454 lines
30 KiB
HTML
454 lines
30 KiB
HTML
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<title>Week 36: Linear Regression and Gradient descent</title>
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{'highest level': 2,
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'sections': [('Plans for week 36', 2, None, 'plans-for-week-36'),
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('Material for lecture Monday September 2',
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2,
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None,
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'material-for-lecture-monday-september-2'),
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('Mathematical Interpretation of Ordinary Least Squares',
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2,
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None,
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('Residual Error', 2, None, 'residual-error'),
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('Linear Regression Problems',
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('Fixing the singularity', 2, None, 'fixing-the-singularity'),
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None,
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('Deriving the Ridge Regression Equations',
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('Basic math of the SVD', 2, None, 'basic-math-of-the-svd'),
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('The SVD, a Fantastic Algorithm',
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2,
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'the-svd-a-fantastic-algorithm'),
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('Economy-size SVD', 2, None, 'economy-size-svd'),
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('Codes for the SVD', 2, None, 'codes-for-the-svd'),
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('Note about SVD Calculations',
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2,
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None,
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'note-about-svd-calculations'),
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('Mathematics of the SVD and implications',
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2,
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None,
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'mathematics-of-the-svd-and-implications'),
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('Example Matrix', 2, None, 'example-matrix'),
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('Setting up the Matrix to be inverted',
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2,
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None,
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'setting-up-the-matrix-to-be-inverted'),
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('Further properties (important for our analyses later)',
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2,
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None,
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'further-properties-important-for-our-analyses-later'),
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('Back to Ridge and LASSO Regression',
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2,
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None,
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'back-to-ridge-and-lasso-regression'),
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('Interpreting the Ridge results',
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2,
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None,
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'interpreting-the-ridge-results'),
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('More interpretations', 2, None, 'more-interpretations'),
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('Deriving the Lasso Regression Equations',
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2,
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None,
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'deriving-the-lasso-regression-equations'),
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('Optimization and gradient descent, the central part of any '
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'Machine Learning algortithm',
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2,
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None,
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'optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm'),
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("Reminder on Newton-Raphson's method",
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2,
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None,
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'reminder-on-newton-raphson-s-method'),
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('The equations', 2, None, 'the-equations'),
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('Simple geometric interpretation',
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2,
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None,
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'simple-geometric-interpretation'),
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('Extending to more than one variable',
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2,
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None,
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'extending-to-more-than-one-variable'),
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('Steepest descent', 2, None, 'steepest-descent'),
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('More on Steepest descent', 2, None, 'more-on-steepest-descent'),
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('The ideal', 2, None, 'the-ideal'),
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('The sensitiveness of the gradient descent',
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2,
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None,
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'the-sensitiveness-of-the-gradient-descent'),
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('Convex functions', 2, None, 'convex-functions'),
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('Convex function', 2, None, 'convex-function'),
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('Conditions on convex functions',
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2,
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None,
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'conditions-on-convex-functions'),
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('More on convex functions', 2, None, 'more-on-convex-functions'),
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('Some simple problems', 2, None, 'some-simple-problems'),
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('Revisiting Ordinary Least Squares',
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2,
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None,
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'revisiting-ordinary-least-squares'),
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('Gradient descent example', 2, None, 'gradient-descent-example'),
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('The derivative of the cost/loss function',
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2,
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None,
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'the-derivative-of-the-cost-loss-function'),
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('The Hessian matrix', 2, None, 'the-hessian-matrix'),
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('Simple program', 2, None, 'simple-program'),
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('Gradient Descent Example', 2, None, 'gradient-descent-example'),
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('Gradient descent and Ridge',
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2,
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None,
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'gradient-descent-and-ridge'),
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('The Hessian matrix for Ridge Regression',
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2,
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None,
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'the-hessian-matrix-for-ridge-regression'),
