FunctionApprox_BrunoVincent.ipynb

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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   "outputs": [],
   "source": [
    "from numpy.random import normal,multivariate_normal\n",
    "import numpy as np\n",
    "from numpy.linalg import inv\n",
    "from scipy import *\n",
    "from scipy.linalg import norm, pinv, det\n",
    "from scipy.misc import derivative\n",
    "from sympy import symbols, diff\n",
    "import matplotlib.pyplot as plt\n",
    "import numdifftools as nd\n",
    "np.random.seed(0)\n",
    "\n",
    "def flatten_function(l):\n",
    "    return [item for sublist in l for item in sublist]\n",
    "\n",
    "def flatten_args(l):\n",
    "    ret = [[l[0]]] #l[0] is k\n",
    "    ret.append(list(l[1]))\n",
    "    return flatten_function(ret)\n",
    "\n",
    "def partial_function(f___,input,pos,value):\n",
    "    tmp  = input[pos]\n",
    "    input[pos] = value\n",
    "    ret = f___(*input)\n",
    "    input[pos] = tmp\n",
    "    return ret\n",
    "\n",
    "def partial_derivative(f,input,wrt): #wrt = with respect to (partial derivative wrt x_k, for instance)\n",
    "    ret = np.empty(len(input))\n",
    "    for i in range(len(input)):\n",
    "        fg = lambda x:partial_function(f,input,i,x)\n",
    "        ret[i] = nd.Derivative(fg)(input[i])\n",
    "#     print(\"ret\",ret)\n",
    "    if wrt == -1:\n",
    "        return ret\n",
    "    else:\n",
    "        return ret[wrt]\n",
    "\n",
    "def partial_derivative_multi(f,input,wrt): #for that, we need f to be a list of functions, representing the compponents\n",
    "    if len(f) == 1: #this \"if\" should make the program crash if f has only one component. \n",
    "                    #use partial_derivative function in this case\n",
    "        print(\"one dimension\")\n",
    "        return\n",
    "    input_flatten = flatten_args(input)\n",
    "#     print(\"more than one dimension\")\n",
    "    ret = []\n",
    "    for f_component in f:\n",
    "        ret.append(partial_derivative(f_component,input_flatten,wrt=-1))\n",
    "    ret = np.array(ret)\n",
    "#     print(ret.shape)\n",
    "#     print(\"entire partial derivative\",ret)\n",
    "#     print(\"returned\",ret[:,wrt:])\n",
    "    return ret[:,wrt:]\n",
    "\n",
    "def evaluate_f(f,value,multiDimensional = False):\n",
    "    k_value = value[0]\n",
    "    x_flattened = [el for el in value[1]]\n",
    "    if multiDimensional == False:\n",
    "        return f(*value)\n",
    "    else:\n",
    "        value_flatten = flatten_args(value)\n",
    "        return [f_component(*value_flatten) for f_component in f] \n",
    "\n",
    "# print(evaluate_f(f,[1,0],multiDimensional))    \n",
    "\n",
    "def kmeans(x,k):\n",
    "    n = x.shape[0]\n",
    "    n_iterations = 1000\n",
    "    epsilon = 1e-8\n",
    "    err = np.inf\n",
    "\n",
    "    #initialize random centroids\n",
    "    c = np.zeros((k,x.shape[1]))\n",
    "    for i in range(k):\n",
    "        idx = np.random.randint(n)\n",
    "        c[i] = x[idx,:]\n",
    "\n",
    "    for iteration in range(n_iterations):\n",
    "        d = np.zeros((n, k))  # distances to centroids square\n",
    "        for i in range(k):\n",
    "            d[:, i] = np.sqrt(np.sum(np.square(np.subtract(x, np.tile(c[i, :], (n, 1)))), 1).flatten())\n",
    "\n",
    "        #assign each point to nearest centroid\n",
    "        l = np.argmin(d,1)\n",
    "\n",
    "\n",
    "        #recompute centroids as center of mass of each cluster\n",
    "        for j in range(k):\n",
    "            if len(l[l==j]):\n",
    "                c[j,:] = x[l==j].sum(0) / len(l[l==j])\n",
    "\n",
    "        err_new = np.sum(np.min(d,1))/n\n",
    "\n",
    "\n",
    "        if np.abs(err_new - err)<epsilon:\n",
    "            break\n",
    "        err = err_new\n",
    "\n",
    "    return c,l,err    \n",
    "\n",
    "def extended_KF(x0,y,T,f,h,R,Q,StateDimension,OutputDimension):\n",
    "\n",
    "    #Initialization\n",
    "    V = np.zeros([T,StateDimension,StateDimension])\n",
    "    V[0] = np.identity(StateDimension)\n",
    "    x_f = np.zeros([T,StateDimension])\n",
    "    x_f[0] = x0\n",
    "    \n",
    "    V_plus_all = np.zeros([T,StateDimension,StateDimension])\n",
    "    A_all = np.zeros([T,StateDimension,StateDimension])\n",
    "    C_all = np.zeros([T,OutputDimension,StateDimension])\n",
    "    x_plus_all = np.zeros([T,StateDimension])\n",
    "\n",
    "    #Propagation\n",
    "    for k in range(1,T):  \n",
    "#         print(x_f[k-1])\n",
    "#         A_all[k-1] = np.matrix(partial_derivative(f,np.array([k-1,x_f[k-1]]))[1])  #derivative of f, wrt x_k\n",
    "        A_all[k-1] = np.matrix(partial_derivative_multi(f,[k-1,x_f[k-1]],wrt=1)) #derivative of f, wrt x_k\n",
    "        x_plus = evaluate_f(f,[k-1,x_f[k-1]],multiDimensional)\n",
    "        x_plus_all[k] = x_plus\n",
    "#         C_all[k] = np.matrix(partial_derivative(h,np.array([k,x_plus]))[1]) #derivative of h, with respect to x_k\n",
    "        C_all[k] = np.matrix(partial_derivative_multi(h,[k,x_plus],wrt=1)) #derivative of h, wrt x_k\n",
    "        V_plus = np.matrix(A_all[k-1].dot(V[k-1])).dot(A_all[k-1].T) + Q \n",
    "        V_plus_all[k] = V_plus\n",
    "                \n",
    "        K = V_plus.dot(C_all[k].T).dot(inv(C_all[k].dot(V_plus.dot(C_all[k].T)) + R))\n",
    "        x_f[k] = x_plus + K.dot(y[k] - evaluate_f(h,[k,x_plus],multiDimensional))\n",
    "        V[k] = (np.identity(K.shape[0]) - K.dot(C_all[k])).dot(V_plus)\n",
    "\n",
    "#     A_all[T-1] = np.matrix(partial_derivative(f,np.array([T-1,x_f[T-1]]))[1]) #derivative of f, with respect to x_k\n",
    "    A_all[T-1] = np.matrix(partial_derivative_multi(f,[T-1,x_f[T-1]],wrt=1)) #derivative of f, with respect to x_k\n",
    "    \n",
    "    #Recursive\n",
    "    #F = A, B = H, P = V\n",
    "\n",
    "    #Initialization\n",
    "    x_b = np.zeros([T,StateDimension])\n",
