Gradient pattern analysis

%Gradient Pattern Analysis (texto principal)

The {\it gradient pattern analysis} (GPA) (\cite{rosa2000},\cite{rosa03}) basically consists in a geometric computing method for characterizing symmetry breaking of an ensemble of asymmetric vectors regularly distributed in a square lattice. Usually, the lattice of vectors represent the first-order gradient of a scalar field, here an $M\times M$ square amplitude matrix. An important property of the gradient representation is the following: A given $M\times M$ matrix where all amplitudes are different results a $M\times M$ gradient lattice containing $N_{V}=M^{2}$ asymmetric vectors. As each vector can be characterized by its norm and phase, variations in the $M^{2}$ amplitudes can modify the respective $M^{2}$ gradient pattern.

The original ideas on GPA was introduced by Rosa, Sharma and Valdivia, 1999 \cite{Rosa99}. Usually GPA is applied for spatio-temporal pattern analysis in physics and environmental sciences operating on time-series and images.

{\bf Contents}

\itemize Calculation \itemize Relation to other methods \itemize References \itemize External Links

{\bf Calculation}

By connecting all vectors using a Delaunay triangulation criterium it is possible to characterize gradient asymetries computing the so-called \textit{gradient asymmetry coefficient}, that has been defined as: \begin{equation} G_A=\frac{|N_C-N_V|}{N_V}, \end{equation} where $N_V>0$ is the total number of asymmetric vectors and $N_C$ is the number of Delaunay connections among them.

As the asymmetry coefficient is very sensitive to small changes in the phase and modulus of each gradient vector, it can distinguish complex variability patterns even when they are very similar but consist of a very fine structural difference. Not that, unlike most of the statistical tools, the GPA does not rely on the statistical properties of the data but depends solely on the local symmetry properties of the correspondent gradient pattern.

For a complex extended pattern (matrix of amplitudes of a spatio-temporal pattern) composed by locally asymmetric fluctuations, $ G_{A}$ is nonzero, defining different classes of irregular fluctuation patterns (1/f noise, chaotic, reactive-diffusive, etc). Besides $G_{A}$ other measurements (called {\it gradient moments}) can be calculated form the gradient lattice \cite{rosa03}. Considering the sets of local norms and phases as discrete compact groups, spatially distributed in a square lattice, the gradient moments have the basic property of being globally invariant (for rotation and modulation).

{\bf Relation to other methods}

When GPA is conjugated with wavelet analysis, then the method is called {\it Gradient Spectral Analysis}, usually applied to short time series analysis \cite{rosa08})

{\bf References}

\bibitem{rosa99} R. R. Rosa, A. S. Sharma and J. Valdivia, {\em Int. J. Mod. Phys. C \/} {\bf 10}, 147 (1999). \bibitem{rosa2000} R. R. Rosa, J. Pontes, C. I. Christov, F. M. Ramos, C. Rodrigues Neto, E. L. Rempel, D. Walgraef, {\em Physica A \/} {\bf 283}, 156 (2000). \bibitem{assireu2002} A. T. Assireu, R. R. Rosa, N. L. Vijaykumar, J. A. Lorenzetti, E. L. Rempel, F. M. Ramos, L. D. Abreu Sá, M. J. A. Bolzan, A. Zanandrea, {\em Physica D \/} {\bf 168} 397 (2002). \bibitem{rosa03} R. R. Rosa, M. R. Campos, F. M. Ramos, N. L. Vijaykumar, S. Fujiwara, T. Sato, {\em Braz. Jour. Phys. \/} {\bf 33}, 605 (2003). \bibitem{baroni06} M. P. M. A. Baroni, R. R. Rosa, A. Ferreira da Silva, I. Pepe, L. S. Roman, F. M. Ramos, R. Ahuja, C. Persson, E. Veje, {\em Microelectronics Journal \/} {\bf 37}, 290 (2006). \bibitem{rosa07} R. R. Rosa, M. P. M. A. Baroni, G. T. Zaniboni, A. Ferreira da Silva, L. S. Roman, J. Pontes and M. J. A. Bolzan, {\em Physica A \/} {\bf 386}, 666 (2007). \bibitem{rosa08} R.R.Rosa et al., {\em Advances in Space Research} {\bf 42}, 844 (2008), doi:10.1016/j.asr.2007.08.015.

{\bf External Links} Lab for Computing and Applied Mathematics (MATLAB code for Gradient Spectral Analysis).