A Maximum Likelihood Estimator for Shape Parameters of S-Distributions

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. is paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. e paper is intended as a guide for investigators entering the fascinating �eld of biological systems analysis and as a resource for practitioners and experts. PreambleBiochemical systems theory (BST) is a mathematical and computational framework for analyzing and simulating systems. It was originally developed for biochemical pathways but by now has become much more widely applied to systems throughout biology and beyond. BST is called "canonical, " which means that model construction, diagnosis, and analysis follow stringent rules, which will be discussed throughout this paper. e key ingredient of BST is the power-law representation of all processes in a system.BST has been the focus of a number of books [1-6], including some in Chinese and Japanese [7][8][9]. e most detailed modern text dedicated speci�cally to BST is [3]. Moreover, since its inception, numerous reviews have portrayed the evolving state of the art in BST. Most of these reviews summarized methodological advances for the analysis of biochemical pathway systems . Others compared BST models with alternative modeling frameworks [32,[51][52][53][54][55][56][57][58][59][60][61][62][63]; some of these comparisons will be discussed later. Yet other reviews focused on specialty areas, such as customized methods for optimizing BST models (e.g., [64][65][66][67]) or estimating their parameters (e.g., [68][69][70][71][72]), strategies for discovering design and operating principles in natural system (e.g., [73][74][75][76][77][78]), and even the use of BST models in statistics [79][80][81]. A historical account of the �rst twenty years of BST was presented in [82].Supporting the methodological developments in the �eld, several soware packages were developed for different aspects of BST analysis. e earliest was ESSYNS [83][84][85], which supported all standard steady-state analyses and also contained a numerical solver that, at the time, was many times faster than any off-the-shelf soware [86,87]; see also [88]. Utilizing advances in computing and an extension of the solver from ESSYNS, Ferreira created the very user-friendly package PLAS, which is openly available and still very widely used [89] (see also [3]). Yamada and colleagues developed soware to translate E-cell models into BST models in PLAS format [90]. Okamoto's group created the soware BestKit, which permits the graphical design and translation of models, along with their analysis [91][92][93]. Vera and colleagues developed the soware tool PLMaddon for analyzing BST models within SBML [94]; see also [95]. It expands the Matlab SBToolbox with speci�c functionalities for power-law repr...

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“…Parameters of the distribution were estimated with leastsquares and maximum likelihood methods [731,740,746].…”

Section: Recastingmentioning

confidence: 90%

Biochemical Systems Theory: A Review

Voit

2013

ISRN Biomathematics

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149

154

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“…Details of this procedure and examples of its performance can be found in reference [9]. Alternatively, the shape parameters g and h of the S-distribution can be obtained by using the corresponding maximum likelihood estimator [23]. In such case, and the initial condition can be obtained using the same procedure stated above.…”

Section: Data Representation Using S-distributionsmentioning

confidence: 99%

A new parametric method based on S‐distributions for computing receiver operating characteristic curves for continuous diagnostic tests

Sorribas

March

Trujillano

2002

Statistics in Medicine

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Receiver operating characteristic (ROC) curves provides a method for evaluating the performance of a diagnostic test. These curves represent the true positive ratio, that is, the true positives among those affected by the disease, as a function of the false positive ratio, that is, the false positives among the healthy, corresponding to each possible value of the diagnostic variable. When the diagnostic variable is continuous, the corresponding ROC curve is also continuous. However, estimation of such curve through the analysis of sample data yields a step-line, unless some assumption is made on the underlying distribution of the considered variable. Since the actual distribution of the diagnostic test is seldom known, it is difficult to select an appropriate distribution for practical use. Data transformation may offer a solution but also may introduce a distortion on the evaluation of the diagnostic test. In this paper we show that the distribution family known as the S-distribution can be used to solve this problem. The S-distribution is defined as a differential equation in which the dependent variable is the cumulative. This special form provides a highly flexible family of distributions that can be used as models for unknown distributions. It has been shown that classical statistical distributions can be represented accurately as S-distributions and that they occur in a definite subspace of the parameter space corresponding to the whole S-distribution family. Consequently, many other distributional forms that do not correspond to known distributions are provided by the S-distribution. This property can be used to model observed data for unknown distributions and is very useful in constructing parametric ROC curves in those cases. After fitting an S-distribution to the observed samples of diseased and healthy populations, ROC curve computation is straightforward. A ROC curve can be considered as the solution of a differential equation in which the dependent variable is the ratio of true positives and the independent variable is the ratio of false positives. This equation can be easily obtained from the S-distributions fitted to observed data. Using these results, we can compute pointwise confidence bands for the ROC curve and the corresponding area under the curve. We shall compare this approach with the empirical and the binormal methods for estimating a ROC curve to show that the S-distribution based method is a useful parametric procedure.

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“…The actual determination of parameter values can be executed in different ways; the most efficient and statistically desirable method is in most cases maximum-likelihood estimation. (26) Since S-distributions are derived from a background of power-law approximation, (20,22) they can be employed in two ways. Either the entire data distribution of interest can be modeled by an S-distribution, or one may focus just on one of the tails or some center portion of the distribution, thereby further improving the (local) data fit.…”

Section: S-distributionmentioning

confidence: 99%

“…The solid line in the background represents the S-distribution that was determined through maximum likelihood estimation. (26) Its parameter values are ␣ ϭ 1.189, g ϭ 1.130, h ϭ 1.702, and X 0 ϭ 0.085. To obtain random numbers with the tables and method proposed previously, the shape parameters are rounded to g ϭ 1.1 and h ϭ 1.7.…”

Section: Examplementioning

confidence: 99%

Random Number Generation from Right‐Skewed, Symmetric, and Left‐Skewed Distributions

Voit

Schwacke

2000

Risk Analysis

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Monte Carlo simulations have become a mainstream technique for environmental and technical risk assessments. Because their results are dependent on the quality of the involved input distributions, it is important to identify distributions that are flexible enough to model all relevant data yet efficient enough to allow thousands of evaluations necessary in a typical simulation analysis. It has been shown in recent years that the S-distribution provides accurate representations for frequency data that are symmetric or skewed to either side. This flexibility makes the S-distribution an ideal candidate for Monte Carlo analyses. To use the distribution effectively, methods must be available for drawing S-distributed random numbers. Such a method is proposed here. It is shown that S-distributed random numbers can be efficiently generated from a simple algebraic formula whose coefficients are tabulated. The method is shown step by step and illustrated with a detailed example. (The tables are accessible in electronic form in the FTP parent directory at http:@www.musc.edu/voiteo/ftp/.)

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A Maximum Likelihood Estimator for Shape Parameters of S-Distributions

Abstract: The S‐distribution is a four‐parameter distribution that is defined in terms of a differential equation, in which the cumulative is represented as the dependent variable: The article proposes a maximum likelihood estimator for the shape parameters of this distribution.

Cited by 8 publications

References 12 publications

Biochemical Systems Theory: A Review

Biochemical Systems Theory: A Review

A new parametric method based on S‐distributions for computing receiver operating characteristic curves for continuous diagnostic tests

Random Number Generation from Right‐Skewed, Symmetric, and Left‐Skewed Distributions

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