    "    x_b[T-1] = x_f[-1]\n",
    "    V_b = np.zeros([T,StateDimension,StateDimension])\n",
    "    V_b[-1] = V[-1]\n",
    "\n",
    "    #Propagation\n",
    "    #Remark: compared to the table 1.2 on page 17, we changed all the indexes by -1\n",
    "    for k in range(T-1, 0, -1): #from T-1 to 1 by -1 increments\n",
    "\n",
    "        A_b = np.matrix(V[k-1].dot(A_all[k].T)).dot(inv(np.matrix(V_plus_all[k])))\n",
    "        V_b[k-1] = V[k-1] - A_b.dot(V_plus_all[k] - V_b[k]).dot(A_b.T)\n",
    "        x_b[k-1] = x_f[k-1] + A_b.dot(x_b[k] - x_plus_all[k])\n",
    "    \n",
    "    return x_f,x_b\n",
    "\n",
    "def factor_analysis(X,m):\n",
    "    #X is a d*n data matrix\n",
    "    #m : dimension of target space (i.e reduced space)\n",
    "    d = X.shape[0]\n",
    "    n = X.shape[1]\n",
    "    print(\"dimension of X\",d)\n",
    "    print(\"number of data in time series\",n)\n",
    "    \n",
    "    mu = np.mean(X, axis = 1)\n",
    "    \n",
    "    \n",
    "    print(d)\n",
    "    print(n)\n",
    "    #Mu = dot(mu, np.ones((n,),\"float\"))\n",
    "    \n",
    "    CenteredData = X\n",
    "\n",
    "    for i in range(n):\n",
    "        for j in range(d):\n",
    "            CenteredData[j,i] = X[j,i]-mu[j]\n",
    "        \n",
    "    \n",
    "    #The tolerance value\n",
    "    tol = 1e-4\n",
    "    \n",
    "    #Maximum number of iteration\n",
    "    maxiter = 50\n",
    "    \n",
    "    #Log-likelihood value\n",
    "    llh = -1\n",
    "    \n",
    "    #Define identity matrix \n",
    "    I = np.identity(m)\n",
    "    \n",
    "    M = dot(CenteredData, np.transpose(CenteredData))\n",
    "    r = M.diagonal()\n",
    "    \n",
    "    #Initialize the weight matrix\n",
    "    W = np.random.rand(d,m)\n",
    "    Wupdated = W\n",
    "    \n",
    "    #Psi is the diagonal covariance matrix, we store only a d-dimensional vector\n",
    "    Psi = np.identity(d)\n",
    "    \n",
    "    for i in  range(d):\n",
    "        Psi[i,i] = random.random()\n",
    "    \n",
    "    Psiupdated = Psi\n",
    "    \n",
    "    for iter in range(maxiter-1):\n",
    "        Psi = Psiupdated\n",
    "        W = Wupdated\n",
    "        G = inv(I + dot(dot(np.transpose(W),inv(Psi)),W))\n",
    "        R = dot(dot(np.transpose(W), inv(Psi)),CenteredData)\n",
    "        \n",
    "        #E-step\n",
    "        Ez = dot(G,R)\n",
    "        Ezz = G + dot(Ez, np.transpose(Ez))\n",
    "        \n",
    "        #M-step\n",
    "        DataCovariance = np.cov(CenteredData)\n",
    "        \n",
    "        #Updated weights\n",
    "        Wupdated = dot(CenteredData, np.transpose(Ez))\n",
    "        Wupdated = dot(Wupdated, inv(Ezz))\n",
    "        \n",
    "        #Updated Psi\n",
    "        V = np.diag(DataCovariance - (1/float(n))**dot(Wupdated,dot(Ez,np.transpose(CenteredData))))\n",
    "        Psiupdated = np.diag(V)\n",
    "        #print(Psiupdated)\n",
    "        \n",
    "        \n",
    "    L = [mu,Wupdated,Psiupdated]\n",
    "    \n",
    "    return L"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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   "outputs": [
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ECEgu57z22urtg7C/QaVigJCUqZY1IMqhBwKSPogf/ABmzYLddsu7muKxv0GlZoCQlKlF\ni5Lf9Pv1y7uSxN57Q/fuMH165QcI+xuUJwOEpEwtWgT9+yeLOJWDDTaAkSPh7rvh7LPzrqZ97G9Q\nOTBASMrUwoXl0//QYsyYZDGppUuhd++8qymc/Q0qJwYISZlatKj8ttAeMwZOPTU5jfHlL+ddTdvs\nb1A5MkBIytTixXDwwXlXsaZ+/ZL+h7vvLt8AYX+Dyp0BQlJmli+HJUvy3URrXcaOTZa1XrkSOnfO\nu5qP2d+gSuFJM0mZeemlZA2IcgwQY8bA66/D44/nXUli6ftLueSRSxhw9QCO+91x9N6gN/eMv4fZ\np89mwtAJhgeVHWcgJGWmsTG5L8cAMWwY9O0Lf/hDstV3XuxvUKUyQEjKTFNTct+/f751tKZz52QW\n4r//G37849Ie2/4GVQMDhKTMNDZCnz7Jwk3l6Jhj4OabYc4cGDw4++PZ36BqYoCQlJnGxvKcfWgx\nahRsuCHccQd873vZHae19RumHTeN0duNdv0GVSwDhKTMNDWVZ/9Di27d4ItfzC5ArN3fMH6X8Uzc\ne6L9DaoKBghJmWlshBEj8q5i/Y45Bn71K5g7F3bYoePjrau/4bQ9T2OLnlt0/ABSmTBASMpMuc9A\nAIweDT16wO23wwUXtH8c+xtUawwQkjLx1luwbFl590AAfPrTcNRRcOutcP75EEK699vfoFplgJCU\niZZLOMt9BgLghBPgllvgiSdgr70Ke4/9Dap1BghJmWhZRKrcZyAA6uthq63gl79cf4Cwv0H6mAFC\nUiaamqBTJ/jMZ/KupG2dO8Pxx8MNN8AVVyRXZ6zO/gbpkzxBJykTjY3Jb/VdKuTXlBNOgDfegKlT\nP36stf0pph03zf0pJJyBkJSRpqbKOH3RYvDgZE+Mn/0M9jjQ/gapLQYISZlobKyMBsoWMUZGnvIw\nP/zjFQy8xv4GqS0GCEmZaGqCPffMu4q2rd3f0HnzQRzw1hSmnWd/g7Q+9kBIKrpVq8p/Eal19Tec\ns9Fs/v5fE1jxvuFBWh9nICQV3b/+BcuXl2cPRFvrN+x6KlxyMVx/PZx9ds7FSmXMACGp6MptEak0\n6zdstRWcdBJcfjl84xvluxW5lDdPYUgqunJZRGrFyhU0zGpg2M+HMfym4cx9bS5Txk6h8YxGJtVP\nWmdz5DnnwOuvJ+tCSGqdMxCSiq6pKdljok+ffI7f0f0pBgyA8ePhkkvga1+DDTYoQdFShTFASCq6\nlks4025M1VHF3J/i/PPhtttg8mQ499wMipUqnAFCUtGVchGprPanGDgQvvlN+NGPkp6ILbcsXs1S\nNbAHQlLRlWIRqfb2N6RxwQXJ6YvzzitCwVKVcQZCUtE1NcGhh2Yzdkf7G9Lo3TuZgTj11KQn4qCD\nijq8VNEMEJKKavlyWLKk+DMQxexvSGPCBPjtb+Hkk2HWLNh440wPJ1UMA4SkonrpJYixOD0QWfU3\npNGpE9x4I+yyC5x+OvzqV6VvDpXKkQFCUlG1rAHRkRmItfenGNR3EFPGTuH4XfPZn6J/f5gyBb78\nZdh9d1eolMAAIanIWlahbM8MRCn7G9L6t3+D2bOTRaa23x6OOCLXcqTcGSAkFVVjY7KAVJoloPPq\nb0hr0iR4/nk49li4/XZDhGqbAUJSUTU2Fjb7UA79DWl16gS33ppckXHMMclS1yefnHdVUj7KYh2I\nEMLXQwgLQwjvhxAeCyHsmXdNktqnrW28S7F+Q5a6dk1WqDzllOT21a/C++/nXZVUerkHiBDCvwGX\nAxcCewBPA9NDCH1zLUxSu6xrEaml7y/lkkcuYcDVAzjud8fRe4PeTDtuGrNPn82EoRNyaY5sry5d\n4L/+C/7P/4FbboHBg+H3v0+uPpFqRe4BApgITIkx/jLG+BxwKvAecEq+ZUlqj7WXsZ7/xny+dc+3\n6D+5PxfcfwGjBozimVOf4b5/v49DBx6ae3NkR5xyCjz9NOy4Y9IPseeeyeyEMxKqBbn2QIQQugJD\ngR+3PBZjjCGEPwH75FaYpHZ56y1Ytgz69Ys8tLiy+hvaa/vtYdo0+NOf4NJLk/6IHj3gsMNgxAjY\nbz8YNAi6dcu7UrWIMbmtWvXxn9e+VetzL7xQvP+PeTdR9gU6A6+s9fgrwA6lL0e1bH3ffJ/6VNJA\np/VbsHgF7HwHP3zlCp67Kf/1G0olBBg1Krm98EKycuXUqTBxIqxYkTz/2c/CtttC377JEtm9eyf7\nbHTpkty6dk2+xlauLPy2alXrt3U9t773tNzK7QdesZ9T8YSY4//REMKWwEvAPjHGx1d7/GJgeIzx\nE7MQIYQhwIzddhtOz5691viCqK8fR339uFa/cPL6c7nUUQ21Zn2MtvTsmdx69YKttkqm6bfdFvbY\nA4YMgX79Ov49Uala1m+49MFreG35ixzwmVGcWzexLNZvyNN778GMGcmln/PmwcKF8MYbyW3p0mTZ\n748++vi2ciV07pzu1qlT27dCXhfCx/dr/3ntm89VxnNTpzYwdWrDGiunvv32Mh5//EGAoTHGmR35\n+s47QHQl6Xc4OsZ412qP3wT0ijEe2cp7hgAzYAYwpFSltqq1D3HtD3Rdz5Xbn8uljnKsLwT44AN4\n5x14++1kiv6ll5JmwRdegNdeS74edtgBvvCFZLXCoUNz/dIsmbXXb/j8p8bz6OSJLG/ahS55z29K\n+oSZM2cyNPkHqsMBItdv8RjjihDCDGAkcBdACCE0//3q9b33t7+FnXbK94eKFGMSJp54IjkPfsst\ncNllsNdecO658MUvVt/XyvrWb/jpxVuwuDOGB6kGlMO3+RXATc1B4gmSqzK6Azet700DBiSNSVKe\nQkhOXfTrB0cdlUxB3303TJ6cdOWPGAE/+1l1fK0Wsj/F2ldgSKpeuZ+cjDHeDnwb+AHwJLArMDrG\n+GquhUnt0LlzMuvwl78kMxJLliSnM6ZMqdwGrjTrN7S1iJSk6lEOMxDEGK8Drsu7DqlYQoBDD01m\nIM48E049FWbNgquuSkJGJWjP/hRNTUlTqaTqVxYBQqpW3bvD9dfD5z8PX/savPpqspdC1655V9a6\njuxPEaOnMKRaYoCQSuArX0l2qDz2WJgwAW68sbyaKwvpb2jLa68lV6sYIKTaYICQSuTII+Gmm+D4\n42HbbeHCC/Ou6OP1G6554hpefOtFRg0YxbTjprVr/YampuTeACHVBgOEVELjxyeLCX3vezBsWNIn\nkYf29De0xQAh1RYDhFRi550Hjz8O//7v8I9/wOabl+a4HelvKERTU7Lk92abFaFYSWXPACGVWKdO\nSQ/ETjvBaafBnXdm2w9RjP6GQjQ1JethuGeIVBsMEFIO+vZNrs446qgkQBxzTPGPUcz+hkJ4BYZU\nWwwQUk6OPBLGjoXvfCe532CD4oybRX9DIRobk+ZQSbXBACHl6PLLk1MZV1yR9Ea0V9b9DYVoaoLh\nw0tyKEllwAAh5Wj77eGb34Qf/xhOOQW2SPmzvlT9DW1ZuTLZVMxlrKXaYbuTlLMLLkiWt77iisLf\nk2Z/ilJYsiQJEfZASLXDGQgpZ717wze+keyTcfbZSYPluuTV39AW14CQao8zEFIZOOOMZC+Jq676\n5HMxRh5a/BBH/uZIBl4zkIbZDZy1z1ksPmMxvzj8F7mHB0gaKMEAIdUSZyCkMrDppsmaEFdfncxC\nbLhh+fQ3FKKpCXr0gI03zrsSSaVigJDKxH/8B1x5JUy5eSmr9ijd+g3F0LIGRDltECYpWwYIqUws\n7zGfbU67irOX/IKu95dPf0Mhmpq8AkOqNQYIKUdrr9+w0RZ9iH86i9vOOY2jR5dm/YZiaGqCXXfN\nuwpJpVSe86FSlVuxcgUNsxoY9vNhDL9pOHNfm8uUsVN4+TuNDHplEg03VE54gKSJ0gZKqbY4AyGV\nUCH7U5x2Gpx5Jrz6atJcWe4+/BBeecUAIdUaZyCkEpj/xny+dc+36D+5PxfcfwGjBozimVOf4b5/\nv49DBx66RnPkuHHJ/a9/nVOxKb30UnJvgJBqizMQUkbauz9F374wZgz88pfJMtflrmURKZsopdpi\ngJCKrBjrN5xwAhx9NMyZA4MHZ1xwB7kKpVSbDBBSkRTS31CoMWNgk03g5pvh4oszKrhIGhuTWrt3\nz7sSSaVkgJA6KIv9Kbp1g2OPhd/8Bn7yk/JeoKllESlJtcUAIbVDe/sb0vjSl+D66+Hvf4c99yzK\nkJkwQEi1yQAhpVDK/SmGD08aKu+4o7wDRGMjHHBA3lVIKjUv45QKsPT9pVzyyCUMuHoAx/3uOHpv\n0Jtpx01j9umzmTB0QiabW3XpAkcemQSIGIs+fFHECIsWwTbb5F2JpFJzBkJajyz6G9I45hi44QZ4\n6inYY4+SHDKVN9+Et9+GrbfOuxJJpWaAkNZSiv6GQtXXQ+/eySxEOQaIxYuTewOEVHsMEFKzUvY3\nFKprVxg7FqZOhR/9KJcS1qslQHgKQ6o9BgjVvDc/eJMbZtzA1U9c3eH1G7IwZgzcckvSrFhuqz0u\nWgQbbACbbZZ3JZJKzQChmpV3f0OhRo+Gzp3h7ruTjbbKyeLFSagp53UqJGXDAKGa0tLfMPmxyfz3\nc/+da39DoTbeGPbfv3wDhP0PUm0yQKgmlGN/QxpjxsD3vgfvvVdeS0YvWgRDh+ZdhaQ85H+CV8rQ\nmx+8yaWPXFrS9RuyMHYsfPAB3H9/3pWsyRkIqXY5A6GqVCn9DYXacUfYdtvkNMaYMXlXk3jnHXj9\ndQOEVKsMEKoaldjfUKgQkuBw113J6o/l0LToGhBSbTNAqOJVen9DoQ45BK69Fl54AbbfPu9qXANC\nqnUGCFWscl+/odhGjEj2x/jjH8snQHTpAp/5TN6VSMqDAUIVp9r6GwrVsyfss08SIL7+9byrSQJE\nv37JGhWSao8BQhWhmvsb0hg1Ci67DD76KPntP0/uwinVtlzneUMIi0IIq1a7rQwhnJ1nTSovK1au\noGFWA8N+PozhNw3nudeeY8rYKTSe0cik+kk1FR4ADjoI3noL/ud/8q7ESzilWpf3DEQEzgduAFr6\nyt/OrxyVi1rrbyjUnntCr17wpz8lpzPytHgxHHxwvjVIyk/eAQLgnRjjq3kXofJQq/0NherSJdni\n+49/hAsuyK+ODz+Ef/7TGQiplpVDgDgnhPA9oBG4DZgcY1yZc00qIfsb0jnoIDjjDHj7bdhww3xq\n8BJOSXkHiKuAmcAbwL7AT4AtgG/nWZRKo1bWbyi2UaOSJsoHH8xvVcr585P7z30un+NLyl/RA0QI\n4SLgu+t5SQQGxRifjzFeudrjs0MIy4EpIYRzY4wr1neciRMn0qtXrzUeGzduHOPGjWtv6SoR+xs6\nZuDAZO2Fv/41vwCxYAF07ZpcximpPDU0NNDQ0LDGY8uWLSva+CHGWLTBAEIIfYA+bbxsQYzxo1be\nOxiYBewYY3xhHeMPAWbMmDGDIUOGdLhelY79DcUzfnyyIuUTT+Rz/LPOgqlT4fnn8zm+pPaZOXMm\nQ5MtdIfGGGd2ZKyiz0DEGF8HXm/n2/cAVgH/Kl5FypP9DdkYMQJ+/evkks6NNir98efPhwEDSn9c\nSeUjtx6IEMLewF7A/SSXbu4LXAHcEmMs3hyLcmF/Q7bq6mDVKnj4YTjssNIff8ECOOCA0h9XUvnI\ns4nyQ+DLwIVAN2AhcDkwOcea1EH2N5TGwIGw5ZZJH0SpA0SMSYA48cTSHldSecktQMQYnwRyXgpH\nxWJ/Q2mFkMxCPPBA6Y/9r3/Bu+96CkOqdXlfxqkKZn9Dvurq4PbbS98HsWBBcu8lnFJtM0AoNfsb\nykNdHaxcCY88AoceWrrjtgSIbbct3TEllR8DhApmf0N5aemDeOCB0gaI+fNh003zWwVTUnkwQKhN\n9jeUp7z6IBYs8PSFJAOE1sH+hsrQ0gdRyn0xFiywgVKSAUJrsb+hsqzeB3HIIaU55vz5yUJWkmqb\nAUKA/Q2VqqUP4v77SxMg3n0XXn4Zttsu+2NJKm8GiBpnf0NlCyGZDfjrX0tzvBead6jZYYfSHE9S\n+TJA1CD7G6pLXR389rel6YOYOze53377bI8jqfwZIGqI/Q3VacSIpA/ib3+D0aOzPdbcudC3L2yy\nSbbHkVTN9bUGAAANO0lEQVT+DBA1wP6G6rbDDrD55snlnFkHiOef9/SFpIQBoorZ31AbWtaDKEUf\nxNy5sOuu2R9HUvkzQFQZ+xtq04gR8K1vwTvvQM+e2RwjxiRAHHNMNuNLqiwGiCphf0Ntq6uDjz5K\n+iAOPjibY7zyStKo6SkMSWCAqHj2Nwhgxx1hs82S0xhZBYiWKzAMEJLAAFGx7G/Q6lrWg8hyX4y5\nc6FTJ5exlpQwQFQQ+xu0PiNGwMSJyWqRPXoUf/znn0+28O7WrfhjS6o8BogKYH+DClFXBytWwKOP\nwkEHFX/8uXM9fSHpYwaIMmZ/g9IYPDhZ5Omvf80mQDz3HHzhC8UfV1JlMkCUIfsb1B5Z9kG8916y\nC+fOOxd/bEmVyQBRJuxvUDHU1cGZZyY/8Lt3L964c+Yk60DsYoaV1MwAkTP7G1RMI0YkfRCPPQYH\nHli8cWfPTu4HDy7emJIqmwEiJ/Y3KAs77QR9+iSnMYodIAYMyObqDkmVyQBRYvY3KEudOiWzEMXe\nF2PWLPsfJK3JAFEC9jeolEaMgO98B95/Hz5dpLNgs2fDSScVZyxJ1cEAkSH7G5SHujpYvjzpg6iv\n7/h4b7wBL79sA6WkNRkgMmB/g/K0886wySZJH0QxAsQzz3w8riS1MEAUkf0NKgedOiWzEH/+M0ya\n1PHxZsxIToXsuGPHx5JUPQwQHWR/g8rR6NFw+unw5puw8cYdG2vGDNh9d+jivxaSVuM/Ce1kf4PK\n2ejRsHJlMgtx9NEdG2vGjGQ8SVqdJ+RTevODN7n0kUsZcPUAjvvdcfTeoDfTjpvG7NNnM2HoBMOD\nysLWWyenHKZP79g4b72V7MI5dGhx6pJUPZyBKJD9Dao0o0fD736XLEEdQvvGePLJ5N4AIWltBoj1\nsL9BleyQQ+Cqq5JdNAcNat8YNlBKWhcDRCvsb1A1GD4cunWDe+9tf4B4/HHYYw8bKCV9kv8srMb1\nG1RNundPVqWcPh0mTkz//hjhoYfghBOKX5ukymeAwP4GVa9DDoFzz4V33oGePdO9d8EC+Oc/4YAD\nsqlNUmWr2V+rY4w8tPghjvrNUQy8ZiANsxs4a5+zWHzGYn5x+C8MD6oKRxwBH34I06alf++DDybN\nl/vtV/y6JFW+mpuBsL9BtWTbbZMehjvvhGOPTffehx6CXXft+EJUkqpTzQQI+xtUq44+Gi66KP3u\nnA8+CIceml1dkipb1QcI+xtU644+Gs4/H+67Dw4/vLD3LFoE8+cne2pIUmsyCxAhhPOAMcDuwIcx\nxk1aeU1/4HqgDngb+CVwToxxVUeO7foN0sd23BEGD05OYxQaIO65J7l086CDsq1NUuXKcgaiK3A7\n8ChwytpPhhA6AdOAl4G9gc8AtwDLgfPbc0D7G6TWHXssXHZZ4VdjTJsG++8PvXplX5ukypTZyf8Y\n46QY41XArHW8ZDSwIzA+xjgrxjgduAD4egghVbBxfwpp/U46Cd59F26/ve3XvvNOsgnXYYdlXpak\nCpZnD8TewKwY42urPTYd+BmwE/B0WwPY3yAVZuut4eCD4YYb4JRPzAeuaerUpOHyS18qTW2SKlOe\nAWIL4JW1HntltefWGyC+fd+3eeDDB+xvkAr0la8koeDpp2G33db9uoYG2Gcf2GabkpUmqQKlOoUR\nQrgohLBqPbeVIYTtsyp2dQvfXMiUsVNoPKORSfWTDA9SGw4/PAkFP/rRul/T1JT0Pxx/fMnKklSh\n0s5AXAbc2MZrFhQ41hJgz7Ue23y159ar/9/6M3XOVKYy9f8/Nm7cOMaNG1fg4aXa0rUrnHcefO1r\n8I9/wE47ffI1110HPXq4/4VUDRoaGmhoaFjjsWXLlhVt/BBjLNpgrR4ghBOByWtfxhlCOASYCmzZ\n0gcRQvgqcDGwWYxxxTrGGwLMmDFjBkOGDMm0dqnaLF8OAwcmK0zedVeyVHWLJUuS5049FS69NL8a\nJWVn5syZDB06FGBojHFmR8bK7CqMEEL/EMJuwNZA5xDCbs23Hs0vuQ+YA9wSQtg1hDAa+CFw7brC\ng6SO+dSn4Mor4Q9/gJtv/vjxGOGss5Lnzz03v/okVY4smyh/AKw+EdqSdOqBB2OMq0IIY0muuvgb\n8C5wE3BhhjVJNe/II+Hkk+GrX4XOnZM1Ir7/fbjtNrj1VtjkE0u+SdInZRYgYownAye38ZomYGxW\nNUhq3ZQpyemME05I1ogIAX7yExg/Pu/KJFWKqt8LQ9Inde2azDacdho88wwMH956U6UkrYsBQqph\n++2X3CQpLfexliRJqRkgJElSagYISZKUmgFCkiSlZoCQJEmpGSAkSVJqBghJkpSaAUKSJKVmgJAk\nSakZICRJUmoGCEmSlJoBQpIkpWaAkCRJqRkgJElSagYISZKUmgFCkiSlZoCQJEmpGSAkSVJqBghJ\nkpSaAUKSJKVmgJAkSakZICRJUmoGCEmSlJoBQpIkpWaAkCRJqRkgJElSagYISZKUmgFCkiSlZoCQ\nJEmpGSAkSVJqBghJkpSaAUKSJKVmgJAkSakZICRJUmoGCEmSlJoBQpIkpWaAkCRJqRkgJElSagYI\n5a6hoSHvElREfp7Vx89UrcksQIQQzgshPBJCeDeE8MY6XrNqrdvKEMKxWdWk8uQ/TtXFz7P6+Jmq\nNV0yHLsrcDvwKHDKel53InAvEJr//maGNUmSpCLILEDEGCcBhBBObOOly2KMr2ZVhyRJKr5y6IH4\naQjh1RDC4yGEk/MuRpIktS3LUxiFuAD4C/AecDBwXQihR4zx2vW8ZwOAZ599tgTlqRSWLVvGzJkz\n8y5DReLnWX38TKvHaj87N+joWCHGWPiLQ7gI+O56XhKBQTHG51d7z4nA5BjjJgWM/33g5Bjj1ut5\nzXHArwouWpIkrW18jPG2jgyQNkD0Afq08bIFMcaPVntPmgBxGDAV2CDGuGI9NYwGFgEfFFi6JElK\nZh62AabHGF/vyECpTmE0H6xDB2zDHsDSdYWH1WroUGqSJKmG/a0Yg2TWAxFC6A9sAmwNdA4h7Nb8\n1LwY47shhLHA5sBjJDMJBwPnApdkVZMkSSqOVKcwUg0cwo3ACa08VR9jfDCEMBq4CPgcyRoQ84Dr\nYow/z6QgSZJUNJkFCEmSVL3KYR0ISZJUYQwQkiQptYoKEAVu0NU/hHB382uWhBAuCSFU1H9nrQoh\nLGplc7Wz865LhQshfD2EsDCE8H4I4bEQwp5516T0QggXtrLZ4Zy861LhQggHhBDuCiG81Pz5fbGV\n1/wghPByCOG9EMIfQwjbpTlGpf1gbdmg62etPdkcFKaRXF2yN8lGXScBPyhRfeqYCJxPcnXOFsCW\nwDW5VqSChRD+DbgcuJDkkuyngekhhL65Fqb2ms3H34tbAPvnW45S6gE8BZxO8m/rGkII3wW+AXwV\nGAa8S/L9+qlCD1CRTZTrWpwqhHAocBewZYzxtebHvgb8BNh09QWuVH5CCAtJPter865F6YUQHgMe\njzH+R/PfA9AEXB1j9PLsChJCuBA4PMY4JO9a1HEhhFXAETHGu1Z77GXg0hjj5Oa/bwS8ApwYY7y9\nkHErbQaiLXsDs1rCQ7PpQC9gp3xKUkrnhBBeCyHMDCF8O4TQOe+C1LYQQldgKPDnlsdi8tvJn4B9\n8qpLHTKwefp7fgjh1ua1fVQFQgjbkswqrf79+hbwOCm+X/PeTKvYtiBJUKt7ZbXnni5tOUrpKmAm\n8AawL8nM0RbAt/MsSgXpC3Sm9e+/HUpfjjroMZLTv3NJTiV+H3gwhLBzjPHdHOtScWxBclqjte/X\nLQodJPcZiBDCRa0066zdSLd93nWqfdJ8vjHGK2OMD8YYZ8cY/ws4E/hm82+3kkokxjg9xnhn8/fi\nH4HDgN7AsTmXpjJSDjMQlwE3tvGaBQWOtQRYu+t789WeU+l15PN9guRrdBvghSLWpOJ7DVjJx99v\nLTbH772KF2NcFkJ4HkjVpa+ytYRkBejNWXMWYnPgyUIHyT1AFHmDrkeB80IIfVfrgzgYWAZ4CVIO\nOvj57gGsAv5VvIqUhRjjihDCDGAkSSNzSxPlSMCm2AoXQuhJEh5+mXct6rgY48IQwhKS789n4P83\nUe4F/LTQcXIPEGm0tUEXcB9JULil+RKVLYEfAteub4dP5S+EsDfJF+/9wNskPRBXALfEGJflWZsK\ndgVwU3OQeAKYCHQHbsqzKKUXQrgUmAosBrYCJgErgIY861LhQgg9SEJfaH5oQPPPzDdijE3AlcD5\nIYR5wCKSn5UvAr8v+BiVdBlnWxt0Nb+mP8k6EXUk17XeBJwbY1xVojLVDiGEPYDrSBruugELSX7b\nmWz4qxwhhNOBs0mmQp8Cvhlj/Hu+VSmtEEIDcADQB3gVeBj4zxjjwlwLU8FCCCNIfiFb+4f8zTHG\nU5pf832SdSA2Bh4Cvh5jnFfwMSopQEiSpPKQ+1UYkiSp8hggJElSagYISZKUmgFCkiSlZoCQJEmp\nGSAkSVJqBghJkpSaAUKSJKVmgJAkSakZICRJUmoGCEmSlNr/AxMHvULuYVbtAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x985bf28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ============================================================================== \n",
    "#                          Create the model \n",
    "# ==============================================================================\n",
    "\n",
    "T = 500 #length of the HMM\n",
    "multiDimensional = False #default value. Do not modify\n",
    "\n",
    "#Specify Dimension of inputs and outputs\n",
    "InputDimension = 0\n",
    "StateDimension = 2\n",
    "OutputDimension = 3\n",
    "\n",
    "\n",
    "\n",
    "if InputDimension == 0:\n",
    "    #We create an artificial input u , that is zero all the time\n",
    "    InputDimension = 1\n",
    "B = np.zeros([StateDimension,InputDimension])\n",
    "D = np.zeros([OutputDimension,InputDimension])\n",
    "mean = [0 for _ in range(StateDimension)] #where x[0] = normal(mean,Q)\n",
    "mean_w = [0 for _ in range(StateDimension)] #do not change\n",
    "mean_v = [0 for _ in range(OutputDimension)] #do not change\n",
    "\n",
    "x = np.zeros((T,StateDimension))\n",
    "y = np.zeros((T,OutputDimension))\n",
    "u = np.zeros((T,InputDimension))\n",
    "  \n",
    "\n",
    "#-----------Define  f (from hidden state to output) and  h (from hidden state to output)----------------------------------\n",
    "\n",
    "#Rmq: h should be of length OutputDimension\n",
    "#Rmq: f should be of length StateDimension\n",
    "choice_function = 4\n",
    "\n",
    "# # function 1 : Gabor filter\n",
    "if choice_function == 1:\n",
    "    Q = np.identity(1) * 0.2\n",
    "    R = np.identity(1) * 0.2\n",
    "    f = lambda k,x_k: 0.01*x_k+20*np.sin(x_k)*np.exp(-x_k**2/2)\n",
    "    C=3\n",
    "    h = lambda k,x_k: C*x_k\n",
    "\n",
    "# # function 2 : Well\n",
    "if choice_function == 2:\n",
    "    Q = np.identity(1) * 1\n",
    "    R = np.identity(1) * 1\n",
    "    f = lambda k,x_k: 0.9*x_k-50*np.exp(-x_k**2/2)+1\n",
    "    C=3\n",
    "    h = lambda k,x_k: C*x_k\n",
    "\n",
    "# # function 3 : Sigmoid\n",
    "if choice_function == 3:\n",
    "    Q = np.identity(1) * 1\n",
    "    R = np.identity(1) * 1\n",
    "    f = lambda k,x_k:7*(1/(1+np.exp(-x_k))-0.5)\n",
    "    C=3\n",
    "    h = lambda k,x_k: C*x_k\n",
    "\n",
    "# function 4 : Gabor filter multidimensional  #test in dimension > 1\n",
    "if choice_function == 4:\n",
    "    multiDimensional = True\n",
    "    Q = np.identity(StateDimension) * 0.2\n",
    "    R = np.identity(OutputDimension) * 0.2\n",
    "    f = [lambda k,x_0,x_1: 0.01*x_0+20*np.sin(x_0)*np.exp(-x_0**2/2), lambda k,x_0,x_1: x_1]\n",
    "    C = 3\n",
    "    h = [lambda k,x_0,x_1: C*x_0, lambda k,x_0,x_1: x_1, lambda k,x_0,x_1: 1]\n",
    "#I do not see any alternative other than f = list of functions (if I try to define f with a function \n",
    "#returning an array, I cannot retrieve the functions representing the components individually, but\n",
    "#we need them in order to calculate their partial derivative. So for the moment we define multicomponent\n",
    "#functions as list of functions of 1 dimension\n",
    "\n",
    "#-----------------------------------------------------------------------------------------\n",
    "\n",
    "\n",
    "# ============================================================================== \n",
    "#                           Print the function \n",
    "# ==============================================================================\n",
    "\n",
    "absc = np.linspace(-10,10,1000)\n",
    "\n",
    "if multiDimensional == False:\n",
    "    ordo = f(1,absc)\n",
    "if multiDimensional == True:\n",
    "    to_plot = 0 #component of f to be plotted\n",
    "    ordo = [f[to_plot](1,el,1) for el in absc] #specific for StateDimension = 2. Here we fix x1 = 1 and look at the evolution of f0 with x0\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(absc,ordo)\n",
