Vision2020-ex01.ipynb

vision2020-ex01.ipynb

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    "colab": {
      "name": "Vision2020-ex01.ipynb",
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      "authorship_tag": "ABX9TyPEw3KydawExSjza5OflOPx",
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      "name": "python3",
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
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      "source": [
        "<a href=\"https://colab.research.google.com/gist/takatakamanbou/55ee0af00777c5595bd46d60afb207b1/vision2020-ex01.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9dZcF12ojSrJ",
        "colab_type": "text"
      },
      "source": [
        "# Vision2020-ex01"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SrIIY0RtpBYO",
        "colab_type": "text"
      },
      "source": [
        "課題の期限や提出の方法などについては，Visionチーム内に書いてます．"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZSx7Ng2FkdUB",
        "colab_type": "text"
      },
      "source": [
        "## 準備"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "BXtjmCk3jNaH",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ioHPZxlrn_Nw",
        "colab_type": "text"
      },
      "source": [
        "## 課題A\n",
        "\n",
        "以下のプログラムは，[PIP/2019](https://www-tlab.math.ryukoku.ac.jp/wiki/?PIP/2020) 第12回講義資料のQ1（の一部）を解くようにできている．その動作を理解しなさい．ただし...\n",
        "- データ行列の要素の並べ方が，授業で説明しているのと違う（転置になってる）ことに注意．\n",
        "- 関数 solve で呼んでる np.linalg.solve は，連立方程式を解く関数である．詳しくは https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.solve.html へどうぞ．\n",
        "- 関数 solve2 については現時点では無視して構わない．\n",
        "\n",
        "提出するもの： Q1を解くプログラムの実行結果にコメントをつけたもの．"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zd9VnEQRjtgq",
        "colab_type": "code",
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      "source": [
        "#####  最小二乗法の正規方程式を解く関数\n",
        "### solving the linear least squares problem (1)\n",
        "#\n",
        "# X.shape is assumed to be (N, D+1)\n",
        "# y.shape is assumed to be (N,)\n",
        "#\n",
        "def solve(X, y):\n",
        "\n",
        "    A = np.dot(X.T, X)\n",
        "    b = np.dot(X.T, y)\n",
        "\n",
        "    # x is the solution of the equation Ax = b\n",
        "    x = np.linalg.solve(A, b)\n",
        "\n",
        "    return x\n",
        "\n",
        "\n",
        "### solving the linear least squares problem (2)\n",
        "#\n",
        "def solve2(X, y):\n",
        "\n",
        "    A = np.dot(X.T, X)\n",
        "    b = np.dot(X.T, y)\n",
        "\n",
        "    # rv[0] is the solution minimizing ||Ax - b||^2\n",
        "    rv = np.linalg.lstsq(A, b)\n",
        "\n",
        "    return rv[0]"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "e62dgLVTkOHR",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#####  PIP2019-12 の Q1 を解くプログラム\n",
        "\n",
        "x = np.array([0, 4, 8, 20])\n",
        "y = np.array([30, 31, 29, 28])\n",
        "\n",
        "y -= y[0]\n",
        "\n",
        "N = x.shape[0]\n",
        "print('# N =', N)\n",
        "\n",
        "X = np.vstack((np.ones(N), x)).T\n",
        "print(X)\n",
        "print('# X.shape =', X.shape)\n",
        "\n",
        "print(y)\n",
        "print('# y.shape =', y.shape)\n",
        "                    \n",
        "w = solve(X, y)\n",
        "#w = solve2(X, y)\n",
        "print('# estimated parameters =', w)\n",
        "\n",
        "b, a = w  # w[0] is b and w[1] is a\n",
        "print('# fitted equation is y =', a, '*x +', b)\n",
        "\n",
        "print(a * 36 + b + 30)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rzAVumnxmCki",
        "colab_type": "text"
      },
      "source": [
        "## 課題B & C 用のデータとその可視化\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pIwsIvDHmEOe",
        "colab_type": "code",
