notebooks/Linear_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear regression\n",
    "\n",
    "We'll solve the problem of linear regression in a few different ways:\n",
    "\n",
    "- First we'll use `sklearn` to find the answer we're looking for\n",
    "- Analytic solution for univariate problems\n",
    "- The normal equation of linear algebra\n",
    "- Singular value decomposition (this is what `sklearn` actually uses)\n",
    "- Batch gradient descent\n",
    "- Stochastic gradient descent\n",
    "- Bayesian linear regression: maximum _a posteriori_ (MAP)\n",
    "- Simulated annealing\n",
    "- Differential evolution (with `scipy`)\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load some data\n",
    "\n",
    "We'll use our 'rocks' data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Vp</th>\n",
       "      <th>Vs</th>\n",
       "      <th>rho</th>\n",
       "      <th>Lithology</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>3045.6</td>\n",
       "      <td>1595.7</td>\n",
       "      <td>2.109121</td>\n",
       "      <td>sandstone</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>3000.6</td>\n",
       "      <td>1517.1</td>\n",
       "      <td>2.090342</td>\n",
       "      <td>sandstone</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3363.6</td>\n",
       "      <td>2041.5</td>\n",
       "      <td>2.131990</td>\n",
       "      <td>sandstone</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>3195.3</td>\n",
       "      <td>1606.2</td>\n",
       "      <td>2.184939</td>\n",
       "      <td>sandstone</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4237.5</td>\n",
       "      <td>2448.6</td>\n",
       "      <td>2.472231</td>\n",
       "      <td>sandstone</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       Vp      Vs       rho  Lithology\n",
       "0  3045.6  1595.7  2.109121  sandstone\n",
       "1  3000.6  1517.1  2.090342  sandstone\n",
       "2  3363.6  2041.5  2.131990  sandstone\n",
       "3  3195.3  1606.2  2.184939  sandstone\n",
       "4  4237.5  2448.6  2.472231  sandstone"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "df = pd.read_csv('https://geocomp.s3.amazonaws.com/data/RPC_simple.csv')\n",
    "\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try to predict `Vs` from `Vp`. Later, you can try adding `Rho_n` as another feature to see how you can use these algorithms on multivariate data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = df[['Vp', 'Vs']].values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "advanced"
    ]
   },
   "source": [
    "<h3>Random data</h3>\n",
    "\n",
    "<p>If you prefer, you can make a dataset from a multivariate normal distribution, so we can control the statistics of the features and target.</p>\n",
    "\n",
    "<b>To do this, run the following code:</b>\n",
    "\n",
    "<pre>\n",
    "import numpy as np\n",
    "\n",
    "num_samples = 100\n",
    "\n",
    "# The desired mean values of the sample.\n",
    "mu = np.array([5.0, 5.0, 10.0])\n",
    "\n",
    "# The desired covariance matrix.\n",
    "r = np.array([\n",
    "        [  2.00,  0.00, -2.00],\n",
    "        [  0.00,  2.00,  2.00],\n",
    "        [ -2.00,  2.00,  4.00]\n",
    "    ])\n",
    "\n",
    "# Generate the random samples.\n",
    "data = np.random.multivariate_normal(mu, r, size=num_samples)\n",
    "</pre>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Make `X` and `y`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can visualize the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot: xlabel='x', ylabel='y'>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "import pandas as pd\n",
    "\n",
    "df = pd.DataFrame(data, columns=['x', 'y'])\n",
    "\n",
    "sns.scatterplot(data=df, x='x', y='y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(400, 1) (400,)\n"
     ]
    }
   ],
   "source": [
    "# Univariate case\n",
    "X = data[:, 0:1]\n",
    "y = data[:, 1]\n",
    "\n",
    "print(X.shape, y.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the multivariate case, we should always standardize the data; in the univariate case, there's usually no need, but in my experience, gradient descent does not perform well on unscaled data, so let's standardize first."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import StandardScaler\n",
