1,720,959 research outputs found
An In-Depth Perspective on the Classical Model
No full textThe Classical Model (CM) or Cooke’s method for performing Structured Expert Judgement (SEJ) is the best-known method that promotes expert performance evaluation when aggregating experts’ assessments of uncertain quantities. Assessing experts’ performance in quantifying uncertainty involves two scores in CM, the calibration score (or statistical accuracy) and the information score. The two scores combine into overall scores, which, in turn, yield weights for a performance-based aggregation of experts’ opinions. The method is fairly demanding, and therefore carrying out a SEJ elicitation with CM requires careful consideration. This chapter aims to address the methodological and practical aspects of CM into a comprehensive overview of the CM elicitation process. It complements the chapter “Elicitation in the Classical Model” in the book Elicitation (Quigley et al. 2018). Nonetheless, we regard this chapter as a stand-alone material, hence some concepts and definitions will be repeated, for the sake of completeness.Applied Probabilit
Decision-Making in Early Internationalization: A Structured Expert Judgement Approach
No full textThe aim of this chapter is to show how a structured approach to elicit expert judgement (SEJ) can guide the practice of early internationalization. We applied SEJ to forecast some critical issues upon which an innovative start-up wished to base their decision of whether to expand their initial operations in Poland and Czech Republic to Brazil. Sixteen participants of an Executive MBA program acted as experts and underwent the procedure for eliciting their judgements. The performance of experts was quantified in terms of statistical accuracy and informativeness, which were combined to provide a performance-based weight for each expert according to Classical Model. The combination of weighted expert judgements led to improved statistical accuracy and informativeness of the forecast. The procedure demonstrates how entrepreneurs can take advantage of expert knowledge in deciding about risky endeavours when lacking their own experiences and reliable data that can guide their choices.Accepted author manuscriptApplied Probabilit
Een analyse van de bootstrap binnen eindige populaties
No full textElectrical Engineering, Mathematics and Computer ScienceDelft Institute of Applied MathematicsTW305
Shape Constrained Nonparametric Estimation in the Cox Model
No full textThe events of interest in any survival analysis study are regularly subject to censoring. There are various censoring schemes, including right or left censoring, and interval censoring. The most frequent censoring scheme is the right censoring, where subjects might drop out of the study or simply because not all events of interest occur before the end of the study. Moreover, for each subject, additional information referred to as covariates is registered at the beginning or throughout the study, such as age, sex, undergoing treatment, etc. The classical model to study the distribution of the events of interest, while accounting for additional information, is the Cox model. The Cox model expresses the hazard function of a subject given a set of covariates in terms of a baseline hazard, for which all covariates are zero, and an exponential function of the covariates and corresponding regression parameters. The baseline hazard can be left completely unspecified while estimating the regression parameters. Nonetheless, in practice, there are numerous studies in which the baseline hazard appears to be monotone. Time to death or to the onset of a disease are observed to have a nondecreasing baseline hazard, while the survival or recovery time after a successful medical treatment usually exhibit a nonincreasing baseline hazard. The aim of this thesis is to study the behavior of nonparametric baseline hazard and baseline density estimators in the Cox model under monotonicity constraints. The event times are assumed to be right censored and the censoring mechanism is assumed to be independent of the event of interest and non-informative. The covariates are assumed to be time-independent, usually recorded at the beginning of the study. In addition to point estimates, interval estimates of a monotone baseline hazard will be provided, based on a likelihood ratio method, along with testing at a fixed point. Furthermore, kernel smoothed estimates of a monotone baseline hazard will be defined and their behavior will be investigated. In Chapter 2, we propose several nonparametric monotone estimators of a baseline hazard or a baseline density within the Cox model. We derive the nonparametric maximum likelihood estimator of a nondecreasing baseline hazard and we consider a Grenander-type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. The two estimators are then shown to be strongly consistent and asymptotically equivalent. Moreover, we derive their common limit distribution at a fixed point. The two equivalent estimators of a nonincreasing baseline hazard and their asymptotic properties are acquired similarly. Furthermore, we introduce a Grenander-type estimator of a nonincreasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function