations should make use of input/output models incorporating... 

Holford, N. H. G. Input-output models. In Kimko, H. C., Duffull, S. B., eds. Simulation for designing clinical trials. A pharmacokinetic-pharmacodynamic modeling perspective. (Drugs and the pharmaceutical sciences, volume 127) Marcel Dekker, New York, 2003. 

Vollenweider, R.A. (1975) Input-output models, with special reference to the phosphorus loading concept in limnology. Schweiz. Z. Hydrobiol. 37, 53-82. 

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