Lez2LinearRegression.ipynb

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      "collapsed_sections": [
        "v_gNEXhIcJJK"
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      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
    }
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  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Regressione Lineare\n",
        "by Elia Cellini\n",
        "\n",
        "La scorsa lezione abbiamo introdotto il machine learning, una tecnica numerica che permette di approssiamare funzioni target $f(x)$ estrapolando la \"conoscenza\" da set di dati, e che si basa su tre ingredienti principali:\n",
        "\n",
        "*   Il dataset $X$\n",
        "*   Lo spazio delle ipotesi, uno spazio di funzioni dipendenti da un set di parametri $W$.\n",
        "*   Algoritmo di learning\n",
        "\n",
        "Durante la procedura di \"addestramento\", l'algoritmo di learning esplora lo spazio delle ipotesi testandolo sul dataset; se la procedura va a buon fine, l'output del programma sarà una funzione $g_W$ che approssima al meglio la funzione target $f$:\n",
        "$$g_W(x)\\simeq f(x)$$\n",
        "\n",
        "In questo laboratorio affronteremo il primo modello di machine learning di questo minicorso: la **regressione lineare**. \n",
        "\n",
        "Consideriamo un dataset $X$ di N campioni $x_i\\in \\mathbb{R}^d$:\n",
        "$$x_i=[x_i^1,x_i^2,...x_i^d]$$\n",
        "e\n",
        "$$X=\\begin{pmatrix}\n",
        "x_1^1 & ... & x_1^d\\\\\n",
        "... & ... & ... \\\\\n",
        "x_i^1 & ... & x_i^d \\\\\n",
        "... & ... & ... \\\\\n",
        "x_N^1 & ... & x_N^d\n",
        "\\end{pmatrix}$$\n"
      ],
      "metadata": {
        "id": "FBKFumVG3Pct"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Testeremo questo modello su un semplice dataset lineare con $d=1$ in presenza di un rumore gaussiano, la funzione target dunque è una retta:\n",
        "$$y=aX+b+\\epsilon$$\n",
        "dove $y, b\\in \\mathbb{R}$, e $\\epsilon$ rappresenta un rumore Gaussiano."
      ],
      "metadata": {
        "id": "EAyAU0Pf46ko"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "-kmwczMaxC54"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Funione per generare dei dati distributi su una retta y=b*x+a con rumore\n",
        "def mk_data(minR=-5,maxR=5, step=0.1, sigma=1):\n",
        "    b = -5.0\n",
        "    a = 3.0\n",
        "    x = np.arange(minR,maxR,step)\n",
        "    y = a*x + b + np.random.normal(0,sigma, len(x))\n",
        "    x = x + np.random.normal(0,sigma, len(x))\n",
        "    return  x, y\n",
        "\n",
        "#Questa funzione aggiunge degli \"outliner\" (dei punti sbagliati) ai dati\n",
        "def add_outliers(y,inc = 20.0):\n",
        "    idx = np.argsort(y)\n",
        "    y[idx[:2]] += inc\n",
        "    y[idx[-2:]] -= inc\n",
        "    return y\n",
        "#Funzione per plottare \n",
        "def plotfit(X,y,W,title):\n",
        "    p= plt.plot (X[:,1], y, 'bx')\n",
        "    p= plt.plot (X[:,1], X.dot(W), 'r-')\n",
        "    plt.title(title)\n",
        "    plt.show()"
      ],
      "metadata": {
        "id": "VXq5hdbFxJ6O"
      },
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Generiamo learning e test set:"
      ],
      "metadata": {
        "id": "qV134wRG5Y8f"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "s = 1.0\n",
        "XL, yL = mk_data() # training sets\n",
        "XT, yT = mk_data(minR=5,maxR=15)  # test sets"
      ],
      "metadata": {
        "id": "0gGNsQJA5UIe"
      },
      "execution_count": 4,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "XL.shape"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "2Cy57WByqHjn",
        "outputId": "19a69b8e-8db6-485c-848c-85de36ae01de"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(100,)"
            ]
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "yL.shape"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9HPQcEFDqI73",
        "outputId": "d0151c73-2866-498f-e3fb-6844d2039327"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(100,)"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(XL,yL,'bx',label='learning set');\n",
        "plt.plot(XT,yT,'rx',label='test set')\n",
        "plt.legend()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "_bc_7i8jxLrX",
        "outputId": "e4624646-90c2-45d3-cb5e-000cd4a656ce"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f7f086d0fd0>"
            ]
          },
          "metadata": {},
          "execution_count": 7
