1. 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.

2. Approaches to uncertainty estimation
Use statistical theory
Bootstrapping
e.g. Standard Error
Confidence Intervals:

3. 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

4. from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

5. Schematic of Bootstrap Process
from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

6. 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

7. from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

8. Bootstrap Algorithm for Standard Error
from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

9. 95% CI and interquartile range from 500 bootstrap samples
Hillsborough River at Zephyr Hills, September flows
Mean = 8621 mgal
S = 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