Authors. Introduction. Introduction

Size: px
Start display at page:

Download "Authors. Introduction. Introduction"

Transcription

1 Auhors Hidden Applied in Agriculural Crops Classificaion Caholic Universiy of Rio de Janeiro (PUC-Rio Paula B. C. Leie Raul Q. Feiosa Gilson A. O. P. Cosa Hidden Applied in Agriculural Crops Classificaion Gilson A. O. P. Cosa Brazilian Naional Insiue for Space Research (INPE Anonio R. Formaggio Ieda D. A. Sanches Leibniz Hannover Universiy (IPI Kian Pakzad Conens Inroducion Inroducion Objecives Plan Phenology Mehodology Experimens Conclusions Who is growing wha and where? Reliable, up-o-dae informaion abou agriculural aciviies: Suppor (global and local sraegic decisions Developmen of commercial plans Decisions regarding subsidies Regulaion of inernal socks Price formaion Inroducion Inroducion How can RS and OBIA be used o classify crops? How can RS and OBIA be used o classify crops? Landsa 7 Feb 3 Landsa 7 Feb 3 Basic problems Differen crops may look similar in a RS image. The same crop may look differen in differen pars of he year. Corn Soybean Proposed approach Objec based image analysis segmens insead of pixels. Muliemporal analysis images of he same region a differen poins in ime. Classificaion model explores knowledge of phenological cycles. Probabilisic ool: Hidden Markov Model.

2 Objecives General Evaluae he poenial of Hidden for classificaion of agriculural crops from RS emporal image sequences. Specific Develop an HMM-based mehod o idenify differen agriculural crops. Evaluae he proposed mehod wih a sequence of medium resoluion saellie images. Plan Phenology The sudy of periodic plan life cycle evens relaive growh Corn Soybean ime Plan Phenology Plan Phenology Plan phenology describes he life cycles of differen species. Can be used o differeniae crops: Some crops ake longer o grow or o achieve mauriy; Some crops have shor cycles: annual crops (corn, soybean; Some crops are semi-perennial (sugar-cane. Plan life cycles can be divided ino phenological sages. In his work we considered four phenological sages: Prepared Soil Growh Adul Phase Pos-Harvesing And five culures: Sugar-cane, Soybean, Corn, Pasure, Riparian Fores Memoryless processes (Markov propery: given he presen sae, fuure saes are independen of he pas saes. Pr( + = x = x, = x,..., = x = Pr( = x = x + Memoryless processes (Markov propery: given he presen sae, fuure saes are independen of he pas saes. Pr( + = x = x, = x,..., = x = Pr( = x = x + Probabiliy of going from one sae o anoher is given by ransiion probabiliies. Pr( + = S j = Si = a ij

3 Memoryless processes (Markov propery: given he presen sae, fuure saes are independen of he pas saes. Pr( + = x = x, = x,..., = x = Pr( = x = x + Probabiliy of going from one sae o anoher is given by ransiion probabiliies. Memoryless processes (Markov propery: given he presen sae, fuure saes are independen of he pas saes. Pr( + = x = x, = x,..., = x = Pr( = x = x + Probabiliy of going from one sae o anoher is given by ransiion probabiliies. a a A = a3 a4 a a a3 a4 a3 a3 a33 a43 a4 a4 a34 a44 a a a 33 S S 3 a a 3 a 34 a 4 a 44 S S S 3 S 4 a A = a4 a a a3 a 34 a a a 33 S S 3 a a 3 a 34 a 4 a 44 S S S 3 S 4 Memoryless processes (Markov propery: given he presen sae, fuure saes are independen of he pas saes. Pr( + = x = x, = x,..., = x = Pr( = x = x + Probabiliy of going from one sae o anoher is given by ransiion probabiliies. Hidden Markov Model (HMM Saes are no direcly observable: hey emi symbols wih b jk probabiliies. a a S S... S n b b m b a a 3 a kn v v v 3 b n... a n bn3 a nn b nm v m S i Saes a ij Transiion probabiliy b jk Symbol emission probabiliy v k Possible observaion a A = a4 a a a3 a 34 PS Prepared Soil a a a 33 GR AP a a 3 a 34 Growh Adul Phase Phase a 4 a 44 PH Pos Harves A Hidden Markov Model is defined as λ = (A,B,π, where π i is he a-priori probabiliy ha he sysem is in a given sae S i a he iniial ime insan. Hidden Markov Model (HMM Saes are no direcly observable: hey emi symbols wih b jk probabiliies. Mehodology General Model Each crop class has is own model. Saes correspond o phenological sages: a n a a a nn S S... S n a a 3 a kn b m b n b b b nm bn3 S i Saes Transiion probabiliy a ij Symbol emission probabiliy b jk PS GR AP PH sugarcane soybean corn v v v 3... v m Possible observaion v k AP pasure riparian fores Given a sequence of observaions O = o, o,..., o T we can calculae he probabiliy P(O, λ ha a given model λ generaes he sequence O. Observable symbols are vecors wih he digial numbers of each specral band plus he NDVI.

