20180403_Finite_Difference_Coding_Challenge.ipynb

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      "name": "Finite Difference Coding Challenge.ipynb",
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      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
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      "source": [
        "[View in Colaboratory](https://colab.research.google.com/github/schwaaweb/aimlds_solutions/blob/master/20180403_Finite_Difference_Coding_Challenge.ipynb)"
      ]
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        "There are two derivatives that we can take programmatically: the first is the analytical, or exact, derivative of a function. This is the 'real' derivative. If we have a nice equation describing the function, we can deduce a nice equation describing the function's derivative. A traditional first course in calculus is mostly about learning to do this by hand. The second type of derivative is the numerical derivative of a function. If we can't or don't want to calculate the exact derivative, we use the numerical derivative as an approximate substitute. This is fine for many purposes. especially when we're working with messy data that has some uncertainty in it to begin with.\n",
        "\n",
        "### The analytical derivative of a function\n",
        "The analytical derivative of a function `f(x) = x^n` is equal to `nx^(n-1)`. You can apply this rule to any function in order to calculate its derivative.\n",
        "\n",
        "[Sympy](http://www.sympygamma.com) can calculate the analytical derivative of a function. \n",
        "\n",
        "\n",
        "### The numerical derivative of a function\n",
        "When we know the function and it's not too complicated, it is usually easiest to find the analytical derivative of the function and then use it to calculate the derivative at some value `x`. When we don't have a nice equation describing our function, we can use the \"finite difference\" method to calculate the derivative at value `x`.\n",
        "\n",
        "To start, let's look at the definition of the exact derivative f'(x). It goes something like this: first, calculate the difference between f(x + 0.1) and f(x), divided by 0.1:\n",
        "\n",
        "`[f(x + 0.1) - f(x)] / 0.1`\n",
        "\n",
        "We think of this as a \"small change in f per small change in x.\" Now make the difference smaller:\n",
        "\n",
        "`[f(x + 0.01) - f(x)] / 0.01`\n",
        "\n",
        "Now even smaller:\n",
        "\n",
        "[f(x + 0.001) - f(x)] / 0.001`\n",
        "\n",
        "If our function has a derivative at all (and not all functions do, although most every function we'll work with does), as we make the finite difference smaller and smaller this ratio will settle down to some definite number. f'(x) is, by definition, the value it settles down to. But this gives us a useful way to approximate f'(x): rather than calculating what the ratio settles down to as the finite difference (or \"step size\") grows infinitely small, we can just use a small *enough* step size. The smaller the step size, the better an approximation we can expect. For example, the ratio\n",
        "\n",
        "`[f(x + 0.000001) - f(x)] / 0.000001`\n",
        "\n",
        "is an excellent approximation to f'(x).\n"
      ]
    },
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      "source": [
        "## Symbolic derivative\n",
        "\n",
        "This graph shows the derivative of $sin(x)$, $cos(x)$. Because we already know the derivative of $sin(x)$ symbolically, it is easy to compute the derivative and graph it.\n",
        "\n",
        "Not the offset in plotting the derivative, `plt.plot(x,d_x*x+d_x-0.67,'r')`: `d_x*x` is the equation for a line slope, and `d_x-0.67` is the `y` intercept so that we can lay the derivative line on top of the `sin` function that it is derivative of."
      ]
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        },
        "outputId": "48f80dc5-243a-46e9-867c-3dc7fa326339"
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      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "x = np.linspace(-4,np.pi,100)\n",
        "sin_x = np.sin(x)\n",
        "plt.plot(x,sin_x)\n",
        "d_x = np.cos(0.67)\n",
        "plt.plot(x,d_x*x+d_x-0.67,'r')\n",
        "plt.axvline(0.67,color=\"gray\")"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.lines.Line2D at 0x7f152ad8dc50>"
            ]
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          "metadata": {
            "tags": []
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          "execution_count": 43
        },
        {
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          "data": {
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WxwPvvEPMqNEYszLJ634dma9Oxl2+AvkFLrbsT2PNzmS27E+j4Pe3gstEhtKmYTwJNctQ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\nvoQQQkiVUfkSQgghVUblSwghhFTZL2Ody9Q+LsygAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f152ad8d630>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "VeF0lNJKCpcD",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "*italicized text*### Finite Difference - Your Assignment\n",
        "\n",
        "When we calculate finite differences using tabular data in an array form, we can only calculate the derivative using the values that exist in the array.\n",
        "\n",
        "The derivative at any point `p` in an array of data `y`, a function of `x`, is simply `(y[p+1]-y[p]) / (x[p+1]-x[p])`.\n"
      ]
    },
    {
      "metadata": {
        "id": "AIiHpEpdHbIZ",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 555
        },
        "outputId": "3b0b6cda-7fa2-4aa6-b474-49f339c8375a"
      },
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "# create your input array\n",
