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"cells": [
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"(Defining latex commands: not to be shown...)\n",
"$$\n",
"\\newcommand{\\norm}[1]{\\left \\| #1 \\right \\|}\n",
"\\DeclareMathOperator{\\minimize}{minimize}\n",
"\\DeclareMathOperator{\\maximize}{maximize}\n",
"\\newcommand{\\real}{\\mathbb{R}}\n",
"\\newcommand{\\blasso}{\\beta^{\\mathrm{LASSO}}}\n",
"\\newcommand{\\bzero}{\\beta^0}\n",
"\\newcommand{\\bLS}{\\hat{\\beta}^{\\mathrm{LS}}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# LASSO and $\\ell_0$ estimator for orthonormal data (30 points)\n",
"\n",
"Let $X \\in \\real^{n \\times p}$ be your training input set, and $Y \\in \\real^n$ your training output.\n",
"In a linear model we predict $\\hat{y}(x) = x^T \\hat{\\beta}$ for some input $x \\in \\real^p$ and some constant $\\hat{\\beta} \\in \\real^p$.\n",
"The so-called $\\ell_0$-penalized estimator $\\bzero(\\lambda)$ tries to find a good $\\beta$ with reslatively few non-zero entries. It is defined as:\n",
"\\begin{equation}\n",
" \\bzero(\\lambda) := \\mathrm{arg \\, min}_{\\beta} \\frac{1}{n} \\norm{Y - X \\beta}_2^2 + \\lambda \\norm{\\beta}_0 ,\n",
"\\end{equation}\n",
"where $\\norm{\\beta}_0 := \\sharp \\{j:\\beta_j \\neq 0\\}$ and $\\lambda > 0$.\n",
"Alternatively, the LASSO estimator $\\blasso(\\lambda)$ is defined as:\n",
"\\begin{equation}\n",
" \\blasso(\\lambda) := \\mathrm{arg \\, min}_{\\beta} \\frac{1}{n} \\norm{Y - X \\beta}_2^2 + \\lambda \\norm{\\beta}_1 ,\n",
"\\end{equation}\n",
"where $\\norm{\\beta}^2_2 := \\sum_{i=1}^p \\beta_i^2$, $\\norm{\\beta}_1 := \\sum_{i=1}^p |\\beta_i|$ and $\\lambda > 0$. \n",
"\n",
"There are in general no analytic formulae for $\\bzero(\\lambda)$ and $\\blasso(\\lambda)$. In practice, they are computed by numerical optimization.\n",
"However, we can compute an analytic expression in the special case of _orthogonal design_, where\n",
"\\begin{equation}\n",
" p=n \\quad \\mathrm{and} \\quad \\frac{1}{n} X^T X = I_{p \\times p} .\n",
"\\end{equation}\n",
"The goal of this exercise is to compute these formulae."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## $\\ell_0$-regularized estimator in the orthonormal design\n",
"\n",
"Let $g_{\\mathrm{hard}, \\lambda}$ be the function:\n",
"\\begin{equation}\n",
" g_{\\mathrm{hard}, \\lambda} : \\real \\rightarrow \\real , \\quad z \\rightarrow z \\, 1_{ \\{|z| > \\lambda\\} } ,\n",
"\\end{equation}\n",
"where $1_{ \\{|z| > \\lambda\\} } = 1$ if $|z| > \\lambda$ and $0$ otherwise.\n",
"Suppose we are in the orthonormal design.\n",
"\n",
"1. Plot $g_{\\mathrm{hard}, \\lambda}$ for $\\lambda = 1$.\n",
"2. Write down the analytical formulae for the least square estimator $\\bLS$. (Not needed for the subsequent calculations)\n",
"3. Prove that $\\bzero_j(\\lambda) = g_{\\mathrm{hard}, \\sqrt{\\lambda}}(z_j)$ where, again, $z_j := \\frac{1}{n}(X^T Y)_j$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## LASSO estimator in the orthonormal design\n",
"\n",
"Let $g_{\\mathrm{soft}, \\lambda}$ be the function:\n",
"\\begin{equation}\n",
" g_{\\mathrm{soft}, \\lambda} : \\real \\rightarrow \\real , \\quad z \\rightarrow \\mathrm{sign}(z) \\, (|z| - \\lambda)_+ .\n",
"\\end{equation}\n",