    "plt.plot(absc,absc)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Entering Extended Kalmann Filter\n",
      "Exiting Extended Kalmann Filter\n",
      "((500L, 2L), (500L, 2L))\n"
     ]
    },
    {
     "data": {
      "image/png": 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Xi4+ZBqwGXnBeJTRr1vAJgr1zY9y3ejUAa++7L+JuoUqp8Ol03kqpmPIO9mR7mFM5gxQ8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\nSGgKECIiIhKaAoSIiIiE9v9mRFin0xPYZwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x41d27f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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s9ANw/vkTg1orJkw4PeEziCqV7jSAUEo12eDBg7EOskeBHSw4zf99+ZbNLAIm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\nIERERCRpCiBEREQkaQogREREJGn/DywHeFSDS05UAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa2b9dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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txEafPv0YMGBAMswNY8eOHZguGL/tB7mSx/kGV7KNPryJf4ItRVFCUQGhKEq7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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9a6dbe0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Extended Kalman filter, with no parameter estimation\n",
    "\n",
    "iterations = 1\n",
    "\n",
    "x_it = np.zeros((iterations,T,StateDimension))\n",
    "y_it = np.zeros((iterations,T,OutputDimension))\n",
    "x_f_it = np.zeros((iterations,T,StateDimension))\n",
    "x_b_it = np.zeros((iterations,T,StateDimension))\n",
    "\n",
    "for i in range(iterations):    \n",
    "#     x[0] = normal(mean,Q)\n",
    "    x[0] = multivariate_normal(mean,Q) #initialisation\n",
    "\n",
    "    #Propagation\n",
    "    for k in range(T-1):\n",
    "#         x[k+1] = f(k,x[k]) + np.dot(B,u[k]) + normal(mean,Q)\n",
    "        x[k+1] = evaluate_f(f,[k,x[k]],multiDimensional) + np.dot(B,u[k]) + multivariate_normal(mean_w, Q)\n",
    "    for k in range(T):\n",
    "#         y[k] = h(k,x[k])  + np.dot(D,u[k]) + normal(mean,R)\n",
    "        y[k] = evaluate_f(h,[k,x[k]],multiDimensional)  + np.dot(D,u[k]) + multivariate_normal(mean_v, R)\n",
    "        \n",
    "#     print(x.shape)\n",
    "#     print(y.shape)\n",
    "    x_it[i] = x #before it was written: x.flatten()\n",
    "    y_it[i] = y\n",
    "    \n",
    "    print(\"Entering Extended Kalmann Filter\")\n",
    "    x_f,x_b = extended_KF(x[0],y,T,f,h,R,Q,StateDimension,OutputDimension)\n",
    "    print(\"Exiting Extended Kalmann Filter\")\n",
    "    print(x_f.shape,x_b.shape)\n",
    "    \n",
    "    x_f_it[i] = x_f\n",
    "    x_b_it[i] = x_b\n",
    "    \n",
    "\n",
    "absc = np.linspace(-10,10,T)\n",
    "if multiDimensional == False:\n",
    "    ordo = f(1,absc)\n",
    "if multiDimensional == True:\n",
    "    to_plot = 0 #component of f to be plotted\n",
    "    ordo = [f[to_plot](1,el,1) for el in absc] #specific for StateDimension = 2. Here we fix x1 = 1 and look at the evolution of f0 with x0\n",
    "\n",
    "    \n",
    "plt.figure()\n",
    "plt.plot(absc,ordo,c='r')\n",
    "for i in range(iterations):\n",
    "    plt.scatter(x_it[i,:-1,0],x_it[i,1:,0]) #looking at the evolution of the first component with time (actually, plotting points (x_t+1,x_t))\n",
    "plt.xlim(-10,10)\n",
    "#plt.savefig(\"graphs/x3.eps\",format = \"eps\")\n",
    "plt.show()\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(absc,ordo,c='r')\n",
    "for i in range(iterations):\n",
    "    plt.scatter(x_f_it[i,:-1,0],x_f_it[i,1:,0]) #looking at the evolution of the first component with time\n",
    "plt.xlim(-10,10)\n",
    "plt.ylim(-15,15)\n",
    "plt.show()\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(absc,ordo,c='r')\n",
    "#plt.plot(absc,absc)\n",
    "for i in range(iterations):\n",
    "    plt.scatter(x_b_it[i,:-1,0],x_b_it[i,1:,0]) #looking at the evolution of the first component with time\n",
    "#plt.savefig(\"graphs/x3_b.eps\",format = \"eps\")\n",
    "plt.xlim(-10,10)\n",
    "plt.ylim(-15,15)\n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "#Initialisation with factor analysis: test\n",
    "    \n",
    "# #Parameters of the sample which has to be inferred\n",
    "# mean = 0\n",
    "# Q=3\n",
    "# R=3\n",
    "# x[0] = normal(mean,Q)\n",
    "\n",
    "# #Propagation\n",
    "# for k in range(T-1):\n",
    "#     x[k+1] = f(k,x[k]) + normal(mean,Q)\n",
    "# for k in range(T):\n",
    "#     y[k] = h(k,x[k]) + normal(mean,R)\n",
    "\n",
    "# print(np.array(y).T.shape)\n",
    "# factor_analysis(np.array(y).T,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "\n",
    "#radial basis function compute exp(-beta * (c-x)^2)\n",
    "def radialbasisfunc(beta, c, x):\n",
    "    return exp(-beta * norm(c-x)**2)\n",
    "\n",
    "def linearfunction(a,b,x):\n",
    "    return a*x+b\n",
    "     \n",
    "#activation fills the Phi (in the pdf)\n",
    "def activation(X, beta, numNeurons, centers):\n",
    "    # calculate activations of RBFs\n",
    "    Phi = zeros((X.shape[0], numNeurons), float)\n",
    "    for i, c in enumerate(centers):\n",
    "        for j, x in enumerate(X):\n",
    "            #Here we fill the element j,i with exp(-beta * (c_i-x_j)^2)\n",
    "            Phi[j,i] = radialbasisfunc(beta, c, x)\n",
    "    return Phi\n",
    "     \n",
    "#trainrbf gives the values of the weights W and the covariance matrix Q\n",
    "def trainrbf(X, Y, beta, numNeurons,linear):\n",
    "    \n",
    "    #Dimensions\n",
    "    #X: n x Inputdimension\n",
    "    #y: n x OutputDimension\n",
    "    \n",
    "    n = X.shape[0]\n",
    "    \n",
    "    #Centers are the neurons in our RBF network\n",
    "    newCenters = kmeans(X,numNeurons)\n",
    "    centers = newCenters[0]\n",
    "    print(\"shape centers\",centers.shape)\n",
    "    #centers = np.linspace(-8, 8, num=numNeurons)\n",
    "    \n",
    "    print \"center\", centers\n",
    "    \n",
    "    # calculate activations of RBFs\n",
