Gen_Data.ipynb 16.1 KB
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{
 "cells": [
  {
   "cell_type": "code",
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   "execution_count": null,
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   "metadata": {},
   "outputs": [],
   "source": [
    "# Imports\n",
    "import sys\n",
    "import os\n",
    "# relative import\n",
    "sys.path.append(os.path.abspath('') + '/../')\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from scipy.stats import uniform, gamma\n",
    "\n",
    "import pyfield\n",
    "from corbass.utils import load, nez2dif"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": null,
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   "metadata": {},
   "outputs": [],
   "source": [
    "# seed for reproducability\n",
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    "seed = 161\n",
    "np.random.seed(seed)"
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generate synthetic data for tests\n",
    "\n",
    "This notebook serves the purpose of generating synthetic data for testing the `CORBASS` algorithm. We therefore generate data from a given set of Gauss coefficients and add synthetic errors from the Fisher-von Mises and the gamma distribution."
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": null,
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   "metadata": {},
   "outputs": [],
   "source": [
    "# some basic parameters\n",
    "# the name of the output file\n",
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    "out = '../dat/synth_data_clean_complete.csv'\n",
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    "# switch for using locations and incompleteness structure from the example data file\n",
    "real_locs = False\n",
    "# the number of records to be generated, is only used if real_locs is False\n",
    "n_points = 412\n",
    "# the fraction of incomplete records, works as a switch if real_locs is False\n",
    "r_inc = 0.\n",
    "# switch for corrupting the data by noise\n",
    "noise = False\n",
    "# the error levels to be stored\n",
    "ddec = 4.5\n",
    "dinc = 4.5\n",
    "dint = 8250\n",
    "# the average concentration parameter from GEOMAGIA for the interval [750, today] is 650\n",
    "kappa = 650\n",
    "\n",
    "header = f\"# This file was produced using the notebook Gen_Data.ipynb with the following parameters:\\n\" \\\n",
    "       + f\"# real_locs={real_locs}, n_points={n_points:d}, r_inc={r_inc:.2f}, noise={noise}, ddec={ddec:.2f}, \" \\\n",
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    "       + f\"dinc={dinc:.2f}, dint={dint:d}, kappa={kappa:.1f}, seed={seed:d}\\n\""
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling from the Fisher-von Mises distribution"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": null,
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   "metadata": {},
   "outputs": [],
   "source": [
    "# The sampling process involves some rotations, thus we first define some convenience functions\n",
    "def angles(vec):\n",
    "    return np.arctan2(vec[1], vec[0]), \\\n",
    "        np.pi/2 - np.arctan2(vec[2], np.sqrt(vec[0]**2 + vec[1]**2))\n",
    "\n",
    "\n",
    "def rot_z(ang):\n",
    "    return np.array([[np.cos(ang), np.sin(ang), 0],\n",
    "                     [-np.sin(ang), np.cos(ang), 0],\n",
    "                     [0, 0, 1]])\n",
    "\n",
    "\n",
    "def rot_y(ang):\n",
    "    return np.array([[np.cos(ang), 0, np.sin(ang)],\n",
    "                     [0, 1, 0],\n",
    "                     [-np.sin(ang), 0,  np.cos(ang)]])\n",
    "\n",
    "\n",
    "def rotator(vec):\n",
    "    vec = np.asarray(vec)\n",
    "    p, t = angles(vec)\n",
    "    return np.dot(rot_y(t).T, rot_z(p))"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": null,
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   "metadata": {},
   "outputs": [],