derived from the Breslow estimator. This estimator is proven to be strongly consistent and its asymptotic distribution at a fixed point is derived. Chapter 3 provides an asymptotic linear representation of the Breslow estimator of the baseline cumulative hazard function in the Cox model. This representation can be used to derive the asymptotic distribution of the Grenander type estimator of a monotone baseline hazard estimator. The representation consists of an average of independent random variables and a term involving the difference between the maximum partial likelihood estimator and the underlying regression parameter. The order of the remainder term is arbitrarily close to n^-1. Chapter 4 focuses on interval estimation and on testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point, via a likelihood ratio method. Nonparametric maximum likelihood estimators under the null hypothesis are defined for both nondecreasing and nonincreasing baseline hazard functions. These characterizations, along with those of the monotone nonparametric maximum likelihood estimators provide the asymptotic distribution of the likelihood ratio test. This asymptotic distribution enables, via inversion, the construction of pointwise confidence intervals. This method of constructing confidence intervals avoids the issue of estimating the nuisance parameters, as in the case of confidence intervals based on the asymptotic distribution of the estimators. Simulations indicate that the two methods yield confidence intervals with comparable coverage probabilities. Nonetheless, the confidence intervals based on the likelihood ratio are smaller, on average. Finally, in chapter 5 we consider smooth baseline hazard estimators. The estimators are obtained by kernel smoothing the maximum likelihood and Grenander-type estimators of a monotone baseline hazard function. Three different estimators are proposed for a nondecreasing baseline hazard, which are provided by the interchange of the smoothing and isotonization step. With this respect, we define a smoothed maximum likelihood estimator (SMLE), as well as a smoothed Grenander type (SG) estimator and a Grenander type smoothed (GS) estimator. All estimators are shown to be strongly pointwise or uniformly consistent.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Calibrating experts’ probabilistic assessments for improved probabilistic predictions
No full textExpert judgement is routinely required to inform critically important decisions. While expert judgement can be remarkably useful when data are absent, it can be easily influenced by contextual biases which can lead to poor judgements and subsequently poor decisions. Structured elicitation protocols aim to: (1) guard against biases and provide better (aggregated) judgements, and (2) subject expert judgements to the same level of scrutiny as is expected for empirical data. The latter ensures that if judgements are to be used as data, they are subject to the scientific principles of review, critical appraisal, and repeatability. Objectively evaluating the quality of expert data and validating expert judgements are other essential elements. Considerable research suggests that the performance of experts should be evaluated by scoring experts on questions related to the elicitation questions, whose answers are known a priori. Experts who can provide accurate, well-calibrated and informative judgements should receive more weight in a final aggregation of judgements. This is referred to as performance-weighting in the mathematical aggregation of multiple judgements. The weights depend on the chosen measures of performance. We are yet to understand the best methods to aggregate judgements, how well such aggregations perform out of sample, or the costs involved, as well as the benefits of the various approaches. In this paper we propose and explore a new measure of experts’ calibration. A sizeable data set containing predictions for outcomes of geopolitical events is used to investigate the properties of this calibration measure when compared to other, well established measures.</p
Exploring the Effect of Model Assumptions on Prediction Performance of Bayesian Networks
No full textThis thesis concerns itself with the effect of the normality assumption, the effects of discretisation choices and other assumptions made by software on prediction performance, when using Gaussian Bayesian Networks. To test these effects, different types of Bayesian Networks are constructed and made to perform predictions, using the same dataset. The dataset used contains records regarding the citations and other bibliometric statistics of articles published by authors aliated with the Delft University of Technology between 2010 and 2014. The first model is a Gaussian Bayesian Network (GBN), which assumes that the conditional probability distributions (CPD) of all variables concerned are Gaussian. The second model is a Multinomial Bayesian Network (MBN), which uses discrete variables. To accommodate to this model, the data is discretized. The third model is the Hybrid Bayesian Network (HBN, which can handle both discrete and continuous data and has no normality assumption on the distribution of the variables. The last model is a non-parametric Bayesian Network (NPBN). To compare the different models, they are used to perform a set of predictions in the form of quantile estimation. The results show that the GBN performs as well as the NPBN, when looking at the bulk of the data. When looking at data, where the mean citation score (mcs), the predicted variable, exceeds the 75-% quantile, the performance of the GBN becomes much worse than that of the NPBN.Applied Mathematic
