diff --git a/is/UB3/ExerciseSheet3Max.ipynb b/is/UB3/ExerciseSheet3Max.ipynb index 536dde8..61d6e1a 100644 --- a/is/UB3/ExerciseSheet3Max.ipynb +++ b/is/UB3/ExerciseSheet3Max.ipynb @@ -32,6 +32,30 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "## Proof\n", + "\\begin{align*}\n", + " f(x) &= \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\cdot e^{\\frac{-(x-\\mu)^2}{2\\sigma^2}}\\cdot \\frac{1}{\\sqrt{2\\pi s^2}}\\cdot e^{\\frac{-(x-m)^2}{2s^2}}\\\\\n", + " &=\\frac{1}{2\\pi s\\sigma}\\cdot e^{-\\alpha}\\\\\n", + " \\text{where }\\alpha &= \\frac{s^2(x-\\mu)^2+\\sigma^2(x-m)^2}{2\\sigma^2 s^2}\\\\\n", + " &=\\frac{(s^2+\\sigma^2)x^2-2(s^2\\mu+\\sigma^2 m)x+s^2\\mu^2+\\sigma^2m^2}{2\\sigma^2s^2}\\\\\n", + " &=\\frac{x^2-2\\frac{s^2\\mu+\\sigma^2 m}{s^2+\\sigma^2}x+\\frac{s^2\\mu^2+\\sigma^2m^2}{s^2+\\sigma^2}}{\\frac{2\\sigma^2s^2}{s^2+\\sigma^2}}\\\\\n", + " &=\\frac{\\left(x-\\frac{s^2\\mu+\\sigma^2 m}{s^2+\\sigma^2}\\right)^2-\\left(\\frac{s^2\\mu+\\sigma^2 m}{s^2+\\sigma^2}\\right)^2+\\frac{s^2\\mu^2+\\sigma^2m^2}{s^2+\\sigma^2}}{\\frac{2\\sigma^2s^2}{s^2+\\sigma^2}}\\\\\n", + " &=\\frac{\\left(x-\\frac{s^2\\mu+\\sigma^2 m}{s^2+\\sigma^2}\\right)^2}{\\frac{2\\sigma^2s^2}{s^2+\\sigma^2}}+\\underbrace{\\frac{\\frac{s^2\\mu^2+\\sigma^2m^2}{s^2+\\sigma^2}-\\left(\\frac{s^2\\mu+\\sigma^2 m}{s^2+\\sigma^2}\\right)^2}{\\frac{2\\sigma^2s^2}{s^2+\\sigma^2}}}_\\beta\\\\\n", + " \\beta&= \\frac{\\frac{s^4\\mu^2+s^2\\sigma^2\\mu^2+s^2\\sigma^2m^2+\\sigma^4m^2-\\left(s^4\\mu^2+2s^2\\sigma^2\\mu m+\\sigma^4m^2\\right)}{\\left(s^2+\\sigma^2\\right)^2}}{\\frac{2\\sigma^2s^2}{s^2+\\sigma^2}}\\\\\n", + " &=\\frac{s^2\\sigma^2\\left(\\mu^2-2\\mu m+m^2\\right)}{2\\sigma^2s^2\\left(s^2+\\sigma^2\\right)}\\\\\n", + " &=\\frac{(\\mu-m)^2}{2(s^2+\\sigma^2)}\\\\\n", + " \\Rightarrow f(x)&=\\frac{1}{2\\pi s\\sigma}\\cdot e^{-(\\alpha-\\beta)-\\beta}\\\\\n", + " &=\\frac{1}{2\\pi s\\sigma}\\cdot e^{-\\frac{\\left(x-\\frac{s^2\\mu+\\sigma^2 m}{s^2+\\sigma^2}\\right)^2}{\\frac{2\\sigma^2s^2}{s^2+\\sigma^2}}}\\cdot e^{-\\frac{(\\mu-m)^2}{2(s^2+\\sigma^2)}}\\\\\n", + " &=\\frac{\\sqrt{2\\pi(s^2+\\sigma^2)}\\sqrt{2\\pi\\frac{\\sigma^2s^2}{s^2+\\sigma^2}}}{2\\pi s\\sigma}\\normal\\left[x;\\frac{s^2\\mu+\\sigma^2 m}{s^2+\\sigma^2},\\frac{\\sigma^2s^2}{s^2+\\sigma^2}\\right]\\normal\\left[\\mu;m,\\sigma^2+s^2\\right]\\\\\n", + " &=\\frac{\\sqrt{2\\pi\\cdot 2\\pi\\sigma^2s^2}}{2\\pi s\\sigma}\\normal\\left[x; \\left(\\frac 1{\\sigma^2}+\\frac 1{s^2}\\right)^{-1}\\left(\\frac \\mu{\\sigma^2}+\\frac m{s^2}\\right),\\left(\\frac 1{\\sigma^2}+\\frac 1{s^2}\\right)^{-1}\\right]\\normal\\left[\\mu;m,\\sigma^2+s^2\\right]\\\\\n", + " &=\\normal\\left[x; \\left(\\frac 1{\\sigma^2}+\\frac 1{s^2}\\right)^{-1}\\left(\\frac \\mu{\\sigma^2}+\\frac m{s^2}\\right),\\left(\\frac 1{\\sigma^2}+\\frac 1{s^2}\\right)^{-1}\\right]\\normal\\left[\\mu;m,\\sigma^2+s^2\\right]\\hspace{2cm} q.e.d.\n", + "\\end{align*}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "# Maximum Likelihood Estimator of Simple Linear Regression (25 Points)\n", "\n", "Derive the formula $\\mathbf{w}_{MLE} = (X^TX)^{-1}X^T\\mathbf{y}$ from the lecture, by calculating the derivative of $p(\\mathbf{y}\\,|X,\\mathbf{w}) = \\normal(\\mathbf{y}\\,|X\\mathbf{w}, \\sigma^2I)$ with respect to $\\mathbf{w}$, setting it to zero and solving it for $\\mathbf{w}$.