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See .\n", "\n", " * [Video on cross validation](https://www.youtube.com/watch?v=fSytzGwwBVw)\n", "\n", " * [Video on Bootstrapping](https://www.youtube.com/watch?v=Xz0x-8-cgaQ)\n", "\n", - " * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)\n", + " * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)" + ] + }, + { + "cell_type": "markdown", + "id": "bf80a276", + "metadata": { + "editable": true + }, + "source": [ + "## Plans for week 37, lab sessions\n", "\n", - "**Material for the active learning sessions on Tuesday and Wednesday.**\n", + "**Material for the lab sessions on Tuesday and Wednesday.**\n", "\n", " * Calculations of expectation values\n", "\n", + " * Discussion of resampling techniques\n", + "\n", " * Exercise set for week 37\n", "\n", " * Work on project 1\n", "\n", - " * See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.\n", - "\n", " * For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended." ] }, { "cell_type": "markdown", - "id": "f7e43b1b", + "id": "0e1f0a90", "metadata": { "editable": true }, "source": [ - "## Material from last week and relevant for the weekly exercises" + "## Material for lecture Monday September 9" ] }, { "cell_type": "markdown", - "id": "af6a447a", - "metadata": { - "editable": true - }, - "source": [ - "## Linking the regression analysis with a statistical interpretation\n", - "\n", - "We will now couple the discussions of ordinary least squares, Ridge\n", - "and Lasso regression with a statistical interpretation, that is we\n", - "move from a linear algebra analysis to a statistical analysis. In\n", - "particular, we will focus on what the regularization terms can result\n", - "in. We will amongst other things show that the regularization\n", - "parameter can reduce considerably the variance of the parameters\n", - "$\\beta$.\n", - "\n", - "The\n", - "advantage of doing linear regression is that we actually end up with\n", - "analytical expressions for several statistical quantities. \n", - "Standard least squares and Ridge regression allow us to\n", - "derive quantities like the variance and other expectation values in a\n", - "rather straightforward way.\n", - "\n", - "It is assumed that $\\varepsilon_i\n", - "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n", - "independent, i.e.:" - ] - }, - { - "cell_type": "markdown", - "id": "b6b7ea38", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\begin{align*} \n", - "\\mbox{Cov}(\\varepsilon_{i_1},\n", - "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n", - "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2. \\end{array} \\right.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "e53d8d41", - "metadata": { - "editable": true - }, - "source": [ - "The randomness of $\\varepsilon_i$ implies that\n", - "$\\mathbf{y}_i$ is also a random variable. In particular,\n", - "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n", - "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n", - "non-random scalar. To specify the parameters of the distribution of\n", - "$\\mathbf{y}_i$ we need to calculate its first two moments. \n", - "\n", - "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n", - "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n", - "row number $i$ and perform a sum over all values $p$." - ] - }, - { - "cell_type": "markdown", - "id": "401644e7", - "metadata": { - "editable": true - }, - "source": [ - "## Assumptions made\n", - "\n", - "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n", - "that there exists a function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n", - "which describe our data" - ] - }, - { - "cell_type": "markdown", - "id": "0603387c", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "eb2b6edf", - "metadata": { - "editable": true - }, - "source": [ - "We approximate this function with our model from the solution of the linear regression equations, that is our\n", - "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with" - ] - }, - { - "cell_type": "markdown", - "id": "f1a08194", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "ad40fa0b", - "metadata": { - "editable": true - }, - "source": [ - "## Expectation value and variance\n", - "\n", - "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$" - ] - }, - { - "cell_type": "markdown", - "id": "d3a4907d", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\begin{align*} \n", - "\\mathbb{E}(y_i) & =\n", - "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n", - "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "d7908c9c", - "metadata": { - "editable": true - }, - "source": [ - "while\n", - "its variance is" - ] - }, - { - "cell_type": "markdown", - "id": "2528f74e", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n", - "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n", - "[\\mathbb{E}(y_i)]^2 \\\\ & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n", - "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n", - "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n", - "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n", - "\\ast} \\, \\beta)^2 \\\\ & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n", - "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n", - "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n", - "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n", - "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2. \n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "525f3a59", - "metadata": { - "editable": true - }, - "source": [ - "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n", - "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD)." - ] - }, - { - "cell_type": "markdown", - "id": "ead1d419", - "metadata": { - "editable": true - }, - "source": [ - "## Expectation value and variance for $\\boldsymbol{\\beta}$\n", - "\n", - "With the OLS expressions for the optimal parameters $\\boldsymbol{\\hat{\\beta}}$ we can evaluate the expectation value" - ] - }, - { - "cell_type": "markdown", - "id": "e471f7ab", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "7886e97f", - "metadata": { - "editable": true - }, - "source": [ - "This means that the estimator of the regression parameters is unbiased.