Ανάπτυξη Αλγορίθμων για Ανθεκτική Ομαδοποίηση, Συνδιακύμανση και Παλινδρόμηση και Εφαρμογές σε Συστήματα Μηχανικού
Creators: name: Παυλίδου, Μερόπη Χ.
Publisher: name: Aristotle University of Thessaloniki
Year: 2020
Subjects: Ανθεκτική στατιστική
Λοξότητα
Έκτοπα
Παλινδρόμηση
Robust statistics
Skewness
Outliers
Regression

description: Η Συνδιακύμανση, η Παλινδρόμηση και η Ομαδοποίηση αποτελούν βασικά προβλήματα της Πολυμεταβλητής Στατιστικής. Τέτοια προβλήματα εμφανίζονται σε πολλούς τομείς όπως και στην επιστήμη της μηχανικής. Σε δεδομένα μεγάλων διαστάσεων, αναζητούμε μαθηματικά μοντέλα που θα μας βοηθήσουν στην περιγραφή τους. Οταν τα δεδομένα ακολουθούν την Κανονική κατανομή, οι βέλτιστες τιμές τους προέρχονται από τις εκτιμήσεις μέγιστης πιθανοφάνειας. Στην πραγματικότητα όμως αυτή η υπόθεση παραβιάζεται και τα δεδομένα ακολουθούν διαφορετικές κατανομές ή/και παρουσιάζουν λοξότητα. Σε αυτή τη διατριβή επικεντρωνόμαστε στην ανθεκτική στατιστική και προτείνουμε καινούριες μεθόδους εντοπισμού εκτόπων, εκτίμησης παραμέτρων παλινδρόμησης και ομαδοποίησης οι οποίες είναι αποτελεσματικές και όταν τα δεδομένα ακολουθούν την κανονική κατανομή και όταν τα δεδομένα παρουσιάζουν λοξότητα ως προς την κατανομή τους.
description: Covariance estimation, Regression as well as Classification problems comprise the basis of mutivariate statistical analysis. such problems appear when parametriza- tion of sata is attempted as in engineering problems.When dealing with big data, numerous or/and with with a considerably large number of dimensions, by convention we seek mathematical models that will assist us in describing them. A first question is whether the data do come from the Gaussian distribution. In that case, the optimal parameter values are attained through maximum likelihood estimation, MLE, relying on least squares solutions. However, in reality, the normality assumption is very often violated resulting in distortions from the assumed models such as for example skewed data. Moreover, even even a small number of outlying observations on the data can have as a consequence the wrong estimations if we use the MLE. This phenomenon is due to the least squares estimation being considerably affected by outliers and therefore the covariance and the regression estimations as well. There is vast bibliography on robust covariance and robust regression estimation which are the main problems in robust statistics. Most methods aim at detecting the outlying observations and subsequently at removing them and applying the MLE estimators. A different approach is to take advantage of various criteria such as the mahalanobis distance or the scale of the residuals so as to distinguish healthy data from outliers. These methods aim at detecting the optimal subset of data that has certain properties such as the smallest covariance determinant, MCD, or the minimum volume ellipsoid , MVE. However, such methods suffer from computational complexity as the size of the problem increases and are mostly based on the assumption of normality. In this Thesis, we are focused on two approaches: In the first approach, we do not assume that the data follows the Gaussian distribution but, instead, we estimate a robust non parametric kernel density model based on our sample. Our aim is to detect the outliers by identifying which observations are assigned with the smallest probabilities in this robust 114, Kernel Density estimation, KDE, which does not require prior knowledge on the data or does not assume certain type of distribution. We test our method on normally distributed as well as skewed data and notice that there is an advantage over state-of-the-art algorithms in efficiency detecting both outliers and inliers. In the second approach, we propose a new formula of mathematical programming based on Support Vector Machines, SVR, where there is no need for prior estimation of the sensitivity parameter. In this new formula, nSVR, the parameter is converted into a decision variable and its estimation is performed simultaneously with the rest of the parameters of the problem through the mathematical programming optimal solution. This new formula is competitive with the rest up-to-date methods in the presence of normality however it seems to be advantageous when the data display skewness.