mhjensen
560 lines
38 KiB
HTML
560 lines
38 KiB
HTML
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'sections': [('Why Linear Regression (aka Ordinary Least Squares and family)',
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2,
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None,
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'___sec0'),
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('Regression analysis, overarching aims', 2, None, '___sec1'),
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('Regression analysis, overarching aims II', 2, None, '___sec2'),
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('Examples', 2, None, '___sec3'),
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('General linear models', 2, None, '___sec4'),
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('Rewriting the fitting procedure as a linear algebra problem',
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2,
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None,
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'___sec5'),
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('Rewriting the fitting procedure as a linear algebra problem, '
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'more details',
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2,
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None,
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'___sec6'),
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('Generalizing the fitting procedure as a linear algebra problem',
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2,
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None,
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'___sec7'),
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('Generalizing the fitting procedure as a linear algebra problem',
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2,
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None,
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'___sec8'),
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('Optimizing our parameters', 2, None, '___sec9'),
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('Our model for the nuclear binding energies',
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2,
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None,
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'___sec10'),
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('Optimizing our parameters, more details', 2, None, '___sec11'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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'___sec12'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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'___sec13'),
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('Some useful matrix and vector expressions',
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2,
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None,
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'___sec14'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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'___sec15'),
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('Own code for Ordinary Least Squares', 2, None, '___sec16'),
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('Adding error analysis and training set up',
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2,
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None,
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'___sec17'),
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('The $\\chi^2$ function', 2, None, '___sec18'),
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('The $\\chi^2$ function', 2, None, '___sec19'),
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('The $\\chi^2$ function', 2, None, '___sec20'),
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('The $\\chi^2$ function', 2, None, '___sec21'),
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('The $\\chi^2$ function', 2, None, '___sec22'),
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('The $\\chi^2$ function', 2, None, '___sec23'),
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('Fitting an Equation of State for Dense Nuclear Matter',
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2,
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None,
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'___sec24'),
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('The code', 2, None, '___sec25'),
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('Splitting our Data in Training and Test data',
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2,
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None,
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'___sec26'),
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('The Boston housing data example', 2, None, '___sec27'),
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('Housing data, the code', 2, None, '___sec28'),
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('Reducing the number of degrees of freedom, overarching view',
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2,
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None,
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'___sec29'),
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('Preprocessing our data', 2, None, '___sec30'),
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('More preprocessing', 2, None, '___sec31'),
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('Simple preprocessing examples, Franke function and regression',
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2,
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None,
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'___sec32'),
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('The singular value decomposition', 2, None, '___sec33'),
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('Linear Regression Problems', 2, None, '___sec34'),
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('Fixing the singularity', 2, None, '___sec35'),
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('Basic math of the SVD', 2, None, '___sec36'),
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('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
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('Economy-size SVD', 2, None, '___sec38'),
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('Codes for the SVD', 2, None, '___sec39'),
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('Mathematical Properties', 2, None, '___sec40'),
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('Ridge and LASSO Regression', 2, None, '___sec41'),
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('More on Ridge Regression', 2, None, '___sec42'),
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('Interpreting the Ridge results', 2, None, '___sec43'),
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('More interpretations', 2, None, '___sec44'),
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('A better understanding of regularization', 2, None, '___sec45'),
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('Decomposing the OLS and Ridge expressions',
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2,
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None,
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'___sec46'),
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('Introducing the Covariance and Correlation functions',
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2,
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None,
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'___sec47'),
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('Correlation Function and Design/Feature Matrix',
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2,
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None,
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'___sec48'),
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('Covariance Matrix Examples', 2, None, '___sec49'),
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('Correlation Matrix', 2, None, '___sec50'),
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('Correlation Matrix with Pandas', 2, None, '___sec51'),
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('Correlation Matrix with Pandas and the Franke function',
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2,