    "    if linear == 0:\n",
    "        Phi = activation(X, beta, numNeurons, centers)\n",
    "        Phi = np.append(Phi, X, axis =1)\n",
    "        Phi = np.append(Phi, np.ones((n,X.shape[1]), \"float\"), axis = 1)\n",
    "    else:\n",
    "        Phi = X\n",
    "        Phi = np.append(Phi, np.ones((n,X.shape[1]), \"float\"), axis = 1)\n",
    "    \n",
    "    #Phi = Phi.T\n",
    "    \n",
    "    print(pinv(Phi).shape[0])\n",
    "    print(pinv(Phi).shape[1])\n",
    "    print(Y.shape[0])\n",
    "        \n",
    "    Y_t = Y.T\n",
    "    \n",
    "    Lambda = 1\n",
    "    \n",
    "    # calculate output weights (pseudoinverse)\n",
    "    #W = dot(pinv(Phi), Y) #written by Victor\n",
    "    W = dot(dot(inv(Lambda*np.identity(numNeurons+2)+dot(Phi.T, Phi)),Phi.T),Y)\n",
    "    \n",
    "    \n",
    "    #Calculate the covariance matrix Q with formula (6.18) \n",
    "    J = 1 #We have J = 1 because of eq (6.15): hyp of gaussians with input u = 0\n",
    "    Q = (1/n) * sum([dot(Y[i] - dot(W,Phi[i,:]),(Y[i] - dot(W,Phi[i,:])).T)]) #written by Victor\n",
    "    \n",
    "    #Q = 1/J * (Y_t.dot(Y_t.T) - W.dot(Phi.dot(Y_t.T)))\n",
    "#     print(\"shape\",Q.shape)\n",
    "    \n",
    "    \n",
    "    L = [W, Q, centers]\n",
    "    \n",
    "   \n",
    "    #print W\n",
    "    print Q\n",
    "    \n",
    "    return L\n",
    "\n",
    "def testrbf(X, W, Q, beta, numNeurons, centers):\n",
    "    \"\"\" X: matrix of dimensions n x indim \"\"\"\n",
    "         \n",
    "    Phi = activation(X, beta, numNeurons, centers)\n",
    "    Phi = np.append(Phi, X, axis=1)\n",
    "    Phi = np.append(Phi, np.ones((n,X.shape[1]), \"float\"), axis=1)\n",
    "    #Phi = Phi.T\n",
    "    \n",
    "    #Y = Phi.T.dot(W.T) + (1/X.shape[0])*random.normal(0, Q, X.shape[0])\n",
    "    Y = dot(Phi, W) + (1/X.shape[0])*random.normal(0, Q, X.shape[0])\n",
    "    \n",
    "    print Phi.shape[1]\n",
    "    print W.shape[1]\n",
    "    return Y\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#Kernel function \n",
    "def MultiRadialbasisfunc(Mean, Sigma, x):\n",
    "    return np.sqrt(1/(2*pi*det(Sigma))) * exp((-1/2)*dot((x-Mean).T,dot(inv(Sigma),x-Mean)))\n",
    "\n",
    "#activation fills the Phi (in the pdf)\n",
    "def Multiactivation(X, Means, Sigma, numNeurons):\n",
    "    # calculate activations of RBFs\n",
    "    Phi = zeros((X.shape[0], numNeurons), float)\n",
    "    for i in range(numNeurons):\n",
    "        for j, x in enumerate(X):\n",
    "            #Here we fill the element j,i with exp(-beta * (c_i-x_j)^2)\n",
    "            Phi[j,i] = MultiRadialbasisfunc(Means[i], Sigma, x)\n",
    "    return Phi\n",
    "\n",
    "\n",
    "def Multitrainrbf(X, Y, Sigma, numNeurons, linear, Lambda):\n",
    "    \n",
    "    #Dimensions\n",
    "    #X: n x Inputdimension\n",
    "    #y: n x OutputDimension\n",
    "    \n",
    "    n = X.shape[0]\n",
    "    d = Y.shape[1]\n",
    "    \n",
    "    #Centers are the neurons in our RBF network\n",
    "    newCenters = kmeans(X,numNeurons)\n",
    "    centers = newCenters[0]\n",
    "    \n",
    "    \n",
    "    # calculate activations of RBFs\n",
    "    if linear == 0:\n",
    "        Phi = Multiactivation(X, centers, Sigma, numNeurons)\n",
    "        Phi = np.append(Phi, X, axis =1)\n",
    "        Phi = np.append(Phi, np.ones((n,1), \"float\"), axis = 1)\n",
    "    else:\n",
    "        Phi = X\n",
    "        Phi = np.append(Phi, np.ones((n,1), \"float\"), axis = 1)\n",
    "    \n",
    "    \n",
    "    # calculate output weights (pseudoinverse)\n",
    "    #W = dot(pinv(Phi), Y) #written by Victor\n",
    "    if linear == 0:\n",
    "        W = dot(dot(inv(Lambda*np.identity(numNeurons+d+1)+dot(Phi.T, Phi)),Phi.T),Y)\n",
    "    else:\n",
    "        W = dot(dot(inv(Lambda*np.identity(d)+dot(Phi.T, Phi)),Phi.T),Y)\n",
    "        \n",
    "   \n",
    "    \n",
    "    #Calculate the covariance matrix Q with formula (6.18) \n",
    "    \n",
    "    #Q = dot(np.transpose(Y),Y) - dot(np.transpose(W), dot(np.transpose(Phi), Y)) #written by Victor\n",
    "    Q = np.zeros((d,d))\n",
    "    for i in range(n):\n",
    "        for k in range(d):\n",
    "            for l in range(d):\n",
    "                Q[k,l] += (1/float(n))*(Y[i,k] - dot(Phi[i,:],W[:,k]))*(Y[i,l] - dot(Phi[i,:],W[:,k]))\n",
    "        \n",
    "    #Q = (1/n)*Q\n",
    "    \n",
    "    return W, Q, centers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Entering in the Kalman Smoother\n",
      "Exiting from the Kalman Smoother\n",
      "[[ 21.57546832   2.86479536]\n",
      " [  2.86479536  11.96942317]]\n",
      "[-1710.43612214]\n",
      "Entering in the Kalman Smoother\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "\n",
    "#EM algorithm\n",
    "\n",
    "# x_k+1 = f(x_k) + w_k\n",
    "# y_k = C*x_k + v_k\n",
    "\n",
    "f = [lambda k,x_0,x_1: 0.01*x_0+20*np.sin(x_0)*np.exp(-x_0**2/2), lambda k,x_0,x_1: 0.01*x_1+20*np.sin(x_1)*np.exp(-x_1**2/2)]\n",
    "Lambda_f = 0\n",
    "\n",
    "C = 3\n",
    "h = [lambda k,x_0,x_1: C*x_0, lambda k,x_0,x_1: x_1, lambda k,x_0,x_1: x_0 + x_1]\n",
    "Lambda_h=0\n",
    "\n",
    "\n",
    "#Parameters of the sample which has to be inferred\n",
    "mean = 0\n",
    "\n",
    "StateDimension =2\n",
    "OutputDimension =3\n",
    "\n",
    "#x and y are the points we need to \n",
    "x = np.zeros((T,StateDimension))\n",
    "y = np.zeros((T,OutputDimension))\n",
    "\n",
    "\n",
    "x_it = np.zeros((iterations,T,StateDimension))\n",
    "y_it = np.zeros((iterations,T,OutputDimension))\n",
    "x_f_it = np.zeros((iterations,T,StateDimension))\n",
    "x_b_it = np.zeros((iterations,T,StateDimension))\n",
    "\n",
    "\n",
    "\n",
    "#We fix the number of kernel centers and the width of Gaussians\n",
    "\n",
    "#Number of Gaussian centers(Neurons)\n",
    "xnumNeurons = 8\n",
    "numNeurons = xnumNeurons**2\n",
    "\n",
