Imperfect Debugging in Software Reliability
|
|
|
- Antony Cain
- 10 years ago
- Views:
Transcription
1 Imperfect Debugging in Software Reliability Tevfik Aktekin and Toros Caglar University of New Hampshire Peter T. Paul College of Business and Economics Department of Decision Sciences and United Health Care July 9, /26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
2 Outline Outline: What is Meant by Software Reliability and Debugging? Motivation and Uncertainties of Interest Very Brief Overview of Software Reliability Models The Proposed Model and Its Bayesian Inference Numerical Applications of the Proposed Model Investigating The Existence of Perfect vs. Imperfect Debugging in Software Data 2/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
3 What is Meant by Software Reliability? Definition Software Reliability: Can be defined as the probability of successful performance of software for a specified time interval under certain conditions. The fact that the software runs without any problems implies that the code is generating the intended output. A failure is said to occur on a piece of software, when a fault (or a so called bug) causes the software to fail. Note: If a failure does not occur for a period of time, this does not necessarily imply that the software is bug free. 3/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
4 Debugging Stage in Software Reliability The goal of software testing (i.e. debugging) is to detect/fix software faults (bugs) inherent in the software code and to decide when to release the software. Software testing is expensive. Thus, there is a trade-off between releasing a reasonably good piece of software and keeping to debug. There are many examples of buggy software being released and negatively affecting sales (first impression in the market in gaming software for instance). The testing stage consists of several consecutive program executions. Whenever a failure occurs, the software engineer attempts to fix the problem. As the faults reveal themselves and are eliminated by the software engineer, the reliability of the software tends to increase if no new faults are introduced to the software during the elimination stage. 4/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
5 Perfect vs. Imperfect Debugging Definition Perfect Debugging: If during the debugging stage the fault which caused the failure has been eliminated permanently and no new faults are introduced, then a perfect debugging is said to have occurred (software reliability gets better). Definition Imperfect Debugging: Loosely speaking, if during the debugging stage a fault is detected and is not eliminated permanently (ex: by introducing new faults), then an imperfect debugging is said to have occurred (software reliability stays the same or worsens). Note that there are many possible definitions of imperfect debugging (dealing with multiple faults vs. a single fault or the type of software being dealt with as in gaming vs. word processing). 5/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
6 What is Meant by Software Reliability? P(T θ) is usually referred to as a software reliability model, where T represents the time to failure of a piece of software. It represents the probability that a particular piece of software will fail in a given interval. P(T t θ) for some t 0 represents the reliability function of T. Note that here the notion of time is based on the mission time t, in other words the time during which the software is executed. 6/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
7 Statistical Motivation and Uncertainties of Interest Carry out inference on the unknown parameters such as the number of faults inherent in the code and fault detection rate after each stage of testing. Obtain the (predictive) reliability function after each debugging stage as a function of the parameters of interest. P(T i+1 t θ, D). (1) Making a decision on whether to stop the testing and release the software based on the reliability assessment of the software at time t. 7/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
8 Software Reliability Models in Literature The following two can be considered to be the building blocks for most of the current software reliability models: Jelinski and Moranda (JM) Model (1972) Each fault is permanently removed upon failure (perfect debugging). Each fault contributes equally to the failure rate at any stage of the testing. Littlewood and Verall (LV) Model (1973) Both assumptions from the JM Model have been relaxed by the LV Model; perfect debugging and equally likely contributions from faults at each stage of debugging. 8/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
