# Approximate Conditional-mean Type Filtering for State-space Models

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Approimate Conditional-mean Tpe Filtering for State-space Models Bernhard Spangl, Universität für Bodenkultur, Wien joint work with Peter Ruckdeschel, Fraunhofer Institut, Kaiserslautern, and Rudi Dutter, Technische Univeristät, Wien UseR! 8, Dortmund, German B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

2 Contents State-space models & Kalman filter Multivariate -tpe filter filter Simulation stud Results R package robkalman Remarks & Outlook B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. /

3 Linear State Space Models State equation: t = Φ t 1 + ε t Observation equation: t = H t + v t Ideal model assumptions: N p (µ,σ ), ε t N p (,Q), v t N q (,R), all independent B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 3/

4 Classical Kalman Filter Initialization (t = ): = µ, P = Σ Prediction (t 1): t t 1 = Φ t 1 t 1 M t = ΦP t 1 Φ + Q = Cov( t t 1 ) Correction (t 1): t t = t t 1 + K t ( t H t t 1 ) P t = M t K t HM t = Cov( t t ) with K t = M t H (HMH + R) 1 (Kalman gain) B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. /

5 Tpes of Outliers Innovational Outliers (IO s): state equation is contaminated not considered here Additive Outliers (AO s): observations are contaminated error process v t is affected possible model: CN q (γ,r,r c ) = (1 γ)n q (,R) + γn q (µ c,r c ) Other Tpes of Outliers: substitutive outliers (SO s) patch outliers B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 5/

6 Masreliez s Theorem (1975) If t Y t 1 N p ( t t 1,M t ), t 1, then t t = E( t Y t ), t 1, is generated b the recursions t t = t t 1 + M t H Ψ t ( t ) P t = M t M t H Ψ t( t )HM t M t+1 = ΦP t Φ + Q, with (Ψ t ()) i = ( / i ) log f t ( Y t 1 ) and (Ψ t ()) ij = ( / j )(Ψ t ()) i. Ψ t () is called the score function. Note: If f t (. Y t 1 ) is Gaussian, Masreliez s filter reduces to the Kalman filter. B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 6/

7 The Score Function Ψ t (a) (b).1.5 (c) (d) 1 d/d d/d 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 7/

8 Multivariate -tpe Filter approimate conditional-mean () tpe filter proposed b B. Spangl and R. Dutter (8) modified correction step: t t = t t 1 + M t H S t ψ(s t ( t H t t 1 )) P t = M t M t H S t ψ (S t ( t H t t 1 ))S t HM t for an S t and a ψ-function appropriatel chosen in the case univariate observations equivalent to Martin s tpe filter (Martin, 1979) B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 8/

9 Huber s Multivariate Psi-function (a) (b) 1 1 coord. coord. 1 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 9/

10 Hampel s Multivariate Psi-function (a) (b) 1 1 coord. coord. 1 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

11 Approimating the Score Function (a) (b) 1 d/d d/d 1 (c) (d) 1 coord. coord. 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 11/

12 Filter proposed b P. Ruckdeschel (1) modified correction step: t t = t t 1 + H b (K t ( t H t t 1 )) with H b (z) = z min{1,b/ z } and. the Euclidean norm optimal for SO s in some sense B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

13 Simulation State Space Process: simulate state space process using two different sets of hper parameters and AO s from two different contamination setups: N (, 1 1 ) or N ( 5 3,.9.9 ). var contamination γ from % to % b 5% each s Filtering: robust filtering (, ) Evaluation: compare with state process via MSE B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 13/

14 Simulation (cont.) Eample I: µ =, Φ = , Q = 3 3, Σ =, H = 1 1 1, R =...5. Eample II: µ =, Φ = 1 1, Q = 9, Σ =, H = , R = 9 9. B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

15 Results contamination level % contamination level 1% 1. coordinate of state process contamination level %. coordinate of state process coordinate of state process 1 1 contamination level 1%. coordinate of state process contamination level % contamination level % coordinate of state process 1. coordinate of state process B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 15/

16 Results (cont.) contamination level % contamination level 1% 1. coordinate of state process contamination level %. coordinate of state process coordinate of state process contamination level 1%. coordinate of state process contamination level % contamination level % coordinate of state process 1. coordinate of state process B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 16/

