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'''Gradient pattern analysis''' ('''GPA''')<ref name=rosa2000>Rosa, R.R., Pontes, J., Christov, C.I., Ramos, F.M., Rodrigues Neto, C., Rempel, E.L., Walgraef, D. ''Physica A'' '''283''', 156 (2000).</ref> is a geometric computing method for characterizing geometrical bilateral [[symmetry breaking]] of an ensemble of
▲'''Gradient pattern analysis''' ('''GPA''')<ref name=rosa2000>Rosa, R.R., Pontes, J., Christov, C.I., Ramos, F.M., Rodrigues Neto, C., Rempel, E.L., Walgraef, D. ''Physica A'' '''283''', 156 (2000).</ref> is 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 x M'' square amplitude [[matrix (mathematics)|matrix]]. An important property of the gradient representation is the following: A given ''M x M'' matrix where all amplitudes are different results in an ''M x M'' gradient lattice containing <math>N_{V} = M^2</math> asymmetric vectors. As each vector can be characterized by its norm and phase, variations in the <math>M^2</math> amplitudes can modify the respective <math>M^2</math> gradient pattern.
The original concept of GPA was introduced by Rosa, Sharma and Valdivia in 1999.<ref name=Rosa99>Rosa, R.R.; Sharma, A.S.and Valdivia, J.A. ''Int. J. Mod. Phys. C'', '''10''', 147 (1999), {{doi|10.1142/S0129183199000103}}.</ref> Usually GPA is applied for spatio-temporal pattern analysis in physics and environmental sciences operating on time-series and digital images.
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is valid for any gradient square lattice.
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 (bilateral asymmetry) even when they are very similar but consist of a very fine structural difference. Note 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.
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Besides <math>G_{A}</math> other measurements (called ''gradient moments'') can be calculated from the gradient lattice.<ref name=rosa03>Rosa, R.R.; Campos, M.R.; Ramos, F.M.; Vijaykumar, N.L.; Fujiwara, S.; Sato, T. ''Braz. J. Phys.'' '''33''', 605 (2003).</ref> 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).
The primary research on gradient lattices applied to characterize [[Wave turbulence|weak wave turbulence]] from X-ray images of [
== Relation to other methods ==
When GPA is conjugated with [[wavelet analysis]], then the method is called ''Gradient spectral analysis'' (GSA), usually applied to short time series analysis.<ref name=rosa08>Rosa, R.R. et al., ''Advances in Space Research'' '''42''', 844 (2008), {{doi|10.1016/j.asr.2007.08.015}}.</ref>
== References ==
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