9 Proposed Model Consider modeling of a multiplicative failure rate model whose components are evolving stochastically over testing stages (an NHPP type model). Proposed model is based on the Jelinski Moranda (JM) model as is the case for most subsequent work in the software reliability literature. Two of the main assumptions of the JM model is that every fault contributes equally to the failure rate at any stage of testing and that each fault is removed permanently upon failure. Consider the case where each fault is removed permanently upon failure, along with the possibility of introducing new faults during debugging. 9/26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
10 Proposed Model and Definitions t i,fori =1,...,N: time until failure during (or prior to) the ith stage of testing. φ i,fori =1,...,N: fault detection rate per fault during (or prior to) the ith stage of testing. λ i,fori =1,...,N: the number of faults present on the software code during (or prior to)the ith stage of testing. λ 1 : the number of faults present on the software code during (or prior to) the first stage of testing (i.e. prior o testing). Note that λ i swillbefunctionsofλ 1 and φ i s. φ i λ i : the failure rate of the software during (or prior to) the ith stage of testing. 10 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
11 Proposed Model Assume that the inter-failure times, t i, are exponentially distributed. The likelihood would be L(Φ, Λ; D) = N φ i λ i exp{ t i φ i λ i }, (2) i=1 where Φ = {φ 1,...,φ N },Λ={λ 1,...,λ N } and D = {t 1,...,t N }. Therefore the joint posterior of Φ and Λ would be given by p(φ, Λ D) N φ i λ i exp{ t i φ i λ i }p(φ)p(λ). (3) i=1 11 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
12 Modeling the fault detection rate per fault, φ i φ i s be given by the following power law relationship: φ i = φ β i 1 ν i, for i =1,...,N, (4) where ν i LN(0, σ 2 ). We can obtain the following the linear model in logarithms: log(φ i )=βlog(φ i 1 )+ɛ i, for i =1,...,N, (5) where ɛ i = log(ν i ). (5) is a first order autoregressive process of the latent fault detection rates per fault in the log scale. The conditional distributions of log(φ i )s can be written as log(φ i ) log(φ i 1 ),β N(βlog(φ i 1 ),σ 2 ), for i =1,...,N, (6) Note the scale of the inter-failure times (all above one in our numerical example). φ i s were all fractions and β was found to be between zero and one. To keep the interpretation of β consistent, it is always possible to rescale the data such that they are all above one. 12 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
13 Modeling the fault detection rate per fault, φ i The relationship implied by (4) also dictates the type of debugging that occurs during the ith debugging stage. If φ i <φ i 1, then perfect debugging is said to have occurred. If φ i φ i 1, then imperfect debugging is said to have occurred. In other words, when a failure is detected at the (i 1)th failure epoch, a fault has been detected and repaired, however a new fault was introduced during the same debugging stage. β determines on average how the fault detection rate per fault is changing from stage to stage. For instance, when 0 <φ i 1 < 1andβ>1then perfect debugging tends to occur, conversely when 0 <β<1 then imperfect debugging tends to occur. 13 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
14 Modeling the total number of faults, λ i Conditional on whether perfect or imperfect debugging has occurred during the previous debugging stage, the total number of faults left in the software code,λ i,isassumedtohavethefollowing structure λ i = λ i 1 γ i, for i =1,...,N (7) where γ i =1,withprobabilityp(φ i <φ i 1 ) =0,withprobabilityp(φ i φ i 1 )fori =1,...,N. In (7), γ i is a Bernoulli process whose probability of success is the probability of perfect debugging, p(φ i <φ i 1 ). When perfect debugging occurs, λ i goes down by one unit, since the fault that has caused the failure has been found and fixed. When imperfect debugging occurs, λ i stays the same, since the fault that has caused the failure has been found and fixed, however a new fault has been introduced while fixing the previous one. 14 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
15 Other Model Priors For the model on φ i s, and σ 2 Gamma(a, b) (8) β U(c, d) (9) and for the initial fault detection rate, φ 1,weassumethefollowing φ 1 LN(e, f ) (10) For the initial number of inherent faults, λ 1,weassumethe following λ 1 Poisson(θ) (11) with θ Gamma(g, h). (12) 15 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
16 Markov Chain Monte Carlo Estimation Goal: To generate samples from N p(φ,λ 1,θ,β,σ 2 D) φ i X i exp{ t i φ i λ i }p(φ β, σ 2 )p(λ 1 θ)p(β)p(σ 2 )p(θ). (13) i=1 In (13), the conditional joint prior distribution for φ i susing the chain rule and dropping terms that are independent can be obtained as p(φ β,σ 2 )=p(φ N φ N 1,β,σ 2 )...p(φ 2 φ 1,β,σ 2 )p(φ 1 ). (14) 16 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