\n", - "\n", - "We can also calculate the variance\n", - "\n", - "The variance of the optimal value $\\boldsymbol{\\hat{\\beta}}$ is" - ] - }, - { - "cell_type": "markdown", - "id": "49c32ca5", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\begin{eqnarray*}\n", - "\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n", - "\\\\\n", - "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} - \\boldsymbol{\\beta}]^{T} \\}\n", - "\\\\\n", - "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} \\, \\mathbf{y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{y} \\, \\mathbf{y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "\\\\\n", - "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n", - "% \\\\\n", - "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n", - "\\\\\n", - "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n", - "\\end{eqnarray*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "9bd27980", - "metadata": { - "editable": true - }, - "source": [ - "where we have used that $\\mathbb{E} (\\mathbf{y} \\mathbf{y}^{T}) =\n", - "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", - "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n", - "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n", - "variance of the estimate of the $j$-th regression coefficient:\n", - "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $. This may be used to\n", - "construct a confidence interval for the estimates.\n", - "\n", - "In a similar way, we can obtain analytical expressions for say the\n", - "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n", - "when we employ Ridge regression, allowing us again to define a confidence interval. \n", - "\n", - "It is rather straightforward to show that" - ] - }, - { - "cell_type": "markdown", - "id": "184f9c3b", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "d0f46be1", - "metadata": { - "editable": true - }, - "source": [ - "We see clearly that \n", - "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n", - "\n", - "We can also compute the variance as" - ] - }, - { - "cell_type": "markdown", - "id": "03e3ac5f", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "13ec2524", - "metadata": { - "editable": true - }, - "source": [ - "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n", - "\n", - "With this, we can compute the difference" - ] - }, - { - "cell_type": "markdown", - "id": "58c3f236", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "bfcfc2a0", - "metadata": { - "editable": true - }, - "source": [ - "The difference is non-negative definite since each component of the\n", - "matrix product is non-negative definite. \n", - "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. \n", - "\n", - "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended." - ] - }, - { - "cell_type": "markdown", - "id": "f3a9f084", - "metadata": { - "editable": true - }, - "source": [ - "## Material for lecture Thursday September 14" - ] - }, - { - "cell_type": "markdown", - "id": "a4338936", + "id": "1b290d64", "metadata": { "editable": true }, @@ -464,7 +114,7 @@ }, { "cell_type": "markdown", - "id": "d7566301", + "id": "ceed3a82", "metadata": { "editable": true }, @@ -476,7 +126,7 @@ }, { "cell_type": "markdown", - "id": "c8af7fae", + "id": "1d7e8f98", "metadata": { "editable": true }, @@ -489,7 +139,7 @@ }, { "cell_type": "markdown", - "id": "959e33e8", + "id": "58722d7d", "metadata": { "editable": true }, @@ -501,7 +151,7 @@ }, { "cell_type": "markdown", - "id": "c847b78b", + "id": "45120edc", "metadata": { "editable": true }, @@ -513,7 +163,7 @@ }, { "cell_type": "markdown", - "id": "75c188ba", + "id": "4f3b680a", "metadata": { "editable": true }, @@ -525,7 +175,7 @@ }, { "cell_type": "markdown", - "id": "60b89ba3", + "id": "08489797", "metadata": { "editable": true }, @@ -536,7 +186,7 @@ }, { "cell_type": "markdown", - "id": "fd5d5a54", + "id": "a1c81f06", "metadata": { "editable": true }, @@ -548,7 +198,7 @@ }, { "cell_type": "markdown", - "id": "56f25582", + "id": "4e092776", "metadata": { "editable": true }, @@ -559,7 +209,7 @@ }, { "cell_type": "markdown", - "id": "9974b303", + "id": "79e5358a", "metadata": { "editable": true }, @@ -571,7 +221,7 @@ }, { "cell_type": "markdown", - "id": "4112e053", + "id": "e5a4f865", "metadata": { "editable": true }, @@ -581,7 +231,7 @@ }, { "cell_type": "markdown", - "id": "7be4a14c", + "id": "10b81ba7", "metadata": { "editable": true }, @@ -612,7 +262,7 @@ }, { "cell_type": "markdown", - "id": "35078f48", + "id": "3349cb27", "metadata": { "editable": true }, @@ -624,7 +274,7 @@ }, { "cell_type": "markdown", - "id": "cb25ead1", + "id": "0fd1502e", "metadata": { "editable": true }, @@ -636,7 +286,7 @@ }, { "cell_type": "markdown", - "id": "5cc80036", + "id": "43cba0e8", "metadata": { "editable": true }, @@ -646,7 +296,7 @@ }, { "cell_type": "markdown", - "id": "8ebc6099", + "id": "d6f4e7dc", "metadata": { "editable": true }, @@ -658,7 +308,7 @@ }, { "cell_type": "markdown", - "id": "3160dff4", + "id": "50c2d673", "metadata": { "editable": true }, @@ -668,7 +318,7 @@ }, { "cell_type": "markdown", - "id": "8a3aaa69", + "id": "30315a10", "metadata": { "editable": true }, @@ -680,7 +330,7 @@ }, { "cell_type": "markdown", - "id": "7af6708c", + "id": "fd8cdf51", "metadata": { "editable": true }, @@ -690,7 +340,7 @@ }, { "cell_type": "markdown", - "id": "c82e095e", + "id": "7323296b", "metadata": { "editable": true }, @@ -702,7 +352,7 @@ }, { "cell_type": "markdown", - "id": "9431eca4", + "id": "e21334ab", "metadata": { "editable": true }, @@ -712,7 +362,7 @@ }, { "cell_type": "markdown", - "id": "1db6c222", + "id": "82974124", "metadata": { "editable": true }, @@ -732,7 +382,7 @@ }, { "cell_type": "markdown", - "id": "98c457ce", + "id": "e053e43c", "metadata": { "editable": true }, @@ -744,7 +394,7 @@ }, { "cell_type": "markdown", - "id": "b6a3893f", + "id": "94098e13", "metadata": { "editable": true }, @@ -754,7 +404,7 @@ }, { "cell_type": "markdown", - "id": "17f0e52e", + "id": "27625b1f", "metadata": { "editable": true }, @@ -766,7 +416,7 @@ }, { "cell_type": "markdown", - "id": "c2bfaea6", + "id": "8f3cc8b4", "metadata": { "editable": true }, @@ -778,7 +428,7 @@ }, { "cell_type": "markdown", - "id": "377eefdf", + "id": "d7d5464d", "metadata": { "editable": true }, @@ -790,7 +440,7 @@ }, { "cell_type": "markdown", - "id": "6da6253e", + "id": "c292afee", "metadata": { "editable": true }, @@ -802,7 +452,7 @@ }, { "cell_type": "markdown", - "id": "e5906b3e", + "id": "cdbbd444", "metadata": { "editable": true }, @@ -814,7 +464,7 @@ }, { "cell_type": "markdown", - "id": "792d292c", + "id": "100a8014", "metadata": { "editable": true }, @@ -826,7 +476,7 @@ }, { "cell_type": "markdown", - "id": "c3bc0674", + "id": "d2e40c4d", "metadata": { "editable": true }, @@ -838,7 +488,7 @@ }, { "cell_type": "markdown", - "id": "0faadf3f", + "id": "1e48456e", "metadata": { "editable": true }, @@ -850,7 +500,7 @@ }, { "cell_type": "markdown", - "id": "a2d478a7", + "id": "38ced5b6", "metadata": { "editable": true }, @@ -860,7 +510,7 @@ }, { "cell_type": "markdown", - "id": "d51ac6ce", + "id": "1163c07b", "metadata": { "editable": true }, @@ -872,7 +522,7 @@ }, { "cell_type": "markdown", - "id": "91d489d9", + "id": "becc087f", "metadata": { "editable": true }, @@ -882,7 +532,7 @@ }, { "cell_type": "markdown", - "id": "ebe26b44", + "id": "5ca2b09d", "metadata": { "editable": true }, @@ -901,7 +551,7 @@ }, { "cell_type": "markdown", - "id": "a67660e2", + "id": "6572c77d", "metadata": { "editable": true }, @@ -922,7 +572,7 @@ }, { "cell_type": "markdown", - "id": "40b7eac0", + "id": "2a3f65e5", "metadata": { "editable": true }, @@ -934,7 +584,7 @@ }, { "cell_type": "markdown", - "id": "16a3b6ea", + "id": "2d264f77", "metadata": { "editable": true }, @@ -945,7 +595,7 @@ }, { "cell_type": "markdown", - "id": "f41ccc6d", + "id": "06b8a13c", "metadata": { "editable": true }, @@ -958,7 +608,7 @@ }, { "cell_type": "markdown", - "id": "ae81305b", + "id": "6d958ed3", "metadata": { "editable": true }, @@ -970,7 +620,7 @@ }, { "cell_type": "markdown", - "id": "f4829b81", + "id": "6330f3f2", "metadata": { "editable": true }, @@ -980,7 +630,7 @@ }, { "cell_type": "markdown", - "id": "812273af", + "id": "ed6a9f67", "metadata": { "editable": true }, @@ -992,7 +642,7 @@ }, { "cell_type": "markdown", - "id": "31ecba0e", + "id": "de42ab8a", "metadata": { "editable": true }, @@ -1002,7 +652,7 @@ }, { "cell_type": "markdown", - "id": "ca6682ec", + "id": "4f36b126", "metadata": { "editable": true }, @@ -1014,7 +664,7 @@ }, { "cell_type": "markdown", - "id": "662eaffb", + "id": "0a7eef00", "metadata": { "editable": true }, @@ -1024,7 +674,7 @@ }, { "cell_type": "markdown", - "id": "08d5551d", + "id": "29940f94", "metadata": { "editable": true }, @@ -1038,7 +688,7 @@ }, { "cell_type": "markdown", - "id": "03f3de9c", + "id": "f2204679", "metadata": { "editable": true }, @@ -1050,7 +700,7 @@ }, { "cell_type": "markdown", - "id": "c57ace49", + "id": "d3f4597f", "metadata": { "editable": true }, @@ -1060,7 +710,7 @@ }, { "cell_type": "markdown", - "id": "cb014b2b", + "id": "c9e9ca01", "metadata": { "editable": true }, @@ -1072,7 +722,7 @@ }, { "cell_type": "markdown", - "id": "1054d47e", + "id": "1c3062b6", "metadata": { "editable": true }, @@ -1082,7 +732,7 @@ }, { "cell_type": "markdown", - "id": "1485d44c", + "id": "b455f707", "metadata": { "editable": true }, @@ -1094,7 +744,7 @@ }, { "cell_type": "markdown", - "id": "f457f909", + "id": "5ac8fdd3", "metadata": { "editable": true }, @@ -1104,7 +754,7 @@ }, { "cell_type": "markdown", - "id": "59447b2a", + "id": "827927db", "metadata": { "editable": true }, @@ -1116,17 +766,17 @@ }, { "cell_type": "markdown", - "id": "4fe26100", + "id": "0c740a13", "metadata": { "editable": true }, "source": [ - "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!" + "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta})$!" ] }, { "cell_type": "markdown", - "id": "d73c2d9c", + "id": "a855fc3c", "metadata": { "editable": true }, @@ -1142,7 +792,7 @@ }, { "cell_type": "markdown", - "id": "8cdf2c97", + "id": "9dda8b9e", "metadata": { "editable": true }, @@ -1154,7 +804,7 @@ }, { "cell_type": "markdown", - "id": "7272ef9f", + "id": "d160205c", "metadata": { "editable": true }, @@ -1164,7 +814,7 @@ }, { "cell_type": "markdown", - "id": "0dc5b2b7", + "id": "1b7501a6", "metadata": { "editable": true }, @@ -1176,7 +826,7 @@ }, { "cell_type": "markdown", - "id": "b6c8c684", + "id": "d75a8c65", "metadata": { "editable": true }, @@ -1189,7 +839,7 @@ }, { "cell_type": "markdown", - "id": "f681ed50", + "id": "c9f718df", "metadata": { "editable": true }, @@ -1201,7 +851,7 @@ }, { "cell_type": "markdown", - "id": "5c3bc898", + "id": "b07b6425", "metadata": { "editable": true }, @@ -1211,7 +861,7 @@ }, { "cell_type": "markdown", - "id": "a3cea046", + "id": "215a8730", "metadata": { "editable": true }, @@ -1223,7 +873,7 @@ }, { "cell_type": "markdown", - "id": "68276410", + "id": "7013d0f0", "metadata": { "editable": true }, @@ -1233,7 +883,7 @@ }, { "cell_type": "markdown", - "id": "dfd55611", + "id": "d56b4ce0", "metadata": { "editable": true }, @@ -1245,7 +895,7 @@ }, { "cell_type": "markdown", - "id": "83656dd8", + "id": "f9c6b14d", "metadata": { "editable": true }, @@ -1257,7 +907,7 @@ }, { "cell_type": "markdown", - "id": "b62e29fd", + "id": "0bde16cb", "metadata": { "editable": true }, @@ -1267,7 +917,7 @@ }, { "cell_type": "markdown", - "id": "1de8855c", + "id": "0a2869fc", "metadata": { "editable": true }, @@ -1279,7 +929,7 @@ }, { "cell_type": "markdown", - "id": "212152f1", + "id": "75b83b9b", "metadata": { "editable": true }, @@ -1291,7 +941,7 @@ }, { "cell_type": "markdown", - "id": "afb042c4", + "id": "1e76d916", "metadata": { "editable": true }, @@ -1303,7 +953,7 @@ }, { "cell_type": "markdown", - "id": "ab1fd3c5", + "id": "09dfe26d", "metadata": { "editable": true }, @@ -1313,7 +963,7 @@ }, { "cell_type": "markdown", - "id": "43979108", + "id": "4037962f", "metadata": { "editable": true }, @@ -1325,7 +975,7 @@ }, { "cell_type": "markdown", - "id": "96622b66", + "id": "39a85ba0", "metadata": { "editable": true }, @@ -1335,7 +985,7 @@ }, { "cell_type": "markdown", - "id": "e17a5b6d", + "id": "b263a1fe", "metadata": { "editable": true }, @@ -1354,7 +1004,7 @@ }, { "cell_type": "markdown", - "id": "24e35401", + "id": "41730165", "metadata": { "editable": true }, @@ -1382,7 +1032,7 @@ }, { "cell_type": "markdown", - "id": "a37f1d06", + "id": "4076cee3", "metadata": { "editable": true }, @@ -1408,7 +1058,7 @@ }, { "cell_type": "markdown", - "id": "58f54a23", + "id": "11139cb3", "metadata": { "editable": true }, @@ -1425,7 +1075,7 @@ }, { "cell_type": "markdown", - "id": "6ed7c809", + "id": "b2bead9b", "metadata": { "editable": true }, @@ -1445,7 +1095,7 @@ }, { "cell_type": "markdown", - "id": "bed9bb2f", + "id": "8df8364e", "metadata": { "editable": true }, @@ -1474,7 +1124,7 @@ }, { "cell_type": "markdown", - "id": "2cafa0a3", + "id": "6b5273f1", "metadata": { "editable": true }, @@ -1499,7 +1149,7 @@ }, { "cell_type": "markdown", - "id": "fefbb980", + "id": "a260b371", "metadata": { "editable": true }, @@ -1519,7 +1169,7 @@ }, { "cell_type": "markdown", - "id": "4560153d", + "id": "2ff1e63c", "metadata": { "editable": true }, @@ -1531,7 +1181,7 @@ }, { "cell_type": "markdown", - "id": "ffcbdd5a", + "id": "ab4b2e8c", "metadata": { "editable": true }, @@ -1541,7 +1191,7 @@ }, { "cell_type": "markdown", - "id": "d055ea55", + "id": "85b6fb62", "metadata": { "editable": true }, @@ -1556,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "004f0161", + "id": "42449344", "metadata": { "editable": true }, @@ -1569,7 +1219,7 @@ }, { "cell_type": "markdown", - "id": "039f4ad5", + "id": "35e4f511", "metadata": { "editable": true }, @@ -1582,7 +1232,7 @@ }, { "cell_type": "markdown", - "id": "413459f1", + "id": "c787234f", "metadata": { "editable": true }, @@ -1594,7 +1244,7 @@ }, { "cell_type": "markdown", - "id": "bef022c8", + "id": "183711dc", "metadata": { "editable": true }, @@ -1607,7 +1257,7 @@ }, { "cell_type": "markdown", - "id": "3d53693d", + "id": "2351078d", "metadata": { "editable": true }, @@ -1618,7 +1268,7 @@ }, { "cell_type": "markdown", - "id": "e682c32a", + "id": "408ba3b5", "metadata": { "editable": true }, @@ -1632,7 +1282,7 @@ }, { "cell_type": "markdown", - "id": "f44053ea", + "id": "3ea79849", "metadata": { "editable": true }, @@ -1642,7 +1292,7 @@ }, { "cell_type": "markdown", - "id": "f10de930", + "id": "69439aad", "metadata": { "editable": true }, @@ -1656,7 +1306,7 @@ }, { "cell_type": "markdown", - "id": "1f91402a", + "id": "e6c29fd8", "metadata": { "editable": true }, @@ -1669,7 +1319,7 @@ }, { "cell_type": "markdown", - "id": "2d7a8e36", + "id": "4bb52549", "metadata": { "editable": true }, @@ -1682,7 +1332,7 @@ }, { "cell_type": "markdown", - "id": "4c756ca2", + "id": "3430d548", "metadata": { "editable": true }, @@ -1692,7 +1342,7 @@ }, { "cell_type": "markdown", - "id": "908f782e", + "id": "9ba636b3", "metadata": { "editable": true }, @@ -1705,7 +1355,7 @@ }, { "cell_type": "markdown", - "id": "f12152da", + "id": "cb19e98f", "metadata": { "editable": true }, @@ -1715,7 +1365,7 @@ }, { "cell_type": "markdown", - "id": "85dd290d", + "id": "3cc840d7", "metadata": { "editable": true }, @@ -1728,7 +1378,7 @@ }, { "cell_type": "markdown", - "id": "57631d10", + "id": "a5062213", "metadata": { "editable": true }, @@ -1740,7 +1390,7 @@ }, { "cell_type": "markdown", - "id": "2d4d5ccb", + "id": "8d555244", "metadata": { "editable": true }, @@ -1759,7 +1409,7 @@ }, { "cell_type": "markdown", - "id": "a29c8551", + "id": "c4356604", "metadata": { "editable": true }, @@ -1772,7 +1422,7 @@ }, { "cell_type": "markdown", - "id": "6509ba36", + "id": "09dc08a1", "metadata": { "editable": true }, @@ -1784,7 +1434,7 @@ }, { "cell_type": "markdown", - "id": "733307b1", + "id": "dab5056a", "metadata": { "editable": true }, @@ -1797,7 +1447,7 @@ }, { "cell_type": "markdown", - "id": "a69b7331", + "id": "8a1f8583", "metadata": { "editable": true }, @@ -1817,7 +1467,7 @@ }, { "cell_type": "markdown", - "id": "ced659da", + "id": "1be08ebb", "metadata": { "editable": true }, @@ -1831,16 +1481,16 @@ "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we found \n", "$\\mathbb{E}(\\boldsymbol{\\beta}) = \\boldsymbol{\\beta}$, which means that the estimator of the regression parameters is unbiased.\n", "\n", - "We found also that the variance of the estimate of the $j$-th regression coefficient is\n", + "In the exercises this week we show that the variance of the estimate of the $j$-th regression coefficient is\n", "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $.\n", "\n", - "This quantity will be used to\n", + "This quantity can be used to\n", "construct a confidence interval for the estimates." ] }, { "cell_type": "markdown", - "id": "2c0d0a71", + "id": "2a368353", "metadata": { "editable": true }, @@ -1855,7 +1505,7 @@ }, { "cell_type": "markdown", - "id": "d3a4d693", + "id": "6fee4a1a", "metadata": { "editable": true }, @@ -1867,7 +1517,7 @@ }, { "cell_type": "markdown", - "id": "3ed3d852", + "id": "1a97c9ef", "metadata": { "editable": true }, @@ -1887,7 +1537,7 @@ }, { "cell_type": "markdown", - "id": "e826e70b", + "id": "0a70afb2", "metadata": { "editable": true }, @@ -1907,7 +1557,7 @@ }, { "cell_type": "markdown", - "id": "10b5d4d5", + "id": "dc32eaea", "metadata": { "editable": true }, @@ -1931,7 +1581,7 @@ }, { "cell_type": "markdown", - "id": "1645ce3c", + "id": "016d0ab3", "metadata": { "editable": true }, @@ -1952,7 +1602,7 @@ }, { "cell_type": "markdown", - "id": "4986483a", + "id": "3f371f0d", "metadata": { "editable": true }, @@ -1982,7 +1632,7 @@ }, { "cell_type": "markdown", - "id": "fe853f3b", + "id": "abc56fe7", "metadata": { "editable": true }, @@ -2006,7 +1656,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "81ec18f8", + "id": "a3d2885b", "metadata": { "collapsed": false, "editable": true @@ -2045,7 +1695,7 @@ }, { "cell_type": "markdown", - "id": "f0b5b588", + "id": "5db67125", "metadata": { "editable": true }, @@ -2055,7 +1705,7 @@ }, { "cell_type": "markdown", - "id": "262a95e8", + "id": "6f1b5417", "metadata": { "editable": true }, @@ -2066,7 +1716,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "d235b516", + "id": "9d98136a", "metadata": { "collapsed": false, "editable": true @@ -2086,7 +1736,7 @@ }, { "cell_type": "markdown", - "id": "c1620bb7", + "id": "c87eac4e", "metadata": { "editable": true }, @@ -2104,7 +1754,7 @@ }, { "cell_type": "markdown", - "id": "fd3bd4b5", + "id": "2f0f4313", "metadata": { "editable": true }, @@ -2116,7 +1766,7 @@ }, { "cell_type": "markdown", - "id": "4a129d86", + "id": "7895ee8b", "metadata": { "editable": true }, @@ -2133,7 +1783,7 @@ }, { "cell_type": "markdown", - "id": "ff57b405", + "id": "be454b28", "metadata": { "editable": true }, @@ -2145,7 +1795,7 @@ }, { "cell_type": "markdown", - "id": "5a13919e", + "id": "7475f8f9", "metadata": { "editable": true }, @@ -2155,7 +1805,7 @@ }, { "cell_type": "markdown", - "id": "45d4a7b6", + "id": "fd472029", "metadata": { "editable": true }, @@ -2167,7 +1817,7 @@ }, { "cell_type": "markdown", - "id": "c837442b", + "id": "b43c8f67", "metadata": { "editable": true }, @@ -2184,7 +1834,7 @@ }, { "cell_type": "markdown", - "id": "4b3c7b8e", + "id": "c47cfb6d", "metadata": { "editable": true }, @@ -2196,7 +1846,7 @@ }, { "cell_type": "markdown", - "id": "7e6f977d", + "id": "593188cd", "metadata": { "editable": true }, @@ -2206,7 +1856,7 @@ }, { "cell_type": "markdown", - "id": "adeab0fd", + "id": "76de4d81", "metadata": { "editable": true }, @@ -2218,7 +1868,7 @@ }, { "cell_type": "markdown", - "id": "63b18db5", + "id": "afc38c97", "metadata": { "editable": true }, @@ -2228,7 +1878,7 @@ }, { "cell_type": "markdown", - "id": "9c7eb00d", + "id": "b0f2971c", "metadata": { "editable": true }, @@ -2240,7 +1890,7 @@ }, { "cell_type": "markdown", - "id": "c3f4530b", + "id": "5a65e495", "metadata": { "editable": true }, @@ -2250,7 +1900,7 @@ }, { "cell_type": "markdown", - "id": "90792236", + "id": "31419298", "metadata": { "editable": true }, @@ -2266,7 +1916,7 @@ }, { "cell_type": "markdown", - "id": "d73c7cd4", + "id": "217ddd53", "metadata": { "editable": true }, @@ -2277,7 +1927,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "58124207", + "id": "ece05002", "metadata": { "collapsed": false, "editable": true @@ -2342,7 +1992,7 @@ }, { "cell_type": "markdown", - "id": "4b4e7bf5", + "id": "de6dc4e2", "metadata": { "editable": true }, @@ -2353,7 +2003,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "47b138a9", + "id": "da218205", "metadata": { "collapsed": false, "editable": true @@ -2410,7 +2060,7 @@ }, { "cell_type": "markdown", - "id": "37b66761", + "id": "5ea5df06", "metadata": { "editable": true }, @@ -2448,7 +2098,7 @@ }, { "cell_type": "markdown", - "id": "5b5c601e", + "id": "40f41d74", "metadata": { "editable": true }, @@ -2459,7 +2109,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "9bc32416", + "id": "03bb50ae", "metadata": { "collapsed": false, "editable": true @@ -2541,7 +2191,7 @@ }, { "cell_type": "markdown", - "id": "eb934dbe", + "id": "cd11e9fd", "metadata": { "editable": true }, @@ -2566,7 +2216,7 @@ }, { "cell_type": "markdown", - "id": "c65f802b", + "id": "6cf68ff2", "metadata": { "editable": true }, @@ -2594,7 +2244,7 @@ }, { "cell_type": "markdown", - "id": "9accabd4", + "id": "c445bfb2", "metadata": { "editable": true }, @@ -2607,7 +2257,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "d5ed3f5c", + "id": "8e26035a", "metadata": { "collapsed": false, "editable": true @@ -2707,7 +2357,7 @@ }, { "cell_type": "markdown", - "id": "db1976f4", + "id": "86a6b012", "metadata": { "editable": true }, @@ -2718,7 +2368,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "3c35540c", + "id": "4a0e1afd", "metadata": { "collapsed": false, "editable": true @@ -2807,7 +2457,7 @@ }, { "cell_type": "markdown", - "id": "e60fabdf", + "id": "bb1e0f15", "metadata": { "editable": true }, @@ -2817,7 +2467,7 @@ }, { "cell_type": "markdown", - "id": "9b6750d6", + "id": "83c87d33", "metadata": { "editable": true }, @@ -2830,7 +2480,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "5e6b91e1", + "id": "74eaac59", "metadata": { "collapsed": false, "editable": true @@ -2908,622 +2558,364 @@ }, { "cell_type": "markdown", - "id": "e83d7b10", + "id": "aa703354", "metadata": { "editable": true }, "source": [ - "## Notes on scaling with examples\n", - "\n", - "The programs here use both ordinrary least squares (OLS) and Ridge\n", - "regression with one value only for the hyperparameter $\\lambda$. The\n", - "first example has no scaling and includes the intercept as well and we\n", - "are trying to fit a second-order polynomial. The second code takes out\n", - "the intercept and subtracts the mean values of each column of the\n", - "design matrix and the mean value of the outputs.\n", - "\n", - "The third and final code uses **Scikit-Learn** as library in order to\n", - "calculate the optimal parameters for OLS and Ridge regression. Note\n", - "that it is highly recommended to not include the intercept in Ridge\n", - "and Lasso regression, in order to avoid penalizing the optimization by\n", - "the intercept. The second and third codes do thus not include the\n", - "intercept. In the second code we do the scaling ourselves while the\n", - "last code uses the standard scaler option included in **Scikit-Learn**, known as centering (where\n", - "we subtract the mean values)." - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "id": "73807393", - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.linear_model import LinearRegression\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.preprocessing import StandardScaler\n", - "\n", - "def MSE(y_data,y_model):\n", - " n = np.size(y_model)\n", - " return np.sum((y_data-y_model)**2)/n\n", - "\n", - "def OLS_fit_beta(X, y):\n", - " return np.linalg.pinv(X.T @ X) @ X.T @ y\n", - "\n", - "def Ridge_fit_beta(X, y,L,d):\n", - " I = np.eye(d,d)\n", - " return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y\n", - "\n", - "# Same random numbers for each test.\n", - "np.random.seed(2018)\n", - "n = 100\n", - "d = 3\n", - "# hyperparameter lambda\n", - "Lambda = 0.01\n", - "\n", - "# Make data set, simple second-order polynomial\n", - "x = np.linspace(-3, 3, n)\n", - "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n", - "\n", - "# The design matrix X includes the intercept and no scaling is made\n", - "X = np.zeros((len(x), d))\n", - "for p in range(d): \n", - " X[:, p] = x ** (p) \n", - "\n", - "\n", - "#Split data, no scaling is used and we include the intercept\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", - "\n", - "\n", - "#Calculate beta, own code\n", - "beta_OLS = OLS_fit_beta(X_train, y_train)\n", - "beta_Ridge = Ridge_fit_beta(X_train, y_train,Lambda,d)\n", - "print(beta_OLS)\n", - "print(beta_Ridge)\n", - "#predict value\n", - "ytilde_test_OLS = X_test @ beta_OLS\n", - "ytilde_test_Ridge = X_test @ beta_Ridge\n", - "\n", - "#Calculate MSE\n", - "print(\" \")\n", - "print(\"test MSE of OLS:\")\n", - "print(MSE(y_test,ytilde_test_OLS))\n", - "print(\" \")\n", - "print(\"test MSE of Ridge\")\n", - "print(MSE(y_test,ytilde_test_Ridge))\n", - "\n", - "plt.scatter(x,y,label='Data')\n", - "plt.plot(x, X @ beta_OLS,'*', label=\"OLS_Fit\")\n", - "plt.plot(x, X @ beta_Ridge, label=\"Ridge_Fit\")\n", - "plt.grid()\n", - "plt.legend()\n", - "plt.show()" + "## Material for the lab sessions" ] }, { "cell_type": "markdown", - "id": "b09d20a2", + "id": "f8bffca6", "metadata": { "editable": true }, "source": [ - "In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the [lecture material](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#more-on-rescaling-data).\n", - "see also the weekly slides [for week 36](https://compphysics.github.io/MachineLearning/doc/pub/week36/html/._week36-bs029.html).\n", - "It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.\n", + "## Linking the regression analysis with a statistical interpretation\n", "\n", - "Before we discuss the code, we repeat some of the basic math from the slides of week 36.\n", + "We will now couple the discussions of ordinary least squares, Ridge\n", + "and Lasso regression with a statistical interpretation, that is we\n", + "move from a linear algebra analysis to a statistical analysis. In\n", + "particular, we will focus on what the regularization terms can result\n", + "in. We will amongst other things show that the regularization\n", + "parameter can reduce considerably the variance of the parameters\n", + "$\\beta$.\n", "\n", - "Let us try to understand what this may imply mathematically when we\n", - "subtract the mean values, also known as *zero centering* or simply *centering*. For\n", - "simplicity, we will focus on ordinary regression, as done in the above example.\n", + "The\n", + "advantage of doing linear regression is that we actually end up with\n", + "analytical expressions for several statistical quantities. \n", + "Standard least squares and Ridge regression allow us to\n", + "derive quantities like the variance and other expectation values in a\n", + "rather straightforward way.\n", "\n", - "The cost/loss function for regression is" + "It is assumed that $\\varepsilon_i\n", + "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n", + "independent, i.e.:" ] }, { "cell_type": "markdown", - "id": "ef1cec97", + "id": "109c07d6", "metadata": { "editable": true }, "source": [ "$$\n", - "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n", + "\\begin{align*} \n", + "\\mbox{Cov}(\\varepsilon_{i_1},\n", + "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n", + "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2. \\end{array} \\right.