    "\n",
    "X = StandardScaler().fit_transform(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Draw a line\n",
    "\n",
    "If you accept that we want the model that minimizes the mean squared error on the estimates, then you could try to manually fit a line that does this. You could print out the figure above and draw a line with a ruler, then calculate the slope (`w`) and the intercept (`b`).\n",
    "\n",
    "Or you could use this widget:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "77e8e739adf34619ac5048a5963bb21f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=0, description='w', max=1000, min=-1000, step=50), IntSlider(value=1500,…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from ipywidgets import interact\n",
    "\n",
    "X_plt = np.arange(-3, 3, 0.1)\n",
    "y_plt_manual = None\n",
    "\n",
    "@interact(w=(-1000, 1000, 50), b=(0, 3000, 50))\n",
    "def draw(w, b):\n",
    "    global y_plt_manual  # So we can inspect it later.\n",
    "    y_plt_manual = w*X_plt + b\n",
    "    plt.scatter(X[:, 0], y)\n",
    "    plt.plot(X_plt, y_plt_manual, color='orange')\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve with `sklearn`\n",
    "\n",
    "Okay, but we have a computer, and we'd like something more robust, more reproducible, and less error prone. \n",
    "\n",
    "We love and trust `sklearn` so let's start with that as a sort of baseline."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MSE: 46591.207\n",
      "\n",
      "w = 431.562\n",
      "b = 2188.855\n"
     ]
    }
   ],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.metrics import mean_squared_error\n",
    "\n",
    "model = LinearRegression().fit(X, y)\n",
    "\n",
    "y_pred = model.predict(X)\n",
    "print(f\"MSE: {mean_squared_error(y, y_pred):.3f}\\n\")\n",
    "\n",
    "w, b = model.coef_, model.intercept_\n",
    "\n",
    "print(f\"w = {w[0].item():.3f}\\nb = {b:.3f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And plot the data with the model we just discovered:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_plt_sk = model.coef_[0]*X_plt + model.intercept_\n",
    "\n",
    "plt.scatter(X[:, 0], y) \n",
    "plt.plot(X_plt, y_plt_manual, color='orange', ls='--', label='manual fit')\n",
    "plt.plot(X_plt, y_plt_sk, color='r', label='scikit-learn')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Closed form solution: algebra\n",
    "\n",
    "For the univariate case, it's possible to derive equations for the parameters based on the partial derivatives of the loss; the equations have a unique solution (equation 18.3 in _Artificial Intelligence: A Modern Approach_ (3rd ed.) by Stuart Russell and Peter Norvig, Pearson, 2016):\n",
    "\n",
    "$$ w = \\frac{N \\Sigma x_i y_i - \\Sigma x_i \\Sigma y_i}{N \\Sigma x^2_i - (\\Sigma x_i)^2} $$\n",
    "\n",
    "$$ b = \\frac{1}{N} \\Sigma y_i - w \\Sigma x_i $$\n",
    "\n",
    "This is not too hard to compute in NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(431.5616664063004, 2188.8553005275007)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = X.shape[0]\n",
    "x = X[:, 0]\n",
    "\n",
    "w = (N * (x*y).sum() - x.sum() * y.sum()) / (N * (x**2).sum() - x.sum()**2)\n",
    "\n",
    "b = (y.sum() - w * x.sum()) / N\n",
    "\n",
    "w, b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These match the numbers we got from `sklearn`, so there's not much point in drawing the plot again."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Closed form solution: linear algebra\n",
    "\n",
    "There's another analytic way to solve this equation that extends more easily to multivariate datasets (i.e. where `X` has more than one column). In principle, this is much faster and easier to do than gradient descent. In practice, it might be expensive or even intractable, e.g. on very large datasets, or with custom loss functions.\n",
    "\n",
    "We need the so-called [**normal equation**](https://en.wikipedia.org/wiki/Linear_least_squares#Derivation_of_the_normal_equations):\n",
