BIOSTATISTICAL METHODS
|
|
- Anissa Whitehead
- 5 years ago
- Views:
Transcription
1 BIOSTATISTICAL METHOS FOR TRANSLATIONAL & CLINICAL RESEARCH ROC Crve: IAGNOSTIC MEICINE
2 iagnostic tests have been presented as alwas having dichotomos otcomes. In some cases, the reslt of the test ma be binar, bt in man cases it is based on the dichotomization of a continos separator or biomarker some factor correlated the absence or presence of the disease. To deal with a continos separator, we need a wellknown graph called the Receiver Operating Characteristic crve or ROC crve.
3 If the idea, in the developmental stage, was to classified people as diseased condition present or health condition absent based on certain continos measrement from blood or rinar components; then we need to dichotomize the measrement: for example, if the measrement is high then he s classified as diseased if it s low, the sbject is health. Bt the basic qestion is How high is high? or How low is low? - i.e. where shold be the ct-point?
4 A SIMPLE PLAUSIBLE MOEL T- ctpoint T+ Separator Y is normall distribted with the same variance, bt different means; no matter where o ct, both errors reslt! More important, specificit & sensitivit are fnctions of the ctpoint.
5 ASSUMPTION In the case of man diseases, the larger vales of the separator Y are associated with the diseased poplation also called poplation of the cases and smaller vales are associated with the control or non-diseased or health poplation e.g. blood glcose for diabetes, PSA for prostate cancer, antibodies for infections, For man others, the smaller vales of the separator Y are associated with the diseased poplation and larger vales are associated with the non-diseased poplation static admittance for Otitis Media, TSH for hperthroidism. We will assme, withot loss of generalit, that larger vales of Y are associated with the diseased poplation.
6 If, in fact, smaller vales of Y are associated with the diseased poplation, methods presented here cold be applied b simpl reversing the roles of cases sbjects with the disease and controls sbjects withot the disease.
7 SENSITIVITY With or assmption that larger vales of Y are associated with the diseased poplation, the sensitivit, PrT+ +, associated with a ctpoint Y is: S + PrY> + tre positive rate - PrY + - F + where F + PrY + is the cmlative distribtion fnction cdf of Y for the diseased poplation or poplation of cases.
8 SPECIFICITY With or assmption that larger vales of Y are associated with the diseased poplation, the specificit, PrT- -, associated with a ctpoint Y is: S - PrY - F -, or - S - - F - false positive rate where F - x is the cmlative distribtion fnction cdf of Y for the non-diseased or health poplation.
9 The sensitivit, S + - F +, and the -specificit, - S - - F - are srvival fnctions. And the vales of the sensitivit and of the specificit both are fnctions of the ct-point which cold be set arbitraril.
10 ROC FUNCTION & ROC CURVE A fnction R from [0,] to [0,] that maps false positive rate to tre positive rate, -F - to -F +, is called the ROC fnction : R[-F - ] -F + or R[-S - ] S + The graph of R. is called the ROC crve The ROC crve, the graph of sensitivit, S +, verss - specificit, -S -, is generated as the ctpoint moves throgh its range of possible vales.
11 The ROC fnction maps sensitivit against -specificit or tre positive rate against false positive rate. It maps a srvival fnction against another srvival fnction.
12 In statistical terms, an ROC crve maps statistical power against tpe I error : It is a tool to present how good a statistical test of significance is. Sa, one can draw an ROC crve for the two-sample t-test and another one for the Wilcoxon test; then compare.
13 , sensitivit -F + S + tre positive rate 0 -specificit, -F - -S - false positive rate ROC Crve
14 STATISTICAL EXPRESSION -cdf is called the Srvival Fnction, St; let ] ] [ [0, ]; [S S R H + t S t S R t F t S t F t S H H
15 Isse #: How to estimate the ROC crve given two independent samples, { 0i ; i,,n 0 } and { j, j,, n } from n 0 controls and n cases?
16 EMPIRICAL ESTIMATE The simplest wa to estimate R. is to replace cdfs F + and F - b their empirical estimates p + and p - ; p + is the proportion of the n observations j s of the cases which are less than or eqal to, and p - is defined similarl. This is a non-parametric estimate and {-p -, -p + } is an nbiased estimator of {-F -, -F + } If there are no ties in the combined sample of 0i s and j s, there n 0 * n points. One cold simpl connect the dots to form a formal graphical estimate of the ROC crve;
17 A RANOM WALK Steck 97 made an statistical attempt to connect the dots, trning them into a step fnction. He combined 2 samples & in the sal increasing order. He described the empirical estimator as a random walk from the bottom-left corner 0,0 to the top-right corner, whose next step is /n p or /n 0 to the right according to whether the next observation in the ordered combined sample is a case s measrement or a control s measrement 0.
18 EXAMPLE #A: < 0 < 2 < 02 < 03 2nd 4th 3rd 5th st 0 It s like an empirical cdf of size 2 with weights /2 at points 0 & /3
19 Index for IAGNOSTIC ACCURACY ROC crve is a graphical device to show all possible combinations of sensitivit and specificit bt, for simplicit, it is desirable to redce an entire crve to a single qantitative index of diagnostic accrac. Possibilities inclde the difference between means of Y for the two poplations, those with disease and those withot; and the ratio of variances. However, the most poplar one has been the area nder the ROC crve. The area nder the crve has a powerfl interpretation and it is related to other well-known statistics making it easier to learn its statistical properties.
