Robust designs for 3D shape analysis with spherical harmonic descriptors

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Robust designs for 3D shape analysis with spherical harmonic descriptors

Spherical harmonic descriptors are frequently used for describing three-dimensional shapes in terms of Fourier coefficients corresponding to an expansion of a function defined on the unit sphere. In a recent paper Dette, Melas and Pepelysheff (2005) determined optimal designs with respect to Kiefer's Phi p-criteria for regression models derived from a truncated Fourier series. In particular it was shown that the uniform distribution on the sphere is Phi p-optimal for spherical harmonic descriptors, for all p > -1. These designs minimize a function of the variance-covariance matrix of the least squares estimate but do not take into account the bias resulting from the truncation of the series. In the present paper we demonstrate that the uniform distribution is also optimal with respect to a minimax criterion based on the mean square error, and as a consequence these designs are robust with respect to the truncation error. Moreover, we also consider heteroscedasticity and possible correlations in the construction of the optimal designs. These features appear naturally in 3D shape analysis, and the uniform design again turns out to be minimax robust against erroneous assumptions of homoscedasticity and independence. AMS 2000 Subject Classi cation: Primary 62K05, 62D32; secondary 62J05

Robust designs for series estimation

We discuss optimal design problems for a popular method of series estimation in regression problems. Commonly used design criteria are based on the generalized variance of the estimates of the coefficients in a truncated series expansion and do not take possible bias into account. We present a general perspective of constructing robust and e±cient designs for series estimators which is based on the integrated mean squared error criterion. A minimax approach is used to derive designs which are robust with respect to deviations caused by the bias and the possibility of heteroscedasticity. A special case results from the imposition of an unbiasedness constraint; the resulting unbiased designs are particularly simple, and easily implemented. Our results are illustrated by constructing robust designs for series estimation with spherical harmonic descriptors, Zernike polynomials and Chebyshev polynomials. --Chebyshev polynomials,direct estimation,minimax designs,robust designs,series estimation,spherical harmonic descriptors,unbiased design,Zernike polynomials

Selecting the number of clusters, clustering models, and algorithms. A unifying approach based on the quadratic discriminant score

Cluster analysis requires many decisions: the clustering method and the implied reference model, the number of clusters and, often, several hyper-parameters and algorithms' tunings. In practice, one produces several partitions, and a final one is chosen based on validation or selection criteria. There exist an abundance of validation methods that, implicitly or explicitly, assume a certain clustering notion. Moreover, they are often restricted to operate on partitions obtained from a specific method. In this paper, we focus on groups that can be well separated by quadratic or linear boundaries. The reference cluster concept is defined through the quadratic discriminant score function and parameters describing clusters' size, center and scatter. We develop two cluster-quality criteria called quadratic scores. We show that these criteria are consistent with groups generated from a general class of elliptically-symmetric distributions. The quest for this type of groups is common in applications. The connection with likelihood theory for mixture models and model-based clustering is investigated. Based on bootstrap resampling of the quadratic scores, we propose a selection rule that allows choosing among many clustering solutions. The proposed method has the distinctive advantage that it can compare partitions that cannot be compared with other state-of-the-art methods. Extensive numerical experiments and the analysis of real data show that, even if some competing methods turn out to be superior in some setups, the proposed methodology achieves a better overall performance.Comment: Supplemental materials are included at the end of the pape

Robust statistical methods: a primer for clinical psychology and experimental psychopathology researchers

This paper reviews and offers tutorials on robust statistical methods relevant to clinical and experimental psychopathology researchers. We review the assumptions of one of the most commonly applied models in this journal (the general linear model, GLM) and the effects of violating them. We then present evidence that psychological data are more likely than not to violate these assumptions. Next, we overview some methods for correcting for violations of model assumptions. The final part of the paper presents 8 tutorials of robust statistical methods using R that cover a range of variants of the GLM (t-tests, ANOVA, multiple regression, multilevel models, latent growth models). We conclude with recommendations that set the expectations for what methods researchers submitting to the journal should apply and what they should report