    "\n",
    "$$ \\mathbf{y} = \\mathbf{w}\\mathbf{X} $$\n",
    "\n",
    "We want the weights $\\mathbf{w}$, so:\n",
    "\n",
    "$$ \\mathbf{w} = \\mathbf{X}^{-1}\\mathbf{y} $$\n",
    "\n",
    "The catch is that $\\mathbf{X}^{-1}$ may not be possible to compute, so we have to use [the pseudoinverse](https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse) of `X`, aka Moore&ndash;Penrose inverse. In our case:\n",
    "\n",
    "$$ \\mathbf{X}^{-1} = (\\mathbf{X}^\\top \\mathbf{X})^{-1} \\mathbf{X}^\\top $$\n",
    "\n",
    "This approach is easier if we add a column of 1's to `X`, and concatenate the weights `w` with the bias `b`, so that we have a single data matrix and a single weight vector. This is sometimes called the **design matrix**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.97030224,  1.        ],\n",
       "       [-1.03277819,  1.        ],\n",
       "       [-0.52880548,  1.        ],\n",
       "       [-0.76246555,  1.        ],\n",
       "       [ 0.68447761,  1.        ]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = X.shape[0]\n",
    "X_ = np.c_[X, np.ones((N, 1))]\n",
    "\n",
    "X_[:5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That [`np.c_`](https://numpy.org/doc/stable/reference/generated/numpy.c_.html) implements a concatenation trick. You can also use `np.hstack`.\n",
    "\n",
    "Now we can implement the 'naive' version of the normal equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.set_printoptions(suppress=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.56166641, 2188.85530053])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy.linalg as la\n",
    "\n",
    "X_inv = la.inv(X_.T @ X_) @ X_.T  # The 'pseudoinverse' of X.\n",
    "\n",
    "X_inv @ y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The same numbers again: we solved it! Pretty cool. But it can be slow for large datasets and besides it won't always work &mdash; so we need a more robust method.\n",
    "\n",
    "We could use `numpy.linalg.pinv()` to get the pseudoinverse directly..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.56166641, 2188.85530053])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.pinv(X_) @ y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... but that's no fun. Let's see what this, and `sklearn` are using internally."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Singular value decomposition\n",
    "\n",
    "The above method is computationally expensive (i.e. slow) and won't always work.\n",
    "\n",
    "Internally, `sklearn` uses `scipy.linalg.lstsq()` and this, in turn, uses one of three algorithms in the Fortran library LAPACK (which NumPy depends on): `gelsd`, `gelsy` and `gelss`. Of these, `gelsd` and `gelss` use the singular value decomposition of our matrix `X_`.\n",
    "\n",
    "In brief, we can solve:\n",
    "\n",
    "$$ \\mathbf{w} = \\mathbf{V} \\mathbf{\\Sigma}^{+} \\mathbf{U}^\\top \\, \\mathbf{y} $$\n",
    "\n",
    "[Read more about this.](https://towardsdatascience.com/why-gradient-descent-and-normal-equation-are-bad-for-linear-regression-928f8b32fa4f)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's how this equation looks in NumPy; the only really tricky part is forming $\\Sigma^{+}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.56166641, 2188.85530053])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, sigma, VT = la.svd(X_, full_matrices=False)\n",
    "\n",
    "VT.T @ np.diag(1/sigma).T @ U.T @ y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "## Gradient descent\n",
    "\n",
    "Let's solve the problem using gradient descent. We clearly don't _need_ gradient descent here, but it's a good optimization method to know about because most problems don't have closed form solutions. In general, if we can form the derivative of the cost function (i.e. it is 'differentiable'), we can use this method.\n",
    "\n",
    "So in order to optimize the weights using gradient descent, we need to know the **gradient** of the **cost function**. The cost $C$ over $N$ samples is given by:\n",
    "\n",
    "$$ C = \\frac{1}{N} \\sum_{i=0}^{N} (\\mathbf{y}_i - \\hat{\\mathbf{y}}_i)^2 $$\n",