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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kSd03Bigr00YgCxbEzu9NrwzCHdBtaioqCz4VMpEn/ymAb4jIZAAXALjKGDMNwCoA94rIXwD4EMB3MnAvQki+YzNTenpUkN2ZKck6te3aBWzdqqtXGxpU5Nev15z4zs7+rT4tlnz3fjJgkY/+WrAFzwdFNwHwDQBPRMcfBjB7oPcihBQQ7jIEe/YA+/fr/v79aq27xd6Kv40RVlQATzwBfPaZM9fChelb4wP5VVEkZGTFqzGmzBjzGoDDAJ4DsB/AURHpjZ5yAMA5mbgXISTPseIOqKj+5Ce6ovQnP9H9sWO1gXZ5dJmO2/1y2WVARwfwhS8AH38MfP45MHMmMGlS/6zxRPnupUKQs74/G4BhALoBXApgn2v8iwB+HXBNI4DdAHa767YTQgoYv5WplZVO8DPRatSuLq0QCYiUl2vuvLvXap6vPg0D5Kp2jYgcjYr8JQCGGWPsitpRAA4GXPOgiNSISM2ZZ56ZycchhISFO9hpjFrgxjjulkWL/IOh3d3Atdeqm6ahAaiu1vcnTsTOXWrW+ADIRHbNmcaYYdH3lQBmAvgPqNh/M3ratwE8NdB7EUIKhO5u4P77gaoqdcsMG6avVVU6vnatbg0Nse6XtWs16LpmDfDII9pjtarKKTVgiUQSl0ggDkEmfqobgEkA9gD4FYBfA2iNjv85gFcA7APwOIAhyeZinjwhOSQbS/tXrXJcMTa/valJXS+2YcesWdrAo6kp1nXT0RHbqzVTz1QCIJsFykTkVwCm+Iy/DeCigc5PCMkSNkBqA5PeAmD9nbO+Xt0wixbpnA8/rC6Z3l7HIv/GN3TfZrtcfbW6dLZujc+csQXESL/I+yqUhJAs4e5+1NSkbpOBLvePRFSo580Djh7VOf2Ee8oUZ8z65uvqKOZZoCCahhBCskQkAkye7B8ATbRoCdDj3lRGW1I40QpTd4OQ7m4tGVxZCeze7dzXb95kz0N8ocgTUgoECeeCBSqulZUqtu7eqcnKBuzfr/1S3d2c5szRVauJctrtL4jZs9W1IwJs26ZB1nnzNECbSpcokhpBzvowNgZeCckSfjXeTztNy/ja8rve3qmpzGnnaGnR16oqZ06/+7qpq5O4/qk2yFoCRcUyCVhPnhASJ5ze6o62aXVdXXpz9qdiZCoiXiBNtPMBijwhRPETTnfao3tFaippi11dar0D+uq2xL3n2flS6RxFSz4tEok8ffKElApB1RjLy7WWTHOzFvFqbo6tLZNovjlz9LyWFn21+4l86snqybCoWEZhCiUhpUCixtW9vbrCtK3NSXtcs0bHE9HZqUFT27UpEtFg6t69iVMz/VaqunPhi6mJdh5AkSekFPATzrlzVagfeEDHjh7VtMe/+IvYPHbASY10C/SXvxzbli8S0X0rxjaNsqUlPXFO9iVA0oI9XgkpZmyvUz/Bdq94BdQKt9b7oEGa0uhdCZtuHfegRVaJnos1adKGPV4JKVWSBTm7ujTlsbIyPp2yokKkoSH+/GQB2XQCqywfnBHA7BpCCohMFw5LlqkSlK9+4YU63tAQO08yIU71+ZlBkzEo8oQUEn5WbnV16tUZ/US2oSFeyN338gqtHW9o0IqRXos+UzAXPiNQ5AkpNKywW3F1l+N1rwj1E13vsY4OR6jdXxb2PO+c9l72evsFYS36TP6NtOQzAkWekELEK65usU4mil5L3Aq7ncMKu1fQu7pErr7a36K3i50yAX3yGYUiT0guyKQvPchdEuR28cO6QrwWuG3OkcyCzqYQZ6NhSQlDkSckF2RKFL3XWWGfOTN190YyV0giX7gVYLcQWwvfW+6AwpwXUOQJyRWZ8DN7xXXECBV420LPPZ6KTz5oP+gZg873c+3QxZIXUOQJySWZyhhxi2iQ/zzV7BpvCd9kQp0s64bB0ryCIk9IrsikCGar0XaqcwZ9WblLEtOvnhdQ5AnJBQP1yfsJcGOjbt77ZFtIU7Hkhw6NbTBC901oJBJ5FigjJFMMtHqiu5aMrRnz2GNa6XH+fJ2nvFyrRdp6M5ms92LryQCxNW0OHdL95mbn3rZg2Jw5WvNm4cLMNAInmSdI/cPYaMmTksfPgg7KefeznAfi4rHz2e5O7vm9+fPua/zKIpCcgmy6awB8EUA3gDcB/AbAwuj4GQCeA/BW9PX0ZHNR5AkRf1+4FVKbL2/dJX6unGQuo1QCs6nGFBiIzQuyLfJnA7gw+v5UAP8bwDgA7QCWRceXAViVbC6KPCl5gix5d5NtmzdfWZk4hTLdFEm7n2p2EFet5g1ZFfm4CYGnAMwEsBfA2eJ8EexNdi1FnpQ0fqJpg5uNjeqmqa7Wf7ZDhuh7ryVvSSbUiQKr7po5idI1uWo1b8iZyAMYA+D/ADgNwFHXuHHve65pBLAbwO7Ro0dn/9MgJB9INZPGjllrfsgQ/Wd74YW6b2vAu+d0C7j9gvDD+0XgXfTkfaWFnrfkROQBnALgVQBzo/tHPcc/TDYHLXlSMvTH1TFrlv6TnT7dCcC6LWe35W/F3pvmKBK7sMpa8h0duSlMRrJC1kUewCAAOwAsco3RXUNIItIJWrpF1x2A9fs1MHRovE/f7UJxV6L027ew1nvBkEjkB5wnb4wxAP4ngP8QkbWuQ08D+DaAe6KvTw30XoQUFek0u+7pcfLUW1o0J725OT4H/4EHgLPOip/TfU5vL7Bmjc519KjOtWaN098V0Pz79eude7GRduESpP6pbgAuBSAAfgXgteh2DYDhAJ6HplDuBHBGsrloyZOSoj+WfH9rzvgRZKkza6bgAMsaEJJnpCukqWSypDNnoi8DZs0UHIlE/gth/HogpKhob1f3hpvubh0PIlEJhP6S6pzd3U7ZghUr9HXePOdvWLo03jUTiWSmdALJPUHqH8ZGS54UJLlwb9iAqjeHPig9MhG01IsOJLDkjR7PD2pqamT37t1hPwYh6WOt46amzBTqssXC7Bzd3cC11wInTgDf+x6wbp0WLnvyydj7eK+z12aqiBnJS4wxr4pIjd8xumsIyQTuTJmmpoFnotiKlG43UFmZCvvKlcDx48Dy5fFivn9/7HX2y8dWlyQlB0WekEzgTTn0+uiDCPLnW//6vHlAa6u+Ll+uQg8AX/gCcNdd8WI+f378dSz/W9oE+XHC2OiTJwXJQHzyqRYLa2hwVq/acgXV1c6+935cyFRSgCmUhGSRgQYyrbDX1cUHVysqtE6NLUhmV7A2NqrAX3hhvJiz/G/JkUjkGXglJB9obVVfe2UlsG2bjs2ZA3z+OTBoEHDppcCuXeqyaWsD5s5V183DDwOLFzsrYPfuBTZvju0uRZdN0cPAKyH5htsXb/35F16o2TOzZwN33+0EWW+8EfjlL4HrrlOf/6ZNwNixwIYNenzFChX4JUuAgwczn39PChr2eCUk17S3a69Wd9/UG24AfvQjHf/0U+D554GGBqenqrceTU9PcP0Zv4VMtOJLFoo8Iblm/35t0N3aqqL9ta+pVT5rlorx4sXAmDHAo4+qcANO5k5HBzBsmJPzfvSoin9dHbBoUWh/EslfKPKE5Jr584HOTnWzTJ0KPPccMHiwHluxQi34jRv1tbVVXTjPPKNfAMOGqVsGAKZMAe6/H6iqUsu+u5sWO4mDIk9IOiRaUQqktto0EtGVqlddpcHU8nINuAIaaH36abXa160DPvsMuPJKZ05rrd95J2CMfjls3apjDLASHxh4JSQd3CtR29uBtWudFaW1tZoRs2CBnptotemePSrggAp1a6sKvjEacAX0taoq3g2zaJG6eD7+GLj9dsfnzgAr8SMotzKMjXnypCBwd2lyd1Syee3V1Yk7M9nzqqqchU1Dh4rU12v+u3shk1++PfPgiQdwMRQhGcC96MkK8cyZsaLe0SFSWRkr0l4htguZ3IuebMPtZALOhh7EB4o8IZnACqptgm0t+ZkzY0V96FAV+qqq+CbaIsErZBsbkws4ywQTHxKJPFe8EpIOa9dqdssttwDbt2t++4YNuv/UU+pT37JF/fErV6pPfevW1IKhLBNM+glXvBKSKXp7NXVx40bg6quBxx/XXPbKSuCrX9Vg6Z49Tl57ebmmS6YCOzKRLECRJ6VJf1r2AWpp79+vov7447pidcoUrRdzzz1aW8aWHlixQlMlN29OvfQwIRmGIk9KE29TjlSaa9juTK2tWkRs8GAV9Pp6LRgWiail73bPMLWRhAx98qR0Sbdln605Y+vJWL/7oEHAjh1chERCI5FPPiMrXo0xDwGoB3BYRCZEx84A8BiAMQDeBTBPRD7MxP1IaZOx+KS7ZZ8t/JUIO/mUKbro6bPPNLBazoXjJH/JlLvmxwCu8owtA/C8iHwFwPPRfUIGTH88Lb4kaNmX1GX/2We64nTxYvW7e/uxpjwRIVkmKLcy3Q1qsf/atb8XwNnR92cD2JtsDubJk1QZ8KLPJIuKEh5ubNRc+ESrWlO8DyGZALlYDOUj8kdd741733NdI4DdAHaPHj066x8GKR6StjFNtHAohUVFvl8k/RFtliEgWSZ0kY/uf5hsDlryJFVS0s1+WtF+1QsaGqL6398Vp2ysTbJIWCJPdw3JCsm0O0aHowffaWiRP1X7C7xXt21lgvp6/zpkaes8LXmSZRKJfDbz5J8G8O3o+28DeCqL9yIlRE+Pk+1o45fuVPTyck1d7+4GEIng3aubMGbjShye2+SbQeMN5AKa7r5tmy5q3b7d6bTX3Z1m4NfdSHvFCn0NCtISkg2C1D+dDcA/A/g9gM8BHADwHQDDoVk1bwHYCeCMZPPQkifpEmTV2xpiDzV0yRGjlrw9cdUqp+CjZdYskUGDROrq9LTGxti6Y3Zua62nbJyzoBjJAWAVSlLMBAnuQw1dchgj5KGG2G+A/9XRJUOHOgUiu7q0WrAWntF5OjrURdPQECzidLOTfIEiT4oer+B2dYn8oGqVPNTQFSvSUSvaWxG4qkqFvqpKZPDg+F4gXqGnm53kExR5UtR4Bde6apIl1dgvBkDFvavLGRs8OF7Uva4apr6TfCGRyLNAGSlo/OKara1aHNLGWHt6dN9dI2ztWmD1ai0mOWiQjrkrBFdWxlYIdlf8dQd+7THWICN5S5D6h7HRki9NvLHJVavUGve2RfWLVaYS1/Ra2tbfbq1365NP5qIhJF8BLXmST3jLudTWar2vBQt0v7xcmy/t2qXnuVMUvWVfUumzYS3tefPUym9pAWbNcioCRyLAzTfrWG9v7DW0zknBE6T+YWy05EsDP5/2aadpILSuTt93dDiB0erq2B7X1lJPZMH7HW9oEGbDkKIEtORJPuG1rOfN00KOt98OPP+802Hv9tu10OOf/qSv1vceiSRfkOQ9vnYt8OijQENDXMFJQoqbIPUPY6MlX1q40x7dGTKnneakM1ZWigwZ4m+BJ0tjtMetBd/UFDvu9fsTUqiAKZQk3/CK+tChsYFRm9rY0KDHq6qcxUtuki1IssdnzowPvtrAKyGFTiKRZ0sbknPcaY+RCHDokKYr2u3gQaC6Ghg7Vl0sF18MTJqk17qv8/b8sEFU933cx5ubY7v9uVuxElK0BKl/GBst+dIgKGja2KjWtU1lbGxUV40xWhHS3ZvDrlhtbHSuT7RAyeu6YfCVFBNg4JXkC4n6sz7wAHDLLboQqaUF2LgRqKjQCpAjR6oVDmh6ZGenOnTmz9cxb8qj34Kl5mbgX/7Ft9sfIcVLkPqHsdGSL35SKQngLjdgLW67QMpbviDVwClLEZBiBrTkSb7gTp+8/HJdBDV3rnPclhsYMkTLDaxb59Rwb2vT+u4rV+prW1vqzbtZioCULEHqH8ZGS750sNZ6ZaVjoTc1SV9xMLsgyqZTNjY65QimT48tQUBIqQNa8iSfcGe9DB4M3HWXWubr1wPnnAN89hmwfDmwaJG+njihGTdtbeqz37VLX22nJkJIMBR5khRvrRkgvoZMqnirRm7ZoitcN24Epk9XMW9oUAFvbdXXZ54BLrtMA6fbt+uXw/bt8ZUlCSHxME+eJMWWCHDnp9v9dPH6xgEtSDZlCvDSSyrw27c7vveWFudc9zNEIv1/BkJKCVryJCl+tWa8Qp0K3vTJ7m4NvF52GfDee5oquX07cMMN8XVmGDglpDczsbwAABD8SURBVH/QkicpEYnoSlGvdZ0O3l8ENtd95MjYXwk/+pEKfm+v5sxfe62WArblg63o+5UZJoTEQkuepIS3REB/Ap7eXwSbN2v1yQcecMR65EgNxk6ZoiI+dqxWoTx40HkOd7VJILMxA0KKjqC0mzA2plDmJ5leSJSsqFhQz9Zk1Sa50ImUKgizCiWAqwDsBbAPwLJE51Lk85NUWuylSrLywBbvF0G6XwwUeFJKhCbyAMoA7Afw5wAGA3gdwLig8yny4dBfEU/3ulQt7nQteUuyLwJCipUwRf4SADtc+80AmoPOp8iHQ3/dHele19+m26k0/KAlT0qZMEX+mwD+ybXfAOCHnnMaAewGsHv06NHZ/ixIAP0VyUyLq/eLYNUqFXh3gw9vww/65Empk0jkQ8+uEZEHRaRGRGrOPPPMsB+nZHGnSDY1BacmejNZIhFn4VKi61LFmxa5dCkwZozOb7Ny2tp0P1FZYebQE6JkW+QPAviia39UdIzkGammSIbRINtbgXLKlNgKlLW18YIeiTh59YSUNEEmfiY26GKrtwGcCyfwOj7ofPrkwyFdd4e7y5K7GmS6bpJ0Are2AuXEif4+erpmSCmDsNw1ItIL4O8A7ADwHwA2ichvsnlPkj5ud4ddQOR2d3gXFkUiwOTJWlTsllu0WmR7O7Bnj9aGD7rOi/dXgd9CJztuK1C+8QYwcyawYQNw6639L7FASMkQpP5hbLTkwycVq972V62s1NeuLsfSdlv1VVXxNd+DsmkSBW79ukLNnClMlyQkCsJcDJXORpHPDxIJr1v0u7q0qUdVldPkwy+/PZkbKNWFTu7sGmPUXURXDSEUedIPgoTX60e359XV+V+XzFJP1ZL3flHYPHn65AmhyJM0STX3PZWVqatWqcXtFX4/gU5FsAdaYiGTJRoIyRco8iRlcUu39IDXheLNtGlqinWtuF04YQguF06RYoQiT1IWN78Vp34lBK6+Ovl5HR3ahNsKuzflMixYAoEUGxR5IiL9E7eBWL7uLwzrq29oyA/XCIuZkWKCIk/66I+4ub8crGUuomLd2BhrwXd16Vg+Fw/Lt+chZKBQ5ImI9F/c3MFTt2+9sVFFH9B9mz9/2mn5Wzws356HkEyQSORDL1BGcoNdTbppE7BihdOGL5VaM+XlTn0a22h7yRLg44+BsjKguhr4/veB+nrt2frkk87q2c7O+BWp7lWxuYbFzEjJEaT+YWy05LNHfzNZ3Hnp7uCpe8WpdQF53UC0mgnJDaC7hljSFXu/4OnMmbrKtaVFXTPV1VriwK589VshS/83IdkjkcjTXVNipFoUzGLru9tSxA0NwM6dWvI3EgFOnAD+9Cfgv/03YOtW4LPPgGuvdebPdL15Qkh6UORLDOuDtg04Uqni6PbnT5gArFmjVSE7O4GbbwY6OoDeXp3j7rv1fWenXpuLevOEkAQEmfhhbHTXpI5NX/S6Rrzpi0Gkk0rZ34bdA603TwhJDdAnX3zYCpC21K93P9m12faTuxc/ee+dD4uhCCkmEok83TV5jrenKqD7PT2aqiiiqYuzZgHGAFu26DG/a665Rt0n7lTK5uZYH3omcLcS3L49vics2/IRkjso8nlOokBpJAIsXAgcP64567ffrmPeaxYsAGbPBi6/XIW3udkZb2tTsc9UnvhA8vEJIVkgyMQPY6O7xp8g94q7aYft0mT99O5rqqpEKir8G30kK1Bm78NSvoTkL6BPvvDxa8bh55OvrnbG6ur0GtuGzwZDBw2SwKArFzARUngkEvnysH9JkOS4fdzr16tLpqcHmD9fN5v++OSTTurinDnAJ5/oe2OAKVOA0aO1+faQIc5cw4ZpyqP1k7tTLJua9Bw2yiakcKHI5zHt7Vo3pq3NEdphwzRQ+rOfOcLc3u746O3CpYcfBj79VPPTn3pKFyR9+qmeX17uzLVkiea9u4lEVOBXrtQvAwo8IYULA695TG2tEyi14u0XKPUGWteu1ZWntqDYxRerwI8cqQuXyso0I+euu1Tge3tj7+v95cCgKSEFTJAfJ5UNwA0AfgPgJIAaz7FmAPsA7AVwZSrz0ScfT7r9Vr0LkDo6pK/ejL3e23zbb55c9l0lhAwMZCvwCuB8AGMBvOAWeQDjALwOYAiAcwHsB1CWbD6KvD+prk71LkByV5C0QmyDtUFfGv0RbAZrCQmXrIl83yTxIt8MoNm1vwPAJcnmocjHk64lb1Mm3cJujzc2OmmW7msyIcasNklIeCQS+Wz55M8B8J5r/0B0LA5jTKMxZrcxZveRI0ey9DiFSaoLi7znrVypAVUbYLXHAV0R+8AD+j6TDTPcwVpWmyQkf0iaXWOM2Qngz3wO3SkiTw30AUTkQQAPAkBNTY0MdL5iwq+L0dy5mibpFtHOTh23Y4sW6WtrK3D0aOI0SJuRM1D80jwp9ISET1JLXkQuF5EJPlsigT8I4Iuu/VHRMZIAb50ad4qkZf58YPPm2DIHmzfruJveXhV+t2Xd3R07V6ZgKQNC8pdsuWueBjDfGDPEGHMugK8AeCVL9yoabCrkggUqkO46NVagU60H7+7Lun69U5gsqDnIQGDfVELymCBnfSobgDlQf/unAA4B2OE6dic0q2YvgKtTma+QAq/ZShvs6tLgqK1F465D476fN+PGG2QdMUKkqUnLHHjTKgkhxQWyFXgVkS0iMkpEhojIWSJypevY3SLyZREZKyLbB3KffMSvOmR9vVrQbtJ1kUQiWk3y4491QZPbFWItZb/FSu7n6enRBVSPP64um40bgVtuiV/0RAgpAYLUP4ytkCx5kfi0QVsEbCALiawlX1enqZDe/PhEOelBz8O0RkKKG7AKZfbwqw6ZjrB6RXroUF2w1NHhXxI4mZvIVp5saIidt7GRQk9IsUKRzxJBgp5O/1T3PHV1jsCnK9CrVul11p9fUaE+efe1LDVASHFCkc8CQW6T/rpI3F8MbmvdvncLtJ9Yd3Q4wdWuLg