SPLecture7

SPLecture7 - Lecture 7 Mariana Olvera-Cravioto Columbia...

This preview shows pages 1–8. Sign up to view the full content.

Lecture 7 Mariana Olvera-Cravioto Columbia University [email protected] February 11th, 2015 IEOR 4106, Intro to OR: Stochastic Models Lecture 7 1/18

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
In the long run I Consider once again the example about a random walk on a graph. 1 2 6 7 3 8 5 4 P = 1 / 4 1 / 4 0 0 0 1 / 4 1 / 4 0 1 / 3 0 0 1 / 3 1 / 3 0 0 0 0 0 0 0 0 0 0 1 0 1 / 3 0 0 0 1 / 3 1 / 3 0 0 1 / 4 0 0 1 / 4 1 / 4 0 1 / 4 1 / 4 0 0 1 / 4 1 / 4 0 1 / 4 0 1 / 3 0 0 1 / 3 0 1 / 3 0 0 0 0 1 / 2 0 1 / 2 0 0 0 I Run the program MarkovChain.m in Matlab, which computes the first n powers of P . I Notice anything special? IEOR 4106, Intro to OR: Stochastic Models Lecture 7 2/18
In the long run I Consider once again the example about a random walk on a graph. 1 2 6 7 3 8 5 4 P = 1 / 4 1 / 4 0 0 0 1 / 4 1 / 4 0 1 / 3 0 0 1 / 3 1 / 3 0 0 0 0 0 0 0 0 0 0 1 0 1 / 3 0 0 0 1 / 3 1 / 3 0 0 1 / 4 0 0 1 / 4 1 / 4 0 1 / 4 1 / 4 0 0 1 / 4 1 / 4 0 1 / 4 0 1 / 3 0 0 1 / 3 0 1 / 3 0 0 0 0 1 / 2 0 1 / 2 0 0 0 I Run the program MarkovChain.m in Matlab, which computes the first n powers of P . I Notice anything special? I P n seems to be converging as n → ∞ ! I Moreover, all the rows of P n look the same! IEOR 4106, Intro to OR: Stochastic Models Lecture 7 2/18

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
In the long run... continued I Let X n be the current vertex in the random walk after n steps. I Run the program MarkovChainSim.m in Matlab, which generates { X 0 , X 1 , X 2 , . . . , X n } for a pre-specified initial value X 0 and a given number of steps n . I In addition to the sequence, the program returns a vector of the form: NumberOfVisits = ( N 1 , N 2 , N 3 , N 4 , N 5 , N 6 , N 7 , N 8 ) , where N i is the number of times { X k : 0 k n } is in state i , and RelativeFrequency = 1 n NumberOfVisits . I Notice anything special? IEOR 4106, Intro to OR: Stochastic Models Lecture 7 3/18
In the long run... continued I Let X n be the current vertex in the random walk after n steps. I Run the program MarkovChainSim.m in Matlab, which generates { X 0 , X 1 , X 2 , . . . , X n } for a pre-specified initial value X 0 and a given number of steps n . I In addition to the sequence, the program returns a vector of the form: NumberOfVisits = ( N 1 , N 2 , N 3 , N 4 , N 5 , N 6 , N 7 , N 8 ) , where N i is the number of times { X k : 0 k n } is in state i , and RelativeFrequency = 1 n NumberOfVisits . I Notice anything special? I The vector RelativeFrequency is very close to the rows of P n ! IEOR 4106, Intro to OR: Stochastic Models Lecture 7 3/18

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Aperiodicity and positive recurrence I We say that state i has period d if P n ii = 0 whenever n is not divisible by d , and d is the largest integer with this property. I A state with period 1 is said to be aperiodic . I Aperiodicity is a class property. IEOR 4106, Intro to OR: Stochastic Models Lecture 7 4/18
Aperiodicity and positive recurrence I We say that state i has period d if P n ii = 0 whenever n is not divisible by d , and d is the largest integer with this property.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern