Performance Analysis of l_0 Norm Constraint Least Mean Square Algorithm

As one of the recently proposed algorithms for sparse system identification, $l_0$ norm constraint Least Mean Square ($l_0$-LMS) algorithm modifies the cost function of the traditional method with a penalty of tap-weight sparsity. The performance of $l_0$-LMS is quite attractive compared with its various precursors. However, there has been no detailed study of its performance. This paper presents all-around and throughout theoretical performance analysis of $l_0$-LMS for white Gaussian input data based on some reasonable assumptions. Expressions for steady-state mean square deviation (MSD) are derived and discussed with respect to algorithm parameters and system sparsity. The parameter selection rule is established for achieving the best performance. Approximated with Taylor series, the instantaneous behavior is also derived. In addition, the relationship between $l_0$-LMS and some previous arts and the sufficient conditions for $l_0$-LMS to accelerate convergence are set up. Finally, all of the theoretical results are compared with simulations and are shown to agree well in a large range of parameter setting.Comment: 31 pages, 8 figure

MAD MQP 2001

This project develops an Attitude Determination and Control (ADC) subsystem for a six-unit CubeSat in an extreme low Earth orbit (eLEO) mission performing atmospheric neutral and ion mass spectrometry. The selection of sensors and actuators were evaluated and updated from a previous mission. The performance of algorithms used for detumble, initial attitude determination, and attitude maintenance was evaluated using MATLAB, Simulink, and Systems Tool Kit (STK) simulations. In order to conduct this evaluation, in-depth Simulink models of spacecraft attitude dynamics and control were developed which consider sensor noise and refresh rates for a GPS receiver, gyroscope, magnetometer, and two-axis sun sensors, as well as actuator limitations for reaction wheels and magnetorquers

部分范数约束的稀疏恢复算法及其在单载波水声数据遥测中的应用

对于单载波水声数据压缩与恢复问题,压缩感知能以较低能耗获得信号压缩与恢复效果。但压缩感知核心目标是直接求最小l0范数,该问题表现为NP难问题,因此,常将其转化为求l1范数约束最小化问题,而求l1范数约束最小化的稀疏解精度有限。基于此,推导出基于部分范数约束的稀疏信号恢复算法,该算法通过部分范数约束在拉格朗日求解中增加一个零吸引项,从而动态分配稀疏抽头的软阈值。同时,该算法用于实际海上数据的遥测,结合离散余弦变换（DCT）,可将单载波水声数据恢复精度提高。国家自然科学基金资助项目(No.61701405);;中央高校基本科研业务费专项资金资助项目(No.3102017OQD007);;中国博士后科学基金资助项目(No.2017M613208)~

Asset liability modelling and pension schemes: the application of robust optimization to USS

This paper uses a novel numerical optimization technique - robust optimization - that is well suited to solving the asset-liability management (ALM) problem for pension schemes. It requires the estimation of fewer stochastic parameters, reduces estimation risk and adopts a prudent approach to asset allocation. This study is the first to apply it to a real-world pension scheme, and the first ALM model of a pension scheme to maximise the Sharpe ratio. We disaggregate pension liabilities into three components - active members, deferred members and pensioners, and transform the optimal asset allocation into the scheme’s projected contribution rate. The robust optimization model is extended to include liabilities and used to derive optimal investment policies for the Universities Superannuation Scheme (USS), benchmarked against the Sharpe and Tint, Bayes-Stein, and Black-Litterman models as well as the actual USS investment decisions. Over a 144 month out-of-sample period robust optimization is superior to the four benchmarks across 20 performance criteria, and has a remarkably stable asset allocation – essentially fix-mix. These conclusions are supported by six robustness checks

Change of support problemへの新たな空間統計モデルの開発

筑波大学 (University of Tsukuba)201

Slowly Varying Regression under Sparsity

We consider the problem of parameter estimation in slowly varying regression models with sparsity constraints. We formulate the problem as a mixed integer optimization problem and demonstrate that it can be reformulated exactly as a binary convex optimization problem through a novel exact relaxation. The relaxation utilizes a new equality on Moore-Penrose inverses that convexifies the non-convex objective function while coinciding with the original objective on all feasible binary points. This allows us to solve the problem significantly more efficiently and to provable optimality using a cutting plane-type algorithm. We develop a highly optimized implementation of such algorithm, which substantially improves upon the asymptotic computational complexity of a straightforward implementation. We further develop a heuristic method that is guaranteed to produce a feasible solution and, as we empirically illustrate, generates high quality warm-start solutions for the binary optimization problem. We show, on both synthetic and real-world datasets, that the resulting algorithm outperforms competing formulations in comparable times across a variety of metrics including out-of-sample predictive performance, support recovery accuracy, and false positive rate. The algorithm enables us to train models with 10,000s of parameters, is robust to noise, and able to effectively capture the underlying slowly changing support of the data generating process.Comment: Submitted to Operations Research. First submission: 02/202