24AvpK652Q4iaRyLMKZT/xSxtsbgaWLdNXd7742rWJg6/eQKotGww4AVVA8+bdaZVuenu1omRbm55z8qSOT5vGRUmElDRB6h/GVkiWvB/WZeK24Ds61K+erOZMomBtOn5++4ugqooBV0JKBdCSzw1Ll2pJAfdipbY2YOtWx5r2rmq16Y524ZDfQqJU68J0dwP33w9UVTl1a7j6lJDShiKfBRKJsje/vrZWvwjc7pdIxClpAKTWxMO6cebP1y+VJ5903DxcfUpICRNk4oexFbq7xuJ2rwwapFkubpqadDyVmu5+NeEzVQeeEFIcgNk12cedBeNOWZw2TT9lK/Q2C2bmTAlMs3TP4efnp3gTQtwkEnmjx/ODmpoa2b17d9iP0S+su2TuXN0fO9ZpwP344+pmOf10Lf1722061tQErFsH3HijU+PdO19TU+JSwYQQYox5VURq/I7RJ58hbJBz82atO7NkidOA+4YbAGOADz/U15/+VM8dNgz45BPgscfi/exswkEIyQQU+QHizpaxwrxxIzBlilryra1auExEM15OnlRh/9GP9Ivg7/9euzV5A6OpBFsJISQpQX6cMLZC9Mn71Z6xZYIbGtTvDjgrUW3RMXfD7URz+u0TQogbME8+e1g3zezZjsW+bZta8I8+CowcqXnrU6bouddfr9edfjqwfbu/hc4mHISQTEGRj+JdpASkXgs+EtE89+PHgYULdaytTcsMLFyoeevz5gHf/a7TramsTH32fguVli6N98F7c+cJISQVKPJR/JqAXHttfBOQBQt0c7N2LfDyy0Bdna447exUy3vRIp3Xrmr9p39S4X/kET3e1ha72pUQQjJNUYj8QKxwi1/v1BUrnIJfds7OzthsmLVrNYC6ciXw5S8DJ07oOfb8OXOA/fu1gNiOHSr87vv19tJCJ4RkkSBnfRhbfwOvmQxUptIExJbybWjQV9s7taND67hXV2tteNsAhAFTQkg2QSmseE23I1M6c/g1AbGZMzZDxl1+oLJSj1VWUuAJIdknkciXJ7P0CwX34qGWlvQXD3kbZkciut/cHJuvbufdvl0DqI8+6uxv2qTvjYl9JYSQ0AhS/zC2bFjyqRbu8jvPWwu+q0vdL0OHOmNui97do7WlJf7cVGChMUJIuiBb7hoAqwH8FsCvAGwBMMx1rBnAPgB7AVyZynzZ8MkPxF/vJ7iNjbq552po0C+DWbNiffD2S8GeP9C/hRBC/MimyF8BoDz6fhWAVdH34wC8DmAIgHMB7AdQlmy+/op8Mus3E/56L35iXFXlBGH9niPdudnZiRCSClkT+ZiJgDkAfiKOFd/sOrYDwCXJ5shmWQO/4OlAyLZbJdPPSwgpXhKJfCbz5P8rgO3R9+cAeM917EB0LBSyUewrm6tSWZyMEJIpkmbXGGN2Avgzn0N3ishT0XPuBNAL4CfpPoAxphFAIwCMHj063cuTEpQ1k6/12QvteQkh+U1SS15ELheRCT6bFfj/AqAewM3Rnw0AcBDAF13TjIqO+c3/oIjUiEjNmWeeOaA/xo9MFvvKxMraZLA4GSEkkwyoM5Qx5ioAawF8XUSOuMbHA/gpgIsAjATwPICviMiJRPOl2xmqvV1rw7gt3O5uFcRslArwWtnefUIICYNsdob6IYBTATxnjHnNGLMBAETkNwA2AXgTwM8B/G0yge8PfkXF5s3T8WzgV9+GAk8IyWcKvsdrGL1QW1udlbUrVmT3XoQQkoyi7vGa616ozHwhhBQSBS/yuRRdtw9+xQrHdUOhJ4TkKwUt8rkWXWa+EEIKjYL2yec6u4YQQvKRRD75ghZ5QgghRR54JYQQEgxFnhBCihiKPCGEFDEUeUIIKWIo8oQQUsTkVXaNMeYIgN+F/RweRgB4P+yHCBl+BvwMAH4Glnz8HL4kIr5lfPNK5PMRY8zuoNSkUoGfAT8DgJ+BpdA+B7prCCGkiKHIE0JIEUORT86DYT9AHsDPgJ8BwM/AUlCfA33yhBBSxNCSJ4SQIoYiTwghRQxFPgWMMT8wxhyM9rF9zRhzTdjPlCuMMVcZY/YaY/YZY5aF/TxhYIx51xjzRvS/fUmUSTXGPGSMOWyM+bVr7AxjzHPGmLeir6eH+YzZJuAzKDgtoMinzr0ickF0eybsh8kFxpgyAP8dwNUAxgG4yRgzLtynCo1I9L99weRHD5AfA7jKM7YMwPMi8hUAz0f3i5kfI/4zAApMCyjyJBEXAdgnIm+LyGcAOgFcF/IzkRwgIi8C+H+e4esAPBx9/zCA2Tl9qBwT8BkUHBT51Pk7Y8yvoj/hivpnqotzALzn2j8QHSs1BMCzxphXjTGNYT9MiJwlIr+Pvv+/AM4K82FCpKC0gCIfxRiz0xjza5/tOgDrAXwZwAUAfg+gI9SHJbnmUhG5EOq2+ltjzGVhP1DYiOZel2L+dcFpQXnYD5AviMjlqZxnjPkfALZm+XHyhYMAvujaHxUdKylE5GD09bAxZgvUjfViuE8VCoeMMWeLyO+NMWcDOBz2A+UaETlk3xeKFtCST4Ho/9CWOQB+HXRukdED4CvGmHONMYMBzAfwdMjPlFOMMdXGmFPtewBXoHT++3t5GsC3o++/DeCpEJ8lFApRC2jJp0a7MeYC6M/TdwEsCPdxcoOI9Bpj/g7ADgBlAB4Skd+E/Fi55iwAW4wxgP57+amI/DzcR8o+xph/BjADwAhjzAEAywHcA2CTMeY70JLg88J7wuwT8BnMKDQtYFkDQggpYuiuIYSQIoYiTwghRQxFnhBCihiKPCGEFDEUeUIIKWIo8oQQUsRQ5AkhpIj5/0d2GoUK94peAAAAAElFTkSuQmCC\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Lo spazio delle ipotesi della regressione lineare è un prodotto:\n",
        "$$\\hat{y}=g_w(x)=XW+b$$\n",
        "dove $W \\in \\mathbb{R}^d$, $b\\in \\mathbb{R}$ e $\\hat{y}\\in \\mathbb{R}$.\n",
        "Dunque:\n",
        "$$\\hat{y}=\\begin{pmatrix}\n",
        "\\hat{y}_1 \\\\\n",
        "...  \\\\\n",
        "\\hat{y}_N\n",
        "\\end{pmatrix}=\\begin{pmatrix}\n",
        "x_1^1 & ... & x_1^d\\\\\n",
        "... & ... & ... \\\\\n",
        "x_i^1 & ... & x_i^d \\\\\n",
        "... & ... & ... \\\\\n",
        "x_N^1 & ... & x_N^d\n",
        "\\end{pmatrix} \\begin{pmatrix}\n",
        "w_1 \\\\\n",
        "...  \\\\\n",
        "w_d\n",
        "\\end{pmatrix}+b$$"
      ],
      "metadata": {
        "id": "YV1wspJdHDIS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Prima di addestrare il modello, è comodo \"assorbire\" il termine noto $b$ nei parametri $w$ aggiungendo una dimensione ad $x$ con valore fissato a $1$. Usando questa trasformazione:\n",
        "$$X=\\begin{pmatrix}\n",
        "1 & x_1 \\\\\n",
        "1 & ... \\\\\n",
        "1 & x_i \\\\\n",
        "1 & ... \\\\\n",
        "1 & x_N \n",
        "\\end{pmatrix}$$\n",
        "e\n",
        "$W \\in \\mathbb{R}^{2}$:\n",
        "$$W=[b,m]^T=\\begin{pmatrix}\n",
        "b \\\\\n",
        "w_1  \\\\\n",
        "\\end{pmatrix}$$\n",
        "Dunque:\n",
        "$$\\hat{y}=XW$$\n"
      ],
      "metadata": {
        "id": "q1ARhAH78cG2"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "Xl=np.c_[np.ones(len(XL)),XL]\n",
        "Xt=np.c_[np.ones(len(XT)),XT]\n",
        "yL=yL.reshape(len(XL), 1)\n",
        "yT=yT.reshape(len(XL), 1)"
      ],
      "metadata": {
        "id": "qyZziZ0m-icp"
