The interactive nature of Scientist leads to a higher likelihood of finding optimal parameter values than if fitting were done in batch mode, without the ability to examine and react to computational results. It also enables users to develop a much greater awareness of the sensitivity of models to parameter values. With Scientist you can develop and fit data to the most complex models involving nonlinear, differential and Laplace transform equations.
Scientist can model phenomena from all scientific and engineering disciplines and is being used in many teaching and research applications including: Physical Chemistry, Organic Chemistry, Pharmaceutical Chemistry, Biophysics, Thermodynamics and Heat Transfer, Kinetics, Genetics, Sociology, Economics, Physics, Mechanical Engineering, Electrical Engineering, Civil Engineering, Applied Mathematics and many others. Models can consist of single functions [defined by several lines of code], multiple functions that can be fit simultaneously, implicit equations or systems of equations, parametric equations (i.e. X and Y both defined in terms of a third variable), differential or integral equations and equations involving Laplace transforms.
Minimization Algorithms
Scientist employs a least squares minimization procedure based on a modification of Powell's algorithm. This algorithm is many times faster than the more common algorithms based on sequential searches involving one parameter at a time. The algorithm is a hybrid that combines the reliability of a steep descent method with the speed of the Gauss-Newton method near convergence. Other minimization methods are also available. These methods may be more effective under certain circumstances. For example, a nonlinear simplex algorithm may be to locate the general location of a minimum (i.e. improving parameter estimates prior to least squares minimization). Steepest descent and Levenberg-Marquardt minimization algorithms are also available.
Solving Differential Equations
Scientist provides comprehensive numerical integration of differential equations. This makes Scientist a powerful, easy-to-use model development tool for Scientists, engineers and graduate and undergraduate students in science and engineering. Scientist allows users to focus on science, not software.
The algorithms used in Scientist are adapted from various sources. Scientist implements four standard methods and a method designed to integrate stiff equations (EPISODE):
- Euler's Method
- Runge-Kutta Method (Fourth Order)
- Error Controled Runge-Kutta Method
- Bulirsch-Stoer Method
Laplace Transform Inversion
The use of Laplace transforms can greatly simplify the solution of models representing very complicated physical systems. The Laplace transform reduces differential equations to algebraic equations in order to solve them. The equations can then be inverted to obtain the solution to the differential equations. Scientist can calculate the numerical inverse of models written as Laplace transforms. This is particularly useful when the inverse transform has no explicit solution.
The algorithms used for the inversion of Laplace transforms are adapted from various sources. Scientist implements both Piessens' method and Weeks' method.
Statistics Output
Scientist provides a broad range of statistical output, including parameter estimates, confidence limits, various measures of goodness-to-fit, variance-covariance and correlation information, and analysis of residuals. Confidence limits for parameter estimates are calculated using the customary approach involving a local linearization of the model or a more rigorous approach that locates various points on constant sum of squares contours.