4 Mehodology Problem deviaes from basic HMM descripion s no available for all epochs observaions are no made a regular inervals. Each crop has preferenial monhs for sowing a-priori probabiliy disribuion (π is no consan along he year. Symbol emission probabiliies (b jk depend on seasonal effecs ha can no be fully compensaed in he image preprocessing phase. Mehodology Fiing he Model o he Applicaion A, B, π are esimaed for each pair of images. We assumed a Gaussian disribuion for he symbol emision probabiliies. Mehodology Mehodology Fiing he Model o he Applicaion Fiing he Model o he Applicaion A, B, π are esimaed for each pair of images. A, B, π are esimaed for each pair of images. Emission probabiliy densiy of a symbol x consising of he specral bands and NDVI: (a vecor Emission probabiliy densiy of a symbol x consising of he specral bands and NDVI: (a vecor T ( x μ ( cs Σcs ( x μcs p = exp d π Σ cs T ( x μ ( cs Σcs ( x μcs p = exp d π Σ cs Where μ cs, and Σ cs denoe he mean vecor and he covariance marix for crop c and sae S i,andd is he dimension of x. Esimaion of ransiion probabiliies urns ino he problem of esimaing he mean μ cs and covariance marix Σ cs for each crop class and phenological sage. Mehodology sequence pre-processing Mehodology Classificaion HMM Geomeric Ahmospheric Correcion Correcion HMM parameers esimaion Radiomeric Normalizaion Crop Class c Segmenaion. Feaure Exracion Probabiliy Calculaion HMM n MA Crop Class Segmenaion Feaure Exracion HHM parameers Esimaion HMM c Segmenaion Feaure Exracion Probabiliy Calculaion Reference Classificaion

5 Experimens Sudy Area: Norhern São Paulo Sae (Brazil Experimens Reference daa: 36 poins. São Joaquim da Barra : Landsa 5/7 images Ipuã N W E S 8 8 Meers N Ipuã W E S São Joaquim da Barra 8 8 Meers Experimens Reference daa: visual inerpreaion plus wo field works Resuls Classificaion of segmens enclosing reference poins. Using mean DN and NDVI of pixels inside segmens for raining/hmm parameer esimaion. Crops Classificaion Raes (% Crops Classificaion Raes (% Soybeans (SB Corn (CO Sugarcane (SC Pasure (PA Riparian fores (RF Soybeans (SB Corn (CO Sugarcane (SC Pasure (PA Riparian fores (RF HMM-based classificaion Monoemporal classificaion (Maximum likelihood Resuls Classificaion of segmens enclosing reference poins. Using mean DN and NDVI of pixels inside segmens for HMM parameer esimaion. Resuls Classificaion of segmens enclosing reference poins. Using DN and NDVI of pixels inside segmens for HMM parameer esimaion. Crops Classificaion Soybeans (SB Corn (CO Sugarcane (SC Pasure (PA Riparian fores (RF Raes (% Confusion Marix (Crops SB CO SC PA RF SB 4 CO SC 8 79 PA 9 4 RF Crops Classificaion Soybeans (SB Corn (CO Sugarcane (SC Pasure (PA Riparian fores (RF Raes (% Confusion Marix (Crops SB CO SC PA RF SB 95 4 CO 7 SC 9 4 PA 3 RF 9 Saes Classificaion Prepared soil (PS Growh phase (GR Adul phase (AD Pos-harvesing (PH Raes (% Confusion Marix (Saes PS GR AD PH PS GR AD PH Saes Classificaion Prepared soil (PS Growh phase (GR Adul phase (AD Pos-harvesing (PH Raes (% Confusion Marix (Saes PS GR AD PH PS GR AD PH 5 35

6 Conclusions The End Remarkable superioriy of he HMM-based mehod over a monoemporal maximum likelihood classificaion approach. The performance of he approach was impaced by he scarciy of raining samples of some crop ypes. The approach also performed well wih respec o recogniion of phenological sages. The excepion was he Growh-Phase symbol vecors used o characerize his sage should also ake ino accoun he variaion of specral values hrough ime. Only sequences of daa associaed o one crop ype were considered. An analysis of he behaviour of he mehod considering sequences wih more han one crop ype is planned for fuure. Thank you! Gilson A. O. P. Cosa gilson@ele.puc-rio.br

