From 4291b46bcbc11e4fa29c31a278725a853770a0de Mon Sep 17 00:00:00 2001 From: mhjensen Date: Fri, 7 Sep 2018 05:40:44 +0200 Subject: [PATCH] update on html slides for regression --- .../Regression/html/._Regression-bs046.html | 333 +++++++++++++++++ .../Regression/html/._Regression-bs047.html | 339 ++++++++++++++++++ .../Regression/html/._Regression-bs048.html | 336 +++++++++++++++++ .../Regression/html/._Regression-bs049.html | 316 ++++++++++++++++ .../Regression/html/._Regression-bs050.html | 324 +++++++++++++++++ .../Regression/html/._Regression-bs051.html | 338 +++++++++++++++++ .../Regression/html/._Regression-bs052.html | 326 +++++++++++++++++ .../Regression/html/._Regression-bs053.html | 320 +++++++++++++++++ .../Regression/html/Regression-reveal.html | 8 +- .../Regression/html/Regression-solarized.html | 8 +- doc/pub/Regression/html/Regression.html | 8 +- doc/pub/Regression/ipynb/Regression.ipynb | 10 +- .../ipynb/ipynb-Regression-src.tar.gz | Bin 210 -> 211 bytes .../pdf/Regression-beamer-handouts2x3.pdf | Bin 411149 -> 411066 bytes doc/pub/Regression/pdf/Regression-beamer.pdf | Bin 366764 -> 366730 bytes doc/pub/Regression/pdf/Regression-minted.pdf | Bin 405955 -> 405967 bytes doc/src/Regression/Regression.do.txt | 10 +- 17 files changed, 2648 insertions(+), 28 deletions(-) create mode 100644 doc/pub/Regression/html/._Regression-bs046.html create mode 100644 doc/pub/Regression/html/._Regression-bs047.html create mode 100644 doc/pub/Regression/html/._Regression-bs048.html create mode 100644 doc/pub/Regression/html/._Regression-bs049.html create mode 100644 doc/pub/Regression/html/._Regression-bs050.html create mode 100644 doc/pub/Regression/html/._Regression-bs051.html create mode 100644 doc/pub/Regression/html/._Regression-bs052.html create mode 100644 doc/pub/Regression/html/._Regression-bs053.html diff --git a/doc/pub/Regression/html/._Regression-bs046.html b/doc/pub/Regression/html/._Regression-bs046.html new file mode 100644 index 000000000..e12f107dc --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs046.html @@ -0,0 +1,333 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
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+ +

 

 

 

+ + + + +

Resampling approaches can be computationally expensive

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+Resampling approaches can be computationally expensive, because they +involve fitting the same statistical method multiple times using +different subsets of the training data. However, due to recent +advances in computing power, the computational requirements of +resampling methods generally are not prohibitive. In this chapter, we +discuss two of the most commonly used resampling methods, +cross-validation and the bootstrap. Both methods are important tools +in the practical application of many statistical learning +procedures. For example, cross-validation can be used to estimate the +test error associated with a given statistical learning method in +order to evaluate its performance, or to select the appropriate level +of flexibility. The process of evaluating a model’s performance is +known as model assessment, whereas the process of selecting the proper +level of flexibility for a model is known as model selection. The +bootstrap is widely used. +

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+ +

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+ + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs047.html b/doc/pub/Regression/html/._Regression-bs047.html new file mode 100644 index 000000000..81367fb0f --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs047.html @@ -0,0 +1,339 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Log-likelihood

+ +

+A popular strategy is to choose a penalty parameter that yields a good +but parsimonious model. Information criteria measure the balance +between model fit and model complexity. One possibility is Aikaike's +information criterion (AIC). +The AIC measures model fit by the log-likelihood +and model complexity is measured by the number of parameters used by +the model. The number of model parameters in regular regression simply +corresponds to the number of covariates in the model. Or, by the +degrees of freedom consumed by the model, which is equivalent to the +trace of the hat matrix. For ridge regression it thus seems natural to +define model complexity analogously by the trace of the ridge hat +matrix. This yields the AIC for the linear regression model with ridge +estimates: + +$$ +\begin{align*} +\mbox{AIC}(\lambda) & = 2 \, p - 2 \log(\hat{L}) +\\ +& = 2 \, \mbox{tr} [\mathbf{H}(\lambda)] - 2 \log\{L[\hat{\beta}(\lambda), \hat{\sigma}^2(\lambda)]\} +\\ +& = 2 \, \sum_{j=1}^p \frac{d_{jj}^2}{d_{jj}^2 + \lambda} ++ 2 n \, \log[\sqrt{2 \, \pi} \, \hat{\sigma}(\lambda)] + \frac{1}{\hat{\sigma}^2(\lambda)} \sum_{i=1}^n [y_i - \mathbf{X}_{i, \ast} \, \hat{\beta}(\lambda)]^2. +\end{align*} +$$ + +The value of \( \lambda \) which minimizes \( \mbox{AIC}(\lambda) \) corresponds to the `optimal' balance of model complexity and overfitting. + +

