From 5d36d12c1dca2aa8b1cd7998013717a8e4f1d026 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Wed, 8 Sep 2021 05:53:54 +0200 Subject: [PATCH] updating week 36 --- doc/pub/week36/html/week36-bs.html | 94 ++++++++++--------- doc/pub/week36/html/week36-reveal.html | 30 +++++- doc/pub/week36/html/week36-solarized.html | 38 ++++++-- doc/pub/week36/html/week36.html | 38 ++++++-- doc/pub/week36/ipynb/ipynb-week36-src.tar.gz | Bin 192 -> 191 bytes doc/pub/week36/ipynb/week36.ipynb | 33 ++++++- doc/src/week36/week36.do.txt | 21 ++++- 7 files changed, 186 insertions(+), 68 deletions(-) diff --git a/doc/pub/week36/html/week36-bs.html b/doc/pub/week36/html/week36-bs.html index 3f7b1668b..239aafe78 100644 --- a/doc/pub/week36/html/week36-bs.html +++ b/doc/pub/week36/html/week36-bs.html @@ -47,7 +47,10 @@ Automatically generated HTML file from DocOnce source 2, None, 'summary-from-last-week-and-examples'), - ('Linking with the SVD', 2, None, 'linking-with-the-svd'), + ('Linear Regression and the SVD', + 2, + None, + 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', 2, @@ -57,6 +60,10 @@ Automatically generated HTML file from DocOnce source 2, None, 'ridge-and-lasso-regression'), + ('From OLS to Ridge and Lasso', + 2, + None, + 'from-ols-to-ridge-and-lasso'), ('Deriving the Ridge Regression Equations', 2, None, @@ -76,6 +83,7 @@ Automatically generated HTML file from DocOnce source None, 'simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression'), ('Ridge Regression', 2, None, 'ridge-regression'), + ('Lasso Rgeression', 2, None, 'lasso-rgeression'), ('Linking the regression analysis with a statistical ' 'interpretation', 2, @@ -219,48 +227,50 @@ MathJax.Hub.Config({
  • Plans for week 36
  • Thursday September 9
  • Summary from last Week and Examples
    • -
  • Linking with the SVD
    • +
  • Linear Regression and the SVD
  • What does it mean?
  • And finally \( \boldsymbol{X}\boldsymbol{X}^T \)
  • Ridge and LASSO Regression
    • -
  • Deriving the Ridge Regression Equations
    • -
  • Interpreting the Ridge results
    • -
  • More interpretations
    • -
  • Deriving the Lasso Regression Equations
    • -
  • Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression
    • -
  • Ridge Regression
    • -
  • Linking the regression analysis with a statistical interpretation
    • -
  • Assumptions made
    • -
  • Expectation value and variance
    • -
  • Expectation value and variance for \( \boldsymbol{\beta} \)
    • -
  • Friday September 10
    • -
  • Why resampling methods
    • -
  • Resampling methods
    • -
  • Resampling approaches can be computationally expensive
    • -
  • Why resampling methods ?
    • -
  • Statistical analysis
    • -
  • Resampling methods
    • -
  • Resampling methods: Jackknife and Bootstrap
    • -
  • Resampling methods: Jackknife
    • -
  • Jackknife code example
    • -
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • Summing up
    • -
  • Another Example from Scikit-Learn's Repository
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • -
  • Cross-validation with Ridge
    • +
  • From OLS to Ridge and Lasso
    • +
  • Deriving the Ridge Regression Equations
    • +
  • Interpreting the Ridge results
    • +
  • More interpretations
    • +
  • Deriving the Lasso Regression Equations
    • +
  • Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression
    • +
  • Ridge Regression
    • +
  • Lasso Rgeression
    • +
  • Linking the regression analysis with a statistical interpretation
    • +
  • Assumptions made
    • +
  • Expectation value and variance
    • +
  • Expectation value and variance for \( \boldsymbol{\beta} \)
    • +
  • Friday September 10
    • +
  • Why resampling methods
    • +
  • Resampling methods
    • +
  • Resampling approaches can be computationally expensive
    • +
  • Why resampling methods ?
    • +
  • Statistical analysis
    • +
  • Resampling methods
    • +
  • Resampling methods: Jackknife and Bootstrap
    • +
  • Resampling methods: Jackknife
    • +
  • Jackknife code example
    • +
  • Resampling methods: Bootstrap
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Various steps in cross-validation
    • +
  • How to set up the cross-validation for Ridge and/or Lasso
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • Summing up
    • +
  • Another Example from Scikit-Learn's Repository
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • @@ -295,7 +305,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 7, 2021

