From 9d8be2597a86963fd2952ae18d0c91158124ea3b Mon Sep 17 00:00:00 2001 From: mhjensen Date: Mon, 2 Sep 2019 21:41:40 +0200 Subject: [PATCH] Corrected exercise 4 and changed strange error --- doc/Projects/2019/hw2/html/._hw2-bs000.html | 10 +++++----- doc/Projects/2019/hw2/html/hw2-bs.html | 10 +++++----- doc/Projects/2019/hw2/html/hw2.html | 10 +++++----- .../2019/hw2/ipynb/ipynb-hw2-src.tar.gz | Bin 208 -> 208 bytes doc/Projects/2019/hw2/pdf/hw2.p.tex | 10 +++++----- doc/Projects/2019/hw2/pdf/hw2.pdf | Bin 220491 -> 220239 bytes doc/Projects/2019/hw2/pdf/hw2.tex | 12 ++++++------ doc/Projects/2019/hw2/pdf/hw2.tex~ | 14 +++++++------- doc/src/Projects/2019/Exercises/hw2.do.txt | 10 +++++----- 9 files changed, 38 insertions(+), 38 deletions(-) diff --git a/doc/Projects/2019/hw2/html/._hw2-bs000.html b/doc/Projects/2019/hw2/html/._hw2-bs000.html index aa7234d27..a7fbc548e 100644 --- a/doc/Projects/2019/hw2/html/._hw2-bs000.html +++ b/doc/Projects/2019/hw2/html/._hw2-bs000.html @@ -145,10 +145,10 @@ y = 5*x<
  1. Write your own code for the Ridge method (see chapter 3.4 of Hastie et al., equations (3.43) and (3.44)) and compute the parametrization for different values of \( \lambda \). Compare and analyze your results with those from exercise 2. Study the dependence on \( \lambda \) while also varying the strength of the noise in your expression for \( y(x) \).
    2. -
  2. Repeat the above but using the functionality of scikit-learn. Compare your code with the results from scikit-learn. Remember to run with the same random numbers for generating \( x \) and \( y \).
    3. -
  3. Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)) or use the functionality of scikit-learn and compute their variances. Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie et al. and their figure 3.11.
    4. -
  4. Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie et al.. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of scikit-learn. We recommend the last since we have not yet discussed how to solve the Lasso equations numerically.
    5. -
  5. Finally, using scikit-learn or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
    6. +
  6. Repeat the above but using the functionality of Scikit-Learn. Compare your code with the results from Scikit-Learn. Remember to run with the same random numbers for generating \( x \) and \( y \).
    7. +
  7. Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parameterizing our function with a second-order polynomial). We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function or using Scikit-Learn to find the parameters \( \beta \). From your results calculate the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)). Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie et al. and their figure 3.11.
    8. +
  8. Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie et al.. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of scikit-learn. We recommend the latter since we have not yet discussed how to solve the Lasso equations numerically.
    9. +
  9. Finally, using Scikit-Learn or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} @@ -171,7 +171,7 @@ Discuss these quantities as functions of the variable \( \lambda \) in the Ridge

Exercise 5

-The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides the singular value decomposition. +The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides that deal with the singular value decomposition.

Using the singular value decomposition, show that the variance of the direction vector diff --git a/doc/Projects/2019/hw2/html/hw2-bs.html b/doc/Projects/2019/hw2/html/hw2-bs.html index aa7234d27..a7fbc548e 100644 --- a/doc/Projects/2019/hw2/html/hw2-bs.html +++ b/doc/Projects/2019/hw2/html/hw2-bs.html @@ -145,10 +145,10 @@ y = 5*x<

  1. Write your own code for the Ridge method (see chapter 3.4 of Hastie et al., equations (3.43) and (3.44)) and compute the parametrization for different values of \( \lambda \). Compare and analyze your results with those from exercise 2. Study the dependence on \( \lambda \) while also varying the strength of the noise in your expression for \( y(x) \).
    2. -
  2. Repeat the above but using the functionality of scikit-learn. Compare your code with the results from scikit-learn. Remember to run with the same random numbers for generating \( x \) and \( y \).
    3. -
  3. Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)) or use the functionality of scikit-learn and compute their variances. Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie et al. and their figure 3.11.
    4. -
  4. Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie et al.. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of scikit-learn. We recommend the last since we have not yet discussed how to solve the Lasso equations numerically.
    5. -
  5. Finally, using scikit-learn or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
    6. +
  6. Repeat the above but using the functionality of Scikit-Learn. Compare your code with the results from Scikit-Learn. Remember to run with the same random numbers for generating \( x \) and \( y \).
    7. +
  7. Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parameterizing our function with a second-order polynomial). We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function or using Scikit-Learn to find the parameters \( \beta \). From your results calculate the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)). Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie et al. and their figure 3.11.
    8. +
  8. Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie et al.. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of scikit-learn. We recommend the latter since we have not yet discussed how to solve the Lasso equations numerically.
    9. +
  9. Finally, using Scikit-Learn or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} @@ -171,7 +171,7 @@ Discuss these quantities as functions of the variable \( \lambda \) in the Ridge

Exercise 5

-The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides the singular value decomposition. +The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides that deal with the singular value decomposition.

