evenly_spaced_random_sampling.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "enormous-singles",
   "metadata": {},
   "source": [
    "# Methods for Evenly-Spaced Random Sampling \n",
    "\n",
    "How should we initialise points to evaluate (expensive) models? We want to randomly sample the allowed region, but ideally not have points too close to one-another. This problem can be solved using [Quasi Monte Carlo (QMC) methods](https://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method). Since July 2021 (i.e. v1.7), `scipy` has a [QMC submodule](https://docs.scipy.org/doc/scipy/reference/stats.qmc.html) with lots of method available. We'll comapre some of these methods here.\n",
    "\n",
    "One complication is that out models have parameter constraints. Namely, we have inequality constraints on linear combinations of parameters, while QMC methods in `scipy` only sample on $[0,1]^d$. We can handle this with rejection sampling: take more points than we need, then exclude those that fall outside the constraints.\n",
    "\n",
    "Before finding out about QMC, another solution I considered was an n-body simulation, where points emit a repulsive force from one another. Inequality constriants could be handled by reflecting off the walls. For comparison (and fun), I've implemented a simple n-body integrator to demo this idea."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ethical-extraction",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "np.seterr(divide='ignore')\n",
    "from scipy.spatial import distance\n",
    "from scipy.stats import qmc\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "np.random.seed(364354)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "horizontal-relations",
   "metadata": {},
   "outputs": [],
   "source": [
    "class EvenlySpacedRandomSampling():\n",
    "    \n",
    "    def __init__(self,\n",
    "                 bounds,\n",
    "                 A_ineq=None,\n",
    "                 b_ineq=None):\n",
    "        '''\n",
    "        bounds = list of (lo,hi) tuples for each parameter x\n",
    "        where x is the paramater vector of length n.        \n",
    "        A_ineq and b_ineq are optionally a (m,n) matrix and (m,) vector\n",
    "        defining inequality constraints `A_ineq x leq b_ineq`\n",
    "        '''\n",
    "        n_dim = len(bounds)\n",
    "        self.n_dim = n_dim\n",
    "        for bound0 in bounds:\n",
    "            assert len(bound0)==2\n",
    "            assert bound0[1]>bound0[0]\n",
    "        self.bounds = bounds\n",
    "        self.lo = np.array([b0[0] for b0 in bounds])\n",
    "        self.hi = np.array([b0[1] for b0 in bounds])\n",
    "        self.range = self.hi-self.lo\n",
    "        A = np.zeros((2*n_dim, n_dim))\n",
    "        b = np.zeros(2*n_dim)\n",
    "        for i in range(n_dim):\n",
    "            A[2*i,i] = -1\n",
    "            b[2*i] = -self.lo[i]\n",
    "            A[2*i+1,i] = 1\n",
    "            b[2*i+1] = self.hi[i]\n",
    "        if (A_ineq is None) and (b_ineq is None):\n",
    "            pass\n",
    "        else:\n",
    "            n_ineq, n_dim2 = A_ineq.shape\n",
    "            self.n_ineq = n_ineq\n",
    "            assert n_dim2==n_dim\n",
    "            A = np.vstack((A, A_ineq))\n",
    "            assert len(b_ineq)==n_ineq\n",
    "            b = np.concatenate((b, b_ineq))\n",
    "        self.A = A\n",
    "        self.b = b\n",
    "        self.Aprime = np.matmul(A, np.diag(self.range))\n",
    "        self.bprime = b - np.dot(A, self.lo)\n",
    "        self.Aprime_nrm = (self.Aprime.T/distance.norm(self.Aprime, axis=1)).T\n",
    "\n",
    "    def xprime_from_x(self, x):\n",
    "        '''\n",
    "        Transform points from co-ordinates with specified-bounds to [0,1]^n  \n",
    "        '''\n",
    "        xprime = np.einsum('ij,nj->nj', np.diag(1./self.range), x)\n",
    "        xprime -= (self.lo/self.range)[np.newaxis,:]\n",
    "        return xprime\n",
    "    \n",
    "    def x_from_xprime(self, xprime):\n",
    "        '''\n",
    "        Transform points from [0,1]^n to co-ordinates with specified-bounds\n",
    "        '''\n",
    "        x = np.einsum('ij,nj->nj', np.diag(self.range), xprime)\n",
    "        x += self.lo[np.newaxis,:]\n",
    "        return x\n",
    "    \n",
    "    def uniform_sample_with_rejection(self, n, factor_per_iter=10, n_max_iter=10):\n",
    "        '''\n",
    "        Uniform sampling in [0,1] with rejection outside the contraints\n",
    "        '''\n",
    "        n_samples = 0\n",
    "        samples = np.zeros((n_samples, self.n_dim))\n",
    "        i_iter = 0\n",
    "        n_tmp = factor_per_iter*n\n",
    "        while (i_iter<n_max_iter) and (n_samples<n):\n",
    "            tmp = np.random.uniform(size=(n_tmp, self.n_dim))\n",
    "            is_valid = self.check_if_valid(tmp)\n",
    "            samples = np.vstack((samples, tmp[is_valid, :]))\n",
    "            i_iter += 1\n",
    "            n_samples = samples.shape[0]\n",
    "        if n_samples<n:\n",
    "            raise ValueError('Not enough samples created. Increase factor_per_iter or n_max_iter')\n",
    "        else:\n",
    "            samples = samples[:n, :]\n",
    "        samples = self.x_from_xprime(samples)\n",
    "        return samples\n",
    "    \n",
    "    def step(self,\n",
    "             xp,\n",
    "             force_power_law=-2,\n",
    "             tstep=0.001,\n",
    "             reflect_boost=2.,\n",
    "             kw_rejection_sample={}):\n",
    "        '''\n",
    "        A step for the n-body integrator\n",
    "        '''\n",
    "        n_particle = xp.shape[0]\n",
    "        # square matrix of mutual (repulsive) forces\n",
    "        dist = distance.pdist(xp)\n",
    "        dist = distance.squareform(dist)\n",
    "        force = dist**force_power_law\n",
    "        np.fill_diagonal(force, 0.)\n",
    "        # (n,n,n_dim) matrix of normed mutual offsets\n",