0.1 MAXIMUM LIKELIHOOD ESTIMATION EXPLAINED

0.1 MAXIMUM LIKELIHOOD ESTIMATION EXPLAINED 0.1 MAXIMUM LIKELIHOOD ESTIMATIO EXPLAIED Maximum likelihood esimaion is a bes-fi saisical mehod for he esimaion of he values of he parameers of a sysem, based on a se of observaions of a random variable

More information

Reliability of Technical Systems

Reliability of Technical Systems eliabiliy of Technical Sysems Main Topics Inroducion, Key erms, framing he problem eliabiliy parameers: Failure ae, Failure Probabiliy, Availabiliy, ec. Some imporan reliabiliy disribuions Componen reliabiliy

More information

Speaker Adaptation Techniques For Continuous Speech Using Medium and Small Adaptation Data Sets. Constantinos Boulis

Speaker Adaptation Techniques For Continuous Speech Using Medium and Small Adaptation Data Sets. Constantinos Boulis Speaker Adapaion Techniques For Coninuous Speech Using Medium and Small Adapaion Daa Ses Consaninos Boulis Ouline of he Presenaion Inroducion o he speaker adapaion problem Maximum Likelihood Sochasic Transformaions

More information

Hidden Markov Models

Hidden Markov Models Hidden Markov Models Probabilisic reasoning over ime So far, we ve mosly deal wih episodic environmens Excepions: games wih muliple moves, planning In paricular, he Bayesian neworks we ve seen so far describe

More information

Isolated-word speech recognition using hidden Markov models

Isolated-word speech recognition using hidden Markov models Isolaed-word speech recogniion using hidden Markov models Håkon Sandsmark December 18, 21 1 Inroducion Speech recogniion is a challenging problem on which much work has been done he las decades. Some of

More information

Hidden Markov Models. Adapted from. Dr Catherine Sweeney-Reed s slides

Hidden Markov Models. Adapted from. Dr Catherine Sweeney-Reed s slides Hidden Markov Models Adaped from Dr Caherine Sweeney-Reed s slides Summary Inroducion Descripion Cenral in HMM modelling Exensions Demonsraion Specificaion of an HMM Descripion N - number of saes Q = {q

More information

Robust estimation based on the first- and third-moment restrictions of the power transformation model

Robust estimation based on the first- and third-moment restrictions of the power transformation model h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,

More information

Stochastic Model for Cancer Cell Growth through Single Forward Mutation

Stochastic Model for Cancer Cell Growth through Single Forward Mutation Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle 31 5-1-2017 Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com

More information

Object tracking: Using HMMs to estimate the geographical location of fish

Object tracking: Using HMMs to estimate the geographical location of fish Objec racking: Using HMMs o esimae he geographical locaion of fish 02433 - Hidden Markov Models Marin Wæver Pedersen, Henrik Madsen Course week 13 MWP, compiled June 8, 2011 Objecive: Locae fish from agging

More information

Testing for a Single Factor Model in the Multivariate State Space Framework

Testing for a Single Factor Model in the Multivariate State Space Framework esing for a Single Facor Model in he Mulivariae Sae Space Framework Chen C.-Y. M. Chiba and M. Kobayashi Inernaional Graduae School of Social Sciences Yokohama Naional Universiy Japan Faculy of Economics

More information

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check

More information

) were both constant and we brought them from under the integral.

) were both constant and we brought them from under the integral. YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha

More information

Application of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing

Application of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing Applicaion of a Sochasic-Fuzzy Approach o Modeling Opimal Discree Time Dynamical Sysems by Using Large Scale Daa Processing AA WALASZE-BABISZEWSA Deparmen of Compuer Engineering Opole Universiy of Technology

More information

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN

STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN 001-004 OBARA, Takashi * Absrac The

More information

5. Stochastic processes (1)

5. Stochastic processes (1) Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly

More information

GMM - Generalized Method of Moments

GMM - Generalized Method of Moments GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................

More information

The electromagnetic interference in case of onboard navy ships computers - a new approach

The electromagnetic interference in case of onboard navy ships computers - a new approach The elecromagneic inerference in case of onboard navy ships compuers - a new approach Prof. dr. ing. Alexandru SOTIR Naval Academy Mircea cel Bărân, Fulgerului Sree, Consanţa, soiralexandru@yahoo.com Absrac.

