A General Framework for Constrained Smoothing

There are a wide array of smoothing methods available for finding structure in data. A general framework is developed which shows that many of these can be viewed as a projection of the data, with respect to appropriate norms. The underlying vector space is an unusually large product space, which allows inclusion of a wide range of smoothers in our setup (including many methods not typically considered to be projections). We give several applications of this simple geometric interpretation of smoothing. A major payoff is the natural and computationally frugal incorporation of constraints. Our point of view also motivates new estimates and it helps to understand the finite sample and asymptotic behaviour of these estimates

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

A comparison of methods for temporal analysis of aoristic crime

Objectives: To test the accuracy of various methods previously proposed (and one new method) to estimate offence times where the actual time of the event is not known. Methods: For 303 thefts of pedal cycles from railway stations, the actual offence time was determined from closed-circuit television and the resulting temporal distribution compared against commonly-used estimated distributions using circular statistics and analysis of residuals. Results: Aoristic analysis and allocation of a random time to each offence allow accurate estimation of peak offence times. Commonly-used deterministic methods were found to be inaccurate and to produce misleading results. Conclusions: It is important that analysts use the most accurate methods for temporal distribution approximation to ensure any resource decisions made on the basis of peak times are reliable

A general projection framwork for constrained smoothing

There are a wide array of smoothing methods available for finding structure in data. A general framework is developed which shows that many of these can be viewed as a projection of the data, with respect to appropriate norms. The underlying vector space is an unusually large product space, which allows inclusion of a wide range of smoothers in our setup (including many methods not typically considered to be projections). We give several applications of this simple geometric interpretation of smoothing. A major payoff is the natural and computationally frugal incorporation of constraints. Our point of view also motivates new estimates and helps understand the finite sample and asymptotic behavior of these estimates

A New Approach to Variable Selection in Least Squares Problems

The title Lasso has been suggested by Tibshirani [7] as a colourful name for a technique of variable selection which requires the minimization of a sum of squares subject to an ll bound r; on the solution. This forces zero components in the minimizing solution for small values of r;. Thus this bound can function as a selection parameter. This paper makes two contributions to computational problems associated with implementing the Lasso: (1) a com- pact descent method for solving the constrained problem for a particular value of r; is formulated, and (2) a homotopy method, in which the constraint bound r; becomes the homotopy parameter, is developed to completely describe the possible selection regimes. Both algorithms have a finite termination property

Knot Selection for Regression Splines via the LASSO

: Tibshirani (1996) proposes the "Least Absolute Shrinkage and Selection Operator" (lasso) as a method for regression estimation which combines features of shrinkage and variable selection. In this paper we present an algorithm that allows efficient calculation of the lasso estimator. In particular our algorithm can also be used when the number of variables exceeds the number of observations. This algorithm is then applied to the problem of knot selection for regression splines. 1 Introduction The performance of regression spline smoothing is governed by the choice of knots used in calculating the estimator, and much research effort has been devoted to the difficult problem of knot selection (see, e.g., Wand, 1997; Denison et al., 1998). Knot selection is not unlike variable selection in linear regression, for which Tibshirani (1996) proposes the least absolute shrinkage and selection operator. The lasso estimator is the solution of the constrained estimation problem minimise fi2R ..

Development and validation of an algorithm to temporally align polysomnography and actigraphy data

Current actigraphic sleep/wake detection algorithms have predominantly been validated against polysomnography, although the accuracy of such validations is dependent on the degree to which the timestamps of these two methods of data collection are synchronised. We created and validated an algorithm to temporally align actigraphy and polysomnography data using a sample of 100 healthy young adults, recruited from a pool of participants in the Western Australian Pregnancy Cohort (Raine) Study. Each participant underwent one night of polysomnography with simultaneous wrist actigraphy (Actigraph GT3X+). Our alignment algorithm incorporates the raw acceleration data and considers the best alignment when the sum of the products of acceleration and polysomnography values are maximised. Segments of the night of various lengths and locations were considered as input values in addition to several values for the maximum allowable discrepancy. The optimal input values were determined by comparing accuracies, sensitivities and specificities calculated from two commonly used sleep/wake classification methods, and then validated using a simulation study. Validation suggested that our alignment algorithm can successfully align polysomnography and actigraphy timestamps. This allows for more accurate and detailed actigraphic sleep/wake detection algorithms to be created, thus strengthening the use of actigraphy as an appropriate method for sleep detection

Incorporating geography into a new generalized theoretical and statistical framework addressing the modifiable areal unit problem

Abstract Background All analyses of spatially aggregated data are vulnerable to the modifiable areal unit problem (MAUP), which describes the sensitivity of analytical results to the arbitrary choice of spatial aggregation unit at which data are measured. The MAUP is a serious problem endemic to analyses of spatially aggregated data in all scientific disciplines. However, the impact of the MAUP is rarely considered, perhaps partly because it is still widely considered to be unsolvable. Results It was originally understood that a solution to the MAUP should constitute a comprehensive statistical framework describing the regularities in estimates of association observed at different combinations of spatial scale and zonation. Additionally, it has been debated how such a solution should incorporate the geographical characteristics of areal units (e.g. shape, size, and configuration), and in particular whether this can be achieved in a purely mathematical framework (i.e. independent of areal units). We argue that the consideration of areal units must form part of a solution to the MAUP, since the MAUP only manifests in their presence. Thus, we present a theoretical and statistical framework that incorporates the characteristics of areal units by combining estimates obtained from different scales and zonations. We show that associations estimated at scales larger than a minimal geographical unit of analysis are systematically biased from a true minimal-level effect, with different zonations generating uniquely biased estimates. Therefore, it is fundamentally erroneous to infer conclusions based on data that are spatially aggregated beyond the minimal level. Instead, researchers should measure and display information, estimate effects, and infer conclusions at the smallest possible meaningful geographical scale. The framework we develop facilitates this. Conclusions The proposed framework represents a new minimum standard in the estimation of associations using spatially aggregated data, and a reference point against which previous findings and misconceptions related to the MAUP can be understood