   "source": [
    "# This cell contains the sampling procedure\n",
    "def sample_Fisher(n, mu=(0, 0, 1), kappa=20):\n",
    "    \"\"\" Generate samples from the Fisher distribution\n",
    "    \n",
    "    Parameters:\n",
    "    -----------\n",
    "    n : int\n",
    "        The number of samples to be generated\n",
    "    mu : array-like of length 3, optional\n",
    "        A vector pointing towards the center of the distribution. Its length is ignored.\n",
    "    kappa : float, optional\n",
    "        The concentration parameter.\n",
    "    \n",
    "    Returns:\n",
    "    --------\n",
    "        numpy array of shape (3, n) containing the sampled vectors\n",
    "    \n",
    "    Reference:\n",
    "    ----------\n",
    "    [1]: W. Jakob, \"Numerically stable sampling of the von Mises \n",
    "         Fisher distribution on S^2 (and other tricks)\",\n",
    "         http://www.mitsuba-renderer.org/~wenzel/files/vmf.pdf,\n",
    "         2015\n",
    "    \"\"\"\n",
    "    if kappa <= 0:\n",
    "        raise ValueError(f\"The concentration parameter has to be positive, but kappa={kappa} was given.\\n\"\n",
    "                         f\"For kappa=0 use a uniform sampler on the sphere.\")\n",
    "    trafo_mat = rotator(mu)\n",
    "    \n",
    "    # sample from the uniform circle, V in [1]\n",
    "    angles = uniform.rvs(scale=2*np.pi, size=n)\n",
    "    vs = np.array([np.cos(angles),\n",
    "                   np.sin(angles)])\n",
    "    # sample W in [1] via inverse cdf sampling\n",
    "    def inv_cdf(x):\n",
    "        return 1 + np.log(x + (1-x)*np.exp(-2*kappa))/kappa\n",
    "    \n",
    "    unis = uniform.rvs(size=n)\n",
    "    ws = inv_cdf(unis)\n",
    "    ret = np.sqrt(1-ws**2)*vs\n",
    "    res = np.einsum(\"i...,ij->j...\", np.array([ret[0], ret[1], ws]), trafo_mat)\n",
    "    if n == 1:\n",
    "        return res.flatten()\n",
    "    else:\n",
    "        return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We draw some samples and plot the result, to eye-check whether the procedure works:"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": null,
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   "metadata": {},
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   "outputs": [],
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   "source": [
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    "SF = sample_Fisher(1000, mu=(-1, 0, 0), kappa=kappa);\n",
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    "fig, ax = plt.subplots(1, 1, figsize=(10, 5))\n",
    "\n",
    "D = np.arctan2(SF[1], SF[0])\n",
    "I = np.arctan2(SF[2], np.sqrt(SF[0]**2 + SF[1]**2))\n",
    "\n",
    "ax.set_aspect(1)\n",
    "ax.set_xlim((-np.pi, np.pi));\n",
    "ax.set_xlabel('D [rad.]')\n",
    "ax.set_ylim((-np.pi/2., np.pi/2.));\n",
    "ax.set_ylabel('I [rad.]')\n",
    "ax.scatter(D, I, alpha=0.4);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generate synthetic data\n",
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    "We first need a set of coefficients for the field. At the time of writing this, IGRF-13 [1] had just been released and we directly download the reported coefficients as a reference model."
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   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
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    "IGRF = pd.read_csv('https://www.ngdc.noaa.gov/IAGA/vmod/coeffs/igrf13coeffs.txt', header=0, delim_whitespace=True, skiprows=3)\n",
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    "coeffs = IGRF[['2020.0']].to_numpy().flatten()\n",
    "# retrieve the maximal degree using pyfield and the index of the last entry in coeffs\n",
    "l_max = pyfield.i2lm_l(len(coeffs)-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we generate random points as points of observation:"