\n", + "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", - "id": "e43c422b", + "id": "1ea48f22", "metadata": { "editable": true }, "source": [ - "Recall also that we use the squared value. This expression can lead to an\n", - "increased penalty for higher differences between predicted and\n", - "output/target values.\n", + "The randomness of $\\varepsilon_i$ implies that\n", + "$\\mathbf{y}_i$ is also a random variable. In particular,\n", + "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n", + "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n", + "non-random scalar. To specify the parameters of the distribution of\n", + "$\\mathbf{y}_i$ we need to calculate its first two moments. \n", "\n", - "What we have done is to single out the $\\beta_0$ term in the\n", - "definition of the mean squared error (MSE). The design matrix $X$\n", - "does in this case not contain any intercept column. When we take the\n", - "derivative with respect to $\\beta_0$, we want the derivative to obey" + "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n", + "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n", + "row number $i$ and perform a sum over all values $p$." ] }, { "cell_type": "markdown", - "id": "852577bc", + "id": "212f90ec", + "metadata": { + "editable": true + }, + "source": [ + "## Assumptions made\n", + "\n", + "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n", + "that there exists a function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n", + "which describe our data" + ] + }, + { + "cell_type": "markdown", + "id": "a019e7cb", "metadata": { "editable": true }, "source": [ "$$\n", - "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n", + "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n", "$$" ] }, { "cell_type": "markdown", - "id": "4edd229a", + "id": "2ca46c43", "metadata": { "editable": true }, "source": [ - "for all $j$. For $\\beta_0$ we have" + "We approximate this function with our model from the solution of the linear regression equations, that is our\n", + "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with" ] }, { "cell_type": "markdown", - "id": "bb93d608", + "id": "c61ed9c8", "metadata": { "editable": true }, "source": [ "$$\n", - "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n", + "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "5ef162e3", + "id": "96d84043", "metadata": { "editable": true }, "source": [ - "Multiplying away the constant $2/n$, we obtain" + "## Expectation value and variance\n", + "\n", + "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$" ] }, { "cell_type": "markdown", - "id": "b7c5467d", + "id": "e53b66df", "metadata": { "editable": true }, "source": [ "$$\n", - "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n", + "\\begin{align*} \n", + "\\mathbb{E}(y_i) & =\n", + "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n", + "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n", + "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", - "id": "493c75d6", + "id": "71d542a3", "metadata": { "editable": true }, "source": [ - "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n", - "Our result for $\\beta_0$ simplifies then to" + "while\n", + "its variance is" ] }, { "cell_type": "markdown", - "id": "862f2222", + "id": "01b18637", "metadata": { "editable": true }, "source": [ "$$\n", - "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n", + "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n", + "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n", + "[\\mathbb{E}(y_i)]^2 \\\\ & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n", + "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n", + "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n", + "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n", + "\\ast} \\, \\beta)^2 \\\\ & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n", + "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n", + "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n", + "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n", + "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2. \n", + "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", - "id": "e0d0a4e0", + "id": "e35bdcc1", "metadata": { "editable": true }, "source": [ - "We obtain then" + "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n", + "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD)." ] }, { "cell_type": "markdown", - "id": "73d327de", + "id": "15b7ea3c", + "metadata": { + "editable": true + }, + "source": [ + "## Expectation value and variance for $\\boldsymbol{\\beta}$\n", + "\n", + "With the OLS expressions for the optimal parameters $\\boldsymbol{\\hat{\\beta}}$ we can evaluate the expectation value" + ] + }, + { + "cell_type": "markdown", + "id": "a4c5f625", "metadata": { "editable": true }, "source": [ "$$\n", - "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n", + "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "8be27cbd", + "id": "450007e4", "metadata": { "editable": true }, "source": [ - "If we define" + "This means that the estimator of the regression parameters is unbiased.\n", + "\n", + "We can also calculate the variance\n", + "\n", + "The variance of the optimal value $\\boldsymbol{\\hat{\\beta}}$ is" ] }, { "cell_type": "markdown", - "id": "d49a9bac", + "id": "3bcf81c1", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n", + "\\begin{eqnarray*}\n", + "\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n", + "\\\\\n", + "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} - \\boldsymbol{\\beta}]^{T} \\}\n", + "\\\\\n", + "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} \\, \\mathbf{y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{y} \\, \\mathbf{y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "\\\\\n", + "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n", + "% \\\\\n", + "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n", + "\\\\\n", + "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n", + "\\end{eqnarray*}\n", "$$" ] }, { "cell_type": "markdown", - "id": "dd964823", + "id": "f69a019e", "metadata": { "editable": true }, "source": [ - "and the mean value of the outputs as" + "where we have used that $\\mathbb{E} (\\mathbf{y} \\mathbf{y}^{T}) =\n", + "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", + "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n", + "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n", + "variance of the estimate of the $j$-th regression coefficient:\n", + "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $. This may be used to\n", + "construct a confidence interval for the estimates.