Type Ia Supernova Light Curve Inference: Hierarchical Bayesian Analysis in the Near Infrared

We present a comprehensive statistical analysis of the properties of Type Ia SN light curves in the near infrared using recent data from PAIRITEL and the literature. We construct a hierarchical Bayesian framework, incorporating several uncertainties including photometric error, peculiar velocities, dust extinction and intrinsic variations, for coherent statistical inference. SN Ia light curve inferences are drawn from the global posterior probability of parameters describing both individual supernovae and the population conditioned on the entire SN Ia NIR dataset. The logical structure of the hierarchical model is represented by a directed acyclic graph. Fully Bayesian analysis of the model and data is enabled by an efficient MCMC algorithm exploiting the conditional structure using Gibbs sampling. We apply this framework to the JHK_s SN Ia light curve data. A new light curve model captures the observed J-band light curve shape variations. The intrinsic variances in peak absolute magnitudes are: sigma(M_J) = 0.17 +/- 0.03, sigma(M_H) = 0.11 +/- 0.03, and sigma(M_Ks) = 0.19 +/- 0.04. We describe the first quantitative evidence for correlations between the NIR absolute magnitudes and J-band light curve shapes, and demonstrate their utility for distance estimation. The average residual in the Hubble diagram for the training set SN at cz > 2000 km/s is 0.10 mag. The new application of bootstrap cross-validation to SN Ia light curve inference tests the sensitivity of the model fit to the finite sample and estimates the prediction error at 0.15 mag. These results demonstrate that SN Ia NIR light curves are as effective as optical light curves, and, because they are less vulnerable to dust absorption, they have great potential as precise and accurate cosmological distance indicators.Comment: 24 pages, 15 figures, 4 tables. Accepted for publication in ApJ. Corrected typo, added references, minor edit

Non-Parametric Learning for Monocular Visual Odometry

This thesis addresses the problem of incremental localization from visual information, a scenario commonly known as visual odometry. Current visual odometry algorithms are heavily dependent on camera calibration, using a pre-established geometric model to provide the transformation between input (optical flow estimates) and output (vehicle motion estimates) information. A novel approach to visual odometry is proposed in this thesis where the need for camera calibration, or even for a geometric model, is circumvented by the use of machine learning principles and techniques. A non-parametric Bayesian regression technique, the Gaussian Process (GP), is used to elect the most probable transformation function hypothesis from input to output, based on training data collected prior and during navigation. Other than eliminating the need for a geometric model and traditional camera calibration, this approach also allows for scale recovery even in a monocular configuration, and provides a natural treatment of uncertainties due to the probabilistic nature of GPs. Several extensions to the traditional GP framework are introduced and discussed in depth, and they constitute the core of the contributions of this thesis to the machine learning and robotics community. The proposed framework is tested in a wide variety of scenarios, ranging from urban and off-road ground vehicles to unconstrained 3D unmanned aircrafts. The results show a significant improvement over traditional visual odometry algorithms, and also surpass results obtained using other sensors, such as laser scanners and IMUs. The incorporation of these results to a SLAM scenario, using a Exact Sparse Information Filter (ESIF), is shown to decrease global uncertainty by exploiting revisited areas of the environment. Finally, a technique for the automatic segmentation of dynamic objects is presented, as a way to increase the robustness of image information and further improve visual odometry results

