DOCSLIB.ORG

Generating Random Variates

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generating Random Variates CHAPTER 8 Generating Random Variates 8.1 Introduction................................................................................................2 8.2 General Approaches to Generating Random Variates ....................................3 8.2.1 Inverse Transform...............................................................................................................3 8.2.2 Composition.....................................................................................................................11 8.2.3 Convolution......................................................................................................................16 8.2.4 Acceptance-Rejection.......................................................................................................18 8.2.5 Special Properties.............................................................................................................26 8.3 Generating Continuous Random Variates....................................................27 8.4 Generating Discrete Random Variates.........................................................27 8.4.3 Arbitrary Discrete Distribution...........................................................................................28 8.5 Generating Random Vectors, Correlated Random Variates, and Stochastic Processes ........................................................................................................34 8.5.1 Using Conditional Distributions ..........................................................................................35 8.5.2 Multivariate Normal and Multivariate Lognormal................................................................36 8.5.3 Correlated Gamma Random Variates ................................................................................37 8.5.4 Generating from Multivariate Families ................................................................................39 8.5.5 Generating Random Vectors with Arbitrarily Specified Marginal Distributions and Correlations ...................................................................................................................................................40 8.5.6 Generating Stochastic Processes........................................................................................41 8.6 Generating Arrival Processes .....................................................................42 8.6.1 Poisson Process................................................................................................................42 8.6.2 Nonstationary Poisson Process..........................................................................................43 8.6.3 Batch Arrivals ...................................................................................................................50 8-1 8.1 Introduction Algorithms to produce observations (“variates”) from some desired input distribution (exponential, gamma, etc.) Formal algorithm—depends on desired distribution But all algorithms have the same general form: Generate one or more Transformation Return X ~ desired IID U(0, 1) random (depends on desired distribution numbers distribution) Note critical importance of a good random-number generator (Chap. 7) May be several algorithms for a desired input distribution form; want: Exact: X has exactly (not approximately) the desired distribution Example of approximate algorithm: Treat Z = U1 + U2 + ... + U12 – 6 as N(0, 1) Mean, variance correct; rely on CLT for approximate normality Range clearly incorrect Efficient: Low storage Fast (marginal, setup) Efficient regardless of parameter values (robust) Simple: Understand, implement (often tradeoff against efficiency) Requires only U(0, 1) input One U ® one X (if possible—for speed, synchronization in variance reduction) 8-2 8.2 General Approaches to Generating Random Variates Five general approaches to generating a univariate RV from a distribution: Inverse transform Composition Convolution Acceptance-rejection Special properties 8.2.1 Inverse Transform Simplest (in principle), “best” method in some ways; known to Gauss Continuous Case Suppose X is continuous with cumulative distribution function (CDF) F(x) = P(X £ x) for all real numbers x that is strictly increasing over all x Algorithm: 1. Generate U ~ U(0, 1) (random-number generator) 2. Find X such that F(X) = U and return this value X Step 2 involves solving the equation F(X) = U for X; the solution is written X = F–1(U), i.e., we must invert the CDF F Inverting F might be easy (exponential), or difficult (normal) in which case numerical methods might be necessary (and worthwhile—can be made “exact” up to machine accuracy) 8-3 Proof: (Assume F is strictly increasing for all x.) For a fixed value x0, –1 (def. of X in algorithm) P(returned X is £ x0) = P(F (U) £ x0) -1 = P(F(F (U)) £ F(x0)) (F is monotone •) (def. of inverse function) = P(U £ F(x0)) = P(0 £ U £ F(x0)) (U ³ 0 for sure) = F(x0) – 0 (U ~ U(0,1)) = F(x0) (as desired) Proof by Picture: x0 Pick a fixed value x0 X1 £ x0 if and only if U1 £ F(x0), so P(X1 £ x0) = P(U1 £ F(x0)) = F(x0), by definition of CDFs 8-4 Example of Continuous Inverse-Transform Algorithm Derivation Weibull (a, b) distribution, parameters a > 0 and b > 0 a ìab -a xa-1e-(x/ b ) if x > 0 Density function is f (x) = í î0 otherwise a x ì1 - e -(x/ b ) if x > 0 CDF is F(x) = ò f (t)dt = í -¥ î0 otherwise Solve U = F(X) for X: a U = 1 - e -(X / b ) a e -( X / b ) = 1 - U - (X / b )a = ln(1 - U ) X / b = [- ln(1 - U )]1 / a X = b[- ln(1 -U )]1 /a Since 1 – U ~ U(0, 1) as well, can replace 1 – U by U to get the final algorithm: 1. Generate U ~ U(0, 1) 2. Return X = b (- ln U )1 / a 8-5 Intuition Behind Inverse-Transform Method (Weibull (a = 1.5, b = 6) example) 8-6 The algorithm in action: 8-7 Discrete Case Suppose X is discrete with cumulative distribution function (CDF) F(x) = P(X £ x) for all real numbers x and probability mass function p(xi) = P(X = xi), where x1, x2, ... are the possible values X can take on Algorithm: 1. Generate U ~ U(0,1) (random-number generator) 2. Find the smallest positive integer I such that U £ F(xI) 3. Return X = xI Step 2 involves a “search” of some kind; several computational options: Direct left-to-right search—if p(xi)’s fairly constant If p(xi)’s vary a lot, first sort into decreasing order, look at biggest one first, ..., smallest one last—more likely to terminate quickly Exploit special properties of the form of the p(xi)’s, if possible Unlike the continuous case, the discrete inverse-transform method can always be used for any discrete distribution (but it may not be the most efficient approach) Proof: From the above picture, P(X = xi) = p(xi) in every case 8-8 Example of Discrete Inverse-Transform Method Discrete uniform distribution on 1, 2, ..., 100 xi = i, and p(xi) = p(i) = P(X = i) = 0.01 for i = 1, 2, ..., 100 F (x ) 1.00 0.99 0.98 0.97 0.03 0.02 0.01 x 0 1 2 3 98 99 100 “Literal” inverse transform search: 1. Generate U ~ U(0,1) 2. If U £ 0.01 return X = 1 and stop; else go on 3. If U £ 0.02 return X = 2 and stop; else go on 4. If U £ 0.03 return X = 3 and stop; else go on . 100. If U £ 0.99 return X = 99 and stop; else go on 101. Return X = 100 Equivalently (on a U-for-U basis): 1. Generate U ~ U(0,1) 2. Return X = ë100 Uû + 1 8-9 Generalized Inverse-Transform Method Valid for any CDF F(x): return X = min{x: F(x) ³ U}, where U ~ U(0,1) Continuous, possibly with flat spots (i.e., not strictly increasing) Discrete Mixed continuous-discrete Problems with Inverse-Transform Approach Must invert CDF, which may be difficult (numerical methods) May not be the fastest or simplest approach for a given distribution Advantages of Inverse-Transform Approach Facilitates variance-reduction techniques; an example: Study effect of speeding up a bottleneck machine Compare current machine with faster one To model speedup: Change service-time distribution for this machine Run 1 (baseline, current system): Use inverse transform to generate service-time variates for existing machine Run 2 (with faster machine): Use inverse transform to generate service-time variates for proposed machine Use the very same uniform U(0,1) random numbers for both variates: Service times will be positively correlated with each other Inverse-transform makes this correlation as strong as possible, in comparison with other variate-generation methods Reduces variability in estimate of effect of faster machine Main reason why inverse transform is often regarded as “the best” method Common random numbers Can have big effect on estimate quality (or computational effort required) Generating from truncated distributions (pp. 447-448 and Problem 8.4 of SMA) Generating order statistics without sorting, for reliability models: Y1, Y2, ..., Yn IID ~ F; want Y(i) — directly, generate Yi’s, and sort –1 Alternatively, return Y(i) = F (V) where V ~ beta(i, n – i + 1) 8-10 8.2.2 Composition Want to generate from CDF F, but inverse transform is difficult or slow Suppose we can find other CDFs F1, F2, ... (finite or infinite list) and weights p1, p2, ... (pj ³ 0 and p1 + p2 + ... = 1) such that for all x, F(x) = p1F1(x) + p2F2(x) + ... (Equivalently, can decompose density f(x) or mass function p(x) into convex combination of other density or mass functions) Algorithm: 1. Generate a positive random integer J such that P(J = j) = pj 2. Return X with CDF FJ (given J = j, X is generated independent of J) Proof: For fixed x, P(returned X £ x) = å P(X £ x | J = j)P(J = j) (condition on J = j) j = å P(X £ x | J = j) p j (distribution of