+

+ +

+ + +
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+ +
+ + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs048.html b/doc/pub/Regression/html/._Regression-bs048.html new file mode 100644 index 000000000..cacd2337a --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs048.html @@ -0,0 +1,336 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Cross-validation

+ +

+Instead of choosing the penalty parameter to balance model fit with +model complexity, cross-validation requires it (i.e. the penalty +parameter) to yield a model with good prediction +performance. Commonly, this performance is evaluated on novel +data. Novel data need not be easy to come by and one has to make do +with the data at hand. The setting of `original' and novel data is +then mimicked by sample splitting: the data set is divided into two +(groups of samples). One of these two data sets, called the training +set, plays the role of `original' data on which the model is +built. The second of these data sets, called the test set, plays the +role of the `novel' data and is used to evaluate the prediction +performance (often operationalized as the log-likelihood or the +prediction error or its square or the R2 score) of the model built on the training data set. This +procedure (model building and prediction evaluation on training and +test set, respectively) is done for a collection of possible penalty +parameter choices. The penalty parameter that yields the model with +the best prediction performance is to be preferred. The thus obtained +performance evaluation depends on the actual split of the data set. To +remove this dependence the data set is split many times into a +training and test set. For each split the model parameters are +estimated for all choices of \( \lambda \) using the training data and +estimated parameters are evaluated on the corresponding test set. The +penalty parameter that on average over the test sets performs best (in +some sense) is then selected. + +

+

+ +

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+ + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs049.html b/doc/pub/Regression/html/._Regression-bs049.html new file mode 100644 index 000000000..1cde9b0fe --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs049.html @@ -0,0 +1,316 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
+ +
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+ + +
+ +

 

 

 

+ + + + +

Computationally expensive

+ +

+The validation set approach is conceptually simple and is easy to implement. But it has two potential drawbacks: + +

+ +

+ +

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+ + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs050.html b/doc/pub/Regression/html/._Regression-bs050.html new file mode 100644 index 000000000..150bbb796 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs050.html @@ -0,0 +1,324 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
+ +
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+ + +
+ +

 

 

 

+ + + + +

Various steps in cross-validation

+ +

+When the repetitive splitting of the data set is done randomly, +samples may accidently end up in a fast majority of the splits in +either training or test set. Such samples may have an unbalanced +influence on either model building or prediction evaluation. To avoid +this \( k \)-fold cross-validation structures the data splitting. The +samples are divided into \( k \) more or less equally sized exhaustive and +mutually exclusive subsets. In turn (at each split) one of these +subsets plays the role of the test set while the union of the +remaining subsets constitutes the training set. Such a splitting +warrants a balanced representation of each sample in both training and +test set over the splits. Still the division into the \( k \) subsets +involves a degree of randomness. This may be fully excluded when +choosing \( k=n \). This particular case is referred to as leave-one-out +cross-validation (LOOCV). + +

+

+ +

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+ + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs051.html b/doc/pub/Regression/html/._Regression-bs051.html new file mode 100644 index 000000000..7542dbe4a --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs051.html @@ -0,0 +1,338 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

How to set up the cross-validation for Ridge and/or Lasso

+ +
    +
  1. Define a range of interest for the penalty parameter.
    2. +
  2. Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
    3. +
  3. Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \hat{\sigma}_{-i}^2(\lambda) \), as
    4. +
+ +$$ +\begin{align*} +\hat{\beta}_{-i}(\lambda) & = ( \hat{X}_{-i, \ast}^{\top} +\hat{X}_{-i, \ast} + \lambda \hat{I}_{pp})^{-1} +\hat{X}_{-i, \ast}^{\top} \hat{y}_{-i} +\end{align*} +$$ + + +
    +
  1. Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \hat{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \hat{X}_{i, \ast} \hat{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
    2. +
  2. Repeat steps 1) to 3) such that each sample plays the role of the test set once.
    3. +
  3. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter by computing the cross-validated log-likelihood. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
    4. +
+ +$$ +\begin{align*} +\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\}. +\end{align*} +$$ + + +
    +
  1. The value of the penalty parameter that maximizes the cross-validated log-likelihood is the value of choice. Or we can use the MSE or the R2 score functions.
    2. +
+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs052.html b/doc/pub/Regression/html/._Regression-bs052.html new file mode 100644 index 000000000..daaad1808 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs052.html @@ -0,0 +1,326 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Predicted Residual Error Sum of Squares

