*** This website is an archived version from 2011 ***
FWF Project: PKSIRT
 
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Department of Psychology
School of Natural Sciences  
University of Graz            
Projects                               
  Overview                         
  Mathematical Modelling     
  Software Development     

Software Development

Software development in this project is aimed to expand and adapt existing tools and to develop
new software. Those tasks range from providing tools for data analyses via model validation to
visualizations of the results. In close teamwork of software developer, mathematician, and
psychologist, existing software has to be evaluated, various tools to support developing
mathematical models (i.e., simulations, data and structure visualizations), tools for validating
models (i.e., parameter estimation and hypotheses testing), and a professional data-bank have to be developed. Furthermore a clear, integrative user interface for the relevant software
components has to be created. The development of a professional website as a platform to
publish the software packages and for presenting the project and its results is another aim.
Finally, an important issue is documentation and dissemination of the newly developed software
packages.

Simulation Tools

  » Resampling Methods
   

The Jackknife and the Bootstrap are nonparametric methods for assessing the errors in a statistical estimation problem. They provide several advantages over the traditional parametric approach: the methods are easy to describe and they apply to arbitrarily complicated situations; distribution assumptions, such as normality, are never made. These methods, however, can be computationally very intensive.
     For the current project, we implemented Jackknife methods (delete-one, delete-d) and Boostrap with different confidence interval methods (i.e. normal approximation and percentile). For the implementations we used the .NET-based Sasquatch-Software as front-end. In the course of the project, revised versions of these resampling methods will be implemented in the ePsyt-platform.

Further readings:
» Biswal B.B., Taylor P.A., & Ulmer J.L. (2001). Journal of Computer Assisted Tomographie,
   25(1),:113-20. [Abstract]
» Cheng, R.C.H. (2000). Analysis of simulation experiments by bootstrap resampling. In B. A.
   Peters, J. S. Smith, D. J. Medeiros, and M. W. Rohrer (Eds.), Proceedings of the 2001 Winter
   Simulation Conference
, WSC 2001, December 9-12, 2001, Arlington, VA, USA. [PDF]
» Dixon, P.M. Bootstrap Resampling. [PDF]
» Efron, B. (1982). The Jackknife, the Bootstrap, and Other Resampling Plans. Philadelphia:
   SIAM.
» Friedl, H., & Stampfer, E. (2000). Resampling Methods. [PS]
» Helmers, R., & Putter, H. Bootstrap Resampling: A survey of recent research in the
   Netherlands. [PDF]
» Shimodaira, H. (2004). Approximately unbiased tests of region using multistep-multiscale
   bootstrap resampling. The Annals of Statistics, 32 (6), 2616–2641. [PDF]
» Wan, J. (1998). A parametric version of Jackknife-after-Bootstrap. In D.J. Medeiros, E.F.
   Watson, J.S. Carson, and M.S. Manivannan (Eds.), Proceedings of the 1998 Winter Simulation
   Conference
, WSC 1998, December 13-16, 1998, Washington DC, USA. [PDF]

     
  » Binary and Model-based Simulations
   

A central issue for generating and validating models in the framework of the current project is simulating data sets. We developed a number of simulation methods based on random binary simulations and on model-based simulations. For the implementations we used the .NET-based Sasquatch-Software as front-end. In the course of the project, revised versions of these simulation methods will be implemented in the ePsyt-platform.

» Sasquatch Simulations Manual (Kickmeier-Rust, M.D., 2005; Draft, restricted access)

     
  » Random Number Generators
   

A central issue of computer-based simulations are (pseudo-) random number generators (RNG). In the context of the current project, we evaluated a number of RNGs, e.g. Rand, MotherofAll, or Mersenne Twister [Summary]. For random number generations in all simulation tools we implemented the Mersenne Twister by Makoto Matsumoto and Takuji Nishimura.