20 Sppose that an observation is randoml sampled from the diseased poplation and another random observation Y 0 is independentl sampled from the non-diseased poplation; and let PrY >Y 0 denote the probabilit of the event that the Y observation is larger than the Y 0 observation; we have: A Pr Y > Y 0 A F + d F A A 0 R d Area nder ROC crve
21 There are man different was to obtain standard error and/or confidence intervals bt nmerical reslts are ver similar. Pick or choice & learn
22 AN ALTERNATIVE INEX ctpoint T- T+ Separator Y is normall distribted with the same variance, bt different means; no matter where o ct, both errors reslt! The sizes of these errors depend on the standardized distance d μ μ H /σ
23 Are the two indices, A and d different? Yes, different nmerical vales bt statisticall eqivalent. If we let Φ -. denote inverse of the standard normal cmlative distribtion fnction, for example Φ , then Simpson and Fitter 973 showed that: d 2 Φ A So, what is special abot index d? It has a ver powerfl interpretation in terms of disease development!
24 LOGISTIC REGRESSION The probabilit of disease development and the vale Y of the separator Y are related b the Logistic Regression Model : ln π π π β0 + β e Pr + Y β0 + + e β 0 + β β, or
25 USE OF BAYES RULE π π Pr Pr + Y Y π π Pr Y Pr Y + Pr Pr + / / Pr Y Pr Y π π Pr Y Pr Y + Pr Pr + ln π π Constant + ln[ PrY PrY + ]
26 RESULT Sppose Y is normall distribted with the same variance, different means for PrY + and PrY -, we have: σ β d Y Y H H Constant ln ] } / exp{ } / exp{ ln[ Constant ln ] Pr Pr ln[ Constant ln σ µ µ π π σ µ σ µ π π π π
27 INTERPRETATION OF d Under logistic model and Sppose Y is normall distribted with the same variance bt different means for PrY + and PrY -, then: d β σ The vale of Index d is eqal to the logodds Ratio de to a change of one S in the vale of the marker Y
28 The Optimization Problem
29 We all know that, for example, high PSA likel indicates prostate cancer; bt how high it is to classif a man as having prostate cancer? If we set the ct-point too high, we wold miss cases that is low sensitivit ; if we set the ct-point too low, we wold have man false positives that is low specificit!
30 For a continos marker/predictor sch as PSA ; the basic qestion is How high is high? or How low is low?. In practice, ctpoints are formed arbitraril becase we fail to form and jstif a criterion or criteria. We need an optimal ctpoint ; bt what do we mean b optimal? Good, bt what it is good for? Ma be more than one soltion becase there are different criteria.
31 Basic Strateg/Criterion: To determine an optimal ctpoint for a continos marker b maximizing the Yoden s Index of the dichotomized test. Using this strateg, when sing the reslting dichotomized test in a prevalence srve we wold obtain an estimate with minimal error. There are other gains too.
32 SOLUTION #: EMPIRICAL Pool the two samples and arrange in increasing order At each midwa between two data points, calclate the Sensitivit S + and Specificit S - ; then the Yoden s Index J S + +S - - Locate the ctpoint corresponding to the maximm vale of J. Ver simple!
33 SOLUTION #2: NON-PARAMETRIC The ROC fnction R. maps U -S - on the horizontal axis to V S + on the vertical axis: V RU The Yoden s Index J S + +S - - RU - U is maximized when: 0 R U -, or R U. Process: i Smooth empirical estimate b an smoothing techniqe eg. Lowess, ii Locate the point with slope to obtain specificit, then iii Go to control sample to get ct-point.
34 , sensitivit -F + S + 0 -specificit, -F - -S - If the ROC crve is smmetric between 0,0 and,, the point on the crve with slope is closest to corner 0,.
35 SOLUTION #3: SEMI-PARAMETRIC Still looking for the point on the crve with Slope bt, first fitting empirical data with a smooth crve Y RU θ becase it wold take less data to do a better job than nonparametric smoothing we need a model bt can check for goodness-of-fit. The two components needed are: i Choosing a meaningfl parameter θ, ii Choosing a fnctional form for R.; one possibilit is the Proportional Hazard model
36 EXAMPLE: PROSTATE CANCER There were 53 patients with prostate cancer; 20 of them with nodal involvement and 33 withot. We examined level of acid phosphatase in blood serm x00. ata are reprodced from Miller et al 980 and are as follows: Patients withot Nodal Involvement: 40, 40, 46, 47, 48, 48, 49, 49, 50, 50, 50, 50, 50, 52, 52, 55, 55, 56, 59, 62, 62, 63, 65, 66, 7, 75, 76, 78, 83, 95, 98, 02, 87. Patients with Nodal Involvement: 486, 499, 56, 562.5, 6730, 6730, 6730, 7032,5, , 7235, , 7840, 84, , , 8445, 8946, 9949, 265, 3652; nmbers in parentheses are the ranks in the combined sample, mid-ranks are sed for tied observations.
