Lecture notes on ridge regression

The linear regression model cannot be fitted to high-dimensional data, as the high-dimensionality brings about empirical non-identifiability. Penalized regression overcomes this non-identifiability by augmentation of the loss function by a penalty (i.e. a function of regression coefficients). The ridge penalty is the sum of squared regression coefficients, giving rise to ridge regression. Here many aspect of ridge regression are reviewed e.g. moments, mean squared error, its equivalence to constrained estimation, and its relation to Bayesian regression. Finally, its behaviour and use are illustrated in simulation and on omics data. Subsequently, ridge regression is generalized to allow for a more general penalty. The ridge penalization framework is then translated to logistic regression and its properties are shown to carry over. To contrast ridge penalized estimation, the final chapter introduces its lasso counterpart

Evaluating the accuracy of diffusion MRI models in white matter

Models of diffusion MRI within a voxel are useful for making inferences about the properties of the tissue and inferring fiber orientation distribution used by tractography algorithms. A useful model must fit the data accurately. However, evaluations of model-accuracy of some of the models that are commonly used in analyzing human white matter have not been published before. Here, we evaluate model-accuracy of the two main classes of diffusion MRI models. The diffusion tensor model (DTM) summarizes diffusion as a 3-dimensional Gaussian distribution. Sparse fascicle models (SFM) summarize the signal as a linear sum of signals originating from a collection of fascicles oriented in different directions. We use cross-validation to assess model-accuracy at different gradient amplitudes (b-values) throughout the white matter. Specifically, we fit each model to all the white matter voxels in one data set and then use the model to predict a second, independent data set. This is the first evaluation of model-accuracy of these models. In most of the white matter the DTM predicts the data more accurately than test-retest reliability; SFM model-accuracy is higher than test-retest reliability and also higher than the DTM, particularly for measurements with (a) a b-value above 1000 in locations containing fiber crossings, and (b) in the regions of the brain surrounding the optic radiations. The SFM also has better parameter-validity: it more accurately estimates the fiber orientation distribution function (fODF) in each voxel, which is useful for fiber tracking

Uncertainty Quantification in Lasso-Type Regularization Problems

Regularization techniques, which sit at the interface of statistical modeling and machine learning, are often used in the engineering or other applied sciences to tackle high dimensional regression (type) problems. While a number of regularization methods are commonly used, the 'Least Absolute Shrinkage and Selection Operator' or simply LASSO is popular because of its efficient variable selection property. This property of the LASSO helps to deal with problems where the number of predictors is larger than the total number of observations, as it shrinks the coefficients of non-important parameters to zero. In this chapter, both frequentist and Bayesian approaches for the LASSO are discussed, with particular attention to the problem of uncertainty quantification of regression parameters. For the frequentist approach, we discuss a refit technique as well as the classical bootstrap method, and for the Bayesian method, we make use of the equivalent LASSO formulation using a Laplace prior on the model parameters

Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors

Penalized regression is an attractive framework for variable selection problems. Often, variables possess a grouping structure, and the relevant selection problem is that of selecting groups, not individual variables. The group lasso has been proposed as a way of extending the ideas of the lasso to the problem of group selection. Nonconvex penalties such as SCAD and MCP have been proposed and shown to have several advantages over the lasso; these penalties may also be extended to the group selection problem, giving rise to group SCAD and group MCP methods. Here, we describe algorithms for fitting these models stably and efficiently. In addition, we present simulation results and real data examples comparing and contrasting the statistical properties of these methods

A Novel Euler's Elastica based Segmentation Approach for Noisy Images via using the Progressive Hedging Algorithm

Euler's Elastica based unsupervised segmentation models have strong capability of completing the missing boundaries for existing objects in a clean image, but they are not working well for noisy images. This paper aims to establish a Euler's Elastica based approach that properly deals with random noises to improve the segmentation performance for noisy images. We solve the corresponding optimization problem via using the progressive hedging algorithm (PHA) with a step length suggested by the alternating direction method of multipliers (ADMM). Technically, all the simplified convex versions of the subproblems derived from the major framework of PHA can be obtained by using the curvature weighted approach and the convex relaxation method. Then an alternating optimization strategy is applied with the merits of using some powerful accelerating techniques including the fast Fourier transform (FFT) and generalized soft threshold formulas. Extensive experiments have been conducted on both synthetic and real images, which validated some significant gains of the proposed segmentation models and demonstrated the advantages of the developed algorithm

The use of the Kalman filter in the automated segmentation of EIT lung images

In this paper, we present a new pipeline for the fast and accurate segmentation of impedance images of the lungs using electrical impedance tomography (EIT). EIT is an emerging, promising, non-invasive imaging modality that produces real-time, low spatial but high temporal resolution images of impedance inside a body. Recovering impedance itself constitutes a nonlinear ill-posed inverse problem, therefore the problem is usually linearized, which produces impedance-change images, rather than static impedance ones. Such images are highly blurry and fuzzy along object boundaries. We provide a mathematical reasoning behind the high suitability of the Kalman filter when it comes to segmenting and tracking conductivity changes in EIT lung images. Next, we use a two-fold approach to tackle the segmentation problem. First, we construct a global lung shape to restrict the search region of the Kalman filter. Next, we proceed with augmenting the Kalman filter by incorporating an adaptive foreground detection system to provide the boundary contours for the Kalman filter to carry out the tracking of the conductivity changes as the lungs undergo deformation in a respiratory cycle. The proposed method has been validated by using performance statistics such as misclassified area, and false positive rate, and compared to previous approaches. The results show that the proposed automated method can be a fast and reliable segmentation tool for EIT imaging