Links and further readings:
» Mersenne Twister Homepage
» pLAB (University of Salzburg)
» A comparison of algorithms.
» Fog, A. Chaotic Random Number Generators with Random Cycle Lengths. [PDF]
» L’Ecuyer, P., & Touzin, R. (2003). On the Deng-Lin Random Number Generators and Related
   Methods
. [PDF]


Optimization Tools

  » Praxis / EPM (Extended Principal Axis Minimization)
   

Numerical optimization (i.e.function minimization resp. maximization) is an impoprtant aim for software development in the framework of PKSIRT. We evaluated common minimization methods and algorithms, e.g. EM-algorithm, the Newton-Raphson method, or Brent's Principal Axis Minimization (PRAXIS).
     In this course, we developed EPM, an extension of the C implementation of PRAXIS by Karl Gegenfurtner. The EPM approach automatically generates and tests multiple sets of start values, randomly drawn from user specified weighted primary and secondary intervals (Round 1), and it refines and stabilizes the smallest Round 1 approximated minimum by means of certain iterative minimization loops and a stopping criterion on the change in approximated smallest minimum found in previous minimization runs and the minimum of the current run (Rounds 2 and 3). EPM is availaible in two versions; one uses the double data type, one long double.

» Ünlü, A., & Kickmeier-Rust, M.D. (2005). EPM: a strategy for principal axis minimization.
   Proceedings of the 14th Hellenic European Conference on Computer Mathematics & its
   Applications, accepted for publication. Athens, Greece: Lea Publishers. [PDF, restricted    access]
» Download EPM (double version) [ZIP, 16kB]
» Download EPM (long double version) [ZIP, 17kB]

Further readings:
» Brent, R.P. (1973). Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ:
   Prentice-Hall.
» Gegenfurtner, K.R. (1992). PRAXIS: Brent's algorithm for function minimization. Behavior
   Research Methods,Instruments, & Computers
, 24, 560-564.
» Powell, M.J.D. (1964). An efficient method of finding the minimum of a function of several
   variables without calculating derivatives. Computer Journal, 7, 155-162.


Software Evaluation

   

To support mathematical modelling in the framework of PKSIRT existing software applications in the fields of Latent Class Analysis, Item Response Theory, and Knowledge Space Theory must be evaluated. The Listing [DOC, 37kB] shows the relevant evaluated applications.


Latent Class Modeling with Random Effects

  Response error (careless error and lucky guess) rates are modeled and estimated using the general class of latent class models with random effects. In this class, we have the general latent class model with random effects (LCMRE), the special LCMRE with item-independent, class-specific random effects parameters (LCMRE_bu), and, as another important special case, the traditional unrestricted two-classes latent class model (TLCM). Related C sources are available for download below. Please note that each of the three model types requires the matrix operation tools!
 
  » General Latent Class Model with Random Effects (LCMRE)
   

Parameter estimation (incl. variances) and goodness-of-fit testing [ZIP, 32 kB]

    Data simulation 1 (Box-Muller implementation of N(0,1)-distribution) [ZIP, 8 kB]
    Data simulation 2 (Gauss-Hermite discretization of N(0,1)-distribution) [ZIP, 9 kB]
    Data simulation 3 (Multinomial probabilities calculated from model parameters) [ZIP, 9 kB]
    Summary statistics (see Info.txt included in the Zip-File) [ZIP, 9 kB]
     
  » LCMRE with item-independent, class-specific random effects parameters (LCMRE_bu)
    Parameter estimation (incl. variances) and goodness-of-fit testing [ZIP, 33 kB]
    Data simulation 1 (Box-Muller implementation of N(0,1)-distribution) [ZIP, 8 kB]
    Data simulation 2 (Gauss-Hermite discretization of N(0,1)-distribution) [ZIP, 9 kB]
    Data simulation 3 (Multinomial probabilities calculated from model parameters) [ZIP, 9 kB]
    Summary statistics (see Info.txt included in the Zip-File) [ZIP, 9 kB]
     
  » Traditional unrestricted two-classes latent class model (TLCM)
    Parameter estimation (incl. variances) and goodness-of-fit testing [ZIP, 31 kB]
    Data simulations using a general basic local independence model (Note: data simulations with the TLCM are obtained as special cases.) [ZIP, 7 kB]
    Summary statistics (see Info.txt included in the Zip-File) [ZIP, 9 kB]
     
  » Matrix operation tools
    Small Matrix Toolbox for C programmers version 0.42 (Support Unix and DOS)
by Patrick Ko Shu-pui. Copyright (c) 1992, 1993, 1994. All Rights Reserved.
Modified to long double data type [ZIP, 17 kB]

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