17 Results (cont.) MSE (a) MSE 1 3 (b) contamination in % contamination in % B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 17/

18 The R package robkalman general function recursivefilter with parameters: observations state-space model (hper parameters) functions for the init./pred./corr. step available filters: KalmanFilter, Filter, filter, mfilter all: wrappers to recursivefilter B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 18/

19 Remarks & Outlook performs better than for both contamination situations ields larger errors in the case of % contamination because it was calibrated to a loss of efficienc δ = 1% all simulations were made with R R-package robkalman for filtering alread eists (but is still under construction!) robkalman/ S classes for state-space models and filtering results B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 19/

20 References R.D. Martin (1979). Approimate Conditional-mean Tpe Smoothers and Interpolators. In Gasser and Rossenblatt (eds.), Smoothing Techniques for Curve Estimation, , Springer, Berlin. C.J. Masreliez (1975). Approimate non-gaussian filtering with linear state and observation relations. IEEE Transactions on Automatic Control,, R Development Core Team (5). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. P. Ruckdeschel (1). Ansätze zur Robustifizierung des Kalman-Filters. PhD thesis, Universität Bareuth, Bareuth. P. Ruckdeschel and B. Spangl (7). robkalman: An R package for robust Kalman filtering. Web: B. Spangl and R. Dutter (8). Approimate Conditional-mean Tpe Filtering for Vector-valued Observations. Technical Report TR-AS-8-1, Universität für Bodenkultur, Vienna. B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. /

### Robust forecasting with exponential and Holt-Winters smoothing

Faculty of Economics and Applied Economics Robust forecasting with exponential and Holt-Winters smoothing Sarah Gelper, Roland Fried and Christophe Croux DEPARTMENT OF DECISION SCIENCES AND INFORMATION

### Robust Forecasting with Exponential and Holt-Winters Smoothing

Robust Forecasting with Exponential and Holt-Winters Smoothing Sarah Gelper 1,, Roland Fried 2, Christophe Croux 1 September 26, 2008 1 Faculty of Business and Economics, Katholieke Universiteit Leuven,

### Iterative Solvers for Linear Systems

9th SimLab Course on Parallel Numerical Simulation, 4.10 8.10.2010 Iterative Solvers for Linear Systems Bernhard Gatzhammer Chair of Scientific Computing in Computer Science Technische Universität München

### On-line outlier detection and data cleaning

Computers and Chemical Engineering 8 (004) 1635 1647 On-line outlier detection and data cleaning Hancong Liu a, Sirish Shah a,, Wei Jiang b a Department of Chemical and Materials Engineering, University

### The GMWM: A New Framework for Inertial Sensor Calibration

The GMWM: A New Framework for Inertial Sensor Calibration Roberto Molinari Research Center for Statistics (GSEM) University of Geneva joint work with S. Guerrier (UIUC), J. Balamuta (UIUC), J. Skaloud

### Moving Least Squares Approximation

Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the

### Section 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions

Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano,

### Time Series Analysis III

Lecture 12: Time Series Analysis III MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis III 1 Outline Time Series Analysis III 1 Time Series Analysis III MIT 18.S096 Time Series Analysis

### An EM algorithm for the estimation of a ne state-space systems with or without known inputs

An EM algorithm for the estimation of a ne state-space systems with or without known inputs Alexander W Blocker January 008 Abstract We derive an EM algorithm for the estimation of a ne Gaussian state-space

### Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations

Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.

### LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

### CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION Vol. III Design of State Space Controllers (Pole Placement) for SISO Systems - Lohmann, Boris

DESIGN OF STATE SPACE CONTROLLERS (POLE PLACEMENT) FOR SISO SYSTEMS Lohmann, Boris Institut für Automatisierungstechnik, Universität Bremen, Germany Keywords: State space controller, state feedback, output

### 15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

### Tolerance of Radial Basis Functions against Stuck-At-Faults

Tolerance of Radial Basis Functions against Stuck-At-Faults Ralf Eickhoff 1 and Ulrich Rückert 1 Heinz Nixdorf Institute System and Circuit Technology University of Paderborn, Germany eickhoff,rueckert@hni.upb.de

### Graphing Nonlinear Systems

10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities

### Marshall-Olkin distributions and portfolio credit risk

Marshall-Olkin distributions and portfolio credit risk Moderne Finanzmathematik und ihre Anwendungen für Banken und Versicherungen, Fraunhofer ITWM, Kaiserslautern, in Kooperation mit der TU München und

### Optimization of Supply Chain Networks

Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal

### x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac

Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square

### 7.1 LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS

7.1 LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the method of substitution to solve systems of linear equations in two variables.