37 SOLUTION #4: PARAMETRIC 0 ; [0,] ]; [ ] [ ' ' ' R R J S R J S S R t S t S R S t F t S S t F t S H H H + +
38 LOG-LOGISTIC ISTRIBUTION If lnx is distribted as logistic, X is distribted as log-logistic; the loglogistic distribtion is similar to log-normal distribtion bt with thicker tails so fits better real non-negative measrements. S t ρ ν + e σ µ,, ρ t ν ; where μ is Mean where σ St eviation
39 BOTH ARE LOG-LOGISTIC ISTRIBUTIONS β σ µ µ σ σ σ ρ ν exp : R Then t t S H H + + +
40 β 0 < µ exp < for β σ µ µ > H µ H
41 exp exp : : ] [ ' 2 ' d where S S Optimal R R R H σ µ µ β β β β β β β β β
42 SCREENING VALUE OF BIOMARKERS d S-S+ 62% 2 73% 3 82% 4 88%
i=1 y i 1fd i = dg= P N i=1 1fd i = dg.
ECOOMETRICS II (ECO 240S) University of Toronto. Department of Economics. Winter 208 Instrctor: Victor Agirregabiria SOLUTIO TO FIAL EXAM Tesday, April 0, 208. From 9:00am-2:00pm (3 hors) ISTRUCTIOS: -
More informationA fundamental inverse problem in geosciences
A fndamental inverse problem in geosciences Predict the vales of a spatial random field (SRF) sing a set of observed vales of the same and/or other SRFs. y i L i ( ) + v, i,..., n i ( P)? L i () : linear
More informationLecture 8: September 26
10-704: Information Processing and Learning Fall 2016 Lectrer: Aarti Singh Lectre 8: September 26 Note: These notes are based on scribed notes from Spring15 offering of this corse. LaTeX template cortesy
More informationBIOSTATISTICS METHODS
BIOTATITIC METHOD FOR TRANLATIONAL & CLINICAL REEARCH DIAGNOTIC MEDICINE Part B: tatistical Issues Predictive Values The design in stage I involves two samples, those with the disease and those without.
More informationBayes and Naïve Bayes Classifiers CS434
Bayes and Naïve Bayes Classifiers CS434 In this lectre 1. Review some basic probability concepts 2. Introdce a sefl probabilistic rle - Bayes rle 3. Introdce the learning algorithm based on Bayes rle (ths
More informationFormal Methods for Deriving Element Equations
Formal Methods for Deriving Element Eqations And the importance of Shape Fnctions Formal Methods In previos lectres we obtained a bar element s stiffness eqations sing the Direct Method to obtain eact
More informationMathematical Analysis of Nipah Virus Infections Using Optimal Control Theory
Jornal of Applied Mathematics and Physics, 06, 4, 099- Pblished Online Jne 06 in SciRes. http://www.scirp.org/jornal/jamp http://dx.doi.org/0.436/jamp.06.464 Mathematical Analysis of Nipah Virs nfections
More informationFEA Solution Procedure
EA Soltion Procedre (demonstrated with a -D bar element problem) EA Procedre for Static Analysis. Prepare the E model a. discretize (mesh) the strctre b. prescribe loads c. prescribe spports. Perform calclations
More informationA New Approach to Direct Sequential Simulation that Accounts for the Proportional Effect: Direct Lognormal Simulation
A ew Approach to Direct eqential imlation that Acconts for the Proportional ffect: Direct ognormal imlation John Manchk, Oy eangthong and Clayton Detsch Department of Civil & nvironmental ngineering University
More information3 2D Elastostatic Problems in Cartesian Coordinates
D lastostatic Problems in Cartesian Coordinates Two dimensional elastostatic problems are discssed in this Chapter, that is, static problems of either plane stress or plane strain. Cartesian coordinates
More informationAn Investigation into Estimating Type B Degrees of Freedom
An Investigation into Estimating Type B Degrees of H. Castrp President, Integrated Sciences Grop Jne, 00 Backgrond The degrees of freedom associated with an ncertainty estimate qantifies the amont of information
More informationLab Manual for Engrd 202, Virtual Torsion Experiment. Aluminum module
Lab Manal for Engrd 202, Virtal Torsion Experiment Alminm modle Introdction In this modle, o will perform data redction and analsis for circlar cross section alminm samples. B plotting the torqe vs. twist
More informationVIBRATION MEASUREMENT UNCERTAINTY AND RELIABILITY DIAGNOSTICS RESULTS IN ROTATING SYSTEMS
VIBRATIO MEASUREMET UCERTAITY AD RELIABILITY DIAGOSTICS RESULTS I ROTATIG SYSTEMS. Introdction M. Eidkevicite, V. Volkovas anas University of Technology, Lithania The rotating machinery technical state
More informationDecision Making in Complex Environments. Lecture 2 Ratings and Introduction to Analytic Network Process
Decision Making in Complex Environments Lectre 2 Ratings and Introdction to Analytic Network Process Lectres Smmary Lectre 5 Lectre 1 AHP=Hierar chies Lectre 3 ANP=Networks Strctring Complex Models with
More informationThis Topic follows on from Calculus Topics C1 - C3 to give further rules and applications of differentiation.
