Morten Hjorth-Jensen
477 lines
29 KiB
HTML
477 lines
29 KiB
HTML
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<title>Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</title>
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Plans for week 35', 2, None, 'plans-for-week-35'),
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('Reading recommendations:', 3, None, 'reading-recommendations'),
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('Why Linear Regression (aka Ordinary Least Squares and family), '
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|
'repeat from last week',
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|
2,
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|
None,
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|
'why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week'),
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('Regression analysis, overarching aims',
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2,
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None,
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'regression-analysis-overarching-aims'),
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('Regression analysis, overarching aims II',
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2,
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'regression-analysis-overarching-aims-ii'),
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('Examples', 2, None, 'examples'),
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('General linear models', 2, None, 'general-linear-models'),
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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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'rewriting-the-fitting-procedure-as-a-linear-algebra-problem'),
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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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'rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details'),
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('Generalizing the fitting procedure as a linear algebra problem',
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'generalizing-the-fitting-procedure-as-a-linear-algebra-problem'),
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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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'generalizing-the-fitting-procedure-as-a-linear-algebra-problem'),
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('Optimizing our parameters',
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2,
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None,
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'optimizing-our-parameters'),
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('Our model for the nuclear binding energies',
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2,
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'our-model-for-the-nuclear-binding-energies'),
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('Optimizing our parameters, more details',
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2,
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None,
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'optimizing-our-parameters-more-details'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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'interpretations-and-optimizing-our-parameters'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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'interpretations-and-optimizing-our-parameters'),
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('Some useful matrix and vector expressions',
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2,
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None,
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'some-useful-matrix-and-vector-expressions'),
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('Meet the Hessian Matrix', 2, None, 'meet-the-hessian-matrix'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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'interpretations-and-optimizing-our-parameters'),
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('Own code for Ordinary Least Squares',
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2,
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None,
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'own-code-for-ordinary-least-squares'),
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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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'adding-error-analysis-and-training-set-up'),
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('Splitting our Data in Training and Test data',
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2,
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('Examples', 2, None, 'examples'),
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('Making your own test-train splitting',
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2,
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None,
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('The Boston housing data example',
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2,
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None,
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'the-boston-housing-data-example'),
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('Housing data, the code', 2, None, 'housing-data-the-code'),
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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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'reducing-the-number-of-degrees-of-freedom-overarching-view'),
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('Preprocessing our data', 2, None, 'preprocessing-our-data'),
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('Functionality in Scikit-Learn',
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2,
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None,
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'functionality-in-scikit-learn'),
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('More preprocessing', 2, None, 'more-preprocessing'),
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('Frequently used scaling functions',
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2,
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None,
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'frequently-used-scaling-functions'),
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('Example of own Standard scaling',
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2,
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None,
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'example-of-own-standard-scaling'),
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('Min-Max Scaling', 2, None, 'min-max-scaling'),
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('Testing the Means Squared Error as function of Complexity',
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2,
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None,
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'testing-the-means-squared-error-as-function-of-complexity'),
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('More preprocessing examples, Franke function and regression',
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2,
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None,
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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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('Simple case', 2, None, 'simple-case'),
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('The singular value decomposition',
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2,
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None,
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'the-singular-value-decomposition'),
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('Linear Regression Problems',
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2,
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'linear-regression-problems'),
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('Fixing the singularity', 2, None, 'fixing-the-singularity'),
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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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None,
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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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'note-about-svd-calculations'),
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('Mathematics of the SVD and implications',
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2,
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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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('Further properties (important for our analyses later)',