Load more

Recommended publications
  • 2.1 Introduction Civil Engineering Systems Deal with Variable Quantities, Which Are Described in Terms of Random Variables and Random Processes
    2.1 Introduction Civil Engineering systems deal with variable quantities, which are described in terms of random variables and random processes. Once the system design such as a design of building is identified in terms of providing a safe and economical building, variable resistances for different members such as columns, beams and foundations which can sustain variable loads can be analyzed within the framework of probability theory. The probabilistic description of variable phenomena needs the use of appropriate measures. In this module, various measures of description of variability are presented. 2. HISTOGRAM AND FREQUENCY DIAGRAM Five graphical methods for analyzing variability are: 1. Histograms, 2. Frequency plots, 3. Frequency density plots, 4. Cumulative frequency plots and 5. Scatter plots. 2.1. Histograms A histogram is obtained by dividing the data range into bins, and then counting the number of values in each bin. The unit weight data are divided into 4- kg/m3 wide intervals from to 2082 in Table 1. For example, there are zero values between 1441.66 and 1505 Kg / m3 (Table 1), two values between 1505.74 kg/m3 and 1569.81 kg /m3 etc. A bar-chart plot of the number of occurrences in each interval is called a histogram. The histogram for unit weight is shown on fig.1. Table 1 Total Unit Weight Data from Offshore Boring Total unit Total unit 2 3 Depth weight (x-βx) (x-βx) Depth weight Number (m) (Kg / m3) (Kg / m3)2 (Kg / m3)3 (m) (Kg / m3) 1 0.15 1681.94 115.33 -310.76 52.43 1521.75 2 0.30 1906.20 2050.36 23189.92 2.29 1537.77
    [Show full text]
  • Stat 345 - Homework 3 Chapter 4 -Continuous Random Variables
    STAT 345 - HOMEWORK 3 CHAPTER 4 -CONTINUOUS RANDOM VARIABLES Problem One - Trapezoidal Distribution Consider the following probability density function f(x). 8 > x+1 ; −1 ≤ x < 0 > 5 > <> 1 ; 0 ≤ x < 4 f(x) = 5 > 5−x ; 4 ≤ x ≤ 5 > 5 > :>0; otherwise a) Sketch the PDF. b) Show that the area under the curve is equal to 1 using geometry. c) Show that the area under the curve is equal to 1 by integrating. (You will have to split the integral into pieces). d) Find P (X < 3) using geometry. e) Find P (X < 3) with integration. Problem Two - Probability Density Function Consider the following function f(x), where θ > 0. 8 < c (θ − x); 0 < x < θ f(x) = θ2 :0; otherwise a) Find the value of c so that f(x) is a valid PDF. Sketch the PDF. b) Find the mean and standard deviation of X. p c) If nd , 2 and . θ = 2 P (X > 1) P (X > 3 ) P (X > 2 − 2) c) Compute θ 3θ . P ( 10 < X < 5 ) Problem Three - Cumulative Distribution Functions For each of the following probability density functions, nd the CDF. The Median (M) of a distribution is dened by the property: 1 . Use the CDF to nd the Median of each distribution. F (M) = P (X ≤ M) = 2 a) (Special case of the Beta Distribution) 8 p <1:5 x; 0 < x < 1 f(x) = :0; otherwise b) (Special case of the Pareto Distribution) 8 < 1 ; x > 1 f(x) = x2 :0; otherwise c) Take the derivative of your CDF for part b) and show that you can get back the PDF.
    [Show full text]
  • STAT 345 Spring 2018 Homework 5 - Continuous Random Variables Name
    STAT 345 Spring 2018 Homework 5 - Continuous Random Variables Name: Please adhere to the homework rules as given in the Syllabus. 1. Trapezoidal Distribution. Consider the following probability density function. 8 x+1 ; −1 ≤ x < 0 > 5 > 1 < 5 ; 0 ≤ x < 4 f(x) = 5−x > ; 4 ≤ x ≤ 5 > 5 :>0; otherwise a) Sketch the PDF. b) Show that the area under the curve is equal to 1 using geometry. c) Show that the area under the curve is equal to 1 by integrating. (You will have to split the integral into peices). d) Find P (X < 3) using geometry. e) Find P (X < 3) with integration. 2. Let X have the following PDF, for θ > 0. ( c (θ − x); 0 < x < θ f(x) = θ2 0; otherwise a) Find the value of c which makes f(x) a valid PDF. Sketch the PDF. b) Find the mean and standard deviation of X. θ 3θ c) Find P 10 < X < 5 . 3. PDF to CDF. For each of the following PDFs, find and sketch the CDF. The Median (M) of a continuous distribution is defined by the property F (M) = 0:5. Use the CDF to find the Median of each distribution. a) (Special case of Beta Distribution) p f(x) = 1:5 x; 0 < x < 1 b) (Special case of Pareto Distribution) 1 f(x) = ; x > 1 x2 4. CDF to PDF. For each of the following CDFs, find the PDF. a) 8 0; x < 0 <> F (x) = 1 − (1 − x2)2; 0 ≤ x < 1 :>1; x ≥ 1 b) ( 0; x < 0 F (x) = 1 − e−x2 ; x ≥ 0 5.