    "\n",
    "where the estimate $\\hat{\\mathbf{y}}$ is given by the matrix product of the weights $\\theta$ and the 'augmented' data $\\bar{\\mathbf{X}}$ (the data plus an extra column of ones):\n",
    "\n",
    "$$ \\hat{\\mathbf{y}} = \\mathbf{w} \\mathbf{X} + b = \\theta \\bar{\\mathbf{X}} $$\n",
    "\n",
    "Using this 'combined' weight matrix is easier than dealing with the weights and bias term separately. Note that $\\theta$ is a vector, although it's displayed with an italic character here.\n",
    "\n",
    "#### Chain rule\n",
    "\n",
    "There are 2 functions here: the expression for $C$ and the one for $\\hat{\\mathbf{y}}$. The chain rule tells us how to differentiate a function of a function:\n",
    "\n",
    "$$ \\frac{\\mathrm{d}}{\\mathrm{d}x}\\big[ f(g(x)) \\big] = f^\\prime(g(x))\\ \\times\\ g^\\prime(x) $$\n",
    "\n",
    "Therefore the gradient $g$ is given by:\n",
    "\n",
    "$$ g = \\frac{\\mathrm{d}C}{\\mathrm{d}\\theta} = \\frac{\\mathrm{d}C}{\\mathrm{d}\\hat{\\mathbf{y}}}\\ \\times\\ \\frac{\\mathrm{d}\\hat{\\mathbf{y}}}{\\mathrm{d}\\theta} $$\n",
    "\n",
    "So our problem now looks like this:\n",
    "\n",
    "$$ g = \\frac{2}{N}\\sum_{i=0}^{N} (\\mathbf{y}_i - \\hat{\\mathbf{y}}_i)\\ \\times\\ -\\mathbf{X}_i $$\n",
    "\n",
    "so if the error $\\mathbf{e} = (\\mathbf{y} - \\hat{\\mathbf{y}})$ then:\n",
    "\n",
    "$$ g = -\\frac{2}{N}\\sum_{i=0}^{N} \\mathbf{e}_i \\mathbf{X}_i $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       0 5023924.206\n",
      "     500 718852.312\n",
      "    1000 137389.832\n",
      "    1500 58854.880\n",
      "    2000 48247.594\n",
      "    2500 46814.926\n",
      "    3000 46621.424\n",
      "    3500 46595.288\n",
      "    4000 46591.758\n",
      "    4500 46591.282\n",
      "     end 46591.217\n"
     ]
    }
   ],
   "source": [
    "alpha = 1e-3   # This must be tuned to the scale of the input.\n",
    "epochs = 5_000\n",
    "\n",
    "# Initialize.\n",
    "cost_history = []\n",
    "theta = np.zeros(X_.shape[1])\n",
    "\n",
    "for i in range(epochs):\n",
    "    # Make estimate with current weights.\n",
    "    y_hat = theta @ X_.T\n",
    "    \n",
    "    # Compute error and cost.\n",
    "    err = y - y_hat\n",
    "    cost = np.sum(err**2) / N\n",
    "    cost_history.append(cost)\n",
    "    if i % (epochs // 10) == 0:\n",
    "        print(f\"{i:>8d} {cost:.3f}\")\n",
    "        \n",
    "    # Compute gradients (one for each feature).\n",
    "    grad = (-2/N) * np.sum(err[:, None] * X_, axis=0)\n",
    "    \n",
    "    # Adjust weights.\n",
    "    theta -= alpha * grad\n",
    "\n",
    "print(f\"{'end':>8s} {cost:.3f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(cost_history)\n",
    "plt.yscale('log')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's make sure it's similar to what we got before, which was"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(431.5616664063004, 2188.8553005275007)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# With SVD...\n",
    "w, b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.54226875, 2188.75691675])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# With gradient descent...\n",
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stochastic gradient descent\n",
    "\n",
    "Here, we're computing the cost over the entire dataset. In practice, this might not be possible, especially on high-dimensional datasets. We can reduce the computational burden by computing the cost over smaller 'batches'. This is known as 'stochastic gradient descent'. \n",
    "\n",
    "We'll also shuffle the data before each epoch (one epoch is one pass through the entire dataset). This helps to avoid getting stuck in a local minimum, because every time we take a step on the cost surface, we're on a slightly different version of the surface."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       0 4664949.291\n",
      "      50 992912.647\n",
      "     100 222309.973\n",
      "     150 112513.943\n",
      "     200 44707.229\n",
      "     250 29450.188\n",
      "     300 80772.023\n",
      "     350 47068.038\n",
