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DEC 8 – 9am FINAL EXAM EN 2007. Today: Quizz 11: review. Last quizz! Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions. Resampling. Resampling Introduction.
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DEC 8 – 9am FINAL EXAMEN 2007 Today: Quizz 11: review. Last quizz! Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions
ResamplingIntroduction We have relied on idealized models of the origins of our data (ε ~N) to make inferences But, these models can be inadequate Resampling techniques allow us to base the analysis of a study solely on the design of that study, rather than on a poorly-fitting model
ResamplingWhy resampling • Fewer assumptions • Ex: resampling methods do not require that distributions be Normal or that sample sizes be large • Generality: Resampling methods are remarkably similar for a wide range of statistics and do not require new formulas for every statistic • Promote understanding: Boostrap procedures build intuition by providing concrete analogies to theoretical concepts
Resampling Collection of procedures to make statistical inferences without relying on parametric assumptions - bias - variance, measures of error - Parameter estimation - hypothesis testing
ResamplingProcedures Permutation (randomization) Bootsrap Jackknife Monte Carlo techniques
Resampling With replacement Without replacement
Resamplingpermutation If number of samples in each group=10 And n groups=2 number of permutations=184756 If n groups = 3 number of permutations > 5 000 billions
ResamplingBootstrap Bradley Efron 1979 "to pull oneself up by one's bootstraps" The Surprising Adventures of Baron Munchausen, (1781) by Rudolf Erich Raspe
ResamplingBootstrap Hypothesis testing, parameter estimation, assigning measures of accuracy to sample estimates e.g.: se, CI Useful when: formulas for parameter estimates are based on assumptions that are not met computational formulas only valid for large samples computational formulas do not exist
ResamplingBootstrap Assume that sample is representative of population Approximate the distribution of the population by repeatedly resampling (with replacement) from the sample
ResamplingBootstrap Non-parametric bootstrap resample observation from original samples Parametric bootstrap fit a particular model to the data and then use this model to produce bootstrap samples
Confidence intervals Non Parametric Bootstrap Large Capelin Small Capelin OtherPrey
Confidence intervals Non Parametric Bootstrap 74 lc %Nlc = 48.3% 76 sc %Nsc = 49.6% 3 ot %Nlc = 1.9% What about uncertainty around the point estimates? Bootstrap
Confidence intervals Non Parametric Bootstrap Bootstrap: - 153 balls, each with a tag: 76 sc, 74 lc, 3 ot - Draw 153 random samples (with replacement) and record tag - Calculate %Nlc*1, %Nsc*1, %Not*1 - Repeat nboot=50 000 times (%Nlc*1, %Nsc*1, %Not*1), (%Nlc*2, %Nsc*2, %Not*2),…, ( %Nlc*nboot, %Nsc*nboot, %Not*nboot) - sort the %Ni*b UCL = 48750th %Ni*b (0.975) LCL = 1250th %Ni*b (0.025)
Confidence intervalsParametric Bootstrap Params β α
Confidence intervalsParametric Bootstrap Params β*1 α*1
Confidence intervalsParametric Bootstrap Params β*2 α*2
Confidence intervalsParametric Bootstrap Params β*nboot α*nboot
Confidence intervalsParametric Bootstrap Params β*1, β*2, …,β*nboot Construct Confidence Interval for β 1. Percentile Method 2. Bias-Corrected Percentile Confidence Limits 3. Accelerated Bias-Corrected Percentile Limits 4 .Bootstrap-t 5, 6, …. , Other methods
BootstrapCaveats Independence Incomplete data Outliers Cases where small perturbations to the data-generating process produce big swings in the sampling distribution
ResamplingJackknife Quenouille 1956 Tukey 1958 Estimate bias and variance of a statistic Concept: Leave one observation out and recompute statistic
Cross-validationJackknife Assess the performance of the model How accurately will the model predict a new observation?
Jackknife Bootstrap Differences Both estimate variability of a statistic between subsamples Jackknife provides estimate of the variance of an estimator Bootstrap first estimates the distribution of the estimator. From this distribution, we can estimate the variance Using the same data set: bootstrap results will always be different (slightly) jackknife results will always be identical
ResamplingMonte Carlo John von Neumann Stanisław Ulam Mid 1940s “The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays.”
ResamplingMonte Carlo Monte Carlo methodS: not just one no clear consensus on how they should be defined Commonality: repeated sampling from populations with known characteristics, i.e. we assume a distribution and create random samples that follow that distribution, then compare our estimated statistic to the distribution of outcomes
Monte Carlo Goal: assess the robustness of constant escapement and constant harvest rate policies with respect to management error for Pacific salmon fisheries
