Dynamic Programming. Lecture 13 (5/31/2017)
|
|
- Ira Collins
- 5 years ago
- Views:
Transcription
1 Dynamc Programmng Lecture 13 (5/31/2017)
2 - A Forest Thnnng Example - Projected yeld (m3/ha) at age 20 as functon of acton taken at age 10 Age 10 Begnnng Volume Resdual Ten-year Volume volume thnned volume growth at age Projected yeld (m3/ha) at age 30 as functon of begnnng volume at age 20 and acton taken at age 20 Age 20 Begnnng Volume Resdual Ten-year Volume volume thnned volume growth at age Age Stage Age Stage Age Stage 3 Age Source: Dykstra s Mathematcal Programmng for Natural Resource Management (1984)
3 Dynamc Programmng Cont. Stages, states and actons (decsons) Backward and forward recurson x decson varable: mmedate destnaton node at stage ; f ( s, x ) the maxmum total volume harvested durng all remanng stages, gven that the stand has reached the state correspondng to node s at stage and the mmedate destnaton node s x ; x the value of x that maxmzes f ( s, x ); and * f ( s) the maxmum value of f ( s, x ),.e., f ( s) f ( s, x ). * * *
4 Solvng Dynamc Programs Recursve relaton at stage (Bellman Equaton): f () s max H f ( x ), * * s, x 1 x where H s the volume harvested by thnnng at the begnnng of stage sx, n order to move the stand from state s at the begnnng of stage to state x at the end of stage.
5 Dynamc Programmng Structural requrements of DP The problem can be dvded nto stages Each stage has a fnte set of assocated states (dscrete state DP) The mpact of a polcy decson to transform a state n a gven stage to another state n the subsequent stage s determnstc Prncple of optmalty: gven the current state of the system, the optmal polcy for the remanng stages s ndependent of any pror polcy adopted
6 Examples of DP The Floyd-Warshall Algorthm (used n the Bucket formulaton of ARM): Let c (, j) denote the length (or weght) of the path between node and j; and Let s (, jk, ) denote the length (or total weght) of the shortest path between nodes and j gong through ntermedate node 1, 2,...,k. Then, the followng recurson wll gve the length of the shortest paths between all pars of nodes n a graph: s (, jk, ) c (, j) f k1 mn s (, jk, 1), skk (,, 1) sk (, jk, 1) otherwse
7 Examples of DP cont. Mnmzng the rsk of losng an endangered speces (non-lnear DP): Probablty of Project Falure Fundng level (state) Project (stage) $1M 1 50% 30% 20% 2 70% 50% 30% 3 80% 50% 40% $1M $1M $1M $1M $1M $1M Stage 1 Stage 2 Stage 3 Source: Buongorno and Glles (2003) Decson Methods for Forest Resource Management
8 Mnmzng the rsk of losng an endangered speces (DP example) Stages (t=1,2,3) represent $ allocaton decsons for Project 1, 2 and 3; States (=0,1,2) represent the budgets avalable at each stage t; Decsons (j=0,1,2) represent the budgets avalable at stage t+1; pt (, j) denotes the probablty of falure of Project t f decson j s made (.e., -j s spent on Project t); and V t* () s the smallest probablty of total falure from stage t onward, startng n state and makng the best decson j*. Then, the recursve relaton can be stated as: V () mn p (, j) V ( j) * * t t t1 j
9 Markov Chans (based on Buongorno and Glles 2003) Volume by stand states State Volume (m3/ha) L <400 M H 700< 20-yr Transton probabltes w/o management Start State L End State j M H L 40% 60% 0% M 0% 30% 70% H 5% 5% 90% Transton probablty matrx P pt = probablty dstrbuton of stand states n perod t p p P t t t-1 Vector p t converges to a vector of steady-state probabltes p*. p* s ndependent of p 0!