    +

    Sep 8, 2021


    @@ -319,7 +329,7 @@ MathJax.Hub.Config({

  • 9
  • 10
  • ...
    • -
  • 46
    • +
  • 48
  • »
    • diff --git a/doc/pub/week36/html/week36-reveal.html b/doc/pub/week36/html/week36-reveal.html index fa24b9fb2..5c79e75b5 100644 --- a/doc/pub/week36/html/week36-reveal.html +++ b/doc/pub/week36/html/week36-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Sep 7, 2021

    +

    Sep 8, 2021


    @@ -179,10 +179,10 @@ MathJax.Hub.Config({

    -

    Linking with the SVD

    +

    Linear Regression and the SVD

    -We saw earlier that +We used the SVD to analyse the matrix to invert in ordinary lineat regression

     
    $$ \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T. @@ -345,6 +345,11 @@ $$ \vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. $$

     
    +

    + + +
    +

    From OLS to Ridge and Lasso

    By minimizing the above equation with respect to the parameters @@ -537,7 +542,7 @@ Similarly, Mehta et a

    Deriving the Lasso Regression Equations

    -Using the matrix-vector expression for Lasso regression and dropping the parameter \( 1/n \) in front of the standard means squared error equation, we have the following cost function +Using the matrix-vector expression for Lasso regression and dropping the parameter \( 1/n \) in front of the standard mean squared error equation, we have the following cost function

     
    $$ @@ -568,7 +573,7 @@ $$ $$

     
    -This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package     CVXOPT. We will discuss this later. +This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package CVXOPT. We will discuss this later.

    @@ -616,6 +621,21 @@ $$ +
    +

    Lasso Rgeression

    + +

    +For Ridge regression our cost function is +

     
    +$$ +C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\beta_i^2,, +$$ +

     
    + +and minimizing we have that +

    + +

    Linking the regression analysis with a statistical interpretation

    diff --git a/doc/pub/week36/html/week36-solarized.html b/doc/pub/week36/html/week36-solarized.html index 836b7a150..6deff0044 100644 --- a/doc/pub/week36/html/week36-solarized.html +++ b/doc/pub/week36/html/week36-solarized.html @@ -67,7 +67,10 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'summary-from-last-week-and-examples'), - ('Linking with the SVD', 2, None, 'linking-with-the-svd'), + ('Linear Regression and the SVD', + 2, + None, + 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', 2, @@ -77,6 +80,10 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'ridge-and-lasso-regression'), + ('From OLS to Ridge and Lasso', + 2, + None, + 'from-ols-to-ridge-and-lasso'), ('Deriving the Ridge Regression Equations', 2, None, @@ -96,6 +103,7 @@ div { text-align: justify; text-justify: inter-word; } None, 'simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression'), ('Ridge Regression', 2, None, 'ridge-regression'), + ('Lasso Rgeression', 2, None, 'lasso-rgeression'), ('Linking the regression analysis with a statistical ' 'interpretation', 2, @@ -240,7 +248,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 7, 2021

    +

    Sep 8, 2021












    @@ -264,10 +272,10 @@ MathJax.Hub.Config({











    -

    Linking with the SVD

    +

    Linear Regression and the SVD

    -We saw earlier that +We used the SVD to analyse the matrix to invert in ordinary lineat regression $$ \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T. $$ @@ -403,6 +411,11 @@ $$ \vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. $$ +

    +









    + +

    From OLS to Ridge and Lasso

    +

    By minimizing the above equation with respect to the parameters \( \boldsymbol{\beta} \) we could then obtain an analytical expression for the @@ -566,7 +579,7 @@ Similarly, Mehta et a

