HW6.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from sympy import Symbol, lambdify\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "train_data = pd.read_csv('Input/training.dat', sep=' ', header=None, names=['x', 'y']);\n",
    "test_data = pd.read_csv('Input/test.dat', sep=' ', header=None, names=['x', 'y']);\n",
    "\n",
    "x_train = np.array(train_data['x'])\n",
    "y_train = np.array(train_data['y'])\n",
    "\n",
    "x_test = np.array(test_data['x'])\n",
    "y_test = np.array(test_data['y'])\n",
    "\n",
    "norm_train = np.array(train_data)\n",
    "norm_train = (norm_train - norm_train.mean(0)) / norm_train.std(0)\n",
    "\n",
    "# comment out below lines to use normalized data\n",
    "\n",
    "#x_train = norm_train[:,0]\n",
    "#y_train = norm_train[:,1]\n",
    "\n",
    "w0 = Symbol(\"w0\")\n",
    "w1 = Symbol(\"w1\")\n",
    "w2 = Symbol(\"w2\")\n",
    "\n",
    "func_a_coefficients = np.array([np.sum(y_train**2), 1*np.size(y_train), np.sum((x_train**2)), np.sum(2*y_train), np.sum(2*y_train*x_train), np.sum(2*x_train)])\n",
    "func_a_variables = np.array([1, w0**2, w1**2, -w0, -w1, w0*w1])\n",
    "\n",
    "func_b_coefficient = np.array([np.sum(y_train**2), 1*np.size(y_train), np.sum(x_train**2), np.sum(x_train**4), np.sum(2*y_train), np.sum(2*y_train*x_train), np.sum(2*y_train*x_train**2), np.sum(2*x_train), np.sum(2*x_train**2), np.sum(2*x_train**3)])\n",
    "func_b_variables = np.array([1, w0**2, w1**2, w2**2, -w0, -w1, -w2, w0*w1, w0*w2, w1*w2])\n",
    "\n",
    "func_a = np.sum(func_a_coefficients*func_a_variables)\n",
    "f_a = lambdify([[w0, w1]], func_a, \"numpy\")\n",
    "gf_a = lambdify([[w0, w1]], func_a.diff([[w0, w1]]), \"numpy\")\n",
    "grad_fa = lambda x_arr : np.array(gf_a(x_arr), 'float64').reshape(1,len(x_arr))\n",
    "\n",
    "func_b = np.sum(func_b_coefficient*func_b_variables)\n",
    "f_b = lambdify([[w0, w1, w2]], func_b, \"numpy\")\n",
    "gf_b = lambdify([[w0, w1, w2]], func_b.diff([[w0, w1, w2]]), \"numpy\")\n",
    "grad_fb = lambda x_arr : np.array(gf_b(x_arr), 'float64').reshape(1,len(x_arr))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Useful Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "regA = lambda w_s, x_arr : x_arr * w_s[1,0] + w_s[0,0]\n",
    "regB = lambda w_s, x_arr : x_arr**2 * w_s[2,0] + x_arr * w_s[1,0] + w_s[0,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plotRegressionGraph(data, regFunc, w_star, title, name=\"graph\"):\n",
    "    xmin = data[:,0].min()\n",
    "    xmax = data[:,0].max()\n",
    "    t1 = np.arange(xmin-1, xmax+1, 0.1)\n",
    "    plt.figure()\n",
    "    plt.plot(t1, regFunc(w_star, t1), 'b-', label='Regression line')\n",
    "    plt.scatter(data[:,0], data[:,1], color=\"black\", label=\"Data points\")\n",
    "    plt.title(title)\n",
    "    plt.legend()\n",
    "    plt.savefig(\"{0}.png\".format(name))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "np_str = lambda x_k : np.array2string(x_k.reshape(len(x_k)), precision=3, separator=',')\n",
    "\n",
    "f_str = lambda x : \"{0:.4f}\".format(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "class OutputTable:    \n",
    "    def __init__(self):\n",
    "        self.table = pd.DataFrame([],columns=['k', 'x^k', 'f(x^k)', 'd^k', 'a^k', 'x^k+1'])\n",
    "    def add_row(self, k, xk, fxk, dk, ak, xkp):\n",
    "        self.table.loc[len(self.table)] = [k, np_str(xk), f_str(fxk.item()), np_str(dk), ak, np_str(xkp)]\n",
    "    def print_latex(self):\n",
    "        print(self.table.to_latex(index=False))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part A : Least Square Method with Steepest Descent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exact Line Search"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def BisectionMethod(f,epsilon, a=-100,b=100) :\n",
    "    iteration=0\n",
    "    while (b - a) >= epsilon:\n",
    "        x_1 = (a + b) / 2\n",
    "        fx_1 = f(x_1)\n",
    "        if f(x_1 + epsilon) <= fx_1:\n",
    "            a = x_1\n",
    "        else:\n",
    "            b = x_1\n",
    "        iteration+=1\n",
    "    x_star = (a+b)/2\n",
    "    return x_star\n",
    "\n",
    "def NewtonsMethod(df, ddf, x_0, epsilon, a, b):\n",
    "    iteration = 0\n",
    "    while True:\n",
    "        dfx0 = df(x_0)\n",
    "        ddfx0 = ddf(x_0)\n",
    "        x_1 = x_0-dfx0/ddfx0\n",
    "        iteration +=1\n",
    "        if abs(x_0-x_1)<epsilon:\n",
    "            break\n",
    "        if x_1<a or x_1>b:\n",
    "            break\n",
    "        x_0 = x_1\n",
    "    x_star = x_0\n",
    "    return x_star\n",
    "\n",
    "def ExactLineSearch(f, x0, d, eps=10**(-10)):\n",
    "    alpha = Symbol('alpha')\n",
    "    function_alpha = f(np.array(x0)+alpha*np.array(d)).item()\n",
    "    f_alp = lambdify(alpha, function_alpha)\n",
    "    bisecEps = 10**(-4)\n",
    "    alp_star = BisectionMethod(f_alp, epsilon=bisecEps)\n",
    "    df_alp = lambdify(alpha, function_alpha.diff(alpha))\n",
    "    ddf_alp = lambdify(alpha, function_alpha.diff(alpha).diff(alpha))\n",
    "    alp_star = NewtonsMethod(df_alp, ddf_alp, alp_star, eps, alp_star-bisecEps, alp_star+bisecEps)\n",
    "    return alp_star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Steepest Descent Method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def steepestDescentMethod(f, grad_f, x_0, descentEpsilon, exactLineEpsilon=10**(-10)):\n",
    "    xk = np.array(x_0).reshape(-1,1)\n",
    "    k = 0\n",
    "    stop = False\n",
    "    output = OutputTable()\n",
    "    while(stop == False):\n",
    "        d = - np.transpose(grad_f(xk))\n",
    "        if(np.linalg.norm(d) < descentEpsilon):\n",
    "            stop = True\n",
    "        else:\n",
    "            a = ExactLineSearch(f,xk,d, exactLineEpsilon)\n",
    "            xkp = xk + a*d\n",
    "            output.add_row(k, xk, f(xk), d, a, xkp)\n",
    "            k += 1\n",
    "            xk = xkp\n",
    "            if(k>100):\n",
    "                break\n",
    "    output.add_row(k,xk,f(xk),d,None,np.array([]))\n",
    "    print(\"Total iteration : {0}\".format(k))\n",
    "    return xk, f(xk).item(), output"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total iteration : 9\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(array([[113.34987312],\n",
       "        [  0.74045582]]),\n",
       " 3835642.7250206247,\n",
       " 50246.68568956938,\n",
       " 10689612467.303635)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ws_a, fs_a, outputs_a = steepestDescentMethod(f_a, grad_fa, [0,0], 0.005)\n",
    "\n",
    "SSE_train_a = fs_a\n",
    "MSE_test_a = np.sum((y_test-regA(ws_a,x_test))**2)/np.size(y_test)\n",
    "var_test_a = np.sum((MSE_test_a-(y_test-regA(ws_a,x_test))**2)**2)/(np.size(y_test)-1)\n",
    "ws_a, SSE_train_a, MSE_test_a, var_test_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotRegressionGraph(np.array(train_data), regA, ws_a, \"Linear Regression and Train Data\", \"part1a_train\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotRegressionGraph(np.array(test_data), regA, ws_a, \"Linear Regression and Test Data\", \"part1a_test\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total iteration : 101\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(array([[ 5.26127578e-03],\n",
       "        [ 1.49525475e+00],\n",
       "        [-7.66019510e-04]]),\n",
       " 3506233.1120957993,\n",
       " 43743.010704807275,\n",
       " 8605111510.643654)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ws_b, fs_b, outputs_b = steepestDescentMethod(f_b, grad_fb, [0,0,0], 0.005)\n",
    "\n",
    "SSE_train_b = fs_b\n",
    "MSE_test_b = np.sum((y_test-regB(ws_b,x_test))**2)/np.size(y_test)\n",
    "var_test_b = np.sum((MSE_test_b-(y_test-regB(ws_b,x_test))**2)**2)/(np.size(y_test)-1)\n",
    "ws_b, SSE_train_b, MSE_test_b, var_test_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotRegressionGraph(np.array(train_data), regB, ws_b, \"Linear Regression and Train Data\", \"part1b_train\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotRegressionGraph(np.array(test_data), regB, ws_b, \"Linear Regression and Test Data\", \"part1b_test\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part B : Neural Network"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigmoidalFunc = lambda output_array : 1 / (1 + np.exp(-output_array))\n",
    "sigmoidalDeriv = lambda hiddenlayer : hiddenlayer * (1 - hiddenlayer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def backpropagation(trainingData, hiddenLayerSize, alpha = 0.5, momentum = 0.9, epsilon = 0.001, seed = 440):\n",
    "    np.random.seed(seed)\n",
    "    t = 0\n",
    "    patterns = np.copy(trainingData)\n",
    "    patterns = np.insert(patterns, 0, -1, axis=1) # x0 = -1 unit is added\n",
    "    P = np.size(patterns, 0) # pattern size\n",
    "    I = 1 # output unit size\n",
    "    K = np.size(patterns, 1) - I # input layer size\n",
    "    J = hiddenLayerSize + 1 # h0 = -1 is added\n",
    "    w_matrix = np.random.rand(J, K) # weights between input and hidden layer (we will exclude first row in the result since h0 is excluded)\n",
    "    W_matrix = np.random.rand(I, J) # weight between hidden and output layer\n",
    "    while(alpha >= epsilon):\n",
    "        np.random.shuffle(patterns)\n",
    "        x = np.transpose(patterns[:,:-1]).reshape(K, -1)\n",
    "        y = patterns[:,-1]\n",
    "        H = np.zeros(J)\n",
    "        H[0] = -1 # h0 is equal to -1\n",
    "        O = np.zeros_like(y)\n",
    "        for p in range(P):\n",
    "            for j in range(1,J):\n",
    "                hj = np.sum(w_matrix[j] * x[:,p])\n",
    "                H[j] = sigmoidalFunc(hj)\n",
    "            for i in range(I):\n",
    "                o = np.sum(W_matrix[i] * H)\n",
    "                O[p] = o # linear function g(x) = x\n",
    "            S_O = 0 # since there is only one output unit\n",
    "            S_H = np.zeros_like(H)\n",
    "            for i in range(I):\n",
    "                S_O = 1 * (y[p] - O[p])\n",
    "            for j in range(1,J):\n",
    "                S_H[j] = sigmoidalDeriv(H[j]) * np.sum(W_matrix[0,j] * S_O)\n",
    "            for j in range(J):\n",
    "                dWj = alpha * S_O * H[j]\n",
    "                W_matrix[0,j] += dWj\n",
    "            for k in range(K):\n",
