lab_classes/gprs/GP differential equation.ipynb

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      "DRAFT\n",
      "==\n",
      "# Differential Equations and Gaussian Processes\n",
      "\n",
      "# Gaussian Process Summer School, Melbourne, Australia\n",
      "### 25th-27th February 2015\n",
      "\n",
      "### Cristian Guarnizo Lemus, Mu Niu and Neil D. Lawrence"
     ]
    },
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Gaussian process models can deal with linear operations and retain their nature as a Gaussian process. One example of this type of linear operation is a convolution. Such convolutions are often the result of driving a differential equaition with a function or setting that function as the initial condition. In this notebook we demonstrate the combination of Gaussian processes with differential equations, an approach that we refer to as *latent force models*. We consider a second order dynamical system and a spatial differential equation\n",
      "\n",
      "## Simulation Study\n",
      "\n",
      "First, we generate some data points from a second order differential equations. We start by importing some libraries"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "from scipy import integrate\n",
      "from IPython.display import display\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
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     "source": [
      "Now we define a function that allow us to simulate a dynamical system"
     ]
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calc_deri(yvec, time, damper, spring, s):\n",
      "    return (yvec[1], 1.*s - damper*yvec[1] - spring*yvec[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
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      "Using the previous function, We simulate the dynamical system regarding the following equation:\n",
      "$$ \\frac{\\text{d}^2 y_d(t)}{\\text{d} t^2} + C_d \\frac{\\text{d} y_d(t)}{\\text{d} t} + B_d y_d(t) = \\sum_{q=1}^{Q} S_{d,q}u_q(t) $$\n",
      "In the function coded before, u(t) is the unit step."
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     "input": [
      "# System parameters:\n",
      "# Spring constants\n",
      "B=[1.,3.]\n",
      "# Damper Constants\n",
      "C=[4.,1.]\n",
      "# Sensitivities\n",
      "S=[[1.],[1.]]\n",
      "# Numer of inputs\n",
      "q=1\n",
      "# Number of outputs\n",
      "d=2\n",
      "# Initialization of output values\n",
      "Nd = 100\n",
      "ti = 0.\n",
      "tf = 10.\n",
      "t = np.linspace(ti, tf, Nd)\n",
      "start = 0\n",
      "Y = np.empty([d*Nd, 1])\n",
      "T = np.empty([d*Nd, 1])\n",
      "index = np.empty(Y.shape, dtype = np.int16)\n",
      "for i in range(d):\n",
      "    T[start:start+Nd, 0] = t\n",
      "    yarr = integrate.odeint(calc_deri, (0, 0), t, args=(C[i], B[i], S[i][0]))\n",
      "    Y[start:start+Nd, 0] = yarr[:, 0] + 0.01*np.random.randn(yarr.shape[0])\n",
      "    index[start:start+Nd, 0] = i*np.ones((Nd,), dtype = np.int16)\n",
      "    start = start + Nd\n",
      "\n",
      "X = np.hstack((T, index))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
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    {
     "cell_type": "markdown",
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     "source": [
      "Using the minimum and maximum value of time inputs we generate the inducing inputs (here we make use of a trick to differentiate between ouput and latent time inputs)"
     ]
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     "cell_type": "code",
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     "input": [
      "Mq = 50\n",
      "indexz = np.empty([Mq*q, 1], dtype = np.int16)\n",
      "z = np.empty([Mq*q, 1])\n",
      "start = 0\n",
      "zi = np.linspace(ti, tf, Mq)\n",
      "for i in range(q):\n",
      "    z[start:start+Mq] = zi.reshape((zi.size, 1))\n",
      "    indexz[start:start+Mq] = i*np.ones((zi.size, 1), dtype = np.int16)\n",
      "    start = start + Mq\n",
      "#Index number greater than the number of outputs are considered as latent time inputs\n",
      "indexz += d\n",
      "Z = np.hstack((z, indexz))"
     ],
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     "prompt_number": 4
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     "source": [
      "## Model identification using GPy's sparse inference\n",
      "\n",
      "Kernel is created using the second order differential equation covariance function. Then, the model is trained using GPy's SparseGPRegression model."
     ]
    },
    {
     "cell_type": "code",
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     "input": [
      "import GPy\n",
      "kern = GPy.kern.EQ_ODE2(2, output_dim=d, rank=q)\n",
      "model = GPy.models.SparseGPRegression(X=X, Y=Y, Z=Z, kernel=kern)\n",
      "model.inducing_inputs.fix()\n",
      "model.optimize('scg', max_iters = 100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
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     "source": [
      "Now we review the outcome from the sparse inference."
     ]
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     "input": [
      "display(model)"
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       "text": [
        "\n",
        "Name                 : sparse gp mpi\n",
        "Log-likelihood       : 586.791497527\n",
        "Number of Parameters : 108\n",
        "Parameters:\n",
        "  sparse_gp_mpi.           |        Value        |  Constraint  |  Prior  |  Tied to\n",
        "  \u001b[1minducing inputs        \u001b[0;0m  |            (50, 2)  |    fixed     |         |         \n",
        "  \u001b[1meq_ode2.lengthscale    \u001b[0;0m  |      1.61296828862  |     +ve      |         |         \n",
        "  \u001b[1meq_ode2.C              \u001b[0;0m  |               (2,)  |     +ve      |         |         \n",
        "  \u001b[1meq_ode2.B              \u001b[0;0m  |               (2,)  |     +ve      |         |         \n",
        "  \u001b[1meq_ode2.W              \u001b[0;0m  |             (2, 1)  |              |         |         \n",
        "  \u001b[1mGaussian_noise.variance\u001b[0;0m  |  0.000112833006706  |     +ve      |         |         \n"
       ]
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     "source": [
      "Additionally we print the parameters."
     ]
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "display(model.kern.B)\n",
      "display(model.kern.C)\n",
      "display(model.kern.W)"
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        "  Index  |  \u001b[1msparse_gp_mpi.eq_ode2.B\u001b[0;0m  |  Constraint  |   Prior   |  Tied to\n",
        "   [0]   |               0.53967786  |     +ve      |           |    N/A    \n",
        "   [1]   |                2.3962038  |     +ve      |           |    N/A       Index  |  \u001b[1msparse_gp_mpi.eq_ode2.C\u001b[0;0m  |  Constraint  |   Prior   |  Tied to\n",
        "   [0]   |                2.0954422  |     +ve      |           |    N/A    \n",
        "   [1]   |                1.0402882  |     +ve      |           |    N/A       Index  |  \u001b[1msparse_gp_mpi.eq_ode2.W\u001b[0;0m  |  Constraint  |   Prior   |  Tied to\n",
        "  [0 0]  |               0.35545259  |              |           |    N/A    \n",
        "  [1 0]  |               0.53738722  |              |           |    N/A    \n"
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     "source": [
      "Finally, we perform predictions from the training data, and compare them with the real output values."
     ]
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     "input": [
      "Yp_mean = m.predict(X)\n",
      "Yp = Yp_mean[0][:, 0]"
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     "metadata": {},
     "outputs": [],
     "prompt_number": 8
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      "%matplotlib inline\n",
      "%config InlineBackend.figure_format = 'svg'\n",
      "plt.plot(X[:100, 0], Yp[:100], X[:100, 0], Y[:100, 0])"
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