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+

+Another approach in the LOOCV scheme is to the use the so-called Predicted Residual Error Sum of Squares (PRESS). + +

+We can define the optimal penalty parameter to minimize +$$ +\begin{align*} +\lambda_{\mbox{{\tiny opt}}} = \arg \min_{\lambda} \frac{1}{n} \sum_{i=1}^n [y_i - \hat{X}_{i, \ast} \hat{\beta}_{-i}(\lambda)]^2. +\end{align*} +$$ + +

+The LOOCV prediction performance can be +expressed analytically in terms of the known quantities derived from +the design matrix and the parameters \( \beta \). +

+
+ + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs053.html b/doc/pub/Regression/html/._Regression-bs053.html new file mode 100644 index 000000000..ad25d53b8 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs053.html @@ -0,0 +1,320 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Bootstrap

+
+
+

+Bootstrapping is a nonparametric approach to statistical inference +that substitutes computation for more traditional distributional +assumptions and asymptotic results. Bootstrapping offers a number of +advantages: + +

    +
  1. The bootstrap is quite general, although there are some cases in which it fails.
    2. +
  2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.
    3. +
  3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.
    4. +
  4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).
    5. +
+
+
+ + +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/Regression/html/Regression-reveal.html b/doc/pub/Regression/html/Regression-reveal.html index ee2d9bcc5..63a356a89 100644 --- a/doc/pub/Regression/html/Regression-reveal.html +++ b/doc/pub/Regression/html/Regression-reveal.html @@ -1907,7 +1907,7 @@ cross-validation (LOOCV).

  1. Define a range of interest for the penalty parameter.

  2. Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
    3. -

  3. Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set as
    4. +

  4. Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \hat{\sigma}_{-i}^2(\lambda) \), as

 
$$ @@ -1919,22 +1919,20 @@ $$ $$

 
-and the corresponding estimate of the error variance \( \hat{\sigma}_{-i}^2(\lambda) \).

  1. Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \hat{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \hat{X}_{i, \ast} \hat{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.

  2. Repeat steps 1) to 3) such that each sample plays the role of the test set once.
    3. -

  3. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter
    4. +

  4. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter by computing the cross-validated log-likelihood. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as

 
$$ \begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[Y_i, \mathbf{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\}. +\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\}. \end{align*} $$

 
-The quantity above is called the cross-validated log-likelihood. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data.