    "        offset = xp[np.newaxis,:,:] - xp[:,np.newaxis,:]\n",
    "        tmp = 1.*dist\n",
    "        np.fill_diagonal(tmp, np.inf)\n",
    "        normed_offset = offset/tmp[:,:,np.newaxis]\n",
    "        # velocity = summed mututal forces\n",
    "        # square matrix counts mutual forces twice, so half this\n",
    "        velocity = normed_offset * force[:,:,np.newaxis]\n",
    "        velocity = 0.5 * np.sum(velocity, axis=0)\n",
    "        # update position \n",
    "        xpnew = xp + tstep * velocity\n",
    "        # find particles with have escaped boundary in this timesep\n",
    "        # these will satisfy A'(x' + f*t_step*v) - b' = 0 for some 0<f<1\n",
    "        # i.e. y + fz = 0 for y = A'x' - b' and z = t_step * A'v\n",
    "        y = np.einsum('cd,nd->nc', self.Aprime, xp) - self.bprime\n",
    "        z = tstep * np.einsum('cd,nd->nc', self.Aprime, velocity)\n",
    "        f = -y/z\n",
    "        # relect particle off whichever boundary is crossed first\n",
    "        # i.e. find the crossing with minimum f for 0<f<1\n",
    "        f[f<0] = np.inf\n",
    "        f[f>1] = np.inf\n",
    "        argmin_f = np.argmin(f, axis=1)\n",
    "        min_f = f[np.arange(n_particle), argmin_f]\n",
    "        idx = np.where(min_f < np.inf)\n",
    "        # for the reflected velocity flip the sign of velocity...\n",
    "        # ... component perpendicular to boundary wall\n",
    "        v_cmp_perp = np.einsum('ij,ij->i',\n",
    "                               self.Aprime_nrm[argmin_f[idx],:],\n",
    "                               velocity[idx])\n",
    "        v_cmp_perp = (v_cmp_perp * self.Aprime_nrm[argmin_f[idx],:].T).T\n",
    "        v_cmp_par = velocity[idx] - v_cmp_perp\n",
    "        # add a boost to the reflected perpendicular velocity to prevent bunching up at edges\n",
    "        v_reflected = v_cmp_par - reflect_boost * v_cmp_perp\n",
    "        # for particles which have crossed boundary (ie subscript `idx`)\n",
    "        # replace their positions with the reflected version which traveled \n",
    "        # with original/reflected velocity for fractions f/(1-f) of the timesetep\n",
    "        xpnew[idx] = xp[idx] + tstep * (min_f[idx]*velocity[idx].T + (1-min_f[idx])*v_reflected.T).T\n",
    "        # corner particles may still not be valid\n",
    "        # since their reflected positions may be outside a second boundary\n",
    "        # replace these (hopefully few) instances with rehection samples\n",
    "        is_valid = self.check_if_valid(xpnew)\n",
    "        idx_not_valid = np.where(is_valid==False)\n",
    "        n_not_valid = len(idx_not_valid[0])\n",
    "        fraction_not_valid = n_not_valid/xp.shape[0]\n",
    "        rejection_sampled = self.uniform_sample_with_rejection(n_not_valid, **kw_rejection_sample)\n",
    "        xpnew[idx_not_valid] = rejection_sampled\n",
    "        return xpnew, fraction_not_valid\n",
    "    \n",
    "    def check_if_valid(self, xp):\n",
    "        '''\n",
    "        Check if a (primed i.e. in [0,1]) poistion is within the constrained region\n",
    "        '''\n",
    "        is_valid = np.matmul(self.Aprime, xp.T).T <= self.bprime\n",
    "        is_valid = np.prod(is_valid, axis=1, dtype=bool)\n",
    "        return is_valid\n",
    "    \n",
    "    def calibrate_tstep(self,\n",
    "                        n_part,\n",
    "                        tstep_init=1e-4,\n",
    "                        n_max_iter=10,\n",
    "                        reflect_boost=2.,\n",
    "                        factor_reduce_tsep_per_init=5.,\n",
    "                        n_steps=30,\n",
    "                        target_invalid_frac=0.01,\n",
    "                        kw_rejection_sample={},\n",
    "                        kw_step={}\n",
    "                        ):\n",
    "        '''\n",
    "        Calibrate the timestep so that only a small fraction `target_invalid_frac`\n",
    "        of particles are sent outside the constrained region\n",
    "        '''\n",
    "        i_iter = 0\n",
    "        avg_fraction_not_valid = 1.\n",
    "        tstep = factor_reduce_tsep_per_init * tstep_init\n",
    "        while (avg_fraction_not_valid > target_invalid_frac) and (i_iter < n_max_iter):\n",
    "            tstep = tstep/factor_reduce_tsep_per_init\n",
    "            tmp = []\n",
    "            xpnew = self.uniform_sample_with_rejection(n_part, **kw_rejection_sample)\n",
    "            for i in range(n_steps):\n",
    "                xpnew, fraction_not_valid = self.step(xpnew,\n",
    "                                                      tstep=tstep,\n",
    "                                                      **kw_step)\n",
    "                tmp += [fraction_not_valid]\n",
    "            avg_fraction_not_valid = np.mean(tmp)\n",
    "            i_iter += 1\n",
    "        return tstep, avg_fraction_not_valid, i_iter\n",
    "        \n",
    "    def sample_nbody(self,\n",
    "                     n_part=100,\n",
    "                     kw_rejection_sample={},\n",
    "                     kw_step={},\n",
    "                     n_steps=100):\n",
    "        '''\n",
    "        Main sampling method \n",
    "        '''\n",
    "        tstep, _, _ = self.calibrate_tstep(n_part)\n",
    "        x_steps = np.zeros((n_steps, n_part, self.n_dim))\n",
    "        xnew = self.uniform_sample_with_rejection(n_part, **kw_rejection_sample)\n",
    "        xpnew = self.xprime_from_x(xnew)\n",
    "        for i in range(n_steps):\n",
    "            x_steps[i] = self.x_from_xprime(xpnew)\n",
    "            xpnew, _ = self.step(xpnew, tstep=tstep, **kw_step)\n",
    "        return x_steps\n",
    "\n",
    "    def get_min_dist(self, xp):\n",
    "        '''\n",
    "        Get array of distances to nearest neighbour\n",
    "        '''\n",
    "        dist = distance.pdist(xp)\n",
    "        dist = distance.squareform(dist)\n",
    "        dist[dist==0] = np.inf\n",
