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Quantifying uncertainty using the bootstrap Reading Efron, B. and R. Tibishirani, (1993), An Introduction to the Bootstrap, Chapman Hall, New York, 436 p. Chapters 1, 2, 6. Approaches to uncertainty estimation • Use statistical theory 2 e.g. Standard Error • Bootstrapping ˆ s(x ) SEx N seboot s(x* ) Confidence Intervals: ˆ z / 2 se boot Bootstrapping • Motivated by the absence of equations for other accuracy measures (bias, prediction error, confidence intervals) for statistics of interest (correlation, regressions, ACF) • Definition: “The bootstrap is a data-based simulation method for statistical inference.” • Principle: resample with replacement from data. After Efron and Tibshirani, An Introduction to the Bootstrap, 1993 from Efron and Tibshirani, An Introduction to the Bootstrap, 1993 Schematic of Bootstrap Process from Efron and Tibshirani, An Introduction to the Bootstrap, 1993 Bootstrapping BOOTSTRAP WORLD REAL WORLD Unknown Probability Distribution F Observed Random Sample x = {x1, x2, …, xn} ˆ s (x) Statistic of Interest Empirical Distribution F* Sampling with replacement Bootstrap Sample x * = {x*1, x * 2, …, x *n} ˆ* s (x* ) Bootstrap Replication After Efron and Tibshirani, An Introduction to the Bootstrap, 1993 from Efron and Tibshirani, An Introduction to the Bootstrap, 1993 Bootstrap Algorithm for Standard Error from Efron and Tibshirani, An Introduction to the Bootstrap, 1993 Hillsborough River at Zephyr Hills, September flows 0 2 4 6 8 Frequency 12 Mean = 8621 mgal S = 8194 mgal N = 31 0 5000 10000 15000 20000 25000 30000 35000 Uncertainty on estimates of the mean 80 40 0 Frequency 120 2 One and two standard errors SEx N 95% CI and interquartile range from 500 bootstrap samples 0 5000 10000 15000 20000 25000 Millions of gallons 30000 35000