More information

Air Traffic Forecast Empirical Research Based on the MCMC Method

Air Traffic Forecast Empirical Research Based on the MCMC Method Compuer and Informaion Science; Vol. 5, No. 5; 0 ISSN 93-8989 E-ISSN 93-8997 Published by Canadian Cener of Science and Educaion Air Traffic Forecas Empirical Research Based on he MCMC Mehod Jian-bo Wang,

More information

Recent Developments In Evolutionary Data Assimilation And Model Uncertainty Estimation For Hydrologic Forecasting Hamid Moradkhani

Recent Developments In Evolutionary Data Assimilation And Model Uncertainty Estimation For Hydrologic Forecasting Hamid Moradkhani Feb 6-8, 208 Recen Developmens In Evoluionary Daa Assimilaion And Model Uncerainy Esimaion For Hydrologic Forecasing Hamid Moradkhani Cener for Complex Hydrosysems Research Deparmen of Civil, Consrucion

More information

Georey E. Hinton. University oftoronto. Technical Report CRG-TR February 22, Abstract

Georey E. Hinton. University oftoronto.   Technical Report CRG-TR February 22, Abstract Parameer Esimaion for Linear Dynamical Sysems Zoubin Ghahramani Georey E. Hinon Deparmen of Compuer Science Universiy oftorono 6 King's College Road Torono, Canada M5S A4 Email: zoubin@cs.orono.edu Technical

More information

Solutions to the Exam Digital Communications I given on the 11th of June = 111 and g 2. c 2

Solutions to the Exam Digital Communications I given on the 11th of June = 111 and g 2. c 2 Soluions o he Exam Digial Communicaions I given on he 11h of June 2007 Quesion 1 (14p) a) (2p) If X and Y are independen Gaussian variables, hen E [ XY ]=0 always. (Answer wih RUE or FALSE) ANSWER: False.

More information

Mechanical Fatigue and Load-Induced Aging of Loudspeaker Suspension. Wolfgang Klippel,

Mechanical Fatigue and Load-Induced Aging of Loudspeaker Suspension. Wolfgang Klippel, Mechanical Faigue and Load-Induced Aging of Loudspeaker Suspension Wolfgang Klippel, Insiue of Acousics and Speech Communicaion Dresden Universiy of Technology presened a he ALMA Symposium 2012, Las Vegas

More information

State-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter

State-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter Sae-Space Models Iniializaion, Esimaion and Smoohing of he Kalman Filer Iniializaion of he Kalman Filer The Kalman filer shows how o updae pas predicors and he corresponding predicion error variances when

More information

Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.

Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate. Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since

More information

Anno accademico 2006/2007. Davide Migliore

Anno accademico 2006/2007. Davide Migliore Roboica Anno accademico 2006/2007 Davide Migliore migliore@ele.polimi.i Today Eercise session: An Off-side roblem Robo Vision Task Measuring NBA layers erformance robabilisic Roboics Inroducion The Bayesian

More information

HIDDEN MARKOV MODELS APPLIED IN AGRICULTURAL CROPS CLASSIFICATION

HIDDEN MARKOV MODELS APPLIED IN AGRICULTURAL CROPS CLASSIFICATION HIDDEN MARKOV MODELS APPLIED IN AGRICULTURAL CROPS CLASSIFICATION P. B. C. Leite a*, R.Q. Feitosa a, A.R. Formaggio b, G. A. O. P. Costa a, K.Pakzad c,, I. D. A. Sanche b, a Catholic University of Rio

More information

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070

More information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in

More information

The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite

The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite American Journal of Operaions Research, 08, 8, 8-9 hp://wwwscirporg/journal/ajor ISSN Online: 60-8849 ISSN Prin: 60-8830 The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process

More information

Estimation of Kinetic Friction Coefficient for Sliding Rigid Block Nonstructural Components

Estimation of Kinetic Friction Coefficient for Sliding Rigid Block Nonstructural Components 7 Esimaion of Kineic Fricion Coefficien for Sliding Rigid Block Nonsrucural Componens Cagdas Kafali Ph.D. Candidae, School of Civil and Environmenal Engineering, Cornell Universiy Research Supervisor:

More information

Tracking. Announcements

Tracking. Announcements Tracking Tuesday, Nov 24 Krisen Grauman UT Ausin Announcemens Pse 5 ou onigh, due 12/4 Shorer assignmen Auo exension il 12/8 I will no hold office hours omorrow 5 6 pm due o Thanksgiving 1 Las ime: Moion

More information

Financial Econometrics Kalman Filter: some applications to Finance University of Evry - Master 2

Financial Econometrics Kalman Filter: some applications to Finance University of Evry - Master 2 Financial Economerics Kalman Filer: some applicaions o Finance Universiy of Evry - Maser 2 Eric Bouyé January 27, 2009 Conens 1 Sae-space models 2 2 The Scalar Kalman Filer 2 21 Presenaion 2 22 Summary

More information

Institute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler

Institute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion

More information

Chapter 2. Models, Censoring, and Likelihood for Failure-Time Data

Chapter 2. Models, Censoring, and Likelihood for Failure-Time Data Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based

More information

Development of a new metrological model for measuring of the water surface evaporation Tovmach L. Tovmach Yr. Abstract Introduction

