1. Quantifying Uncertainty
Two approaches
Use statistical theory
Bootstrapping

2. Statistical Theory
Uncorrelated
Correlated
Effective
sample size

3. Significance Statistics
Significance statistics use the standard error:
Confidence Intervals:

4. 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

5. 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

6. 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

7. Box-Cox Normality Plot for Monthly September Flows on Alafia R.
Using Kolmogorov-Smirnov (KS) Statistic
Peak at
 = -0.39
What is the range of uncertainty on this?

8. Example for the ks.test How?
Produce 500 new datasets (x*) of the same length as x by sampling with replacement from x
Find the optimal  value for each
Determine the 10th and 90th percentiles to cover 80% of the  values calculated.

9. 80% confidence interval 10% 50% 90% -0.425 -0.250 -0.068
Look back at the original plot and verify that the original “optimal” value was at the far left of the broad top, which is reflected in this confidence interval.
80% confidence interval
10% % %