  1. The value of the penalty parameter that maximizes the cross-validated log-likelihood is the value of choice. Or we can use the MSE or the R2 score functions.
    2. diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html index afaf59f99..88feb089c 100644 --- a/doc/pub/Regression/html/Regression-solarized.html +++ b/doc/pub/Regression/html/Regression-solarized.html @@ -1866,7 +1866,7 @@ cross-validation (LOOCV).
    1. Define a range of interest for the penalty parameter.
    2. Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
      3. -
    3. Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set as
      4. +
    4. Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \hat{\sigma}_{-i}^2(\lambda) \), as
    $$ @@ -1877,21 +1877,19 @@ $$ \end{align*} $$ -and the corresponding estimate of the error variance \( \hat{\sigma}_{-i}^2(\lambda) \).
    1. Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \hat{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \hat{X}_{i, \ast} \hat{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
    2. Repeat steps 1) to 3) such that each sample plays the role of the test set once.
      3. -
    3. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter
      4. +
    4. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter by computing the cross-validated log-likelihood. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
    $$ \begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[Y_i, \mathbf{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\}. +\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\}. \end{align*} $$ -The quantity above is called the cross-validated log-likelihood. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data.
    1. The value of the penalty parameter that maximizes the cross-validated log-likelihood is the value of choice. Or we can use the MSE or the R2 score functions.
      2. diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html index 366f94425..a53c0e587 100644 --- a/doc/pub/Regression/html/Regression.html +++ b/doc/pub/Regression/html/Regression.html @@ -1871,7 +1871,7 @@ cross-validation (LOOCV).
      1. Define a range of interest for the penalty parameter.
      2. Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
        3. -
      3. Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set as
        4. +
      4. Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \hat{\sigma}_{-i}^2(\lambda) \), as
      $$ @@ -1882,21 +1882,19 @@ $$ \end{align*} $$ -and the corresponding estimate of the error variance \( \hat{\sigma}_{-i}^2(\lambda) \).
      1. Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \hat{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \hat{X}_{i, \ast} \hat{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
      2. Repeat steps 1) to 3) such that each sample plays the role of the test set once.
        3. -
      3. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter
        4. +
      4. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter by computing the cross-validated log-likelihood. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
      $$ \begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[Y_i, \mathbf{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\}. +\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \hat{\beta}_{-i}(\lambda), \hat{\sigma}_{-i}^2(\lambda)]\}. \end{align*} $$ -The quantity above is called the cross-validated log-likelihood. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data.
      1. The value of the penalty parameter that maximizes the cross-validated log-likelihood is the value of choice. Or we can use the MSE or the R2 score functions.
        2. diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb index e36087719..09033bf6f 100644 --- a/doc/pub/Regression/ipynb/Regression.ipynb +++ b/doc/pub/Regression/ipynb/Regression.ipynb @@ -2271,7 +2271,7 @@ "\n", "2. Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n", "\n", - "3. Fit the linear regression model by means of ridge estimation for each $\\lambda$ in the grid using the training set as" + "3. Fit the linear regression model by means of ridge estimation for each $\\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\\hat{\\sigma}_{-i}^2(\\lambda)$, as" ] }, { @@ -2291,13 +2291,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "and the corresponding estimate of the error variance $\\hat{\\sigma}_{-i}^2(\\lambda)$.\n", - "\n", "1. Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\hat{X}_{i, \\ast}; \\hat{\\beta}_{-i}(\\lambda), \\hat{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\hat{X}_{i, \\ast} \\hat{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", "\n", "2. Repeat steps 1) to 3) such that each sample plays the role of the test set once.\n", "\n", - "3. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter" + "3. Average the prediction performances of the test sets at each grid point of the penalty bias/parameter by computing the *cross-validated log-likelihood*. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as" ] }, { @@ -2306,7 +2304,7 @@ "source": [ "$$\n", "\\begin{align*}\n", - "\\frac{1}{n} \\sum_{i = 1}^n \\log\\{L[Y_i, \\mathbf{X}_{i, \\ast}; \\hat{\\beta}_{-i}(\\lambda), \\hat{\\sigma}_{-i}^2(\\lambda)]\\}.\n", + "\\frac{1}{n} \\sum_{i = 1}^n \\log\\{L[y_i, \\mathbf{X}_{i, \\ast}; \\hat{\\beta}_{-i}(\\lambda), \\hat{\\sigma}_{-i}^2(\\lambda)]\\}.\n", "\\end{align*}\n", "$$" ] @@ -2315,8 +2313,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The quantity above is called the *cross-validated log-likelihood*. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data.\n", - "\n", "1. The value of the penalty parameter that maximizes the cross-validated log-likelihood is the value of choice. Or we can use the MSE or the R2 score functions.\n", "\n", "## Predicted Residual Error Sum of Squares\n", diff --git a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz index 39e17ea5b7a69d4723e727ac0958db4b5e9f976c..c0e43e8017395b8a747bd43e4e15c180a6fcb7ef 100644 GIT binary patch literal 211 zcmb2|=3tQiJTaPq`R)0$Sxkl^ZHd?8j_zzyOkT8(dsjiAz(+PE;kiF1-%x2P{@%y+ zcGX=^&D!~Ak1NVQoOAV0?l-&jr?&FMxoMgjy1lcU+atfy?flP$a#w@0b9?4pbIT6= zYY@g9xIX&jX;tyrzd~>A<-XaPUtOEK`|rc~&ow{ItUt2rYoyrebz2NQI*#Th*_=4_ zcGBsa#{+&Ai<@0?ng0I#q&vrts;iroB+WiHeP43_;o`2{DgHa9{(W;~Kn5TD?=Y;( L>zvJ?!N33j*}P>4 literal 210 zcmb2|=3rp>I5C=m`R)0GSxklk#|qEe9kn@V^meWE(_7YwcE5UuZ05IFc30(Ne6Y_N4SD{NJ}Joi&-^1Sin~Mq zU!G7I=X7kAa%%rEE#l$ZV_3Xoii`;?RaE3S^djBoww^T+2Ae`P73j0wQzC35`F#|b>xATQ*1f|zsyh&(1C zDxC%rk7qX6073G|f2x#u0Dwd}CT?T^71aM53nq5w0>$Kw{VD#%sER<>#_p%L91G z_ylU%fKlwN03N9w^NEk@ymufFbOwV+6TOleb`u-sGTCY42HAB4W9BIWA8Vc7cACi? 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