Development of a new metrological model for measuring of the water surface evaporation Tovmach L. Tovmach Yr. Abstract Introduction Developmen of a new merological model for measuring of he waer surface evaporaion Tovmach L. Tovmach Yr. Sae Hydrological Insiue 23 Second Line, 199053 S. Peersburg, Russian Federaion Telephone (812) 323

More information

Excel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand

Excel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338

More information

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time. Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike

More information

Scholar Journal of Applied Sciences and Research

Scholar Journal of Applied Sciences and Research Scholar Journal of Applied Sciences and Research Oscillaion Energy of Plan Biological Time in Onogenesis Volume : 5 Mikael Makarovych Naumov * Odessa Sae Environmenal Universiy, Ukraine Absrac Background:

More information

Learning Naive Bayes Classifier from Noisy Data

Learning Naive Bayes Classifier from Noisy Data UCLA Compuer Science Deparmen Technical Repor CSD-TR No 030056 1 Learning Naive Bayes Classifier from Noisy Daa Yirong Yang, Yi Xia, Yun Chi, and Richard R Munz Universiy of California, Los Angeles, CA

More information

Sensors, Signals and Noise

Sensors, Signals and Noise Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1 Noise Descripion Noise Waveforms and Samples Saisics of Noise

More information

Evaluation of Mean Time to System Failure of a Repairable 3-out-of-4 System with Online Preventive Maintenance

Evaluation of Mean Time to System Failure of a Repairable 3-out-of-4 System with Online Preventive Maintenance American Journal of Applied Mahemaics and Saisics, 0, Vol., No., 9- Available online a hp://pubs.sciepub.com/ajams/// Science and Educaion Publishing DOI:0.69/ajams--- Evaluaion of Mean Time o Sysem Failure

More information

Data assimilation for local rainfall near Tokyo on 18 July 2013 using EnVAR with observation space localization

Data assimilation for local rainfall near Tokyo on 18 July 2013 using EnVAR with observation space localization Daa assimilaion for local rainfall near Tokyo on 18 July 2013 using EnVAR wih observaion space localizaion *1 Sho Yokoa, 1 Masaru Kunii, 1 Kazumasa Aonashi, 1 Seiji Origuchi, 2,1 Le Duc, 1 Takuya Kawabaa,

More information

Modal identification of structures from roving input data by means of maximum likelihood estimation of the state space model

Modal identification of structures from roving input data by means of maximum likelihood estimation of the state space model Modal idenificaion of srucures from roving inpu daa by means of maximum likelihood esimaion of he sae space model J. Cara, J. Juan, E. Alarcón Absrac The usual way o perform a forced vibraion es is o fix

More information

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid

More information

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon 3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of

More information

Recursive Least-Squares Fixed-Interval Smoother Using Covariance Information based on Innovation Approach in Linear Continuous Stochastic Systems

Recursive Least-Squares Fixed-Interval Smoother Using Covariance Information based on Innovation Approach in Linear Continuous Stochastic Systems 8 Froniers in Signal Processing, Vol. 1, No. 1, July 217 hps://dx.doi.org/1.2266/fsp.217.112 Recursive Leas-Squares Fixed-Inerval Smooher Using Covariance Informaion based on Innovaion Approach in Linear

More information

Fractional Method of Characteristics for Fractional Partial Differential Equations

Fractional Method of Characteristics for Fractional Partial Differential Equations Fracional Mehod of Characerisics for Fracional Parial Differenial Equaions Guo-cheng Wu* Modern Teile Insiue, Donghua Universiy, 188 Yan-an ilu Road, Shanghai 51, PR China Absrac The mehod of characerisics

More information

CS 4495 Computer Vision Hidden Markov Models

CS 4495 Computer Vision Hidden Markov Models CS 4495 Compuer Vision Aaron Bobick School of Ineracive Compuing Adminisrivia PS4 going OK? Please share your experiences on Piazza e.g. discovered somehing ha is suble abou using vl_sif. If you wan o

More information

Time series Decomposition method

Time series Decomposition method Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,

More information

A new flexible Weibull distribution

A new flexible Weibull distribution Communicaions for Saisical Applicaions and Mehods 2016, Vol. 23, No. 5, 399 409 hp://dx.doi.org/10.5351/csam.2016.23.5.399 Prin ISSN 2287-7843 / Online ISSN 2383-4757 A new flexible Weibull disribuion

More information

Vectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1

Vectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1 Vecorauoregressive Model and Coinegraion Analysis Par V Time Series Analysis Dr. Sevap Kesel 1 Vecorauoregression Vecor auoregression (VAR) is an economeric model used o capure he evoluion and he inerdependencies