        "x = [-7, -5, -3, -1, 1, 3, 5, 7, 8]\n",
        "\n",
        "# fill it with values from a hand drawn -sigmoid-ish function\n",
        "y = [4, 3, 1.5, 0.3, -1.8, -3, -4, -4.1, -4]\n",
        "\n",
        "# tranpose it so it looks like the lecture matrix\n",
        "y = np.matrix([[-7, -5, -3, -1, 1, 3, 5, 7, 8],[4, 3, 1.5, 0.3, -1.8, -3, -4, -4.1, -4]])\n",
        "print(y)\n",
        "print(y.T)\n",
        "y = y.T\n",
        "\n",
        "# plot it as a sanity test:\n",
        "plt.plot(y[::,0:1:],y[::,1:2:],'ro')\n",
        "\n"
      ],
      "execution_count": 24,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[-7.  -5.  -3.  -1.   1.   3.   5.   7.   8. ]\n",
            " [ 4.   3.   1.5  0.3 -1.8 -3.  -4.  -4.1 -4. ]]\n",
            "[[-7.   4. ]\n",
            " [-5.   3. ]\n",
            " [-3.   1.5]\n",
            " [-1.   0.3]\n",
            " [ 1.  -1.8]\n",
            " [ 3.  -3. ]\n",
            " [ 5.  -4. ]\n",
            " [ 7.  -4.1]\n",
            " [ 8.  -4. ]]\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f6bce390e10>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 24
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f6bce01c278>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "4I992J5Ij176",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "# Linear interpolation between two values\n",
        "\n",
        "Filling in for data that is missing\n",
        "\n"
      ]
    },
    {
      "metadata": {
        "id": "usQ1YRkEj9W6",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 386
        },
        "outputId": "80da8d1d-9f4f-477c-ae09-7c314f40071e"
      },
      "cell_type": "code",
      "source": [
        "def interpolate(x,x_prime,fraction):\n",
        "  # fraction must be between 0 and 1\n",
        "  return x+(x_prime*fraction - x*(1.0-fraction))\n",
        "  \n",
        "m = np.matrix([[1,3],[-1.8,-3]]).T\n",
        "plt.plot(m[::,0:1:],m[::,1:2:],'ro')\n",
        "plt.plot(m[::,0:1:],m[::,1:2:],'black')\n",
        "plt.xlim((-7,8))\n",
        "plt.ylim((-5,4))\n",
        "x_at = 2.0\n",
        "plt.axvline(x_at,color=\"gray\")\n",
        "\n",
        "x_values = m[::,0:1:]\n",
        "y_values = m[::,1:2:]\n",
        "\n",
        "\n",
        "fraction = (x_at-x_values[0])/(x_values[1]-x_values[0])\n",
        "y_interpolated = interpolate(y_values[0],y_values[1],fraction)\n",
        "\n",
        "\n",
        "\n",
        "print(y_interpolated)\n",
        "plt.axhline(y_interpolated,color=\"gray\")\n"
      ],
      "execution_count": 68,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[-2.4]]\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.lines.Line2D at 0x7f6bcd4ccba8>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 68
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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QamtrVV3d+J9PD1hotDNiiw/5dtOHi0vsG83csmttba0KC2fp5MlKFRQUaPDgdE2fnq/E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\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f6bcd3e6d30>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "xJ4Htp4qX2Lt",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 364
        },
        "outputId": "f78d0305-b0c1-467e-cd4e-ad82e4352033"
      },
      "cell_type": "code",
      "source": [
        "# Lets actually compute the finite difference\n",
        "\n",
        "def d_f_x_finite_difference(x_array,f_x,at_x):\n",
        "  upper = np.argmax(x_array>at_x)\n",
        "  lower = upper-1\n",
        "  if(upper==0):\n",
        "    upper = 1\n",
        "    lower = 0\n",
        "  return ((f_x[upper]-f_x[lower])/(x_array[upper]-x_array[lower]))\n",
        "\n",
        "\n",
        "\n",
        "# choose a value along x to compute the derivative\n",
        "at_x = 6\n",
        "\n",
        "# compute the derivative with d_f_x_finite_difference\n",
        "d_x = d_f_x_finite_difference(y[::,0:1:],y[::,1:2:],at_x)\n",
        "\n",
        "# interpolate the value of x at `at_x` because we don't have \n",
        "# a sample there\n",
        "x_samples = y[::,0:1:]\n",
        "y_samples = y[::,1:2:]\n",
        "\n",
        "x_argm = np.argmax(x_samples>at_x)\n",
        "x_upper = x_samples[x_argm]\n",
        "x_lower = x_samples[x_argm-1]\n",
        "\n",
        "fraction = (at_x-x_lower)/(x_upper-x_lower)\n",
        "y_interpolated = interpolate(y_samples[x_argm-1],y_samples[x_argm],fraction)\n",
        "\n",
        "# graph the derivative as a line of the form y = mx+b\n",
        "mxb = d_x*x_samples.T - d_x*at_x+y_interpolated\n",
        "mxb = np.matrix(mxb).T\n",
        "\n",
        "plt.plot(y[::,0:1:],y[::,1:2:],'r')\n",
        "plt.plot(y[::,0:1:],mxb,'b')\n",
        "\n",
        "# getting `b` right is tricky, but the mx part is easy"
      ],
      "execution_count": 86,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f6bcd256438>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 86
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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fy/YC3L26kTFxnC7EEomRqMo3GAwyfPhw+vbtS58+ffjoI90bKJJ0LBbKhg7D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\n7iOSpavvEp3KV0SitushJU2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            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f6bcd61ff60>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "tlye6rREY3Qc",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Hypothetically:\n",
        "\n",
        "# Hey, we measured hypoconfabulator after configuring at 1.2 bar pressure,\n",
        "# here are the values:\n",
        "\n",
        "y = [9, 2, 3, 9, 1, 4, 4, 2] \n",
        "x = [0, 1, 2, 3, 4, 5, 6, 7] # We were not given x, x is simply the\n",
        "# indices of the y array"
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}