10 Markov Chans cont. Berger-Parker Landscape Index BP t plt pmt pht max( p, p, p ) Lt Mt Ht,where s the probablty that the stand p t s n state L, M or H n perod t. Mean Resdence Tmes m D (1 p ),where D s the length of each perod, and p s the probablty that a stand n state n the begnnng of perod t stays there tll the end of perod t. Mean Recurrence Tmes m D,where s the steady-state probablty of state.
11 Markov Chans cont. Forest dynamcs (expected revenues or bodversty wth vs. w/o management: 20-yr Transton probabltes w/o management Start State L End State j M H L 40% 60% 0% M 0% 30% 70% H 5% 5% 90% 20-yr Transton probabltes w/ management Start State L End State j M H L 40% 60% 0% M 0% 30% 70% H 40% 60% 0% Expected long-term bodversty: Expected long-term perodc ncome: B B B B L L M M H H R R R R L L M M H H
12 Markov Chans cont. Present value of expected returns 1 V R p V p V p V (1 r) t, 1 20 L Lt M Mt H Ht where V s the present value of the expected return from t, a stand n state n state = L, M or H managed wth a specfc harvest polcy wth t perods to go before the end of the plannng horzon. R s the mmedate return from managng a stand n state wth the gven harvest polcy. p, p, p are the probabltes that a stand n state moves to L M H state L, M or H, respectvely.
13 Markov Decson Processes 20-yr Transton probabltes dependng on the harvest decson No Cut Cut Start End State j Start End State j State L M H State L M H L 40% 60% 0% L 40% 60% 0% M 0% 30% 70% M 40% 60% 0% H 5% 5% 90% H 40% 60% 0% 1 V max R p V p V p V (1 r) * * * *, t1 j 20 Lj Lt Mj Mt Hj Ht j where V s the hghest present value of the expected return from * t, 1 a stand n state n state = L, M or H managed wth a specfc harvest polcy wth t perods to go before the end of the plannng horzon. R s the mmedate return from j managng a stand n state wth harvest polcy j. p, p, p are the probabltes that a stand n state moves to Lj Mj Hj state L, M or H, respectvely f the harvest polcy s j.
Suggested solutions for the exam in SF2863 Systems Engineering. June 12,
Suggested solutons for the exam n SF2863 Systems Engneerng. June 12, 2012 14.00 19.00 Examner: Per Enqvst, phone: 790 62 98 1. We can thnk of the farm as a Jackson network. The strawberry feld s modelled
More informationCS 331 DESIGN AND ANALYSIS OF ALGORITHMS DYNAMIC PROGRAMMING. Dr. Daisy Tang
CS DESIGN ND NLYSIS OF LGORITHMS DYNMIC PROGRMMING Dr. Dasy Tang Dynamc Programmng Idea: Problems can be dvded nto stages Soluton s a sequence o decsons and the decson at the current stage s based on the
More informationComputational issues surrounding the management of an ecological food web
Computatonal ssues surroundng the management of an ecologcal food web Wllam J M Probert, Eve McDonald-Madden, Nathale Peyrard, Régs Sabbadn AIGM 12, ECAI2012 Montpeller, France Ratonale Ecology has many
More informationAnalysis of Discrete Time Queues (Section 4.6)
Analyss of Dscrete Tme Queues (Secton 4.6) Copyrght 2002, Sanjay K. Bose Tme axs dvded nto slots slot slot boundares Arrvals can only occur at slot boundares Servce to a job can only start at a slot boundary
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationHidden Markov Models & The Multivariate Gaussian (10/26/04)
CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models
More informationWinter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan
Wnter 2008 CS567 Stochastc Lnear/Integer Programmng Guest Lecturer: Xu, Huan Class 2: More Modelng Examples 1 Capacty Expanson Capacty expanson models optmal choces of the tmng and levels of nvestments
More informationContinuous Time Markov Chains