      },
      "execution_count": 10,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "La loss function che minimizzeremo per addestrare il nostro algoritmo lineare è l'errore quadratico medio (MSE):\n",
        "$$E(W)=L_{MSE}=\\frac{1}{N}\\sum_i^N(y_i-g_W(x_i))^2$$\n",
        "In questo seplice caso, è possibile trovare una soluzione analitica:\n",
        "$$w^*=(X^TX)^{-1}X^Ty$$\n",
        "dove $X$ rappresenta il dataset, per noi Xl.\n"
      ],
      "metadata": {
        "id": "D46seK_K9BeO"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def NormReg(X,y):\n",
        "    return np.linalg.inv(X.T.dot(X)).dot(X.T).dot(y)"
      ],
      "metadata": {
        "id": "4G_JfJa15RMK"
      },
      "execution_count": 11,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "Wnorm=NormReg(Xl,yL)\n",
        "print(Wnorm)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "iKTZBytoqh5Y",
        "outputId": "ae1a5762-9555-4eb0-d98b-b8775eb2bc7a"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[-4.78191404]\n",
            " [ 2.78668638]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plotfit(Xl,yL,Wnorm, 'train set')\n",
        "plotfit(Xt,yT,Wnorm, 'test set')  "
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 545
        },
        "id": "VdRXvg5Cqe6j",
        "outputId": "b48fd169-7ded-43a5-dd50-08c440cc0485"
      },
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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Ba1O+pkjqmvlwMPGbyvPPxx+2rlwZ9O5T0MxvQFJcqoHe3YeBbwA3Ag8Di919WZrXFElb9GHgvHn5qy3Vq1x6I9Gbihm8972j22uvHfTi11qrgZNW1qxvQFJc6jl6d/+Nu3/Q3d/v7j9M+3oiaUvj4WCp9MbYsQndVG69Nd6LHxmBFSvqb3QNjjoKzjor/2al4mbNoxWmRNpEsdWnli5NYChnYYDfd1+44YbE2l3JwADsv3/QjCVLgn3RbfXuG6cVpkQ6RLHVp4oFwVL7Y849F445Jn9fCzp2S5fC1VcHP4c3MjPYdtv4sZmYj9CGWj68UkQCiefio0H+q1+tKcgnOX4/HB4ZvZEddxycfrpG4zRNufoIzX6p1o10q8Rqwhx2WFVFyJrWniLnjBYmU7GyZKAyxSLN0UgvOPqANzw++oC3qvOYwSWXjG5fcEHdqZqkh0SWGqkEGo3TFOXuAs1+qUcvnSypXnA158lbjKOwB19nL76YpCpGllo8ZMYM9eiTQKvLFNfyUqCXTpdUKqLSefr73Sdu/E4swN97xm/zjmtkdaa00ypppIe6lQK9SJMl1Qsue54ivfhiAbLeYNqMIKwlApOjQC/SRKn36F9+ORbgt+LxsjeVetqkINxZFOhFmiTtHH2xXny1AVyrM2VbpUCvUTciCUmqNELsPBvew9BL+bNbb73+NSZO8KpKI2h1JlEJBJF2Vli+AMC96lWutDpTd2j1ClMiUo+f/rR4EbJcx6zaxTiWLoUDD8w/ZvFiWLRIBcW6iQK9dK2mLdNXKzM44oj8fe7Fe/cVnHgiTJ8eT+tcdZVKDbTtv38KFOila7Xdykc/+lE8mIePXRughT+Ka7t//zSVe1Lb7JdG3UiztU2tlcIRNR/5SOKX0MibuLb5928QGnUjUlozVz4qlioY2vPA4r34P/850Wtr5E1x3bLylQK9dLVmBsDCVAFmTPzjktEDLrkklXrxaSx9mBXdcgNUoJeu1ewAGObKX9zv8OK9+N7eVK6bxtKHWdBNN0CNo5euVe1Y9MSMjMCYMfn77ryz4tO/prezS2Tp77XSOHoFepFm2HVXuOeevF3hzNYw0BQLPEcdBc89B7ffPtorX7gwSDVcd112c8pSm5ZNmDKzuWa23Mzuzb0+m9a1RNrWG28EaZpIkL/12lfBPZYqKDbcb9EiuO02mD07eO+ww2DWrODhoYK8VCvtxcH/w93PTPkaIu2pIA//3+9+H3f/55MsXQrvrBfPlYe59HAB7fPOy19Ue7vt4NJLg1T+8cc3+bNIR0s70It0n+XLYdKk/H1vv834sWMJO+GFk5ai2+Fwv76+0ff32y8I8nvuCTfcEPT21aOXaqU96uYbZna/mV1sZu8udoCZzTCzQTMbHBoaSrk5Iikzyw/y06YFI2rGjvapys1ULTbcb+FCuOyyoCf/8MOjaZxqRod00zR/KaPcbKpKL+AW4MEir6nApsAYgpvJD4GLK51PM2OlY/X3x2e3VlA4U7VYHfoNNnBfay33BQvyj1mwoLblALVcX7ZRYWZsU0bdmNlk4Dp337HccRp1Ix2pcEz8V74SVJ8sIxzDHebiw1x9sVE3ABdckP+7tQwBLHYtpX2ypdKom9Tq1gCbRX7+38CiSr+jHr20Sl1L5/3ylzX34sPzNruXrTo32UYLa93MN7MHzOx+oCcX7EXaUs2VDM3goINGt88+u+ryBc2eqdot0/yljHJ3gWa/1KOXVqqqkuGpp9bVi09TuW8jytF3B1S9UqQ6FSsZmsF3vjO6fccdqRQhq1W5byOqcyOgEggiq5V8aHnIIXDFFfkHt9H/N6AHrt1Oa8ZKJqQ9HrxYJcMvHTQc9OKjQf7pp9suyEP31FWX+ijQS1sLA3w0PTEwEAw7THLZt1iK42sf4O8vvyv/IHfYYotkLpgwPXCVclQCQdpaGOAXLw5eBxwAq1bBmmvCkiXJ9VxXj0n/xz9g/Pj8N197DdZbL5kLpSD6baSnJ3hpXViJUo9e2lq0XMDAQBDkV66E445LIYiZ5Qf5HXYIevFtHORBD1ylMgV6aap6cu3R/LNZdemJmq5zzz3x2a3Dw/Dgg2U/S7s48cT4Ta+np/MWz5D0KNBLU9U8MSl3zFlnwTrrBLXBCnv5tV4n7yZgFiwKEtptN3Bn4NYxKvwlmaFAL01VrnJjMWGAnj49WFHp6quDbSifnih3nd13hwv2vyHWi1+4wGFwsKqbj0hHKTebqtkvzYztHtXWXqmlBk2xY3t7i1ynYGbrT9Y6xhcsqGJWbAV11csRSQAVZsa2PLhHXwr03aGqUgMNnDc834IF7mZBsJ8wwf3RmQtiQT56E2i08JfKDUirKNBLW0k7GIbn6+0NgnxYx70wwD886yd5N5skevTR6yd9ExMpp1KgV45emirtoYDhCJ1LL4VDD4Xj7/pyLBc/0O/s+bOvMnt2MHJy9uxgwe3Zs0dnxVa7glOp62uGqrQT1bqRTFld8+VoZ94P8vsxnx9/K8cv2ZOlS4PRO6edNnqTGTs2GFEZDkmsdXGP2PVVc0aaqFKtG82MlcwIg+zQSwY/KHjTneOrXNWp0SCvGarSbpS6kaZoxiLV9/x5ZRDkI+78yf3MPyP41losrVLPuP5SNENV2la5BH6zX3oYm11JPoQtNowxthhIkQVBSj0o1QNU6XRo1I20i6QCat5N4tFH4wH+lVfK/06Rba2pKp2sUqBXjl6aJpo66eurP28dpkR6Pm3xN0sMLqiUVomW+A3z6yJZoRy9NE1iNdP7++NBfni47IIgpQp/RcsgNzq0shma8axDsqehQG9mB5nZMjMbMbMpBe/NNrPHzOwRM9unsWZKpyu2glNdAdUM9torb9fECUERsnp02gPUJB8eSxcpl9ep9AK2A7YFfg9MiezfHrgPGAdsBTwOjKl0PuXos6vhOjAXXxzLxXdrqQE9PJZCpJmjd/eHAaywljdMBRa5+1vAE2b2GLAH8JdGriedq9iY9Kpz4QX/fT20y5d4YeEvivbCuyG3ntSzDukeaeXoNweeiWw/m9sXY2YzzGzQzAaHhoZSao50pB/8IL4giDvb3/OLovn28GaS9Ty21oeVWlUM9GZ2i5k9WOQ1NYkGuPuF7j7F3adMnDgxiVNKFoRLSYUuu6zsw9aoLOexE3vWIV2lYurG3feu47zLgS0i25Ny+0TKmzoVrr02f1+VAT4UXXQkazVnyj08zsLnk3SkNY7+WuAXZrYQeC+wDXBnSteSLHCHNQq+YN5xB+yxR12ny2oeu6FnHdK1Gh1eeYCZPQt8BLjezG4EcPdlwGLgIeC3wDHu/k6jjZWMeve740HefXWQrye/rjy2yKiGAr27L3H3Se4+zt03dfd9Iu/90N3f7+7buvsNjTdVMmflyiAX/+qrq3f96cpng3HxDeTXlccWyaeZsZKYmka7mME66+Tvc+dj0zavafHwYqJ57PDa0UlQWRqBI1INBXpJTOFol6OOggMOyO+N/+mXz8WHTK5YkffAtdFVmqLlDsI2hfuzNAJHpFoqaiaJKRztcuWVBQNmzPhY4S8VGVFTmF9v5GFjlkfgiFRLPfoO0gkTgaK98eOOg6uvhu8deF+8Fz8yUjLIJ51f1zqu0u0U6DtIJ0wEivXGP238/tVdRg844YQgwMfLZgD1FRmrdAPUCBzpeuUK4TT7paJmlVUqaNVw8bAE2tbf7+5nnx0rQpZWAa5yi4okubKVSLtCK0xlT7nVkFoZ2FbfZAoC/LW7zU29LaVugK288Yk0iwJ9xlRTojbtMralgudfPnVSLMg3M8hqOUDpVgr0GVJLbz3NoFesHbF1W3/1q+QvXEWbVKNdulGlQK+HsR2k2geVaT98jA5Z/NtWn44v6+cOX/xishctI3wofeCB+W0bGGi/UUkirWDBzaA9TJkyxQcHB1vdjI4WHZ7Y0xPfTszICIwpWL7vnntgl12KH5+i+fNHRx6FnxVg0SK46iqNm5fsM7O73H1Kqfc1YSpjmlLGdo01YmPgB/qdnubH+NVBPvxsixcHs3GnTIH77lOQFwGNo8+c6PT/UHT1pYa8/now/j0a5P/+dwb6vWVFwwrnFgCsWgW/+50mR4mEFOilOmaw/vr5+9xh002rmtQUSnp2bzQnP2cO7L8/rLmmJkeJRCnQZ1giQfWpp+KzWN98M5a6qfZbQxqze6MlDoaHYckSlScWiVKgz7CGg6oZTJ48ur355kGAHzeu7jYV9sCTeFAcjjLaay8YG3nqVMs3DZFMKzf2stkvjaNPXl3jy//4x/i4+JGRRNuV1Dh/lTgQ0Tj6rldz5UYz+PjHR7dPOaVsEbJ6JDnOv54iaCJdp9xdoNkv9eiTV3WP/vTT4734FNujHrhIclCPvntVXdvdDE46afXm7z87n4F+j50riRmm6oGLNF9Dgd7MDjKzZWY2YmZTIvsnm9lKM7s39zq/8aZKrSoG1Z12iqdk3PFZ306t7n2q4/xFpKiGSiCY2XbACHABMMvdB3P7JwPXufuOtZxPJRCaqDDAX345HHLI6s0wuGv5PZH2V6kEQkM9end/2N0faeQc0mRmRXvxA5sdwlFHjaZnklh+rxOWPhTpBmnm6Lcys3vM7A9mtmepg8xshpkNmtng0NBQis3pcsPDsQC/77q3MdDvDAwEM0qvvHI0PZPEyJhOWPpQpCuUe1KbS+vcAjxY5DU1cszvgSmR7XHAxrmfdwOeATaodC2NuqlPxVWUCkfT5BYE2WAD93XWcV97bffx40uPhGlkZIzqxIukj0ZH3bj73u6+Y5HXNWV+5y13fzn3813A48AH67sVSSWles4f2faVeJpm+XJwp6cHvvlNWLECVq6E444L0jPz5wflfQtz8gceWN/ImCRSQCLSoHJ3gWpfxHv0E4ExuZ+3BpYDG1U6j3r09SvsORfrxUePnTGjeI8+6XHu6tGLpI80lxIEDgCeBd4CXgBuzO3/IrAMuBe4G/h8NedToG9MX5/7+3k0FuA32/itvMC9wQbu666bH9w32CAe7BsNzpocJdIclQK9VpjKiIEB4kv6TZsGV14ZGyp54IHB29Onj6Zrxo6FRx6B978/GNN+2GFw6aXBw9h58+prU+GiIGE7ly7VuHmRJFUaXplI6iapl3r09bnnzFtivfjCnnO5ImKFPe0FC9zN3Ht71QMX6QSoBEJ2zZ8PmLHLrL1X73tuvyOYf4bnzYCtNFQyWjr4sMNg1iw480y45BLVdBfJhHJ3gWa/1KOvwY9/HOvFF8uB15InD3v9vb35+/OGaopI26FCj16Lg3eigiGTJ637f1nz+G9wXpFFPKpdLLxYrz/6OxoWKdK5FOg7yZFHwkUX5e9zZ805wTj1vr54QC720LMwcEerXIbvJbHyk4i0B+XoO4VZfpBfujSoUZNAqQKVDhbJNg2vbHc77QQPPJC/L/dvVtgTL9wWke6QavVKiUusYmNYhCwa5B99dHWQB/XERaQ6CvQJq1SxsaobgRm86135B7kz8MwH8o4rtYhHeM6y1xCRrqFAn7DomPQ5c+KplLI3glfiRchuvf611bn4akv8qjywiOQpN/ay2a9GxtFXLNXbZNXMRM2rJVOilHAtNWeifwfh7/b2BnVtNLtVJLtIs6hZ0q9GAn07FdCqJkCHN4Kzjn4oHuSHh2PHFbthlLpueL3e3uIToEQkW7om0Lu3R0ncam444b5YgJ8ypei5+vqCXvmCBfFrFX5jifbkVa9GpDt0VaB3r60HnIZKKaT+fvd9x/85FuSL/U6xQmNhsC/3jaWwJ6/ywCLZ1lWBvh169BUV9uKPPbZoz7zYDWPBgmChkHKfr78/OKawJ696NSLZ1TWBvp1y9EX94hfxIF+HWsoNt93fgYikolKgz8zwyraePGQGhxwyuv3rX+dNfKpWpXIHbf13ICItoxIIaZqTqzYWVeffdyeVO9DKUiLNpRIIVUisbEHIPejFR4P8fffVHeShs3rrmrAl0mbK5XWa/WrVwiOJ5rbnzk0kF9/pOuLBuEhGoIVHKouWLQgX0K45JTI8HK9P8/LLsNFGiba1U/T0BH+Xperki0jzNJS6MbMfmdlfzex+M1tiZhtG3pttZo+Z2SNmtk/jTU1XNFle3eIAAAcASURBVDDNnFljYDrkkPwgv88+QV++S4M8VH5wLCLN02iO/mZgR3ffCfgvYDaAmW0PTAd2APYFzjWzMQ1eK1V1BabXXw9y8VdcMbrvzTfht79NrZ2dIPqgeN48LTAu0moNBXp3v8ndh3ObtwOTcj9PBRa5+1vu/gTwGLBHI9dKU12BaaedYP31R7e/9a2gFz9uXOrtbXed9OBYpBskmaM/Argy9/PmBIE/9GxuX4yZzQBmAGy55ZYJNqd61S6gDcBzz8HmBR9lZCRWXribVbNOrYg0T8VAb2a3AO8p8tbJ7n5N7piTgWHg8lob4O4XAhdCMI6+1t9PQtWBqTCYn3MOfP3rqbVLRCQJFQO9u+9d7n0z+wrwOWCv3DAfgOXAFpHDJuX2dab774edd87f10YTzUREyml01M2+wInAF9x9ReSta4HpZjbOzLYCtgHubORaLWOWH+Svu05BXkQ6SqM5+h8D44CbLUhr3O7uR7v7MjNbDDxEkNI5xt3fafBazXXTTcEwySgFeBHpQA0Fenf/QJn3fgj8sJHzt0xhLv6uu2DXXVvTFhGRBqnWTdRFF8WDvHtTg3zidXdEpOt1fKBPJDCGRciOPHJ031NPtSRVo4JgIpK0jg/0DQfGJUtgjchfw4c/HAT4Fo3pj9bdmTOnfUsRi0jn6PiiZnUXJHvnHRhb8PFXrIC1106trdVSQTARSVLH9+ihjoJk552XH+TPPTfoxbdBkAcVBBORZHV8jx7igbHkdPuVK2GddfL3DQ/DmPapt1a4clRPj9I3ItKYju/RV12Q7JRT8oP8VVcFvfg2CvKggmAikryOXzO24vqkr7wCG2+c/0sqQiYiGVJpzdiOD/Rl9fbCZZeNbt92G3z848mdX0SkDVQK9JnI0cc89RRMnjy6/b73wZNPtqo1IiIt1fE5+piZM/OD/LJlCvIi0tWyE+hffRW+9jU4//xgu6cneNi6/fatbZeISItlI9APDsIOO8DFFwdPYFesgP7+VrdKRKQtZCNHv/XWQaC/5hqYUvJ5hIhIV8pGoN9oo6B+vIiIxGQjdSMiIiUp0IuIZJwCvYhIxinQi4hknAK9iEjGKdCLiGScAr2ISMYp0IuIZFxblSk2syHgqRY2YQLwUguvnxR9jvaSlc8B2fksWfsc73P3iaUOaqtA32pmNliupnOn0OdoL1n5HJCdz9Jtn0OpGxGRjFOgFxHJOAX6fBe2ugEJ0edoL1n5HJCdz9JVn0M5ehGRjFOPXkQk4xToRUQyToG+gJkda2Z/NbNlZja/1e1plJmdYGZuZhNa3ZZ6mNmPcv8e95vZEjPbsNVtqoWZ7Wtmj5jZY2Z2UqvbUw8z28LMBszsodz/F99sdZsaYWZjzOweM7uu1W1phJltaGa/yv3/8bCZfaTUsQr0EWbWA0wFdnb3HYAzW9ykhpjZFsC/AE+3ui0NuBnY0d13Av4LmN3i9lTNzMYA5wD7AdsDXzKzTlytfhg4wd23Bz4MHNOhnyP0TeDhVjciAWcBv3X3fwJ2psxnUqDPNxM43d3fAnD3F1vcnkb9B3Ai0LFP3N39Jncfzm3eDkxqZXtqtAfwmLv/zd1XAYsIOhIdxd2fd/e7cz+/RhBQNm9tq+pjZpOAfwV+0uq2NMLMxgOfAC4CcPdV7v5qqeMV6PN9ENjTzO4wsz+Y2e6tblC9zGwqsNzd72t1WxJ0BHBDqxtRg82BZyLbz9KhATJkZpOBDwF3tLYldfs/BJ2fkVY3pEFbAUPAT3NpqJ+Y2bqlDs7G4uA1MLNbgPcUeetkgr+PjQi+nu4OLDazrb1Nx6BW+CzfIUjbtL1yn8Pdr8kdczJBCuHyZrZNRpnZesB/At9y93+0uj21MrPPAS+6+11m9qlWt6dBY4FdgWPd/Q4zOws4CegrdXBXcfe9S71nZjOBq3KB/U4zGyEoGjTUrPbVotRnMbP/RXDHv8/MIEh33G1me7j735vYxKqU+zcBMLOvAJ8D9mrXm24Jy4EtItuTcvs6jpm9iyDIX+7uV7W6PXX6GPAFM/sssBawgZld5u6Htrhd9XgWeNbdw29WvyII9EUpdZPvaqAHwMw+CKxJB1a4c/cH3H0Td5/s7pMJ/qPYtR2DfCVmti/BV+0vuPuKVrenRkuBbcxsKzNbE5gOXNviNtXMgt7CRcDD7r6w1e2pl7vPdvdJuf8npgP9HRrkyf2//IyZbZvbtRfwUKnju65HX8HFwMVm9iCwCji8w3qQWfRjYBxwc+7bye3ufnRrm1Qddx82s28ANwJjgIvdfVmLm1WPjwG9wANmdm9u33fc/TctbJPAscDluU7E34B/K3WgSiCIiGScUjciIhmnQC8iknEK9CIiGadALyKScQr0IiIZp0AvIpJxCvQiIhn3P4LwXkF4DbgtAAAAAElFTkSuQmCC\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Gradient Descent**\n",
        "Addestriamo il modello usando una discesa del gradiente:\n",
        "$$\n",
        "W(i+1) = W(i) - \\eta\\cdot \\nabla_{W}E(W) \\\\\n",
        "$$\n",
        "\n",
        "\n",
        "*   Implementa in gradiente $\\nabla_{W} E(W)=X^T(XW-y)$\n",
        "*   Implementa un ciclo per addestrare il modello usando la discesa del gradiente\n",
        "*   (Opzionale: Implementa lo \"stochastic gradient descent\" SGD, suddividi il dataset in \"mini batch\", ad es. 10 campioni, ed usali per aggionare i parametri. Al posto delle iterazioni ti conviene ragionare in \"epoche\", ovvero il numero di step di SGD necessari per \"vedere\" l'intero dataset) \n",
        "\n"
      ],
      "metadata": {
        "id": "LWGBuAaTAoXG"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def gradL(X,y,w):\n",
        "  \n",
        "  return dw"
      ],
      "metadata": {
        "id": "ZmbAHhlGNJdV"
      },
      "execution_count": 26,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def GD(X,y,i_max,eta):\n",
        "\n",
        "  return w"
      ],
      "metadata": {
        "id": "VmMSQ2CyNJZL"
      },
      "execution_count": 27,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "w=GD(Xl,yL,10,0.001)\n",
        "print(w)"
      ],
      "metadata": {
        "id": "D0BDvIOusL5u"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(Xt[:,1], yT, 'bx')\n",
        "plt.plot(Xt[:,1], Xt.dot(w), 'g-')\n",
        "plt.title('Prediction on Test Set')\n",
        "plt.show()"
      ],
      "metadata": {
        "id": "kO8tNFweN1P6"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Soluzione"
      ],
      "metadata": {
        "id": "i5ZIz307NeUN"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def gradL2(X,y,w):\n",
        "    return X.T.dot((X.dot(w)-y))"
      ],
      "metadata": {
        "id": "0N-qyeLGGJdc"
      },
      "execution_count": 11,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def GD(grad, X, y,epochs,eta):\n",
        "    n,d = X.shape\n",
        "    w = np.zeros(d).reshape(d, 1)\n",
        "    for e in range(epochs):\n",
        "        w = w-eta*grad(X,y,w)\n",
        "    return w"
      ],
      "metadata": {
        "id": "U-AqAT0mGJZi"
      },
      "execution_count": 12,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "w=GD(gradL2,Xl,yL,5,0.001)\n",
        "print(w)"
      ],
      "metadata": {
        "id": "Zy6FLD7iGNd-",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "46ff53da-4ed5-4515-a814-fefee47ca3f7"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[-2.09279279]\n",
            " [ 2.70112728]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#results on test set\n",
        "plt.show()\n",
        "plt.plot(Xt[:,1], yT, 'bx')\n",
        "plt.plot(Xt[:,1], Xt.dot(w), 'g-',label='GD')\n",
        "plt.plot(Xt[:,1], Xt.dot(Wnorm), 'r-',label='Soluzione Analitica')\n",
        "plt.legend()\n",
        "plt.title('Prediction on Test Set')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 281
        },
        "id": "rj_y71ipGRqJ",
        "outputId": "efef391c-6cd0-4acf-8452-ae38a3b1a479"
      },
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
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        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#Loss function e inferenza statistica\n",
        "https://arxiv.org/pdf/1803.08823.pdf\n",
        "\n",
        "Usando l'inferenza statistica, possiamo dedurre la loss function $L_{MSE}$ studiando la posterior: \n",
        "$$p(W|X)$$\n",
        "ovvero la distribuzione di probabibilità che condiziona i parametri W ai dati X.\n",
        "Usando il teorema di Bayes:\n",
        "$$p(W|X)=\\frac{p(X|W)p(W)}{c}$$\n",