    "                dwk = alpha * S_H * x[k,p]\n",
    "                w_matrix[:,k] += dwk\n",
    "        alpha *= momentum\n",
    "        t += 1\n",
    "        actualHiddens = sigmoidalFunc(w_matrix @ x)\n",
    "        actualHiddens[0,:] = -1 # h1, ..., hj\n",
    "        actualOutputMatrix = W_matrix @ actualHiddens # o1, ..., oi\n",
    "        error = np.sum(np.square(y - actualOutputMatrix))\n",
    "        #print(\"Iteration {0} : error = {1}\".format(t,error))\n",
    "    w_matrix = w_matrix[1:] # first row is removed since it corresponds to H0\n",
    "    return w_matrix, W_matrix, error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def backpropagationWithMatrix(patterns, hiddenLayerSize, alpha = 0.5, momentum = 0.9, epsilon = 0.001, seed = 440):\n",
    "    np.random.seed(seed)\n",
    "    t = 0\n",
    "    P = np.size(patterns, 0)\n",
    "    w_matrix = np.random.rand(hiddenLayerSize, np.size(patterns,1))*1 # patterns data includes y values, its column size is selected since we will add x0 to input layer\n",
    "    W_matrix = np.random.rand(1, hiddenLayerSize+1)*1 # we will add h0 to hidden layer\n",
    "    while(alpha > epsilon):\n",
    "        np.random.shuffle(patterns)\n",
    "        desiredOutputs = patterns[:,-1].reshape(-1,1)\n",
    "        inputLayers = np.transpose(np.insert(patterns, 0, -1, axis=1)[:,:-1]) # x0 is added to all patterns and its value is -1, output values are excluded\n",
    "        hiddenLayer = np.zeros((hiddenLayerSize+1, 1)) # hiddenlayersize doesn't include h0 so it's added\n",
    "        hiddenLayer[0,:] = -1 # h0 is equal to -1\n",
    "        actualOutput = np.zeros_like(desiredOutputs)\n",
    "        for p in range(P):\n",
    "            hiddenLayer[1:] = sigmoidalFunc(w_matrix @ inputLayers[:,p].reshape(-1,1))\n",
    "            actualOutput[p] = W_matrix @ hiddenLayer\n",
    "            # since the function is linear, net output is equal to actual output\n",
    "            S_output = (1 * (desiredOutputs[p] - actualOutput[p])).reshape(-1,1)\n",
    "            S_hidden = (sigmoidalDeriv(hiddenLayer[1:]) * (np.transpose(W_matrix[:,1:]) @ S_output)).reshape(-1,1)\n",
    "            delta_W = alpha * S_output @ np.transpose(hiddenLayer)\n",
    "            W_matrix += delta_W\n",
    "            delta_w = alpha * S_hidden @ np.transpose(inputLayers[:,p].reshape(-1,1))\n",
    "            w_matrix += delta_w\n",
    "        alpha = momentum * alpha\n",
    "        t += 1\n",
    "        actualHiddens = sigmoidalFunc(w_matrix @ inputLayers) # h1, ..., hj\n",
    "        actualOutputMatrix = W_matrix @ np.insert(actualHiddens, 0, -1, axis=0) # o1, ..., oi\n",
    "        error = np.sum(np.square(desiredOutputs - np.transpose(actualOutputMatrix)))\n",
    "        #print(\"Iteration {0} : error = {1}\".format(t,error))\n",
    "    return w_matrix, W_matrix, error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[ 8.47339514e-01,  3.97995080e-01],\n",
       "        [ 6.16147498e-01,  6.29039082e-01],\n",
       "        [ 3.13448399e+02, -3.69468170e+04]]),\n",
       " array([[-127.77235014,  128.42128315,  128.78880676,  249.38554076]]),\n",
       " 7954415.602024415)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "patterns = np.array(train_data)\n",
    "backpropagation(patterns, 3, seed=440)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[  0.62495165, -21.52694768],\n",
