Efficient Second-Order Shape-Constrained Function Fitting

We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted-$L_{\infty}$ norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first- and/or second-order differences. Our algorithm computes an approximation with additive error $\varepsilon$ in $O\left(n \log \frac{U}{\varepsilon} \right)$ time, where $U$ captures the range of input values. We also give a simple greedy algorithm that runs in $O(n)$ time for the special case of unweighted $L_{\infty}$ convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming.Comment: accepted for WADS 2019; (v2 fixes various typos

Efficient Second-Order Shape-Constrained Function Fitting

We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted-$L_{\infty}$ norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first- and/or second-order differences. Our algorithm computes an approximation with additive error $\varepsilon$ in $O\left(n \log \frac{U}{\varepsilon} \right)$ time, where $U$ captures the range of input values. We also give a simple greedy algorithm that runs in $O(n)$ time for the special case of unweighted $L_{\infty}$ convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming

Topics on Threshold Estimation, Multistage Methods and Random Fields.

We consider the problem of identifying the threshold at which a one-dimensional regression function leaves its baseline value. This is motivated by applications from dose-response studies and environmental statistics. We develop a novel approach that relies on the dichotomous behavior of p-value statistics around this threshold. We study the asymptotic behavior of our estimate in two different sampling settings for constructing confidence intervals. The multi-dimensional version of the threshold estimation problem has connections to fMRI studies, edge detection and image processing. Here, interest centers on estimating a region (equivalently, its complement) where a function is at its baseline level. In certain applications, this set corresponds to the background of an image; hence, identifying this region from noisy observations is equivalent to reconstructing the image. We study the computational and theoretical aspects of an extension of the p-value procedure to this setting, primarily under a convex shape-constraint in two dimensions, and explore its applicability to other situations as well. Multistage procedures, obtained by splitting the sampling budget across stages, and designing the sampling at a particular stage based on information obtained from previous stages, are often advantageous as they typically accelerate the rate of convergence of the estimates, relative to one-stage procedures. The step-by-step process, though, induces dependence across stages and complicates the analysis in such problems. We develop a generic framework for M-estimation in a multistage setting and apply empirical process techniques to describe the asymptotic behavior of the resulting M-estimates. Applications to change-point estimation, inverse isotonic regression and mode estimation are provided. In a departure from the more statistical components of the dissertation, we consider a central limit question for linear random fields. Random fields -- real valued stochastic processes indexed by a multi-dimensional set -- arise naturally in spatial data analysis and thus, have received considerable interest. We prove a Central Limit Theorem (CLT) for linear random fields that allows sums to be taken over sets as general as the disjoint union of rectangles. A simple version of our result provides a complete analogue of a CLT for linear processes with no extra assumptions.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/102290/1/atulm_1.pd

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Untangling hotel industry’s inefficiency: An SFA approach applied to a renowned Portuguese hotel chain

The present paper explores the technical efficiency of four hotels from Teixeira Duarte Group - a renowned Portuguese hotel chain. An efficiency ranking is established from these four hotel units located in Portugal using Stochastic Frontier Analysis. This methodology allows to discriminate between measurement error and systematic inefficiencies in the estimation process enabling to investigate the main inefficiency causes. Several suggestions concerning efficiency improvement are undertaken for each hotel studied.info:eu-repo/semantics/publishedVersio

Lipschitz Unimodal and Isotonic Regression on Paths and Trees

We describe algorithms for finding the regression of t, a sequence of values, to the closest sequence s by mean squared error, so that s is always increasing (isotonicity) and so the values of two consecutive points do not increase by too much (Lipschitz). The isotonicity constraint can be replaced with a unimodular constraint, where there is exactly one local maximum in s. These algorithm are generalized from sequences of values to trees of values. For each scenario we describe near-linear time algorithms.