mhjensen
262 lines
11 KiB
HTML
262 lines
11 KiB
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'sections': [('Plans for week 37', 2, None, '___sec0'),
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('Thursday September 10', 2, None, '___sec1'),
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('A Bayesian approach to develop intuition about skrinkage '
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'methods',
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2,
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None,
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('The singular value decomposition', 2, None, '___sec3'),
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('Fixing the singularity', 2, None, '___sec5'),
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('The SVD, a Fantastic Algorithm', 2, None, '___sec7'),
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('Economy-size SVD', 2, None, '___sec8'),
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('Codes for the SVD', 2, None, '___sec9'),
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('Mathematical Properties', 2, None, '___sec10'),
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('Friday September 12', 2, None, '___sec11'),
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('Ridge and LASSO Regression', 2, None, '___sec12'),
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('More on Ridge Regression', 2, None, '___sec13'),
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('Interpreting the Ridge results', 2, None, '___sec14'),
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('More interpretations', 2, None, '___sec15'),
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('A better understanding of regularization', 2, None, '___sec16'),
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('Decomposing the OLS and Ridge expressions',
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('Introducing the Covariance and Correlation functions',
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<a class="navbar-brand" href="week37-bs.html">Week 37: Ridge and Lasso Regression</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week37-bs001.html#___sec0" style="font-size: 80%;">Plans for week 37</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs002.html#___sec1" style="font-size: 80%;">Thursday September 10</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs003.html#___sec2" style="font-size: 80%;">A Bayesian approach to develop intuition about skrinkage methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs004.html#___sec3" style="font-size: 80%;">The singular value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">Linear Regression Problems</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs006.html#___sec5" style="font-size: 80%;">Fixing the singularity</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs007.html#___sec6" style="font-size: 80%;">Basic math of the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.html#___sec7" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.html#___sec8" style="font-size: 80%;">Economy-size SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs010.html#___sec9" style="font-size: 80%;">Codes for the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs011.html#___sec10" style="font-size: 80%;">Mathematical Properties</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs012.html#___sec11" style="font-size: 80%;">Friday September 12</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#___sec12" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs014.html#___sec13" style="font-size: 80%;">More on Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs015.html#___sec14" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs016.html#___sec15" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#___sec16" style="font-size: 80%;">A better understanding of regularization</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#___sec17" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#___sec18" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#___sec19" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#___sec20" style="font-size: 80%;">Covariance Matrix Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs022.html#___sec21" style="font-size: 80%;">Correlation Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs023.html#___sec22" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs024.html#___sec23" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs025.html#___sec24" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.html#___sec25" style="font-size: 80%;">Linking with SVD</a></li>
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</li>
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<a name="part0005"></a>
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<!-- !split -->
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<h2 id="___sec4" class="anchor">Linear Regression Problems </h2>
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<p>
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One of the typical problems we encounter with linear regression, in particular
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when the matrix \( \boldsymbol{X} \) (our so-called design matrix) is high-dimensional,
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are problems with near singular or singular matrices. The column vectors of \( \boldsymbol{X} \)
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may be linearly dependent, normally referred to as super-collinearity.
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This means that the matrix may be rank deficient and it is basically impossible to
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to model the data using linear regression. As an example, consider the matrix
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$$
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\begin{align*}
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\mathbf{X} & = \left[
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\begin{array}{rrr}
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1 & -1 & 2
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\\
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1 & 0 & 1
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\\
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1 & 2 & -1
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\\
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1 & 1 & 0
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\end{array} \right]
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\end{align*}
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$$
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<p>
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The columns of \( \boldsymbol{X} \) are linearly dependent. We see this easily since the
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the first column is the row-wise sum of the other two columns. The rank (more correct,
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the column rank) of a matrix is the dimension of the space spanned by the
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column vectors. Hence, the rank of \( \mathbf{X} \) is equal to the number
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of linearly independent columns. In this particular case the matrix has rank 2.
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<p>
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Super-collinearity of an \( (n \times p) \)-dimensional design matrix \( \mathbf{X} \) implies
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that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
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$$
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\begin{align*}
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\boldsymbol{X} & = \left[
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\begin{array}{rr}
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1 & -1
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\\
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1 & -1
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\end{array} \right].
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\end{align*}
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$$
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We see easily that \( \mbox{det}(\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0 \). Hence, \( \mathbf{X} \) is singular and its inverse is undefined.
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This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least an eigenvalue which is zero.
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<p>
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<p>
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