17 Markov Chain Monte Carlo Estimation To generate the full conditionals p(φ i...,d)fori =1,...,N: Use Metropolis-Hastings. p(λ 1...,D): Discrete. In addition, one can compute λ j as λ j 1 1(φ j <φ j 1 )forj =2,...,N once we have the required samples. p(θ...,d): Gamma p(β...,d): Normal p(σ 2...,D): Use Metropolis-Hastings. 17 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
18 Markov Chain Monte Carlo Estimation To generate samples from p(φ,λ 1,θ,β,σ 2 D N ) 1 Assume the starting points (λ (0) 1,φ(0) 2,θ(0), (σ 2 ) (0),β (0) )andsetl=1. 2 Generate φ (l) 1 using λ (l 1) 1, φ (l) 2, β(l 1) and (σ 2 ) (l 1) from (φ 1...,D). 3 Sequentially generate φ (l) i φ (l) i 1 from (φ i...,d). for i =2,...,N using λ (l 1) 1, β (l 1), (σ 2 ) (l 1) and 4 Generate β (l) using (σ 2 ) (l 1) and φ (l) i for i =1,...,N from (β...,d). 5 Generate (σ 2 ) (l) using β (l) and φ (l) i for i =1,...,N from (σ 2...,D). 6 Generate λ (l) 1 using φ (l) i for i =1,...,N and θ (l 1) from (X 1...,D. 7 Sequentially compute λ (l) i for i =2,...,N using λ (l) i 1 via λ i = λ i 1 1(φ i <φ i 1 ). 8 Generate θ (l) using λ (l) 1 from (θ...,d N ). 9 Set l=l+1 and go back to step 1. and φ(l) i for i =1,...,N 18 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
19 Numerical Example Dataset 1: The numerical application of our model is carried out on the well known dataset first reported in JM The dataset consists of 31 software inter-failure times, 26 of which were obtained during the production stage of debugging and the remaining 5 during the rest of the testing stage. In our example, all 31 inter-failure times were used (most inference is based on this one). Dataset 2: The military systems application data (data 17) of John Musa of Bell Telephone Laboratories 1 with 38 inter-failure times. (only used for comparison purposes). Model Comparison: Use the harmonic mean estimator of the marginal likelihood and the DIC. Models to Compare Against: MS88-M1, MS88-M2, KY96-GOS-W, KY96-RVS-P and a simple perfect debugging (PD) model / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
20 Numerical Example ID PD MS88-M1 MS88-M2 KY96-GOS-W KY96-RVS-P log{p(d)} DIC Table: log{p(d)} and DIC for the JM dataset ID PD MS88-M1 MS88-M2 KY96-GOS-W KY96-RVS-P log{p(d)} DIC Table: log{p(d)} and DIC for the MUSA dataset 20 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
21 Summary of Findings Probability of Perfect Debugging φ i Debugging Stages Debugging Stages Figure: Boxplots of φ i for i =1,...,31 (left) and the probability of perfect debugging vs. debugging stages (right) 21 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
22 Summary of Findings Density Density β φ 1 Density Density θ X 1 Figure: Posterior distribution plots of β,φ 1,θ and λ 1 22 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
23 Summary of Findings X i Debugging Stages Figure: Boxplots of λ i for i =1,...,31 23 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
24 Summary of Findings Failure Rate Debugging Stages Figure: Boxplot of the failure rates, φ i λ i,fori =1,...,31 24 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
25 Predictive reliability function estimation The predictive reliability function during the i th testing stage (given that the software has gone through (i 1)th stages of testing) can be computed via R(t i D (i 1) )=1 1 S S j=1 F (t i φ (j) i,λ (j) i, D (i 1) ). (15) Once (15) is estimated it can easily be used as part of a software reliability optimal release scheme. Predictive Reliability Function After 28 Stages of Testing Predictive Reliability Function After 29 Stages of Testing Predictive Reliability Function After 30 Stages of Testing R(t D) R(t D) R(t D) t t t Figure: Predictive reliability functions for i =29,...,31 25 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
26 Concluding Remarks and Future Work Introduce a Markov chain type of structure on the number of bugs repaired or introduced during the debugging stage instead of the current Bernoulli setup. Investigate the possibility of state space evolution of the β coefficient in the power law relationship between the inter-failure times. Note: Both extensions would be challenging from an MCMC estimation point of view. 26 / 26 Tevfik Aktekin and Toros Caglar Imperfect Debugging in Software Reliability
Bayesian Statistics in One Hour. Patrick Lam
Bayesian Statistics in One Hour Patrick Lam Outline Introduction Bayesian Models Applications Missing Data Hierarchical Models Outline Introduction Bayesian Models Applications Missing Data Hierarchical
