chapter_rnn/1_markov_model.ipynb

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      "provenance": []
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    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
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    {
      "cell_type": "markdown",
      "source": [
        "<a target=\"_blank\" href=\"https://colab.research.google.com/github/ashkan-mokarian/d2l-pytorch/blob/main/chapter_rnn/1_markov_model.ipynb\">\n",
        "  <img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/>\n",
        "</a>"
      ],
      "metadata": {
        "id": "GW-oPfi8s2Kv"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "kUds9XZ-s0PT"
      },
      "outputs": [],
      "source": [
        "%pip install d2l==1.0.0a0"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import torch\n",
        "from d2l import torch as d2l"
      ],
      "metadata": {
        "id": "Y7sOxFBes-OC"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Markov Model"
      ],
      "metadata": {
        "id": "mnYVVnBUtd5K"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Synthetic sequence data"
      ],
      "metadata": {
        "id": "krQY5vnstizO"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class Data(d2l.DataModule):\n",
        "    def __init__(self, batch_size=16, T=1000, num_train=600, tau=4):\n",
        "        super().__init__()\n",
        "        self.save_hyperparameters()\n",
        "        self.time = torch.arange(1, T+1, dtype=torch.float32)\n",
        "        self.x = torch.sin(0.01 * self.time) + torch.randn(T) * 0.2\n",
        "\n",
        "data = Data()\n",
        "d2l.plot(data.time, data.x, 'time', 'x', xlim=[1, 1000], figsize=(6, 3))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
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        "id": "lxmgk9MZtcr0",
        "outputId": "3cc87309-8a6c-431a-9db9-85f3ae189a6d"
      },
      "execution_count": 5,
      "outputs": [
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          "output_type": "display_data",
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          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "@d2l.add_to_class(Data)\n",
        "def get_dataloader(self, train):\n",
        "    self.features = [self.x[i: self.T-self.tau+i] for i in range(self.tau)]\n",
        "    self.features = torch.stack(self.features, 1)\n",
        "    self.labels = self.x[self.tau:].reshape((-1, 1))\n",
        "    i = slice(0, self.num_train) if train else slice(self.num_train, None)\n",
        "    return self.get_tensorloader([self.features, self.labels], train, i)\n",
        "\n",
        "batch_x, batch_y = next(iter(data.get_dataloader(True)))\n",
        "batch_x.shape, batch_y.shape"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "1eJajke9udhJ",
        "outputId": "f3f01006-5191-4b56-8034-7f5f5b54dfae"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(torch.Size([16, 4]), torch.Size([16, 1]))"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "model = d2l.LinearRegression(lr=0.01)\n",
        "trainer = d2l.Trainer(max_epochs=5)\n",
        "trainer.fit(model, data)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "geFTpF49wRyG",
        "outputId": "3394ce0c-3cb1-4273-9833-fb8d0e300efb"
      },
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 350x250 with 1 Axes>"
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    },
    {
      "cell_type": "code",
      "source": [
        "one_step_pred = model(data.features).detach()\n",
