A Symbolic Numeric Environment for Analyzing Measurement Data in Multi-Model Settings (Extended Abstract) Christoph Richard 1 and Andreas Weber 2? 1 Institut für Theoretische Physik, Universität Tübingen, 72076 Tübingen, Germany christoph.richard@uni-tuebingen.de 2 Arbeitsbereich Symbolisches Rechnen, Universität Tübingen, 72076 Tübingen, Germany weber@informatik.uni-tuebingen.de http://www-sr.informatik.uni-tuebingen.de/ Abstract. We have built a complete system which allows the analysis of measurement data arising from scientific experiments. Within the system, it is possible to fit parameter-dependent curves to given data points numerically in order to obtain estimates of experimental quantities. The system provides moreover a convenient tool to test different theoretical models against a given experiment: We use the computer algebra system Maple not only as a graphical interface to visualize the data but mainly as a symbolic calculator to investigate and to implement solutions of the underlying theory. The system has been used successfully in a project with researchers from the department of chemistry. 1 Introduction An important issue in the daily work of a scientist is the analysis of measurement data of an experiment. Very often mathematical models in form of parametric functions are available that describe the outcome of an ideal experiment. The task is then to determine the correct model and to estimate parameter values from the given data. The least square method invented by Gauß [4] is a major tool for (linear) curve fitting and is available in many software systems. Many experiments, however, cannot be described using linear functions, and there are indeed generalizations of the method non-linear least-square fit algorithms which work for a much more general setting. Although much less common than algorithms for the Gaussian least square method, several implementations of non-linear least-square fit algorithms are available. E. g., the so called non-linear tool-box of the Matlab system [3] contains an implementation of the Levenberg-Marquardt algorithm for non-linear least-square fit. The functions whose parameters have to be estimated by the nonlinear least-square algorithm have to be given as Matlab programs. Using the Matlab system alone will leave some tedious work to the user, since the functions which are supposed to describe the experiment have to be coded as Matlab functions for each of the competing models.? Supported by Deutsche Forschungsgemeinschaft under grant Ku 966/6-1.
Using a combination of symbolic and numeric systems we have built an environment that allows a comparatively easy numerical analysis of measurement data arising from scientific experiments with respect to various mathematical models. The system will have its main use in situations where different theoretic descriptions are available which should be tested against the available empiric data, such that the best model for a given experiment can be picked without much effort. The symbolic part of our system is currently implemented on top of the computer algebra system Maple [1], the numeric part on top of a library based on algorithms described in [4]. 2 Building Blocks of the System All steps of the data analysis are done within the Maple graphical user interface environment. A Maple worksheet has been designed for the visualization and manipulation of data. Typing one-line commands, the user can perform all steps of data analysis such as reading data from experiment, visualizing data curves, selecting data points for numerical data fitting, perform fits, comparing numerical and experimental results. The numerical data analysis is done by fitting parameter-dependent functions, which are suggested from theory, against the given data curves. These functions are commonly characterized by (sets of) ordinary differential equations for the quantity of measurement. The Maple environment is well suited to clarify the theoretical description since Maple has a number of specialized tools which allow an extensive analysis of such functions. The example discussed in Sec. 3 shows prototypically such a multi-step formulation: A chemical reaction can be described by a differential equation. A solution for the differential equation has to be found and verified. Simpler models for specialized cases can be developed. The chemical reaction cannot be observed directly, but only some resulting effect. The resulting effect can be described by some other mathematical means (e. g. as the integral of some simpler effect). The computer algebra system can be used to obtain solutions to tasks like symbolic integration, finding symbolic solutions for differential equations and simplifying results for the final parameter dependent function. Having obtained a parameter dependent function which is supposed to describe the process under consideration correctly, this function can be tested against the experimental data using the Levenberg-Marquardt algorithm for non-linear leastsquare fit [4]. In order to perform the numerical analysis with appropriate speed, this algorithm is implemented as an external C-routine which is called from the data-worksheet. The routine requires a C-function as one of its arguments, which returns values for the function to be fitted and its Jacobian. Thus we have to provide the C-Code of the function and its Jacobian and compile it together with a driver