      "     400 59305.019\n",
      "     450 54844.389\n",
      "     end 60063.054\n"
     ]
    }
   ],
   "source": [
    "alpha = 1e-3\n",
    "epochs = 500\n",
    "batch_size = 50\n",
    "\n",
    "splits = np.arange(batch_size, N, batch_size)\n",
    "idx = np.arange(N)\n",
    "\n",
    "# Initialize.\n",
    "cost_history = []\n",
    "theta = np.random.random(X_.shape[1])\n",
    "\n",
    "for i in range(epochs):\n",
    "    \n",
    "    # Shuffle the data.\n",
    "    np.random.shuffle(idx)\n",
    "    X_shuffle = X_[idx]\n",
    "    y_shuffle = y[idx]\n",
    "    X_batches = np.split(X_shuffle, splits)\n",
    "    y_batches = np.split(y_shuffle, splits)\n",
    "\n",
    "    for j, (X_batch, y_batch) in enumerate(zip(X_batches, y_batches)):\n",
    "\n",
    "        # Make estimate with current weights.\n",
    "        y_hat = theta @ X_batch.T\n",
    "\n",
    "        # Compute error and (once per epoch) cost.\n",
    "        err = y_batch - y_hat\n",
    "        \n",
    "        if j == 0:\n",
    "            cost = np.sum(err**2) / batch_size\n",
    "            cost_history.append(cost)\n",
    "\n",
    "        # Compute gradients (one for each feature).\n",
    "        grad = (-2/batch_size) * np.sum(err[:, None] * X_batch, axis=0)\n",
    "\n",
    "        # Adjust weights.\n",
    "        theta -= alpha * grad\n",
    "\n",
    "    if i % (epochs // 10) == 0:\n",
    "        print(f\"{i:>8d} {cost:.3f}\")\n",
    "\n",
    "print(f\"{'end':>8s} {cost:.3f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(cost_history)\n",
    "plt.yscale('log')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that this converges much faster than using the entire dataset: only 500 epochs, compared to 5000."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.41794573, 2188.12743804])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This compares well with what we got from the SVD method above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(431.5616664063004, 2188.8553005275007)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w, b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And let's check the new parameters against the previous solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Two parameters:\n",
    "w_sgd, b_sgd = theta\n",
    "\n",
    "y_plt = w_sgd*X_plt + b_sgd\n",
    "\n",
    "plt.scatter(X[:, 0], y) \n",
    "plt.plot(X_plt, y_plt_sk, color='r')  # regression line\n",
    "plt.plot(X_plt, y_plt, '--', color='c')  # regression line\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It matches!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Compare with `sklearn`'s own SGD regressor\n",
    "\n",
    "Let's try with [`sklearn.linear_model.SGDRegressor`](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.SGDRegressor.html), which implements a generic SGD regressor that can solve linear regression (with or without regularization)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MSE: 46593.118\n",
      "\n",
      "w0 = 430.388\n",
      "b = 2189.586\n"
     ]
    }
   ],
   "source": [
    "from sklearn.linear_model import SGDRegressor\n",
    "\n",
    "model = SGDRegressor(alpha=0).fit(X, y)  # Turn off regularization.\n",
    "\n",
    "y_pred = model.predict(X)\n",
    "print(f\"MSE: {mean_squared_error(y, y_pred):.3f}\\n\")\n",
    "\n",
    "(w,), (b, ) = model.coef_, model.intercept_\n",
    "print(f\"w0 = {w:.3f}\\nb = {b:.3f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It matches too!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bayesian linear regression\n",
    "\n",
    "In general, Bayesian methods try to account for uncertainty by treating everything as a distribution. Correspondingly, instead of finding _best fit_ parameters, a Bayesian approach will yield _posterior distributions_ for our parameters.\n",
    "\n",
    "Broadly, there are two Bayesian approaches: \"maximum likelihood estimation\" (MLE), which is broadly equivalent to ordinary least squares (under the assumption that the data contains Gaussian noise), and \"maximum a posteriori\" (MAP). These approach the problem in different ways:\n",
    "\n",