    "#Define the size of the kernels \n",
    "Sigma = np.identity(StateDimension)\n",
    "Q=np.identity(StateDimension)\n",
    "R=np.identity(OutputDimension)\n",
    "multi = 1\n",
    "\n",
    "\n",
    "#Initialize the parameters \n",
    "Q_f=np.identity(StateDimension)\n",
    "R_h=np.identity(OutputDimension)\n",
    "\n",
    "#Define the initial rbf with all weights equal to 1\n",
    "W_k = np.ones((numNeurons+3,StateDimension))\n",
    "\n",
    "#Define the centers \n",
    "Means_f = np.zeros((numNeurons,StateDimension))\n",
    "centers = linspace(-8,8,xnumNeurons)\n",
    "for i in range(xnumNeurons):\n",
    "    for j in range(xnumNeurons):\n",
    "        Means_f[i*(xnumNeurons-1)+j] = np.array([centers[i],centers[j]])\n",
    "\n",
    "Means_h = Means_f\n",
    "\n",
    "x[0] = multivariate_normal(np.array([0,0]),Q)\n",
    "\n",
    "#Propagation\n",
    "for k in range(T-1):\n",
    "    x[k+1] =  evaluate_f(f,[k,x[k]],multiDimensional) + multivariate_normal(np.array([0,0]),Q)\n",
    "for k in range(T):\n",
    "    y[k] = evaluate_f(h,[k,x[k]],multiDimensional) + multivariate_normal(np.array([0,0,0]),R)\n",
    "    \n",
    "\n",
    "#mu, W, Psi = factor_analysis(y,StateDimension)\n",
    "\n",
    "\n",
    "\n",
    "#sum([MultiRadialbasisfunc(Mean[:,i], Sigma, np.array([x_0, x_1]))*W_k[i,0] for i in range(numNeurons)])\n",
    "f_rbf = [lambda k,x_0,x_1:  sum([MultiRadialbasisfunc(Means_f[i], Sigma, np.array([x_0, x_1]))*W_k[i,0] for i in range(numNeurons)])+x_0*W_k[numNeurons,0] + x_1*W_k[numNeurons+1,0] + W_k[numNeurons+2,0], lambda k,x_0,x_1: sum([MultiRadialbasisfunc(Means_f[i], Sigma, np.array([x_0, x_1]))*W_k[i,0] for i in range(numNeurons)])+x_0*W_k[numNeurons,1] + x_1*W_k[numNeurons+1,1] + W_k[numNeurons+2,1]] \n",
    "h_rbf = [lambda k,x_0,x_1:  2*x_0 , lambda k,x_0,x_1: x_1, lambda k,x_0,x_1: x_0+ x_1] \n",
    "\n",
    "#Number of iterations \n",
    "maxIter = 3\n",
    "\n",
    "#Log-likelihood\n",
    "llh = np.zeros((maxIter,1)) \n",
    "\n",
    "\n",
    "#Start the EM algorithm\n",
    "for t in range(maxIter):\n",
    "    \n",
    "    #E-Step\n",
    "    \n",
    "    #Input : f,h,Q,R, y_1, ..., y_T\n",
    "    #Output : x_1,...,x_T\n",
    "    \n",
    "    \n",
    "    iterations = 1\n",
    "\n",
    "    for i in range(iterations):    \n",
    "        \n",
    "        #Start with the initial mean=0\n",
    "        x[0] = multivariate_normal(np.array([0,0]),Q)\n",
    "        \n",
    "        #Infer the states with non-linear Kalman filter\n",
    "        print(\"Entering in the Kalman Smoother\")\n",
    "        x_f,x_b = extended_KF(x[0],y,T,f_rbf,h_rbf,R,Q,StateDimension,OutputDimension)\n",
    "        print(\"Exiting from the Kalman Smoother\")\n",
    "        \n",
    "        x_f_it[i] = x_f\n",
    "        x_b_it[i] = x_b\n",
    "        \n",
    "    #M-Step\n",
    "    \n",
    "    #Input : x_1, ..., x_T, y_1,..., y_T\n",
    "    #Output : f,h,Q,R\n",
    "    \n",
    "    #############################Fitting Input dynamics######################## \n",
    "    \n",
    "    #Learn the state parameters\n",
    "    \n",
    "    #X represents x_k\n",
    "    #Y represents x_{k+1}\n",
    "    \n",
    "    X = x_f[0:T-2,:]\n",
    "    Y = x_f[1:T-1,:]\n",
    "    \n",
    "    #Train the rbf and learn the new parameters\n",
    "    W_f, Q_f, Means_f = Multitrainrbf(X,Y, Sigma, numNeurons, 0, Lambda_f)\n",
    "    print(Q_f)\n",
    "    \n",
    "    \n",
    "    #Update the Radial basis function with the new centers and the new weights\n",
    "    f_rbf = [lambda k,x_0,x_1: sum([np.sqrt(1/(2*pi*det(Sigma))) * exp((-1/2)*dot((np.array([x_0, x_1])-Means_f[i]).T,dot(inv(Sigma),np.array([x_0, x_1])-Means_f[i])))*W_f[i,0] for i in range(numNeurons)])+x_0*W_f[numNeurons,0] + x_1*W_f[numNeurons+1,0] + W_f[numNeurons+2,0], lambda k,x_0,x_1: sum([np.sqrt(1/(2*pi*det(Sigma))) * exp((-1/2)*dot((np.array([x_0, x_1])-Means_f[i]).T,dot(inv(Sigma),np.array([x_0, x_1])-Means_f[i])))*W_f[i,1] for i in range(numNeurons)])+x_0*W_f[numNeurons,1] + x_1*W_f[numNeurons+1,1] + W_f[numNeurons+2,1]]\n",
    "    \n",
    "    ############################################################################\n",
    "    \n",
    "    #############################Fitting Output dynamics######################## \n",
    "    \n",
    "    X = x_f[0:T-1,:]\n",
    "    Y = y[0:T-1,:]\n",
    "    \n",
    "    #Train the rbf and learn the new parameters\n",
    "    W_h, R_h, Means_h  = Multitrainrbf(X,Y, Sigma, numNeurons,1, Lambda_h)\n",
    "    \n",
    "    h_rbf = [lambda k,x_0,x_1: x_0*W_h[0,0] + x_1*W_h[1,0] + W_h[2,0], lambda k,x_0,x_1: x_0*W_h[0,1] + x_1*W_h[1,1] + W_h[2,1], lambda k,x_0,x_1:  x_0*W_h[0,2] + x_1*W_h[1,2] + W_h[2,2]] \n",
    "    \n",
    "    #############################################################################\n",
    "    \n",
    "    #llh[t] = log(np.sum([MultiRadialbasisfunc(x[i,:], Q, x_f[i,:]) for i in range(T)])) - (1/2) * log(det(Q))\n",
    "    llh[t] = np.sum([(-1.0/2.0)*dot((x_f[i,:]-x[i,:]).T,dot(inv(Q),x_f[i,:] - x[i,:])) for i in range(T)]) - (float(T)/2.0) * log(det(Q))\n",
    "    print(llh[t])\n",
    "\n",
    "\n",
    "absc = np.linspace(-10,10,T)\n",
    "if multiDimensional == False:\n",
    "    ordo = f(1,absc)\n",
    "if multiDimensional == True:\n",
    "    to_plot = 0 #component of f to be plotted\n",
    "    ordo = [f[to_plot](1,el,1) for el in absc] #specific for StateDimension = 2. Here we fix x1 = 1 and look at the evolution of f0 with x0\n",
    "    ordorbf = [f_rbf[to_plot](1,el,1) for el in absc]\n",
    "    \n",
    "plt.figure()\n",
    "plt.plot(absc,ordo,c='r')\n",
    "for i in range(iterations):\n",
    "    plt.scatter(x_it[i,:-1,0],x_it[i,1:,0]) #looking at the evolution of the first component with time (actually, plotting points (x_t+1,x_t))\n",
    "plt.xlim(-10,10)\n",
    "#plt.savefig(\"graphs/x3.eps\",format = \"eps\")\n",
    "plt.show()\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(absc,ordo)\n",