"Suppose we are in the orthonormal design.\n",
"\n",
"1. Plot $g_{\\mathrm{soft}, \\lambda}$ for $\\lambda = 1$.\n",
"2. Prove that $\\blasso_j(\\lambda) = g_{\\mathrm{soft}, \\frac{\\lambda}{2}}(z_j)$, where $z_j := \\frac{1}{n}(X^T Y)_j$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Application of LASSO on a gene data set (70 points)\n",
"\n",
"(This exercise is independent of the first one.)\n",
"\n",
"The goal of this exercise is to apply both ridge regression and LASSO on real data.\n",
"The data can be found on the homepage of the course, files _xtrain_xxxx.csv_ and _xtest_xxxx.csv_ , and comes from the paper <cite data-cite=\"Kemmeren2014\"> Kemmeren et al.</cite>.\n",
"It consists of the logarithm of the gene expression levels of yeast cells.\n",
"\n",
"More specifically, we consider two datasets corresponding to two different targets: say gene 4710 for the first and gene 3290 for the second.\n",
"In both cases, we would like to predict the gene expression levels of the target gene, given the gene expression levels of the other 6170 genes.\n",
"For each dataset, we are given a training set $X_\\mathrm{train} \\in \\real^{140 \\, \\times \\, 6170}$, $y_\\mathrm{train} \\in \\real^{140}$ consisting of 140 yeast cells, and a test set $X_\\mathrm{test} \\in \\real^{20 \\, \\times \\, 6170}$ with 20 cells. To evaluate the performance of our model, we also provide the target values $y_\\mathrm{test} \\in \\real^{20}$ of the test set.\n",
"\n",
"You are asked, to hand in:\n",
"\n",
"1. A printout of the code of the function _printOutput_, that you will have appropriately modified (see section [_The printOutput function_](#The-printOutput-function)).\n",
"2. Answer questions a. and b. of the section [_Linear regression with no regularizer_](#Linear-regression-with-no-regularizer).\n",
"3. A printout of all the lines marked _### CHANGE THIS LINE ###_. On the printout, all regressions should be done using a regularizing parameter $\\lambda$ that you will have optimized using 10-fold cross-validation on the training set. However, you do not need to hand in the printout of the lines corresponding to these cross-validations.\n",
"4. A printout of the output of all lines marked _### PROVIDE THE OUTPUT ###_.\n",
"5. The same printouts as for questions 3. and 4., but with target gene 3290 instead of 4710 (i.e. with the second dataset).\n",
"\n",
"_Hint_: For all the regression tasks, we recommend to use python functions of the type _linear_model.xxx(xxx)_, of the pacakge _sklearn_. The code asked to hand in should look pretty similar to the code of the section [_Linear regression with no regularizer_](#Linear-regression-with-no-regularizer). Cross-validation can be easily done using ipython functions of the type _linear_model.xxxCV(xxx)_. Do not hesitate to consult the documentation of these functions. Finally, note that, what we call $\\lambda$ corresponds to the parameter _alpha_ of the python functions."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Loading the required packages."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"from numpy import *\n",
"from scipy import *\n",
"from matplotlib import pyplot as plt\n",
"from sklearn import linear_model\n",