    "- MLE: How likely is the data, given the parameters?\n",
    "- MAP: How likely are the parameters, given the data?\n",
    "\n",
    "We'll apply MAP here, guided by Chris Bishop's book, _Pattern Recognition and Machine Learning_ (Springer, 2006, [PDF](https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf)), and by [this blog post](https://brunomaga.github.io/Bayesian-Linear-Regression)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.56165971, 2188.85526658])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N, M = X_.shape\n",
    "alpha = 1  # Init.\n",
    "beta = 1   # Initialize as 1/variance.\n",
    "mean = np.zeros(M)\n",
    "lamda = 1e-6  # Regularization.\n",
    "\n",
    "snaps = []  # Capture the first guess in each loop.\n",
    "\n",
    "for iteration in range(3):\n",
    "    weights = []\n",
    "    S_0_inv = np.identity(M) / alpha\n",
    "    for i, (xi, yi) in enumerate(zip(X_, y)):\n",
    "\n",
    "        # Update the (inverse) covariance, Bishop's equation 3.51.\n",
    "        S_N_inv = S_0_inv + beta * np.outer(xi, xi)\n",
    "\n",
    "        # Update the mean, Bishop's equation 3.50.\n",
    "        S_N = la.inv(S_N_inv)\n",
    "        mean = S_N @ (S_0_inv @ mean + beta * xi * yi)\n",
    "        S_0_inv = S_N_inv\n",
    "\n",
    "        weights.append((mean, S_N_inv))\n",
    "        \n",
    "        # Snapshot early iterations for plotting later.\n",
    "        if iteration == 0 and i % 10 == 0:\n",
    "            snaps.append((i, ) + weights[-1])\n",
    "\n",
    "# Final coefficients:\n",
    "weights[-1][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we step over the early data points, there are dramatic updates to the weights. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x500 with 5 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "\n",
    "fig, axs = plt.subplots(ncols=5, sharey=True, figsize=(15, 5))\n",
    "\n",
    "for ax, (i, mean, S_N_inv) in zip(axs, snaps[:5]):\n",
    "\n",
    "    S_N = np.linalg.inv(S_N_inv)\n",
    "    dist = stats.multivariate_normal(mean=mean, cov=S_N_inv)\n",
    "\n",
    "    w = np.linspace(-1000, 1000, 100)\n",
    "    b = np.linspace(-1000, 3000, 100)\n",
    "    W = np.dstack(np.meshgrid(w, b))\n",
    "    pdf = dist.pdf(W)\n",
    "\n",
    "    im = ax.imshow(pdf**0.0025, origin='lower',\n",
    "                   extent=[-1000, 1000, -1000, 3000],\n",
    "                  )\n",
    "    ax.scatter(*weights[-1][0], marker='x', c='w')\n",
    "    ax.set_title(f'{i+1} points')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The resulting model is comparable to the one we found with `sklearn`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w, b = weights[-1][0]\n",
    "y_plt = w*X_plt + b\n",
    "\n",
    "plt.scatter(X[:, 0], y) \n",
    "plt.plot(X_plt, y_plt_sk, color='r')  # regression line\n",
    "plt.plot(X_plt, y_plt, '--', color='c')  # regression line\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They match! Nice."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulated annealing\n",
    "\n",
    "From [this Cross Validated post](https://stats.stackexchange.com/a/340687/74162):\n",
    "\n",
    "> Simulated annealing's fundamental mechanism is to compare the current state with a proposed state, then (probabilistically) decide whether to move to the proposed state or not.\n",
    "\n",
    "The same post has some R code, translated to Python below. It's not very sophisticated (e.g., no guesstimate of the required `scale` hyperparameter, and no early stopping), but it gets the job done."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import t\n",
    "\n",
    "\n",
    "def mean_squared_error(X, y, w):\n",
    "    \"\"\"Mean square error for a given guess.\"\"\"\n",
    "    degrees_of_freedom = len(y)\n",
    "    y_hat = X @ w.reshape(-1, 1)\n",
    "    return np.sum((y - y_hat.T)**2) / degrees_of_freedom\n",
    "\n",
    "\n",
    "def simulated_annealing(X, y, scale=1, max_iter=1000, step=0.05):\n",
    "    \"\"\"\n",
    "    Implements simple simulated annealing algorithm.\n",
    "    Starting from: https://stats.stackexchange.com/a/340687/74162\n",
    "    \"\"\"\n",
