Nonlinear_simulation.ipynb

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    "# See how sensitive the prediction at s[51] is on Y[50]"
   ]
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   "execution_count": 1,
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      "/Users/liy0h/anaconda/lib/python3.6/site-packages/h5py/__init__.py:34: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.\n",
      "  from ._conv import register_converters as _register_converters\n",
      "Using TensorFlow backend.\n"
     ]
    }
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   "source": [
    "import numpy as np\n",
    "import scipy.stats as stats\n",
    "##Library neural nets\n",
    "from keras.models import Sequential\n",
    "from keras.layers import Dense, Dropout, BatchNormalization\n",
    "from keras.optimizers import Adam\n",
    "# Library for Gaussian process\n",
    "import GPy\n",
    "##Library for visualization\n",
    "import matplotlib as mpl\n",
    "mpl.style.use(\"seaborn\")\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib;matplotlib.rcParams['figure.figsize'] = (8,6)\n",
    "import pylab "
   ]
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   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "scrolled": true
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   "source": [
    "# Generate a dataset\n",
    "N = 100\n",
    "s = np.linspace(0,1,N).reshape(-1,1)\n",
    "Y = 10 * np.cos(20 * s)\n",
    "Y[Y<0] = 0\n",
    "np.random.seed(1)\n",
    "Z = np.random.normal(Y,1, size=(N,1))\n",
    "plt.plot(s,Z,\".\")\n",
    "plt.plot(s,Y)\n",
    "plt.title(\"(a)\")\n",
    "plt.xlabel(\"s\")\n",
    "plt.ylabel(\"Y(s) and Z(s)\")\n",
    "#plt.show()\n",
    "plt.savefig(\"nonlinear_z.pdf\")"
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  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_mlp(feature_dim):\n",
    "    # create model\n",
    "    model = Sequential()\n",
    "    model.add(Dense(100, input_dim = feature_dim, kernel_initializer='normal', activation='relu'))\n",
    "    model.add(Dense(100, activation='relu'))\n",
    "    model.add(Dense(100, activation='relu'))\n",
    "    model.add(Dense(1, activation='linear'))\n",
    "    # Compile model\n",
    "    model.compile(loss='mse', optimizer='adam', metrics=['mse','mae'])\n",
    "    return model"
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  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_basis = [10,19,37,73]\n",
    "knots = [np.linspace(0,1,i) for i in num_basis]\n",
    "##Wendland kernel\n",
    "K = 0 ## basis size\n",
    "phi = np.zeros((N, sum(num_basis)))\n",
    "for res in range(len(num_basis)):\n",
    "    theta = 1/num_basis[res]*2.5\n",
    "    for i in range(num_basis[res]):\n",
    "        d = np.absolute(s-knots[res][i])/theta\n",
    "        for j in range(len(d)):\n",
    "            if d[j] >= 0 and d[j] <= 1:\n",
    "                phi[j,i + K] = (1-d[j])**6 * (35 * d[j]**2 + 18 * d[j] + 3)/3\n",
    "            else:\n",
    "                phi[j,i + K] = 0\n",
    "    K = K + num_basis[res]"
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  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
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     "data": {
      "text/plain": [
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     "execution_count": 16,
     "metadata": {},
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   "source": [
    "phi.shape"
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  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = create_mlp(feature_dim = K)\n",
    "model.fit(phi,Z, validation_split = 0, epochs = 200, batch_size = 32, verbose = 0)\n",
    "pred = model.predict(phi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "kernel = GPy.kern.Exponential(1,1,0.1) ##Covariance Function\n",
    "m = GPy.models.GPRegression(s, Z, kernel)\n",
    "m.optimize()\n",
    "GP_pred,GP_var=m.predict(s)"
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   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
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   "source": [
    "plt.plot(s,Z,\".\",label=\"Z(s)\")\n",
    "plt.plot(s,Y,label=\"Y(s)\")\n",
    "plt.plot(s,pred,label=\"DeepKriging\")\n",
    "plt.plot(s,GP_pred,label=\"Kriging\")\n",
    "plt.legend(loc='best')\n",
    "plt.title(\"(a)\")\n",
    "plt.xlabel(\"s\")\n",
    "plt.ylabel(\"Y(s),Z(s), and $\\hatY$(s)\")\n",
    "#plt.show()\n",
    "plt.savefig(\"pred_y.pdf\")"
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  {
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   "execution_count": 104,
   "metadata": {},
   "outputs": [],
   "source": [
    "i = 50\n",
    "j = 51\n",
    "M = 50\n",
    "z_50 = np.linspace(Z.min(),Z.max(),M).reshape(-1,1)\n",
    "pred_GP = np.zeros(M)\n",
    "pred_DK = np.zeros(M)\n",
    "nr = 20"
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  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "for k in range(M):\n",
    "    Z_2 = np.delete(Z,j,0)\n",
    "    phi_2 = np.delete(phi,j,0)\n",
    "    s_2 = np.delete(s,j,0)\n",
    "    Z_2[i] = z_50[k]\n",
    "    for rep in range(nr):\n",
    "        ## Calculate the average of nr deepKriging results to avoid the random batches.\n",
    "        model.fit(phi_2,Z_2, epochs = 200, batch_size = 32, verbose = 0)\n",
    "        pred_DK[k] = pred_DK[k] + model.predict(phi)[j]/nr\n",
    "    m = GPy.models.GPRegression(s_2, Z_2, kernel)\n",
    "    m.optimize()\n",
    "    mean,var = m.predict(s)\n",
    "    pred_GP[k] = mean[j]"
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   "execution_count": 106,
   "metadata": {},
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   "source": [
    "plt.plot(z_50, pred_DK,\"b\",label=\"DeepKriging\")\n",
    "plt.plot(z_50, pred_GP,\"r\",label=\"Kriging\")\n",
    "pylab.legend(loc='best')\n",
    "plt.title(\"(b)\")\n",
    "plt.xlabel(\"Z(50)\")\n",
    "plt.ylabel(\"$\\hat{Y}$(51)\")\n",
    "#plt.show()\n",
    "plt.savefig(\"pred_z.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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