CALCULUS C Topic Overview C FURTHER DIFFERENTIATION This Topic follows on from Calcls Topics C - C to give frther rles applications of differentiation. Yo shold be familiar with Logarithms (Algebra Topic
More informationUNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL
8th International DAAAM Baltic Conference "INDUSTRIAL ENGINEERING - 19-1 April 01, Tallinn, Estonia UNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL Põdra, P. & Laaneots, R. Abstract: Strength analysis is a
More informationOn Multiobjective Duality For Variational Problems
The Open Operational Research Jornal, 202, 6, -8 On Mltiobjective Dality For Variational Problems. Hsain *,, Bilal Ahmad 2 and Z. Jabeen 3 Open Access Department of Mathematics, Jaypee University of Engineering
More informationCHARACTERIZATIONS OF EXPONENTIAL DISTRIBUTION VIA CONDITIONAL EXPECTATIONS OF RECORD VALUES. George P. Yanev
Pliska Std. Math. Blgar. 2 (211), 233 242 STUDIA MATHEMATICA BULGARICA CHARACTERIZATIONS OF EXPONENTIAL DISTRIBUTION VIA CONDITIONAL EXPECTATIONS OF RECORD VALUES George P. Yanev We prove that the exponential
More informationStudy on the impulsive pressure of tank oscillating by force towards multiple degrees of freedom
EPJ Web of Conferences 80, 0034 (08) EFM 07 Stdy on the implsive pressre of tank oscillating by force towards mltiple degrees of freedom Shigeyki Hibi,* The ational Defense Academy, Department of Mechanical
More informationMath 116 First Midterm October 14, 2009
Math 116 First Midterm October 14, 9 Name: EXAM SOLUTIONS Instrctor: Section: 1. Do not open this exam ntil yo are told to do so.. This exam has 1 pages inclding this cover. There are 9 problems. Note
More informationWorkshop on Understanding and Evaluating Radioanalytical Measurement Uncertainty November 2007
1833-3 Workshop on Understanding and Evalating Radioanalytical Measrement Uncertainty 5-16 November 007 Applied Statistics: Basic statistical terms and concepts Sabrina BARBIZZI APAT - Agenzia per la Protezione
More information4.2 First-Order Logic
64 First-Order Logic and Type Theory The problem can be seen in the two qestionable rles In the existential introdction, the term a has not yet been introdced into the derivation and its se can therefore
More informationTechnical Note. ODiSI-B Sensor Strain Gage Factor Uncertainty
Technical Note EN-FY160 Revision November 30, 016 ODiSI-B Sensor Strain Gage Factor Uncertainty Abstract Lna has pdated or strain sensor calibration tool to spport NIST-traceable measrements, to compte
More information1 Differential Equations for Solid Mechanics
1 Differential Eqations for Solid Mechanics Simple problems involving homogeneos stress states have been considered so far, wherein the stress is the same throghot the component nder std. An eception to
More informationResearch Article Permanence of a Discrete Predator-Prey Systems with Beddington-DeAngelis Functional Response and Feedback Controls
Hindawi Pblishing Corporation Discrete Dynamics in Natre and Society Volme 2008 Article ID 149267 8 pages doi:101155/2008/149267 Research Article Permanence of a Discrete Predator-Prey Systems with Beddington-DeAngelis
More informationEXPT. 5 DETERMINATION OF pk a OF AN INDICATOR USING SPECTROPHOTOMETRY
EXPT. 5 DETERMITIO OF pk a OF IDICTOR USIG SPECTROPHOTOMETRY Strctre 5.1 Introdction Objectives 5.2 Principle 5.3 Spectrophotometric Determination of pka Vale of Indicator 5.4 Reqirements 5.5 Soltions
More informationModelling by Differential Equations from Properties of Phenomenon to its Investigation
Modelling by Differential Eqations from Properties of Phenomenon to its Investigation V. Kleiza and O. Prvinis Kanas University of Technology, Lithania Abstract The Panevezys camps of Kanas University
More informationMath 263 Assignment #3 Solutions. 1. A function z = f(x, y) is called harmonic if it satisfies Laplace s equation:
Math 263 Assignment #3 Soltions 1. A fnction z f(x, ) is called harmonic if it satisfies Laplace s eqation: 2 + 2 z 2 0 Determine whether or not the following are harmonic. (a) z x 2 + 2. We se the one-variable
More informationConditions for Approaching the Origin without Intersecting the x-axis in the Liénard Plane
Filomat 3:2 (27), 376 377 https://doi.org/.2298/fil7276a Pblished by Faclty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Conditions for Approaching
More informationPhysicsAndMathsTutor.com
. Two smooth niform spheres S and T have eqal radii. The mass of S is 0. kg and the mass of T is 0.6 kg. The spheres are moving on a smooth horizontal plane and collide obliqely. Immediately before the
More informationEssentials of optimal control theory in ECON 4140
Essentials of optimal control theory in ECON 4140 Things yo need to know (and a detail yo need not care abot). A few words abot dynamic optimization in general. Dynamic optimization can be thoght of as
More informationBLOOM S TAXONOMY. Following Bloom s Taxonomy to Assess Students
BLOOM S TAXONOMY Topic Following Bloom s Taonomy to Assess Stdents Smmary A handot for stdents to eplain Bloom s taonomy that is sed for item writing and test constrction to test stdents to see if they
