UCSB ECE 240A - Lecture Twelve (53 pages)

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Lecture Twelve



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Lecture Twelve

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Lecture Notes


Pages:
53
School:
University of California, Santa Barbara
Course:
Ece 240a - Optimal Estimation and Filtering
Optimal Estimation and Filtering Documents

Unformatted text preview:

Lecture Twelve 1 Outline Failure Time Analysis Linear Probability Model Poisson Distribution 2 Failure Time Analysis Example Duration of Expansions Issue does the probability of an expansion ending depend on how long it has lasted Exponential distribution assumes the answer since the hazard rate is constant Weibull distribution allows a test to be performed 3 Part II Failure Time Analysis Exponential survival function hazard rate Weibull Exploratory Data Analysis Lab Seven 4 Duration of Post War Economic Expansions in Months 5 Trough Oct 1945 Oct 1949 May 1954 April 1958 Feb 1961 Nov 1970 March 1975 July 1980 Nov 1982 March 1991 Peak Nov 1948 July 1953 August 1957 April 1960 Dec 1969 Nov 1973 January 1980 July 1981 July 1990 March 2000 Duration 37 45 39 24 106 36 58 12 92 120 6 Estimated Survivor Function for Ten Post War Expansions 7 Duration 0 12 24 36 37 39 45 58 92 106 120 Ending 0 1 1 1 1 1 1 1 1 1 1 At Risk 10 10 9 8 7 6 5 4 3 2 1 F t 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 Survivor 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 8 Figure 2 Estimated Survivor Function for Post War Expansions 1 2 Survivor Function 1 0 8 0 6 0 4 0 2 0 0 20 40 60 80 100 120 140 Duration in Months 9 Figure 3 Exponential Trendline Fitted to Estimated Survivor Function 1 S t e 0 9 t 0 8 0 0217x y 1 1972e 2 R 0 9533 Survivor Function 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 20 40 60 80 100 120 Duration in Months 10 Figure 4 Constrained Expontial trendline Fitted to Estimated Survivor Function 1 0 9 0 8 Survivor Function 0 7 0 019x y e 2 R 0 9313 0 6 0 5 0 4 0 3 0 2 0 1 0 0 20 40 60 80 100 120 Duration in Months 11 Exponential Distribution Hazard rate ratio of density function to the survivor function h t f t S t measure of probability of failure at time t given that you have survived that long for the exponential it is a constant h t exp t exp t Duration 0 12 24 36 37 39 45 58 92 106 120 Ending 0 1 1 1 1 1 1 1 1 1 1 At Risk Inter Haz 10 0 10 0 1000 9 0 1111 8 0 1250 7 0 1429 6 0 1667 5 0 2000 4 0



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