Isotonic Regression Algorithm Research Articles

Estimation of Ordinary Differential Equation Models with Discretization Error Quantification

We consider parameter estimation of ordinary differential equation (ODE) models from noisy observations. For this problem, one conventional approach is to fit numerical solutions (e.g., Euler, Runge--Kutta) of ODEs to data. However, such a method does not account for the discretization error in numerical solutions and has limited estimation accuracy. In this study, we develop an estimation method that quantifies the discretization error based on data. The key idea is to model the discretization error as random variables and estimate their variance simultaneously with the ODE parameter. The proposed method has the form of iteratively reweighted least squares, where the discretization error variance is updated with the isotonic regression algorithm and the ODE parameter is updated by solving a weighted least squares problem using the adjoint system. Experimental results demonstrate that the proposed method attains robust estimation with at least comparable accuracy to the conventional method by successfully quantifying the reliability of the numerical solutions.

Better synonyms for enriching biomedical search.

In a biomedical literature search, the link between a query and a document is often not established, because they use different terms to refer to the same concept. Distributional word embeddings are frequently used for detecting related words by computing the cosine similarity between them. However, previous research has not established either the best embedding methods for detecting synonyms among related word pairs or how effective such methods may be. In this study, we first create the BioSearchSyn set, a manually annotated set of synonyms, to assess and compare 3 widely used word-embedding methods (word2vec, fastText, and GloVe) in their ability to detect synonyms among related pairs of words. We demonstrate the shortcomings of the cosine similarity score between word embeddings for this task: the same scores have very different meanings for the different methods. To address the problem, we propose utilizing pool adjacent violators (PAV), an isotonic regression algorithm, to transform a cosine similarity into a probability of 2 words being synonyms. Experimental results using the BioSearchSyn set as a gold standard reveal which embedding methods have the best performance in identifying synonym pairs. The BioSearchSyn set also allows converting cosine similarity scores into probabilities, which provides a uniform interpretation of the synonymy score over different methods. We introduced the BioSearchSyn corpus of 1000 term pairs, which allowed us to identify the best embedding method for detecting synonymy for biomedical search. Using the proposed method, we created PubTermVariants2.0: a large, automatically extracted set of synonym pairs that have augmented PubMed searches since the spring of 2019.

Resolutions to flip-over credit risk and beyond-least squares estimates and maximum likelihood estimates with monotonic constraints

Given a risk outcome y over a rating system $\{R_i\}^k_{i=1}$ for a portfolio, we show in this paper that the maximum likelihood estimates with monotonic constraints, when y is binary (the Bernoulli likelihood) or takes values in the interval 0≤y≤1 (the quasi-Bernoulli likelihood), are each given by the average of the observed outcomes for some consecutive rating indexes. These estimates are in average equal to the sample average risk over the portfolio and coincide with the estimates by least squares with the same monotonic constraints. These results are the exact solution of the corresponding constrained optimization. A non-parametric algorithm for the exact solution is proposed. For the least squares estimates, this algorithm is compared with “pool adjacent violators” algorithm for isotonic regression. The proposed approaches provide a resolution to flip-over credit risk and a tool to determine the fair risk scales over a rating system.

Consistent estimation of survival functions under uniform stochastic ordering; thek-sample case

Let S 1 , S 2 , … , S k be survival functions of life distributions. They are said to be uniformly stochastically ordered, S 1 ≤ u s o S 2 ≤ u s o ⋯ ≤ u s o S k , if S i / S i + 1 is a survival function for 1 ≤ i ≤ k − 1 . The nonparametric maximum likelihood estimators of the survival functions subject to this ordering constraint are known to be inconsistent in general. Consistent estimators were developed only for the case of k = 2 . In this paper we provide consistent estimators in the k -sample case, with and without censoring. In proving consistency, we needed to develop a new algorithm for isotonic regression that may be of independent interest.