More informationCurves - Foundation of Free-form Surfaces
Crves - Fondation of Free-form Srfaces Why Not Simply Use a Point Matrix to Represent a Crve? Storage isse and limited resoltion Comptation and transformation Difficlties in calclating the intersections
More informationDepartment of Industrial Engineering Statistical Quality Control presented by Dr. Eng. Abed Schokry
Department of Indstrial Engineering Statistical Qality Control presented by Dr. Eng. Abed Schokry Department of Indstrial Engineering Statistical Qality Control C and U Chart presented by Dr. Eng. Abed
More informationChapter 3. Preferences and Utility
Chapter 3 Preferences and Utilit Microeconomics stdies how individals make choices; different individals make different choices n important factor in making choices is individal s tastes or preferences
More informationArtemisa. edigraphic.com. The uncertainty concept and its implications for laboratory medicine. medigraphic. en línea. Reporte breve Metrología
medigraphic rtemisa en línea Reporte breve Metrología The ncertainty concept and its implications for laboratory medicine nders Kallner, PhD MD* MESUREMENT PERFORMNE * Department of linical hemistry Karolinska
More informationThe Dual of the Maximum Likelihood Method
Department of Agricltral and Resorce Economics University of California, Davis The Dal of the Maximm Likelihood Method by Qirino Paris Working Paper No. 12-002 2012 Copyright @ 2012 by Qirino Paris All
More informationFormulas for stopped diffusion processes with stopping times based on drawdowns and drawups
Stochastic Processes and their Applications 119 (009) 563 578 www.elsevier.com/locate/spa Formlas for stopped diffsion processes with stopping times based on drawdowns and drawps Libor Pospisil, Jan Vecer,
More informationSources of Non Stationarity in the Semivariogram
Sorces of Non Stationarity in the Semivariogram Migel A. Cba and Oy Leangthong Traditional ncertainty characterization techniqes sch as Simple Kriging or Seqential Gassian Simlation rely on stationary
More informationLinear System Theory (Fall 2011): Homework 1. Solutions
Linear System Theory (Fall 20): Homework Soltions De Sep. 29, 20 Exercise (C.T. Chen: Ex.3-8). Consider a linear system with inpt and otpt y. Three experiments are performed on this system sing the inpts
More informationSensitivity Analysis in Bayesian Networks: From Single to Multiple Parameters
Sensitivity Analysis in Bayesian Networks: From Single to Mltiple Parameters Hei Chan and Adnan Darwiche Compter Science Department University of California, Los Angeles Los Angeles, CA 90095 {hei,darwiche}@cs.cla.ed
More informationDynamic Optimization of First-Order Systems via Static Parametric Programming: Application to Electrical Discharge Machining
Dynamic Optimization of First-Order Systems via Static Parametric Programming: Application to Electrical Discharge Machining P. Hgenin*, B. Srinivasan*, F. Altpeter**, R. Longchamp* * Laboratoire d Atomatiqe,
More informationPubH 7405: REGRESSION ANALYSIS INTRODUCTION TO LOGISTIC REGRESSION
PubH 745: REGRESSION ANALYSIS INTRODUCTION TO LOGISTIC REGRESSION Let Y be the Dependent Variable Y taking on values and, and: π Pr(Y) Y is said to have the Bernouilli distribution (Binomial with n ).
More informationANOVA INTERPRETING. It might be tempting to just look at the data and wing it
Introdction to Statistics in Psychology PSY 2 Professor Greg Francis Lectre 33 ANalysis Of VAriance Something erss which thing? ANOVA Test statistic: F = MS B MS W Estimated ariability from noise and mean
More information1 Introduction. r + _
A method and an algorithm for obtaining the Stable Oscillator Regimes Parameters of the Nonlinear Sstems, with two time constants and Rela with Dela and Hsteresis NUŢU VASILE, MOLDOVEANU CRISTIAN-EMIL,
More informationSTEP Support Programme. STEP III Hyperbolic Functions: Solutions
STEP Spport Programme STEP III Hyperbolic Fnctions: Soltions Start by sing the sbstittion t cosh x. This gives: sinh x cosh a cosh x cosh a sinh x t sinh x dt t dt t + ln t ln t + ln cosh a ln ln cosh
More informationClassify by number of ports and examine the possible structures that result. Using only one-port elements, no more than two elements can be assembled.
Jnction elements in network models. Classify by nmber of ports and examine the possible strctres that reslt. Using only one-port elements, no more than two elements can be assembled. Combining two two-ports
More informationControl Using Logic & Switching: Part III Supervisory Control
Control Using Logic & Switching: Part III Spervisor Control Ttorial for the 40th CDC João P. Hespanha Universit of Sothern California Universit of California at Santa Barbara Otline Spervisor control overview
More informationThe Linear Quadratic Regulator
10 The Linear Qadratic Reglator 10.1 Problem formlation This chapter concerns optimal control of dynamical systems. Most of this development concerns linear models with a particlarly simple notion of optimality.