\n", @@ -54,19 +78,11 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 15, "metadata": { "collapsed": false }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - ":0: FutureWarning: IPython widgets are experimental and may change in the future.\n" - ] - } - ], + "outputs": [], "source": [ "# Loading the required packages\n", "%matplotlib inline\n", @@ -78,7 +94,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 16, "metadata": { "collapsed": false, "scrolled": false @@ -102,7 +118,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -126,7 +142,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 18, "metadata": { "collapsed": false, "scrolled": false @@ -135,144 +151,122 @@ { "data": { "text/plain": [ - "" + "" ] }, - "execution_count": 6, + "execution_count": 18, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAYYAAAEPCAYAAABGP2P1AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG1tJREFUeJzt3X285nOdx/HX2wxRhFQMxtJETDGjZZasHCsaYmdnVXIT\n", - "IjfRNCuVu12mKCpJW9RaN7EMKy2LSiaczCqDjDGMCWW2QSYKyza2YT77x/dH5zrnzJzrOue6ru/v\n", - "d13v5+NxPeZc51w378PM73N97xURmJmZvWqV3AHMzKxcXBjMzKyGC4OZmdVwYTAzsxouDGZmVsOF\n", - "wczMamQtDJIulrRE0vw+35sh6XFJc4vb5JwZzcy6Te4WwyVA/wt/AOdExLbF7aYMuczMulbWwhAR\n", - "s4FnB/mR2p3FzMyS3C2GFZkmaZ6kiyStkzuMmVk3KWNh+DawGTAR+C3wtbxxzMy6y+jcAfqLiN+9\n", - 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"metadata": {}, @@ -281,9 +275,10 @@ ], "source": [ "# Plot the input data here\n", - "plt.plot(x,y)\n", - "plt.xlabel('x')\n", - "plt.ylabel('y') " + "plt.plot(x,y,'o')\n", + "plt.title('Input data')\n", + "plt.ylabel('Output y')\n", + "plt.xlabel('Input x') " ] }, { @@ -296,7 +291,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -324,7 +319,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -344,7 +339,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 24, "metadata": { "collapsed": false, "scrolled": true @@ -354,204 +349,154 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAXcAAAEKCAYAAADpfBXhAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3XecHVX5x/HPN40E6U1CR6nBSBERRDT0gBRBigkdBAQp\n", - "Ih2UyQgIghQLqEjvIE2KCAESRGkqAYKh5SclgPTeE/L8/jgT2Gx2k93bZu/d7/v12lf23pk58+wq\n", - 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" plt.plot(x,X.dot(w))\n", + " plt.title(\"MSE %f\" % MSE(X.dot(w),y))\n", + " plt.plot(x,y,'yo')" ] }, { @@ -578,74 +523,7 @@ }, { "cell_type": "code", - "execution_count": 308, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "array([[ 1.01010000e+04, 1.01010000e+03, 1.01010000e+02],\n", - " [ 1.01010000e+03, 1.01010000e+02, 1.01010000e+01],\n", - " [ 1.01010000e+02, 1.01010000e+01, 1.01010000e+00]])" - ] - }, - "execution_count": 308, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A.T.dot(A)" - ] - }, - { - "cell_type": "code", - "execution_count": 309, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import scipy\n", - "from scipy import linalg\n", - "import numpy as np" - ] - }, - { - "cell_type": "code", - "execution_count": 310, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "1\n", - "[[ 9.80101960e-03 9.80101960e-04 9.80101960e-05]\n", - " [ 9.80101960e-04 9.80101960e-05 9.80101960e-06]\n", - " [ 9.80101960e-05 9.80101960e-06 9.80101960e-07]]\n", - "True\n" - ] - } - ], - "source": [ - "A = np.array([[10,1,0.1],[10,1,0.2],[10,1.1,0.7]])\n", - "a = np.array([[10,1,0.1]])\n", - "A = a.T.dot(a)\n", - "\n", - "B,rank = linalg.pinv(A,return_rank=True)\n", - "print rank\n", - "print B\n", - "print np.allclose(A,A.dot(B.dot(A)))" - ] - }, - { - "cell_type": "code", - "execution_count": 