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('Program example for gradient descent with Ridge Regression',
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2,
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None,
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'program-example-for-gradient-descent-with-ridge-regression'),
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('Using gradient descent methods, limitations',
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2,
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None,
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'using-gradient-descent-methods-limitations'),
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('Material for lab sessions sessions Tuesday and Wednesday',
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2,
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None,
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'material-for-lab-sessions-sessions-tuesday-and-wednesday'),
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('Linear Regression and the SVD',
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2,
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None,
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'linear-regression-and-the-svd'),
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('What does it mean?', 2, None, 'what-does-it-mean'),
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('Ridge and LASSO Regression',
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2,
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None,
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'ridge-and-lasso-regression'),
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('From OLS to Ridge and Lasso',
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2,
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None,
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'from-ols-to-ridge-and-lasso'),
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('Deriving the Ridge Regression Equations',
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2,
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None,
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'deriving-the-ridge-regression-equations'),
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('Note on Scikit-Learn', 2, None, 'note-on-scikit-learn'),
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('Comparison with OLS', 2, None, 'comparison-with-ols'),
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('SVD analysis', 2, None, 'svd-analysis'),
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('Interpreting the Ridge results',
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2,
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None,
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'interpreting-the-ridge-results'),
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('More interpretations', 2, None, 'more-interpretations'),
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('Deriving the Lasso Regression Equations',
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2,
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None,
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'deriving-the-lasso-regression-equations'),
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('Simple example to illustrate Ordinary Least Squares, Ridge and '
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'Lasso Regression',
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2,
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None,
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'simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression'),
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('Ridge Regression', 2, None, 'ridge-regression'),
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('Lasso Regression', 2, None, 'lasso-regression'),
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('Yet another Example', 2, None, 'yet-another-example'),
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('The OLS case', 2, None, 'the-ols-case'),
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('The Ridge case', 2, None, 'the-ridge-case'),
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('Writing the Cost Function',
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2,
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None,
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'writing-the-cost-function'),
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('Lasso case', 2, None, 'lasso-case'),
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('The first Case', 2, None, 'the-first-case'),
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('Simple code for solving the above problem',
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2,
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None,
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'simple-code-for-solving-the-above-problem'),
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('With Lasso Regression', 2, None, 'with-lasso-regression'),
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('Another Example, now with a polynomial fit',
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2,
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None,
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'another-example-now-with-a-polynomial-fit')]}
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end of tocinfo -->
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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="._week36-bs001.html#plans-for-week-36" style="font-size: 80%;">Plans for week 36</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs002.html#material-for-lecture-monday-september-2" style="font-size: 80%;">Material for lecture Monday September 2</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs003.html#mathematical-interpretation-of-ordinary-least-squares" style="font-size: 80%;">Mathematical Interpretation of Ordinary Least Squares</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.html#residual-error" style="font-size: 80%;">Residual Error</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs005.html#simple-case" style="font-size: 80%;">Simple case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs006.html#the-singular-value-decomposition" style="font-size: 80%;">The singular value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#linear-regression-problems" style="font-size: 80%;">Linear Regression Problems</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs008.html#fixing-the-singularity" style="font-size: 80%;">Fixing the singularity</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs053.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs011.html#basic-math-of-the-svd" style="font-size: 80%;">Basic math of the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#the-svd-a-fantastic-algorithm" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#economy-size-svd" style="font-size: 80%;">Economy-size SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#codes-for-the-svd" style="font-size: 80%;">Codes for the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.html#note-about-svd-calculations" style="font-size: 80%;">Note about SVD Calculations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.html#mathematics-of-the-svd-and-implications" style="font-size: 80%;">Mathematics of the SVD and implications</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.html#example-matrix" style="font-size: 80%;">Example Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#setting-up-the-matrix-to-be-inverted" style="font-size: 80%;">Setting up the Matrix to be inverted</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#further-properties-important-for-our-analyses-later" style="font-size: 80%;">Further properties (important for our analyses later)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#back-to-ridge-and-lasso-regression" style="font-size: 80%;">Back to Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs057.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs059.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization and gradient descent, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#reminder-on-newton-raphson-s-method" style="font-size: 80%;">Reminder on Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs030.