A pseudolikelihood method for analyzing interval censored data

We introduce a method based on a pseudolikelihood ratio for estimating the distribution function of the survival time in a mixed-case interval censoring model. In a mixed-case model, an individual is observed a random number of times, and at each time it is recorded whether an event has happened or not. One seeks to estimate the distribution of time to event. We use a Poisson process as the basis of a likelihood function to construct a pseudolikelihood ratio statistic for testing the value of the distribution function at a fixed point, and show that this converges under the null hypothesis to a known limit distribution, that can be expressed as a functional of different convex minorants of a two-sided Brownian motion process with parabolic drift. Construction of confidence sets then proceeds by standard inversion. The computation of the confidence sets is simple, requiring the use of the pool-adjacent-violators algorithm or a standard isotonic regression algorithm. We also illustrate the superiority of the proposed method over competitors based on resampling techniques or on the limit distribution of the maximum pseudolikelihood estimator, through simulation studies, and illustrate the different methods on a dataset involving time to HIV seroconversion in a group of haemophiliacs.

Algorithms for a Class of Isotonic Regression Problems

The isotonic regression problem has applications in statistics, operations research, and image processing. In this paper a general framework for the isotonic regression algorithm is proposed. Under this framework, we discuss the isotonic regression problem in the case where the directed graph specifying the order restriction is a directed tree with n vertices. A new algorithm is presented for this case, which can be regarded as a generalization of the PAV algorithm of Ayer et al. Using a simple tree structure such as the binomial heap, the algorithm can be implemented in O(n log n) time, improving the previously best known O(n 2 ) time algorithm. We also present linear time algorithms for special cases where the directed graph is a path or a star.

Estimating Survival Curves with Left-Truncated and Interval-Censored Data under Monotone Hazards

We show that the nonparametric maximum likelihood estimator (NPMLE) of a survival function may severely underestimate the survival probabilities at very early times for left-truncated and interval-censored data. As an alternative, we propose to compute the (nonparametric) MLE under a nondecreasing hazard assumption, the monotone MLE, by a gradient projection algorithm when the assumption holds. The projection step is accomplished via an isotonic regression algorithm, the pool-adjacent-violators algorithm. This gradient projection algorithm is computationally efficient and converges globally. Monte Carlo simulations show superior performance of the monotone MLE over that of the NPMLE in terms of either bias or variance, even for large samples. The methodology is illustrated with the application to the Wisconsin Epidemiological Study of Diabetic Retinopathy data to estimate the probability of incidence of retinopathy.

An Algorithm for Isotonic Regression on Ordered Rectangular Grids

An Algorithm for Isotonic Regression on Ordered Rectangular Grids

An Algorithm for Isotonic Regression on Ordered Rectangular Grids

Abstract In this article, we give an algorithm for isotonic regressions on ordered rectangular grids. The running time of the algorithm is no more than cubic in the number of grid points. This algorithm makes bivariate isotonic regression a practical choice for some data analysis.

Structure Algorithms for Partially Ordered Isotonic Regression

Abstract An algorithm for isotonic regression is called a structure algorithm if it searches for a “solution partition”—that is, a class of sets on each of which the isotonic regression is a constant. We discuss structure algorithms for partially ordered isotonic regression. In this article we provide a new class of structure algorithms called the isotonic block class (IBC) type algorithms. One of these is called the isotonic block class with recursion method (IBCR) algorithm, which works for partially ordered isotonic regression. It is a generalization of the pooled adjacent violators algorithm and is simpler than the min-max algorithm. We also give a polynomial time algorithm—the isotonic block class with stratification (IBCS) algorithm for matrix-ordered isotonic regression. We demonstrate the efficiency of our IBCR algorithm by using simulation to estimate the relative frequencies of the numbers of level sets of isotonic regressions on certain two-dimensional grids with the matrix order.

An algorithm for isotonic regression with arbitrary convex distance function

In the present paper we consider the isotonic regression problem with an arbitrary convex distance function d(·), and the main purpose being to present an algorithm for obtaining all isotonic regression under this reasonable assumption on d(·). Further, we consider a piece-wise linear distance function d(·) of the type d( t)= C −| t| for t<0 and d( t)= C +| t| for t⩾0 and get an isotonic pth fractile regression by choosing p=C + (C −+C +)