Ao modelar-se depósitos minerais é comum enfrentarmos o problema de estimar múltiplos atributos possivelmente correlacionados, onde algumas variáveis são amostradas menos densamente do que outras. A falta de dados impõe um problema que requer atenção antes de qualquer modelagem subsequente. Precisamos, ao final, de modelos que sejam estatisticamente representativos. A maioria dos conjuntos de dados de problemas práticos são amostrados de maneira heterotópica e, para obter resultados coerentes, é preciso entender os motivos pelos quais alguns dados faltam e quais são os mecanismos que influenciaram a ausência de informações. A teoria de dados faltantes relaciona as amostras ausentes com aquelas medidas através de três mecanismos distintos: Faltante Completamente Aleatório (Missing Completely At Random - MCAR), Faltante Aleatório (Missing At Random - MAR) e Faltante Não Aleatório (Missing Not At Random - MNAR). O último mecanismo é extremamente complexo e a literatura recomenda ser tratado inicialmente como um mecanismo MAR. E após uma transformação fixa deve ser aplicada aos valores complementados para que estes se transformem em valores MNAR Embora existam métodos estatísticos clássicos para lidar com dados faltantes, tais abordagens ignoram a correlação espacial, uma característica que ocorre naturalmente em dados geológicos. A metodologia adequada para tratar com a falta de dados geológicos é a atualização bayesiana, em que se inserem valores sob mecanismo MAR considerando a correlação espacial. No presente estudo, a atualização bayesiana foi combinada com transformações fixas para tratar o mecanismo de falta de dados MNAR em dados geológicos. A transformação fixa aqui empregada é baseada no erro de inserção gerado em um cenário MAR no conjunto de dados. Assim, com o conjunto completo resultante foi utilizado em uma simulação sequencial gaussiana dos teores de uma base de dados multivariada, apresentando resultados satisfatórios, superiores aos obtidos por meio da cossimulação sequencial gaussiana, não inserindo qualquer viés no modelo final. / When modeling mineral deposits, it is common to face the problem of estimating multiple attributes possibly correlated where some variables are more densely sampled then others. Missing data imposes a problem that requires attention prior to any subsequent modeling. The later requires estimation models statistically representative. Most practical data sets are often heterotopically sampled, and to obtain coherent results one must understand the reasons why there are missing data and what are the mechanisms that cause the absence of information. The theory of missing data relates the missing samples to those measured through three different mechanisms: Missing Completely At Random (MCAR), Missing At Random (MAR), and Missing Not At Random (MNAR). The last mechanism is quite complex to deal with, and the literature recommends being treated as a MAR mechanism and after a fixed transform should be applied to the imputed values so that these turn into MNAR imputed values. Even though there are classical statistical methods to deal with missing data, such approaches ignore spatial correlation, a feature that occurs naturally in geological data. The adequate methodology to deal with missing geologic data is Bayesian Updating, which approaches the MAR mechanism and accounts for spatial correlation. In the present study, bayesian updating was used combined with fixed transforms to treat MNAR missing data mechanism in geologic data. The fixed transform herein used is based on the error of MAR imputation on the data set. The resulting complete set was then used on a sequential gaussian simulation of the grades on a multivariate data set, presenting satisfactory results, superior to those obtained through sequential gaussian cossimulation, not inserting any biases on the final model.

Many tasks in machine learning require learning some kind of input-output relation (function), for example, recognising handwritten digits (from image to number) or learning the motion behaviour of a dynamical system like a pendulum (from positions and velocities now to future positions and velocities). We consider this problem using the Bayesian framework, where we use probability distributions to represent the state of uncertainty that a learning agent is in. In particular, we will investigate methods which use Gaussian processes to represent distributions over functions. Gaussian process models require approximations in order to be practically useful. This thesis focuses on understanding existing approximations and investigating new ones tailored to specific applications. We advance the understanding of existing techniques first through a thorough review. We propose desiderata for non-parametric basis function model approximations, which we use to assess the existing approximations. Following this, we perform an in-depth empirical investigation of two popular approximations (VFE and FITC). Based on the insights gained, we propose a new inter-domain Gaussian process approximation, which can be used to increase the sparsity of the approximation, in comparison to regular inducing point approximations. This allows GP models to be stored and communicated more compactly. Next, we show that inter-domain approximations can also allow the use of models which would otherwise be impractical, as opposed to improving existing approximations. We introduce an inter-domain approximation for the Convolutional Gaussian process - a model that makes Gaussian processes suitable to image inputs, and which has strong relations to convolutional neural networks. This same technique is valuable for approximating Gaussian processes with more general invariance properties. Finally, we revisit the derivation of the Gaussian process State Space Model, and discuss some subtleties relating to their approximation. We hope that this thesis illustrates some benefits of non-parametric models and their approximation in a non-parametric fashion, and that it provides models and approximations that prove to be useful for the development of more complex and performant models in the future.

