Zembowitz, R., and Zytkow, J. (1992). Discovery of equations: experimental evaluation of convergence. In Proc. Tenth National Conference on Artificial Intelligence. Morgan Kaufmann, San Mateo, CA.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....distinguishes Lagramge from other system identification methods like neural networks, which can be used for obtaining black box models, i.e. models with incomprehensible structure. On the equation discovery side, the presented work is related to equation discovery systems, such as Bacon [9] EF [21], E [19] Lagrange [2] and GoldHorn [8] However, none of them was applied to the task of time series prediction. Various architectures of neural networks have already been used for system identification and prediction. Some of them are closely related to the architectures used in this paper [4, ....

Zembowitz, R., and Zytkow, J. (1992). Discovery of equations: experimental evaluation of convergence. In Proc. Tenth National Conference on Artificial Intelligence. Morgan Kaufmann, San Mateo, CA.

....discovery systems use di erent approaches to allow the user to restrict the space of the possible equations. In equation discovery systems that are based on genetic programming, the user is allowed to specify a set of algebraic operators that can be used. A similar approach has been used in the EF [10] equation discovery system. The equation discovery system SDS [7] e ectively uses user provided scale type information about the dimensions of the system variables and is capable of discovering complex equations from noisy data. Finally, the equation discovery system Lagramge [6] allows the user ....

....the equation discovery system Lagramge [6] allows the user to specify the space of possible equations using a context free grammar. Note that grammars are a more general and powerful mechanism for tailoring the space of the equations to the domain of use than the ones used in SDS [7] and EF [10]. In the rest of this section we will describe this grammar based approach to equation discovery used in Lagramge. 2.1 Grammar Based Equation Discovery The problem of grammar based equation discovery can be formalized as follows. a grammar G; and Find a model E in the form of one or more ....

R. Zembowicz and J. M. _ Zytkow. Discovery of equations: Experimental evaluation of convergence. In Proceedings of the Tenth National Conference on Arti cial Intelligence, pages 70-75, San Jose, CA, 1992. Morgan Kaufmann.

....observed system, mentioned above. Several equation discovery systems make use of domain speci c knowledge. In equation discovery systems that are based on genetic programming, the user is allowed to specify a set of algebraic operators that can be used. A similar approach has been used in the EF [10] equation discovery system. The equation discovery system SDS [8] e ectively uses scale type information about the dimensions of the system variables and is capable of discovering complex equations from noisy data. However, expert users can usually provide much more modeling knowledge about the ....

....bias formalism. In Lagramge [7] the formalism of context free grammars has been used to specify the space of possible equations. Note here that context free grammars are a far more general and powerful mechanism for incorporating domain speci c knowledge than the ones used in SDS [8] and EF [10]. The use of declarative bias in the form of context free grammar was crucial for modeling the phytoplankton growth in Lake Glumsoe in Denmark from realworld sparse noisy measurements [3] However, one can argue that it is dicult for the users of Lagramge to express their knowledge about the ....

R. Zembowicz and J. M. _ Zytkow. Discovery of equations: Experimental evaluation of convergence. In Proc. of the Tenth National Conference on Arti cial Intelligence, pages 70-75. Morgan Kaufmann, 1992.

....the variables and the model of regression. Current function finding systems using regression can be categorised in three families. The first one uses a list of predefined models, like KEPLER [12] or E [10] the second one systematically explores an ordered but huge space of models, like EF [13] and LAGRANGE [2] These two categories use regression in a classical way, applying it to complete models, a statistical criterion selecting the best law. The third category are BACON s like systems [5] which use heuristics in order to iteratively build the model. Using regression in such an ....

....are quite difficult to discover, especially when many variables are involved in the model. The simplest solution in order to find a polynomial is, when the model is already known, to apply regression and if needed, to test the pertinence of each term in the polynomial in order to simplify it. EF [13] and LAGRANGE[2] apply regression on a family of models, that they iteratively build, starting with the simplest ones. Since they do not have any additional knowledge on the model, they test all possible models inside a limit the user sets. 3.1 BACON.3: Iterative Construction of the Model A ....

R. Zembowicz and J.M. Zytkow. Discovery of Equations : Experimental Evaluation of Convergence. In Proceedings of the 10th National Conference on Artificial Intelligence, pages 70--75, 1992. Machine Learning 452 M. Moulet

....subspaces of the domain. There have been some ideas about discovering a set of simple laws from observational data with BACON, but they have not actually been incorporated into BACON. Thus, none of the above system meets both of the above criteria in a satisfactory way. The Equation Finder [Zembowitz and Zytkow 1992], used as a module in FAHRENHEIT, comes closest to meeting the above requirements. Given a set of real valued pairs (x i ; y i ) it tries to find a formula Y = f(X) without asking for additional data. The main problem with Equation Finder is that it is looking for a function of one variable ....