Contnuous Tme Markov Chans Brth and Death Processes,Transton Probablty Functon, Kolmogorov Equatons, Lmtng Probabltes, Unformzaton Chapter 6 1 Markovan Processes State Space Parameter Space (Tme) Dscrete
More informationOutline and Reading. Dynamic Programming. Dynamic Programming revealed. Computing Fibonacci. The General Dynamic Programming Technique
Outlne and Readng Dynamc Programmng The General Technque ( 5.3.2) -1 Knapsac Problem ( 5.3.3) Matrx Chan-Product ( 5.3.1) Dynamc Programmng verson 1.4 1 Dynamc Programmng verson 1.4 2 Dynamc Programmng
More informationHidden Markov Models
CM229S: Machne Learnng for Bonformatcs Lecture 12-05/05/2016 Hdden Markov Models Lecturer: Srram Sankararaman Scrbe: Akshay Dattatray Shnde Edted by: TBD 1 Introducton For a drected graph G we can wrte
More information6. Stochastic processes (2)
Contents Markov processes Brth-death processes Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuous-tme and dscrete-state stochastc process X(t) wth state space
More information6. Stochastic processes (2)
6. Stochastc processes () Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 6. Stochastc processes () Contents Markov processes Brth-death processes 6. Stochastc processes () Markov process
More informationEquilibrium with Complete Markets. Instructor: Dmytro Hryshko
Equlbrum wth Complete Markets Instructor: Dmytro Hryshko 1 / 33 Readngs Ljungqvst and Sargent. Recursve Macroeconomc Theory. MIT Press. Chapter 8. 2 / 33 Equlbrum n pure exchange, nfnte horzon economes,
More informationAmiri s Supply Chain Model. System Engineering b Department of Mathematics and Statistics c Odette School of Business
Amr s Supply Chan Model by S. Ashtab a,, R.J. Caron b E. Selvarajah c a Department of Industral Manufacturng System Engneerng b Department of Mathematcs Statstcs c Odette School of Busness Unversty of
More informationEEL 6266 Power System Operation and Control. Chapter 3 Economic Dispatch Using Dynamic Programming
EEL 6266 Power System Operaton and Control Chapter 3 Economc Dspatch Usng Dynamc Programmng Pecewse Lnear Cost Functons Common practce many utltes prefer to represent ther generator cost functons as sngle-
More informationGoogle PageRank with Stochastic Matrix
Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d
More informationIntroduction to Algorithms
Introducton to Algorthms 6.046J/8.40J LECTURE 6 Shortest Paths III All-pars shortest paths Matrx-multplcaton algorthm Floyd-Warshall algorthm Johnson s algorthm Prof. Charles E. Leserson Shortest paths
More informationSimultaneous Optimization of Berth Allocation, Quay Crane Assignment and Quay Crane Scheduling Problems in Container Terminals
Smultaneous Optmzaton of Berth Allocaton, Quay Crane Assgnment and Quay Crane Schedulng Problems n Contaner Termnals Necat Aras, Yavuz Türkoğulları, Z. Caner Taşkın, Kuban Altınel Abstract In ths work,
More informationIntroduction to Algorithms
Introducton to Algorthms 6.046J/8.40J/SMA5503 Lecture 9 Prof. Erk Demane Shortest paths Sngle-source shortest paths Nonnegate edge weghts Djkstra s algorthm: OE + V lg V General Bellman-Ford: OVE DAG One
More information8. Modelling Uncertainty
8. Modellng Uncertanty. Introducton. Generatng Values From Known Probablty Dstrbutons. Monte Carlo Smulaton 4. Chance Constraned Models 5 5. Markov Processes and Transton Probabltes 6 6. Stochastc Optmzaton
More informationLecture 14: Bandits with Budget Constraints
IEOR 8100-001: Learnng and Optmzaton for Sequental Decson Makng 03/07/16 Lecture 14: andts wth udget Constrants Instructor: Shpra Agrawal Scrbed by: Zhpeng Lu 1 Problem defnton In the regular Mult-armed