       "        [  0.25547392,   0.39632991],\n",
       "        [  0.3773151 ,   0.99657423]]),\n",
       " array([[-128.24385088, 1238.84623456,  129.41258477,  128.96205743]]),\n",
       " 7953714.347670076)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "patterns = np.array(train_data)\n",
    "backpropagationWithMatrix(patterns, 3, seed=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[0.39799508, 0.6161475 , 0.62903908],\n",
       "        [0.15285732, 0.17501496, 0.38176556],\n",
       "        [0.26716745, 0.63469106, 0.28434213]]),\n",
       " array([[-96.51737743,  98.39875203,  98.06407988,  97.76263623]]),\n",
       " 7954322.33853856)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "patterns2 = np.insert(np.array(train_data), 1, np.square(train_data['x']), axis=1)\n",
    "backpropagation(patterns2, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def averageError(w_matrix, W_matrix, test_data):\n",
    "    inputLayers = np.transpose(np.insert(test_data, 0, -1, axis=1)[:,:-1]) # h1, ..., hj\n",
    "    desiredOutputs = test_data[:,-1].reshape(-1,1)\n",
    "    actualHiddens = sigmoidalFunc(w_matrix @ inputLayers)\n",
    "    actualOutputMatrix = W_matrix @ np.insert(actualHiddens, 0, -1, axis=0) # o1, ..., oi\n",
    "    squareResiduals = np.square(desiredOutputs - np.transpose(actualOutputMatrix))\n",
    "    sse = np.sum(squareResiduals)\n",
    "    mse = sse / np.size(desiredOutputs)\n",
    "    variance = np.sum(np.square(mse-squareResiduals)) / (np.size(desiredOutputs) - 1)\n",
    "    return mse, variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hiddenUnit(train_data, test_data, Jq = 3, epsilon = 0.001, seed = 440):\n",
    "    train = np.array(train_data)\n",
    "    test = np.array(test_data)\n",
    "    q = 1\n",
    "    Et = np.infty\n",
    "    while(True):\n",
    "        patterns = np.copy(train)\n",
    "        w, W, total_error = backpropagation(patterns, Jq, epsilon=epsilon, seed = seed)\n",
    "        Etp, var = averageError(w, W, test)\n",
    "        print(\"{0} hidden units : MSE = {1} , variance = {2}\".format(Jq,Etp,var))\n",
    "        if(Etp >= Et):\n",
    "            break\n",
    "        Jq += 1\n",
    "        q += 1\n",
    "        Et = Etp\n",
    "    return Jq-1, Et"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 hidden units : MSE = 99809.23449953625 , variance = 21303014605.258526\n",
      "4 hidden units : MSE = 99769.40473595812 , variance = 21230429635.30169\n",
      "5 hidden units : MSE = 99539.03958209316 , variance = 20795803720.511967\n",
      "6 hidden units : MSE = 99572.37714493924 , variance = 20860392979.14613\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(5, 99539.03958209316)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hiddenUnit(train_data, test_data, epsilon=0.001, seed = 440)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 hidden units : MSE = 99377.58914272355 , variance = 20473670331.01753\n",
      "4 hidden units : MSE = 99756.81719579452 , variance = 21207344327.7232\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(3, 99377.58914272355)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "train_d = np.insert(np.array(train_data), 1, np.square(train_data['x']), axis=1)\n",
    "test_d = np.insert(np.array(test_data), 1, np.square(test_data['x']), axis=1)\n",
    "hiddenUnit(train_d, test_d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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