More information

Semi-Competing Risks on A Trivariate Weibull Survival Model

Semi-Competing Risks on A Trivariate Weibull Survival Model Semi-Compeing Risks on A Trivariae Weibull Survival Model Jenq-Daw Lee Graduae Insiue of Poliical Economy Naional Cheng Kung Universiy Tainan Taiwan 70101 ROC Cheng K. Lee Loss Forecasing Home Loans &

More information

Internet Traffic Modeling for Efficient Network Research Management Prof. Zhili Sun, UniS Zhiyong Liu, CATR

Internet Traffic Modeling for Efficient Network Research Management Prof. Zhili Sun, UniS Zhiyong Liu, CATR Inerne Traffic Modeling for Efficien Nework Research Managemen Prof. Zhili Sun, UniS Zhiyong Liu, CATR UK-China Science Bridge Workshop 13-14 December 2011, London Ouline Inroducion Background Classical

More information

UNIVERSITY OF TRENTO MEASUREMENTS OF TRANSIENT PHENOMENA WITH DIGITAL OSCILLOSCOPES. Antonio Moschitta, Fabrizio Stefani, Dario Petri.

UNIVERSITY OF TRENTO MEASUREMENTS OF TRANSIENT PHENOMENA WITH DIGITAL OSCILLOSCOPES. Antonio Moschitta, Fabrizio Stefani, Dario Petri. UNIVERSIY OF RENO DEPARMEN OF INFORMAION AND COMMUNICAION ECHNOLOGY 385 Povo reno Ialy Via Sommarive 4 hp://www.di.unin.i MEASUREMENS OF RANSIEN PHENOMENA WIH DIGIAL OSCILLOSCOPES Anonio Moschia Fabrizio

More information

Stochastic Structural Dynamics. Lecture-6

Stochastic Structural Dynamics. Lecture-6 Sochasic Srucural Dynamics Lecure-6 Random processes- Dr C S Manohar Deparmen of Civil Engineering Professor of Srucural Engineering Indian Insiue of Science Bangalore 560 0 India manohar@civil.iisc.erne.in

More information

1 Differential Equation Investigations using Customizable

1 Differential Equation Investigations using Customizable Differenial Equaion Invesigaions using Cusomizable Mahles Rober Decker The Universiy of Harford Absrac. The auhor has developed some plaform independen, freely available, ineracive programs (mahles) for

More information

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance

More information

UNIVERSITÀ DI PISA DIPARTIMENTO DI INGEGNERIA MECCANICA, NUCLEARE E DELLA PRODUZIONE VIA DIOTISALVI 2, PISA

UNIVERSITÀ DI PISA DIPARTIMENTO DI INGEGNERIA MECCANICA, NUCLEARE E DELLA PRODUZIONE VIA DIOTISALVI 2, PISA G r u p p o R I c e r c a N u c le a r e S a n P I e r o a G r a d o N u c l e a r a n d I n d u s r I a l E n g I n e e r I n g UNIVERSIÀ DI PISA DIPARIMENO DI INGEGNERIA MECCANICA NUCLEARE E DELLA PRODUZIONE

More information

USP. Surplus-Production Models

USP. Surplus-Production Models USP Surplus-Producion Models 2 Overview Purpose of slides: Inroducion o he producion model Overview of differen mehods of fiing Go over some criique of he mehod Source: Haddon 2001, Chaper 10 Hilborn and

More information

18 Biological models with discrete time

18 Biological models with discrete time 8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so

More information

Article from. Predictive Analytics and Futurism. July 2016 Issue 13

Article from. Predictive Analytics and Futurism. July 2016 Issue 13 Aricle from Predicive Analyics and Fuurism July 6 Issue An Inroducion o Incremenal Learning By Qiang Wu and Dave Snell Machine learning provides useful ools for predicive analyics The ypical machine learning

More information

ADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91

ADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91 ddiional Problems 9 n inverse relaionship exiss beween he ime-domain and freuency-domain descripions of a signal. Whenever an operaion is performed on he waveform of a signal in he ime domain, a corresponding

More information

An introduction to the theory of SDDP algorithm

An introduction to the theory of SDDP algorithm An inroducion o he heory of SDDP algorihm V. Leclère (ENPC) Augus 1, 2014 V. Leclère Inroducion o SDDP Augus 1, 2014 1 / 21 Inroducion Large scale sochasic problem are hard o solve. Two ways of aacking

More information

Lecture 20: Riccati Equations and Least Squares Feedback Control

Lecture 20: Riccati Equations and Least Squares Feedback Control 34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he

More information

A quantum method to test the existence of consciousness

A quantum method to test the existence of consciousness A quanum mehod o es he exisence of consciousness Gao Shan The Scieniss Work Team of Elecro-Magneic Wave Velociy, Chinese Insiue of Elecronics -0, NO.0 Building, YueTan XiJie DongLi, XiCheng Disric Beijing