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "def ran_sph(npoints, r=1., ndim=3, nocluster=True):\n",
    "    \"\"\" Sample random points on the ndim-sphere.\n",
    "\n",
    "    Parameters:\n",
    "    -----------\n",
    "    npoints : int\n",
    "        The number of points to sample\n",
    "    r : float, optional\n",
    "        The radius of the sphere to be sampled on\n",
    "    ndim : int, optional\n",
    "        The dimension of the sphere\n",
    "    nocluster : bool, optional\n",
    "        Whether toexclude points sampled outside of the ball, i.e. sample \n",
    "        uniformly distributed\n",
    "\n",
    "    Returns:\n",
    "    --------\n",
    "        Array of shape (ndim, npoints) including the sampled points.\n",
    "\n",
    "    References:\n",
    "    -----------\n",
    "    [1] http://mathworld.wolfram.com/SpherePointPicking.html\n",
    "    [2] http://mathworld.wolfram.com/HyperspherePointPicking.html\n",
    "    \"\"\"\n",
    "\n",
    "    vec = np.random.randn(ndim, npoints)\n",
    "    if nocluster:\n",
    "        n = np.linalg.norm(vec, axis=0)\n",
    "        vec = vec[:, n <= 1.]\n",
    "        while vec.shape[1] < npoints:\n",
    "            ad = np.random.randn(ndim, npoints)\n",
    "            n = np.linalg.norm(ad, axis=0)\n",
    "            ad = ad[:, n <= 1.]\n",
    "            vec = np.concatenate((vec, ad), axis=1)\n",
    "        if vec.shape[1] > npoints:\n",
    "            vec = vec[:, 0:npoints]\n",
    "    vec /= np.linalg.norm(vec, axis=0)\n",
    "    if npoints == 1:\n",
    "        vec = vec.flatten()\n",
    "    return vec*r"
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "if real_locs:\n",
    "    # load reference data\n",
    "    pars = load('../examples/Example_Parfile.py')\n",
    "    # ungroup the data\n",
    "    ref_data = pars.data.filter(lambda x: True)\n",
    "    ref_data.reset_index(inplace=True)\n",
    "    n_points = len(ref_data)\n",
    "\n",
    "    x_obs = np.zeros((3, n_points), order='F')\n",
    "    x_obs[0] = ref_data['co-lat']\n",
    "    x_obs[1] = ref_data['lon']\n",
    "    x_obs[2] = ref_data['rad']\n",
    "else:\n",
    "    x_obs = ran_sph(n_points, r=pyfield.REARTH)\n",
    "    # transform x_obs from cartesian coordinates to co-lat, lon, rad\n",
    "    pyfield.mapLoc(fromSys=pyfield.SYS_GEO,\n",
    "                   fromForm=pyfield.COOR_CAR,\n",
    "                   toSys=pyfield.SYS_GEO,\n",
    "                   toForm=pyfield.COOR_CLR,\n",
    "                   t=0,\n",
    "                   x=np.asfortranarray(x_obs))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we use the `pyfield` library to get the basis functions and generate a field from the coefficients."
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "dspharm = np.empty((len(coeffs), 3*n_points), order='F')\n",
    "pyfield.dspharm(src=pyfield.SOURCE_INTERNAL,\n",
    "                gSys=pyfield.SYS_GEO,\n",
    "                atSys=pyfield.SYS_GEO,\n",
    "                atForm=pyfield.COOR_CLR,\n",
    "                bSys=pyfield.SYS_GEO,\n",
    "                bForm=pyfield.FIELD_NED,\n",
    "                lmax=l_max,\n",
    "                R=pyfield.REARTH,\n",
    "                t=0.,\n",
    "                at=x_obs,\n",
    "                B=dspharm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fields $N, E, Z$ components are then easily calculated using a dot product with the coefficients."
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "field_obs = np.dot(coeffs, dspharm).reshape(-1, 3).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using a utility function from `CORBASS`, we can transform these to $D,I,F$."
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "decs, incs, ints = nez2dif(*field_obs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we set up error levels for the components and add some synthetic noise."