    "        min_dist = np.min(dist, axis=0)\n",
    "        return min_dist\n",
    "    \n",
    "    def sample_with_scipy_qmc(self, n_part=100, method='Sobol', max_iter=100):\n",
    "        '''\n",
    "        Use scipy's QMC methods to sample in [0,1]^n, rejecting those \n",
    "        outside constraints, increasing the number of trials till the \n",
    "        desired number of samples remains\n",
    "        '''        \n",
    "        n_valid = 0\n",
    "        n_trial = 1*n_part\n",
    "        i_iter = 0\n",
    "        while (n_valid<n_part) and (i_iter < max_iter):\n",
    "            if method=='Sobol':\n",
    "                sampler = qmc.Sobol(d=self.n_dim, scramble=True)\n",
    "            elif method=='Halton':\n",
    "                sampler = qmc.Halton(d=self.n_dim, scramble=True)\n",
    "            elif method=='LatinHypercube':\n",
    "                sampler = qmc.LatinHypercube(d=self.n_dim)\n",
    "            sample = sampler.random(n=n_trial)\n",
    "            is_valid = self.check_if_valid(sample)\n",
    "            sample = sample[is_valid]\n",
    "            n_valid = sample.shape[0]\n",
    "            n_trial = int(n_trial*1.1)\n",
    "            i_iter += 1\n",
    "        # if there are extra points, remove the ones with nearest neighbours\n",
    "        min_dist = self.get_min_dist(sample)\n",
    "        sample = sample[np.argsort(min_dist)[-n_part:], :]\n",
    "        sample = self.x_from_xprime(sample)\n",
    "        return sample"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "economic-roller",
   "metadata": {},
   "source": [
    "## A 2D example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "damaged-breeding",
   "metadata": {},
   "outputs": [],
   "source": [
    "esrs = EvenlySpacedRandomSampling(\n",
    "    [(0,0.5), (0,1)],\n",
    "    A_ineq=np.array([[1,-1]]),\n",
    "    b_ineq=np.array([0])\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "considerable-filename",
   "metadata": {},
   "source": [
    "First the simplest random sampling: uniform in $[0,1]^n$ with rejection sampling to exclude points outside the constriants,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "tight-douglas",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_particles = 300\n",
    "x = esrs.uniform_sample_with_rejection(n_particles)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "covered-salvation",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_2d_example(x, isgif=False):\n",
    "    plt.plot(*x.T, '.k')\n",
    "    plt.axvline(0, ls=':')\n",
    "    plt.axvline(0.5, ls=':')\n",
    "    plt.plot([0,0.5], [0,0.5], ls=':')\n",
    "    plt.axhline(1, ls=':')\n",
    "    return"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "fallen-emerald",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_2d_example(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acute-protection",
   "metadata": {},
   "source": [
    "Next let's compare some of the methods available in `scipy.stats.qmc`,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "sufficient-measurement",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/pjethwa/miniconda3/envs/base_py39/lib/python3.9/site-packages/scipy/stats/_qmc.py:1078: UserWarning: The balance properties of Sobol' points require n to be a power of 2.\n",
      "  warnings.warn(\"The balance properties of Sobol' points require\"\n"
     ]
    },
    {
     "data": {
      "image/png": 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bwEZgD/Av+a4b1EP/n6ffkM7Hj41tAWiTvQqTvTeT8s4kWzKRyz6//RZ+6kNc9ce12FI05nf7Y78snY8fq8VD9yLo/5pB0P/Q7JgLgdsBBQwEPgY6ZrjW1UAVUNWnT59AiRl14xNy+MW3xNLhacPyaqaLVdKY/NLzitcy9iuccQ4e8FsGhVqvTYihVwCvpm1PAiY1O+ZF4NS07WnAyFzXDeqhv/HWu1LWoVPBeRUmC4/ND1dL8g5N9dAb7+W1DiX1LMRVz/cdrJV9B2sDnx9W0FsBK4H+HOoUHdLsmHuAnzf8vwewHuiW67pBBV2kMBcK9iI8SQiryS8aL5hkf1z11u/Xlqa9rJN4CQd5GSaVb6EEPXU+ZwFLSY12+WnD364Brmn4f0/gn8BHwHzgknzXDCroz76/Vp59f22wnBD/nkJchZavQiUlTKa+aPxggn2F7nVGSRJ55XeocBj7wmpYaEHX8UtiLheTvLVM5Hr4TJuPwut+nXbZJFQ2eJ0mEXf5+smrsGXp5nJJo7q2Tqpr6wKdG7YgkhSRJB9O0140YfOipYSuWlL/QTpBy9freWHLMoyGiRSYoIchTEGY4O2Y6JXaJlSmvhh13U9nWltyfbQ6hq7jF1TQn5q9Rp6avSbQuSLBC6KlejtesEmobG6lBUGXvSY4OJnw8wl+UuUYVsMKStCTmg/d1ApsI1E8TGGa1aa20mx6WZjq4Hgpo6TL0cXQDcGmB85UTHgxmthKMyFf/GCyvfnK1/ZyzCXo1s2Hng+dc1PYPD+MKXidm8TEctQ5R40J+eIHk5cLzFe+JpSjNrIpve5fUA/9sVmr5bFZqzPui+Lt6LxwvcTVJNYZO04qJq3T+2tp9T7JcsylYV6gkEIuF/+pUi7+U2XGfVF0eJnajCwkdDeJbS3HpEIFJueXjS+afDbn0jAvFJSg5yJsxTS1oycJbB5zn+RHWDrzTJfwmlrvW9IQUz+0GEEXCVcQpnoqNg0LjNIGm8rR7z3DjNKJui6YUN6ZMPXr6KQpKEF/eMbH8vCMjwOd6wXT3sxJVC4bxvLmI27b4pwLJAj58sPEskxKWP2Wpd98C6thBSXol/15llz251m+zjGxsnrF1E/rTfdi4sZPfripmQ9h4ovGa34FzdcgGpZOQQm6X0yuzF5Iyv4kx/ImSdhQTxxzgfjF1LIKmg9xiLyXeySVry1a0OPKdJ2VzJQK3Px4m1+UmYgzTXF6nqYKZ5Bn06R6l5QtBSXof357pfz57ZWej48j002qZEEw9YGPG1M92Sgw8YUd5B6649t+CXIPvxrWnFyCbt2XojNWbAHgO6P7ezo+ji/a/HwdZsqXfukE/bqt0L6c1fkFYdL4Las4vngM8mx6LaPKykrGjh3bdNzUqVO1PJNBngG/GuaLbEqv+2fjXC7ZiLoTpSUOUzQFP3lfaC2UdEyuE1HGt01OZzYopJCLqURVyUztBLWdqNNnoxA0x8QRJlHd12v52BhmKyhBn/Lmcpny5nIRMTdGluta+SqZjRXMdHSIr+3lZOoLKUq7vDy7SeRDuoYFIZegWxdDf3/1DsBfjCwoUd/DS8ywkOO42dDdr6AjHmx7OcU1K6Dfso3SLi/x7SRmjWzUMC1kU3rdv7Ahlzg8JFvmBDE1XGKKh6TrHra1EJtf18R8N7XlYBIUUsilEVMrZPq5cYisqQ+AaTHMsOWRxEtTd9maOM48DruSIMo0FZSg3/XGMrnrjWUiYq6HFKfI+n1o4npYCmmUQVI2ujh9YdA8HyY+OLVJw4KQS9Cti6Ev3LCr6f9xjIPOdI98ccE4Vy3xE8uNo9/Br10mr3zTSFKr0Jgap/caF7ehbOOgef2ZvXwje0o667lZNqXX/bN12KIXryNuz8Sr153EpFAmtqD8npukp2la+MFkr9u0vGok6jyjkEIuSWPz1LImP4xBCNvHEcf85Umhy96wToHNnbxhcDH0LNz5+lK58/Wlgc6NAtMrTj50ClPcohdGXGyPT+dCZx2N8yXqB1s610XCa1guQbcuhr5y855E7297XFBXv0Oc8flGwsSYTY1PR4HOmH+Y+q/TrjjKM6o6rlXDsim97p+tIRdHZmwZsx/VuSYTdPx3HP0dOu0ydRhm1FBIIZc4KNQHPQg2dBw6Po9fITRxHvgk6lQu+0yp4wUl6P/z6mL5n1cXBzrXC0kVmokvkULvOIwSm9NuiufZHBOX6ouinMNqWC5B9xRDV0qNA+4EioH7ReTWDMeMAe4ASoAtIvLl8AGhz7Nh5wEdl20iiTHHScSfveA3L5rH502c+10HppafV0ztT4jbLi/1PYo+KJ0allfQlVLFwF3AV4F1wGyl1HMisjDtmM7A3cA4EVmjlCrXZC///a/DdF0aSKZyJ/XhSj7C5IXtIucHU8vPK6Z29Mdtl+5nv9HB+abOtGRz3Rt/QAXwatr2JGBSs2N+APwq37UkgpBLHMTdfDYlNpeJoHlh2lJhOjGh/EzNQ1PtyoYN4+QJE0MHLiQVZmncvhT4Y7Nj7iDlxU8H5gCXZbnW1UAVUNWnT59Aibn15UVy68uLAmZFcHRXTNsqfj68VmATxDAKkiw/U79kLZSyjYJ0B+ewMZfLuT97IPC1cgm6lxi6yuTYN9tuBQwHxgJlQKVSaqaILG3WGrgPuA9gxIgRza/hiR37qoOcFgqv4YO41yY0Ga/N5bjDFVHE9TNdI8nySyrkk++5sD0UFSXp4ZySdp3pcviRWu7jRdDXAel37w1syHDMFhHZC+xVSr0FDAOWEjG3XHB81JfMi5eKaXLMOKnOSS8iF2efRRRlZGI5J9Wpme+5MLWztZE4n4vY+gOyue5yKEzSClgJ9AdKgbnAkGbHDAamNhzbFpgPDM11XZNj6M3x0nQ0dehXXM1eGz7wiaKMTC5nE+drNzWUGPS5MCE9hB2HDpxFytteAfy04W/XANekHXM9sLBBzH+c75pBBf1XLyyQX72wINC5YchXkKbGC+MQIFvG7kdhp81CoANb0xXkuYiqnofVsFyC7mkcuoi8BLzU7G/3Ntv+LfBbry2DoByoqQ99jSBNrXzhA1OHfsXR7LVl7H4UZRTkGkmFaeIIKQTpOzDh+4Qgz0VU9TwKDctKNqXX/Ys65OI+Uc9OHCN0osxTL/aaGvrIRBK2mtpqMun5S6KFFwUU0qf/mfCT0TYJgU34eaFG8cCb8nB5QYet+fLR1JeI7c+fCSGmghL0nz83X37+3PzP/M3vRyy2CEFQTKh0mYj6gTc1nZmI6oXXeIxpq2aJeCu7lvD85SOThvkhl6BbNx96JvzEw5KOdeuOH5o4rK4RLzFIv2VpStqykV7ekyZNynusl7LzOudI3PXcS9kl/fwFxYS4vyeyKb3uX1Ix9CSJwzsJ2qSNI//8hFNML0sv+C1vP8sbmjoUtVDKLh3TWhUUUsjFZkwdQhhnhdX1wJsoJH7L20852NbRbTOmxf1zCbp1IZeb/jEfgJvPG5r3WNOaSXEMIQzSpI1z2KGOMImpYSa/5e2n7ILmo9dnwuTP9uN+rqN+bv1omF+sE/Q2JUWejjPxIfcrtkErrt+H3fZPtE0VnyAvV539An6eibjrhNe6nsRzHXXc36uGBSKb6677pzvkYlozyS9xN3lNDFmImDuiw0aChIDiqBNu2LE/KKSQi1dM9zrzodvrbO4RmTpixNQRHTYSJAQUR176qethnmvTQrBayKb0un9BPfQb/jZXbvjbXE/HmuB1BrVBp9dpk0drgq0m1KOoMDEtfss4Uxps+kLVj4ZlgkIa5ZLUAhdBCFuJdD18tjVbkxQhk4SgkAlTxrZ9oRpWw3IJunUhl4njjknaBM+EDZvoavKGDUfF3XRtiYtHtDTClHHUH6zpRqeGWSfoNmFqvC9MzNnE0UM6MUkIHJmx5QvVTbsP8Lc567noi0dyWLtSLfewTtCve3ouAP/9r8MStiQ/QStRHKIZ1CNKaqrcpB5EE4SgpRBmmK6XMkqipbfs0920a92Knp3L2LTrILe9spi3lm6m12FlWjTMOkHv2alN0ib4IkglMrmZn8T45KRbBKaOAApKEOE0fQ4iU8pIRNhbXUf71q3Yub+GM+98m++eNoCJ445hSM+OvHvDV3jyvTXa7m+doP/frx2dtAnaMbmZH7fHavLLzUaCCGccL9Wg5WzaUMRvTamkW/vW3HPJcDqVlXD3hJM44cjOACil6NW5TKuGWSfoNuK30pnezI/TGzL55WYjQYQzjpdqkHI2ofX2SOUqpi3exF+uGAnAOSf0on3r4qb9XxtyeKz2WCfoP37iAwDuuOjEhC3xRtBKF0Y0/bxATPNwmmP6y802gghnHC9V0+cgamTpp7t5bNYaJp11DK1bFYNSFBcVcaCmjjYlxVw6qm/ea+jUMOsEfUD39kmb4Iu4K52fF0jcHk5cc9PYjmkjnOJ6qZo4B1F1bT3vrtjCkJ4dKe/QhvXb9/Pk7LVcOLw3Q3t14tJRfT2JeDpaNSzbAHXdv5YyH3rcH6b4+YAizo8t3Ac63nD55A8dz/3+6lrZuuegiIis2LRb+k58QR54Z6WIiFTX1sm+g7WR3SsIFNKHRZkwIZaWjbhDBn68ljjj015bKqaHgHRTCJ3AcZZhVK03EUEpRW1dPaNvm8aZQ4/g5vOGMqB7e/561cmM6HcYACXFRZQU57lYkmRTet2/oB76D/86R3741zmf+VtYT9NU7z4oftJj0ox6UXintpelCR667s/wTePnz82Xqx6a3bT9SOUqmblii7b7ZdIwP1BIHvqxPTt+7m9hv8g01bsPih+vJa74tJeWSljvtBDKMulO4LB5aEML4+1lm/nfDzfw2wuPRylFz05lFCnV5KVf4jMm7pdMGhYV1gn6D8YM/NzfwjwESVXAlhhayPfyCBsCMklMwpSvzXPXmDjMdPveal78aCPnndiL9q1bsWHHfmau3Mrm3Qcp79iG7542IFZ7MmlYZGRz3XX/TFlTNOomopfmahLNUltCEYXQ3DfFjiAUSthr/fZ9snn3ARERmbVyq/Sd+IK8Mn+jiIjU1NZJfX19YraFhUKaPvd7D1fJ9x6uCnRuNqKqgF4fhrin8rRZYPzi9YWqU3BMmqo1CCYIchCqa+tERGTrnoPS/4YX5PevLxURkdq6elm0cacxIh5Ww3IJunUhl5P6dg58brZmsNcmblRrW8bdLDUpFKGbfGUZR5zdxLCDH6J6HuLkkvtnUd6xNb/71gl0aVfKbd88npH9uwBQXKQ45nB9cWu/hNGwvGRTet2/uEMuUSw2EeUojTi9oJbkoecjLu/ZJC9Xhy1Jh2Yen7VafvLkB03bf5y2TB6ducr3dWyEQvLQgxLWS/Vyvp/OWb8dX2E72dzn8yni8p6zlW/cXq2uFkncI5I+3rKX//1wPdd+ZRDFRYpt+6rZsGM/1bX1lLYq4oena+xotIlsSq/7F9RDv/LB9+TKB9/zfV4cHrouTPawTfJEvZKUzUmUo64WSdi05LOrrq5eqlZtlR17q0VE5IW5G2TApBdlwfqdIiLGxMODEFTDGiGsh66UGgfcCRQD94vIrVmO+yIwExgvIs9E88r5LKcc1S3QeWG91CS9XFNj4LaO+456WKBXrzuJctTVIgn7PGSyq65e2F+Tmkt88Se7+eY9ldx6wXFcNLIPYweXU/XTM5pW+lFKRZKOJAiqYZ7IpvSNP1IivgIYAJQCc4Fjsxw3DXgJuDDfdU0ZtpgNkzzPpFsH2fLB9tEcUeC33ySJcsxXl5NssTTe92BNnZz869dl8osLRSTlgb84b4Ps3F8dq002QJhhi0AF8Gra9iRgUobjfgz8EHjQdkE3McSRxEOXLx9MzKfmmDZE0SRHQSTZMrzlpUVy3VMfNm3f9cYymb5kU2z3t5Vcgu4l5NILWJu2vQ44Of0ApVQv4HzgK8AXs11IKXU1cDVAnz59PNz683z7gfcAeOg7IwOd7wUTQxxJfD2YLx/CNrtNX9bMC35DGqZNBRxnXZ+1citTF2/ixrMGA1BSrGhVXNS0X+sXlAahU8O8CHqmYJU0274DmCgidbliWyJyH3AfwIgRI5pfwxNnDC5n5ccfc8stt2gTAtvHEUeFl3wIKlAmL2vmB9NGEPl9Seqs63sO1jJt8Sa+dmwP2pQUs2jjLp6uWsv3ThtA1/at+Y8WsJxkJs4YXK7v4tlcd/ERcgE+BlY1/PYAm4Dzcl03aMglriaiaU3jpAiSD17OiSP+bkNIKEqCpjfKur5jb3VT3PuNxZ9K34kvyNRFn4hIap7xmoavOR3BIWQMvRWwEujPoU7RITmOfxCNMXTXEWc2XkXFhBezTS9tU16SmWgcQvjprv1y1KQX5U9vrRARkYM1dfLex1ulrs7eIYYmkkvQ84ZcRKRWKfUj4FVSI1keEJEFSqlrGvbfG7aV4Id3S06i/Fs3s+mpmxIJh5j0ubOJeA1zJL2smU1DLr3aGneoUES46qEqeh9Wxi/OHUp5hzbccOYxnDqoOwClrYr4Yr8uWm2wkQn3zwTgr1eNivzansahi8hLpIYjpv8to5CLyOXhzcrOhNMGM6pXKcXH3Ry7qPpdr9NE4ddtlx9RsXma2Dgx6SX59w/WMX/9Lm46+1iUUgwsb0/3Dq2b9l91arxT0drI2cf31HfxbK677p/Jwxaz4bVJa2rs1oQwhykEyQtd/Qk6bI2Kddv3yQPvrGwKq/zmlUVy9u/fbprZ0BE/FNL0uUli6vS4XjHVrqTwI7ZBXwBRCXFcL8n6+npZ+sku2XOgRkREnpq9RvpOfEEWb9wlIqmpaB3JkkvQrZuca/yUSgCe/F745qTf8IPXJq2pwx5NtSsp/IR8goRoogzr+LHVb70WEarr6mndqph563Zy7l3vcudFJ3DuCb0487gjOGVgN3p1LgNSU9E6whGlhn2ObEqv+xfUQ39q9hp5avaaQOemo7sZa2rYwVS7whBHmpL20HXZub+6Vr78m2ly+2tLRCQ1KdZjs1bLpl0HtNvaUgmrYbiQy+dx4YfCwM8wyShi2UnE0P3gpV7f/toS+dULCw6d8+LCpuXZHOaTS9CtC7nU1NUDUJL2yXAQTA0/tNTRMUHxEtaIaohikFE5cY/kyVSvP1y7gxkrtjR9Wr9tbzV7DtQ2nTOp4VN8RzxEpWGZsE7QL7l/FhA+/mTaJ9tg7thoU+0Cby9mm4YohqWiooJXX3udx6bOYcLY4VRUVHD39OX8cdpyLh7Zh85tS/nluUOTNrNFE5WGZcI6Qb9o5JGRXavQJkrS5UWbahd4ezGb2hrLhd88O1BThwiUlRazt1N/Xt63nUt7fAGAS0f15YpT+lNWWqzbbIcHotSwz5EtFqP7l3QM3UTCdKLp7ICLyy6d8WabOoP95vf67ftk8E0vN62pue9grUxd9IkcqKmNw1xHzFBIMfT91XUAn/E2TI3v+iVMGEhnWCEOu3SHdUxrjeUiX56JCNc+/gH9urbjuq8fzRGd2nDFl/pxXK9OQOrZ+MoxPZIy35GHTBoWFdYJ+uV/Sc0l3Bh/8ioEtoh+UOHRHVbQbZcf4behHMOQKc9e/mgjyzft4dqxg1BK0b51qyZBUEpx/dePSdhqh1eaa1iUWCfol4zq+5ntOEc5mEwSnbxexDXKj7EKpRzz5VtFRQXPvvQaT745j6u/dgIVFRX84vkFvL1sCz84fSDFRYpbv3l8ApY7oqC5hkVKtliM7l9UMXQv8UYTxpzbFMP1go6Yfb48MqEcw5Ir3zbs2Cf7q1Nx74dmfCx9J74gyzftFpHUB0A2r3TviA4KKYa+60ANAB3blAB2jHIoFM8yHR0x+3xhnaTLMQqa59sbb6Ty7f0127ng7hncd+lwvjbkcM4Z1pOT+3dlQLd2ALQpcSNUCoXmGhYl1gn6dx+qAj4bf8onBDrDEV7CDoU4DjoJcTXx2wG/NOZbjSjK/+1W9h45BIDjenVi4rhjGNLQsdm5bSmd25YmaapDE5k0LCpUyoOPnxEjRkhVVZXv816ZvxGAcUOPiNok3/jpkC00Dx1aRgdllNz75gpq6+oZXraF6dOns6TTCC740hDOGaZxfmyHcYTVMKXUHBEZkWmfdR66l0yIS2hMWnggCXQPBbT9hbH4k13MWb2dCSenOsEWbdxFTV09P5pgzxBKR/TodEatE/Rte6sB6NIuc3M0Tm/YhtV5bBVFG1s19fXCh+t2cELvzhQVKV6ct5Epb63knGE96dCmhNu/dQJFbvrZFk8+DQtD9LPDaOb7j87h+4/Oybo/k9esi0bP++abbzZScBpF8aabbmLs2LFUVlZ6Pu+WW27xfHxQ23LdI85yDENdvTRNtvT8vA1ccPcM5q7bAcDlp/Rj1qSxdGjo/HJi7oD8GhYG6zz07+ZZszDuzjqTv0AM0hkbh2fs5R42jGhZu20f5931Lv959mDOP7E3Y44u547xJzCoRwcAurZvnecKjpZIPg0Lg3WCfsaxuT9ptilebdKCzY3EMSLHyz1MLMf6euGGZ+cxsLw9V592FL06l/G1IYdz5GFtAehUVsJ5J/ZK2EqH6eTTsDBYJ+ibdh8AoLxDm6zHmOw1NxKHJxxEFOPwjL3ew4RyfGPJJtZs3ce3T+lHUZFix76aprnEi4oUt1xwXKL2OezDi4YFxTpBv/axDwBN6/HFSFxj0/2KYhyecRT30NW62X2ghqpV2zn9mHIA/rngE95dvpVLR/WlqEhx32UZR4s5HJ7RqWHWCfr3xxyVtAmRYHKMOIhnHGTB7aBCHHXrZue+Gtq1LqZVcRF/nbWGW19ezNv/73SO7NKWG8YNpl3rYteh6YgMnRpmnaCPObo8aRMiIaiXauIwxLiHGEbZuqlatY2L7pvJg1eMZPSgblxwYi9G9D2saZX7Tm2j/zzb0bLRqWHWCfqGHfsB6NnwwJmEbi/V1LHZcU9tEKZ1s/tADd99uIpvDOvJhJP7MrRXJ7572gB6H5aqT+Ud21DeMfrYpsPRiE4Ns07Qf/Lkh4B5MfQ4xDaocJo4miYMfls3D1euAuCyin60b92K9q1LaN0qNdlVm5JiJo5zc4k74kOnhlkn6Nd+ZVDs9zRlAq4gwmnqaJoo7pntPqu37uWj9Ts5+/jUHClvLtkMpARdKcX933Ydm47k0Klh1gn66EHdgPhiyV4FMQ4vNYhwmjqaJmpWbN7DgG7tUErxcOVqHpm5mq8cU07b0lbcNeEkN/2swxgaNUwH1gn6mq37eP+D97n4nK/FEks2bQIuv8Jp8mgaCP5iTk3onxoL/vcP1vGTJ+fy6o9P4+jDO3DVqf35zuj+tC1NVW8n5g6TWLN1HwB9uraN/NqeBF0pNQ64EygG7heRW5vtnwBMbNjcA3xfROZGaWgj1z8zlzVr1sfWCWfDBFy5MPGLy0aChoPWbN3HJX+exY1nDWbc0MM5dVB3bj53CD06pj61P6KTeR3mDkcj1z+TksZEYuhKqWLgLuCrwDpgtlLqORFZmHbYx8CXRWS7UupM4D7g5MitBX7y1S+wYEENc2PyOk0WRK8EfdGEDWvlO99r66euXvjViwv5Qo8O/NvIPvTs3IahvTrSsSxVfbu1b82lFf182+dwJMFPvvoFbdf24qGPBJaLyEoApdQTwLlAk6CLyIy042cCvaM0Mp1RA7oyasBpHBujyLbEeb/DdqaGnYBr1sqtbNi5n/NP7E1xkWLB+l2UtkpNDtqquIi7JwyPJJ0OR9yMGtBV27W9CHovYG3a9jpye99XAi9n2qGUuhq4GqBPnz4eTfwsKzbvAfSKbJwCW6hjy/1OwDX6tC9T2vPQ8MG/zlrDnNXbOe+EXiileOLqUe5rTUdB0KhhR3VvH/m1vQh6pqco47p1SqnTSQn66Ez7ReQ+UuEYRowYEWjtuxuf/QjQNw7d5q8eoyRsZ6qX8w/U1HHyyaOoqKjg91OXcfu9M3jvxjPo3qE1P/2XwXRsU4JSqernxNxRKOjUMC+Cvg44Mm27N7Ch+UFKqeOB+4EzRWRrNOZ9nv837mhdlwbs+upRJ2H7DvKd/97H27j8L+/xyJUnM7zvYZx/Yi+G9upIp7LUp/Y93NeajgJFp4blXSRaKdUKWAqMBdYDs4GLRWRB2jF9gGnAZc3i6VkJuki0bqL20L2Eb5KIocd9z537a7j+6bmcPawn5wzrya4DNdz68mK+86X+DCyPvunpcBQqoRaJFpFapdSPgFdJDVt8QEQWKKWuadh/L/AzoCtwd0MTuTbbDcOy5JPdABx9eAcdl490VIvXl0Pcwx3jCis9M2cdxUVw/om96dimFVv2HGTfwdRc4h3blDD5fDeXuKPloVPDPI1DF5GXgJea/e3etP9fBVwVrWmZ+dn/zgf0zuUSlcBGEb7R4UnrsmvT7gMs3rib077QHYBn319HcZHi/BN7o5Ti2R98KRL7HQ6b0alh1n0peuNZg30dn+SQwLDxcV2edJR2telczmvP/42KigrueH0Z//hgPe/f9FXalBRzz4ThTWPFHQ5HCr8a5gfrnrZhR3b2fKwXQdQp+GHDN7o6aKOyq83Rp9LtG9fx3LQZVFRUcPWpA/h2RT9aN4wXd3OJOxyfx4+G+cU6QV+wYScAQ3p2yntsPkGMaybCoNfUOQImiF2rtuzlh4+9zznHfonS0lJqNixiz4y/8uVf/hCAft3aRWafw1Go+NEwv1gn6L98PvWBqpf4Uz5BTGoMuNdWQdzTDjS3q7aunt9PW86g8vZ8Y1hPDu/Uhk5lJQwefJT10yE4HEnhR8N8k5q1Lv7f8OHDJQjz1++Q+et3eD5+xowZMnnyZJkxY0bGfWVlZVJcXCxlZWUZj4maJO7phUa72hwxUDodO7rJrq/f/qb88vkFCVvncBQOfjWsOUCVZNFV6zx0v82UXKGFJCbeiqtV4LUVUF8vrNi8p8muLqPGU1renzfeSNn13I9GN82h4nA4wqMj1NKIdYI+d+0OILqOhbjHgMfxZWi+voG6eqFIgVKKO6Yu457py7nn66dRWlrKzjcfpFV9Nae/9L8ATswdjoiJWsPSsU7QJ7+0CDBvTVGvxNEqyNUKeO/jbXz/0Tk8etXJDD6iI+cMO4Kjurdj9JDDXVzc4YgBnRpmnaD/8tyhSZsQGt2tgvRWQOsOXVjUcQTTFn/KV47pwYDu7ag4qitFDZNeDSzvwMDyDrHY5XA49GqYdYKu65P/QmJnh/7c9ujL7Fkyg1NP+zI3VVazfscBILUYxB8vPilhCx2OlotODbNO0Oes3gbA8L5dErbkEEkvULH7QA3LN+3hxD6HAXDvmyto17o1f500CYBpp0jTNLQOhyNZdGqYdYL+m1eWAObE0JNaoGLvwVratU4V3389t4Bpizcx+6dnUFJcxD2XnET39q2bjnVi7nCYg04Ns07QJ19g1gx9uochZvL+n31/HTf87SPemXg65R3b8N1TB3DxyD4UNwi3WyTZ4TAXnRpmnaDrWLYpDDqHITZ6/3Vtu9B1QRGTt9Zw+dmnMezIzlwxul/TcYOP6BjZPR0Oh150aph1gj5zZWoxJJ0Lrfoh7DDETB54dW09D1euYs4bc6iurqa+bhuUtGXW+/O4/OzTOKp7eyadqW/GNofDoQ+dGmadoN/+2lJAT/wpaOdm0OF+6fH3su59uOO+B7nyG6dRUqx4cMYqju1xTJP3v+PJiVw2darvezgcDrPQqWHWCfpvLxym5bpxd26KCC9Oe6cp/t7uy1fy+8otXPmNVCfmS/9+Kh3blHDF8e5jH4ejkNClYWChoPfp2lbLdeOYY0Xk0PDB219fxtP7jqW0bQeq9+1m77uP8Kvv3t90bMc2qbnE3cc+DkdhoUvDwEJBf2fZFgBGD+qWcX/QsInuOVbe+3gb1z09l4e/M5J+3drx1cE96N6+lN7jXmTmO286D9zhaCHk07AwWCfof5i2DMicGWHCJlHPsbJjXzV3vL6McUMPZ9SArvQ6rIz+3dqxr7oOgON6d+K43p2Afpx+6imh7uVwOOwhl4aFxTpBv338CVn3hQ2bhAlviAgzVmxFAacM7EZZaTEvzNtAv65tU4LeuYyHvjMy0LUdDkfhkEvDwmKdoPfsnP2jmTimpk2nurae1Vv3MqhHB5RS/PL5hXTv0JpTBnajdatiKieNpaTYTT/rcDgOkUvDwmKdoE9fsgmAMUeXf26fn7BJ0Fh7TV19k0hf/8xcKldsZeaksRQVKe6+5CR6pn2l6cTc4XA0J5eGhcU6Qb9n+goge2Z4CZsEjbX/44P1/NdzC3jr+tPp1LaEb5/Sj3NP6Ik07DftK1aHw2Ee+TQsDNYJ+h8uPjH0NbzG2ldt2cvklxbx72cMYkjPTgwsb8+/HH8E+2vq6EQJJzXMbuhwOBxeiULDsmFdTKC8QxvKO7QJdY3GWHtxcfFnYu3VtfU8NXstsxo+ze1YVsKCDbv4ZGdqLvGhvTox+fzjOLxTuPs7HI6WSxQalg3rPPTXF34KwBnH9gh8jfRY+wmjTqNDn2MBKC5S/ObVJXx9SA9OHtCVLu1KeWfi6W76WYfDERlRaFg2rBP0P729EgiXGbsO1DTF2i+8Zwb75s3jpX8/leIixQvXjqZHRzeXuMPh0EMUGpYNJSL5j9LAiBEjpKqqyvd52/ZWA9ClXWmg+97x+lL+8u4qZv/0DEpbFTFn9Xbat27llrZzOByxEFbDlFJzRGREpn3Weeh+M6Fq1TZ+/vwC/nTZCI7oVMYpR3WjpLiImrp6SlsVMbyv69h0OBzxEVTIvWBdp+gr8zfyyvyNWffv3FfD715byrx1OwDo3LaU0uIitu5JvRVH9u/CD08f2LR8m8PhcMRJPg0LgydBV0qNU0otUUotV0rdkGG/Ukr9vmH/PKWUtmXl//LuKv7y7qqmbRFh3rodfLh2BwCtihX3v72SqlXbARhY3p5nf/AlhvbqpMskh8Ph8ExzDYuSvDF0pVQxsBT4KrAOmA38m4gsTDvmLOBa4CzgZOBOETk513WDxtB3Haihvl7YV11Hz85liAin/fYNBnZvz1+uSM2VsudgLe2dB+5wOAxk14Ea4NAU2X4JG0MfCSwXkZUNF3sCOBdYmHbMucDDkno7zFRKdVZKHSEikbcrOrYp4f88/gFz1+1g+nVjUEpx18Un0afLoTmGnZg7HA5TCSrkXvCifL2AtWnb60h54fmO6QV8RtCVUlcDVwP06dPHr60APD93A0d2KWPs4HLqBYoVHN+7c6BrORwOR9w8P3cDAN8Y1jPya3uJoWcaiN08TuPlGETkPhEZISIjunfv7sW+z/HozNVUrdrOuSf0orjIjRF3OBx28ejM1Tw6c7WWa3vx0NcBR6Zt9wY2BDgmEh68ws0p7nA47EWnhnnx0GcDg5RS/ZVSpcBFwHPNjnkOuKxhtMsoYKeO+DlAWWkxZaXFOi7tcDgc2tGpYXk9dBGpVUr9CHgVKAYeEJEFSqlrGvbfC7xEaoTLcmAfcIUWa4G/f7AOgPNP7K3rFg6Hw6ENnRrmaTiIiLxESrTT/3Zv2v8F+GG0pmXmifdSfa9O0B0Oh43o1DDr5nKpqasH3GpADofDTsJqWEHN5eKE3OFw2IxODbNOHZ+uWsvTVWvzH+hwOBwGolPDrBP0Z+as45k565I2w+FwOAKhU8MSi6ErpTYDQUfXdwO2RGiODbg0twxcmlsGYdLcV0QyfpmZmKCHQSlVla1ToFBxaW4ZuDS3DHSl2bqQi8PhcDgy4wTd4XA4CgRbBf2+pA1IAJfmloFLc8tAS5qtjKE7HA6H4/PY6qE7HA6HoxlO0B0Oh6NAMFrQTVqcOi48pPkYpVSlUuqgUuq6JGyMGg9pntBQvvOUUjOUUsOSsDNKPKT53Ib0fqiUqlJKjU7CzijJl+a0476olKpTSl0Yp31R46GMxyildjaU8YdKqZ+FvqmIGPkjNVXvCmAAUArMBY5tdsxZwMukVkwaBcxK2u4Y0lwOfBH4NXBd0jbHlOZTgMMa/n9mCynn9hzq4zoeWJy03brTnHbcNFKzu16YtN2ay3gM8EKU9zXZQ29anFpEqoHGxanTaVqcWkRmAp2VUkfEbWiE5E2ziGwSkdlATRIGasBLmmeIyPaGzZmkVsSyGS9p3iMNTz3QjgxLOlqGl+cZ4Frgb8CmOI3TgNf0RorJgp5t4Wm/x9hEoaXHC37TfCWpVpnNeEqzUup8pdRi4EXgOzHZpou8aVZK9QLOB+7FfrzW6wql1Fyl1MtKqSFhb2qyoEe2OLVFFFp6vOA5zUqp00kJ+kStFunH66LqfxeRY4DzgJt1G6UZL2m+A5goInX6zdGOl/S+T2pelmHAH4B/hL2pyYJu1OLUMVFo6fGCpzQrpY4H7gfOFZGtMdmmC1/lLCJvAUcppbrpNkwjXtI8AnhCKbUKuBC4Wyl1XizWRU/e9IrILhHZ0/D/l4CSsGVssqAbtTh1THhJc6GRN81KqT7As8ClIrI0ARujxkuaByqlVMP/TyLVsWbziyxvmkWkv4j0E5F+wDPAD0TkH7FbGg1eyvjwtDIeSUqPQ5WxsSsWiWGLU8eBlzQrpQ4HqoCOQL1S6sekes93JWV3GDyW88+ArqQ8NoBasXh2Po9p/iYpZ6UG2A+MT+sktQ6PaS4YPKb3QuD7SqlaUmV8Udgydp/+OxwOR4FgcsjF4XA4HD5wgu5wOBwFghN0h8PhKBCcoDscDkeB4ATd4XA4CgQn6A6Hw1EgOEF3OByOAuH/AzpKddAjeN+HAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for method in ['Sobol', 'Halton', 'LatinHypercube']:\n",