More informationSimplified Identification Scheme for Structures on a Flexible Base
Simplified Identification Scheme for Strctres on a Flexible Base L.M. Star California State University, Long Beach G. Mylonais University of Patras, Greece J.P. Stewart University of California, Los Angeles
More informationQUANTILE ESTIMATION IN SUCCESSIVE SAMPLING
Jornal of the Korean Statistical Society 2007, 36: 4, pp 543 556 QUANTILE ESTIMATION IN SUCCESSIVE SAMPLING Hosila P. Singh 1, Ritesh Tailor 2, Sarjinder Singh 3 and Jong-Min Kim 4 Abstract In sccessive
More informationMATH2715: Statistical Methods
MATH275: Statistical Methods Exercises VI (based on lectre, work week 7, hand in lectre Mon 4 Nov) ALL qestions cont towards the continos assessment for this modle. Q. The random variable X has a discrete
More information10.2 Solving Quadratic Equations by Completing the Square
. Solving Qadratic Eqations b Completing the Sqare Consider the eqation ( ) We can see clearl that the soltions are However, What if the eqation was given to s in standard form, that is 6 How wold we go
More informationChapter 2 Difficulties associated with corners
Chapter Difficlties associated with corners This chapter is aimed at resolving the problems revealed in Chapter, which are cased b corners and/or discontinos bondar conditions. The first section introdces
More informationThe Oscillatory Stable Regime of Nonlinear Systems, with two time constants
6th WSES International Conference on CIRCUITS SYSTEMS ELECTRONICSCONTROL & SIGNL PROCESSING Cairo Egpt Dec 9-3 7 5 The Oscillator Stable Regime of Nonlinear Sstems with two time constants NUŢU VSILE *
More informationChapter 1: Differential Form of Basic Equations
MEG 74 Energ and Variational Methods in Mechanics I Brendan J. O Toole, Ph.D. Associate Professor of Mechanical Engineering Howard R. Hghes College of Engineering Universit of Nevada Las Vegas TBE B- (7)
More informationThe Replenishment Policy for an Inventory System with a Fixed Ordering Cost and a Proportional Penalty Cost under Poisson Arrival Demands
Scientiae Mathematicae Japonicae Online, e-211, 161 167 161 The Replenishment Policy for an Inventory System with a Fixed Ordering Cost and a Proportional Penalty Cost nder Poisson Arrival Demands Hitoshi
More informationCHANNEL SELECTION WITH RAYLEIGH FADING: A MULTI-ARMED BANDIT FRAMEWORK. Wassim Jouini and Christophe Moy
CHANNEL SELECTION WITH RAYLEIGH FADING: A MULTI-ARMED BANDIT FRAMEWORK Wassim Joini and Christophe Moy SUPELEC, IETR, SCEE, Avene de la Bolaie, CS 47601, 5576 Cesson Sévigné, France. INSERM U96 - IFR140-
More informationFEA Solution Procedure
EA Soltion Procedre (demonstrated with a -D bar element problem) MAE 5 - inite Element Analysis Several slides from this set are adapted from B.S. Altan, Michigan Technological University EA Procedre for
More informationsin u 5 opp } cos u 5 adj } hyp opposite csc u 5 hyp } sec u 5 hyp } opp Using Inverse Trigonometric Functions
13 Big Idea 1 CHAPTER SUMMARY BIG IDEAS Using Trigonometric Fnctions Algebra classzone.com Electronic Fnction Library For Yor Notebook hypotense acent osite sine cosine tangent sin 5 hyp cos 5 hyp tan
More informationDiscussion of The Forward Search: Theory and Data Analysis by Anthony C. Atkinson, Marco Riani, and Andrea Ceroli
1 Introdction Discssion of The Forward Search: Theory and Data Analysis by Anthony C. Atkinson, Marco Riani, and Andrea Ceroli Søren Johansen Department of Economics, University of Copenhagen and CREATES,
More informationCalculations involving a single random variable (SRV)
Calclations involving a single random variable (SRV) Example of Bearing Capacity q φ = 0 µ σ c c = 100kN/m = 50kN/m ndrained shear strength parameters What is the relationship between the Factor of Safety
More informationEXCITATION RATE COEFFICIENTS OF MOLYBDENUM ATOM AND IONS IN ASTROPHYSICAL PLASMA AS A FUNCTION OF ELECTRON TEMPERATURE
EXCITATION RATE COEFFICIENTS OF MOLYBDENUM ATOM AND IONS IN ASTROPHYSICAL PLASMA AS A FUNCTION OF ELECTRON TEMPERATURE A.N. Jadhav Department of Electronics, Yeshwant Mahavidyalaya, Ned. Affiliated to
More informationEarthquake Simulation by Restricted Random Walks
1 Earthqake Simlation by Restricted Random Walks Steven N. Ward Institte of Geophysics and Planetary Physics University of California, Santa Crz Abstract. This article simlates earthqake slip distribtions
More informationPrandl established a universal velocity profile for flow parallel to the bed given by
EM 0--00 (Part VI) (g) The nderlayers shold be at least three thicknesses of the W 50 stone, bt never less than 0.3 m (Ahrens 98b). The thickness can be calclated sing Eqation VI-5-9 with a coefficient
More informationB-469 Simplified Copositive and Lagrangian Relaxations for Linearly Constrained Quadratic Optimization Problems in Continuous and Binary Variables
B-469 Simplified Copositive and Lagrangian Relaxations for Linearly Constrained Qadratic Optimization Problems in Continos and Binary Variables Naohiko Arima, Snyong Kim and Masakaz Kojima October 2012,
More information1 Undiscounted Problem (Deterministic)
Lectre 9: Linear Qadratic Control Problems 1 Undisconted Problem (Deterministic) Choose ( t ) 0 to Minimize (x trx t + tq t ) t=0 sbject to x t+1 = Ax t + B t, x 0 given. x t is an n-vector state, t a
More informationPropagation of measurement uncertainty in spatial characterisation of recreational fishing catch rates using logistic transform indicator kriging
st International Congress on Modelling and Simlation, Gold Coast, Astralia, 9 Nov to 4 Dec 05 www.mssan.org.a/modsim05 Propagation of measrement ncertainty in spatial characterisation of recreational fishing
More informationRegression techniques provide statistical analysis of relationships. Research designs may be classified as experimental or observational; regression
LOGISTIC REGRESSION Regression techniques provide statistical analysis of relationships. Research designs may be classified as eperimental or observational; regression analyses are applicable to both types.