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "if noise:\n",
    "    mus = np.array([np.cos(np.deg2rad(incs))*np.cos(np.deg2rad(decs)),\n",
    "                    np.cos(np.deg2rad(incs))*np.sin(np.deg2rad(decs)),\n",
    "                    np.sin(np.deg2rad(incs))])\n",
    "\n",
    "    # the angular components retrieve an error from the Fisher-vonMises distribution\n",
    "    # the intensities retrieve an error from the gamma distribution\n",
    "    for it, mu, d, i, f in zip(np.arange(n_points), mus.T, decs, incs, ints):\n",
    "        samp = sample_Fisher(1, mu=mu, kappa=kappa)\n",
    "        decs[it] = np.rad2deg(np.arctan2(samp[1], samp[0]))\n",
    "        incs[it] = np.rad2deg(np.arctan2(samp[2], np.sqrt(samp[0]**2 + samp[1]**2)))\n",
    "        b = ints[it]/dint**2\n",
    "        a = ints[it] * b\n",
    "        ints[it] = gamma.rvs(a=a, scale=1./b)\n",
    "\n",
    "# mimic some incompleteness in the data\n",
    "if r_inc != 0.:\n",
    "    if real_locs:\n",
    "        # incompleteness of reference data\n",
    "        decs[ref_data.query('not (D==D)').index] = np.nan\n",
    "        incs[ref_data.query('not (I==I)').index] = np.nan\n",
    "        ints[ref_data.query('not (F==F)').index] = np.nan\n",
    "    else:\n",
    "        if r_inc != 0.:\n",
    "            # generate indices of missing points\n",
    "            ind = np.arange(n_points)\n",
    "            np.random.shuffle(ind)\n",
    "            mnum = np.int(r_inc*n_points)\n",
    "            mind = ind[0:mnum]\n",
    "            # generate a boolean array of triples of True and False for missing\n",
    "            # values at each point\n",
    "            mdist = np.zeros((3, n_points), dtype=bool)\n",
    "            bools = [True, True, False, False]\n",
    "            for j in mind:\n",
    "                np.random.shuffle(bools)\n",
    "                mdist[:, j] = bools[0:3]\n",
    "            # set the missing values to nan\n",
    "            decs[mdist[0]] = np.nan\n",
    "            incs[mdist[1]] = np.nan\n",
    "            ints[mdist[2]] = np.nan"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we build a pandas `DataFrame` from the synthetic data and store it."
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "synth_data = pd.DataFrame({'co-lat': x_obs[0], \n",
    "                           'lat': 90-x_obs[0],\n",
    "                           'lon': x_obs[1], \n",
    "                           'rad': x_obs[2],\n",
    "                           't': 2020,\n",
    "                           'dt': 0,\n",
    "                           'D': decs, \n",
    "                           'I': incs, \n",
    "                           'F': ints,\n",
    "                           'dD': ddec,\n",
    "                           'dI': dinc,\n",
    "                           'dF': dint})\n",
    "\n",
    "out_frame = synth_data.to_csv()\n",
    "outfile = open(out, \"w\")\n",
    "outfile.write(header + out_frame)\n",
    "outfile.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References\n",
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    "[1] P. Alken, E. Thebault, C. Beggan, H. Amit, J. Aubert, J. Baerenzung, T.N. Bondar,  \n",
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    "      W. Brown, S. Cali, A. Chambodut, A. Chulliat, G. Cox, C. C. Finlay, A. Fournier,  \n",
    "      N. Gillet, A. Grayver, M. Hammer, M. Holschneider, L. Huder, G. Hulot, T. Jager,  \n",
    "      C. Kloss, M. Korte, W. Kuang, A. Kuvshinov, B. Langlais, J.-M. Leger, V. Lesur,  \n",
    "      P. W. Livermore, F. J. Lowes, S. Macmillan, W. Magnes, M. Mandea, S. Marsal,  \n",
    "      J. Matzka, M. C. Metman, T. Minami, A. Morschhauser, J. E. Mound, M. Nair,  \n",
    "      S. Nakano, N. Olsen, F. J. Pavon-Carrasco, V. G. Petrov, G. Ropp, M. Rother,  \n",
    "      T. J. Sabaka, S. Sanchez, D. Saturnino, N. Schnepf, X. Shen, C. Stolle,  \n",
    "      A. Tangborn, L. Tner-Clausen, H. Toh, J. M. Torta, J. Varner, P. Vigneron,  \n",
    "      F. Vervelidou, I. Wardinski, J. Wicht, A. Woods, Y. Yang, Z. Zeren and B. Zhou,  \n",
    "      \"International Geomagnetic Reference Field: the thirteenth generation\",  \n",
    "      submitted to Earth, Planets and Space, see also:  \n",
    "      https://www.ngdc.noaa.gov/IAGA/vmod/igrf.html"
   ]
  }
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