    "    # Initialize everything.\n",
    "    rng = np.random.default_rng(42)\n",
    "    k_best = 1\n",
    "    temp = 1\n",
    "    w = w_best = np.ones(X.shape[-1])\n",
    "    mse = mse_best = mean_squared_error(X, y, w)\n",
    "\n",
    "    for k in range(max_iter):\n",
    "        \n",
    "        # Update the scaling factor to control learning rate.\n",
    "        temp *= (1 - step)\n",
    "        \n",
    "        # Make and score a new proposal.\n",
    "        w_prop = t.rvs(size=2, df=3, loc=w, scale=scale)\n",
    "        mse_prop = mean_squared_error(X, y, w_prop)\n",
    "        test = np.exp(-(mse_prop - mse) / temp)\n",
    "        \n",
    "        # Use the test, and a random element, to decide\n",
    "        # if we're updating or not.\n",
    "        if (mse_prop < mse) or (rng.uniform(0, 1) < test):\n",
    "            w = w_prop\n",
    "            mse = mse_prop\n",
    "            \n",
    "            if mse_prop < mse_best:\n",
    "                w_best = w\n",
    "                mse_best = mse\n",
    "\n",
    "    return w_best"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/tmp/ipykernel_294098/3801482789.py:31: RuntimeWarning: overflow encountered in exp\n",
      "  test = np.exp(-(mse_prop - mse) / temp)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 431.55809991, 2189.1815819 ])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simulated_annealing(X_, y, scale=10, max_iter=2000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ah, the familiar parameters, very cool. \n",
    "\n",
    "There's a simulated annealing algorithm in `scipy` too. It's more sophisticated than the algorithm implemented above, but of course yields roughly the same parameters. [Read about it.](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.dual_annealing.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.561584  , 2188.85511069])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import dual_annealing\n",
    "\n",
    "def sum_squares(w, X, y):\n",
    "    \"\"\"\n",
    "    The function providing the cost. `w` contains the parameters.\n",
    "    \"\"\"\n",
    "    y_hat = X @ w\n",
    "    return np.sum((y - y_hat)**2)\n",
    "\n",
    "bounds = [(-1e4, 1e4), (-1e4, 1e4)]\n",
    "result = dual_annealing(sum_squares, bounds, args=(X_, y))\n",
    "result.x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Differential evolution\n",
    "\n",
    "Rather than implementing more stochastic optimizers ourselves, let's have a look in the `scipy.optimize` module. In there, apart from linear least squares and simulated annealing, anod some other exotic solvers, we'll find [differential evolution](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.differential_evolution.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 431.56209855, 2188.85513769])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import differential_evolution\n",
    "\n",
    "bounds = [(-1e4, 1e4), (-1e4, 1e4)]\n",
    "result = differential_evolution(sum_squares, bounds, args=(X_, y))\n",
    "result.x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It works, sweet.\n",
    "\n",
    "Other stochastic optimizers include particle swarms (e.g. see [PySwarms](https://pyswarms.readthedocs.io/en/latest/)) and genetic algorithms. [There are lots](https://en.wikipedia.org/wiki/Evolutionary_algorithm) of related algorithms. Thanks to Leo Cahya Dinendra for suggesting I add some to this notebook."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More...\n",
    "\n",
    "- You should dig around some more in [`scipy.optimize`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html), it's fast.\n",
    "- Take a look at [`statsmodels`](https://www.statsmodels.org/dev/generated/statsmodels.regression.linear_model.OLS.html) too, it's also fast.\n",
    "- Check out [`pylops`](https://pylops.readthedocs.io/en/latest/gallery/plot_linearregr.html) for linear operators.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "\n",
    "&copy; 2023 Matt Hall / [https://github.com/kwinkunks](https://github.com/kwinkunks) / MIT licence"
   ]
  }
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