From 5ee2934f9aa967dca3ffda5046865b45e8fbff58 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Tue, 26 Oct 2021 22:58:15 +0200 Subject: [PATCH] update book chapters --- doc/BookChapters/chapter3.do.txt | 40 +++- .../_build/.doctrees/chapter3.doctree | Bin 1156562 -> 1158428 bytes .../_build/.doctrees/environment.pickle | Bin 186418 -> 187159 bytes .../_build/html/_images/chapter3_107_1.png | Bin 0 -> 15299 bytes .../_build/html/_images/chapter3_115_1.png | Bin 0 -> 11379 bytes .../_build/html/_images/chapter3_117_1.png | Bin 11379 -> 13640 bytes .../_build/html/_images/chapter3_148_1.png | Bin 0 -> 20156 bytes .../_build/html/_images/chapter3_166_1.png | Bin 0 -> 31681 bytes .../_build/html/_images/chapter3_169_1.png | Bin 20063 -> 31007 bytes .../_build/html/_images/chapter3_173_1.png | Bin 0 -> 20063 bytes .../_build/html/_images/chapter3_175_13.png | Bin 0 -> 197708 bytes .../_build/html/_images/chapter3_177_0.png | Bin 29129 -> 72871 bytes .../_build/html/_images/chapter3_181_0.png | Bin 0 -> 29129 bytes .../_build/html/_images/chapter3_47_0.png | Bin 11282 -> 11376 bytes .../_build/html/_images/chapter3_65_9.png | Bin 0 -> 16384 bytes .../_build/html/_sources/chapter3.ipynb | 64 ++++-- doc/LectureNotes/_build/html/chapter3.html | 154 ++++++++------ doc/LectureNotes/_build/html/searchindex.js | 2 +- .../_build/jupyter_execute/chapter3.ipynb | 201 +++++++++++------- .../_build/jupyter_execute/chapter3.py | 44 ++-- .../_build/jupyter_execute/chapter3_107_1.png | Bin 0 -> 15299 bytes .../_build/jupyter_execute/chapter3_115_1.png | Bin 0 -> 11379 bytes .../_build/jupyter_execute/chapter3_117_1.png | Bin 11379 -> 13640 bytes .../_build/jupyter_execute/chapter3_148_1.png | Bin 0 -> 20156 bytes .../_build/jupyter_execute/chapter3_166_1.png | Bin 0 -> 31681 bytes .../_build/jupyter_execute/chapter3_169_1.png | Bin 20063 -> 31007 bytes .../_build/jupyter_execute/chapter3_173_1.png | Bin 0 -> 20063 bytes .../jupyter_execute/chapter3_175_13.png | Bin 0 -> 197708 bytes .../_build/jupyter_execute/chapter3_177_0.png | Bin 29129 -> 72871 bytes .../_build/jupyter_execute/chapter3_181_0.png | Bin 0 -> 29129 bytes .../_build/jupyter_execute/chapter3_47_0.png | Bin 11282 -> 11376 bytes .../_build/jupyter_execute/chapter3_65_9.png | Bin 0 -> 16384 bytes doc/LectureNotes/chapter3.ipynb | 64 ++++-- 33 files changed, 367 insertions(+), 202 deletions(-) create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_107_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_115_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_148_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_166_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_173_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_175_13.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_181_0.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_65_9.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_107_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_115_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_148_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_166_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_173_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_175_13.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_181_0.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_65_9.png diff --git a/doc/BookChapters/chapter3.do.txt b/doc/BookChapters/chapter3.do.txt index f831f9344..68153e5e4 100644 --- a/doc/BookChapters/chapter3.do.txt +++ b/doc/BookChapters/chapter3.do.txt @@ -1202,7 +1202,7 @@ standard deviation. Most machine learning libraries do this as a default. This m the results may differ. The -"Standadscaler":"https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html" +"Standardscaler":"https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html" function in _Scikit-Learn_ does this for us. For the data sets we have been studying in our various examples, the data are in many cases already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a @@ -1256,12 +1256,16 @@ The cost/loss function for regression is 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,. \] !et -Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values. -What we have done is to single out the $\beta_0$ term in the definition of the mean squared error (MSE). -The design matrix -$X$ does in this case not contain any intercept column. -When we take the derivative with respect to $\beta_0$, we want the derivative to obey +Recall also that we use the squared value. This expression can lead to an +increased penalty for higher differences between predicted and +output/target values. + +What we have done is to single out the $\beta_0$ term in the +definition of the mean squared error (MSE). The design matrix $X$ +does in this case not contain any intercept column. When we take the +derivative with respect to $\beta_0$, we want the derivative to obey + !bt \[ \frac{\partial C}{\partial \beta_j} = 0, @@ -1282,7 +1286,7 @@ Multiplying away the constant $2/n$, we obtain \] !et -Let us special first to the case where we have only two parameters $\beta_0$ and $\beta_1$. +Let us specialize first to the case where we have only two parameters $\beta_0$ and $\beta_1$. Our result for $\beta_0$ simplifies then to !bt \[ @@ -1298,10 +1302,10 @@ We obtain then If we define !bt \[ -\mu_1=\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1}, +\mu_{\bm{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}, \] !et -and if we define the mean value of the outputs as +and the mean value of the outputs as !bt \[ \mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i, @@ -1310,16 +1314,30 @@ and if we define the mean value of the outputs as we have !bt \[ -\beta_0 = \mu_y - \beta_1\mu_{1}. +\beta_0 = \mu_y - \beta_1\mu_{\bm{x}_1}. \] !et -In the general case withmore parameters than $\beta_0$ and $\beta_1$, we have +In the general case with more parameters than $\beta_0$ and $\beta_1$, we have !bt \[ \beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j. \] !et +We can rewrite the latter equation as +!bt +\[ +\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\bm{x}_j}\beta_j, +\] +!et +where we have defined +!bt +\[ 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