<p>Mapping is a central and common task in robotics research. Building an accurate map without human assistance provides several applications such as space missions, search and rescue, surveillance and can be used in dangerous areas. One application for robotic mapping is to measure changes in terrain volume. In Sweden there are over a hundred landfills that are regulated by laws that says that the growth of the landfill has to be measured at least once a year.</p><p>In this thesis, a preliminary study of methods for measuring terrain volume by the use of an Unmanned Aerial Vehicle (UAV) and a Light Detection And Ranging (LIDAR) sensor is done. Different techniques are tested, including data merging strategies and regression techniques by the use of Gaussian Processes. In the absence of real flight scenario data, an industrial robot has been used fordata acquisition. The result of the experiment was successful in measuring thevolume difference between scenarios in relation to the resolution of the LIDAR. However, for more accurate volume measurements and better evaluation of the algorithms, a better LIDAR is needed.</p> / <p>Kartering är ett centralt och vanligt förekommande problem inom robotik. Att bygga en korrekt karta av en robots omgivning utan mänsklig hjälp har en mängd tänkbara användningsområden. Exempel på sådana är rymduppdrag, räddningsoperationer,övervakning och användning i områden som är farliga för människor. En tillämpning för robotkartering är att mäta volymökning hos terräng över tiden. I Sverige finns det över hundra soptippar, och dessa soptippar är reglerade av lagar som säger att man måste mäta soptippens volymökning minst en gång om året.</p><p>I detta exjobb görs en undersökning av möjligheterna att göra dessa volymberäkningarmed hjälp av obemannade helikoptrar utrustade med en Light Detectionand Ranging (LIDAR) sensor. Olika tekniker har testats, både tekniker som slår ihop LIDAR data till en karta och regressionstekniker baserade på Gauss Processer. I avsaknad av data inspelad med riktig helikopter har ett experiment med en industri robot genomförts för att samla in data. Resultaten av volymmätningarnavar goda i förhållande till LIDAR-sensorns upplösning. För att få bättre volymmätningaroch bättre utvärderingar av de olika algoritmerna är en bättre LIDAR-sensor nödvändig.</p>

A Multiple-Input Multiple Output (MIMO) radar can be broadly defined as a radar system employing multiple transmit waveforms and having the ability to jointly process signals received at multiple receive antennas. In terms of configurations, the antennas can be widely separated or co-located. The first part of the thesis is on detection under Gaussian and non-Gaussian environments using a MIMO radar which contains several widely separated subarrays with one or more elements each. Two different situations are considered. Firstly, we consider that the interference is Gaussian but correlation between subarrays can arise due to insufficient spacing and the imperfect orthogonality of waveforms. Secondly, we consider that the interference is non-Gaussian, a situation which arises under sea and ground clutter and when the resolution is very high. The second part is on the application of MIMO techniques to Space-Time Adaptive Processing (STAP). The coherent MIMO configuration is studied in terms of antenna element distribution and inter-element spacing to improve detection and estimation performance. A preliminary study is also done on the use of spatial diversity to improve detection stability w.r.t. target Radar Cross Section (RCS) fluctuations and velocity direction changes.

There are several methods to measure surface evenness for car roads, but almost none for bike paths. Accordingly, VTI (the Swedish National Road and Transport Research Institute) have created a smartphone application which uses the accelerometers in the phone to measure the vibration from the road. This report’s aim is to analyze the data collected using this application, investigate if the data is repeatable, to find factors that are important for evenness and perform classification of bike paths as even or wiggly. Two main methods were used, Gaussian process and wavelets. Gaussian process was used to classify bike paths and wavelets to investigate the repeatability and see how many trips are needed to get a consistent result. The results show that the two different smartphones gave quite different results; one smartphone indicated almost twice as high RMS values (measure of vibration) than the other. The GPS positions of smartphones were quite good, except under a tunnel and close to high buildings. Some short section of the road gave very high or very low RMS values, but the general standard of all investigated bike paths were too even to detect any significant differences between the paths. The results show that there’s some unexplained variance in the turns, but the effect of the turns hasn’t been tested. The wavelets analysis show that around 15 trips were needed to get a consistent result. The report contains a description of a designed experiment that will continue this project. This new data will be collected in a more carefully to make a better separation between good and bad cycle routes by the RMS value. / <p>Uppdragsgivare: VTI (Anna Niska och Leif Sjögren)</p>