Advances in robust clustering methods with applications

Robust methods in statistics are mainly concerned with deviations from model assumptions. As already pointed out in Huber (1981) and in Huber & Ronchetti (2009) \these assumptions are not exactly true since they are just a mathematically convenient rationalization of an often fuzzy knowledge or belief". For that reason \a minor error in the mathematical model should cause only a small error in the nal conclusions". Nevertheless it is well known that many classical statistical procedures are \excessively sensitive to seemingly minor deviations from the assumptions". All statistical methods based on the minimization of the average square loss may suer of lack of robustness. Illustrative examples of how outliers' in uence may completely alter the nal results in regression analysis and linear model context are provided in Atkinson & Riani (2012). A presentation of classical multivariate tools' robust counterparts is provided in Farcomeni & Greco (2015). The whole dissertation is focused on robust clustering models and the outline of the thesis is as follows. Chapter 1 is focused on robust methods. Robust methods are aimed at increasing the eciency when contamination appears in the sample. Thus a general denition of such (quite general) concept is required. To do so we give a brief account of some kinds of contamination we can encounter in real data applications. Secondly we introduce the \Spurious outliers model" (Gallegos & Ritter 2009a) which is the cornerstone of the robust model based clustering models. Such model is aimed at formalizing clustering problems when one has to deal with contaminated samples. The assumption standing behind the \Spurious outliers model" is that two dierent random mechanisms generate the data: one is assumed to generate the \clean" part while the another one generates the contamination. This idea is actually very common within robust models like the \Tukey-Huber model" which is introduced in Subsection 1.2.2. Outliers' recognition, especially in the multivariate case, plays a key role and is not straightforward as the dimensionality of the data increases. An overview of the most widely used (robust) methods for outliers detection is provided within Section 1.3. Finally, in Section 1.4, we provide a non technical review of the classical tools introduced in the Robust Statistics' literature aimed at evaluating the robustness properties of a methodology. Chapter 2 is focused on model based clustering methods and their robustness' properties. Cluster analysis, \the art of nding groups in the data" (Kaufman & Rousseeuw 1990), is one of the most widely used tools within the unsupervised learning context. A very popular method is the k-means algorithm (MacQueen et al. 1967) which is based on minimizing the Euclidean distance of each observation from the estimated clusters' centroids and therefore it is aected by lack of robustness. Indeed even a single outlying observation may completely alter centroids' estimation and simultaneously provoke a bias in the standard errors' estimation. Cluster's contours may be in ated and the \real" underlying clusterwise structure might be completely hidden. A rst attempt of robustifying the k- means algorithm appeared in Cuesta-Albertos et al. (1997), where a trimming step is inserted in the algorithm in order to avoid the outliers' exceeding in uence. It shall be noticed that k-means algorithm is ecient for detecting spherical homoscedastic clusters. Whenever more exible shapes are desired the procedure becomes inecient. In order to overcome this problem Gaussian model based clustering methods should be adopted instead of k-means algorithm. An example, among the other proposals described in Chapter 2, is the TCLUST methodology (Garca- Escudero et al. 2008), which is the cornerstone of the thesis. Such methodology is based on two main characteristics: trimming a xed proportion of observations and imposing a constraint on the estimates of the scatter matrices. As it will be explained in Chapter 2, trimming is used to protect the results from outliers' in uence while the constraint is involved as spurious maximizers may completely spoil the solution. Chapter 3 and 4 are mainly focused on extending the TCLUST methodology. In particular, in Chapter 3, we introduce a new contribution (compare Dotto et al. 2015 and Dotto et al. 2016b), based on the TCLUST approach, called reweighted TCLUST or RTCLUST for the sake of brevity. The idea standing behind such method is based on reweighting the observations initially agged as outlying. This is helpful both to gain eciency in the parameters' estimation process and to provide a reliable estimation of the true contamination level. Indeed, as the TCLUST is based on trimming a xed proportion of observations, a proper choice of the trimming level is required. Such choice, especially in the applications, can be cumbersome. As it will be claried later on, RTCLUST methodology allows the user to overcome such problem. Indeed, in the RTCLUST approach the user is only required to impose a high preventive trimming level. The procedure, by iterating through a sequence of decreasing trimming levels, is aimed at reinserting the discarded observations at each step and provides more precise estimation of the parameters and a nal estimation of the true contamination level ^. The theoretical properties of the methodology are studied in Section 3.6 and proved in Appendix A.1, while, Section 3.7, contains a simulation study aimed at evaluating the properties of the methodology and the advantages with respect to some other robust (reweigthed and single step procedures). Chapter 4 contains an extension of the TCLUST method for fuzzy linear clustering (Dotto et al. 2016a). Such contribution can be viewed as the extension of Fritz et al. (2013a) for linear clustering problems, or, equivalently, as the extension of Garca-Escudero, Gordaliza, Mayo-Iscar & San Martn (2010) to the fuzzy clustering framework. Fuzzy clustering is also useful to deal with contamination. Fuzziness is introduced to deal with overlapping between clusters and the presence of bridge points, to be dened in Section 1.1. Indeed bridge points may arise in case of overlapping between clusters and may completely alter the estimated cluster's parameters (i.e. the coecients of a linear model in each cluster). By introducing fuzziness such observations are suitably down weighted and the clusterwise structure can be correctly detected. On the other hand, robustness against gross outliers, as in the TCLUST methodology, is guaranteed by trimming a xed proportion of observations. Additionally a simulation study, aimed at comparing the proposed methodology with other proposals (both robust and non robust) is also provided in Section 4.4. Chapter 5 is entirely dedicated to real data applications of the proposed contributions. In particular, the RTCLUST method is applied to two dierent datasets. The rst one is the \Swiss Bank Note" dataset, a well known benchmark dataset for clustering models, and to a dataset collected by Gallup Organization, which is, to our knowledge, an original dataset, on which no other existing proposals have been applied yet. Section 5.3 contains an application of our fuzzy linear clustering proposal to allometry data. In our opinion such dataset, already considered in the robust linear clustering proposal appeared in Garca-Escudero, Gordaliza, Mayo-Iscar & San Martn (2010), is particularly useful to show the advantages of our proposed methodology. Indeed allometric quantities are often linked by a linear relationship but, at the same time, there may be overlap between dierent groups and outliers may often appear due to errors in data registration. Finally Chapter 6 contains the concluding remarks and the further directions of research. In particular we wish to mention an ongoing work (Dotto & Farcomeni, In preparation) in which we consider the possibility of implementing robust parsimonious Gaussian clustering models. Within the chapter, the algorithm is briefly described and some illustrative examples are also provided. The potential advantages of such proposals are the following. First of all, by considering the parsimonious models introduced in Celeux & Govaert (1995), the user is able to impose the shape of the detected clusters, which often, in the applications, plays a key role. Secondly, by constraining the shape of the detected clusters, the constraint on the eigenvalue ratio can be avoided. This leads to the removal of a tuning parameter of the procedure and, at the same time, allows the user to obtain ane equivariant estimators. Finally, since the possibility of trimming a xed proportion of observations is allowed, then the procedure is also formally robust