Probabilistic Models for Big Data. Alex Davies and Roger Frigola University of Cambridge 13th February 2014
Probabilistic Models for Big Data Alex Davies and Roger Frigola University of Cambridge 13th February 2014 The State of Big Data Why probabilistic models for Big Data? 1. If you don t have to worry about
11. Time series and dynamic linear models
11. Time series and dynamic linear models Objective To introduce the Bayesian approach to the modeling and forecasting of time series. Recommended reading West, M. and Harrison, J. (1997). models, (2 nd
Gaussian Processes to Speed up Hamiltonian Monte Carlo
Gaussian Processes to Speed up Hamiltonian Monte Carlo Matthieu Lê Murray, Iain http://videolectures.net/mlss09uk_murray_mcmc/ Rasmussen, Carl Edward. "Gaussian processes to speed up hybrid Monte Carlo
Markov Chain Monte Carlo Simulation Made Simple
Markov Chain Monte Carlo Simulation Made Simple Alastair Smith Department of Politics New York University April2,2003 1 Markov Chain Monte Carlo (MCMC) simualtion is a powerful technique to perform numerical
Modeling and Analysis of Call Center Arrival Data: A Bayesian Approach
Modeling and Analysis of Call Center Arrival Data: A Bayesian Approach Refik Soyer * Department of Management Science The George Washington University M. Murat Tarimcilar Department of Management Science
α α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =
Tutorial on Markov Chain Monte Carlo
Tutorial on Markov Chain Monte Carlo Kenneth M. Hanson Los Alamos National Laboratory Presented at the 29 th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Technology,
Message-passing sequential detection of multiple change points in networks
Message-passing sequential detection of multiple change points in networks Long Nguyen, Arash Amini Ram Rajagopal University of Michigan Stanford University ISIT, Boston, July 2012 Nguyen/Amini/Rajagopal
PS 271B: Quantitative Methods II. Lecture Notes
PS 271B: Quantitative Methods II Lecture Notes Langche Zeng [email protected] The Empirical Research Process; Fundamental Methodological Issues 2 Theory; Data; Models/model selection; Estimation; Inference.
Simple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
CHAPTER 2 Estimating Probabilities
CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a
Web-based Supplementary Materials for Bayesian Effect Estimation. Accounting for Adjustment Uncertainty by Chi Wang, Giovanni
1 Web-based Supplementary Materials for Bayesian Effect Estimation Accounting for Adjustment Uncertainty by Chi Wang, Giovanni Parmigiani, and Francesca Dominici In Web Appendix A, we provide detailed
Big Data, Statistics, and the Internet
Big Data, Statistics, and the Internet Steven L. Scott April, 4 Steve Scott (Google) Big Data, Statistics, and the Internet April, 4 / 39 Summary Big data live on more than one machine. Computing takes
Supplement to Call Centers with Delay Information: Models and Insights
Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page
Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8
C. Wohlin, M. Höst, P. Runeson and A. Wesslén, "Software Reliability", in Encyclopedia of Physical Sciences and Technology (third edition), Vol.
C. Wohlin, M. Höst, P. Runeson and A. Wesslén, "Software Reliability", in Encyclopedia of Physical Sciences and Technology (third edition), Vol. 15, Academic Press, 2001. Software Reliability Claes Wohlin
Basics of Statistical Machine Learning
CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu [email protected] Modern machine learning is rooted in statistics. You will find many familiar
Credit Risk Models: An Overview
Credit Risk Models: An Overview Paul Embrechts, Rüdiger Frey, Alexander McNeil ETH Zürich c 2003 (Embrechts, Frey, McNeil) A. Multivariate Models for Portfolio Credit Risk 1. Modelling Dependent Defaults:
Inference on Phase-type Models via MCMC
Inference on Phase-type Models via MCMC with application to networks of repairable redundant systems Louis JM Aslett and Simon P Wilson Trinity College Dublin 28 th June 202 Toy Example : Redundant Repairable
A Latent Variable Approach to Validate Credit Rating Systems using R
A Latent Variable Approach to Validate Credit Rating Systems using R Chicago, April 24, 2009 Bettina Grün a, Paul Hofmarcher a, Kurt Hornik a, Christoph Leitner a, Stefan Pichler a a WU Wien Grün/Hofmarcher/Hornik/Leitner/Pichler
A Bootstrap Metropolis-Hastings Algorithm for Bayesian Analysis of Big Data
A Bootstrap Metropolis-Hastings Algorithm for Bayesian Analysis of Big Data Faming Liang University of Florida August 9, 2015 Abstract MCMC methods have proven to be a very powerful tool for analyzing
Hedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15
Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.
