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  {
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   "source": [
    "# 1D parameter estimation using maximum likelihood estimation\n",
    "\n",
    "This example will cover:\n",
    "\n",
    " - Use maximum likelihood estimation to optimise kernel parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ba0576fc",
   "metadata": {},
   "outputs": [],
   "source": [
    "from gptide import cov\n",
    "from gptide import GPtideScipy\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e300d22",
   "metadata": {},
   "source": [
    "## Generate some data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bef3bf50",
   "metadata": {},
   "source": [
    "Start off with the same kernel as Example 1 and generate some data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "0b153f56",
   "metadata": {},
   "outputs": [],
   "source": [
    "####\n",
    "# These are our kernel input parameters\n",
    "np.random.seed(1)\n",
    "noise = 0.5\n",
    "η = 1.5\n",
    "ℓ = 100\n",
    "covfunc = cov.expquad_1d\n",
    "\n",
    "###\n",
    "# Domain size parameters\n",
    "dx = 25.\n",
    "N = 100\n",
    "covparams = (η, ℓ)\n",
    "\n",
    "# Input data points\n",
    "xd = np.arange(0,dx*N,dx)[:,None]\n",
    "\n",
    "GP = GPtideScipy(xd, xd, noise, covfunc, covparams)\n",
    "\n",
    "# Use the .prior() method to obtain some samples\n",
    "yd = GP.prior(samples=1)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e66b5d8e",
   "metadata": {},
   "source": [
    "## Inference"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0197a84e",
   "metadata": {},
   "source": [
    "We now use the `gptide.mle` function do the parameter estimation. This calls the `scipy.optimize.minimize` routine.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "bbd4da06",
   "metadata": {},
   "outputs": [],
   "source": [
    "# mle function import\n",
    "from gptide import mle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "561a3727",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Noise (true): 0.50, |Noise| (mle): 0.45\n",
      "ℓ (true): 1.50, ℓ (mle): 1.21\n",
      "η (true): 100.00, η (mle): 95.02\n"
     ]
    }
   ],
   "source": [
    "# Initial guess of the noise and covariance parameters (these can matter)\n",
    "noise_ic = 0.01\n",
    "covparams_ic = [1., 10.]\n",
    "\n",
    "# There is no mean function in this case\n",
    "# meanfunc = None\n",
    "# meanparams_ic = ()\n",
    "\n",
    "\n",
    "soln = mle(\n",
    "    xd, yd, \n",
    "    covfunc, \n",
    "    covparams_ic,\n",
    "    noise_ic,\n",
    "    verbose=False)\n",
    "\n",
    "print('Noise (true): {:3.2f}, |Noise| (mle): {:3.2f}'.format(noise, abs(soln['x'][0])))\n",
    "print('ℓ (true): {:3.2f}, ℓ (mle): {:3.2f}'.format(covparams[0], soln['x'][1]))\n",
    "print('η (true): {:3.2f}, η (mle): {:3.2f}'.format(covparams[1], soln['x'][2]))"
   ]
  }
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