13, + "execution_count": 53, "metadata": { "collapsed": false }, @@ -653,283 +531,335 @@ { "data": { "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAXYAAAEKCAYAAAAGvn7fAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3Xd4FOX2wPHvCYQOIlKliAj2q1iwITLYG6jYRUERLFgQ\n", - "uPayWXu5oohXvYoi4hUVsaGCgDKAP71WimADlV6V3kvO7493AktI2SS7mc3mfJ4nT7I7s++c3U1O\n", - "3j3zzvuKqmKMMSZ9ZIQdgDHGmMSyxG6MMWnGErsxxqQZS+zGGJNmLLEbY0yascRujDFpxhK7iYuI\n", - 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- "" + "" ] }, "metadata": {}, @@ -945,15 +875,27 @@ "w = wRidge(X,y,1.0)\n", "@interact(n=[1,16])\n", "def plot(n):\n", - " X = construct_poly(x,n) \n", - " Xt = construct_poly(xt,n) #n=d\n", - " w = wRidge(X,y,1.0)\n", - " plt.plot(x,X.dot(w))\n", - " plt.plot(xt,Xt.dot(w))\n", - " plt.title(\"MSE train: %f test %f\" % (MSE(X.dot(w),y),MSE(Xt.dot(w),yt)))\n", - " plt.plot(xt,yt)\n", - " plt.plot(x,y)\n", - "# was ist mit fehler kurven gemeint?" + " X = construct_poly(x,n) \n", + " Xt = construct_poly(xt,n) #n=d\n", + " w = wRidge(X,y,1.0)\n", + " plt.plot(x,X.dot(w),'g',label='regression training')\n", + " plt.plot(xt,Xt.dot(w),'b',label='regression test')\n", + " plt.title(\"MSE train: %f MSE test: %f\" % (MSE(X.dot(w),y),MSE(Xt.dot(w),yt)))\n", + " plt.plot(xt,yt,'b*',label='data test')\n", + " plt.plot(x,y,'go',label='data trainig')\n", + " plt.ylabel('Output y') \n", + " plt.xlabel('Input x') \n", + " plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Conclusion:\n", + "We get suitable solutions for n from 5 to 7 for the test regression. For higher values we get overfitting. For lower ones underfitting.\n", + "The MSE of the test regression is always higher than the one of the training regression.\n", + "That's evident because the regression function is fitted to the values of the training data and so doesn't provide an optimal fitting for the test data.\n" ] }, { @@ -965,7 +907,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 54, "metadata": { "collapsed": false }, @@ -974,259 +916,324 @@ "name": "stdout", "output_type": "stream", "text": [ - "1.75123654094\n", - "0.781528050024\n", - "11.0428184273\n" + "Minimum MSE: 0.6275\n" ] }, { "data": { "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAXYAAAEKCAYAAAAGvn7fAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3XmclXP7wPHPNUt7qbQoIbLTIyHRk06lKImESJKtpEg8\n", - "1sfPmWPfd1lL9ihC2QrdlaJoI4R4ilJpT/sy1++P7z3NaUw1M2e5z5y53q/XeTVnu+/rzPO45nuu\n", - "+/p+v6KqGGOMSR8ZQQdgjDEmviyxG2NMmrHEbowxacYSuzHGpBlL7MYYk2YssRtjTJqxxG6MMWnG\n", - 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MSE(X.dot(w),y))\n", + " \n", " \n", " w = wRidge(X,y,0.1)\n", - " plt.plot(xt,Xt.dot(w))\n", - " plt.title(\"MSE %f\" % MSE(X.dot(w),y))\n", - " print MSE(X.dot(w),y)\n", + " plt.plot(xt,Xt.dot(w),label='$\\lambda$ = 0.1'+ \" MSE: %f\" % MSE(X.dot(w),y))\n", + " \n", " \n", " w = wRidge(X,y,10.0)\n", - " plt.plot(xt,Xt.dot(w))\n", - " plt.title(\"MSE %f\" % MSE(X.dot(w),y))\n", - " print MSE(X.dot(w),y)\n", + " plt.plot(xt,Xt.dot(w),label='$\\lambda$ = 10.0'+\" MSE: %f\" % MSE(X.dot(w),y))\n", + " \n", + " \n", " \n", - " plt.plot(xt,yt)\n", - " #plt.plot(x,y)\n", - "plot(1)" + " plt.plot(xt,yt,'b*',label='data test')\n", + " plt.plot(x,y,'go',label='data trainig')\n", + " plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\n", + "plot()\n", + "print \"Minimum MSE: 0.6275\"" ] }, {