More information

arxiv:cond-mat/ May 2002

arxiv:cond-mat/ May 2002 -- uadrupolar Glass Sae in para-hydrogen and orho-deuerium under pressure. T.I.Schelkacheva. arxiv:cond-ma/5538 6 May Insiue for High Pressure Physics, Russian Academy of Sciences, Troisk 49, Moscow Region,

More information

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his

More information

in Engineering Prof. Dr. Michael Havbro Faber ETH Zurich, Switzerland Swiss Federal Institute of Technology

in Engineering Prof. Dr. Michael Havbro Faber ETH Zurich, Switzerland Swiss Federal Institute of Technology Risk and Saey in Engineering Pro. Dr. Michael Havbro Faber ETH Zurich, Swizerland Conens o Today's Lecure Inroducion o ime varian reliabiliy analysis The Poisson process The ormal process Assessmen o he

More information

( ) = b n ( t) n " (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2.

( ) = b n ( t) n  (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2. Andrei Tokmakoff, MIT Deparmen of Chemisry, 3/14/007-6.4 PERTURBATION THEORY Given a Hamilonian H = H 0 + V where we know he eigenkes for H 0 : H 0 n = E n n, we can calculae he evoluion of he wavefuncion

More information

Kinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.

Kinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8. Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages

More information

SEIF, EnKF, EKF SLAM. Pieter Abbeel UC Berkeley EECS

SEIF, EnKF, EKF SLAM. Pieter Abbeel UC Berkeley EECS SEIF, EnKF, EKF SLAM Pieer Abbeel UC Berkeley EECS Informaion Filer From an analyical poin of view == Kalman filer Difference: keep rack of he inverse covariance raher han he covariance marix [maer of

More information

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec

More information

AN603 APPLICATION NOTE

AN603 APPLICATION NOTE AN603 APPLICAION NOE URBOSWICH IN A PFC BOOS CONVERER INRODUCION SMicroelecronics offers wo families of 600V ulrafas diodes (URBOSWICH"A" and "B" ) having differen compromises beween he forward characerisics

More information

A Bayesian Approach to Spectral Analysis

A Bayesian Approach to Spectral Analysis Chirped Signals A Bayesian Approach o Specral Analysis Chirped signals are oscillaing signals wih ime variable frequencies, usually wih a linear variaion of frequency wih ime. E.g. f() = A cos(ω + α 2

More information

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts?

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts? AP Chemisry Tes (Chaper 12) Muliple Choice (40%) 1) Which of he following is a kineic quaniy? A) Enhalpy B) Inernal Energy C) Gibb s free energy D) Enropy E) Rae of reacion 2) Of he following quesions,

More information

STATE-SPACE MODELLING. A mass balance across the tank gives:

STATE-SPACE MODELLING. A mass balance across the tank gives: B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing

More information

References are appeared in the last slide. Last update: (1393/08/19)

References are appeared in the last slide. Last update: (1393/08/19) SYSEM IDEIFICAIO Ali Karimpour Associae Professor Ferdowsi Universi of Mashhad References are appeared in he las slide. Las updae: 0..204 393/08/9 Lecure 5 lecure 5 Parameer Esimaion Mehods opics o be

More information

2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS

2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/2007 2-17 2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described

More information

CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XI Control of Stochastic Systems - P.R. Kumar

CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XI Control of Stochastic Systems - P.R. Kumar CONROL OF SOCHASIC SYSEMS P.R. Kumar Deparmen of Elecrical and Compuer Engineering, and Coordinaed Science Laboraory, Universiy of Illinois, Urbana-Champaign, USA. Keywords: Markov chains, ransiion probabiliies,

More information

Stability and Bifurcation in a Neural Network Model with Two Delays

Stability and Bifurcation in a Neural Network Model with Two Delays Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy

More information

CS 4495 Computer Vision Tracking 1- Kalman,Gaussian

CS 4495 Computer Vision Tracking 1- Kalman,Gaussian CS 4495 Compuer Vision A. Bobick CS 4495 Compuer Vision - KalmanGaussian Aaron Bobick School of Ineracive Compuing CS 4495 Compuer Vision A. Bobick Adminisrivia S5 will be ou his Thurs Due Sun Nov h :55pm

More information

Multi-Frequency Sheath Dynamics

Multi-Frequency Sheath Dynamics Muli-Frequency Sheah Dynamics Seven Shannon, Alex Paerson, Theodoros Panagopoulos, Daniel Hoffman, John Holland, Dennis Grimard (Universiy of Michigan) Purpose of research RF plasmas wih muliple frequency