More informationDesign and Analysis of Algorithms
Desgn and Analyss of Algorthms CSE 53 Lecture 4 Dynamc Programmng Junzhou Huang, Ph.D. Department of Computer Scence and Engneerng CSE53 Desgn and Analyss of Algorthms The General Dynamc Programmng Technque
More informationHidden Markov Models
Note to other teachers and users of these sldes. Andrew would be delghted f you found ths source materal useful n gvng your own lectures. Feel free to use these sldes verbatm, or to modfy them to ft your
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experments- MODULE LECTURE - 6 EXPERMENTAL DESGN MODELS Dr. Shalabh Department of Mathematcs and Statstcs ndan nsttute of Technology Kanpur Two-way classfcaton wth nteractons
More informationResource Allocation with a Budget Constraint for Computing Independent Tasks in the Cloud
Resource Allocaton wth a Budget Constrant for Computng Independent Tasks n the Cloud Wemng Sh and Bo Hong School of Electrcal and Computer Engneerng Georga Insttute of Technology, USA 2nd IEEE Internatonal
More informationDepartment of Computer Science Artificial Intelligence Research Laboratory. Iowa State University MACHINE LEARNING
MACHINE LEANING Vasant Honavar Bonformatcs and Computatonal Bology rogram Center for Computatonal Intellgence, Learnng, & Dscovery Iowa State Unversty honavar@cs.astate.edu www.cs.astate.edu/~honavar/
More informationCS : Algorithms and Uncertainty Lecture 17 Date: October 26, 2016
CS 29-128: Algorthms and Uncertanty Lecture 17 Date: October 26, 2016 Instructor: Nkhl Bansal Scrbe: Mchael Denns 1 Introducton In ths lecture we wll be lookng nto the secretary problem, and an nterestng
More informationIntra-Domain Traffic Engineering
Intra-Doman Traffc Engneerng Traffc Engneerng (TE) MPLS and traffc engneerng (wll go over very brefly) traffc engneerng as networ-wde optmzaton problem TE through ln weght assgnments Traffc Matrx Estmaton
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More informationA LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,
A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare
More informationMarkov decision processes
IMT Atlantque Technopôle de Brest-Irose - CS 83818 29238 Brest Cedex 3 Téléphone: +33 0)2 29 00 13 04 Télécope: +33 0)2 29 00 10 12 URL: www.mt-atlantque.fr Markov decson processes Lecture notes therry.chonavel@mt-atlantque.fr
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More informationProblem Set 9 - Solutions Due: April 27, 2005
Problem Set - Solutons Due: Aprl 27, 2005. (a) Frst note that spam messages, nvtatons and other e-mal are all ndependent Posson processes, at rates pλ, qλ, and ( p q)λ. The event of the tme T at whch you
More informationA NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT. Hirotaka Matsumoto and Yoshio Tabata
Scentae Mathematcae Japoncae Onlne, Vol. 9, (23), 419 429 419 A NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT Hrotaka Matsumoto and Yosho Tabata Receved September
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationCS-433: Simulation and Modeling Modeling and Probability Review
CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown
More informationCIE4801 Transportation and spatial modelling Trip distribution
CIE4801 ransportaton and spatal modellng rp dstrbuton Rob van Nes, ransport & Plannng 17/4/13 Delft Unversty of echnology Challenge the future Content What s t about hree methods Wth specal attenton for
More informationRevenue comparison when the length of horizon is known in advance. The standard errors of all numbers are less than 0.1%.
