mhjensen
480 lines
31 KiB
HTML
480 lines
31 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 singular value decomposition', 2, None, '___sec27'),
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('The Ising model', 2, None, '___sec28'),
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('Reformulating the problem to suit regression',
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2,
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None,
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'___sec29'),
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('Linear regression', 2, None, '___sec30'),
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('Singular Value decomposition', 2, None, '___sec31'),
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('Linear Regression Problems', 2, None, '___sec32'),
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('Fixing the singularity', 2, None, '___sec33'),
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('Basic math of the SVD', 2, None, '___sec34'),
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('The SVD, a Fantastic Algorithm', 2, None, '___sec35'),
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('Another Example', 2, None, '___sec36'),
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('Economy-size SVD', 2, None, '___sec37'),
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('Mathematical Properties', 2, None, '___sec38'),
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('Ridge and LASSO Regression', 2, None, '___sec39'),
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('More on Ridge Regression', 2, None, '___sec40'),
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('Interpreting the Ridge results', 2, None, '___sec41'),
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('More interpretations', 2, None, '___sec42'),
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('Where are we going?', 2, None, '___sec43'),
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('Resampling methods', 2, None, '___sec44'),
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('Resampling approaches can be computationally expensive',
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2,
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None,
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'___sec45'),
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('Why resampling methods ?', 2, None, '___sec46'),
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('Statistical analysis', 2, None, '___sec47'),
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('Statistics', 2, None, '___sec48'),
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('Statistics, moments', 2, None, '___sec49'),
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('Statistics, central moments', 2, None, '___sec50'),
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('Statistics, covariance', 2, None, '___sec51'),
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('Statistics, more covariance', 2, None, '___sec52'),
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('Statistics, independent variables', 2, None, '___sec53'),
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('Statistics, more variance', 2, None, '___sec54'),
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('Statistics and stochastic processes', 2, None, '___sec55'),
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('Statistics and sample variables', 2, None, '___sec56'),
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('Statistics, sample variance and covariance',
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2,
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None,
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'___sec57'),
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('Statistics, law of large numbers', 2, None, '___sec58'),
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('Statistics, more on sample error', 2, None, '___sec59'),
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('Statistics', 2, None, '___sec60'),
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('Statistics, central limit theorem', 2, None, '___sec61'),
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('Statistics, more technicalities', 2, None, '___sec62'),
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('Statistics', 2, None, '___sec63'),
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('Statistics and sample variance', 2, None, '___sec64'),
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('Statistics, uncorrelated results', 2, None, '___sec65'),
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('Statistics, computations', 2, None, '___sec66'),
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('Statistics, more on computations of errors',
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2,
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None,
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'___sec67'),
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('Statistics, wrapping up 1', 2, None, '___sec68'),
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('Statistics, final expression', 2, None, '___sec69'),
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('Statistics, effective number of correlations',
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2,
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None,
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'___sec70'),
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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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'___sec71'),
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('Assumptions made', 2, None, '___sec72'),
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('Expectation value and variance', 2, None, '___sec73'),
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('Expectation value and variance for $\\boldsymbol{\\beta}$',
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None,
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'___sec74'),
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('Cross-validation', 2, None, '___sec75'),
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('Computationally expensive', 2, None, '___sec76'),
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('Various steps in cross-validation', 2, None, '___sec77'),
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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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'___sec78'),
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('Resampling methods: Jackknife and Bootstrap',
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2,
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None,
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'___sec79'),
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('Resampling methods: Jackknife', 2, None, '___sec80'),
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('Jackknife code example', 2, None, '___sec81'),
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('Resampling methods: Bootstrap', 2, None, '___sec82'),
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('Resampling methods: Bootstrap background', 2, None, '___sec83'),
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('Resampling methods: More Bootstrap background',
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2,
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None,
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'___sec84'),
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('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
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('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
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('Code example for the Bootstrap method', 2, None, '___sec87'),
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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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'___sec88'),
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('The bias-variance tradeoff', 2, None, '___sec89'),
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('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
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('Understanding what happens', 2, None, '___sec91'),
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('Summing up', 2, None, '___sec92'),
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("Another Example rom Scikit-Learn's Repository",
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2,
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None,
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'___sec93'),
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('The one-dimensional Ising model', 2, None, '___sec94'),
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('Ridge regression', 2, None, '___sec95'),
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|
('LASSO regression', 2, None, '___sec96'),
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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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'___sec97'),
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('Finding the optimal value of $\\lambda$', 2, None, '___sec98'),
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('Further Exercises', 2, None, '___sec99'),
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('Exercise 1', 3, None, '___sec100'),
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('Exercise 2, variance of the parameters $\\beta$ in linear '
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'regression',
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3,
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None,
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'___sec101'),
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('Exercise 3', 3, None, '___sec102'),
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('Exercise 4', 3, None, '___sec103')]}