### Introduction Bilateral Filtering Results. Bilateral Filtering. Mathias Eitz. TU Berlin. November, 21st 2006

Introduction TU Berlin November, 21st 2006 About Me Introduction Student at TU Berlin since 2002 eitz@cs.tu-berlin.de Outline Introduction 1 Introduction Smoothing Filters Comparison 2 Intuition Mathematical

### Cross Validation. Dr. Thomas Jensen Expedia.com

Cross Validation Dr. Thomas Jensen Expedia.com About Me PhD from ETH Used to be a statistician at Link, now Senior Business Analyst at Expedia Manage a database with 720,000 Hotels that are not on contract

### Reasoning with Equations and Inequalities

Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.

### Ensemble-based reservoir history matching for complex geology and seismic data. Yanhui Zhang

Ensemble-based reservoir history matching for complex geology and seismic data Yanhui Zhang Outline 1 Overview 2 Part I: Updating of geologic facies models with ensemble-based methods 3 Part II: Data assimilation

### Modelling electricity market data: the CARMA spot model, forward prices and the risk premium

Modelling electricity market data: the CARMA spot model, forward prices and the risk premium Formatvorlage des Untertitelmasters Claudia Klüppelberg durch Klicken bearbeiten Technische Universität München

### SAS Fraud Framework for Banking

SAS Fraud Framework for Banking Including Social Network Analysis John C. Brocklebank, Ph.D. Vice President, SAS Solutions OnDemand Advanced Analytics Lab SAS Fraud Framework for Banking Agenda Introduction

### Nonlinear Systems of Ordinary Differential Equations

Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally

### Software and Hardware Solutions for Accurate Data and Profitable Operations. Miguel J. Donald J. Chmielewski Contributor. DuyQuang Nguyen Tanth

Smart Process Plants Software and Hardware Solutions for Accurate Data and Profitable Operations Miguel J. Bagajewicz, Ph.D. University of Oklahoma Donald J. Chmielewski Contributor DuyQuang Nguyen Tanth

### Exponential and Logarithmic Functions

Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course

### indicate how accurate the is and how we are that the result is correct. We use intervals for this purpose.

Sect. 6.1: Confidence Intervals Statistical Confidence When we calculate an, say, of the population parameter, we want to indicate how accurate the is and how we are that the result is correct. We use

### Robust Neural Network Regression for Offline and Online Learning

Robust Neural Network Regression for Offline and Online Learning homas Briegel* Siemens AG, Corporate echnology D-81730 Munich, Germany thomas.briegel@mchp.siemens.de Volker resp Siemens AG, Corporate

### A Comparative Study of the Pickup Method and its Variations Using a Simulated Hotel Reservation Data

A Comparative Study of the Pickup Method and its Variations Using a Simulated Hotel Reservation Data Athanasius Zakhary, Neamat El Gayar Faculty of Computers and Information Cairo University, Giza, Egypt

### C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)}

C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)} 1. EES 800: Econometrics I Simple linear regression and correlation analysis. Specification and estimation of a regression model. Interpretation of regression

### Matching Investment Strategies in General Insurance Is it Worth It? Aim of Presentation. Background 34TH ANNUAL GIRO CONVENTION

Matching Investment Strategies in General Insurance Is it Worth It? 34TH ANNUAL GIRO CONVENTION CELTIC MANOR RESORT, NEWPORT, WALES Aim of Presentation To answer a key question: What are the benefit of

### 9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES

86 CHAPTER 9 LINEAR PROGRAMMING 9. LINEAR PROGRAMMING INVOLVING TWO VARIABLES Figure 9.0 Feasible solutions Man applications in business and economics involve a process called optimization, in which we

### Particle Filter-Based On-Line Estimation of Spot (Cross-)Volatility with Nonlinear Market Microstructure Noise Models. Jan C.