    [Show full text]
  • Hand-Book on STATISTICAL DISTRIBUTIONS for Experimentalists
    Internal Report SUF–PFY/96–01 Stockholm, 11 December 1996 1st revision, 31 October 1998 last modification 10 September 2007 Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists by Christian Walck Particle Physics Group Fysikum University of Stockholm (e-mail: [email protected]) Contents 1 Introduction 1 1.1 Random Number Generation .............................. 1 2 Probability Density Functions 3 2.1 Introduction ........................................ 3 2.2 Moments ......................................... 3 2.2.1 Errors of Moments ................................ 4 2.3 Characteristic Function ................................. 4 2.4 Probability Generating Function ............................ 5 2.5 Cumulants ......................................... 6 2.6 Random Number Generation .............................. 7 2.6.1 Cumulative Technique .............................. 7 2.6.2 Accept-Reject technique ............................. 7 2.6.3 Composition Techniques ............................. 8 2.7 Multivariate Distributions ................................ 9 2.7.1 Multivariate Moments .............................. 9 2.7.2 Errors of Bivariate Moments .......................... 9 2.7.3 Joint Characteristic Function .......................... 10 2.7.4 Random Number Generation .......................... 11 3 Bernoulli Distribution 12 3.1 Introduction ........................................ 12 3.2 Relation to Other Distributions ............................. 12 4 Beta distribution 13 4.1 Introduction .......................................
    [Show full text]
  • MARLAP Manual Volume III: Chapter 19, Measurement Uncertainty
    19 MEASUREMENT UNCERTAINTY 19.1 Overview This chapter discusses the evaluation and reporting of measurement uncertainty. Laboratory measurements always involve uncertainty, which must be considered when analytical results are used as part of a basis for making decisions.1 Every measured result reported by a laboratory should be accompanied by an explicit uncertainty estimate. One purpose of this chapter is to give users of radioanalytical data an understanding of the causes of measurement uncertainty and of the meaning of uncertainty statements in laboratory reports. The chapter also describes proce- dures which laboratory personnel use to estimate uncertainties. This chapter has more than one intended audience. Not all readers are expected to have the mathematical skills necessary to read and completely understand the entire chapter. For this reason the material is arranged so that general information is presented first and the more tech- nical information, which is intended primarily for laboratory personnel with the required mathe- matical skills, is presented last. The general discussion in Sections 19.2 and 19.3 requires little previous knowledge of statistical metrology on the part of the reader and involves no mathe- matical formulas; however, if the reader is unfamiliar with the fundamental concepts and terms of probability and statistics, he or she should read Attachment 19A before starting Section 19.3. The technical discussion in Sections 19.4 and 19.5 requires an understanding of basic algebra and at least some familiarity with the fundamental concepts of probability and statistics. The discus- sion of uncertainty propagation requires knowledge of differential calculus for a com- Contents plete understanding.
    [Show full text]
  • Tumor Detection Using Trapezoidal Function and Rayleigh Distribution
    IOSR Journal of Electronics and Communication Engineering (IOSR-JECE) e-ISSN: 2278-2834,p- ISSN: 2278-8735.Volume 12, Issue 3, Ver. IV (May - June 2017), PP 27-32 www.iosrjournals.org Tumor Detection Using Trapezoidal Function and Rayleigh Distribution. *Mohammad Aslam. C1 Satya Narayana. D2 Padma Priya. K3 1 Vaagdevi Institute of Technology and Science, Proddatur, A.P 2 Rajiv Gandhi Memorial College Of Engineering And Technology, Nandyal, A.P 3jntuk College of Engineering, Kakinada, A.P Corresponding Author: Mohammad Aslam. C1 Abstract: Breast cancer is one of the most common cancers worldwide. In developed countries, among one in eight women develop breast cancer at some stage of their life. Early diagnosis of breast cancer plays a very important role in treatment of the disease. With the goal of identifying genes that are more correlated with the prognosis of breast cancer. An algorithm based on Trapezoidal and Rayleigh distributions. Property is used for the detection of the cancer tumors. Fuzzy logic and markov random fields work in real time. We also try to show its performance by comparing it with the existing real time algorithm that uses only markov random fields. This is not only suitable for the detection of the tumors, but also suitable for various general applications like in the detection of object in an image etc. Keywords: Tumors, Rayleigh distribution texture, Segmentation, Fuzzy Logic. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 19-06-2017 Date of acceptance: 26-07-2017 ----------------------------------------------------------------------------------------------------------------------------- ---------- I. Introduction Breast Cancer is one of the main sources of malignancy related passings in ladies. There are accessible distinctive strategies for the discovery of malignancy tumors, these techniques are utilized for the finding of disease in ladies at early stages.