More informationA Single Species in One Spatial Dimension
Lectre 6 A Single Species in One Spatial Dimension Reading: Material similar to that in this section of the corse appears in Sections 1. and 13.5 of James D. Mrray (), Mathematical Biology I: An introction,
More informationFEA Solution Procedure
EA Soltion rocedre (demonstrated with a -D bar element problem) MAE - inite Element Analysis Many slides from this set are originally from B.S. Altan, Michigan Technological U. EA rocedre for Static Analysis.
More informationLeast-squares collocation with covariance-matching constraints
J Geodesy DOI 1.17/s19-7-133-5 ORIGINAL ARTICLE Least-sqares collocation with covariance-matching constraints Christopher Kotsakis Received: 18 Agst 26 / Accepted: 28 December 26 Springer-Verlag 27 Abstract
More informationChapter 4 Supervised learning:
Chapter 4 Spervised learning: Mltilayer Networks II Madaline Other Feedforward Networks Mltiple adalines of a sort as hidden nodes Weight change follows minimm distrbance principle Adaptive mlti-layer
More informationA Characterization of Interventional Distributions in Semi-Markovian Causal Models
A Characterization of Interventional Distribtions in Semi-Markovian Casal Models Jin Tian and Changsng Kang Department of Compter Science Iowa State University Ames, IA 50011 {jtian, cskang}@cs.iastate.ed
More informationINPUT-OUTPUT APPROACH NUMERICAL EXAMPLES
INPUT-OUTPUT APPROACH NUMERICAL EXAMPLES EXERCISE s consider the linear dnamical sstem of order 2 with transfer fnction with Determine the gain 2 (H) of the inpt-otpt operator H associated with this sstem.
More informationCubic graphs have bounded slope parameter
Cbic graphs have bonded slope parameter B. Keszegh, J. Pach, D. Pálvölgyi, and G. Tóth Agst 25, 2009 Abstract We show that every finite connected graph G with maximm degree three and with at least one
More informationComplex Variables. For ECON 397 Macroeconometrics Steve Cunningham
Comple Variables For ECON 397 Macroeconometrics Steve Cnningham Open Disks or Neighborhoods Deinition. The set o all points which satis the ineqalit
More informationLecture Notes: Finite Element Analysis, J.E. Akin, Rice University
9. TRUSS ANALYSIS... 1 9.1 PLANAR TRUSS... 1 9. SPACE TRUSS... 11 9.3 SUMMARY... 1 9.4 EXERCISES... 15 9. Trss analysis 9.1 Planar trss: The differential eqation for the eqilibrim of an elastic bar (above)
More information1 The space of linear transformations from R n to R m :
Math 540 Spring 20 Notes #4 Higher deriaties, Taylor s theorem The space of linear transformations from R n to R m We hae discssed linear transformations mapping R n to R m We can add sch linear transformations
More informationm = Average Rate of Change (Secant Slope) Example:
Average Rate o Change Secant Slope Deinition: The average change secant slope o a nction over a particlar interval [a, b] or [a, ]. Eample: What is the average rate o change o the nction over the interval
More informationPartial Differential Equations with Applications
Universit of Leeds MATH 33 Partial Differential Eqations with Applications Eamples to spplement Chapter on First Order PDEs Eample (Simple linear eqation, k + = 0, (, 0) = ϕ(), k a constant.) The characteristic
More informationCharacterizations of probability distributions via bivariate regression of record values
Metrika (2008) 68:51 64 DOI 10.1007/s00184-007-0142-7 Characterizations of probability distribtions via bivariate regression of record vales George P. Yanev M. Ahsanllah M. I. Beg Received: 4 October 2006
More informationDEFINITION OF A NEW UO 2 F 2 DENSITY LAW FOR LOW- MODERATED SOLUTIONS (H/U < 20) AND CONSEQUENCES ON CRITICALITY SAFETY
DEFINITION OF A NEW UO 2 F 2 DENSITY LAW FOR LOW- MODERATED SOLUTIONS ( < 20) AND CONSEQUENCES ON CRITICALITY SAFETY N. Leclaire, S. Evo, I.R.S.N., France Introdction In criticality stdies, the blk density
More informationA Model-Free Adaptive Control of Pulsed GTAW
A Model-Free Adaptive Control of Plsed GTAW F.L. Lv 1, S.B. Chen 1, and S.W. Dai 1 Institte of Welding Technology, Shanghai Jiao Tong University, Shanghai 00030, P.R. China Department of Atomatic Control,