    "plt.plot(absc,ordorbf, 'r-', linewidth=2)\n",
    "for i in range(iterations):\n",
    "    plt.scatter(x_f_it[i,:-1,0],x_f_it[i,1:,0]) #looking at the evolution of the first component with time\n",
    "plt.xlim(-10,10)\n",
    "plt.ylim(-15,15)\n",
    "\n",
    "        \n",
    "plt.show()\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy import *\n",
    "\n",
    "#Specify Dimension of inputs and outputs\n",
    "InputDimension = 0\n",
    "StateDimension = 2\n",
    "OutputDimension = 3\n",
    "\n",
    "np.random.seed(0)\n",
    "\n",
    "df = pd.read_csv('data/weather_madrid_LEMD_1997_2015.csv')\n",
    "df = df[['CET', 'Max TemperatureC', 'Mean TemperatureC', 'Min TemperatureC']]\n",
    "df_usable = df.loc[df.index > 2459] #starts on 2004-02-01\n",
    "data = df_usable[['Max TemperatureC', 'Min TemperatureC']]\n",
    "data = np.array(data.as_matrix())\n",
    "# print(data.shape)\n",
    "month = np.reshape(np.array([(31. + i)%366 / 366 * 12 for i in range(data.shape[0])]),(T,1))\n",
    "# print(month)\n",
    "\n",
    "#Parameters of the sample which has to be inferred\n",
    "mean = np.zeros((1,2))\n",
    "Q=np.identity(2)\n",
    "R=np.identity(2)\n",
    "\n",
    "#Define the size of the kernels \n",
    "beta = 1\n",
    "Q_k=np.identity(2)\n",
    "R_h=np.identity(2)\n",
    "\n",
    "T = data.shape[0]\n",
    "\n",
    "#We fix the number of kernel centers and the width of Gaussians\n",
    "\n",
    "#Number of Gaussian centers(Neurons)\n",
    "numNeurons = 15\n",
    "\n",
    "#Define the initial rbf with all weights equal to 1\n",
    "W_k = np.ones(numNeurons+2)\n",
    "centers = linspace(-8,8,numNeurons)\n",
    "\n",
    "def f_rbf(k, x_k):\n",
    "        u=0\n",
    "        for i in range(numNeurons):\n",
    "            u += exp(-beta * norm(centers[i]-x_k)**2) * W_k[i]\n",
    "        u += x_k * W_k[numNeurons] + W_k[numNeurons+1]\n",
    "        return u\n",
    "    \n",
    "def h_rbf(k, y_k):\n",
    "        return y_k * W_k[numNeurons] + W_k[numNeurons+1]\n",
    "\n",
    "    \n",
    "x = np.zeros((T,StateDimension))\n",
    "x[0] = np.ones((1,StateDimension))\n",
    "print(x)\n",
    "y = np.concatenate([data,month],axis=1)\n",
    "print(y.shape)\n",
    "\n",
    "\n",
    "#Number of iterations \n",
    "maxIter = 8\n",
    "\n",
    "#Start the EM algorithm\n",
    "for t in range(maxIter):\n",
    "    \n",
    "    #E-Step\n",
    "    \n",
    "    #Input : f,h,Q,R, y_1, ..., y_T\n",
    "    #Output : x_1,...,x_T\n",
    "    \n",
    "    \n",
    "    iterations = 1\n",
    "\n",
    "    x_f_it = np.zeros((iterations,T))\n",
    "    x_b_it = np.zeros((iterations,T))\n",
    "\n",
    "    for i in range(iterations):    \n",
    "        \n",
    "        #Start with the initial mean=0\n",
    "        x[0] = multivariate_normal(mean[0],Q_k)\n",
    "        \n",
    "        #Infer the states with non-linear Kalman filter\n",
    "        x_f,x_b = extended_KF(np.array([x[0]]),y,T,f_rbf,h_rbf,R_h,Q_k)\n",
    "        \n",
    "        x_it[i] = x.flatten()\n",
    "        y_it[i] = y.flatten()\n",
    "    \n",
    "        x_f_it[i] = x_f.flatten()\n",
    "        x_b_it[i] = x_b.flatten()\n",
    "        \n",
    "    #M-Step\n",
    "    \n",
    "    #Input : x_1, ..., x_T, y_1,..., y_T\n",
    "    #Output : f,h,Q,R\n",
    "    \n",
    "    #############################Fitting Input dynamics######################## \n",
    "    \n",
    "    #Learn the state parameters\n",
    "    \n",
    "    #X represents x_k\n",
    "    #Y represents x_{k+1}\n",
    "    \n",
    "    X = x_f[0:T-2]\n",
    "    Y = x_f[1:T-1]\n",
    "    \n",
    "    #Train the rbf and learn the new parameters\n",
    "    StateParameters = trainrbf(X,Y, beta, numNeurons,0)\n",
    "    \n",
    "    W_k = StateParameters[0].T\n",
    "    Q_k = StateParameters[1] #estimation of Q\n",
    "    centers = StateParameters[2]\n",
    "    print(\"shape\",W_k.shape,centers.shape)\n",
    "    \n",
    "    \n",
    "    #Update the Radial basis function with the new centers and the new weights\n",
    "    def f_rbf(k, x_k):\n",
    "        u=0\n",
    "        for i in range(numNeurons):\n",
    "            u += exp(-beta * norm(centers[i]-x_k)**2) * W_k[i]\n",
    "        u += x_k * W_k[numNeurons] + W_k[numNeurons+1]\n",
    "        return u\n",
    "    #mean = W_k[numNeutrons+1]\n",
    "    \n",
    "    ############################################################################\n",
    "    \n",
    "    #############################Fitting Output dynamics######################## \n",
    "    \n",
    "    X = x_f[0:T-1]\n",
    "    Y = y[0:T-1]\n",
    "    \n",
    "    #Train the rbf and learn the new parameters\n",
    "    OutputParameters = trainrbf(X,Y, beta, numNeurons,1)\n",
    "    \n",
    "    W_h = OutputParameters[0]\n",
    "    R_h = OutputParameters[1] #Estimation of R\n",
    "    \n",
    "    def h_rbf(k, y_k):\n",
    "        return y_k * W_h[0] + W_h[1]\n",
    "    \n",
    "    #############################################################################\n",
    "\n",
    "absc = np.linspace(-15,15,T)\n",
    "ordof = f(1,absc)\n",
    "ordorbf = np.zeros(T)\n",
    "for i in range(T):\n",
    "    ordorbf[i] = f_rbf(1,absc[i])\n",
    "    \n",
    "#Plot the real states and the real function\n",
    "plt.figure()\n",
    "plt.plot(absc,ordof)\n",
    "for i in range(iterations):\n",
    "    plt.scatter(x_it[i,:-1],x_it[i,1:])\n",
    "plt.xlim(-10,10)\n",
    "#plt.ylim(-25,25)\n",
    "\n",
    "#Plot the infered states and the corresponding radial basis function\n",
    "plt.figure()\n",
    "plt.plot(absc,ordof)\n",
    "plt.plot(absc,ordorbf, 'r-', linewidth=2)\n",
    "for i in range(iterations):\n",
    "    plt.scatter(x_f_it[i,:-1],x_f_it[i,1:])\n",
    "plt.xlim(-10,10)\n",
    "#plt.ylim(-25,25)\n",
    "\n",
    "        \n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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