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

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

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?

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

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

Computational statistics, lecture 3 Data-driven inference - inference based on resampling observed data Sampling with replacement Resampled data Observed data

Computational statistics, lecture3 Nonparametric bootstrap - empirical cdf

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

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

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

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

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

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

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