Particle Filter-Based On-Line Estimation of Spot (Cross-)Volatility with Nonlinear Market Microstructure Noise Models Jan C. Neddermeyer (joint work with Rainer Dahlhaus, University of Heidelberg) DZ BANK

### Exponential Functions

Eponential Functions In this chapter we will study the eponential function and its inverse the logarithmic function. These important functions are indispensable in working with problems that involve population

### 6.1 Exponential and Logarithmic Functions

Section 6.1 Eponential and Logarithmic Functions 1 6.1 Eponential and Logarithmic Functions We start our review o eponential and logarithmic unctions with the deinition o an eponential unction. Deinition

### Lesson 5.4 Exercises, pages

Lesson 5.4 Eercises, pages 8 85 A 4. Evaluate each logarithm. a) log 4 6 b) log 00 000 4 log 0 0 5 5 c) log 6 6 d) log log 6 6 4 4 5. Write each eponential epression as a logarithmic epression. a) 6 64

### Density nowcasts of euro area real GDP growth: does pooling matter?

Density nowcasts of euro area real GDP growth: does pooling matter? Marta Bańbura 1 and Lorena Saiz 2 Abstract In this paper we review different strategies to combine and evaluate individual density forecasts

### SECTION 1-4 Absolute Value in Equations and Inequalities

1-4 Absolute Value in Equations and Inequalities 37 SECTION 1-4 Absolute Value in Equations and Inequalities Absolute Value and Distance Absolute Value in Equations and Inequalities Absolute Value and

### Solving Linear Recurrence Relations. Niloufar Shafiei

Solving Linear Recurrence Relations Niloufar Shafiei Review A recursive definition of a sequence specifies Initial conditions Recurrence relation Example: a 0 =0 and a 1 =3 Initial conditions a n = 2a

### Option Valuation Using Intraday Data

Option Valuation Using Intraday Data Peter Christoffersen Rotman School of Management, University of Toronto, Copenhagen Business School, and CREATES, University of Aarhus 2nd Lecture on Thursday 1 Model

### Stock Option Pricing Using Bayes Filters

Stock Option Pricing Using Bayes Filters Lin Liao liaolin@cs.washington.edu Abstract When using Black-Scholes formula to price options, the key is the estimation of the stochastic return variance. In this

### Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

### 1. (16 pts) D = 5000/yr, C = 600/unit, 1 year = 300 days, i = 0.06, A = 300 Current ordering amount Q = 200

HW 2 Solution 1. (16 pts) D = 5000/yr, C = 600/unit, 1 year = 300 days, i = 0.06, A = 300 Current ordering amount Q = 200 (a) T * = (b) Total(Holding + Setup) cost would be (c) The optimum cost would be

### Symbolic Determinants: Calculating the Degree

Symbolic Determinants: Calculating the Degree Technical Report by Brent M. Dingle Texas A&M University Original: May 4 Updated: July 5 Abstract: There are many methods for calculating the determinant of

### Lecture 3: DC Analysis of Diode Circuits.

Whites, EE 320 Lecture 3 Page 1 of 10 Lecture 3: DC Analysis of Diode Circuits. We ll now move on to the DC analysis of diode circuits. Applications will be covered in following lectures. Let s consider

### 11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems

a b Objectives Solve maimum minimum problems involving quadratic functions. Fit a quadratic function to a set of data to form a mathematical model, and solve related applied problems. 11.7 MATHEMATICAL

### Extreme Value Modeling for Detection and Attribution of Climate Extremes

Extreme Value Modeling for Detection and Attribution of Climate Extremes Jun Yan, Yujing Jiang Joint work with Zhuo Wang, Xuebin Zhang Department of Statistics, University of Connecticut February 2, 2016

### ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems

Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed

### Applied Spatial Statistics in R, Section 5

Applied Spatial Statistics in R, Section 5 Geostatistics Yuri M. Zhukov IQSS, Harvard University January 16, 2010 Yuri M. Zhukov (IQSS, Harvard University) Applied Spatial Statistics in R, Section 5 January

### Geostatistics Exploratory Analysis

Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa Master of Science in Geospatial Technologies Geostatistics Exploratory Analysis Carlos Alberto Felgueiras cfelgueiras@isegi.unl.pt

### Example 1: Calculate and compare RiskMetrics TM and Historical Standard Deviation Compare the weights of the volatility parameter using,, and.