3. Regression & Exponential Smoothing
3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a
The Dirichlet-Multinomial and Dirichlet-Categorical models for Bayesian inference
The Dirichlet-Multinomial and Dirichlet-Categorical models for Bayesian inference Stephen Tu [email protected] 1 Introduction This document collects in one place various results for both the Dirichlet-multinomial
Binomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
Financial Market Microstructure Theory
The Microstructure of Financial Markets, de Jong and Rindi (2009) Financial Market Microstructure Theory Based on de Jong and Rindi, Chapters 2 5 Frank de Jong Tilburg University 1 Determinants of the
Model Selection and Claim Frequency for Workers Compensation Insurance
Model Selection and Claim Frequency for Workers Compensation Insurance Jisheng Cui, David Pitt and Guoqi Qian Abstract We consider a set of workers compensation insurance claim data where the aggregate
STA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! [email protected]! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct
A Bayesian Antidote Against Strategy Sprawl
A Bayesian Antidote Against Strategy Sprawl Benjamin Scheibehenne ([email protected]) University of Basel, Missionsstrasse 62a 4055 Basel, Switzerland & Jörg Rieskamp ([email protected])
A LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA
REVSTAT Statistical Journal Volume 4, Number 2, June 2006, 131 142 A LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA Authors: Daiane Aparecida Zuanetti Departamento de Estatística, Universidade Federal de São
Part 2: One-parameter models
Part 2: One-parameter models Bernoilli/binomial models Return to iid Y 1,...,Y n Bin(1, θ). The sampling model/likelihood is p(y 1,...,y n θ) =θ P y i (1 θ) n P y i When combined with a prior p(θ), Bayes
Maximum Likelihood Estimation
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for
6.0 RELIABILITY ALLOCATION
6.0 RELIABILITY ALLOCATION Reliability Allocation deals with the setting of reliability goals for individual subsystems such that a specified reliability goal is met and the hardware and software subsystem
Spatial Statistics Chapter 3 Basics of areal data and areal data modeling
Spatial Statistics Chapter 3 Basics of areal data and areal data modeling Recall areal data also known as lattice data are data Y (s), s D where D is a discrete index set. This usually corresponds to data
Bayesian Statistics: Indian Buffet Process
Bayesian Statistics: Indian Buffet Process Ilker Yildirim Department of Brain and Cognitive Sciences University of Rochester Rochester, NY 14627 August 2012 Reference: Most of the material in this note
Factorial experimental designs and generalized linear models
Statistics & Operations Research Transactions SORT 29 (2) July-December 2005, 249-268 ISSN: 1696-2281 www.idescat.net/sort Statistics & Operations Research c Institut d Estadística de Transactions Catalunya
Using SAS PROC MCMC to Estimate and Evaluate Item Response Theory Models
Using SAS PROC MCMC to Estimate and Evaluate Item Response Theory Models Clement A Stone Abstract Interest in estimating item response theory (IRT) models using Bayesian methods has grown tremendously
Statistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
Christfried Webers. Canberra February June 2015
c Statistical Group and College of Engineering and Computer Science Canberra February June (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 829 c Part VIII Linear Classification 2 Logistic
Bayesian coalescent inference of population size history
Bayesian coalescent inference of population size history Alexei Drummond University of Auckland Workshop on Population and Speciation Genomics, 2016 1st February 2016 1 / 39 BEAST tutorials Population
Package zic. September 4, 2015
Package zic September 4, 2015 Version 0.9 Date 2015-09-03 Title Bayesian Inference for Zero-Inflated Count Models Author Markus Jochmann Maintainer Markus Jochmann
Centre for Central Banking Studies
Centre for Central Banking Studies Technical Handbook No. 4 Applied Bayesian econometrics for central bankers Andrew Blake and Haroon Mumtaz CCBS Technical Handbook No. 4 Applied Bayesian econometrics
Model-based Synthesis. Tony O Hagan
Model-based Synthesis Tony O Hagan Stochastic models Synthesising evidence through a statistical model 2 Evidence Synthesis (Session 3), Helsinki, 28/10/11 Graphical modelling The kinds of models that
Recent Developments of Statistical Application in. Finance. Ruey S. Tsay. Graduate School of Business. The University of Chicago
Recent Developments of Statistical Application in Finance Ruey S. Tsay Graduate School of Business The University of Chicago Guanghua Conference, June 2004 Summary Focus on two parts: Applications in Finance:
Basic Bayesian Methods
6 Basic Bayesian Methods Mark E. Glickman and David A. van Dyk Summary In this chapter, we introduce the basics of Bayesian data analysis. The key ingredients to a Bayesian analysis are the likelihood
1 Prior Probability and Posterior Probability
Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which
Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback
Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback Juho Kanniainen Tampere University of Technology New Thinking in Finance 12 Feb. 2014, London Based on J. Kanniainen and R. Piche,
CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
![CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.](/thumbs/40/21160284.jpg "CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.")