Non-Parametric Learning for Monocular Visual Odometry

This thesis addresses the problem of incremental localization from visual information, a scenario commonly known as visual odometry. Current visual odometry algorithms are heavily dependent on camera calibration, using a pre-established geometric model to provide the transformation between input (optical flow estimates) and output (vehicle motion estimates) information. A novel approach to visual odometry is proposed in this thesis where the need for camera calibration, or even for a geometric model, is circumvented by the use of machine learning principles and techniques. A non-parametric Bayesian regression technique, the Gaussian Process (GP), is used to elect the most probable transformation function hypothesis from input to output, based on training data collected prior and during navigation. Other than eliminating the need for a geometric model and traditional camera calibration, this approach also allows for scale recovery even in a monocular configuration, and provides a natural treatment of uncertainties due to the probabilistic nature of GPs. Several extensions to the traditional GP framework are introduced and discussed in depth, and they constitute the core of the contributions of this thesis to the machine learning and robotics community. The proposed framework is tested in a wide variety of scenarios, ranging from urban and off-road ground vehicles to unconstrained 3D unmanned aircrafts. The results show a significant improvement over traditional visual odometry algorithms, and also surpass results obtained using other sensors, such as laser scanners and IMUs. The incorporation of these results to a SLAM scenario, using a Exact Sparse Information Filter (ESIF), is shown to decrease global uncertainty by exploiting revisited areas of the environment. Finally, a technique for the automatic segmentation of dynamic objects is presented, as a way to increase the robustness of image information and further improve visual odometry results

Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes

The application of Stochastic Differential Equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a non-parametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudo-samples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behaviour of complex systems

Stochastic Variational Inference with Gradient Linearization

Variational inference has experienced a recent surge in popularity owing to stochastic approaches, which have yielded practical tools for a wide range of model classes. A key benefit is that stochastic variational inference obviates the tedious process of deriving analytical expressions for closed-form variable updates. Instead, one simply needs to derive the gradient of the log-posterior, which is often much easier. Yet for certain model classes, the log-posterior itself is difficult to optimize using standard gradient techniques. One such example are random field models, where optimization based on gradient linearization has proven popular, since it speeds up convergence significantly and can avoid poor local optima. In this paper we propose stochastic variational inference with gradient linearization (SVIGL). It is similarly convenient as standard stochastic variational inference - all that is required is a local linearization of the energy gradient. Its benefit over stochastic variational inference with conventional gradient methods is a clear improvement in convergence speed, while yielding comparable or even better variational approximations in terms of KL divergence. We demonstrate the benefits of SVIGL in three applications: Optical flow estimation, Poisson-Gaussian denoising, and 3D surface reconstruction.Comment: To appear at CVPR 201

Most Likely Separation of Intensity and Warping Effects in Image Registration

This paper introduces a class of mixed-effects models for joint modeling of spatially correlated intensity variation and warping variation in 2D images. Spatially correlated intensity variation and warp variation are modeled as random effects, resulting in a nonlinear mixed-effects model that enables simultaneous estimation of template and model parameters by optimization of the likelihood function. We propose an algorithm for fitting the model which alternates estimation of variance parameters and image registration. This approach avoids the potential estimation bias in the template estimate that arises when treating registration as a preprocessing step. We apply the model to datasets of facial images and 2D brain magnetic resonance images to illustrate the simultaneous estimation and prediction of intensity and warp effects

The Eccentricity Distribution of Short-Period Planet Candidates Detected by Kepler in Occultation

We characterize the eccentricity distribution of a sample of ~50 short-period planet candidates using transit and occultation measurements from NASA's Kepler Mission. First, we evaluate the sensitivity of our hierarchical Bayesian modeling and test its robustness to model misspecification using simulated data. When analyzing actual data assuming a Rayleigh distribution for eccentricity, we find that the posterior mode for the dispersion parameter is $\sigma=0.081 \pm^{0.014}_{0.003}$. We find that a two-component Gaussian mixture model for $e \cos \omega$ and $e \sin \omega$ provides a better model than either a Rayleigh or Beta distribution. Based on our favored model, we find that $\sim90\%$ of planet candidates in our sample come from a population with an eccentricity distribution characterized by a small dispersion ($\sim0.01$), and $\sim10\%$ come from a population with a larger dispersion ($\sim0.22$). Finally, we investigate how the eccentricity distribution correlates with selected planet and host star parameters. We find evidence that suggests systems around higher metallicity stars and planet candidates with smaller radii come from a more complex eccentricity distribution.Comment: Accepted for publication in Ap