Morten Hjorth-Jensen
390 lines
16 KiB
HTML
390 lines
16 KiB
HTML
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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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('Thursday September 1', 2, None, 'thursday-september-1'),
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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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None,
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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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2,
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None,
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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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None,
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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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None,
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'splitting-our-data-in-training-and-test-data'),
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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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'making-your-own-test-train-splitting'),
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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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'more-preprocessing-examples-franke-function-and-regression'),
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('Mathematical Interpretation of Ordinary Least Squares',
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2,
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None,
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'mathematical-interpretation-of-ordinary-least-squares'),
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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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None,
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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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None,
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'note-about-svd-calculations'),
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('Friday September 2', 2, None, 'friday-september-2'),
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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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('Meet the Covariance Matrix',
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2,
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None,
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'meet-the-covariance-matrix'),
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('Introducing the Covariance and Correlation functions',
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2,
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None,
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'introducing-the-covariance-and-correlation-functions'),
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('Covariance and Correlation Matrix',
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2,
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None,
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'covariance-and-correlation-matrix'),
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('Correlation Function and Design/Feature Matrix',
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2,
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None,
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'correlation-function-and-design-feature-matrix'),
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('Covariance Matrix Examples',
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2,
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None,
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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,
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'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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end of tocinfo -->
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<body>
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<script type="text/x-mathjax-config">
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MathJax.Hub.Config({
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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="part0065"></a>
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<!-- !split -->
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<h2 id="deriving-the-ridge-regression-equations" class="anchor">Deriving the Ridge Regression Equations </h2>
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<p>Using the matrix-vector expression for Ridge regression and dropping the parameter \( 1/n \) in front of the standard means squared error equation, we have</p>
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$$
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C(\boldsymbol{X},\boldsymbol{\beta})=\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta},
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$$
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<p>and
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taking the derivatives with respect to \( \boldsymbol{\beta} \) we obtain then
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a slightly modified matrix inversion problem which for finite values
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of \( \lambda \) does not suffer from singularity problems. We obtain
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the optimal parameters
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</p>
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$$
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\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
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$$
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<p>with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that</p>
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$$
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\sum_{i=0}^{p-1} \beta_i^2 \leq t,
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$$
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<p>with \( t \) a finite positive number. </p>
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<p>If we keep the \( 1/n \) factor, the equation for the optimal \( \beta \) changes to</p>
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$$
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\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+n\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
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$$
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<p>In many textbooks the \( 1/n \) term is often omitted. Note that a library like <b>Scikit-Learn</b> does not include the \( 1/n \) factor in the setup of the cost function.</p>
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<p>When we compare this with the ordinary least squares result we have</p>
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$$
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\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
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$$
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<p>which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \).</p>
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<p>We see that Ridge regression is nothing but the standard OLS with a
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modified diagonal term added to \( \boldsymbol{X}^T\boldsymbol{X} \). The consequences, in
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particular for our discussion of the bias-variance tradeoff are rather
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interesting. We will see that for specific values of \( \lambda \), we may
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even reduce the variance of the optimal parameters \( \boldsymbol{\beta} \). These topics and other related ones, will be discussed after the more linear algebra oriented analysis here.
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</p>
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<p>Using our insights about the SVD of the design matrix \( \boldsymbol{X} \)
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We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \( \boldsymbol{U} \) as
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</p>
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$$
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\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta} =\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}.
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$$
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<p>For Ridge regression this becomes</p>
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$$
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\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
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$$
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<p>with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \) from the SVD of the matrix \( \boldsymbol{X} \). </p>
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<p>
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