Mapping is a central and common task in robotics research. Building an accurate map without human assistance provides several applications such as space missions, search and rescue, surveillance and can be used in dangerous areas. One application for robotic mapping is to measure changes in terrain volume. In Sweden there are over a hundred landfills that are regulated by laws that says that the growth of the landfill has to be measured at least once a year. In this thesis, a preliminary study of methods for measuring terrain volume by the use of an Unmanned Aerial Vehicle (UAV) and a Light Detection And Ranging (LIDAR) sensor is done. Different techniques are tested, including data merging strategies and regression techniques by the use of Gaussian Processes. In the absence of real flight scenario data, an industrial robot has been used fordata acquisition. The result of the experiment was successful in measuring thevolume difference between scenarios in relation to the resolution of the LIDAR. However, for more accurate volume measurements and better evaluation of the algorithms, a better LIDAR is needed. / Kartering är ett centralt och vanligt förekommande problem inom robotik. Att bygga en korrekt karta av en robots omgivning utan mänsklig hjälp har en mängd tänkbara användningsområden. Exempel på sådana är rymduppdrag, räddningsoperationer,övervakning och användning i områden som är farliga för människor. En tillämpning för robotkartering är att mäta volymökning hos terräng över tiden. I Sverige finns det över hundra soptippar, och dessa soptippar är reglerade av lagar som säger att man måste mäta soptippens volymökning minst en gång om året. I detta exjobb görs en undersökning av möjligheterna att göra dessa volymberäkningarmed hjälp av obemannade helikoptrar utrustade med en Light Detectionand Ranging (LIDAR) sensor. Olika tekniker har testats, både tekniker som slår ihop LIDAR data till en karta och regressionstekniker baserade på Gauss Processer. I avsaknad av data inspelad med riktig helikopter har ett experiment med en industri robot genomförts för att samla in data. Resultaten av volymmätningarnavar goda i förhållande till LIDAR-sensorns upplösning. För att få bättre volymmätningaroch bättre utvärderingar av de olika algoritmerna är en bättre LIDAR-sensor nödvändig.

This study presents the results of an extraction of the 2nd-order nonlinear responses from model test data. Emphasis is given on the effects of assumptions made for the Gaussian and non-Gaussian input on the estimation of the 2nd-order response, employing the quadratic Volterra model. The effects of sea severity and data length on the estimation of response are also investigated at the same time. The data sets used in this study are surge forces on a fixed barge, a surge motion of a compliant mini TLP (Tension Leg Platform), and surge forces on a fixed and truncated column. Sea states are used from rough sea (Hs=3m) to high sea (Hs=9m) for a barge case, very rough sea (Hs=3.9m) for a mini TLP, and phenomenal sea (Hs=15m) for a truncated column. After the estimation of the response functions, the outputs are reconstructed and the 2nd order nonlinear responses are extracted with all the QTF distributed in the entire bifrequency domain. The reconstituted time series are compared with the experiment in both the time and frequency domains. For the effects of data length on the estimation of the response functions, 3, 15, and 40- hour data were investigated for a barge, but 3-hour data was used for a mini TLP and a fixed and truncated column due to lack of long data. The effects of sea severity on the estimation of the response functions are found in both methods. The non-Gaussian method for estimation is more affected by data length than the Gaussian method.

This thesis describes the implementation of metamodeling approaches as a solution to approximate multivariate, stochastic and dynamic simulations. In the area of statistics, metamodeling (or ``model of a model") refers to the scenario where an empirical model is build based on simulated data. In this thesis, this idea is exploited by using pre-recorded dynamic simulations as a source of simulated dynamic data. Based on this simulated dynamic data, an empirical model is trained to map the dynamic evolution of the system from the current discrete time step, to the next discrete time step. Therefore, it is possible to approximate the dynamics of the complex dynamic simulation, by iteratively applying the trained empirical model. The rationale in creating such approximate dynamic representation is that the empirical models / metamodels are much more affordable to compute than the original dynamic simulation, while having an acceptable prediction error. The successful implementation of metamodeling approaches, as approximations of complex dynamic simulations, requires understanding of the propagation of error during the iterative process. Prediction errors made by the empirical model at earlier times of the iterative process propagate into future predictions of the model. The propagation of error means that the trained empirical model will deviate from the expensive dynamic simulation because of its own errors. Based on this idea, Gaussian process model is chosen as the metamodeling approach for the approximation of expensive dynamic simulations in this thesis. This empirical model was selected not only for its flexibility and error estimation properties, but also because it can illustrate relevant issues to be considered if other metamodeling approaches were used for this purpose.