        "d2l.plot(data.time[data.tau:], [data.labels, one_step_pred], 'time', 'x', legend=['labels', 'one_step_pred'], figsize=(6, 3))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 302
        },
        "id": "vB525KNfwuct",
        "outputId": "e14e96ea-435f-4399-b16a-178c8c412f40"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
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146.605433 65.404971 \nL 146.911326 64.378664 \nL 147.217219 64.964379 \nL 147.523112 50.539688 \nL 147.829005 68.224223 \nL 148.440791 70.382673 \nL 148.746684 68.510032 \nL 149.052578 64.71659 \nL 149.358471 76.901448 \nL 149.664364 75.903639 \nL 149.970257 78.923228 \nL 150.27615 73.943661 \nL 150.582043 82.763497 \nL 150.887936 50.108718 \nL 151.193829 81.59763 \nL 151.499722 79.003586 \nL 151.805615 82.028432 \nL 152.111509 72.521383 \nL 152.417402 78.913802 \nL 152.723295 58.96528 \nL 153.029188 70.190096 \nL 153.335081 60.174961 \nL 153.640974 74.833238 \nL 153.946867 66.216102 \nL 154.25276 71.49791 \nL 154.558653 73.38988 \nL 154.864546 77.755761 \nL 155.17044 74.470626 \nL 155.476333 86.779902 \nL 155.782226 88.132193 \nL 156.088119 83.694395 \nL 156.394012 82.144533 \nL 156.699905 87.003902 \nL 157.005798 74.309254 \nL 157.311691 101.234032 \nL 157.617584 77.27966 \nL 157.923477 98.192543 \nL 158.229371 75.151483 \nL 158.535264 83.766253 \nL 158.841157 88.864708 \nL 159.14705 72.402678 \nL 159.452943 101.340299 \nL 159.758836 83.036033 \nL 160.064729 93.588179 \nL 160.370622 76.681582 \nL 160.676515 85.366113 \nL 160.982408 72.07772 \nL 161.288302 82.164572 \nL 161.594195 86.454452 \nL 161.900088 88.441591 \nL 162.205981 79.315924 \nL 162.817767 89.848537 \nL 163.12366 74.401243 \nL 163.429553 94.644835 \nL 163.735446 91.143645 \nL 164.04134 100.453345 \nL 164.347233 84.017408 \nL 164.653126 103.943473 \nL 164.959019 87.387882 \nL 165.264912 99.366974 \nL 165.570805 106.264393 \nL 165.876698 97.007267 \nL 166.182591 99.221061 \nL 166.488484 90.812175 \nL 166.794377 86.925339 \nL 167.100271 97.861521 \nL 167.406164 95.631788 \nL 167.712057 109.947124 \nL 168.01795 98.463575 \nL 168.323843 103.326478 \nL 168.629736 96.322639 \nL 168.935629 92.72824 \nL 169.241522 97.058946 \nL 169.547415 82.877652 \nL 169.853308 110.405878 \nL 170.159202 83.928025 \nL 170.465095 111.225878 \nL 170.770988 95.520614 \nL 171.076881 108.641302 \nL 171.382774 104.515254 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119.040944 \nL 183.618498 109.817109 \nL 183.924391 131.172006 \nL 184.230284 113.930579 \nL 184.536177 141.790995 \nL 184.84207 124.640709 \nL 185.147964 142.659632 \nL 185.453857 125.201492 \nL 185.75975 137.9633 \nL 186.065643 117.865878 \nL 186.371536 125.710243 \nL 186.677429 123.238119 \nL 186.983322 105.217583 \nL 187.289215 129.788972 \nL 187.595108 109.135424 \nL 187.901001 118.81732 \nL 188.206895 119.168178 \nL 188.512788 124.622612 \nL 188.818681 135.232825 \nL 189.124574 128.56534 \nL 189.73636 139.012197 \nL 190.042253 131.818672 \nL 190.348146 150.061939 \nL 190.654039 140.619261 \nL 190.959932 141.71292 \nL 191.265826 143.391156 \nL 191.571719 133.849443 \nL 191.877612 135.854459 \nL 192.183505 135.534564 \nL 192.489398 145.428508 \nL 192.795291 147.033539 \nL 193.101184 154.254421 \nL 193.407077 152.423009 \nL 193.71297 137.414337 \nL 194.018863 146.8684 \nL 194.324757 118.172471 \nL 194.63065 137.95829 \nL 194.936543 135.27026 \nL 195.242436 128.563119 \nL 195.548329 146.933804 \nL 195.854222 133.802862 \nL 196.160115 131.996872 \nL 196.466008 138.962497 \nL 196.771901 116.499487 \nL 197.077795 136.096974 \nL 197.383688 118.025678 \nL 197.689581 149.958584 \nL 197.995474 127.183427 \nL 198.301367 148.23813 \nL 198.60726 138.638186 \nL 198.913153 143.925926 \nL 199.219046 146.235809 \nL 199.524939 146.719231 \nL 199.830832 145.669262 \nL 200.136726 143.069773 \nL 200.442619 126.486 \nL 200.748512 137.880957 \nL 201.054405 128.825703 \nL 201.360298 131.855004 \nL 201.666191 146.399072 \nL 201.972084 139.536144 \nL 202.277977 135.25281 \nL 202.58387 135.649783 \nL 202.889763 134.357868 \nL 203.195657 123.135164 \nL 203.50155 131.638949 \nL 203.807443 131.167155 \nL 204.113336 139.80619 \nL 204.419229 123.875477 \nL 204.725122 140.456173 \nL 205.031015 125.609012 \nL 205.336908 144.826694 \nL 205.642801 135.809878 \nL 205.948694 156.109492 \nL 206.254588 140.935209 \nL 206.560481 150.263407 \nL 206.866374 147.792969 \nL 207.172267 141.110966 \nL 207.47816 143.324062 \nL 207.784053 149.026909 \nL 208.089946 136.553601 \nL 208.395839 156.774387 \nL 208.701732 146.781983 \nL 209.007625 148.612469 \nL 209.313519 149.473535 \nL 209.619412 147.639502 \nL 209.925305 135.280065 \nL 210.231198 135.533586 \nL 210.537091 144.643356 \nL 210.842984 133.597785 \nL 211.148877 132.236841 \nL 211.45477 145.611125 \nL 211.760663 122.320108 \nL 212.066557 153.382094 \nL 212.37245 135.770983 \nL 212.678343 141.484718 \nL 212.984236 129.829103 \nL 213.290129 126.438875 \nL 213.596022 131.313591 \nL 213.901915 132.580885 \nL 214.207808 139.865057 \nL 214.513701 132.704849 \nL 214.819594 144.044952 \nL 215.125488 133.869869 \nL 215.431381 146.850313 \nL 215.737274 141.21983 \nL 216.043167 146.632807 \nL 216.34906 143.398703 \nL 216.654953 145.005317 \nL 216.960846 141.357158 \nL 217.266739 140.777257 \nL 217.572632 132.727092 \nL 217.878525 147.775826 \nL 218.184419 145.31414 \nL 218.490312 147.409946 \nL 218.796205 142.096918 \nL 219.102098 152.066806 \nL 219.407991 140.588146 \nL 219.713884 146.591813 \nL 220.019777 143.678613 \nL 220.32567 147.971363 \nL 220.631563 134.994533 \nL 220.937456 138.028381 \nL 221.24335 134.855824 \nL 221.549243 129.477813 \nL 221.855136 130.729386 \nL 222.161029 131.037458 \nL 222.466922 132.400107 \nL 223.078708 125.946981 \nL 223.384601 154.0785 \nL 223.690494 120.684726 \nL 223.996387 151.362277 \nL 224.302281 133.23465 \nL 224.608174 131.10262 \nL 224.914067 137.206915 \nL 225.21996 125.745215 \nL 225.831746 133.154544 \nL 226.137639 133.175153 \nL 226.443532 126.782211 \nL 226.749425 140.515096 \nL 227.055318 121.710806 \nL 227.361212 132.42261 \nL 227.667105 127.372504 \nL 227.972998 131.33379 \nL 228.278891 119.186306 \nL 228.584784 124.211263 \nL 228.890677 134.628424 \nL 229.19657 108.710759 \nL 229.502463 138.023915 \nL 229.808356 124.019913 \nL 230.11425 118.737506 \nL 230.420143 138.056792 \nL 230.726036 123.37408 \nL 231.031929 145.230717 \nL 231.337822 128.669434 \nL 231.643715 121.275224 \nL 231.949608 141.719205 \nL 232.255501 127.955588 \nL 232.561394 126.371148 \nL 232.867287 138.539437 \nL 233.173181 125.612499 \nL 233.479074 122.934065 \nL 233.784967 130.770365 \nL 234.09086 135.696796 \nL 234.396753 127.114966 \nL 234.702646 124.262726 \nL 235.008539 128.530858 \nL 235.314432 110.588888 \nL 235.620325 122.192467 \nL 235.926218 117.726547 \nL 236.232112 127.346693 \nL 236.538005 123.287066 \nL 236.843898 128.573868 \nL 237.149791 129.607398 \nL 237.761577 122.593409 \nL 238.06747 109.166893 \nL 238.373363 120.643701 \nL 238.679256 103.562482 \nL 238.985149 117.52129 \nL 239.291043 121.312954 \nL 239.596936 107.472958 \nL 239.902829 129.00628 \nL 240.208722 116.933007 \nL 240.514615 123.226625 \nL 240.820508 112.129262 \nL 241.126401 124.17061 \nL 241.432294 110.431474 \nL 241.738187 131.389356 \nL 242.04408 114.263462 \nL 242.349974 124.562984 \nL 242.655867 118.053079 \nL 242.96176 105.234905 \nL 243.267653 121.490658 \nL 243.879439 107.979357 \nL 244.185332 113.166052 \nL 244.491225 101.580861 \nL 244.797118 109.033754 \nL 245.103012 111.149128 \nL 245.408905 106.433335 \nL 245.714798 123.961925 \nL 246.020691 109.305089 \nL 246.326584 109.993755 \nL 246.632477 121.610721 \nL 246.93837 101.853482 \nL 247.244263 131.746034 \nL 247.550156 93.053786 \nL 247.856049 116.302835 \nL 248.161943 87.016598 \nL 248.467836 110.03253 \nL 248.773729 97.037243 \nL 249.079622 108.657751 \nL 249.385515 107.785019 \nL 249.691408 96.890014 \nL 249.997301 101.286895 \nL 250.303194 100.366019 \nL 250.609087 103.245279 \nL 250.91498 110.739636 \nL 251.220874 92.110197 \nL 251.526767 117.982609 \nL 251.83266 90.494033 \nL 252.138553 109.543834 \nL 252.444446 95.565796 \nL 252.750339 103.772139 \nL 253.056232 90.065035 \nL 253.362125 112.098186 \nL 253.668018 83.984594 \nL 253.973911 104.851736 \nL 254.279805 80.073835 \nL 254.585698 90.393948 \nL 254.891591 94.915552 \nL 255.197484 86.967898 \nL 255.503377 90.481428 \nL 255.80927 103.178352 \nL 256.115163 84.516798 \nL 256.421056 115.581281 \nL 256.726949 98.105323 \nL 257.032842 93.355471 \nL 257.338736 100.282502 \nL 257.644629 82.970892 \nL 257.950522 95.196016 \nL 258.256415 79.61447 \nL 258.562308 101.305403 \nL 258.868201 76.185643 \nL 259.174094 92.099598 \nL 259.479987 83.672509 \nL 259.78588 93.379136 \nL 260.091773 83.058449 \nL 260.397667 96.503521 \nL 260.70356 72.290127 \nL 261.009453 88.79985 \nL 261.315346 78.215259 \nL 261.621239 82.823481 \nL 261.927132 81.126065 \nL 262.233025 93.255384 \nL 262.538918 84.013286 \nL 263.150705 97.931492 \nL 263.456598 82.79726 \nL 263.762491 90.456304 \nL 264.068384 69.494945 \nL 264.374277 83.541911 \nL 264.68017 70.866534 \nL 264.986063 76.601255 \nL 265.291956 77.030136 \nL 265.597849 73.17821 \nL 265.903742 78.404732 \nL 266.209636 70.805007 \nL 266.515529 79.950221 \nL 266.821422 73.175558 \nL 267.127315 92.122177 \nL 267.433208 64.739572 \nL 267.739101 83.305898 \nL 268.044994 78.34028 \nL 268.350887 79.247195 \nL 268.65678 71.461211 \nL 268.962673 89.625498 \nL 269.268567 82.77 \nL 269.57446 71.172385 \nL 269.880353 76.779332 \nL 270.186246 79.049587 \nL 270.492139 61.084335 \nL 270.798032 78.992021 \nL 271.103925 69.206802 \nL 271.409818 77.096372 \nL 271.715711 73.464041 \nL 272.021604 66.377291 \nL 272.327498 73.124817 \nL 272.633391 65.256845 \nL 272.939284 62.617483 \nL 273.245177 73.482332 \nL 273.55107 57.100365 \nL 273.856963 74.647969 \nL 274.162856 54.181177 \nL 274.468749 78.900991 \nL 274.774642 54.697673 \nL 275.080535 75.497799 \nL 275.386429 71.045112 \nL 275.692322 62.241578 \nL 275.998215 86.026336 \nL 276.304108 67.040883 \nL 276.610001 71.881683 \nL 276.915894 64.045086 \nL 277.221787 51.474444 \nL 277.52768 49.790444 \nL 277.833573 40.15028 \nL 278.139467 51.010799 \nL 278.44536 44.882131 \nL 278.751253 67.580296 \nL 279.057146 57.840361 \nL 279.363039 58.478331 \nL 279.668932 68.521254 \nL 279.974825 53.467178 \nL 280.280718 56.367869 \nL 280.586611 51.026355 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293.128229 46.525128 \nL 293.434122 40.881856 \nL 293.740015 48.72268 \nL 294.045908 34.272102 \nL 294.351801 47.044339 \nL 294.657694 36.732204 \nL 294.963587 52.122209 \nL 295.26948 40.693043 \nL 295.881266 42.767862 \nL 296.18716 43.05952 \nL 296.493053 43.588641 \nL 297.104839 52.848926 \nL 297.410732 46.691285 \nL 297.716625 49.253016 \nL 298.022518 43.103884 \nL 298.328411 40.099214 \nL 298.634304 31.493859 \nL 299.551984 48.170432 \nL 299.857877 37.279719 \nL 300.16377 36.969828 \nL 300.469663 29.713317 \nL 300.775556 36.179632 \nL 301.081449 23.753698 \nL 301.387342 38.647089 \nL 