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
Confidence Intervals for Exponential Reliability
Chapter 408 Confidence Intervals for Exponential Reliability Introduction This routine calculates the number of events needed to obtain a specified width of a confidence interval for the reliability (proportion
The Exponential Distribution
21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding
Bayesian Approaches to Handling Missing Data
Bayesian Approaches to Handling Missing Data Nicky Best and Alexina Mason BIAS Short Course, Jan 30, 2012 Lecture 1. Introduction to Missing Data Bayesian Missing Data Course (Lecture 1) Introduction to
CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York
BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal - the stuff biology is not
Introduction to Markov Chain Monte Carlo
Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem
Analysis of Bayesian Dynamic Linear Models
Analysis of Bayesian Dynamic Linear Models Emily M. Casleton December 17, 2010 1 Introduction The main purpose of this project is to explore the Bayesian analysis of Dynamic Linear Models (DLMs). The main
Multiple Choice Models II
Multiple Choice Models II Laura Magazzini University of Verona [email protected] http://dse.univr.it/magazzini Laura Magazzini (@univr.it) Multiple Choice Models II 1 / 28 Categorical data Categorical
Math 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
Section 5. Stan for Big Data. Bob Carpenter. Columbia University
Section 5. Stan for Big Data Bob Carpenter Columbia University Part I Overview Scaling and Evaluation data size (bytes) 1e18 1e15 1e12 1e9 1e6 Big Model and Big Data approach state of the art big model
CHAPTER 3 EXAMPLES: REGRESSION AND PATH ANALYSIS
Examples: Regression And Path Analysis CHAPTER 3 EXAMPLES: REGRESSION AND PATH ANALYSIS Regression analysis with univariate or multivariate dependent variables is a standard procedure for modeling relationships
MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...
MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................
The Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014
Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA 1.
3F3: Signal and Pattern Processing
3F3: Signal and Pattern Processing Lecture 3: Classification Zoubin Ghahramani [email protected] Department of Engineering University of Cambridge Lent Term Classification We will represent data by
GLMs: Gompertz s Law. GLMs in R. Gompertz s famous graduation formula is. or log µ x is linear in age, x,
Computing: an indispensable tool or an insurmountable hurdle? Iain Currie Heriot Watt University, Scotland ATRC, University College Dublin July 2006 Plan of talk General remarks The professional syllabus
Generating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010
Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 Quasi-Monte
Chapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 )
Chapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 ) and Neural Networks( 類 神 經 網 路 ) 許 湘 伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) LR Chap 10 1 / 35 13 Examples
Monte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
I. Basic concepts: Buoyancy and Elasticity II. Estimating Tax Elasticity III. From Mechanical Projection to Forecast
Elements of Revenue Forecasting II: the Elasticity Approach and Projections of Revenue Components Fiscal Analysis and Forecasting Workshop Bangkok, Thailand June 16 27, 2014 Joshua Greene Consultant IMF-TAOLAM
More details on the inputs, functionality, and output can be found below.