More information

Localization and Map Making

Localization and Map Making Localiaion and Map Making My old office DILab a UTK ar of he following noes are from he book robabilisic Roboics by S. Thrn W. Brgard and D. Fo Two Remaining Qesions Where am I? Localiaion Where have I

More information

Réseaux de neurones récurrents Handwriting Recognition with Long Short-Term Memory Networks

Réseaux de neurones récurrents Handwriting Recognition with Long Short-Term Memory Networks Réseaux de neurones récurrens Handwriing Recogniion wih Long Shor-Term Memory Neworks Dr. Marcus Eichenberger-Liwicki DFKI, Germany Marcus.Liwicki@dfki.de Handwriing Recogniion (Sae of he Ar) Transform

More information

The expectation value of the field operator.

The expectation value of the field operator. The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining

More information

Biol. 356 Lab 8. Mortality, Recruitment, and Migration Rates

Biol. 356 Lab 8. Mortality, Recruitment, and Migration Rates Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese

More information

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number

More information

Ensamble methods: Boosting

Ensamble methods: Boosting Lecure 21 Ensamble mehods: Boosing Milos Hauskrech milos@cs.pi.edu 5329 Senno Square Schedule Final exam: April 18: 1:00-2:15pm, in-class Term projecs April 23 & April 25: a 1:00-2:30pm in CS seminar room

More information

ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING

ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING Chung-ki Min Professor

More information

Statistical Machine Learning Methods for Bioinformatics I. Hidden Markov Model Theory

Statistical Machine Learning Methods for Bioinformatics I. Hidden Markov Model Theory Saisical Machine Learning Mehods for Bioinformaics I. Hidden Markov Model Theory Jianlin Cheng, PhD Informaics Insiue, Deparmen of Compuer Science Universiy of Missouri 2009 Free for Academic Use. Copyrigh

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

MANY FACET, COMMON LATENT TRAIT POLYTOMOUS IRT MODEL AND EM ALGORITHM. Dimitar Atanasov

MANY FACET, COMMON LATENT TRAIT POLYTOMOUS IRT MODEL AND EM ALGORITHM. Dimitar Atanasov Pliska Sud. Mah. Bulgar. 20 (2011), 5 12 STUDIA MATHEMATICA BULGARICA MANY FACET, COMMON LATENT TRAIT POLYTOMOUS IRT MODEL AND EM ALGORITHM Dimiar Aanasov There are many areas of assessmen where he level

More information

Affine term structure models

Affine term structure models Affine erm srucure models A. Inro o Gaussian affine erm srucure models B. Esimaion by minimum chi square (Hamilon and Wu) C. Esimaion by OLS (Adrian, Moench, and Crump) D. Dynamic Nelson-Siegel model (Chrisensen,

More information

Energy Storage Benchmark Problems

Energy Storage Benchmark Problems Energy Sorage Benchmark Problems Daniel F. Salas 1,3, Warren B. Powell 2,3 1 Deparmen of Chemical & Biological Engineering 2 Deparmen of Operaions Research & Financial Engineering 3 Princeon Laboraory

More information

Anti-Disturbance Control for Multiple Disturbances

Anti-Disturbance Control for Multiple Disturbances Workshop a 3 ACC Ani-Disurbance Conrol for Muliple Disurbances Lei Guo (lguo@buaa.edu.cn) Naional Key Laboraory on Science and Technology on Aircraf Conrol, Beihang Universiy, Beijing, 9, P.R. China. Presened

More information

Kalman filtering for maximum likelihood estimation given corrupted observations.

Kalman filtering for maximum likelihood estimation given corrupted observations. alman filering maimum likelihood esimaion given corruped observaions... Holmes Naional Marine isheries Service Inroducion he alman filer is used o eend likelihood esimaion o cases wih hidden saes such

More information

Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, , 2016

Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, , 2016 Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 1, 358-363, 216 doi:1.21311/1.39.1.41 Face Deecion and Recogniion Based on an Improved Adaboos Algorihm and Neural Nework Haoian Zhang*, Jiajia Xing, Muian Zhu,

More information

Lab #2: Kinematics in 1-Dimension

Lab #2: Kinematics in 1-Dimension Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion

More information

PET467E-Analysis of Well Pressure Tests/2008 Spring Semester/İTÜ Midterm Examination (Duration 3:00 hours) Solutions

PET467E-Analysis of Well Pressure Tests/2008 Spring Semester/İTÜ Midterm Examination (Duration 3:00 hours) Solutions M. Onur 03.04.008 PET467E-Analysis of Well Pressure Tess/008 Spring Semeser/İTÜ Miderm Examinaion (Duraion 3:00 hours) Soluions Name of he Suden: Insrucions: Before saring he exam, wrie your name clearly

More information