Golrezae, Nazerzadeh, and Rusmevchentong: Real-tme Optmzaton of Personalzed Assortments Management Scence 00(0, pp. 000 000, c 0000 INFORMS 37 Onlne Appendx Appendx A: Numercal Experments: Appendx to Secton
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationk t+1 + c t A t k t, t=0
Macro II (UC3M, MA/PhD Econ) Professor: Matthas Kredler Fnal Exam 6 May 208 You have 50 mnutes to complete the exam There are 80 ponts n total The exam has 4 pages If somethng n the queston s unclear,
More informationHopfield networks and Boltzmann machines. Geoffrey Hinton et al. Presented by Tambet Matiisen
Hopfeld networks and Boltzmann machnes Geoffrey Hnton et al. Presented by Tambet Matsen 18.11.2014 Hopfeld network Bnary unts Symmetrcal connectons http://www.nnwj.de/hopfeld-net.html Energy functon The
More informationRandom Partitions of Samples
Random Parttons of Samples Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract In the present paper we construct a decomposton of a sample nto a fnte number of subsamples
More informationEngineering Risk Benefit Analysis
Engneerng Rsk Beneft Analyss.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, 22.82, ESD.72, ESD.72 RPRA 2. Elements of Probablty Theory George E. Apostolaks Massachusetts Insttute of Technology Sprng 2007
More informationPortfolios with Trading Constraints and Payout Restrictions
Portfolos wth Tradng Constrants and Payout Restrctons John R. Brge Northwestern Unversty (ont wor wth Chrs Donohue Xaodong Xu and Gongyun Zhao) 1 General Problem (Very) long-term nvestor (eample: unversty
More informationAn Integrated Asset Allocation and Path Planning Method to to Search for a Moving Target in in a Dynamic Environment
An Integrated Asset Allocaton and Path Plannng Method to to Search for a Movng Target n n a Dynamc Envronment Woosun An Mansha Mshra Chulwoo Park Prof. Krshna R. Pattpat Dept. of Electrcal and Computer
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationAnnexes. EC.1. Cycle-base move illustration. EC.2. Problem Instances
ec Annexes Ths Annex frst llustrates a cycle-based move n the dynamc-block generaton tabu search. It then dsplays the characterstcs of the nstance sets, followed by detaled results of the parametercalbraton
More informationThe Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationContinuous Time Markov Chain
Contnuous Tme Markov Chan Hu Jn Department of Electroncs and Communcaton Engneerng Hanyang Unversty ERICA Campus Contents Contnuous tme Markov Chan (CTMC) Propertes of sojourn tme Relatons Transton probablty
More informationECE 534: Elements of Information Theory. Solutions to Midterm Exam (Spring 2006)
ECE 534: Elements of Informaton Theory Solutons to Mdterm Eam (Sprng 6) Problem [ pts.] A dscrete memoryless source has an alphabet of three letters,, =,, 3, wth probabltes.4,.4, and., respectvely. (a)
More informationAn Experiment/Some Intuition (Fall 2006): Lecture 18 The EM Algorithm heads coin 1 tails coin 2 Overview Maximum Likelihood Estimation
An Experment/Some Intuton I have three cons n my pocket, 6.864 (Fall 2006): Lecture 18 The EM Algorthm Con 0 has probablty λ of heads; Con 1 has probablty p 1 of heads; Con 2 has probablty p 2 of heads
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationHidden Markov Models
Hdden Markov Models Namrata Vaswan, Iowa State Unversty Aprl 24, 204 Hdden Markov Model Defntons and Examples Defntons:. A hdden Markov model (HMM) refers to a set of hdden states X 0, X,..., X t,...,
