**ABSTRACT:**

When solving real problems there is often missing a reliable theory behind

them. In such situations the ideas about a choice of an appropriate model are

very vague and produce models where it is hard to balance

the requirement on sufficient regularity of the model

(as few parameters as possible to guarantee numerical stability)

and feasible precision which forces the analyst to increase the number of

model components typically leading to overparametrization

accompanied with non-uniqueness and numerical instability of solutions.

The standard estimation algorithms use to fail due to numerical

instability caused by strong overparametrization.

In~\cite{Ves05} there was implemented

a computationally intensive sparse parameter estimation

technique based on BPA4 --- a four-step modification

of the Basis Pursuit Algorithm originally suggested

by Chen et al~\cite{CDS98} for time-scale analysis of digital

signals and utilizing numerical procedure~\cite{Sau01}.

\cite{Ves05} is a collection of functions allowing one to construct

and manipulate big finite frames in any abstract Hilbert space $\mathcal{H}$ with

arbitrarily parametrized user-defined frame atoms going far beyond

the common shift/scale/modulation schemes widely used for spectral

representation of signals.

Then BPA4 serves as a universal tool both for finding

a stable sparse frame expansion approximating any object from or outside of

$\mathcal{H}$ and possibly establishing the appropriate dual frame atoms

if necessary.

In addition to some minimal theoretical background this contribution demonstrates

performance and flexibility of BPA4 on

four problems coming from completely diverse application

fields: kernel approximation and smoothing (denoising)~\cite{ZVH04},

improved time series forecasting within an overcomplete stochastic frame

of type ARMA~\cite{VT06},

analysis of air pollution by suspended particulate matter~\cite{VTMK06}

and ROC curve estimation~\cite{MV05b}.

This new computationally intensive approach allowed us to

reliably identify nearly zero parameters in the respective model and

thus to find numerically stable sparse solutions.

\begin{thebibliography}{1}

\bibitem{CDS98}

{S. S.} Chen, {D. L.} Donoho, and {M. A.} Saunders.

\newblock Atomic decomposition by basis pursuit.

\newblock {\em SIAM J. Sci. Comput.}, 20(1):33--61, 1998.

\newblock reprinted in SIAM Review, {\bf 43} (2001), no. 1, pp. 129--159.

\bibitem{Sau01}

{M.~A.} Saunders.

\newblock {\it pdsco.m\/}: {MATLAB} code for minimizing convex separable

objective functions subject to \mbox{$Ax=b,x\geq 0$},

{\rm http://www-stat.stanford.edu/\~{}atomizer/\/}, 2001.

\bibitem{Ves05}

V.~Vesel\'y.

\newblock {\it framebox\/}: {MATLAB} toolbox for overcomplete modeling and sparse

parameter estimation, (C) 2001--2007.

\bibitem{ZVH04}

J.~Zelinka, V.~Vesel\'y, and I.~Horov\'a, \emph{Comparative study of two kernel

smoothing techniques}, {Proceedings of the summer school DATASTAT'2003,

Svratka} (I.~Horov\'a, ed.), Folia Fac. Sci. Nat. Univ. Masaryk. Brunensis,

Mathematica, vol.~15, Dept. of Appl. Math., Masaryk University, Brno, Czech

Rep., 2004, pp.~419--436.

\bibitem{VT06}

V.~Vesel\'y and J.~Tonner.

\newblock {Sparse Parameter Estimation in Overcomplete Time Series Models}.

\newblock {\em {Austrian Journal of Statistics}}, 35(2\&3): 371-378, 2006.

\bibitem{VTMK06}

V.~Vesel\'y, J.~Tonner, J.~Mich\'alek, and M.~Kol\'a\v{r}, \emph{{Air pollution

analysis based on sparse estimates from an overcomplete model}}, {TIES 2006,

18-22 June 2006, Kalmar, Sweden}, 2006.

\bibitem{MV05b}

J.~Mich\'alek and V.~Vesel\'y, \emph{{Comparison of the ROC curve estimators}},

{25th European Meeting of Statisticians, 24--28 July 2005, Oslo Norway, Final

Programme and Abstracts}, University of Oslo, Norway, 2005, p.~499.

\end{thebibliography}

\bigskip

\cite{ZVH04}-\cite{MV05b} are available in electronic

form at author's web site

\vfill

Research supported by M\v{S}MT

(Ministery of Education of the Czech Republic):

research contract MSM0021622418