Affine Invariant Functions Research Articles

On masking and swamping robustness of leading nonparametric outlier identifiers for multivariate data

For any outlier detection procedure, a key concern is robustness with respect to possible misclassification errors, masking (Type I) and swamping (Type II). Although parametric model-based simulation results are informative, one also desires nonparametric masking and robustness measures that are more broadly applicable. To this effect, notions of finite-sample masking and swamping breakdown points formulated abstractly for outlyingness functions in arbitrary data settings (Serfling and Wang, 2014) are introduced in the present paper into the multivariate data setting. Formulas for the measures are derived for three important affine invariant nonparametric multivariate outlyingness functions: Mahalanobis distance, Mahalanobis spatial, and projection. Using the formulas, favorable masking and swamping breakdown points, balanced equally, are seen for the Mahalanobis distance outlyingness using minimum covariance determinant (MCD) location and scatter estimators, and likewise for the projection outlyingness with median and MAD for univariate location and scale. Also, Mahalanobis spatial outlyingness with MCD standardization is competitive when swamping robustness is given higher priority than masking robustness. A small simulation study with bivariate contaminated standard normal and contaminated exponential models yields results consistent with the theoretical formulas. Some practical recommendations are discussed.

Construction of Affine Invariant Functions in Spatial Domain

Affine invariant functions are constructed in spatial domain. Unlike the previous affine representation functions in transform domain, these functions are constructed directly on the object contour without any transformation. To eliminate the effect of the choice of points on the contour, an affine invariant function using seven points on the contour is constructed. For objects with several separable components, a closed curve is derived to construct the affine invariant functions. Several experiments have been conducted to evaluate the performance of the proposed method. Experimental results show that the constructed affine invariant functions can be used for object classification.

Synthesized affine invariant function for 2D shape recognition

By defining the weighted wavelet synthesis, the synthesized feature signals of an interesting shape are extracted to derive the innovative synthesized affine invariant function (SAIF). The synthesized feature signals hold the shape information with minimum loss by excluding simply the translation dependent and noise-contaminated bands. The SAIF is shown excellent in the invariance property and representative in describing the original shape for automated recognition. Experimental results demonstrate that automated shape recognition based on the SAIF achieves high correctness and significantly outperforms those using conventional wavelet affine invariant functions.

Wavelet approximation-based affine invariant shape representation functions

In this paper, new wavelet-based affine invariant functions for shape representation are presented. Unlike the previous representation functions, only the approximation coefficients are used to obtain the proposed functions. One of the derived functions is computed by applying a single wavelet transform; the other function is calculated by applying two different wavelet transforms with two different wavelet families. One drawback of the previously derived detail-based invariant representation functions is that they are sensitive to noise at the finer scale levels, which limits the number of scale levels that can be used. The experimental results in this paper demonstrate that the proposed functions are more stable and less sensitive to noise than the detail-based functions.

Computationally efficient wavelet affine invariant functions for shape recognition

An affine invariant function for object recognition is constructed from wavelet coefficients of the object boundary. In previous works, undecimated dyadic wavelet transform was used to construct affine invariant functions. In this paper, an algorithm based on decimated wavelet transform is developed to compute an affine invariant function. As a result computational complexity is reduced without decreasing recognition performance. Experimental results are presented.

Some Extensions of Tukey's Depth Function

As the extensions of Tukey's depth, a family of affine invariant depth functions are introduced for multivariate location and dispersion. The location depth functions can be used for the purpose of multivariate ordering. Such kind ordering can retain more information from the original data than that based on Tukey's depth. The dispersion depth functions provide some additional view of the dispersion of the data set. It is shown that these sample depth functions converge to their population versions uniformly on any compact subset of the parameter space. The deepest points of these depth functions are affine equivariant estimates of multivariate location and dispersion. Under some general conditions these estimates are proved to have asymptotic breakdown points at least 1/3 and convergence rates of 1/n. Their asymptotic distributions are also obtained under some regularity conditions. A new algorithm based on the idea of thresholding is presented for computing these kinds of estimates and realized in the bivariate case. Simulations indicate that some of them could have the empirical mean squared errors smaller than those based on Tukey's depth function or Donoho's depth function.

Affine invariants for object recognition using the wavelet transform

Dyadic wavelet transform has been used to derive affine invariant functions. The invariant functions are based on the dyadic wavelet transform of the object boundary. Three invariant functions have been calculated using different numbers of dyadic levels. Experimental results show that these invariant functions outperform some traditional invariant functions. The stability of these invariant functions have been tested for a large perspective transformation.