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2,
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('Meet the Covariance Matrix',
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('Introducing the Covariance and Correlation functions',
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('Covariance and Correlation Matrix',
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('Correlation Function and Design/Feature Matrix',
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('Covariance Matrix Examples',
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('Correlation Matrix', 2, None, 'correlation-matrix'),
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('Correlation Matrix with Pandas',
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2,
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None,
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'correlation-matrix-with-pandas'),
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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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'correlation-matrix-with-pandas-and-the-franke-function'),
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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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'rewriting-the-covariance-and-or-correlation-matrix'),
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('Linking with the SVD', 2, None, 'linking-with-the-svd'),
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('What does it mean?', 2, None, 'what-does-it-mean'),
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('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$',
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2,
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None,
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'and-finally-boldsymbol-x-boldsymbol-x-t'),
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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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('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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|
('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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|
('Exercises for week 35', 2, None, 'exercises-for-week-35'),
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|
('Exercise 1: Setting up various Python environments',
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2,
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None,
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'exercise-1-setting-up-various-python-environments'),
|
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('Exercise 2: making your own data and exploring scikit-learn',
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2,
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None,
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'exercise-2-making-your-own-data-and-exploring-scikit-learn'),
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('Exercise 3: Normalizing our data',
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2,
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|
None,
|
|
'exercise-3-normalizing-our-data'),
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|
('Exercise 4: Adding Ridge Regression',
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2,
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|
None,
|
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'exercise-4-adding-ridge-regression'),
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('Exercise 5: Analytical exercises',
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2,
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None,
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'exercise-5-analytical-exercises')]}
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<a class="navbar-brand" href="week35-bs.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a>
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<ul class="nav navbar-nav navbar-right">
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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="._week35-bs001.html#plans-for-week-35" style="font-size: 80%;"><b>Plans for week 35</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#reading-recommendations" style="font-size: 80%;"> Reading recommendations:</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs002.html#why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week" style="font-size: 80%;"><b>Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs003.html#regression-analysis-overarching-aims" style="font-size: 80%;"><b>Regression analysis, overarching aims</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs004.html#regression-analysis-overarching-aims-ii" style="font-size: 80%;"><b>Regression analysis, overarching aims II</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs022.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs006.html#general-linear-models" style="font-size: 80%;"><b>General linear models</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem" 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="._week35-bs008.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details" 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="._week35-bs010.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" 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="._week35-bs010.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" 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="._week35-bs011.html#optimizing-our-parameters" style="font-size: 80%;"><b>Optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#our-model-for-the-nuclear-binding-energies" style="font-size: 80%;"><b>Our model for the nuclear binding energies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#optimizing-our-parameters-more-details" style="font-size: 80%;"><b>Optimizing our parameters, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#some-useful-matrix-and-vector-expressions" style="font-size: 80%;"><b>Some useful matrix and vector expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs017.html#meet-the-hessian-matrix" style="font-size: 80%;"><b>Meet the Hessian Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs019.html#own-code-for-ordinary-least-squares" style="font-size: 80%;"><b>Own code for Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs020.html#adding-error-analysis-and-training-set-up" style="font-size: 80%;"><b>Adding error analysis and training set up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs021.html#splitting-our-data-in-training-and-test-data" 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="._week35-bs022.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#making-your-own-test-train-splitting" style="font-size: 80%;"><b>Making your own test-train splitting</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs024.html#the-boston-housing-data-example" style="font-size: 80%;"><b>The Boston housing data example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs025.html#housing-data-the-code" style="font-size: 80%;"><b>Housing data, the code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs026.html#reducing-the-number-of-degrees-of-freedom-overarching-view" style="font-size: 80%;"><b>Reducing the number of degrees of freedom, overarching view</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.html#preprocessing-our-data" style="font-size: 80%;"><b>Preprocessing our data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs028.html#functionality-in-scikit-learn" style="font-size: 80%;"><b>Functionality in Scikit-Learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs029.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs030.html#frequently-used-scaling-functions" style="font-size: 80%;"><b>Frequently used scaling functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs031.html#example-of-own-standard-scaling" style="font-size: 80%;"><b>Example of own Standard scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs032.html#min-max-scaling" style="font-size: 80%;"><b>Min-Max Scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs033.html#testing-the-means-squared-error-as-function-of-complexity" style="font-size: 80%;"><b>Testing the Means Squared Error as function of Complexity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs034.html#more-preprocessing-examples-franke-function-and-regression" style="font-size: 80%;"><b>More preprocessing examples, Franke function and regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs035.html#mathematical-interpretation-of-ordinary-least-squares" style="font-size: 80%;"><b>Mathematical Interpretation of Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs037.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs038.html#the-singular-value-decomposition" style="font-size: 80%;"><b>The singular value decomposition</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs039.html#linear-regression-problems" style="font-size: 80%;"><b>Linear Regression Problems</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs040.html#fixing-the-singularity" style="font-size: 80%;"><b>Fixing the singularity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs041.html#basic-math-of-the-svd" style="font-size: 80%;"><b>Basic math of the SVD</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs042.html#the-svd-a-fantastic-algorithm" style="font-size: 80%;"><b>The SVD, a Fantastic Algorithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs043.html#economy-size-svd" style="font-size: 80%;"><b>Economy-size SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs044.html#codes-for-the-svd" style="font-size: 80%;"><b>Codes for the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs045.html#note-about-svd-calculations" style="font-size: 80%;"><b>Note about SVD Calculations</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs046.html#mathematics-of-the-svd-and-implications" style="font-size: 80%;"><b>Mathematics of the SVD and implications</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs047.html#example-matrix" style="font-size: 80%;"><b>Example Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs048.html#setting-up-the-matrix-to-be-inverted" style="font-size: 80%;"><b>Setting up the Matrix to be inverted</b></a></li>