    "    x_sobol = esrs.sample_with_scipy_qmc(n_particles, method=method)\n",
    "    plot_2d_example(x_sobol)\n",
    "    plt.gca().set_title(method)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "blessed-energy",
   "metadata": {},
   "source": [
    "The LatinHypercube does not look as good as Sobol or Halton.\n",
    "\n",
    "Next try the n-body simulation,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "closed-boring",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_steps = 30\n",
    "xsteps = esrs.sample_nbody(\n",
    "    n_part=n_particles,\n",
    "    n_steps=n_steps\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "innovative-extraction",
   "metadata": {},
   "outputs": [],
   "source": [
    "import imageio\n",
    "import os\n",
    "\n",
    "def gif_2d_example(xsteps, outfile='mygif.gif'):\n",
    "    n_steps, _, _ = xsteps.shape\n",
    "    filenames = []\n",
    "    for i in range(n_steps):\n",
    "        x = xsteps[i]\n",
    "        plot_2d_example(x)\n",
    "        fig = plt.gcf()\n",
    "        filename = f'{i}.png'\n",
    "        filenames.append(filename)\n",
    "        plt.savefig(filename)\n",
    "        plt.close()\n",
    "    # build gif\n",
    "    with imageio.get_writer(outfile, mode='I') as writer:\n",
    "        for f0 in filenames:\n",
    "            image = imageio.imread(f0)\n",
    "            writer.append_data(image)\n",
    "    # Remove files\n",
    "    for f0 in set(filenames):\n",
    "        os.remove(f0)\n",
    "    return"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "oriental-enlargement",
   "metadata": {},
   "outputs": [],
   "source": [
    "gif_2d_example(xsteps, outfile='example_2d.gif')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "881c00b1",
   "metadata": {},
   "source": [
    "<img src=\"example_2d.gif\" align=\"center\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08b11cb8",
   "metadata": {},
   "source": [
    "The particles bunch up against the walls. Though they are reflected, they seem to be pinned to the edges by the repulsion of all other particles. We could maybe fix this by boosting the velocities of the particles when they are reflected off the walls. Here by a factor of ten,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "4a11122d",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_steps = 30\n",
    "xsteps = esrs.sample_nbody(\n",
    "    n_part=n_particles,\n",
    "    n_steps=n_steps,\n",
    "    kw_step={'reflect_boost':10.}    \n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "974d2840",
   "metadata": {},
   "outputs": [],
   "source": [
    "gif_2d_example(xsteps, outfile='example_2d_boost.gif')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7389fe21",
   "metadata": {},
   "source": [
    "<img src=\"example_2d_boost.gif\" align=\"center\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "518da152",
   "metadata": {},
   "source": [
    "To compare these quantitatively, we can look at the distribution of distances from each particles to its nearest neighbour,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "lined-anderson",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/pjethwa/miniconda3/envs/base_py39/lib/python3.9/site-packages/scipy/stats/_qmc.py:1078: UserWarning: The balance properties of Sobol' points require n to be a power of 2.\n",
      "  warnings.warn(\"The balance properties of Sobol' points require\"\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Distribution of Distance to Nearest Neighbours')"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = esrs.uniform_sample_with_rejection(n_particles)\n",
    "min_dist_unif = esrs.get_min_dist(x)\n",
    "x = esrs.sample_with_scipy_qmc(n_particles, method='Sobol')\n",
    "min_dist_sobol = esrs.get_min_dist(x)\n",
    "x = esrs.sample_with_scipy_qmc(n_particles, method='Halton')\n",
    "min_dist_halton = esrs.get_min_dist(x)\n",
    "x = esrs.sample_with_scipy_qmc(n_particles, method='LatinHypercube')\n",
    "min_dist_lhcs = esrs.get_min_dist(x)\n",
    "# for n-body take the last snapshot of the previous simulation\n",
    "min_dist_nbody = esrs.get_min_dist(xsteps[-1])\n",
    "\n",
    "x = [min_dist_unif,\n",
    "     min_dist_sobol,\n",
    "     min_dist_halton,\n",
    "     min_dist_lhcs,\n",
    "     min_dist_nbody]\n",
    "labels=['Uniform',\n",
    "        'Sobol',\n",
    "        'Halton',\n",
    "        'LHCS',\n",
    "        'n-body']\n",
    "_ = plt.boxplot(x, labels=labels, vert=False, meanline=False)\n",
    "plt.axvline(np.median(min_dist_unif), ls=':', color='k')\n",
    "plt.gca().set_title('Distribution of Distance to Nearest Neighbours')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "886eaf2c",
   "metadata": {},
   "source": [
    "As we saw visually above, it appears that latin hypercube sampling is no better than uniform random. This may be a cosequence of using rejection sampling to impose the ineuqality constraints.\n",
    "\n",
    "Note there is a paper about [Latin hypercube sampling with inequality constraints](https://arxiv.org/abs/0909.0329), which presents a method to permute a LHC sample to satisfy linear inequalities while retaining the LHCS property. How well would this work? Figure 4 of that paper suggests to me that the perutations will lead to samples bunching up against the inequality.\n",
    "\n",
    "I don't think it is worth looking into LHCS further, as Halton and Sobal produce better results. The best result here is from the n-body code.\n",
    "\n",
    "## A 7D example\n",
    "\n",
    "Our modelling problem has 7 parameters which must satisfy 4 inequality constraints. The exact parameter ranges and the form of the inqualites depend on other hyper-parameters, which can change from problem instance to instance. Hence here, we will test the sampling methods with a *representative* 7D space with 4 constraints,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "7e1b6a91",