The Globular Cluster System in the Inner Region of M87

1057 globular cluster candidates have been identified in a WFPC2 image of the inner region of M87. The Globular Cluster Luminosity Function (GCLF) can be well fit by a Gaussian profile with a mean value of m_V^0=23.67 +/- 0.07 mag and sigma=1.39 +/- 0.06 mag (compared to m_V^0=23.74 mag and sigma=1.44 mag from an earlier study using the same data by Whitmore it et al. 1995). The GCLF in five radial bins is found to be statistically the same at all points, showing no clear evidence of dynamical destruction processes based on the luminosity function (LF), in contradiction to the claim by Gnedin (1997). Similarly, there is no obvious correlation between the half light radius of the clusters and the galactocentric distance. The core radius of the globular cluster density distribution is R_c=56'', considerably larger than the core of the stellar component (R_c=6.8''). The mean color of the cluster candidates is V-I=1.09 mag which corresponds to an average metallicity of Fe/H = -0.74 dex. The color distribution is bimodal everywhere, with a blue peak at V-I=0.95 mag and a red peak at V-I=1.20 mag. The red population is only 0.1 magnitude bluer than the underlying galaxy, indicating that these clusters formed late in the metal enrichment history of the galaxy and were possibly created in a burst of star/cluster formation 3-6 Gyr after the blue population. We also find that both the red and the blue cluster distributions have a more elliptical shape (Hubble type E3.5) than the nearly spherical galaxy. The average half light radius of the clusters is ~2.5 pc which is comparable to the 3 pc average effective radius of the Milky Way clusters, though the red candidates are ~20% smaller than the blue ones.Comment: 40 pages, 17 figures, 4 tables, latex, accepted for publication in the Ap