3.6 Compare and contrast different parametric and non-parametric approaches for estimating conditional volatility. 3.7 Calculate conditional volatility using parametric and non-parametric approaches. Parametric

### Curriculum vitae. July 2007 present Professor of Mathematics (W3), Technische

Peter Bank Institut für Mathematik, Sekr. MA 7-1 Straße des 17. Juni 136 10623 Berlin Germany Tel.: +49 (30) 314-22816 Fax.: +49 (30) 314-24413 e-mail: bank@math.tu-berlin.edu URL: www.math.tu-berlin.de/

### On exponentially ane martingales. Johannes Muhle-Karbe

On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes Muhle-Karbe Joint work with Jan Kallsen HVB-Institut für Finanzmathematik, Technische Universität München 1 Outline 1. Semimartingale characteristics

### Inference of the Nugget Effect and Variogram Range with Sample Compositing

Inference of the Nugget Effect and Variogram Range with Sample Compositing David F. Machuca Mory and Clayton V. Deutsch Centre for Computational Geostatistics Department of Civil and Environmental Engineering

### How can you construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot

9. Scatter Plots How can ou construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot Work with a partner. The weights (in ounces) and circumferences C (in inches) of several sports

### MATH Area Between Curves

MATH - Area Between Curves Philippe Laval September, 8 Abstract This handout discusses techniques used to nd the area of regions which lie between two curves. Area Between Curves. Theor Given two functions

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### Uintah Framework. Justin Luitjens, Qingyu Meng, John Schmidt, Martin Berzins, Todd Harman, Chuch Wight, Steven Parker, et al

Uintah Framework Justin Luitjens, Qingyu Meng, John Schmidt, Martin Berzins, Todd Harman, Chuch Wight, Steven Parker, et al Uintah Parallel Computing Framework Uintah - far-sighted design by Steve Parker

### The Master of Science in Finance (English Program) - MSF. Department of Banking and Finance. Chulalongkorn Business School. Chulalongkorn University

The Master of Science in Finance (English Program) - MSF Department of Banking and Finance Chulalongkorn Business School Chulalongkorn University Overview of Program Structure Full Time Program: 1 Year

### 5. Linear regression and correlation

Statistics for Engineers 5-1 5. Linear regression and correlation If we measure a response variable at various values of a controlled variable, linear regression is the process of fitting a straight line

### Transformations and Expectations of random variables

Transformations and Epectations of random variables X F X (): a random variable X distributed with CDF F X. Any function Y = g(x) is also a random variable. If both X, and Y are continuous random variables,

### Discrete Frobenius-Perron Tracking

Discrete Frobenius-Perron Tracing Barend J. van Wy and Michaël A. van Wy French South-African Technical Institute in Electronics at the Tshwane University of Technology Staatsartillerie Road, Pretoria,

### MATH 590: Meshfree Methods

MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter

### z 0 and y even had the form

Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z

### State Space Time Series Analysis

State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman http://staff.feweb.vu.nl/koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011 State

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### Classifying Manipulation Primitives from Visual Data

Classifying Manipulation Primitives from Visual Data Sandy Huang and Dylan Hadfield-Menell Abstract One approach to learning from demonstrations in robotics is to make use of a classifier to predict if

### Customer Intimacy Analytics

Customer Intimacy Analytics Leveraging Operational Data to Assess Customer Knowledge and Relationships and to Measure their Business Impact by Francois Habryn Scientific Publishing CUSTOMER INTIMACY ANALYTICS

### The Distance Formula and the Circle

10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

### Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

### ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

### Regression Modeling Strategies

Frank E. Harrell, Jr. Regression Modeling Strategies With Applications to Linear Models, Logistic Regression, and Survival Analysis With 141 Figures Springer Contents Preface Typographical Conventions

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.