    [Show full text]
  • Determining the 95% Confidence Interval of Arbitrary Non-Gaussian Probability Distributions
    XIX IMEKO World Congress Fundamental and Applied Metrology September 6−11, 2009, Lisbon, Portugal DETERMINING THE 95% CONFIDENCE INTERVAL OF ARBITRARY NON-GAUSSIAN PROBABILITY DISTRIBUTIONS France Pavlovčič, Janez Nastran, David Nedeljković University of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia E-mail: [email protected], [email protected], [email protected] Abstract − Measurements are nowadays permanent at- drift, the time drift, and the resolution, all have the rectan- tendants of scientific research, health, medical care and gular distribution. Sometimes also the A-type uncertainties treatment, industrial development, safety and even global do not match with the normal distribution as they are result economy. All of them depend on accurate measurements of regulated quantities, or are affected by the resolution of and tests, and many of these fields are under the legal me- measuring device. In the latter case the probability distribu- trology because of their severity. How the measurement is tion consists of two Dirac functions, one at the lower meas- accurate, is expressed by uncertainty, which is obtained by urement reading and the other at the upper measurement multiplication of standard deviations by coverage factors to reading regarding the resolution. If the distribution is un- increase trustfulness in the measured results. These coverage known the coverage factor is calculated from Chebyshev's factors depend on degree of freedom, which is the function inequality [2] and is 4.472 for the 95% confidence interval. of the number of implied repetitions of measurements, and The uncertainty, the extended or the standard one, does not therefore the reliability of the results is increased.
    [Show full text]
  • Generalized Trapezoidal Distributions
    1 Generalized Trapezoidal Distributions J. René van Dorp, Samuel Kotz Department of Engineering Management & Systems Engineering, The George Washington University, 1776 G Street, N.W., Washington D.C. 20052 (e-mail: [email protected]). Submitted April 24, 2002; Revised, March 2003 Abstract. We present a construction and basic properties of a class of continuous distributions of an arbitrary form defined on a compact (bounded) set by concatenating in a continuous manner three probability density functions with bounded support using a modified mixture technique. These three distributions may represent growth, stability and decline stages of a physical or mental phenomenon. Keywords: Mixtures of Distributions, Risk AnalysisÄ Applied Physics 1. INTRODUCTION Trapezoidal distributions have been advocated in risk analysis problems by Pouliquen (1970) and more recently by Powell and Wilson (1997). They have also found application as membership functions in fuzzy set theory (see, e.g. Chen and Hwang (1992)). Our interest in trapezoidal distributions and their modifications stems mainly from the conviction that many physical processes in nature and human body and mind (over time) reflect the form of the trapezoidal distribution. In this context, trapezoidal distributions have been used in the screening and detection of cancer (see, e.g. Flehinger and Kimmel, (1987) and Brown (1999)). Trapezoidal distributions seem to be appropriate for modeling the duration and the form of a phenomenon which may be represented by three stages. The first stage can be To appear in Metrika, Vol. 58, Issue 1, July 2003 2 viewed as a growth-stage, the second corresponds to a relative stability and the third represents a decline (decay).
    [Show full text]
  • RCEM – Reclamation Consequence Estimating Methodology Interim
    RCEM – Reclamation Consequence Estimating Methodology Interim Examples of Use U.S. Department of the Interior Bureau of Reclamation June 2014 Mission Statements The U.S. Department of the Interior protects America’s natural resources and heritage, honors our cultures and tribal communities, and supplies the energy to power our future. The mission of the Bureau of Reclamation is to manage, develop, and protect water and related resources in an environmentally and economically sound manner in the interest of the American public. Interim RCEM – Reclamation Consequence Estimating Methodology Examples of Use U.S. Department of the Interior Bureau of Reclamation June 2014 RCEM – Examples of Use Interim Table of Contents Introduction ................................................................................................................... 1 General Usage ........................................................................................................... 1 Description of Examples ........................................................................................... 1 Format of Examples .................................................................................................. 3 Graphical Approach ...................................................................................................... 3 Uncertainty and Variability ........................................................................................... 8 Examples ......................................................................................................................