        "dove $p(X|W)$ è generalmente chiamata likelihood, $p(W)$ è detta prior mentre $c$ è una costante di normalizzazione. Supponiamo che i target $y$ siano delle fluttuazioni i.i.d centrate sulle predizioni del modello, ovvero immaginiamo che la likelihood sia una produttoria di Gaussiane:\n",
        "$$p(X|W)=∏_{i}^N\\frac{1}{Z}e^{-\\frac{\\beta}{2}(y_i-\\hat{y}_i)^2}=\\frac{1}{Z}e^{-\\frac{\\beta}{2}\\sum_{i}^N(y_i-\\hat{y}_i)^2}=\\frac{1}{Z}e^{-\\frac{\\beta N}{2}E(W)}$$\n",
        "massimizzare questo oggetto equivale a minimizzare:\n",
        "$$-\\log p(X|W)={\\frac{\\beta}{2}\\sum_{i}^N(y_i-\\hat{y}_i)^2}+c=\\frac{\\beta N}{2}E(W)+c$$\n",
        "ovvero la loss MSE. Nel caso in cui la prior è costante e dunque trascurabile, massimizzare la posterior equivale al principio di \"Maximun Likelihood\". \n",
        "$$w^*=max_Wp(X|W)$$\n",
        "\n",
        "Ricordate il tradeoff tra bias e varianza? Le fluttuazioni misurate dal termine di varianza sono legate alla prior p(W) e possiamo tentare di controllarle durante la costruzione della loss. Immaginiamo che i parametri siano i.i.d:\n",
        "$$p(W)=\\frac{1}{Z}\\prod_k e^{-\\frac{\\alpha}{2}w_k^2}$$\n",
        "Dunque, se minimiziamo $-\\log p(W|X)$ otteniamo una nuova loss \"regolarizzata\":\n",
        "$$E_{\\lambda}(W)=\\frac{1}{N}\\sum_i^N(y_i-\\hat{y}_i)^2+\\lambda\\sum_k w_k^2$$\n",
        "dove $\\lambda=\\frac{\\alpha}{\\beta}$ è un iperparametro. Il termine che abbiamo aggiunto si chiama regolarizzatore e, in questo caso, ha l'effetto di tenere \"piccoli\" i parametri.\n"
      ],
      "metadata": {
        "id": "kqJZDJ3KLi-O"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Esercizi\n",
        "Aggiungiamo rumore e degli outliner ai dati."
      ],
      "metadata": {
        "id": "mmNYQPMxVR7i"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "XL, yL = mk_data() # training sets\n",
        "yL = add_outliers(yL)\n",
        "XT, yT = mk_data(minR=5,maxR=15)  # test sets\n",
        "Xl=np.c_[np.ones(len(XL)),XL]\n",
        "Xt=np.c_[np.ones(len(XT)),XT]\n",
        "yL=yL.reshape(len(XL), 1)\n",
        "yT=yT.reshape(len(XL), 1)\n",
        "plt.plot(XL,yL,'bx',label='learning set');\n",
        "plt.plot(XT,yT,'rx',label='test set')\n",
        "plt.legend()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "T7mdcfawLiYi",
        "outputId": "c43555cb-d7c5-4f8b-8757-39d15b967cd7"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f7f060b5460>"
            ]
          },
          "metadata": {},
          "execution_count": 14
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Implementa e testa le seguenti loss:\n",
        "   * $L_{MAE} = \\frac{1}{N}\\sum_{i=1}^N | y_i -g_W(x_i)| $\n",
        "\n",
        "   * $L_{LC}= \\frac{1}{N}\\sum_{i=1}^N \\ln [ \\cosh (y_i -g_W(x_i)) ] $\n",
        "\n",
        "e confrontane i risultati con $L_{MSE}$.\n"
      ],
      "metadata": {
        "id": "wPubgSdEBAFT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Esercizio** Implementa i seguenti regolarizzatori:\n",
        "\n",
        "*   $L_1=\\lambda\\sum_{k}|w_k|$ (Lasso)\n",
        "*   $L_2=\\lambda\\sum_kw_k^2$ (Ridge)\n",
        "\n",
        "e plotta la delle componenti di $w^*$ al variare di $\\lambda$."
      ],
      "metadata": {
        "id": "4NnFwhzHZjFH"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Soluzioni"
      ],
      "metadata": {
        "id": "v_gNEXhIcJJK"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#Gradient of loss function\n",
        "def gradL1(X,y,w):\n",
        "    return X.T.dot(np.sign(X.dot(w)-y))\n",
        "                   \n",
        "def gradLC(X,y,w):\n",
        "    D=y-X.dot(w)\n",
        "    return (-X).T.dot((1/np.cosh(D))*np.sinh(D))\n",
        "#training\n",
        "wl1=GD(gradL1,Xl,yL,1000,0.0001)\n",
        "wl2=GD(gradL2,Xl,yL,15,0.001)\n",
        "wllc=GD(gradLC,Xl,yL,60,0.001)\n",
        "#results on test set\n",
        "plt.plot(Xt[:,1], yT, 'bx')\n",
        "plt.plot(Xt[:,1], Xt.dot(wl1), 'k-', label='L1')\n",
        "plt.plot(Xt[:,1], Xt.dot(wl2), 'g-',label='L2')\n",
        "plt.plot(Xt[:,1], Xt.dot(wllc), 'r-',label='LLC')\n",
        "plt.legend()\n",
        "plt.title('Prediction on Test Set')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 281
        },
        "id": "sbZElMj_xgyk",
        "outputId": "6e10102b-9301-4937-db9e-68ccdd3baf9e"
      },
      "execution_count": 20,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "MExqDK44wthB"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}