More informationVectors in Rn un. This definition of norm is an extension of the Pythagorean Theorem. Consider the vector u = (5, 8) in R 2
MATH 307 Vectors in Rn Dr. Neal, WKU Matrices of dimension 1 n can be thoght of as coordinates, or ectors, in n- dimensional space R n. We can perform special calclations on these ectors. In particlar,
More informationDecision Oriented Bayesian Design of Experiments
Decision Oriented Bayesian Design of Experiments Farminder S. Anand*, Jay H. Lee**, Matthew J. Realff*** *School of Chemical & Biomoleclar Engineering Georgia Institte of echnology, Atlanta, GA 3332 USA
More informationPubH 7470: STATISTICS FOR TRANSLATIONAL & CLINICAL RESEARCH
PubH 7470: STATISTICS FOR TRANSLATIONAL & CLINICAL RESEARCH The First Step: SAMPLE SIZE DETERMINATION THE ULTIMATE GOAL The most important, ultimate step of any of clinical research is to do draw inferences;
More informationOptimization via the Hamilton-Jacobi-Bellman Method: Theory and Applications
Optimization via the Hamilton-Jacobi-Bellman Method: Theory and Applications Navin Khaneja lectre notes taken by Christiane Koch Jne 24, 29 1 Variation yields a classical Hamiltonian system Sppose that
More informationChapter 3 MATHEMATICAL MODELING OF DYNAMIC SYSTEMS
Chapter 3 MATHEMATICAL MODELING OF DYNAMIC SYSTEMS 3. System Modeling Mathematical Modeling In designing control systems we mst be able to model engineered system dynamics. The model of a dynamic system
More informationHome Range Formation in Wolves Due to Scent Marking
Blletin of Mathematical Biology () 64, 61 84 doi:1.16/blm.1.73 Available online at http://www.idealibrary.com on Home Range Formation in Wolves De to Scent Marking BRIAN K. BRISCOE, MARK A. LEWIS AND STEPHEN
More informationFOUNTAIN codes [3], [4] provide an efficient solution
Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design Francisco Lázaro, Stdent Member, IEEE, Gianligi Liva, Senior Member, IEEE, Gerhard Bach, Fellow, IEEE arxiv:176.5814v1 [cs.it 19 Jn
More informationOptimal Control, Statistics and Path Planning
PERGAMON Mathematical and Compter Modelling 33 (21) 237 253 www.elsevier.nl/locate/mcm Optimal Control, Statistics and Path Planning C. F. Martin and Shan Sn Department of Mathematics and Statistics Texas
More informationTheorem (Change of Variables Theorem):
Avance Higher Notes (Unit ) Prereqisites: Integrating (a + b) n, sin (a + b) an cos (a + b); erivatives of tan, sec, cosec, cot, e an ln ; sm/ifference rles; areas ner an between crves. Maths Applications:
More informationDiscussion Papers Department of Economics University of Copenhagen
Discssion Papers Department of Economics University of Copenhagen No. 10-06 Discssion of The Forward Search: Theory and Data Analysis, by Anthony C. Atkinson, Marco Riani, and Andrea Ceroli Søren Johansen,
More informationON THE SHAPES OF BILATERAL GAMMA DENSITIES
ON THE SHAPES OF BILATERAL GAMMA DENSITIES UWE KÜCHLER, STEFAN TAPPE Abstract. We investigate the for parameter family of bilateral Gamma distribtions. The goal of this paper is to provide a thorogh treatment
More information3. Several Random Variables
. Several Random Variables. To Random Variables. Conditional Probabilit--Revisited. Statistical Independence.4 Correlation beteen Random Variables Standardied (or ero mean normalied) random variables.5
More informationLatent Differential Equation Modeling with Multivariate Multi-Occasion Indicators
Latent Differential Eqation Modeling with Mltivariate Mlti-Occasion Indicators Steven M. Boker University of Notre Dame Michael C. Neale Medical College of Virginia Joseph R. Rasch University of Notre
More informationGeometric Image Manipulation. Lecture #4 Wednesday, January 24, 2018
Geometric Image Maniplation Lectre 4 Wednesda, Janar 4, 08 Programming Assignment Image Maniplation: Contet To start with the obvios, an image is a D arra of piels Piel locations represent points on the
More information1. State-Space Linear Systems 2. Block Diagrams 3. Exercises
LECTURE 1 State-Space Linear Sstems This lectre introdces state-space linear sstems, which are the main focs of this book. Contents 1. State-Space Linear Sstems 2. Block Diagrams 3. Exercises 1.1 State-Space
More information