routine. The code-generation is done by a special Maple-worksheet which guides the user through the necessary tasks step by step. The code generated by Maple is highly optimized with respect to common subexpression elimination the methods implemented in Maple can compete with the ones used by the best optimizing compilers. 3 Example: An Antigen-Antibody-Reaction Our system was used to analyze a special type of antigen-antibody reaction. This example arose from a collaboration with researchers from the department of chemistry, when we tried to analyze the setting giving in [2]. The reaction between antigen and antibody is an equilibrium reaction. The change of concentration during the reaction is described by the following differential equation (cf. [2, p. 23 eqn.(18)]): @c AgAk (t) = k ass (c 0 Ak ; c AgAk (t))(c 0 Ag ; c AgAk (t)) ; k diss c AgAk (t) (1) @t with positive constants k ass : association rate, k diss : dissociation rate, c AgAk : concentration of connected antibody binding points, c 0 Ag : starting concentration of antigen, c 0 Ak : starting concentration of antibody. We are mainly interested in estimates of the reaction constants k ass and k diss.for a complete determination, it is necessary to analyze the reaction away from its stationary state. It is therefore useful to implement nonlinear techniques for parameter estimation. Solution of the differential equation. We use the computer algebra system Maple to analyze and to solve the given differential equation. Maple contains specialized tools for the analysis of ordinary differential equations (ODE s). Given an ODE, the odeadvisor command classifies it according to standard text books and displays a help page including related information for solving it. Using these symbolic tools of Maple, it was possible to find an explicit solution of the ODE without consulting textbooks on differential equations. Form of the signal curve. As explained in [2], the concentration c AgAk of connected antibody binding points cannot be observed directly, but only the one of a derivate, c Ak bind. Moreover, only the cumulative concentration is measured by the experiment. However, we know that c0 Ak ; c AgAk(t) 2 (2) c Ak bind (t) = 1 2 signal = off + F Z c 0 Ak c Ak bind (t) dt (3)
off and F being (unknown) scaling constants of the experimental setup, cf. [2]. The symbolic expression for c AgAk (t) 2, which was found by explicitly solving the corresponding ODE, can be integrated symbolically by Maple. Thus we can derive a closed expression for the signal curve. Since this expressions depends on the parameters we are interested in, a least square fit of signal to the empiric measurement data will provide estimates for the parameters of interest. A limiting case: the integrated first order time law. Although it is possible to analyze the antigen-antibody reaction using the full solution, we also considered a limiting case in order to have a comparison between the most general description and a simpler one, which had been considered previously, cf. [2]. This limiting case refers to experiments where the antibody concentration is much lower than the concentration of antigen. In this case, the reaction can be described by the differential equation @c AgAk (t) = k ass c 0 Ag (c 0 Ak ; c AgAk (t)) ; k diss c AgAk (t): (4) @t This ODE can be solved quite easily in symbolic form by Maple and also the integration which is required for the signal curve (cf. equation 3) could be done symbolically. Results. In the figure below the measurement data of one of the experiments described in [2] are given, together with the theoretic signal curves of the general law and the law for the limiting case, whose parameters have been fitted by the Levenberg-Marquardt algorithm. Both functions can be fitted quite well, and their graphs almost coincide. The parameter estimates we got are k ass = 5:02 10 6 and k diss = 0:00034. Measurement data and fitted functions 0.09 0.08 0.07 0.06 0.05 sign. 0.04 0.03 0.02 0.01 40 60 80 100 120 140 160 180 200 t Acknowledgement. We are grateful for the possibility to collaborate with Alexander Jung from the department of chemistry. References 1. CHAR, B.W.,GEDDES, K. O., GONNET, G. H., BENTON, L. L., MONAGAN, M. B., AND WATT, S. M. Maple V Language Reference Manual. Springer-Verlag, New York, 1991.
2. JUNG, A. Markierungsfreie Untersuchung der Kinetik von Antigen-Antikörper- Wechselwirkungen in homogener Phase. Diplomarbeit, Universität Tübingen, 1998. 3. MATHWORKS INC. Matlab 5, 1997. http://www.mathworks.com/products/ matlab/. 4. PRESS, W. H., TEUKOLSKY, S. A., VETTERLING, W.T.,AND FLANNERY, B.P. Numerical Recipes in C: The Art of Scientific Computing, second ed. Cambridge University Press, 1992.