    Deriving the Lasso Regression Equations

    -Using the matrix-vector expression for Lasso regression and dropping the parameter \( 1/n \) in front of the standard means squared error equation, we have the following cost function +Using the matrix-vector expression for Lasso regression and dropping the parameter \( 1/n \) in front of the standard mean squared error equation, we have the following cost function $$ C(\boldsymbol{X},\boldsymbol{\beta})=\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1, @@ -589,7 +602,7 @@ $$ \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta})+\lambda sgn(\boldsymbol{\beta})=2\boldsymbol{X}^T(\boldsymbol{y}. $$ -This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package CVXOPT. We will discuss this later. +This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package CVXOPT. We will discuss this later.











    @@ -627,6 +640,19 @@ $$ \hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \frac{\boldsymbol{y}}{1+\lambda}. $$ +

    +









    + +

    Lasso Rgeression

    + +

    +For Ridge regression our cost function is +$$ +C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\beta_i^2,, +$$ + +and minimizing we have that +

    diff --git a/doc/pub/week36/html/week36.html b/doc/pub/week36/html/week36.html index 358a9abf0..dde24d44d 100644 --- a/doc/pub/week36/html/week36.html +++ b/doc/pub/week36/html/week36.html @@ -72,7 +72,10 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'summary-from-last-week-and-examples'), - ('Linking with the SVD', 2, None, 'linking-with-the-svd'), + ('Linear Regression and the SVD', + 2, + None, + 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', 2, @@ -82,6 +85,10 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'ridge-and-lasso-regression'), + ('From OLS to Ridge and Lasso', + 2, + None, + 'from-ols-to-ridge-and-lasso'), ('Deriving the Ridge Regression Equations', 2, None, @@ -101,6 +108,7 @@ div { text-align: justify; text-justify: inter-word; } None, 'simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression'), ('Ridge Regression', 2, None, 'ridge-regression'), + ('Lasso Rgeression', 2, None, 'lasso-rgeression'), ('Linking the regression analysis with a statistical ' 'interpretation', 2, @@ -245,7 +253,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 7, 2021

    +

    Sep 8, 2021












    @@ -269,10 +277,10 @@ MathJax.Hub.Config({











    -

    Linking with the SVD

    +

    Linear Regression and the SVD

    -We saw earlier that +We used the SVD to analyse the matrix to invert in ordinary lineat regression $$ \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T. $$ @@ -408,6 +416,11 @@ $$ \vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. $$ +

    +









    + +

    From OLS to Ridge and Lasso

    +

    By minimizing the above equation with respect to the parameters \( \boldsymbol{\beta} \) we could then obtain an analytical expression for the @@ -571,7 +584,7 @@ Similarly, Mehta et a

    Deriving the Lasso Regression Equations

    -Using the matrix-vector expression for Lasso regression and dropping the parameter \( 1/n \) in front of the standard means squared error equation, we have the following cost function +Using the matrix-vector expression for Lasso regression and dropping the parameter \( 1/n \) in front of the standard mean squared error equation, we have the following cost function $$ C(\boldsymbol{X},\boldsymbol{\beta})=\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1, @@ -594,7 +607,7 @@ $$ \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta})+\lambda sgn(\boldsymbol{\beta})=2\boldsymbol{X}^T(\boldsymbol{y}. $$ -This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package CVXOPT. We will discuss this later. +This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package CVXOPT. We will discuss this later.











    @@ -632,6 +645,19 @@ $$ \hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \frac{\boldsymbol{y}}{1+\lambda}. $$ +

    +









    + +

    Lasso Rgeression

    + +

    +For Ridge regression our cost function is +$$ +C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\beta_i^2,, +$$ + +and minimizing we have that +

    diff --git a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz index 8d24c23d6299695de871c5182dd149419c31ffac..1a865aee4a9ffda4fd2e9b9800a1c48a8ef3ce32 100644 GIT binary patch literal 191 zcmV;w06_mAiwFS5GdN)Y1MSbv3c@f92k@Qu6nTP??xQ;kdhj5K_yQfvT%ESscIe*T zeSn@6FGGa>E`LHoNOr5maB>xAh~IoC&0mejkseyplK?3rGHXa0$OEiJ6KovXD1r5zPoH*h0r5+2D*rwS-F t*`Wo7uB|i#LUka@QaY)XIE5FZ4@R7A6#n`d&+|O*YY(7l8IAx5004JiR<8g6 literal 192 zcmV;x06+g9iwFR`&NpEI1MSaC3c@fD2H>uHia9|^Y}%{^yKo_hc!88oZLCddlA^u6 zeSoeMH${Yei}?u|hM7aQ-t4l--CeL4LP*LOjLFkvO0t^Y6G{`HY^a>^6=y(boWwi; zvfN59opr(tt2EUal|}WgZ)hva4|C>I;F*8oSV;@peeWu*Kxv1$)-~J^>o|>M+p8Q3 ujduJ3gV# \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Sep 7, 2021**\n", + "Date: **Sep 8, 2021**\n", "\n", "Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -28,9 +28,9 @@ "\n", "## Summary from last Week and Examples\n", "\n", - "## Linking with the SVD\n", + "## Linear Regression and the SVD\n", "\n", - "We saw earlier that" + "We used the SVD to analyse the matrix to invert in ordinary lineat regression" ] }, { @@ -299,6 +299,8 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "## From OLS to Ridge and Lasso\n", + "\n", "By minimizing the above equation with respect to the parameters\n", "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n", "parameters $\\boldsymbol{\\beta}$. We can add a regularization parameter $\\lambda$ by\n", @@ -576,7 +578,7 @@ "\n", "## Deriving the Lasso Regression Equations\n", "\n", - "Using the matrix-vector expression for Lasso regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have the following **cost** function" + "Using the matrix-vector expression for Lasso regression and dropping the parameter $1/n$ in front of the standard mean squared error equation, we have the following **cost** function" ] }, { @@ -640,7 +642,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package [CVXOPT](https://cvxopt.org/). We will discuss this later. \n", + "This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package [CVXOPT](https://cvxopt.org/). We will discuss this later. \n", "\n", "\n", "\n", @@ -716,6 +718,27 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "## Lasso Rgeression\n", + "\n", + "For Ridge regression our cost function is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\beta_i^2,,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and minimizing we have that\n", + "\n", + "\n", "\n", "## Linking the regression analysis with a statistical interpretation\n", "\n", diff --git a/doc/src/week36/week36.do.txt b/doc/src/week36/week36.do.txt index 02d6bbf19..fd9b5cbd3 100644 --- a/doc/src/week36/week36.do.txt +++ b/doc/src/week36/week36.do.txt @@ -18,9 +18,9 @@ DATE: today ===== Summary from last Week and Examples ===== !split -===== Linking with the SVD ===== +===== Linear Regression and the SVD ===== -We saw earlier that +We used the SVD to analyse the matrix to invert in ordinary lineat regression !bt \[ \bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{U}\bm{\Sigma}\bm{V}^T=\bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T. @@ -163,6 +163,9 @@ where we have used the definition of a norm-2 vector, that is \] !et +!split +===== From OLS to Ridge and Lasso ===== + By minimizing the above equation with respect to the parameters $\bm{\beta}$ we could then obtain an analytical expression for the parameters $\bm{\beta}$. We can add a regularization parameter $\lambda$ by @@ -325,7 +328,7 @@ Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also re !split ===== Deriving the Lasso Regression Equations ===== -Using the matrix-vector expression for Lasso regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have the following _cost_ function +Using the matrix-vector expression for Lasso regression and dropping the parameter $1/n$ in front of the standard mean squared error equation, we have the following _cost_ function !bt \[ @@ -352,7 +355,7 @@ and reordering we have \bm{X}^T\bm{X}\bm{\beta})+\lambda sgn(\bm{\beta})=2\bm{X}^T(\bm{y}. \] !et -This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package "CVXOPT":"https://cvxopt.org/". We will discuss this later. +This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package "CVXOPT":"https://cvxopt.org/". We will discuss this later. @@ -393,6 +396,16 @@ and minimizing we have that !et +!split +===== Lasso Rgeression ===== + +For Ridge regression our cost function is +!bt +\[ +C(\bm{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\beta_i^2,, +\] +!et +and minimizing we have that !split