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Scientist® 3.0 - Overview
Scientist® is designed to provide a comprehensive solution to the
problem of fitting experimental data on the PC. It includes the
capability of solving systems of model equations that can
include nonlinear equations, ordinary differential equations and
Laplace transforms. Scientist is an application for researchers
who "know what's going on" with their data and need to establish
solid parameter values to model real-world phenomena. Scientist
provides remarkably simple model equation entry, data
management, control of initial parameter estimates and
constraints on parameter values, as well as complete statistical
analysis and publication quality graphical output.
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.
This technique can be applied to a broad range of scientific,
engineering and other technical problems. It also allows the
solution of problems that might otherwise be impractical. The
inverse Laplace transform may be calculated for a single point,
for a curve representing a range of time values, or for a family
of curves in situations dependent on both space and time
coordinates. Equations involving Laplace transforms can be
directly fitted to data, freeing the user from the time
consuming iterative parameter refinement process that would
otherwise be required.
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.
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