The robustness of Bayesian networks: Robustheid van Bayesiaanse netwerken
No full textApplying multiple structure learning algorithms like hill-climbing, incremental association and grow-shrink, to investigate their robustness, predictive capabiliy and goodness of fit on multiple discrete Bayesian networks and Gaussian Bayesian networks
Dating blood traces using Bayesian Networks
No full textAt crime scenes, various methods can be used to determine how much time has passed since the act happened, for example a body left at the scene, camera footage or eye witnesses. In this thesis, however, a different method shall be used to determine the time of the crime. The time will be determined using blood evidence. To be exact, the goal of this thesis is to analyse the aging of bloodstains through colour analysis, and constructing a Bayesian network (BN) which can accurately make predictions on the time of deposition of a bloodstain. The data used for the construction of the BNs was obtained from images provided by the Leiden Institute of Physics (LION). To obtain the data, the bloodstains in the images first need to be isolated from the background. Afterwards, the bloodstains are split into two parts: the inner part of the bloodstain and the complete bloodstain. Both the isolation and the splitting are done using a method called masking. Afterwards they can be converted to RGB (red, green, blue) values and using these RGB values, the following data can be collected for each colour channel of both the inner part of the bloodstain and the whole bloodstain: mean, min, max, variance, and the 5%, 20%, 50%, 80% and 95% quantiles.Using various subsets of the data, BNs can be constructed. The structure of the BN is determined using a structure-learning algorithm called hill-climbing . To determine the validity of a given structure k-fold cross-validation can be performed using a given loss function. In this thesis, k=5 has been used, and the Mean Squared Error (MSE) has been taken as the loss function. Upon comparison of the MSE, it seems that the best model is given by the red values of a bloodstain. However, even the best performing model found in this thesis still has a considerably poor performance as the BN for the red variables has an MSE of 15149.81.Applied Mathematic
Optimize the indescribable: A Look at the Unification between Machine Learning and Optimization
No full textPackages to encode Machine Learned models into optimization problems is an underdeveloped area, despite the advantages is could provide. The main draw of implementing Machine Learned models into optimization models, is that it allows the optimizer to better account for the human experience.Maragno D., Wiberg H. et al. constructed an implementation of the encoding with their package OptiCL. In order to verify their implementation and provide principles for (re)designing packages with similar functions, an amount of components of OptiCL were replicated within this paper. The requirements forthe program were first constructed before detailing the implementation process. After the program was implemented, both OptiCL and the found program were tested in order to compare performances. Using the results and an investigation of the two implementations, a framework for encoding similar packageswas provided using the insights gained. Using mathematical formulations supplied by Maragno D., Wiberg H. et al., design principles outlined in this report and research into the encoding of other Machine Learned models, other developers could construct robust packages that allow for easy integration ofvaluable information gained from Machine Learning into optimization problems. This in turn allows for frequently used optimization models to account for more human understanding.Computer Scienc
Selections of vine structures and their applications
No full textCopulas are important models that allow to capture the dependence among variables. There are many types of bivariate parametric copula families, which allow to model data sets with different properties: symmetric and asymmetric dependence, upper (lower) tail dependence. In higher dimensions popular families of copulas, e.g., Gaussian, Student-t and canonical Archimedean are not sufficiently flexible in representing different types of dependence that they can realize. By decomposing the multivariate copula into a sequence of bivariate (conditional) copulas, based on a graph called vine (which is a nested set of trees), one is able to construct a n dimensional copula with the bivariate copulas that can have different types of dependence (e.g., tail behavior and asymmetries). The model constructed this way is called the vine copulamodel...Applied Probabilit
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