Using the singular value decomposition, show that the variance of the direction vector diff --git a/doc/Projects/2019/hw2/html/hw2.html b/doc/Projects/2019/hw2/html/hw2.html index 3ba5761ce..183b95c13 100644 --- a/doc/Projects/2019/hw2/html/hw2.html +++ b/doc/Projects/2019/hw2/html/hw2.html @@ -110,10 +110,10 @@ y = 5*x<

  1. Write your own code for the Ridge method (see chapter 3.4 of Hastie et al., equations (3.43) and (3.44)) and compute the parametrization for different values of \( \lambda \). Compare and analyze your results with those from exercise 2. Study the dependence on \( \lambda \) while also varying the strength of the noise in your expression for \( y(x) \).
    2. -
  2. Repeat the above but using the functionality of scikit-learn. Compare your code with the results from scikit-learn. Remember to run with the same random numbers for generating \( x \) and \( y \).
    3. -
  3. Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)) or use the functionality of scikit-learn and compute their variances. Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie et al. and their figure 3.11.
    4. -
  4. Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie et al.. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of scikit-learn. We recommend the last since we have not yet discussed how to solve the Lasso equations numerically.
    5. -
  5. Finally, using scikit-learn or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
    6. +
  6. Repeat the above but using the functionality of Scikit-Learn. Compare your code with the results from Scikit-Learn. Remember to run with the same random numbers for generating \( x \) and \( y \).
    7. +
  7. Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parameterizing our function with a second-order polynomial). We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function or using Scikit-Learn to find the parameters \( \beta \). From your results calculate the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)). Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie et al. and their figure 3.11.
    8. +
  8. Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie et al.. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of scikit-learn. We recommend the latter since we have not yet discussed how to solve the Lasso equations numerically.
    9. +
  9. Finally, using Scikit-Learn or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} @@ -136,7 +136,7 @@ Discuss these quantities as functions of the variable \( \lambda \) in the Ridge

Exercise 5

-The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides the singular value decomposition. +The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides that deal with the singular value decomposition.

Using the singular value decomposition, show that the variance of the direction vector diff --git a/doc/Projects/2019/hw2/ipynb/ipynb-hw2-src.tar.gz b/doc/Projects/2019/hw2/ipynb/ipynb-hw2-src.tar.gz index b4095c320780f442e0eaa85289d76c0df69d419f..5a9ba93bcdcf2c58c42f8ab891595ec863efeb31 100644 GIT binary patch literal 208 zcmb2|=3uDE&y8hZetZ6W7L%dC@xt?VM{N#Hy1Qd{`a-!XjgB8o6MOS+&B_qESo`}I z`>o);o|<*`)8&nM4xdRftF4?Z`g4jq|KcfGYQj5HZeO_`a&_%Ktya6 HgMk46Z;}PzX8GV=Sb|sgy8*e|) zJ!$H@rjGyjpDpfMw4y5N&OKY5S-aM5ULEN5HCMZR&&%D(TE(mSx0o%;O<()2$8@ho zZtm^#F0t>Ibo;KathUHTRzU zkveO`yhlN8Pfjn?zgeH>^zG3jl{^V2Ud>(Ke_Ab{{>IMB`2Smf24wJ{{wCw;uVU*N HG#D5Fk|tzT diff --git a/doc/Projects/2019/hw2/pdf/hw2.p.tex b/doc/Projects/2019/hw2/pdf/hw2.p.tex index 39ddf5ad0..97bbfc181 100644 --- a/doc/Projects/2019/hw2/pdf/hw2.p.tex +++ b/doc/Projects/2019/hw2/pdf/hw2.p.tex @@ -184,13 +184,13 @@ y = 5*x*x+0.1*np.random.randn(100,1) \begin{enumerate} \item Write your own code for the Ridge method (see chapter 3.4 of Hastie \emph{et al.}, equations (3.43) and (3.44)) and compute the parametrization for different values of $\lambda$. Compare and analyze your results with those from exercise 2. Study the dependence on $\lambda$ while also varying the strength of the noise in your expression for $y(x)$. -\item Repeat the above but using the functionality of \textbf{scikit-learn}. Compare your code with the results from \textbf{scikit-learn}. Remember to run with the same random numbers for generating $x$ and $y$. +\item Repeat the above but using the functionality of \textbf{Scikit-Learn}. Compare your code with the results from \textbf{Scikit-Learn}. Remember to run with the same random numbers for generating $x$ and $y$. -\item Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})+\lambda\hat{I})^{-1}$) or use the functionality of \textbf{scikit-learn} and compute their variances. Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie \emph{et al.} and their figure 3.11. +\item Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parameterizing our function with a second-order polynomial). We will use standard linear regression and the Ridge regression. You can now opt for either writing your own function or using \textbf{Scikit-Learn} to find the parameters $\beta$. From your results calculate the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})+\lambda\hat{I})^{-1}$). Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie \emph{et al.} and their figure 3.11. -\item Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie \emph{et al.}. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of \textbf{scikit-learn}. We recommend the last since we have not yet discussed how to solve the Lasso equations numerically. +\item Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie \emph{et al.}. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of \textbf{scikit-learn}. We recommend the latter since we have not yet discussed how to solve the Lasso equations numerically. -\item Finally, using \textbf{scikit-learn} or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as +\item Finally, using \textbf{Scikit-Learn} or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as \end{enumerate} \noindent @@ -210,7 +210,7 @@ Discuss these quantities as functions of the variable $\lambda$ in the Ridge and \subsection{Exercise 5} -The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides the singular value decomposition. +The theory behind this exercise will be covered during the lectures of week 36. It requires reading chapter three of Hastie et al, in particular the derivations preceeding equation (3.49) as the well as the material in the Regression slides that deal with the singular value decomposition. 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