\n", + "\n", + "In a similar way, we can obtain analytical expressions for say the\n", + "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n", + "when we employ Ridge regression, allowing us again to define a confidence interval. \n", + "\n", + "It is rather straightforward to show that" ] }, { "cell_type": "markdown", - "id": "09a87e67", + "id": "95b37da0", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n", + "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "9b8797a0", + "id": "241c2335", "metadata": { "editable": true }, "source": [ - "we have" + "We see clearly that \n", + "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n", + "\n", + "We can also compute the variance as" ] }, { "cell_type": "markdown", - "id": "30aeafdb", + "id": "cf794384", "metadata": { "editable": true }, "source": [ "$$\n", - "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n", + "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", "$$" ] }, { "cell_type": "markdown", - "id": "dc867cbb", + "id": "4cd5260a", "metadata": { "editable": true }, "source": [ - "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have" + "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n", + "\n", + "With this, we can compute the difference" ] }, { "cell_type": "markdown", - "id": "f154321e", + "id": "7361e846", "metadata": { "editable": true }, "source": [ "$$\n", - "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n", + "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "6015f267", + "id": "6a07837b", "metadata": { "editable": true }, "source": [ - "We can rewrite the latter equation as" - ] - }, - { - "cell_type": "markdown", - "id": "f181e88f", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "a6e6c0e7", - "metadata": { - "editable": true - }, - "source": [ - "where we have defined" - ] - }, - { - "cell_type": "markdown", - "id": "708652ea", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "b360788c", - "metadata": { - "editable": true - }, - "source": [ - "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n", + "The difference is non-negative definite since each component of the\n", + "matrix product is non-negative definite. \n", + "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. \n", "\n", - "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)" - ] - }, - { - "cell_type": "markdown", - "id": "8f473602", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "70f518a7", - "metadata": { - "editable": true - }, - "source": [ - "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then" - ] - }, - { - "cell_type": "markdown", - "id": "674d02d9", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "6715261c", - "metadata": { - "editable": true - }, - "source": [ - "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n", - "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n", - "\n", - "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then" - ] - }, - { - "cell_type": "markdown", - "id": "17600ee3", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "a8c6f068", - "metadata": { - "editable": true - }, - "source": [ - "Now we try to implement this." - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "id": "1d89f80d", - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "np.random.seed(2018)\n", - "n = 100\n", - "# we do not include the intercept\n", - "d = 2\n", - "Lambda = 0.01\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n)\n", - "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n", - "\n", - "#Design matrix X does not include the intercept. \n", - "X = np.zeros((len(x), d))\n", - "for p in range(d): \n", - " X[:, p] = x ** (p+1)\n", - "\n", - "\n", - "#Split data in train and test\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", - "\n", - "# Scale data by subtracting mean value,own implementation\n", - "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n", - "X_train_mean = np.mean(X_train,axis=0)\n", - "#Center by removing mean from each feature\n", - "X_train_scaled = X_train - X_train_mean\n", - "X_test_scaled = X_test - X_train_mean\n", - "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered, note)\n", - "y_scaler = np.mean(y_train)\n", - "y_train_scaled = y_train - y_scaler\n", - "\n", - "\n", - "#Calculate beta\n", - "beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)\n", - "beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,Lambda,d)\n", - "print(beta_OLS)\n", - "print(beta_Ridge)\n", - "# calculate intercepts and print them\n", - "interceptOLS = y_scaler - X_train_mean @ beta_OLS\n", - "interceptRidge = y_scaler - X_train_mean @ beta_Ridge\n", - "print(interceptOLS)\n", - "print(interceptRidge)\n", - "\n", - "#predict value with intercept\n", - "ytilde_test_OLS = X_test_scaled @ beta_OLS+y_scaler\n", - "ytilde_test_Ridge = X_test_scaled @ beta_Ridge+y_scaler\n", - "\n", - "\n", - "#Calculate MSE\n", - "\n", - "print(\" \")\n", - "print(\"test MSE of OLS:\")\n", - "print(MSE(y_test,ytilde_test_OLS))\n", - "print(\" \")\n", - "print(\"test MSE of Ridge\")\n", - "print(MSE(y_test,ytilde_test_Ridge))\n", - "\n", - "plt.scatter(x,y,label='Data')\n", - "plt.plot(x, X @ beta_OLS+interceptOLS,'*', label=\"OLS_Fit\")\n", - "plt.plot(x, X @ beta_Ridge+interceptRidge, label=\"Ridge_Fit\")\n", - "plt.grid()\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "id": "522668c3", - "metadata": { - "editable": true - }, - "source": [ - "Finally, instead of using our own function we repeat the same example\n", - "using the **standardscaler** functionality of the library\n", - "**Scikit-Learn**. Here we limit ourselves to Ridge regression only." - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "id": "9954bcc1", - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn import linear_model\n", - "np.random.seed(2018)\n", - "n = 10\n", - "d = 2\n", - "Lambda = 0.01\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n)\n", - "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n", - "\n", - "# Design matrix X does not include the intercept. \n", - "X = np.zeros((n, d))\n", - "for p in range(d): \n", - " X[:, p] = x ** (p+1)\n", - "\n", - "#Split data in train and test\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", - "# Scale data by subtracting mean value of the input using scikit-learn\n", - "scaler = StandardScaler(with_std=False)\n", - "scaler.fit(X_train)\n", - "X_train_mean = np.mean(X_train,axis=0)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# We scale also the output, here by our own code\n", - "y_scaler = np.mean(y_train)\n", - "y_train_scaled = y_train - y_scaler\n", - "y_test_scaled = y_test- y_scaler\n", - "\n", - "#Calculate beta\n", - "OLS = LinearRegression()\n", - "betaOLS=OLS.fit(X_train_scaled,y_train_scaled)\n", - "ypredictOLS = OLS.predict(X_test_scaled)\n", - "linear_model.Ridge(Lambda)\n", - "RegRidge.fit(X_train_scaled,y_train_scaled)\n", - "ypredictRidge = RegRidge.predict(X_test_scaled)\n", - "betaOLS = OLS.coef_\n", - "betaRidge = RegRidge.coef_\n", - "print(betaOLS)\n", - "print(betaRidge)\n", - "interceptOLS = np.mean(y_train) - X_train_mean @ betaOLS\n", - "interceptRidge = y_scaler - X_train_mean @ betaRidge\n", - "print(interceptOLS)\n", - "print(interceptRidge)\n", - "#predict value \n", - "ytilde_test_Ridge = X_test_scaled @ betaRidge+y_scaler\n", - "ytilde_test_OLS = X_test_scaled @ betaOLS+y_scaler\n", - "\n", - "#Calculate MSE\n", - "print(\" \")\n", - "print(\"test MSE of OLS\")\n", - "print(MSE(y_test,ytilde_test_OLS))\n", - "print(\" \")\n", - "print(\"test MSE of Ridge\")\n", - "print(MSE(y_test,ytilde_test_Ridge))\n", - "plt.scatter(x,y,label='Data')\n", - "plt.plot(x, X @ RegRidge.coef_ + RegRidge.intercept_ , label=\"Ridge_Fit\")\n", - "plt.grid()\n", - "plt.legend()\n", - "plt.show()" + "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended." ] } ], diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js index 5f95f4674..63fe57975 100644 --- a/doc/LectureNotes/_build/html/searchindex.js +++ b/doc/LectureNotes/_build/html/searchindex.js @@ -1 +1 @@ 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 7 (midnight), 2024","Teaching schedule with links to material","1. 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 7 (midnight), 2024","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Linear Regression and Statistical interpretations","Week 37: Statistical interpretations and Resampling Methods"],titleterms:{"1":[0,15,16,17,18,21,26,27],"14":[],"17":[],"18":[],"2":[0,15,16,17,18,26,27,28],"2023":24,"2024":21,"26":27,"3":[0,15,16,26,27],"34":[15,26],"35":[16,27],"36":[17,28],"37":[18,29],"4":[0,27],"5":0,"7":21,"9":29,"case":[8,10,23,27,28],"do":[1,28,29],"final":[12,27,28],"function":[0,1,6,7,8,10,11,12,13,21,23,26,27,28,29],"import":[5,20,26,27,28],"new":[4,28,29],A:[0,1,4,8,9,26,28,29],And:[26,27,28],For:27,In:24,Ising:6,The:[0,1,2,3,5,6,7,8,9,11,12,17,19,26,27,28,29],To:[26,27],With:[4,28],about:[26,27],abov:28,activ:[1,12,28],ad:[0,6,17,21,26,27],adaboost:10,adagrad:13,adam:13,adapt:10,adjust:1,adversari:4,again:[3,9],ai:[21,26],aim:[8,9,17,18,26],aka:[26,27],algebra:[20,26],algorithm:[9,10,11,12,27],algortithm:13,all:8,altern:[],an:[0,4,10,26],analys:[5,27],analysi:[0,5,6,11,19,21,23,26,27,28,29],analyt:[0,16,17],analyz:[],ani:13,anoth:[9,28,29],appli:19,approach:[0,8,14,26,29],approxim:12,architectur:1,arrai:[20,26],assist:24,assumpt:[28,29],august:27,autocorrel:23,autograd:[2,13],automat:13,b:[17,21],back:[1,11,12],background:[19,21,29],bag:10,base:[13,29],basic:[0,5,7,9,10,11,20,27,28,29],batch:1,bay:[5,28,29],befor:11,beta:[28,29],better:8,bia:[6,21,29],binari:1,bind:26,bird:10,boldsymbol:[27,28,29],boost:10,bootstrap:[6,10,29],boston:0,breast:1,brief:[26,29],bring:12,build:[1,3,9],c:[21,26],calcul:27,can:[26,29],cancer:[1,7,9,11],cart:9,center:27,central:[13,19,23,29],chain:12,chang:10,channel:26,chi:[0,26],choos:1,cifar01:3,classic:11,classif:[1,9,10],classifi:8,clip:1,cluster:14,cnn:3,code:[0,1,2,5,9,11,12,13,14,26,27,28,29],collect:[1,3],commun:26,compar:[2,10],comparison:28,complet:27,complex:[0,6,21,27],complic:6,compon:11,comput:9,computation:29,computerlab:26,con:9,concept:23,condit:[28,29],confid:29,conjug:13,contn:26,convex:[8,13],convolut:[3,12],correctli:[28,29],correl:[11,27],cost:[1,10,27,28,29],cours:[19,25,26],covari:[5,11,23,27],cover:26,critic:[],cross:[6,21,29],cython:26,d:21,data:[0,1,3,6,7,9,11,15,16,19,21,23,26,27,28],dataset:[1,3],deadlin:[21,26],decai:2,decemb:[],decis:[9,10],decomposit:[5,11,17,20,27],deep:[1,2,26],defin:[1,26],degre:[0,27],deliveri:21,delta:29,dens:[0,26],deriv:[5,12,27,28,29],descent:[2,10,13],descript:21,design:27,detail:[3,26],develop:1,diagon:11,differ:8,differenti:[2,13],diffus:2,dimension:[2,3,8,21,27],disadvantag:9,discret:23,distribut:[5,23,28,29],distrubut:29,doe:[27,28],domain:23,down:1,dropout:1,e:21,economi:27,electron:21,element:[0,23,26],elimin:20,energi:26,ensembl:10,entropi:9,environ:[0,15,26],equat:[0,2,12,26,27,28],error:[0,10,26,27,29],essenti:26,estim:[28,29],etc:26,euler:2,evalu:1,exampl:[0,1,2,3,4,6,7,8,9,10,26,27,28,29],exercis:[0,6,15,16,17,18,26,27],expect:[18,23,28,29],expens:29,experi:23,explor:[0,15,16,26],exponenti:2,express:[17,18,27],extrapol:4,extrem:[10,26],ey:10,f:21,fall:24,famili:[1,26,27],famou:20,fantast:27,featur:[9,20,27],feed:[1,12],find:29,fine:1,first:[4,12,26,27,28],fit:[0,10,26,28],fix:27,fold:29,forc:3,forest:10,format:[21,26],forward:[1,2,12],fourier:3,frank:[6,21,27],freedom:[0,27],frequent:27,frequentist:[0,26],from:[5,10,12,27,28,29],full:2,further:[3,5,27],g:21,gan:4,gaussian:20,gd:13,gener:[4,9,26],geometr:11,gini:9,good:[0,26],grade:[24,26],gradient:[1,2,10,13],growth:2,ha:19,handl:[20,26,27],happen:[28,29],hessian:27,hidden:2,histogram:29,hous:0,hyperparamet:1,hyperplan:8,i:1,id3:9,idea:11,ident:[28,29],identifi:29,ii:26,iid:[28,29],illustr:28,implement:1,implic:[5,27],improv:1,includ:13,increment:11,independ:[28,29],index:9,inform:24,input:2,instal:[19,21,26],instructor:24,intercept:27,interpret:[5,11,26,27,28,29],interv:29,introduc:[11,13,27],introduct:[0,6,19,20,21,26],invers:[5,20,28],invert:27,iter:10,its:27,jacobian:27,jax:13,julia:26,jungl:10,k:29,kera:[1,3],kernel:[8,11],lab:29,lagrangian:8,lasso:[5,6,21,27,28,29],last:27,later:[5,27],layer:[1,2,3,12],learn:[0,1,2,11,13,14,15,16,19,21,26,27,28,29],least:[5,6,18,21,26,27,28],lectur:[26,27,28,29],level:10,librari:[19,26],likelihood:[7,28,29],limit:[1,13,23,29],linear:[0,8,13,20,26,27,28],link:[5,11,22,25,27,28,29],literatur:21,logist:[7,26],loss:27,lu:20,machin:[0,8,13,19,21,26],made:[28,29],main:[23,26],make:[0,9,10,15,16,26,27],mani:[10,12],manipul:27,margin:[28,29],mass:26,materi:[21,22,26,27,28,29],math:[5,27],mathemat:[3,5,8,27],matric:[5,20,26,28],matrix:[1,5,11,12,20,26,27,28],matter:[0,26],max:27,maximum:[28,29],mean:[0,27,28],meet:[5,10,23,26,27],mercer:8,method:[6,9,10,13,21,26,29],midnight:21,min:27,minim:26,ml:26,mle:[28,29],mlp:12,mnist:[3,4],model:[0,1,4,6,12,26],momentum:13,mondai:[27,28,29],moon:[8,9],more:[3,6,20,21,26,27,28,29],multilay:12,multipl:[1,3],multipli:8,need:[21,26],network:[1,2,3,4,7,12,26],neural:[1,2,3,4,7,12,26],non:8,normal:[0,1,29],notat:12,note:[21,27,28],novemb:[],now:[1,9,13,28,29],nuclear:[0,26],numba:26,number:[0,2,23,27],numer:[2,21,23],numpi:[20,26],object:3,obtain:11,octob:21,od:2,off:[6,21],ol:[5,6,21,28,29],one:[2,12],oper:20,optim:[1,8,13,19,26,27],order:13,ordinari:[5,6,18,21,26,27,2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\ No newline at end of file diff --git a/doc/LectureNotes/_build/html/week37.html b/doc/LectureNotes/_build/html/week37.html index 32a4cc488..cd478a242 100644 --- a/doc/LectureNotes/_build/html/week37.html +++ b/doc/LectureNotes/_build/html/week37.html @@ -372,41 +372,18 @@ const thebe_selector_output = ".output, .cell_output"