....One of the important features of Equation Finder is the ability to handle errors in the input data. Such a feature is necessary for handling real life data. It is also important that the convergence of Equation Finder and its sensitivity to errors in the input data have been thoroughly analysed [Zembowitz and Zytkow 1992]. IDS [Nordhausen and Langley 1990] touches upon the problem of discovering dynamics. The qualitative schemata generated by IDS can be considered qualitative states of a system. The transitions between states (schemata) capture in a way the qualitative change of the system over time, i.e. its ....

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Zembowitz, R. and Zytkow, J. (1992). Discovery of equations: experimental evaluation of convergence. In Proc. Tenth National Conference on Artificial Intelligence. Morgan Kaufmann, San Mateo, CA.

....the coefficient values of chosen templates, and then determine the probability of fit for each data point to the equation. This is used to define contiguous homogeneous regions, and each region forms a context in the domain of interest. In the model refinement step, 49er invokes Equation Finder [ Zembowicz and Zytkow, 1992 ] to uncover analytic relations between the goal variable and the control variable. Additional relations between the two variables can be explored by considering transformations like log(x) and 1 x , iteratively for the goal and control variables, enabling the derivation of complex, non linear ....

R. Zembowicz and J.M. Zytkow. Discovery of equations: Experimental evaluation of convergence. In Proceedings of the Tenth Conference on Artificial Intelligence, 1992.

....which subsumes equation discovery, also deals with large hypothesis spaces. Given a fixed set of operators and variables, the number of equations that can be constructed is infinite. To restrict the size of the search space, equation discovery systems typically employ a depth limit (see e.g. [Zembowitz and Zytkow 1992], Dzeroski and Todorovski 1993] Few, if any, use declarative bias to specify and restrict their hypothesis space. Declarative bias can limit the search in a potentially huge space to a manageable level. Also, declarative language bias can be viewed as a way of providing domain knowledge to the ....

....Related is also the genetic programming system of [Whigham 1995] which uses grammars to bias its search, but does not deal with the problem of equation discovery. On the equation discovery side, Lagramge is related to machine discovery systems. These include BACON [Langley et al. 1987] EF [Zembowitz and Zytkow 1992], E [Schaffer 1993] LAGRANGE [Dzeroski and Todorovski 1993] and GOLDHORN [Krizman et. al 1995] Except for LAGRANGE and GOLDHORN, none of the mentioned systems can deal with differential equations. All the mentioned systems include some form of bias: the particular biases used are however either ....

Zembowitz, R., and Zytkow, J. (1992). Discovery of equations: experimental evaluation of convergence. In Proc. Tenth National Conference on Artificial Intelligence. Morgan Kaufmann, San Mateo, CA.

....system [3] Coper [5] Schaffer s E algorithm [12] KEDS [11] Kepler [14] In the latter approach, possible solutions are constructed that satisfy certain qualitative features detected in the data. Examples of this approach include the Bacon discovery systems [7] Abacus [2] and Fahrenheit [6, 15]. Since the features are detected by analyzing the data, both approaches can be viewed as extremes of a single spectrum. The degree of sophistication in the tools used for data analysis indicates whether a system should be categorized as data analysis or formula construction, or a combination of ....

Zembowicz, R. and Zytkow, J.M. (1991). "Discovery of Equations: Experimental Evaluation of Convergence." Proc. AAAI Conference. Massachusetts: MIT Press.

....be discovered automatically with the aid of computer. Langley et al. 1981,1983a,1983b) developed the BACON system for discovering conservation laws in scientific fields. In the paper (Falkenhainer Michalski, 1986) the ABACUS system was invented to realize quantitative and qualitative discovery. Zembowicz Zytkow (1992) constructed a system for finding equations. The present paper proposes a method of discovering some kinds of differential equations with interval coefficients, which characterize or explain numerical data obtained by scientific observations and experiments. Such numerical data inevitably involve ....

Zembowicz, R. & Zytkow, J. M. 1992. Discovery of equations: experimental evaluation of convergence. In Proceedings Tenth National Conference on Artificial Intelligence, pages 70--75.

....is a compromise between computational complexity and variety of forms of equations which can be discovered. This compromise is achieved due to choice of more or less narrow set of formulae in which search is concentrated. For example, the Equation Finder, function finding module of 49er system [9] searches for the best formula expressing dependence of variable y on variable x among equations of the form a i f i (x ,y) F(x ,y) In the KDD system PolyAnalyst [10, 11] designed by us rational expressions (polynoms divided by polynoms) play such a special role. In the present paper we ....

Zembowicz, R., and Zytkow J.M., "Discovery of Equations: Experimental Evaluation of Convergence," Proceedings of AAAI-92, Menlo Park, CA, pp 70-75. (1992).<E-48>

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Zembowicz, R. and Zytkow, J. (1992). Discovery of equations: experimental evaluation of convergence. In Proc. Tenth National Conference on Artificial Intelligence, pages 70--75. MIT Press, Cambridge, MA.

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R. Zembowicz and J. M. Zytkow, "Discovery of equations: Experimental evaluation of convergence," in Proc. of AAAI-92, Menro Park, CA: AAAI Press, 1992, pp. 70--75.

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