1. Computational statistics, lecture3 Resampling and the bootstrap  Generating random processes  The bootstrap  Some examples of bootstrap techniques

2. Computational statistics, lecture3 Process-based model of the flow of nitrogen from land to sea Coastal model Anthropo- genic inputs Primary outputs (nutrient concentrations, chlorophyll, oxygen, etc.) Open-sea boundary conditions Watershed model Physio- graphic inputs Meteoro- logical forcings Atmos- pheric inputs Physio- graphic inputs Waterborne inputs Derived outputs Meteoro- logical forcings

3. Computational statistics, lecture3 Decomposing outputs of process-based models driven by meteorological inputs Observed forcing Weather-dependent model output Synthetic forcing Synthetic model output Weather-normalised mean output Weather-specific (random) component of the model output How can we use resampling to better understand model outputs?

4. Computational statistics, lecture3 Resampling daily temperatures  Split observed data into periods of duration one month  Generate new temperature series by resampling 1-month pieces and combining them so that the seasonal pattern is preserved

5. Computational statistics, lecture3 Observed and resampled daily temperatures Observed dataResampled data

6. Computational statistics, lecture 3 Data-driven inference - inference based on resampling observed data 34 67 7988 39 41 85 70 62 905844 60 73 22 58 7988 41 88 85 70 90 223444 60 41 60 Sampling with replacement Resampled data Observed data

7. Computational statistics, lecture3 Nonparametric bootstrap - empirical cdf

8. Computational statistics, lecture3 The bootstrap  Let (X 1, …, X n ) be a sample and  a parameter of the underlying distribution  Suppose  is estimated by  The underlying idea of the bootstrap is to first use the sample to estimate the unknown distribution F of the data. Then this estimated distribution F* is used in place of the unknown true distribution in calculating the distribution of

9. Computational statistics, lecture3 Nonparametric bootstrap - histogram of sample means of bootstrap samples

10. Computational statistics, lecture3 Nonparametric bootstrap - histogram of sample means of bootstrap samples

11. Computational statistics, lecture3 Nonparametric bootstrap - histogram of standard deviations of bootstrap samples

12. Computational statistics, lecture3 Nonparametric bootstrap - confidence intervals by computing percentiles

13. Computational statistics, lecture3 Parametric bootstrap - empirical cdf  Assume that a sample is drawn from an exponential distribution with cdf F( , x) = 1 – exp(-  x)  Use the estimator  Determine the distribution of using the estimated distribution

14. Computational statistics, lecture3 Residual resampling  Consider the linear regression model  Estimate the beta coefficients and determine the residuals  Generate new bootstrap samples  Make inference about the model parameters by fitting linear regression models to bootstrap samples