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationSingle-Facility Scheduling over Long Time Horizons by Logic-based Benders Decomposition
Sngle-Faclty Schedulng over Long Tme Horzons by Logc-based Benders Decomposton Elvn Coban and J. N. Hooker Tepper School of Busness, Carnege Mellon Unversty ecoban@andrew.cmu.edu, john@hooker.tepper.cmu.edu
More informationAbsorbing Markov Chain Models to Determine Optimum Process Target Levels in Production Systems with Rework and Scrapping
Archve o SID Journal o Industral Engneerng 6(00) -6 Absorbng Markov Chan Models to Determne Optmum Process Target evels n Producton Systems wth Rework and Scrappng Mohammad Saber Fallah Nezhad a, Seyed
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationTopics in Probability Theory and Stochastic Processes Steven R. Dunbar. Classes of States and Stationary Distributions
Steven R. Dunbar Department of Mathematcs 203 Avery Hall Unversty of Nebraska-Lncoln Lncoln, NE 68588-0130 http://www.math.unl.edu Voce: 402-472-3731 Fax: 402-472-8466 Topcs n Probablty Theory and Stochastc
More informationRyan (2009)- regulating a concentrated industry (cement) Firms play Cournot in the stage. Make lumpy investment decisions
1 Motvaton Next we consder dynamc games where the choce varables are contnuous and/or dscrete. Example 1: Ryan (2009)- regulatng a concentrated ndustry (cement) Frms play Cournot n the stage Make lumpy
More informationRandomness and Computation
Randomness and Computaton or, Randomzed Algorthms Mary Cryan School of Informatcs Unversty of Ednburgh RC 208/9) Lecture 0 slde Balls n Bns m balls, n bns, and balls thrown unformly at random nto bns usually
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationOther NN Models. Reinforcement learning (RL) Probabilistic neural networks
Other NN Models Renforcement learnng (RL) Probablstc neural networks Support vector machne (SVM) Renforcement learnng g( (RL) Basc deas: Supervsed dlearnng: (delta rule, BP) Samples (x, f(x)) to learn
More informationCourse 395: Machine Learning - Lectures
Course 395: Machne Learnng - Lectures Lecture 1-2: Concept Learnng (M. Pantc Lecture 3-4: Decson Trees & CC Intro (M. Pantc Lecture 5-6: Artfcal Neural Networks (S.Zaferou Lecture 7-8: Instance ased Learnng
More informationSolutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution.
Solutons HW #2 Dual of general LP. Fnd the dual functon of the LP mnmze subject to c T x Gx h Ax = b. Gve the dual problem, and make the mplct equalty constrants explct. Soluton. 1. The Lagrangan s L(x,
More informationLecture 3: Shannon s Theorem
CSE 533: Error-Correctng Codes (Autumn 006 Lecture 3: Shannon s Theorem October 9, 006 Lecturer: Venkatesan Guruswam Scrbe: Wdad Machmouch 1 Communcaton Model The communcaton model we are usng conssts
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationINTRODUCTION TO MACHINE LEARNING 3RD EDITION
ETHEM ALPAYDIN The MIT Press, 2014 Lecture Sldes for INTRODUCTION TO MACHINE LEARNING 3RD EDITION alpaydn@boun.edu.tr http://www.cmpe.boun.edu.tr/~ethem/2ml3e CHAPTER 3: BAYESIAN DECISION THEORY Probablty
More informationComplement of Type-2 Fuzzy Shortest Path Using Possibility Measure
Intern. J. Fuzzy Mathematcal rchve Vol. 5, No., 04, 9-7 ISSN: 30 34 (P, 30 350 (onlne Publshed on 5 November 04 www.researchmathsc.org Internatonal Journal of Complement of Type- Fuzzy Shortest Path Usng
More informationANSWERS CHAPTER 9. TIO 9.2: If the values are the same, the difference is 0, therefore the null hypothesis cannot be rejected.
