Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

Newton based Stochastic Optimization using q-Gaussian Smoothed Functional Algorithms

We present the first q-Gaussian smoothed functional (SF) estimator of the Hessian and the first Newton-based stochastic optimization algorithm that estimates both the Hessian and the gradient of the objective function using q-Gaussian perturbations. Our algorithm requires only two system simulations (regardless of the parameter dimension) and estimates both the gradient and the Hessian at each update epoch using these. We also present a proof of convergence of the proposed algorithm. In a related recent work (Ghoshdastidar et al., 2013), we presented gradient SF algorithms based on the q-Gaussian perturbations. Our work extends prior work on smoothed functional algorithms by generalizing the class of perturbation distributions as most distributions reported in the literature for which SF algorithms are known to work and turn out to be special cases of the q-Gaussian distribution. Besides studying the convergence properties of our algorithm analytically, we also show the results of several numerical simulations on a model of a queuing network, that illustrate the significance of the proposed method. In particular, we observe that our algorithm performs better in most cases, over a wide range of q-values, in comparison to Newton SF algorithms with the Gaussian (Bhatnagar, 2007) and Cauchy perturbations, as well as the gradient q-Gaussian SF algorithms (Ghoshdastidar et al., 2013).Comment: This is a longer of version of the paper with the same title accepted in Automatic

Mathematical modelling in systems biology : cell cycle regulation during leaf development in Arabidopsis

In this thesis, we studied a mathematical modelling approach of systems biology in plants. We have concentrated on two different issues related to the cell cycle and cell division (especially in the plant Arabidopsis). The first issue is that of the epidermal cell population in the Arabidopsis leaf and the second issue deals with gene networks which play an important role during the cell cycle. The chapters are grouped into four parts. In Part I, we described the cell cycle as the series of events that takes place in a cell leading to its division and duplication. We also stated the general objective of the study. We addressed the various aspects of the problems and the key factors that are assumed to influence or cause the problems. We provided a comprehensive mathematical framework in Part II to be used in the other chapters for the modelling, simulation and analysing purposes. Here we introduced and studied Michaelis-Menten kinetics (a model of enzyme kinetics) and the quasi-steady state assumption to reduce the complexity of the model. We also introduced two basic mathematical models for the growth of cell size in plants. In Part III, we considered two case studies related to the cell cycle and cell division in Arabidopsis. The first case study is the temporal control of epidermal cell divisions in the Arabidopsis leaf. The growth of plant organs is the result of two processes acting on the cellular level, namely cell division and cell expansion. The precise nature of the interaction between these two processes is still largely unknown as it is experimentally challenging to disentangle them. The lower epidermal tissue layer of the Arabidopsis leaf is composed of two cell types, puzzle shaped pavement cells and guard cells, which build the stomata. We determined the cell number and the individual cell areas separately for both cell types during development. To dissect the rules whereby different cell types divide and expand, the experimental data were fit into a computational model that describes all possible changes a cell can undergo from a given day to the next day. The model allows to calculate the probabilities for a precursor cell to become a guard or pavement cell, the maximum size at which it can divide into two pavement cells or two guard cells, the cell cycle duration and two different growth rates for two kinds of cells (pavement and guard cells) in one population. The second case study deals with the fact that atypical E2F activity restrains APC/CCCS52A2 function obligatory for endocycle onset. We have demonstrated that the atypical E2F transcription factor E2Fe/DEL1 controls the onset of the endocycle through a direct transcriptional control of APC/C activity. Because E2Fe/DEL1 represses the CCS52A2 promoter, we hypothesize that its level must drop below a critical threshold to allow sufficient accumulation of CCS52A2 during late S and G2 phase for cells to proceed from division to endoreduplication. We built a mathematical model to analyse the above hypothesis. The model is based on an ODE model used for the binding of ligands to proteins with the help of Hill functions. This mathematical model helps to understand mechanistically how decreasing E2Fe/DEL1 levels can account for the division-to-endoreduplication transition. Finally, in Part IV, we discussed some future work to extend the above research. We suggested several ideas to design new experiments and increase the value of the models. The term ”we” is used throughout the text to underline the fact that every single result of the author’s work as represented in the thesis was only possible because of the provision of experiments, equipment, materials and scientific input from others

Principal Component Analysis in an Asymmetric Norm

Principal component analysis (PCA) is a widely used dimension reduction tool in the analysis of many kind of high-dimensional data. It is used in signal processing, mechanical engineering, psychometrics, and other fields under different names. It still bears the same mathematical idea: the decomposition of variation of a high dimensional object into uncorrelated factors or components. However, in many of the above applications, one is interested in capturing the tail variables of the data rather than variation around the mean. Such applications include weather related event curves, expected shortfalls, and speeding analysis among others. These are all high dimensional tail objects which one would like to study in a PCA fashion. The tail character though requires to do the dimension reduction in an asymmetric norm rather than the classical $L_2$-type orthogonal projection. We develop an analogue of PCA in an asymmetric norm. These norms cover both quantiles and expectiles, another tail event measure. The difficulty is that there is no natural basis, no `principal components', to the $k$-dimensional subspace found. We propose two definitions of principal components and provide algorithms based on iterative least squares. We prove upper bounds on their convergence times, and compare their performances in a simulation study. We apply the algorithms to a Chinese weather dataset with a view to weather derivative pricingComment: 31 pages, 5 figure

Smoothed Functional Algorithms for Stochastic Optimization using q-Gaussian Distributions

Smoothed functional (SF) schemes for gradient estimation are known to be efficient in stochastic optimization algorithms, specially when the objective is to improve the performance of a stochastic system. However, the performance of these methods depends on several parameters, such as the choice of a suitable smoothing kernel. Different kernels have been studied in literature, which include Gaussian, Cauchy and uniform distributions among others. This paper studies a new class of kernels based on the q-Gaussian distribution, that has gained popularity in statistical physics over the last decade. Though the importance of this family of distributions is attributed to its ability to generalize the Gaussian distribution, we observe that this class encompasses almost all existing smoothing kernels. This motivates us to study SF schemes for gradient estimation using the q-Gaussian distribution. Using the derived gradient estimates, we propose two-timescale algorithms for optimization of a stochastic objective function in a constrained setting with projected gradient search approach. We prove the convergence of our algorithms to the set of stationary points of an associated ODE. We also demonstrate their performance numerically through simulations on a queuing model