301.693235 36.914181 \nL 301.999128 32.76042 \nL 302.305022 44.259139 \nL 302.610915 41.45917 \nL 302.916808 43.177748 \nL 303.222701 41.805466 \nL 303.528594 52.261674 \nL 303.834487 47.333112 \nL 304.14038 55.342711 \nL 304.446273 49.803506 \nL 304.752166 50.74943 \nL 305.058059 43.74678 \nL 305.363953 42.302546 \nL 305.669846 44.746409 \nL 305.975739 35.802442 \nL 306.281632 45.027741 \nL 306.587525 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          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## k-step ahead prediction\n",
        "In the example above, we used the observed data to make ahead predictions. However, in realistic scenarios, we don't have observations beyond the train dataset and hence need to make use of the predicted values to condition fo predicting steps ahead."
      ],
      "metadata": {
        "id": "4UhR6XdyyFfu"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "multi_step_pred = torch.zeros(data.T)\n",
        "multi_step_pred[:] = data.x\n",
        "for i in range(data.num_train+data.tau, data.T):\n",
        "    multi_step_pred[i] = model(\n",
        "        multi_step_pred[i-data.tau: i].reshape((1, -1))\n",
        "        )\n",
        "multi_step_pred = multi_step_pred.detach().numpy()\n",
        "d2l.plot([data.time[data.tau:], data.time[data.num_train+data.tau:]],\n",
        "         [one_step_pred, multi_step_pred[data.num_train+data.tau:]],\n",
        "         'time', 'x', legend=['one_step_pred', 'multi_step_pred'], figsize=(6, 3))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 302
        },
        "id": "Q0pXeqgfxcwi",
        "outputId": "a9bf4ded-2ee8-484b-8e63-24373e0d48db"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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    {
      "cell_type": "markdown",
      "source": [
        "### Why?\n",
        "why does the multi-step prediction diverge and result to this large error? This is because of cumulative errors. In the first prediction, we will have some error $ϵ_1=\\bar{ϵ}$ and for the second we'll have $ϵ_1=\\bar{ϵ} + cϵ_1$ and this cumulates over time.\n",
        "\n",
        "### Experiment with predicting k-step ahead predictions"
      ],
      "metadata": {
        "id": "Uvf8rzjR1Gai"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def k_step_pred(k):\n",
        "    features = []\n",
        "    for i in range(data.tau):\n",
        "        features.append(data.x[i : i+data.T-data.tau-k+1])\n",
        "    # the (i+tau)-th element stores vaules for the (i+1)-step ahead predictions\n",
        "    for i in range(k):\n",
        "        pred = model(torch.stack(features[i : i+data.tau], 1))\n",
        "        features.append(pred.reshape(-1))\n",
        "    return features[data.tau:]\n",
        "\n",
        "steps = (1, 4, 16, 64)\n",
        "preds = k_step_pred(steps[-1])\n",
        "d2l.plot(data.time[data.tau+steps[-1]-1:],\n",
        "         [preds[k-1].detach().numpy() for k in steps],\n",
        "         'time', 'x', legend=[f'{k}-step preds' for k in steps], figsize=(6, 3))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 302
        },
        "id": "8rQasq-q0SZl",
        "outputId": "e7de6444-3ea4-42d9-918d-51bd8500f231"
      },
      "execution_count": 20,
      "outputs": [
        {
          "output_type": "display_data",
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    },
    {
      "cell_type": "markdown",
      "source": [
        "As seen above, with higher ahead step predictions, the predictions get useless."
      ],
      "metadata": {
        "id": "-KeTzv4G9_Nh"
      }
    }
  ]
}