   "metadata": {},
   "outputs": [],
   "source": [
    "esrs = EvenlySpacedRandomSampling(\n",
    "    [(0,1)]*7,\n",
    "    A_ineq=np.array([\n",
    "        [1,-1,0,0,0,0,0],\n",
    "        [1,0,-1,0,0,0,0],\n",
    "        [0,0,0,0,0,1,-1],\n",
    "        [0,0,0,0,1,0,-1],\n",
    "    ]),\n",
    "    b_ineq=np.array([0,0,0,0])\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3acdad69",
   "metadata": {},
   "source": [
    "Get the nearest neighbour distances for each of the methods,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "8868f6db",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/pjethwa/miniconda3/envs/base_py39/lib/python3.9/site-packages/scipy/stats/_qmc.py:1078: UserWarning: The balance properties of Sobol' points require n to be a power of 2.\n",
      "  warnings.warn(\"The balance properties of Sobol' points require\"\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Distribution of Distance to Nearest Neighbours')"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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PAR8CHt3DmAtm5syZzJ07l+uuu44333yT6667jrlz5zJz5sykQ5MMjY2NNDY2Jh2GSMGkUimam5tZsWIF3d3drFixgubmZlKpVFHXW8jbaje6e3PPEzOrA8aY2Ufc/XFgOtHYxNNAnZkd7u7PAjOAVXH5WDM7zN1/G9dPdwfRWcfd7r69gHHvkZtuugmAL33pS1x00UXU1NRwzjnn7CiX8rF69WoAjj766IQjESmM6dOjw2NLSwtr166lvr6eBQsW7CgvFtuT0684GTzs7o3x8znAfu4+P6POI8APgaOJBq5nuPubZnYscC1RwnoSONfd/2JmxwHXAy8TJZdGd/943N4Q4BXgg+7+dF8xNjU1eUdHR79fWyUws4o4bS6l3vpk0qRJQALfw5g/HOa/VpJVaZsorMHcn2b2lLs3ZZu3R2cY7r4BaEx7fm2OOkfmWP4HwN9nKf8e0VhGNu8Hfp5PshBJ941vfCPpEESCUBHf9Dazi4FzgdOSjkUqz7hx45IOQSQIFfHFPXe/2t0Pcff2vmuHb7CeKu+pVatWsWrVqqTDkAqifSy7ijjDEBmIefPmAfotKZGBUsKQ4N15551JhyASBCUMCd6hhx6adAgiQVDCkGCU2/8w8HnDShZTbW1tSdYjg5sShgSht0HKxL6HAez8ZpJI5VPCkODdfffdSYcgEgQlDAne6NGjkw5BJAgV8T0MkYH43ve+x/e+972kwxCpeDrDkOBdffXVABx33HEJRyJS2ZQwJHjLli1LOgSRIChhSPAOPPDApEMQCYLGMCR4Dz30EA899FDSYYhUPJ1hSPAWLlwIwNSpUxOORKSyKWFI8O67776kQxAJghKGBG/kyJFJhyASBI1hSPCWL1/O8uXLkw5DpOLpDEOCd+ONNwLwyU9+MuFIRCqbEoYE74EHHkg6BJEgKGFI8IYPH550CCJB0BiGBO/ee+/l3nvvTToMkYqnMwwJ3q233grAKaecknAkIpVNCUOC98gjjyQdgkgQlDAkePvuu2/SIYgEQWMYErylS5eydOnSpMMQqXg6w5Dg3XHHHQCcfvrpCUciUtmUMKSgRowYQVdXVyLr9nnDsMtfzznfzIq27traWjZt2lS09kXKgRKGFFRXVxfunszK5w9PbN3FTEYi5UJjGBK8JUuWsGTJkqTDEKl4ShgSPCUMkcLQJSkJ3sqVK5MOQSQIOsMoY7ouLrKT9ofkKWFI8G6//XZuv/32pMMQqXhFSxhmtjlL2Xwzm5NRtsHMRsbTB5rZMjP7rZn92sweMbMjzKzKzG40szVm9ksze9LMxhYrdgmLfnxQpDDK5gzDovPN/wBWuvth7n4k8CXgXcApwEHA+9z974BPAK8mFatUlscee4zHHnss6TCkCFpbW2lsbKS6uprRo0czevRoqquraWxspLW1NZE4Sr3uUiqnQe/JQLe7L+opcPdOADP7N+BFd387Ln8+kQhFpGy0traSSqVYvHgxzz//PHPnzsXdWbJkCQcffDDNzc0ATJ8+vWRxTJw4kfb29pKtu+TcvSgPYHOWsvnAH4DOtMc2YCRwAfDvOdo6GNgQ118I/H1f658wYYJXuujtqSyJxjxvWNbiW265xW+55ZairroS36tKk9nHDQ0N3tbWtst0W1ubNzQ0uLvvMl1M6XH0KNW6iwHo8BzHVfMifTPWzDa7+34ZZfPjRHJtWtkGoAn4NDDW3f81R3s1wMfiRzPwz+7+g4w6s4BZAGPGjJnw3HPPFez1JKFS7wop1jbVp/nDYf5ruxUff/zxADz66KNFW3WlvleVJn3bqq6uZuvWrQwZMmTHNMDQoUPZvn073d3dO6aLKT2OHqVadzGY2VPu3pRtXjldkvoVcHKume7+F+BR4FEz+xMwDfhBRp3bgNsAmpqaEjpqFVZiB989VI4HzmIminSV9l5Vmsxtq76+nvb2diZPnrxjuqccoL29fcd0MaXH0aNU6y61shn0BtqAGjOb2VNgZkeZ2TFm9gEzOyguqwLeB1T26YOIDEgqlaK5uZkVK1Ywd+5cTjvtND796U8zd+5cVqxYQXNzM6lUqqRxdHd3l3TdpVbMM4x9zSx9cPq63iq7u5vZJ4DrzexiYCvRuMVs4DDg9viyFMATwM0Fj1iCdMMNNwBw4YUXJhyJFFLPgHJLSwtr167loIMOAuCMM86gvr6eBQsWlGTQOTOOUq671Io2hpG0pqYm7+joSDqMATGzirvMkWjMOcYwTjzxRAAefPDBoq26Et+rSqM+Lo1KGcMQKYpiJgqRwaScxjAkgz5Nieyk/SF5ShgSvGuvvZZrr72274oi0itdkpLgPf7440mHIBIEJQwJ3v333590CCJBUMKQgkvqy3s+b1hi666trU1kvSKlpIQhBZX0wKTP373s6quvBuDiiy8ubTAigVHCkOB1dnYmHYJIEJQwJHjLli1LOgSRIOi2WhERyYsShgTviiuu4Iorrkg6DJGKp0tSErx169YlHYJIEJQwJHhLly5NOgSRIOiSlIiI5EUJQ4J32WWXcdlllyUdhkjF0yUpCd7GjRuTDkEkCEoYEry77ror6RBEgqBLUiIikhclDAneJZdcwiWXXJJ0GCIVT5ekJHivvPJK0iGIBEEJQ4J32223JR2CSBB0SUpERPKihCHBmzNnDnPmzEk6DJGKp0tSEry33nor6RBEgqCEIcG75ZZbkg5BJAi6JCUiInlRwpDgzZ49m9mzZycdhkjF0yUpqSgjRoygq6trj5a94YYbep3v84Zhl7++R233pba2lk2bNhWlbZFSUcKQitLV1YW7F6fx+cOL1raZFaVdkVLSJSkREcmLEoYE77zzzuO8885LOgyRiqdLUhK8ffbZJ+kQRIKghCHBu/baa5MOQSQIuiQ1yGkwVkpB21kYlDAkeLNmzWLWrFlJhyFS8QaUMMxsc8bzM8zs5j6W2VHHzKaZ2ZEDiUGkLwcccAAHHHBA0mFIibW2ttLY2Eh1dTWNjY20trYmHVLFS3oMYxrwMPDrhOOQgF111VVJhyAl1traSiqVYvHixUycOJH29naam5sBmD59esLRVa6iXZIys6lm9hMz+5mZPWZm78qYfzRwInCNmXWa2WFmNt7MfmxmvzCz/zCz2rjuSjP7mpk9YWbPmNlHixW3iFS+BQsWsHjxYiZPnsyQIUOYPHkyixcvZsGCBUmHVtEGeoaxj5l1pj0fATwYT7cDH3Z3N7PPAV8ELuqp6O6rzexB4GF3vw/AzH4BtLj7KjP7CjAPmN0Tq7t/0MxOiMunZAZjZrOAWQBjxowZ4EsbPEIfkDzzzDMBuOuuuxKNI/R+Lidr165l4sSJu5RNnDiRtWvXJhRRGAaaMN5y9/E9T8zsDKApfnowcK+Z/S2wN/C73hoys+HAO919VVz0TeA7aVWWx3+fAuqyteHutwG3ATQ1NRXp9yPCU7Sf2iiCPTnojh49ugiR9F8l9XOhlTpZ1tfX097ezuTJk3eUtbe3U19fX9I4QlPMMYybgOvc/UEzmwTMH2B7f4n/bif5sRepIF/5yleSDkFKLJVK0dzcvNsYhi5JDUwxD7zDgT/E05/NUecNYH8Ad3/NzLrM7KPu/iNgBrAqx3IiIjn1DGy3tLSwdu1a6uvrWbBggQa8B6iYCWM+8B0z+wPwY2BsljrLgNvN7ALgZKLEssjM9gXWA2cWMT4ZJE4//XQAli5dmnAkUkrTp09XgiiwASUMd98v4/kSYEk8/QDwQJZl0uv8D5D5PYwPZ1lmUtr0y+QYwxDJZty4cUmHIBIEjQVI8L785S8nHYJIEPTTIIPcYL5zR0pH21kYlDAkeKeeeiqnnnpq0mGIVDxdkpLgjR8/PukQRIKghCHBu/jii5MOQSQIuiQlFcfMivIoZtu1tbUJ95rIwOkMQyrKngyefupTnwLg/vvv77v9+f1uXmTQUMKQ4H3kIx9JOgSRIChhSPDmzJmTdAgiQdAYhoiI5EUJQ4J34okncuKJJyYdhkjF0yUpCd6xxx6bdAgiQVDCkOBdeOGFSYcgEgRdkhIRkbwoYUjwjj/+eI4//vikwxCpeLokJcGbOnVq0iGIBEEJQ4L3+c9/PukQRIKgS1IiIpIXJQwJ3pQpU5gyZUrSYYhUPF2SkuCdcsopSYcgEgQlDAnezJkzkw5BJAi6JCUiInlRwpDgTZo0iUmTJiUdhkjF0yUpCd4ZZ5yRdAgiQVDCkOApYYgUhi5JSfC6u7vp7u5OOgyRiqczDAneP/7jPwKwcuXKZAMRqXBKGBKMESNG0NXVlXO+mfWrPZ83DLv89YGGtYva2lo2bdpU0DZFSkUJQ4LR1dWFuxeuwfnDC9se/U9aIuVEYxgSvDfffJM333wz6TBEKp7OMCR4J5xwAqAxDJGBUsKQ4J177rlJhyASBCUMGTAzK/i1/kLSjw8mp9y3DekfjWFI8F577TVee+21pMMQqXhFSRhmljKzX5nZL8ys08w+1EvdDWY2sh9tzzezOYWJVAaDk046iZNOOinpMEQqXsEThpl9BPg48AF3fx8wBdhY6PWI5OuCCy7gggsuSDoMyaK1tZXGxkaqqqoYOnQoVVVVNDY20traumNedXU1jY2NtLS07PK8tbW1z3bT62Yry1VXcnD3gj6ATwIPZSk/FvgZ8EvgTqAmLt8AfA14In4cHpcfAvwA+EX8d0xcPh+Y01ccEyZMcCmNaDNKXsHjmDessO15+fRVqfT2eu+55x4fO3asp1IpHzt2rC9cuNDr6uo8lUr5qFGjfNSoUd7W1ubbtm3zVCrle+21l6dSKd+2bZu3tbX52LFj/Z577snZbs+ybW1tu7XXs/z555+/W91c7Q4WQIfnOr7nmrGnD2A/oBN4Bvg6cAwwlOgs44i4zreA2b4zYaTi6c8AD8fTDwGfjafPAr7rShhlqVwOgrnieOmll/yll17qf4NKGAPW2+ttaGjwtra2HX/dfcfzuro6r6ur26XuwoULvaGhYUdZT91c7abLbK9n+Zqamt3q5mp3sOgtYZgX4Q4GM6sGPgpMBs4GrgI+5e7/EM8/FjjP3T9pZhuAj7n7ejMbAvzR3Q8ws5eBv3X37rj8RXcfaWbzgc3ufm2W9c4CZgGMGTNmwnPPPVfw1ya7K6dvL2fbnnv+F0a/v4cxfzjML+xgeTn1VankOsZUV1ezdetWhg4dytatWxkyZAjd3d0MHTp0x3Jvv/32jrpvvPEG+++/P9u3bwfYUbfneWa7Q4YM2VFWVVWFme1St7u7m7333ptt27btUjdXu4OFmT3l7k3Z5hXltlp33w6sBFaa2S+Bz/a1SI7pXHVyrfc24DaApqYm3ctXQsX44NFfuQ7GF110UYkj6V059FWp9JYg6+vraW9v3/F38uTJO55v2bJlt7qLFi2ivr5+R1lP3VztTp48eUfZIYccslu99vZ2ampqdqubq10pzqD3ODN7T1rReOBPQJ2ZHR6XzQBWpdU5Je3v4/H0auDUePo0oL3QscrgMHXqVKZOnZp0GJIhlUrR3NzMtGnTaG5u5rrrruOss85i2rRpbNmyhS1btrBixQq6u7uZNm0ac+fOZdq0aXR3d7NixQqam5tJpVI52+1ZdsWKFbu117P8zJkzd6ubq12hKGMYE4gO9r8mGrBeDoyk90HvecBPgCfZOehdB7ShQe+yR5lcl88Vx4svvugvvvhi/xvUGMaA9fV677nnHm9oaHAz85qaGjczb2ho8HvuuWfHvKqqKm9oaPDzzz9/l+e9DUxnLputvZ7lc5UPVpR6DKMcNDU1eUdHR9JhDArl8m3eXHGU2xhGOfRVqQy21xuCko9hiJSTiy++OOkQRIKghCEDVu6fII877rikQxi0yn3bkP7Rb0lJ8DZu3MjGjfqxAZGB0hmGBG/GjBmA/h+GyEApYUjwLr300qRDEAmCEoYEpZDfpPZ5wwr+zeza2tqCtidSSkoYEoxcA6zr168H4NBDD+1/m/MHEpFIWJQwJHhnnXUWoDEMkYFSwpDgXX755UmHIBIEJQwJ3jHHHJN0CCJB0PcwJHjr1q1j3bp1SYchUvF0hiHBO/vsswGNYYgMlBKGBO/KK69MOgSRIChhSPCOPvropEMQCYLGMCR4a9asYc2aNUmHIVLxdIYhwTv//PMBjWGIDJQShgTvmmuuSToEkSAoYUjwjjrqqKRDEAmCxjAkeJ2dnXR2diYdhkjF0xmGBG/27NmAxjBEBkoJQ4J3/fXXJx2CSBCUMCR448ePTzoEkSBoDEOC9+STT/Lkk08mHYZIxdMZhgTvC1/4AqAxDJGBUsKQ4N18881JhyASBCUMCV5jY2PSIYgEQWMYErzVq1ezevXqpMMQqXg6w5CiGDFiBF1dXUmHsYPPG4b9r9cTjaG2tpZNmzYlGoPIQChhSFF0dXXh7kmHAUT/cY/WDyYej5klun6RgdIlKQneuHHjkg5BJAhKGBK8VatWJR2CSBCUMCR48+bNSzoEkSAoYUjw7rzzzqRDEAmCEkYWGpwMy6GHHpp0CFIE2k9LTwlDgvfYY48lHYJIEPpMGGZWZ2ZrMsrmm9mcXpZpMrMb4+kaM3vMzDrN7JSBhyzSP1/96leTDkEGmdbWVhobG6murqaxsZHW1tYg1luU72G4ewfQET/9e2CIu4/Pd3kzq3b37cWITQafu+++Gxbr50GkNFpbW0mlUixevJiJEyfS3t5Oc3MzANOnT6/s9bp7rw+gDliTUTYfmAOsBL4GPAE8A3w0nj8JeBj4G+BZ4DWgEzgMOBb4GfBL4E6gJl5mA3AZ0A6cGj+/EnicKPl8APgv4LfAOX3FPWHCBN9TUbfIQJRdH84blnQE5dcnFa5c+7OhocHb2tp2KWtra/OGhoaKWC/Q4bnyQa4Znn/CWBiXnQA85mkJI8v0UGAjcET8/FvAbN+ZML6Yto4NwLnx9L8DvwD2B0YBf84R66w4uXSMGTOmX52U0WF6FOBRLh599NGySRh6hLmNpauqqvJt27btUrZt2zavqqqqiPXSS8LIZ9Db+yhfHv99iii59GYc8Dt3fyZ+/k3gH9Lm35tR/8H47y+Bn7j7G+7+ErDVzN65W0Dut7l7k7s3jRo1qo9Qeperw/TI71FOrr766qRD2CHp9yWkR7mqr6+nvb19l7L29nbq6+srfr35JIxXgNqMshHAy/H0X+K/2+l7TKSv++C2ZDzvafvttOme5/odLMnLsmXLkg5BBpFUKkVzczMrVqygu7ubFStW0NzcTCqVqvj19nnQdffNZvaimR3r7j8wsxHAccANwJn9XN/TQJ2ZHe7uzwIzAP1ugxTVgQcemHQIMoj0DDC3tLSwdu1a6uvrWbBgQVEHvEu13nw/pX8GuMXMFsbPL3f33/b3izPuvtXMzgS+Y2Z7AU8Ci/rViEg/PfTQQ0xNOggZVKZPn170BJHEeq2crwUORFNTk3d0dPRdMQszK+trpJWgnPpw0qRJrJz0M5j/WqJxlFOfhED9WRxm9pS7N2Wbp296Z6GNMCz33Xdf0iFIEWg/LT0lDAneyJEjkw5BJAhKGBK85cuX911JRPqkhCHBu/HGG5MOQSQIShgSvAceeCDpEESCoC+/SdGU0/8r8HnDEo+ntjbz+68ilUUJQ4qinO5guffee7kXcNev64sMhBKGBO/WW28F4JRTlDBEBkIJQ4L3yCOPJB2CSBCUMCR4++67b9IhiARBd0lJ8JYuXcrSpUuTDkOk4ukMQ4J3xx13AHD66acnHIlIZVPCkOB9//vfTzoEkSAoYUjwhgwZknQIIkHQGIYEb8mSJSxZsiTpMEQqnhKGBE8JQ6Qwgv0HSmb2EvBc0nEkbCQ7//e67E790zf1Ud9C66ND3H1UthnBJgwBM+vI9Z+zRP2TD/VR3wZTH+mSlIiI5EUJQ0RE8qKEEbbbkg6gzKl/+qY+6tug6SONYYiISF50hiEiInlRwhARkbwoYVQ4MzvOzNaZ2bNmdnGW+aeZ2S/ix2oze38ScSYpjz46Ke6fTjPrMLOJScSZpL76KK3eUWa23cxOLmV85SCP7WiSmb0Wb0edZnZZEnEWlbvrUaEPoBr4LXAosDfwc+DIjDpHA7Xx9PHAT5KOuwz7aD92jue9D3g66bjLrY/S6rUBjwAnJx13ufURMAl4OOlYi/nQGUZl+yDwrLuvd/dtwDLgpPQK7r7a3bvipz8GDi5xjEnLp482e7zHA+8ABtudIH32UawFuB/4cymDKxP59lHQlDAq27uBjWnPn4/LcmkGHi1qROUnrz4ys0+Y2dPAfwJnlSi2ctFnH5nZu4FPAItKGFc5yXdf+4iZ/dzMHjWzhtKEVjpKGJXNspRl/XRsZpOJEsbcokZUfvLqI3f/D3d/LzANuKLYQZWZfProemCuu28vfjhlKZ8++inR7zC9H7gJ+G6xgyo1JYzK9jwwOu35wcALmZXM7H3AHcBJ7v5KiWIrF3n1UQ93/yFwmJmNLHZgZSSfPmoClpnZBuBk4OtmNq0k0ZWHPvvI3V93983x9CPAkNC2IyWMyvYk8B4zG2tmewOnAg+mVzCzMcByYIa7P5NAjEnLp48ONzOLpz9ANKg5mBJrn33k7mPdvc7d64D7gM+7+3dLHmly8tmODkzbjj5IdHwNajvSf9yrYO7+VzM7H/gvors47nT3X5nZOfH8RcBlwAFEnwgB/uqD5Jc1Ie8++hTwGTPrBt4CTkkbBA9enn00qOXZRycD55rZX4m2o1ND24700yAiIpIXXZISEZG8KGGIiEhelDBERCQvShgiIpIXJQwREcmLEoaIiORFCUNERPLy/wGMkw5itPlnyQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = esrs.uniform_sample_with_rejection(n_particles)\n",
    "min_dist_unif = esrs.get_min_dist(x)\n",
    "x = esrs.sample_with_scipy_qmc(n_particles, method='Sobol')\n",
    "min_dist_sobol = esrs.get_min_dist(x)\n",
    "x = esrs.sample_with_scipy_qmc(n_particles, method='Halton')\n",
    "min_dist_halton = esrs.get_min_dist(x)\n",
    "x = esrs.sample_with_scipy_qmc(n_particles, method='LatinHypercube')\n",
    "min_dist_lhcs = esrs.get_min_dist(x)\n",
    "\n",
    "# For the n-body simulation, let's take more steps\n",
    "# to let things diffuse suffieciently,\n",
    "\n",
    "n_steps = 1000\n",
    "xsteps = esrs.sample_nbody(\n",
    "    n_part=n_particles,\n",
    "    n_steps=n_steps,\n",
    "    kw_step={'reflect_boost':10.}    \n",
    ")\n",
    "min_dist_nbody = esrs.get_min_dist(xsteps[-1])\n",
    "\n",
    "x = [min_dist_unif,\n",
    "     min_dist_sobol,\n",
    "     min_dist_halton,\n",
    "     min_dist_lhcs,\n",
    "     min_dist_nbody]\n",
    "labels=['Uniform',\n",
    "        'Sobol',\n",
    "        'Halton',\n",
    "        'LHCS',\n",
    "        'n-body']\n",
    "_ = plt.boxplot(x, labels=labels, vert=False, meanline=False)\n",
    "plt.axvline(np.median(min_dist_unif), ls=':', color='k')\n",
    "plt.gca().set_title('Distribution of Distance to Nearest Neighbours')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45f2bee0",
   "metadata": {},
   "source": [
    "Again, LHCS looks like it is hardly better than random. N-body produces the best median result, however it gives a long left-tail compared to Sobol and Halton. This is quite sensitive to how we pick the *releftion boost* parameter. While N-body has perfrmed well overall, with relatively little tuning, perhaps it is better to go with more tried-and-trusted approaches.\n",
    "\n",
    "Sobol and Halton are somewhat better than uniform in their median nearest neighbour diatnces, however with significantly shorter left tails. Eihter would be a reasonable choice."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}