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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%;"><b>Why Linear Regression (aka Ordinary Least Squares and family)</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;"><b>Regression analysis, overarching aims</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;"><b>Regression analysis, overarching aims II</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;"><b>Examples</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;"><b>General linear models</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem, more details</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;"><b>Optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;"><b>Our model for the nuclear binding energies</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;"><b>Optimizing our parameters, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;"><b>Some useful matrix and vector expressions</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;"><b>Own code for Ordinary Least Squares</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;"><b>Adding error analysis and training set up</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;"><b>Fitting an Equation of State for Dense Nuclear Matter</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;"><b>The code</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;"><b>Splitting our Data in Training and Test data</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;"><b>The singular value decomposition</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;"><b>The Ising model</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;"><b>Reformulating the problem to suit regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;"><b>Linear regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;"><b>Singular Value decomposition</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;"><b>Linear Regression Problems</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;"><b>Fixing the singularity</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;"><b>Basic math of the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;"><b>The SVD, a Fantastic Algorithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;"><b>Another Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;"><b>Economy-size SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;"><b>Mathematical Properties</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;"><b>Ridge and LASSO Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;"><b>More on Ridge Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;"><b>Interpreting the Ridge results</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;"><b>More interpretations</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;"><b>Where are we going?</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;"><b>Resampling methods</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;"><b>Resampling approaches can be computationally expensive</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;"><b>Why resampling methods ?</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;"><b>Statistical analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;"><b>Statistics</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;"><b>Statistics, moments</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;"><b>Statistics, central moments</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;"><b>Statistics, covariance</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;"><b>Statistics, more covariance</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;"><b>Statistics, independent variables</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;"><b>Statistics, more variance</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;"><b>Statistics and stochastic processes</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;"><b>Statistics and sample variables</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;"><b>Statistics, sample variance and covariance</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;"><b>Statistics, law of large numbers</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;"><b>Statistics, more on sample error</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;"><b>Statistics</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;"><b>Statistics, central limit theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;"><b>Statistics, more technicalities</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;"><b>Statistics</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;"><b>Statistics and sample variance</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;"><b>Statistics, uncorrelated results</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;"><b>Statistics, computations</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;"><b>Statistics, more on computations of errors</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;"><b>Statistics, wrapping up 1</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;"><b>Statistics, final expression</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;"><b>Statistics, effective number of correlations</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;"><b>Linking the regression analysis with a statistical interpretation</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;"><b>Assumptions made</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;"><b>Expectation value and variance</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;"><b>Expectation value and variance for \( \boldsymbol{\beta} \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;"><b>Cross-validation</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;"><b>Computationally expensive</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;"><b>Various steps in cross-validation</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;"><b>How to set up the cross-validation for Ridge and/or Lasso</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;"><b>Resampling methods: Jackknife and Bootstrap</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;"><b>Resampling methods: Jackknife</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;"><b>Jackknife code example</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;"><b>Resampling methods: Bootstrap</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;"><b>Resampling methods: Bootstrap background</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;"><b>Resampling methods: More Bootstrap background</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;"><b>Resampling methods: Bootstrap approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;"><b>Resampling methods: Bootstrap steps</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;"><b>Code example for the Bootstrap method</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;"><b>Code Example for Cross-validation and \( k \)-fold Cross-validation</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;"><b>The bias-variance tradeoff</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;"><b>Example code for Bias-Variance tradeoff</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;"><b>Understanding what happens</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;"><b>Summing up</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;"><b>Another Example rom Scikit-Learn's Repository</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;"><b>The one-dimensional Ising model</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;"><b>Ridge regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;"><b>LASSO regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;"><b>Performance as function of the regularization parameter</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec98" style="font-size: 80%;"><b>Finding the optimal value of \( \lambda \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;"><b>Further Exercises</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec100" style="font-size: 80%;"> Exercise 1</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec101" style="font-size: 80%;"> Exercise 2, variance of the parameters \( \beta \) in linear regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec102" style="font-size: 80%;"> Exercise 3</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec103" style="font-size: 80%;"> Exercise 4</a></li>
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</ul>
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</div>
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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">Finding the optimal value of \( \lambda \) </h2>
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<p>
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To determine which value of \( \lambda \) is best we plot the accuracy of
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the models when predicting the training and the testing set. We expect
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the accuracy of the training set to be quite good, but if the accuracy
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of the testing set is much lower this tells us that we might be
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subject to an overfit model. The ideal scenario is an accuracy on the
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testing set that is close to the accuracy of the training set.
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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>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
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colors <span style="color: #666666">=</span> {
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<span style="color: #BA2121">"ols_sk"</span>: <span style="color: #BA2121">"r"</span>,
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<span style="color: #BA2121">"ridge_sk"</span>: <span style="color: #BA2121">"y"</span>,
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<span style="color: #BA2121">"lasso_sk"</span>: <span style="color: #BA2121">"c"</span>
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}
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<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> train_errors:
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plt<span style="color: #666666">.</span>semilogx(
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lambdas,
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train_errors[key],
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colors[key],
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label<span style="color: #666666">=</span><span style="color: #BA2121">"Train {0}"</span><span style="color: #666666">.</span>format(key),
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linewidth<span style="color: #666666">=4.0</span>
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)
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<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> test_errors:
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plt<span style="color: #666666">.</span>semilogx(
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lambdas,
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test_errors[key],
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colors[key] <span style="color: #666666">+</span> <span style="color: #BA2121">"--"</span>,
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label<span style="color: #666666">=</span><span style="color: #BA2121">"Test {0}"</span><span style="color: #666666">.</span>format(key),
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linewidth<span style="color: #666666">=4.0</span>
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)
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plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"best"</span>, fontsize<span style="color: #666666">=18</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$\lambda$"</span>, fontsize<span style="color: #666666">=18</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$R^2$"</span>, fontsize<span style="color: #666666">=18</span>)
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plt<span style="color: #666666">.</span>tick_params(labelsize<span style="color: #666666">=18</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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From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
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achieves a very good accuracy on the test set. This by far surpasses the
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other models for all values of \( \lambda \).
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<p>
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<p>
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