Many real-world datasets have spatial components that provide valuable information about characteristics of the data. In this dissertation, a novel framework for adaptive models that exploit spatial information in data is proposed. The proposed framework is mainly based on development and applications of Gaussian processes. First, a supervised learning method is proposed for the classification of hyperspectral data with spatially adaptive model parameters. The proposed algorithm models spatially varying means of each spectral band of a given class using a Gaussian process regression model. For a given location, the predictive distribution of a given class is modeled by a multivariate Gaussian distribution with spatially adjusted parameters obtained from the proposed algorithm. The Gaussian process model is generally regarded as a good tool for interpolation, but not for extrapolation. Moreover, the uncertainty of the predictive distribution increases as the distance from the training instances increases. To overcome this problem, a semi-supervised learning algorithm is presented for the classification of hyperspectral data with spatially adaptive model parameters. This algorithm fits the test data with a spatially adaptive mixture-of-Gaussians model, where the spatially varying parameters of each component are obtained by Gaussian process regressions with soft memberships using the mixture-of-Gaussian-processes model. The proposed semi-supervised algorithm assumes a transductive setting, where the unlabeled data is considered to be similar to the training data. This is not true in general, however, since one may not know how many classes may existin the unexplored regions. A spatially adaptive nonparametric Bayesian framework is therefore proposed by applying spatially adaptive mechanisms to the mixture model with infinitely many components. In this method, each component in the mixture has spatially adapted parameters estimated by Gaussian process regressions, and spatial correlations between indicator variables are also considered. In addition to land cover and land use classification applications based on hyperspectral imagery, the Gaussian process-based spatio-temporal model is also applied to predict ground-based aerosol optical depth measurements from satellite multispectral images, and to select the most informative ground-based sites by active learning. In this application, heterogeneous features with spatial and temporal information are incorporated together by employing a set of covariance functions, and it is shown that the spatio-temporal information exploited in this manner substantially improves the regression model. The conventional meaning of spatial information usually refers to actual spatio-temporal locations in the physical world. In the final chapter of this dissertation, the meaning of spatial information is generalized to the parametrized low-dimensional representation of data in feature space, and a corresponding spatial modeling technique is exploited to develop a nearest-manifold classification algorithm. / text

The primate hand, a biomechanical structure with over twenty kinematic degrees of freedom, has an elaborate anatomical architecture. Although the hand requires complex, coordinated neural control, it endows its owner with an astonishing range of dexterous finger movements. Despite a century of research, however, the neural mechanisms that enable finger and grasping movements in primates are largely unknown. In this thesis, we investigate statistical models of finger movement that can provide insights into the mechanics of the hand, and that can have applications in neural-motor prostheses, enabling people with limb loss to regain natural function of the hands. There are many challenges associated with (1) the understanding and modeling of the kinematics of fingers, and (2) the mapping of intracortical neural recordings into motor commands that can be used to control a Brain-Machine Interface. These challenges include: potential nonlinearities; confounded sources of variation in experimental datasets; and dealing with high degrees of kinematic freedom. In this work we analyze kinematic and neural datasets from repeated-trial experiments of hand motion, with the following contributions: We identified static, nonlinear, low-dimensional representations of grasping finger motion, with accompanying evidence that these nonlinear representations are better than linear representations at predicting the type of object being grasped over the course of a reach-to-grasp movement. In addition, we show evidence of better encoding of these nonlinear (versus linear) representations in the firing of some neurons collected from the primary motor cortex of rhesus monkeys. A functional alignment of grasping trajectories, based on total kinetic energy, as a strategy to account for temporal variation and to exploit a repeated-trial experiment structure. An interpretable model for extracting dynamic synergies of finger motion, based on Gaussian Processes, that decomposes and reduces the dimensionality of variance in the dataset. We derive efficient algorithms for parameter estimation, show accurate reconstruction of grasping trajectories, and illustrate the interpretation of the model parameters. Sound evidence of single-neuron decoding of interpretable grasping events, plus insights about the amount of grasping information extractable from just a single neuron. The Laplace Gaussian Filter (LGF), a deterministic approximation to the posterior mean that is more accurate than Monte Carlo approximations for the same computational cost, and that in an off-line decoding task is more accurate than the standard Population Vector Algorithm.