ANSWERS CHAPTER 9 THINK IT OVER thnk t over TIO 9.: χ 2 k = ( f e ) = 0 e Breakng the equaton down: the test statstc for the ch-squared dstrbuton s equal to the sum over all categores of the expected frequency
More informationMIMA Group. Chapter 2 Bayesian Decision Theory. School of Computer Science and Technology, Shandong University. Xin-Shun SDU
Group M D L M Chapter Bayesan Decson heory Xn-Shun Xu @ SDU School of Computer Scence and echnology, Shandong Unversty Bayesan Decson heory Bayesan decson theory s a statstcal approach to data mnng/pattern
More informationCS 3750 Machine Learning Lecture 6. Monte Carlo methods. CS 3750 Advanced Machine Learning. Markov chain Monte Carlo
CS 3750 Machne Learnng Lectre 6 Monte Carlo methods Mlos Haskrecht mlos@cs.ptt.ed 5329 Sennott Sqare Markov chan Monte Carlo Importance samplng: samples are generated accordng to Q and every sample from
More informationResource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis
Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationPROBABILITY PRIMER. Exercise Solutions
PROBABILITY PRIMER Exercse Solutons 1 Probablty Prmer, Exercse Solutons, Prncples of Econometrcs, e EXERCISE P.1 (b) X s a random varable because attendance s not known pror to the outdoor concert. Before
More informationP R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /
Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons
More informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More information8.592J: Solutions for Assignment 7 Spring 2005
8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1
More informationText S1: Detailed proofs for The time scale of evolutionary innovation
Text S: Detaled proofs for The tme scale of evolutonary nnovaton Krshnendu Chatterjee Andreas Pavloganns Ben Adlam Martn A. Nowak. Overvew and Organzaton We wll present detaled proofs of all our results.
More informationReview: Fit a line to N data points
Revew: Ft a lne to data ponts Correlated parameters: L y = a x + b Orthogonal parameters: J y = a (x ˆ x + b For ntercept b, set a=0 and fnd b by optmal average: ˆ b = y, Var[ b ˆ ] = For slope a, set
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationLecture 10 Support Vector Machines. Oct
Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron
More informationRobust Adaptive Markov Decision Processes in Multivehicle
Robust Adaptve Markov Decson Processes n Multvehcle Applcatons The MIT Faculty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher
More informationA Bayesian Approach to Arrival Rate Forecasting for Inhomogeneous Poisson Processes for Mobile Calls
A Bayesan Approach to Arrval Rate Forecastng for Inhomogeneous Posson Processes for Moble Calls Mchael N. Nawar Department of Computer Engneerng Caro Unversty Caro, Egypt mchaelnawar@eee.org Amr F. Atya
More informationAn Optimization Model for Routing in Low Earth Orbit Satellite Constellations
An Optmzaton Model for Routng n Low Earth Orbt Satellte Constellatons A. Ferrera J. Galter P. Mahey Inra Inra Inra Afonso.Ferrera@sopha.nra.fr Jerome.Galter@nra.fr Phlppe.Mahey@sma.fr G. Mateus A. Olvera
More informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for
More informationPROBLEM SET 7 GENERAL EQUILIBRIUM
PROBLEM SET 7 GENERAL EQUILIBRIUM Queston a Defnton: An Arrow-Debreu Compettve Equlbrum s a vector of prces {p t } and allocatons {c t, c 2 t } whch satsfes ( Gven {p t }, c t maxmzes βt ln c t subject
More informationSampling Theory MODULE VII LECTURE - 23 VARYING PROBABILITY SAMPLING
Samplng heory MODULE VII LECURE - 3 VARYIG PROBABILIY SAMPLIG DR. SHALABH DEPARME OF MAHEMAICS AD SAISICS IDIA ISIUE OF ECHOLOGY KAPUR he smple random samplng scheme provdes a random sample where every
More informationFUZZY FINITE ELEMENT METHOD
FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments
More informationDynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)
/24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler sub-problems. It s applcable to problems exhbtng the propertes
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationReinforcement learning
Renforcement learnng Nathanel Daw Gatsby Computatonal Neuroscence Unt daw @ gatsby.ucl.ac.uk http://www.gatsby.ucl.ac.uk/~daw Mostly adapted from Andrew Moore s tutorals, copyrght 2002, 2004 by Andrew
More informationCredit Card Pricing and Impact of Adverse Selection
Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n
More information