Extreme deconvolution: Inferring complete distribution functions from noisy, heterogeneous and incomplete observations

We generalize the well-known mixtures of Gaussians approach to density estimation and the accompanying Expectation--Maximization technique for finding the maximum likelihood parameters of the mixture to the case where each data point carries an individual $d$-dimensional uncertainty covariance and has unique missing data properties. This algorithm reconstructs the error-deconvolved or "underlying" distribution function common to all samples, even when the individual data points are samples from different distributions, obtained by convolving the underlying distribution with the heteroskedastic uncertainty distribution of the data point and projecting out the missing data directions. We show how this basic algorithm can be extended with conjugate priors on all of the model parameters and a "split-and-merge" procedure designed to avoid local maxima of the likelihood. We demonstrate the full method by applying it to the problem of inferring the three-dimensional velocity distribution of stars near the Sun from noisy two-dimensional, transverse velocity measurements from the Hipparcos satellite.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS439 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

The velocity distribution of nearby stars from Hipparcos data I. The significance of the moving groups

We present a three-dimensional reconstruction of the velocity distribution of nearby stars (<~ 100 pc) using a maximum likelihood density estimation technique applied to the two-dimensional tangential velocities of stars. The underlying distribution is modeled as a mixture of Gaussian components. The algorithm reconstructs the error-deconvolved distribution function, even when the individual stars have unique error and missing-data properties. We apply this technique to the tangential velocity measurements from a kinematically unbiased sample of 11,865 main sequence stars observed by the Hipparcos satellite. We explore various methods for validating the complexity of the resulting velocity distribution function, including criteria based on Bayesian model selection and how accurately our reconstruction predicts the radial velocities of a sample of stars from the Geneva-Copenhagen survey (GCS). Using this very conservative external validation test based on the GCS, we find that there is little evidence for structure in the distribution function beyond the moving groups established prior to the Hipparcos mission. This is in sharp contrast with internal tests performed here and in previous analyses, which point consistently to maximal structure in the velocity distribution. We quantify the information content of the radial velocity measurements and find that the mean amount of new information gained from a radial velocity measurement of a single star is significant. This argues for complementary radial velocity surveys to upcoming astrometric surveys

Riemannian metrics for neural networks II: recurrent networks and learning symbolic data sequences

Recurrent neural networks are powerful models for sequential data, able to represent complex dependencies in the sequence that simpler models such as hidden Markov models cannot handle. Yet they are notoriously hard to train. Here we introduce a training procedure using a gradient ascent in a Riemannian metric: this produces an algorithm independent from design choices such as the encoding of parameters and unit activities. This metric gradient ascent is designed to have an algorithmic cost close to backpropagation through time for sparsely connected networks. We use this procedure on gated leaky neural networks (GLNNs), a variant of recurrent neural networks with an architecture inspired by finite automata and an evolution equation inspired by continuous-time networks. GLNNs trained with a Riemannian gradient are demonstrated to effectively capture a variety of structures in synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov-type relations, or long-distance relationships such as the distant-XOR problem. This method does not require adjusting the network structure or initial parameters: the network used is a sparse random graph and the initialization is identical for all problems considered.Comment: 4th version: some changes in notation, more experiment

PENERAPAN GRAY LEVEL CO-OCCURRENCE MATRIX (GLCM) DAN LEARNING VECTOR QUANTIZATION (LVQ) UNTUK KLASIFIKASI PENYAKIT RETINA MATA

Mata sebagai indera penglihatan merupakan organ vital yang mempunyai peranan sangat penting untuk menerima informasi visual, sebagai pengembangan diri dan kualitas hidup manusia, Hasil survei yang dilakukan tahun 1993 di Indonesia penderita penyakit mata yang mengakibatkan kebutaan mencapai 1,5%, sedangkan tahun 2003 mencapai angka 2,2% dan pada tahun 2007 dengan angka 1,67%. Angka-angka tersebut membuat Indonesia menjadi negara dengan tingkat kebutaan tertinggi kedua setelah Ethophia, Identifikasi diabetic retinophaty secara manual cenderung membutuhkan waktu yang lama dan memungkinkan terjadinya kesalahan dalam pengamatan, Penerapan Metode Gray Level Co-Occurrence Matrix (GLCM) sebagai ekstraksi ciri orde dua dapat mengambil ciri dari citra retina diabetic retinophaty dan penggunaan metode Generalized Learning Vector Quantization (GLVQ) dapat mengelompokkan hasil-hasil citra sehingga mempermudah klasifikasi penyakit diabetic retinophaty. Pada penelitian ini hasil yang di peroleh menunjukkan pembelajaran terbaik dilakukan dengan menggunakan α=0,01, maksimum epoch = 1000 dan mendapat akurasi terbesar pada persentasi 90%

Rejection and online learning with prototype-based classifiers in adaptive metrical spaces

Fischer L. Rejection and online learning with prototype-based classifiers in adaptive metrical spaces. Bielefeld: Universität Bielefeld; 2016.The rising amount of digital data, which is available in almost every domain, causes the need for intelligent, automated data processing. Classification models constitute particularly popular techniques from the machine learning domain with applications ranging from fraud detection up to advanced image classification tasks. Within this thesis, we will focus on so-called prototype-based classifiers as one prominent family of classifiers, since they offer a simple classification scheme, interpretability of the model in terms of prototypes, and good generalisation performance. We will face a few crucial questions which arise whenever such classifiers are used in real-life scenarios which require robustness and reliability of classification and the ability to deal with complex and possibly streaming data sets. Particularly, we will address the following problems: - Deterministic prototype-based classifiers deliver a class label, but no confidence of the classification. The latter is particularly relevant whenever the costs of an error are higher than the costs to reject an example, e.g. in a safety critical system. We investigate ways to enhance prototype-based classifiers by a certainty measure which can efficiently be computed based on the given classifier only and which can be used to reject an unclear classification. - For an efficient rejection, the choice of a suitable threshold is crucial. We investigate in which situations the performance of local rejection can surpass the choice of only a global one, and we propose efficient schemes how to optimally compute local thresholds on a given training set. - For complex data and lifelong learning, the required classifier complexity can be unknown a priori. We propose an efficient, incremental scheme which adjusts the model complexity of a prototype-based classifier based on the certainty of the classification. Thereby, we put particular emphasis on the question how to adjust prototype locations and metric parameters, and how to insert and/or delete prototypes in an efficient way. - As an alternative to the previous solution, we investigate a hybrid architecture which combines an offline classifier with an online classifier based on their certainty values, thus directly addressing the stability/plasticity dilemma. While this is straightforward for classical prototype-based schemes, it poses some challenges as soon as metric learning is integrated into the scheme due to the different inherent data representations. - Finally, we investigate the performance of the proposed hybrid prototype-based classifier within a realistic visual road-terrain-detection scenario