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<!-- navigation toc: --> <li><a href="#further-properties-important-for-our-analyses-later" style="font-size: 80%;"><b>Further properties (important for our analyses later)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs050.html#meet-the-covariance-matrix" style="font-size: 80%;"><b>Meet the Covariance Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs051.html#introducing-the-covariance-and-correlation-functions" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs052.html#covariance-and-correlation-matrix" style="font-size: 80%;"><b>Covariance and Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs053.html#correlation-function-and-design-feature-matrix" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs054.html#covariance-matrix-examples" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs055.html#correlation-matrix" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs056.html#correlation-matrix-with-pandas" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs057.html#correlation-matrix-with-pandas-and-the-franke-function" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs058.html#rewriting-the-covariance-and-or-correlation-matrix" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs059.html#linking-with-the-svd" style="font-size: 80%;"><b>Linking with the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs060.html#what-does-it-mean" style="font-size: 80%;"><b>What does it mean?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs061.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;"><b>And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs062.html#ridge-and-lasso-regression" style="font-size: 80%;"><b>Ridge and LASSO Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs063.html#deriving-the-ridge-regression-equations" style="font-size: 80%;"><b>Deriving the Ridge Regression Equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs064.html#interpreting-the-ridge-results" style="font-size: 80%;"><b>Interpreting the Ridge results</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs065.html#more-interpretations" style="font-size: 80%;"><b>More interpretations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs066.html#deriving-the-lasso-regression-equations" style="font-size: 80%;"><b>Deriving the Lasso Regression Equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs067.html#exercises-for-week-35" style="font-size: 80%;"><b>Exercises for week 35</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs067.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs067.html#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs067.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs067.html#exercise-4-adding-ridge-regression" style="font-size: 80%;"><b>Exercise 4: Adding Ridge Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs067.html#exercise-5-analytical-exercises" style="font-size: 80%;"><b>Exercise 5: Analytical exercises</b></a></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="part0049"></a>
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<!-- !split -->
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<h2 id="further-properties-important-for-our-analyses-later" class="anchor">Further properties (important for our analyses later) </h2>
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<p>Let us study again \( \boldsymbol{X}^T\boldsymbol{X} \) in terms of our SVD,</p>
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$$
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\boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T.
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$$
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<p>If we now multiply from the right with \( \boldsymbol{V} \) (using the orthogonality of \( \boldsymbol{V} \)) we get</p>
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$$
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\left(\boldsymbol{X}^T\boldsymbol{X}\right)\boldsymbol{V}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}.
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$$
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<p>This means the vectors \( \boldsymbol{v}_i \) of the orthogonal matrix \( \boldsymbol{V} \) are the eigenvectors of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \)
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with eigenvalues given by the singular values squared, that is
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</p>
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$$
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\left(\boldsymbol{X}^T\boldsymbol{X}\right)\boldsymbol{v}_i=\boldsymbol{v}_i\sigma_i^2.
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$$
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<p>Similarly, if we use the SVD decomposition for the matrix \( \boldsymbol{X}\boldsymbol{X}^T \), we have</p>
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$$
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\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{\Sigma}^T\boldsymbol{U}^T.
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$$
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<p>If we now multiply from the right with \( \boldsymbol{U} \) (using the orthogonality of \( \boldsymbol{U} \)) we get</p>
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$$
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\left(\boldsymbol{X}\boldsymbol{X}^T\right)\boldsymbol{U}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{\Sigma}^T.
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$$
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<p>This means the vectors \( \boldsymbol{u}_i \) of the orthogonal matrix \( \boldsymbol{U} \) are the eigenvectors of the matrix \( \boldsymbol{X}\boldsymbol{X}^T \)
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with eigenvalues given by the singular values squared, that is
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</p>
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$$
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\left(\boldsymbol{X}\boldsymbol{X}^T\right)\boldsymbol{u}_i=\boldsymbol{u}_i\sigma_i^2.
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$$
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<p><b>Important note</b>: we have defined our design matrix \( \boldsymbol{X} \) to be an
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\( n\times p \) matrix. In most supervised learning cases we have that \( n
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\ge p \), and quite often we have \( n >> p \). For linear algebra based methods like ordinary least squares or Ridge regression, this leads to a matrix \( \boldsymbol{X}^T\boldsymbol{X} \) which is small and thereby easier to handle from a computational point of view (in terms of number of floating point operations).
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</p>
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<p>In our lectures, the number of columns will
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always refer to the number of features in our data set, while the
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number of rows represents the number of data inputs. Note that in
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other texts you may find the opposite notation. This has consequences
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for the definition of for example the covariance matrix and its relation to the SVD.
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</p>
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<p>
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