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None,
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'___sec52'),
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('Rewriting the Covariance and/or Correlation Matrix',
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2,
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None,
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'___sec53'),
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('Linking with SVD', 2, None, '___sec54'),
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('Where are we going?', 2, None, '___sec55'),
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('Resampling methods', 2, None, '___sec56'),
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('Resampling approaches can be computationally expensive',
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2,
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None,
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'___sec57'),
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('Why resampling methods ?', 2, None, '___sec58'),
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('Statistical analysis', 2, None, '___sec59'),
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('Statistics', 2, None, '___sec60'),
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('Statistics, moments', 2, None, '___sec61'),
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('Statistics, central moments', 2, None, '___sec62'),
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('Statistics, covariance', 2, None, '___sec63'),
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('Statistics, more covariance', 2, None, '___sec64'),
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('Covariance example', 2, None, '___sec65'),
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('Covariance in numpy', 2, None, '___sec66'),
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('Statistics, independent variables', 2, None, '___sec67'),
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('Statistics, more variance', 2, None, '___sec68'),
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('Statistics and stochastic processes', 2, None, '___sec69'),
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('Statistics and sample variables', 2, None, '___sec70'),
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('Statistics, sample variance and covariance',
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2,
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'___sec71'),
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('Statistics, law of large numbers', 2, None, '___sec72'),
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('Statistics, more on sample error', 2, None, '___sec73'),
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('Statistics', 2, None, '___sec74'),
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('Statistics, central limit theorem', 2, None, '___sec75'),
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('Statistics, more technicalities', 2, None, '___sec76'),
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('Statistics', 2, None, '___sec77'),
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('Statistics and sample variance', 2, None, '___sec78'),
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('Statistics, uncorrelated results', 2, None, '___sec79'),
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('Statistics, computations', 2, None, '___sec80'),
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('Statistics, more on computations of errors',
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2,
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None,
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'___sec81'),
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('Statistics, wrapping up 1', 2, None, '___sec82'),
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('Statistics, final expression', 2, None, '___sec83'),
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('Statistics, effective number of correlations',
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2,
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None,
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'___sec84'),
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('Linking the regression analysis with a statistical '
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'interpretation',
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2,
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None,
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'___sec85'),
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('Assumptions made', 2, None, '___sec86'),
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('Expectation value and variance', 2, None, '___sec87'),
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('Expectation value and variance for $\\boldsymbol{\\beta}$',
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2,
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None,
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'___sec88'),
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('Resampling methods', 2, None, '___sec89'),
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('Resampling methods: Jackknife and Bootstrap',
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2,
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None,
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'___sec90'),
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('Resampling methods: Jackknife', 2, None, '___sec91'),
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('Jackknife code example', 2, None, '___sec92'),
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('Resampling methods: Bootstrap', 2, None, '___sec93'),
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('Resampling methods: Bootstrap background', 2, None, '___sec94'),
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('Resampling methods: More Bootstrap background',
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2,
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None,
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'___sec95'),
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('Resampling methods: Bootstrap approach', 2, None, '___sec96'),
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('Resampling methods: Bootstrap steps', 2, None, '___sec97'),
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('Code example for the Bootstrap method', 2, None, '___sec98'),
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('Various steps in cross-validation', 2, None, '___sec99'),
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('How to set up the cross-validation for Ridge and/or Lasso',
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2,
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None,
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'___sec100'),
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|
('Cross-validation in brief', 2, None, '___sec101'),
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('Code Example for Cross-validation and $k$-fold '
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'Cross-validation',
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2,
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None,
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'___sec102'),
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|
('The bias-variance tradeoff', 2, None, '___sec103'),
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|
('Example code for Bias-Variance tradeoff', 2, None, '___sec104'),
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|
('Understanding what happens', 2, None, '___sec105'),
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|
('Summing up', 2, None, '___sec106'),
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|
("Another Example from Scikit-Learn's Repository",
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2,
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|
None,
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'___sec107'),
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|
('More examples on bootstrap and cross-validation and errors',
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2,
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None,
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'___sec108'),