"from pandas import * # for easy import of data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Target: gene 4710"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Loading the data"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"('Xtrain size: ', (140, 6170), 'ytrain size: ', (140,))\n",
"('Xtest size: ', (20, 6170), 'ytest size: ', (20,))\n"
]
}
],
"source": [
"### CHANGE THE PATHS ###\n",
"Xtrain = read_csv(\"./datasets/Xtrain_4710.csv\", header = False)\n",
"Xtest = read_csv(\"./datasets/Xtest_4710.csv\", header = False)\n",
"ytrain = read_csv(\"./datasets/ytrain_4710.csv\", header = False, names = [\"gene\",\"y\"])\n",
"ytest = read_csv(\"./datasets/ytest_4710.csv\", header = False, names = [\"gene\",\"y\"])\n",
"\n",
"Xtrain = Xtrain.drop(Xtrain.columns[[0]],axis = 1)\n",
"ytrain = squeeze(ytrain.drop(ytrain.columns[[0]],axis = 1))\n",
"Xtest = Xtest.drop(Xtest.columns[[0]],axis = 1)\n",
"ytest = squeeze(ytest.drop(ytest.columns[[0]],axis = 1))\n",
"\n",
"print('Xtrain size: ', Xtrain.shape, 'ytrain size: ', ytrain.shape);\n",
"print('Xtest size: ', Xtest.shape, 'ytest size: ', ytest.shape);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### The printOutput function\n",
"\n",
"The purpose of this function is to plot the regression weights, the predicted values yhat of y against ytest, the (number of the) gene with the strongest coefficient and the so-called _coefficient of determination_ $R^2$, defined as:\n",
"\\begin{equation}\n",
" R^2 := 1 - \\frac{\\sum_{(x_i,y_i) \\, \\in \\mathrm{\\, test \\ set}} (\\hat{y}(x_i) - y_i)^2}{\\sum_i (y_i - \\overline{y})^2},\n",
"\\end{equation}\n",
"where $\\overline{y}$ is the emperical mean of the $y_i$'s in the test set."
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def printOutput(lm_, Xtest_, ytest_): # lm_ = instance of linear_model.xxx(xxx)\n",
" yhat = lm_.predict(Xtest_) ### we want to predict the test set\n",
" plt.figure(1)\n",
" plt.title(\"Regression Weights\")\n",
" plt.plot(lm_.coef_.T)\n",
" plt.figure(2)\n",
" plt.title('yhat vs ytest')\n",
" plt.plot(ytest_, yhat, 'ro')\n",
" plt.show()\n",
" print 'R2 :', lm_.score(Xtest_, ytest_) # we're interested in the score of the prediction. 1 is best, 0 worst\n",
" print 'Gene with Strongest Coefficient :' , argmax(np.absolute(lm_.coef_)) # not sure whether it's supposed to be like that\n",
" if hasattr(lm_, 'alpha'): print 'Used Lambda :', lm.alpha # if using linear_model.xxx\n",
" if hasattr(lm_, 'alpha_'): print 'Lambda_ResultOfCV :', lm.alpha_ # if using linear_model.xxxCV\n",
" # print 'Regression Coefs :', lm_.coef_ # Can be printed out, if wanted\n",
" # print 'Regression Intercept :', lm_.intercept_ # Can be printed out, if wanted"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Linear regression with no regularizer\n",
"\n",
"a. What should happen if you tried to apply linear regression without any regularizer on this data set? \n",
"b. Does this happen with the function _linear_model.LinearReagression()_ ?"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"a) The MSE should increase when more and more test data is taken into account since we assume the data to be iid, however, it is highly expectable that a lot of genes correlate to each other.\n",
"b) In this case not, but it's very likely that it happens if more test data is taken into account."
