updating typos

doc/LectureNotes/_build/html/_sources/exercisesweek37.ipynb

@@ -2,7 +2,7 @@
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@@ -14,7 +14,7 @@
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@@ -27,25 +27,34 @@
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    "source": [
     "## Overarching aims of the exercises this week\n",
     "\n",
-    "This exercise deals with various mean values and variances in  linear regression method (here it may be useful to look up chapter 3, equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer](https://www.springer.com/gp/book/9780387848570)). The exercise is also a part of project 1 and can be reused in the theory part of the project.\n",
+    "This exercise deals with various mean values and variances in linear\n",
+    "regression method (here it may be useful to look up chapter 3,\n",
+    "equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome\n",
+    "H. Friedman, The Elements of Statistical Learning,\n",
+    "Springer](https://www.springer.com/gp/book/9780387848570)). The\n",
+    "exercise is also a part of project 1 and can be reused in the theory\n",
+    "part of the project.\n",
     "\n",
-    "For more discussions on Ridge regression and calculation of expectation values, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n",
+    "For more discussions on Ridge regression and calculation of\n",
+    "expectation values, [Wessel van\n",
+    "Wieringen's](https://arxiv.org/abs/1509.09169) article is highly\n",
+    "recommended.\n",
     "\n",
-    "The assumption we have made is \n",
-    "that there exists a continuous function $f(\\boldsymbol{x})$ and  a normal distributed error $\\boldsymbol{\\varepsilon}\\sim N(0, \\sigma^2)$\n",
-    "which describes our data"
+    "The assumption we have made is that there exists a continuous function\n",
+    "$f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim N(0,\n",
+    "\\sigma^2)$ which describes our data"
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@@ -57,7 +66,7 @@
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@@ -68,7 +77,7 @@
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@@ -80,7 +89,7 @@
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@@ -90,7 +99,7 @@
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@@ -102,7 +111,7 @@
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@@ -114,7 +123,7 @@
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@@ -125,7 +134,7 @@
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@@ -137,7 +146,7 @@
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@@ -150,7 +159,7 @@
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@@ -162,7 +171,7 @@
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@@ -172,7 +181,7 @@
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@@ -184,7 +193,7 @@
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@@ -195,7 +204,7 @@
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@@ -207,20 +216,20 @@
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    "source": [
     "$$\n",
-    "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\\n",
+    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\\n",
     "\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
     "$$"
    ]
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@@ -233,19 +242,19 @@
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    "source": [
     "$$\n",
-    "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T}\\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
+    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T}\\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
     "$$"
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