Dimension reduction is a vital tool in many areas of applied statistics in which the dimensionality of the predictors can be large. In such cases, many statistical methods will fail or yield unsatisfactory results. However, many data sets of high dimensionality actually contain a much simpler, low-dimensional structure. Classical methods such as principal components analysis are able to detect linear structures very effectively, but fail in the presence of nonlinear structures. In the first part of this thesis, we investigate the asymptotic behavior of two nonlinear dimensionality reduction algorithms, LTSA and HLLE. In particular, we show that both algorithms, under suitable conditions, asymptotically recover the true generating coordinates up to an isometry. We also discuss the relative merits of the two algorithms, and the effects of the underlying probability distributions of the coordinates on their performance. Model selection is a fundamental problem in nearly all areas of applied statistics. In particular, a balance must be achieved between good in-sample performance and out-of-sample prediction. It is typically very easy to achieve good fit in the sample data, but empirically we often find that such models will generalize poorly. In the second part of the thesis, we propose a new procedure for the model selection problem which generalizes traditional methods. Our algorithm allows the combination of existing model selection criteria via a ranking procedure, leading to the creation of new criteria which are able to combine measures of in-sample fit and out-of-sample prediction performance into a single value. We then propose an algorithm which provably finds the optimal combination with a specified probability. We demonstrate through simulations that these new combined criteria can be substantially more powerful than any individual criterion.

Šiame darbe nagrinėjami netiesiniai daugdarų atpažinimo metodai ir daugiamačių duomenų projekcijos metodai. Siūloma jais spręsti keliais laisvės laipsniais besisukančio objekto orientacijos radimo problemą. Aprašomi MDS, Trianguliacijos, Sammon, RPM, mRPM, CCA, PCA, LLE, LE, HLLE, LTSA, SMACOF ir Isomap metodai. Su kai kuriais iš jų atliekami web-kamera gautų galvos atvaizdų tyrimai. Isomap algoritmo pagrindu sukuriama programinė įranga ir su ja taipogi atliekami galvos orientacijos tyrimai. / This paper deals with Analysis of non-linear manifold learning methods and multidimensional data projection methods. It is proposed use them in solving problem of detection of orientation of object, moving in few degrees of freedom. Methods described: MDS, triangulation, Sammon, RPM, mRPM, CCA, PCA, LLE, LE, HLLE, LTSA, SMACOF and Isomap. Some of them are used to analyze head images acquired by webcam.. Application is created which is then used to analyze head orientation by implementing Isomap method.