Some Statistical Properties of Spectral Regression Estimators

In this thesis we explore different Spectral Regression Estimators in order to solve the prob- lem in regression where we have multiple columns that are linearly dependent: We explore two scenarios • Scenario 1: p \u3c\u3c n where there exists at least two columns; xj and xk that are nearly linearly dependent which indicates co-linearity and X⊤X becomes near singular. • Scenario 2: n \u3c\u3c p since there are more predictors than observations so some columns must be a linear combination of another column which indicates linear dependence. The scenarios give us an ill conditioned matrix of X⊤X (when solving the normal equa- tion) due to collinearity issues and the matrix becomes singular and makes the least squares estimate unstable and impossible to compute. In the paper, we explore different methods (variable selection, regularization, compression and dimensionality reduction) that solves the above issue. For variable selection techniques, we use Stepwise Selection Regression as well as the method of Best Subset Selection regression. Two approaches for Stepwise Se- lection regression are assessed in the paper: Forward Selection and Backward Elimination. Performance assessment of our regression models will be made based on criterion based procedures like AIC,BIC,R2,R2 adjusted and the Mallow’s CP statistic. In chapter three of this paper we introduce the concepts of General Regularization, Ridge Regression as well as subsequent shrinkage methods such as the Lasso, Bayesian Lasso and the Elastic net. Chapter five will look at Compression and Dimensionality reduction procedures which are outlined via SVD (Singular Value Decomposition) and Eigenvector Decomposition. Hard thresholding is subsequently introduced via SPCA (Sparse Principle Component Analysis) and a novel approach using RPCA (Robust Principle Component Analysis). Furthermore, RPCA also shows how it can aid with data and image compression. The basis of this study is concluded with an empirical exploration of all the methods outlined above using several performance indicators on simulated data and real data sets. Assessment of the data sets is done via cross-validation. We determine the optimal values of the settings and then evalu- ate the predictive and explanatory performance

Computational Modeling of Stability and Laxity in the Natural and Implanted Knee Joint

The knee joint plays a central role in human motion for its dual function: providing a large range of motion in flexion/extension and stability in the other degrees of freedom. Computational modeling is a powerful tool to deepen our understanding of the joint mechanics, overcoming the main limitations of experimental investigations, i.e. time, cost and impracticability, and providing valuable insights for prosthetic design, rehabilitation and surgical planning. Within this background, the specific aim of this dissertation is threefold: to develop a sequentially-defined kinetostatic model of the knee, comparing the performance of spherical and anatomical surfaces; to develop a dynamic model of the knee to predict the quadriceps force during the squat activity; to estimate the compressive force that the implanted knee joint needs in order to reproduce natural stability. This dissertation presents novel and efficient procedures to model and evaluate the behavior of the natural and implanted knee under the effect of static and dynamic loading conditions, extending the current knowledge in the field of musculoskeletal computational modeling

Claim Models: Granular Forms and Machine Learning Forms

This collection of articles addresses the most modern forms of loss reserving methodology: granular models and machine learning models. New methodologies come with questions about their applicability. These questions are discussed in one article, which focuses on the relative merits of granular and machine learning models. Others illustrate applications with real-world data. The examples include neural networks, which, though well known in some disciplines, have previously been limited in the actuarial literature. This volume expands on that literature, with specific attention to their application to loss reserving. For example, one of the articles introduces the application of neural networks of the gated recurrent unit form to the actuarial literature, whereas another uses a penalized neural network. Neural networks are not the only form of machine learning, and two other papers outline applications of gradient boosting and regression trees respectively. Both articles construct loss reserves at the individual claim level so that these models resemble granular models. One of these articles provides a practical application of the model to claim watching, the action of monitoring claim development and anticipating major features. Such watching can be used as an early warning system or for other administrative purposes. Overall, this volume is an extremely useful addition to the libraries of those working at the loss reserving frontier