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 Bootstrapping e.g. Standard Error Confidence Intervals:

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 REAL WORLD BOOTSTRAP WORLD Sampling with replacement F x = {x1, x2, …, xn} BOOTSTRAP WORLD F * x * = {x*1, x * 2, …, x *n} Empirical Distribution Bootstrap Sample Bootstrap Replication Unknown Probability Distribution Observed Random Sample Sampling with replacement Statistic of Interest 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

95% CI and interquartile range from 500 bootstrap samples Hillsborough River at Zephyr Hills, September flows Mean = 8621 mgal S = 8194 mgal N = 31 Uncertainty on estimates of the mean One and two standard errors 95% CI and interquartile range from 500 bootstrap samples Millions of gallons