    [Show full text]
  • Selecting and Applying Error Distributions in Uncertainty Analysis1
    SELECTING AND APPLYING ERROR DISTRIBUTIONS 1 IN UNCERTAINTY ANALYSIS Dr. Howard Castrup Integrated Sciences Group 14608 Casitas Canyon Road Bakersfield, CA 93306 1-661-872-1683 [email protected] ABSTRACT Errors or biases from a number of sources may be encountered in performing a measurement. Such sources include random error, measuring parameter bias, measuring parameter resolution, operator bias, environmental factors, etc. Uncertainties due to these errors are estimated either by computing a Type A standard deviation from a sample of measurements or by forming a Type B estimate based on experience. In this paper, guidelines are given for selecting and applying various statistical error distribution that have been found useful for Type A and Type B analyses. Both traditional and Monte Carlo methods are outlined. Also discussed is the estimation of the degrees of freedom for both Type A or Type B estimates. Where appropriate, concepts will be illustrated using spreadsheets, freeware and commercially available software. INTRODUCTION THE GENERAL ERROR MODEL The measurement error model employed in this paper is given in the expression x = xtrue + (1) where x = a value obtained by measurement xtrue = the true value of the measured quantity = the measurement error. With univariate direct measurements, the variable is composed of a linear sum of errors from various measurement error sources. With multivariate measurements consists of a weighted sum of error components, each of which is make up of errors from measurement error sources. More will be said on this later. APPLICABLE AXIOMS In order to use Eq. (1) to estimate measurement uncertainty, we invoke three axioms [HC95a]: Axiom 1 Measurement errors are random variables that follow statistical distributions.
    [Show full text]
  • Uncertainty-Aware Principal Component Analysis
    Uncertainty-Aware Principal Component Analysis Jochen Gortler,¨ Thilo Spinner, Dirk Streeb, Daniel Weiskopf, and Oliver Deussen 0 0 0 0 0 0 0 0 0 0 0 0 (a) Regular PCA (b) s = 0.5 (c) s = 0.8 (d) s = 1.0 Fig. 1. Data uncertainty can have a significant influence on the outcome of dimensionality reduction techniques. We propose a generalization of principal component analysis (PCA) that takes into account the uncertainty in the input. The top row shows the dataset with varying degrees of uncertainty and the corresponding principal components, whereas the bottom row shows the projection of the dataset, using our method, onto the first principal component. In Figures (a) and (b), with relatively low uncertainty, the blue and the orange distributions are comprised by the red and the green distributions. In Figures (c) and (d), with a larger amount of uncertainty, the projection changes drastically: now the orange and blue distributions encompass the red and the green distributions. Abstract—We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data.
    [Show full text]
  • Trapezoidal and Triangular Distributions for Type B Evaluation of Standard Uncertainty
    IOP PUBLISHING METROLOGIA Metrologia 44 (2007) 117–127 doi:10.1088/0026-1394/44/2/003 Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty Raghu N Kacker1 and James F Lawrence1,2 1 Mathematical and Computational Science Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 2 Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA E-mail: [email protected] Received 6 September 2006 Published 26 February 2007 Online at stacks.iop.org/Met/44/117 Abstract The Guide to the Expression of Uncertainty in Measurement (GUM), published by the International Organization for Standardization (ISO), recognizes Type B state-of-knowledge probability distributions specified by scientific judgment as valid means to quantify uncertainty. The ISO-GUM discusses symmetric probability distributions only. Sometimes an asymmetric distribution is needed. We describe a trapezoidal distribution which may be asymmetric depending on the settings of its parameters. We describe the probability density function, cumulative distribution function (cdf), inverse function of the cdf, moment generating function, moments about origin (zero), expected value and variance of a trapezoidal distribution. We show that triangular and rectangular distributions are special cases of the trapezoidal distribution. Then we derive the moment generating functions, moments, expected values and variances of various special cases of the trapezoidal distribution. Finally, we illustrate through a real life example how a Type B asymmetric trapezoidal distribution may be useful in quantifying a correction for bias (systematic error) in a result of measurement and in quantifying the standard uncertainty associated with the correction. 1. Introduction trapezoidal distribution, which may be asymmetric depending on the settings of its parameters.
    [Show full text]