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.html#convex-functions" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#convex-function" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.html#conditions-on-convex-functions" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs036.html#more-on-convex-functions" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs037.html#some-simple-problems" style="font-size: 80%;">Some simple problems</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs038.html#revisiting-ordinary-least-squares" style="font-size: 80%;">Revisiting Ordinary Least Squares</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs043.html#gradient-descent-example" style="font-size: 80%;">Gradient descent example</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs040.html#the-derivative-of-the-cost-loss-function" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs041.html#the-hessian-matrix" style="font-size: 80%;">The Hessian matrix</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs042.html#simple-program" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs043.html#gradient-descent-example" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.html#gradient-descent-and-ridge" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs045.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;">The Hessian matrix for Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs046.html#program-example-for-gradient-descent-with-ridge-regression" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs047.html#using-gradient-descent-methods-limitations" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.html#material-for-lab-sessions-sessions-tuesday-and-wednesday" style="font-size: 80%;">Material for lab sessions sessions Tuesday and Wednesday</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs049.html#linear-regression-and-the-svd" style="font-size: 80%;">Linear Regression and the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#what-does-it-mean" style="font-size: 80%;">What does it mean?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs052.html#from-ols-to-ridge-and-lasso" style="font-size: 80%;">From OLS to Ridge and Lasso</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs053.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs054.html#note-on-scikit-learn" style="font-size: 80%;">Note on Scikit-Learn</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs055.html#comparison-with-ols" style="font-size: 80%;">Comparison with OLS</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs056.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs057.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs058.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs059.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs060.html#simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression" style="font-size: 80%;">Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs061.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs062.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
|
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<!-- navigation toc: --> <li><a href="._week36-bs063.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<h2 id="with-lasso-regression" class="anchor">With Lasso Regression </h2>
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</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: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> linear_model
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
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<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
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n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
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<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
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<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ [ <span style="color: #666666">2</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">0</span>,<span style="color: #666666">0</span>]])
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [<span style="color: #666666">4</span>, <span style="color: #666666">2</span>, <span style="color: #666666">3</span>])
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<span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
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OLSbeta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
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<span style="color: #008000">print</span>(OLSbeta)
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<span style="color: #408080; font-style: italic"># and then make the prediction</span>
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ytildeOLS <span style="color: #666666">=</span> X <span style="color: #666666">@</span> OLSbeta
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Training MSE for OLS"</span>)
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<span style="color: #008000">print</span>(MSE(y,ytildeOLS))
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ypredictOLS <span style="color: #666666">=</span> X <span style="color: #666666">@</span> OLSbeta
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<span style="color: #408080; font-style: italic"># Repeat now for Ridge regression and various values of the regularization parameter</span>
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I <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">2</span>,<span style="color: #666666">2</span>)
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<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
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nlambdas <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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MSERidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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MSELassoPredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">4</span>, nlambdas)
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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>(nlambdas):
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lmb <span style="color: #666666">=</span> lambdas[i]
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Ridgebeta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #666666">+</span>lmb<span style="color: #666666">*</span>I) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
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<span style="color: #008000">print</span>(Ridgebeta)
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<span style="color: #408080; font-style: italic"># and then make the prediction</span>
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ypredictRidge <span style="color: #666666">=</span> X <span style="color: #666666">@</span> Ridgebeta
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MSERidgePredict[i] <span style="color: #666666">=</span> MSE(y,ypredictRidge)
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RegLasso <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>Lasso(lmb,fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
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RegLasso<span style="color: #666666">.</span>fit(X,y)
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ypredictLasso <span style="color: #666666">=</span> RegLasso<span style="color: #666666">.</span>predict(X)
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<span style="color: #008000">print</span>(RegLasso<span style="color: #666666">.</span>coef_)
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MSELassoPredict[i] <span style="color: #666666">=</span> MSE(y,ypredictLasso)
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<span style="color: #408080; font-style: italic"># Now plot the results</span>
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plt<span style="color: #666666">.</span>figure()
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSERidgePredict, <span style="color: #BA2121">'r--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE Ridge Train'</span>)
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSELassoPredict, <span style="color: #BA2121">'r--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE Lasso Train'</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MSE'</span>)
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plt<span style="color: #666666">.</span>legend()
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plt<span style="color: #666666">.</span>show()
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</pre>
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