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|
('The same example but now with cross-validation',
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2,
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None,
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'___sec109'),
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|
('Cross-validation with Ridge', 2, None, '___sec110'),
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('The Ising model', 2, None, '___sec111'),
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|
('Reformulating the problem to suit regression',
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2,
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|
None,
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'___sec112'),
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('Linear regression', 2, None, '___sec113'),
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('Singular Value decomposition', 2, None, '___sec114'),
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('The one-dimensional Ising model', 2, None, '___sec115'),
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('Ridge regression', 2, None, '___sec116'),
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|
('LASSO regression', 2, None, '___sec117'),
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('Performance as function of the regularization parameter',
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2,
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None,
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'___sec118'),
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('Finding the optimal value of $\\lambda$',
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2,
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|
None,
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'___sec119')]}
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end of tocinfo -->
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<a class="navbar-brand" href="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Economy-size SVD</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Codes for the SVD</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">A better understanding of regularization</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Covariance Matrix Examples</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Correlation Matrix</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Linking with SVD</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, moments</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics and sample variance</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Jackknife code example</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="#___sec98" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">The same example but now with cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">Cross-validation with Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">The Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs117.html#___sec116" style="font-size: 80%;">Ridge regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs118.html#___sec117" style="font-size: 80%;">LASSO regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs119.html#___sec118" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs120.html#___sec119" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
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</li>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0099"></a>
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<!-- !split -->
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<h2 id="___sec98" class="anchor">Code example for the Bootstrap method </h2>
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<p>
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The following code starts with a Gaussian distribution with mean value
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\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data
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used in the bootstrap analysis. The bootstrap analysis returns a data
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set after a given number of bootstrap operations (as many as we have
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data points). This data set consists of estimated mean values for each
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bootstrap operation. The histogram generated by the bootstrap method
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shows that the distribution for these mean values is also a Gaussian,
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centered around the mean value \( \mu=100 \) but with standard deviation
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\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in
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this case the same as the number of original data points). The value
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of the standard deviation is what we expect from the central limit
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theorem.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples </span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
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<span style="color: #008000; font-weight: bold">return</span> mean(data)
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<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
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t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
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<span style="color: #408080; font-style: italic"># non-parametric bootstrap </span>
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(R):
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|
t[i] <span style="color: #666666">=</span> statistic(data[randint(<span style="color: #666666">0</span>,n,n)])
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|
<span style="color: #408080; font-style: italic"># analysis </span>
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|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec"</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000">print</span>(<span style="color: #BA2121">"Bootstrap Statistics :"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"original bias std. error"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> (statistic(data), std(data),mean(t),std(t)))
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<span style="color: #008000; font-weight: bold">return</span> t
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|
|
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|
mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
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datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
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|
x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
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<span style="color: #408080; font-style: italic"># bootstrap returns the data sample </span>
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t <span style="color: #666666">=</span> bootstrap(x, stat, datapoints)
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<span style="color: #408080; font-style: italic"># the histogram of the bootstrapped data </span>
|
|
n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>, alpha<span style="color: #666666">=0.75</span>)
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|
|
<span style="color: #408080; font-style: italic"># add a 'best fit' line </span>
|
|
y <span style="color: #666666">=</span> mlab<span style="color: #666666">.</span>normpdf( binsboot, mean(t), std(t))
|
|
lt <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>plot(binsboot, y, <span style="color: #BA2121">'r--'</span>, linewidth<span style="color: #666666">=1</span>)
|
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Smarts'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'Probability'</span>)
|
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plt<span style="color: #666666">.</span>axis([<span style="color: #666666">99.5</span>, <span style="color: #666666">100.6</span>, <span style="color: #666666">0</span>, <span style="color: #666666">3.0</span>])
|
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plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<p>
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