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      },
      "source": [
        " ##### 課題B & C 用のデータ\n",
        " dat = np.array(\n",
        "     [[0, 0.7550326945263522],\n",
        "      [.05, 1.007767522485821],\n",
        "      [0.1, 0.6361953553367321],\n",
        "      [0.15, 0.6959238424062661],\n",
        "      [0.2, 0.595485181368133],\n",
        "      [0.25, 0.4831892240046355],\n",
        "      [0.3, 0.159575705011379],\n",
        "      [0.35, 0.186501916036992],\n",
        "      [0.4, 0.1567768077057974],\n",
        "      [0.45, 0.04182438372911931],\n",
        "      [0.5, -0.06982713339289053],\n",
        "      [0.55, 0.08544102980639576],\n",
        "      [0.6, -0.1007904952533244],\n",
        "      [0.65, -0.1277469644723259],\n",
        "      [0.7, -0.02542025832150743],\n",
        "      [0.75, -0.411322875976739],\n",
        "      [0.8, 0.01309389848625916],\n",
        "      [0.85, -0.1196592829342933],\n",
        "      [0.9, 0.07144698650328071],\n",
        "      [0.95, -0.1983213666516035]]\n",
        " )\n",
        "\n",
        " print(dat.shape)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hSb0K-S7nZJA",
        "colab_type": "code",
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      },
      "source": [
        "#####  データの可視化\n",
        "\n",
        "fig, ax = plt.subplots(facecolor=\"white\", figsize=(8, 6))\n",
        "ax.set_xlim(-0.1, 1.1)  # X軸の範囲\n",
        "ax.set_ylim(-1, 2)  # Y軸の範囲\n",
        "ax.scatter(dat[:, 0], dat[:, 1], color=\"red\", label=\"data\")\n",
        "ax.legend()\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yac7v3I7tW5o",
        "colab_type": "text"
      },
      "source": [
        "## 課題B\n",
        "\n",
        "以下のプログラムは，上記のデータに対して最小二乗法によって D 次元の多項式をあてはめたときのパラメータ（多項式の係数）を求めるものである．このプログラムの動作を理解しなさい．\n",
        "\n",
        "D = 1 として実行すると，直線のあてはめとなる．このときの結果を提出しなさい．提出するもの:  得られるグラフと，当てはめられた直線の式．"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_0b1CPSMnLjJ",
        "colab_type": "code",
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      },
      "source": [
        "##### dat に対する多項式あてはめ\n",
        "\n",
        "N = dat.shape[0]\n",
        "print('# N =', N)\n",
        "\n",
        "x = dat[:, 0]\n",
        "D = 13   # D is the degree of the polynomial to be fitted\n",
        "X = np.zeros((N, D+1))\n",
        "X[:, 0] = 1\n",
        "for i in range(1, D+1):\n",
        "    X[:, i] = x**i\n",
        "print(X)\n",
        "print('# X.shape =', X.shape)\n",
        "\n",
        "y = dat[:, 1]\n",
        "print(y)\n",
        "print('# y.shape =', y.shape)\n",
        "\n",
        "w = solve(X, y)\n",
        "print(w)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1Ehs5FAzpwN0",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#####  曲線上の点の座標を求める\n",
        "\n",
        "xmin, xmax = -0.1, 1.1\n",
        "xx = np.linspace(xmin, xmax, 1000)\n",
        "XX = np.zeros((xx.shape[0], D+1))\n",
        "XX[:, 0] = 1\n",
        "print(\"# XX.shape =\", XX.shape)\n",
        "for i in range(1, D+1):\n",
        "    XX[:, i] = xx**i\n",
        "\n",
        "yy = XX @ w"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7LeCYpJlvqAn",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#####  結果の可視化\n",
        "\n",
        "fig, ax = plt.subplots(facecolor=\"white\", figsize=(8, 6))\n",
        "ax.set_xlim(xmin, xmax)  # X軸の範囲\n",
        "ax.set_ylim(-1, 2)  # Y軸の範囲\n",
        "ax.scatter(dat[:, 0], dat[:, 1], color=\"red\", label=\"data\")\n",
        "ax.plot(xx, yy, color=\"blue\", label=\"D={}\".format(D))\n",
        "ax.legend()\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jsxIIys6uRcR",
        "colab_type": "text"
      },
      "source": [
        "## 課題C\n",
        "\n",
        "上記のプログラムで D を何通りか変えて，課題Bと同様にグラフを提出しなさい．考察を添えること．"
      ]
    }
  ]
}