Overview: The SMEEACT (Software for More Efficient, Ethical, and Affordable Clinical Trials) web interface (http://research.mdacc.tmc.edu/smeeactweb) implements a single analysis of a two-armed trial comparing
Linda K. Muthén Bengt Muthén. Copyright 2008 Muthén & Muthén www.statmodel.com. Table Of Contents
Mplus Short Courses Topic 2 Regression Analysis, Eploratory Factor Analysis, Confirmatory Factor Analysis, And Structural Equation Modeling For Categorical, Censored, And Count Outcomes Linda K. Muthén
2. Linear regression with multiple regressors
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions
Parallelization Strategies for Multicore Data Analysis
Parallelization Strategies for Multicore Data Analysis Wei-Chen Chen 1 Russell Zaretzki 2 1 University of Tennessee, Dept of EEB 2 University of Tennessee, Dept. Statistics, Operations, and Management
Course 4 Examination Questions And Illustrative Solutions. November 2000
Course 4 Examination Questions And Illustrative Solutions Novemer 000 1. You fit an invertile first-order moving average model to a time series. The lag-one sample autocorrelation coefficient is 0.35.
BayesX - Software for Bayesian Inference in Structured Additive Regression
BayesX - Software for Bayesian Inference in Structured Additive Regression Thomas Kneib Faculty of Mathematics and Economics, University of Ulm Department of Statistics, Ludwig-Maximilians-University Munich
Bayesian Analysis for the Social Sciences
Bayesian Analysis for the Social Sciences Simon Jackman Stanford University http://jackman.stanford.edu/bass November 9, 2012 Simon Jackman (Stanford) Bayesian Analysis for the Social Sciences November
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I February
Statistics Graduate Courses
Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
Principle of Data Reduction
Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then
Electronic Theses and Dissertations UC Riverside
Electronic Theses and Dissertations UC Riverside Peer Reviewed Title: Bayesian and Non-parametric Approaches to Missing Data Analysis Author: Yu, Yao Acceptance Date: 01 Series: UC Riverside Electronic
A General Approach to Variance Estimation under Imputation for Missing Survey Data
A General Approach to Variance Estimation under Imputation for Missing Survey Data J.N.K. Rao Carleton University Ottawa, Canada 1 2 1 Joint work with J.K. Kim at Iowa State University. 2 Workshop on Survey
Final Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin
Final Mathematics 51, Section 1, Fall 24 Instructor: D.A. Levin Name YOU MUST SHOW YOUR WORK TO RECEIVE CREDIT. A CORRECT ANSWER WITHOUT SHOWING YOUR REASONING WILL NOT RECEIVE CREDIT. Problem Points Possible
Alessandro Birolini. ineerin. Theory and Practice. Fifth edition. With 140 Figures, 60 Tables, 120 Examples, and 50 Problems.
Alessandro Birolini Re ia i it En ineerin Theory and Practice Fifth edition With 140 Figures, 60 Tables, 120 Examples, and 50 Problems ~ Springer Contents 1 Basic Concepts, Quality and Reliability Assurance
Lecture notes: single-agent dynamics 1
Lecture notes: single-agent dynamics 1 Single-agent dynamic optimization models In these lecture notes we consider specification and estimation of dynamic optimization models. Focus on single-agent models.
Integrated Resource Plan
Integrated Resource Plan March 19, 2004 PREPARED FOR KAUA I ISLAND UTILITY COOPERATIVE LCG Consulting 4962 El Camino Real, Suite 112 Los Altos, CA 94022 650-962-9670 1 IRP 1 ELECTRIC LOAD FORECASTING 1.1
Package EstCRM. July 13, 2015
Version 1.4 Date 2015-7-11 Package EstCRM July 13, 2015 Title Calibrating Parameters for the Samejima's Continuous IRT Model Author Cengiz Zopluoglu Maintainer Cengiz Zopluoglu
Reject Inference in Credit Scoring. Jie-Men Mok
Reject Inference in Credit Scoring Jie-Men Mok BMI paper January 2009 ii Preface In the Master programme of Business Mathematics and Informatics (BMI), it is required to perform research on a business
Chapter 2: Binomial Methods and the Black-Scholes Formula
Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the
The Kelly criterion for spread bets
IMA Journal of Applied Mathematics 2007 72,43 51 doi:10.1093/imamat/hxl027 Advance Access publication on December 5, 2006 The Kelly criterion for spread bets S. J. CHAPMAN Oxford Centre for Industrial