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Applied 40S February 10, 2009

by Darren Kuropatwa, Educator at ∞ß on Feb 10, 2009

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Introduction to transition matrices.

Introduction to transition matrices.

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Applied 40S February 10, 2009Presentation Transcript

• Transition Matrices 366 • 103 • 20 years later by ﬂickr user Pragmagraphr
• A matrix describing the connections between ﬁve websites (call them 1, 2, 3, 4, 5) is given below. HOMEWORK (a) How many links are there on website 1? (b) How many sites have links to website 5? (c) Suppose that you are on website 1. Describe a path that would take you to website 3.
• Google maintains a connection matrix that tracks links between all the websites that are indexed by Google. Does the matrix contain HOMEWORK mostly 1’s or mostly 0’s? Explain.
• The diagram shows a network that connects computer terminals in 5 ofﬁces. The arrows indicate the directions in which messages can be sent. HOMEWORK B A E C D (a) Write the matrix, C, for the communication network. (b) Find the matrix that represents the number of ways a message can be sent from one ofﬁce to another, using exactly one relay. (c) Find the matrix that represents the number of ways a message can be sent from one ofﬁce to another, using at most one relay.
• Air freight is transported among several cities in Western Canada, with Calgary as the hub. This means all air freight either originates in Calgary or is transported through Calgary. HOMEWORK Edmonton Whitehorse Calgary Vancouver (a) Write a matrix, A, to represent the network. 2 (b) Calculate the matrix A . What do the elements in this matrix represent? (c) Explain why there is one element equal to 3, and why there are nine elements equal to 1, and six elements equal to 0 in the matrix A2 .
• Transition Matrices 366 • 103 • 20 years later by ﬂickr user Pragmagraphr
• An Example A small community has a workforce of 1800 people, with 1680 employed and 120 unemployed. During the course of one year, 10% of the employed workers will lose their jobs and 60% of the unemployed will ﬁnd jobs. The quot;Transition Diagramquot; The quot;Initial State Matrixquot; E U S= EU E U EU T= E S= U The quot;Transition Matrixquot; E U E U T= E E T= U U
• Predicting the future ... A small community has a workforce of 1800 people, with 1680 employed and 120 unemployed. During the course of one year, 10% of the employed workers will lose their jobs and 60% of the unemployed will ﬁnd jobs. After 1 Year After 2 years After 5 Years
• After 50 years Demonstrate stabilization ...
• HOMEWORK You Try ... The Sudzmore Soap Company sells laundry detergent in two-litre and ﬁve-litre packages. Their research shows that 34% of the people buying the small package will switch to the large package for their next purchase, and 12% of the buyers of the large package will switch to the small package for their next purchase. The original market share was 55% for the small package and 45% for the large package. (a) Determine the market share for each size in the next small: 42% large: 58% round of purchases. (b) Does the market share for each size ever stabilize? If so, ﬁnd out when that occurs and what the market share forStabilizes after 7 rounds of purchasing. Market share is small: 26% and large: 74% each size will be.