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": false
},
"outputs": [
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6dc0d3ed0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6acfcffd0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"R2 : 0.897396797403\n",
"Gene with Strongest Coefficient : 1\n"
]
}
],
"source": [
"lm = linear_model.LinearRegression()\n",
"lm.fit(Xtrain,ytrain)\n",
"\n",
"printOutput(lm,Xtest,ytest) ### PROVIDE THE OUTPUT ###"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Ridge regression"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad1871d0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad187150>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"R2 : 0.884001000152\n",
"Gene with Strongest Coefficient : 250\n",
"Used Lambda : 1.0\n"
]
}
],
"source": [
"lm = linear_model.Ridge() ### using the default parameters since nothing different was requested. ###\n",
"lm.fit(Xtrain, ytrain);\n",
"\n",
"printOutput(lm, Xtest, ytest) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### LASSO regression"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad5b14d0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad5c4e50>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"R2 : -0.0360596061469\n",
"Gene with Strongest Coefficient : 0\n",
"Used Lambda : 0.1\n",
"Regression Coefficient of Gene 5954 : 0.0\n",
"-0.0\n"
]
}
],
"source": [
"lm = linear_model.Lasso(alpha=0.1) ### using the default parameters since nothing different was requested. ###\n",
"lm.fit(Xtrain, ytrain)\n",
"\n",
"printOutput(lm, Xtest, ytest) ### PROVIDE THE OUTPUT ###\n",
"\n",
"print 'Regression Coefficient of Gene 5954 : ', lm.coef_[5954] ### CHANGE THIS LINE ### ### PROVIDE THE OUTPUT ###"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Target: Gene 3290"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"('Xtrain size: ', (140, 6170), 'ytrain size: ', (140,))\n",
"('Xtest size: ', (20, 6170), 'ytest size: ', (20,))\n"
]
}
],
"source": [
"### CHANGE THE PATHS ###\n",
"Xtrain = read_csv(\"./datasets/Xtrain_3290.csv\", header = False)\n",
"Xtest = read_csv(\"./datasets/Xtest_3290.csv\", header = False)\n",
"ytrain = read_csv(\"./datasets/ytrain_3290.csv\", header = False, names = [\"gene\",\"y\"])\n",
"ytest = read_csv(\"./datasets/ytest_3290.csv\", header = False, names = [\"gene\",\"y\"])\n",
"\n",
"Xtrain = Xtrain.drop(Xtrain.columns[[0]],axis = 1)\n",
"ytrain = squeeze(ytrain.drop(ytrain.columns[[0]],axis = 1))\n",
"Xtest = Xtest.drop(Xtest.columns[[0]],axis = 1)\n",
"ytest = squeeze(ytest.drop(ytest.columns[[0]],axis = 1))\n",
"\n",
"print('Xtrain size: ', Xtrain.shape, 'ytrain size: ', ytrain.shape);\n",
"print('Xtest size: ', Xtest.shape, 'ytest size: ', ytest.shape);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Linear Regression"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad0c8d50>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad1cb690>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"R2 : 0.568841483776\n",
"Gene with Strongest Coefficient : 3152\n"
]
}
],
"source": [
"lm = linear_model.LinearRegression()\n",
"lm.fit(Xtrain,ytrain)\n",
"\n",
"printOutput(lm,Xtest,ytest) ### PROVIDE THE OUTPUT ###"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Ridge Regression"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad887250>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
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"QmCC\n"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad723ad0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"R2 : 0.530270080734\n",
"Gene with Strongest Coefficient : 3152\n",
"Used Lambda : 1.0\n"
]
}
],
"source": [
"lm = linear_model.Ridge() ### using the default parameters since nothing different was requested. ###\n",
"lm.fit(Xtrain, ytrain);\n",
"\n",
"printOutput(lm, Xtest, ytest) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Lasso"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad6e1050>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x7fb6ad9c1090>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"R2 : -0.036002074828\n",
"Gene with Strongest Coefficient : 0\n",
"Used Lambda : 0.1\n",
"Regression Coefficient of Gene 5954 : -0.0\n"
]
}
],
"source": [
"lm = linear_model.Lasso(alpha=0.1) ### using the default parameters since nothing different was requested. ###\n",
"lm.fit(Xtrain, ytrain)\n",
"\n",
"printOutput(lm, Xtest, ytest) ### PROVIDE THE OUTPUT ###\n",
"\n",
"print 'Regression Coefficient of Gene 5954 : ', lm.coef_[5954] ### CHANGE THIS LINE ### ### PROVIDE THE OUTPUT ###"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"annotations": {
"author": "",
"categories": [
"intelligent-systems-1-2015"
],
"date": "2015-05-07",
"location": "Beginning of next lecture",
"parent": "IS_SS2015",
"submission_date": "2015-05-21",
"subtitle": "Exercise Sheet 4, Regularized Linear Regression",
"tags": [
"IntelligenSystems",
"Course"
],
"title": "Intelligent Systems 1 - Summer Semester 2015"
},
"celltoolbar": "Edit Metadata",
"kernelspec": {
"display_name": "Python 2",
"language": "python",
"name": "python2"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.9"
}
},
"nbformat": 4,
"nbformat_minor": 0
}
Jan-Peter Hohloch