2021/Week5/HitAndRun.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hit-and-run is a Markov chain Monte Carlo (MCMC)\n",
    "sampling technique that iteratively generates\n",
    "a sequence of points in a set by taking steps of\n",
    "random length in randomly generated directions.\n",
    "Hit-and-run can be applied to virtually any bounded"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(1) Generate a random direction uniformly distributed over the surface of a unit hypersphere centered around the current point."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(2) Generate a random point (...)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[1] https://link.springer.com/referenceworkentry/10.1007%2F978-1-4419-1153-7_1145"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distance to a linear constraint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LazySets, Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "_unitary_random_direction (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "# compute a unitary random direction in the given dimensions\n",
    "function _unitary_random_direction(n)\n",
    "    return normalize!(randn(n))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "_distance (generic function with 2 methods)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# distance we have to travel in direction dir,\n",
    "# from point p, to reach a given hyperplane a^T x = b\n",
    "function _distance(a, b, p, dir)\n",
    "    α = dot(a, dir)\n",
    "    LazySets.isapproxzero(α) && throw(ArgumentError(\"the direction is parallel to the hyperplane\"))\n",
    "    λ = (b - dot(a, p)) / α\n",
    "    return λ\n",
    "end\n",
    "\n",
    "function _distance(H::Hyperplane, p, dir)\n",
    "    return _distance(H.a, H.b, p, dir)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " 0.24990147139578237\n",
       " 0.9682712711808726"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = [1, 1.]\n",
    "dir = _unitary_random_direction(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hyperplane{Float64,Array{Float64,1}}([2.5, 1.0], 5.0)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H = Hyperplane([2.5, 1.0], 5.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9416048382107387"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "λ = _distance(H, p, dir)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5930249496703286"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dot(H.a, dir)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig = plot(xlims=(0, 3), ylims=(0, 5), xlab=\"x\", ylab=\"y\")\n",
    "\n",
    "plot!(fig, Singleton(p))\n",
    "plot!(fig, H, lw=3.0, c=:black, style=:dash)\n",
    "plot!(fig, LineSegment(p + zeros(2), p + λ*dir), marker=:x, lw=4.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig = plot(xlims=(0, 3), ylims=(0, 5), xlab=\"x\", ylab=\"y\")\n",
    "\n",
    "for _ in 1:20\n",
    "    dir = _unitary_random_direction(2)\n",
    "    λ = _distance(H, p, dir)\n",
    "    \n",
    "    # keep short distances\n",
    "    λ > 5 && continue\n",
    "\n",
    "    plot!(fig, Singleton(p))\n",
    "    plot!(fig, H, lw=3.0, c=:black, style=:dash)\n",
    "    plot!(fig, LineSegment(p + zeros(2), p + λ*dir), style=:dot, lw=5.0)\n",
    "end\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "hitandrun! (generic function with 1 method)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function _sample!(D, X; thin=1)\n",
    "    \n",
    "    # initialization\n",
    "    p = an_element(X)\n",
    "    n = dim(X)\n",
    "    clist = constraints_list(X)\n",
    "    D[1] = p\n",
    "\n",
    "    if thin == 1\n",
    "        for i in 2:length(D)\n",
    "            D[i] = hitandrun(clist, D[i-1])\n",
    "        end\n",
    "\n",
    "    else\n",
    "        aux = similar(p)\n",
    "        for i in 2:length(D)\n",
    "            D[i] = similar(p)\n",
    "            copy!(aux, D[i-1])\n",
    "            for _ in 1:thin\n",
    "                hitandrun!(D[i], clist, aux)\n",
    "                copy!(aux, D[i])\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    return D\n",
    "end\n",
    "\n",
    "hitandrun(clist::Vector{<:HalfSpace}, p) = hitandrun!(similar(p), clist, p)\n",
    "\n",
    "function hitandrun!(q, clist::Vector{HT}, p) where {N, VT, HT<:HalfSpace{N, VT}}\n",
    "\n",
    "    n = dim(first(clist))\n",
    "    dir = _unitary_random_direction(n)\n",
    "    m = length(clist)\n",
    "    λ = Vector{N}(undef, m)\n",
    "    @inbounds for (i, c) in enumerate(clist)\n",
    "        λ[i] = _distance(c.a, c.b, p, dir)\n",
    "    end\n",
    "\n",
    "    λ₊ = minimum(λ[λ .> 0])\n",
    "    λ₋ = maximum(λ[λ .< 0])\n",
    "\n",
    "    @assert (λ₊ > 0) && (λ₋ < 0)\n",
    "\n",
    "    U = LazySets.DefaultUniform(λ₋, λ₊)\n",
    "    μ = rand(U)\n",
    "\n",
    "    q .= p + μ * dir\n",
    "    return q\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.020072 seconds (200.80 k allocations: 18.079 MiB)\n"
     ]
    }
   ],
   "source": [
    "D = Vector{Vector{Float64}}(undef, 1000)\n",
    "X = rand(HPolygon)\n",
    "@time _sample!(D, X, thin=20);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(X)\n",
    "plot!(Singleton(an_element(X)), c=:red)\n",
    "plot!(Singleton.(D))"
   ]
  }
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