### Stability of the LMS Adaptive Filter by Means of a State Equation

Stability of the LMS Adaptive Filter by Means of a State Equation Vítor H. Nascimento and Ali H. Sayed Electrical Engineering Department University of California Los Angeles, CA 90095 Abstract This work

### 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

### 2. Filling Data Gaps, Data validation & Descriptive Statistics

2. Filling Data Gaps, Data validation & Descriptive Statistics Dr. Prasad Modak Background Data collected from field may suffer from these problems Data may contain gaps ( = no readings during this period)

### SERVO CONTROL SYSTEMS 1: DC Servomechanisms

Servo Control Sstems : DC Servomechanisms SERVO CONTROL SYSTEMS : DC Servomechanisms Elke Laubwald: Visiting Consultant, control sstems principles.co.uk ABSTRACT: This is one of a series of white papers

### A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking

174 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 2, FEBRUARY 2002 A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking M. Sanjeev Arulampalam, Simon Maskell, Neil

### Research Memorandum. In this memorandum we describe how we have used monthly series to construct monthly estimates of GDP and GDI.

Research Memorandum From: James H. Stock and Mark W. Watson Date: September 19, 2010 Subj: Distribution of quarterly values of GDP/GDI across months within the quarter Background For many purposes it is

### Towards a Performance Model Management Repository for Component-based Enterprise Applications

Austin, TX, USA, 2015-02-04 Towards a Performance Model Management Repository for Component-based Enterprise Applications Work-in-Progress Paper (WiP) International Conference on Performance Engineering

### 4F7 Adaptive Filters (and Spectrum Estimation) Kalman Filter. Sumeetpal Singh Email : sss40@eng.cam.ac.uk

4F7 Adaptive Filters (and Spectrum Estimation) Kalman Filter Sumeetpal Singh Email : sss40@eng.cam.ac.uk 1 1 Outline State space model Kalman filter Examples 2 2 Parameter Estimation We have repeated observations

### PTE505: Inverse Modeling for Subsurface Flow Data Integration (3 Units)

PTE505: Inverse Modeling for Subsurface Flow Data Integration (3 Units) Instructor: Behnam Jafarpour, Mork Family Department of Chemical Engineering and Material Science Petroleum Engineering, HED 313,

### A Reliability Point and Kalman Filter-based Vehicle Tracking Technique

A Reliability Point and Kalman Filter-based Vehicle Tracing Technique Soo Siang Teoh and Thomas Bräunl Abstract This paper introduces a technique for tracing the movement of vehicles in consecutive video

### Systems of Equations. from Campus to Careers Fashion Designer

Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.

### Chapter 13 Introduction to Linear Regression and Correlation Analysis

Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing

### Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

### 2.1 Three Dimensional Curves and Surfaces

. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

### Master s thesis tutorial: part III

for the Autonomous Compliant Research group Tinne De Laet, Wilm Decré, Diederik Verscheure Katholieke Universiteit Leuven, Department of Mechanical Engineering, PMA Division 30 oktober 2006 Outline General

### Introduction to Time Series Analysis. Lecture 1.

Introduction to Time Series Analysis. Lecture 1. Peter Bartlett 1. Organizational issues. 2. Objectives of time series analysis. Examples. 3. Overview of the course. 4. Time series models. 5. Time series

### 1. χ 2 minimization 2. Fits in case of of systematic errors

Data fitting Volker Blobel University of Hamburg March 2005 1. χ 2 minimization 2. Fits in case of of systematic errors Keys during display: enter = next page; = next page; = previous page; home = first

### 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

### Capturing Real-Time Power System Dynamics: Opportunities and Challenges. Zhenyu(Henry) Huang, PNNL, Zhenyu.huang@pnnl.gov

1 Capturing Real-Time Power System Dynamics: Opportunities and Challenges Zhenyu(Henry) Huang, PNNL, Zhenyu.huang@pnnl.gov Frequency trending away from nominal requires tools capturing real-time dynamics

### Chapter 5 Applications of Integration

MA111 Application of Integration Asst.Prof.Dr.Supranee Lisawadi 1 Chapter 5 Applications of Integration Section 5.1 Area Between Two Curves In this section we use integrals to find areas of regions that

### Deconvolution of Atomic Force Measurements in Special Modes Methodology and Application

1 Deconvolution of Atomic Force Measurements in Special Modes Methodolog and Application Dipl.-Ing. T. Machleidt PD Dr.-Ing. habil. K.-H. Franke Dipl.-Ing. E. Sparrer Computer Graphics Group / TU-Ilmenau