混合ガウスモデルのギプスサンプリングのJulia実装

GS_GMM.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 混合ガウスモデルのギプスサンプリングのJulia実装\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ground Truthから訓練データをサンプリングする"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using PDMats\n",
    "using Distributions\n",
    "using LinearAlgebra\n",
    "using Plots\n",
    "using ProgressBars"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MixtureModel{MvNormal}(K = 3)\n",
       "components[1] (prior = 0.4000): FullNormal(\n",
       "dim: 2\n",
       "μ: [0.0, 5.0]\n",
       "Σ: [1.0 0.0; 0.0 2.0]\n",
       ")\n",
       "\n",
       "components[2] (prior = 0.3500): FullNormal(\n",
       "dim: 2\n",
       "μ: [-5.0, 2.0]\n",
       "Σ: [0.5 0.75; 0.75 2.0]\n",
       ")\n",
       "\n",
       "components[3] (prior = 0.2500): FullNormal(\n",
       "dim: 2\n",
       "μ: [5.0, -2.0]\n",
       "Σ: [0.5 0.1; 0.1 0.5]\n",
       ")\n",
       "\n"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "μ1_gt = [0.; 5.]\n",
    "μ2_gt = [-5.; 2.]\n",
    "μ3_gt = [5.; -2.]\n",
    "\n",
    "Σ1_gt = [1. 0.; 0. 2.]\n",
    "Σ2_gt = [.5 .75; .75 2.]\n",
    "Σ3_gt = [.5 .1; .1 .5]\n",
    "\n",
    "π_gt = [.4, .35, .25]\n",
    "\n",
    "mixture_components = MvNormal[\n",
    "    MvNormal(μ1_gt, Σ1_gt), \n",
    "    MvNormal(μ2_gt, Σ2_gt), \n",
    "    MvNormal(μ3_gt, Σ3_gt)\n",
    "];\n",
    "\n",
    "model_gt = MixtureModel(mixture_components, π_gt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 300\n",
    "\n",
    "x_train = rand(model_gt, N)\n",
    "scatter(x_train[1, :], x_train[2, :])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# GMMの構造体を作る"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "struct Gauss\n",
    "    μ::Vector{Float64} # Mean Vector\n",
    "    Λ::Matrix{Float64} # Precision Matrix\n",
    "end\n",
    "\n",
    "struct GMM\n",
    "    D::Int\n",
    "    K::Int\n",
    "    mixing_coeff::Vector{Float64}\n",
    "    components::Vector{Gauss}\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "GMMの可視化用の関数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "plot_contours (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function plot_contours(gmm::GMM)\n",
    "    mvnormals = MvNormal[\n",
    "        MvNormal(\n",
    "            gmm.components[k].μ, \n",
    "            PDMat(Symmetric(inv(gmm.components[k].Λ)))\n",
    "        ) for k in 1:gmm.K]\n",
    "    \n",
    "    for k in 1:gmm.K\n",
    "        contour!(-10:0.05:10, -10:0.05:10, (x,y)->pdf(mvnormals[k], [x,y]), color=k, colorbar=:none)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ハイパーパラメタの準備"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Number of Classes\n",
    "K = 3;\n",
    "\n",
    "# Number of Dimensions\n",
    "D = 2;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "クラスタごとに別々の事前分布を設定することもできるが，ここでは全てのクラスタに同じ事前分布を設定している．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Float64,2}:\n",
       " 1.0  0.0\n",
       " 0.0  1.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "α0 = ones(K)\n",
    "μ0 = zeros(D)\n",
    "β0 = 1.0\n",
    "ν0 = convert(Float64, D)\n",
    "W0 = Matrix{Float64}(I, D, D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 潜在変数の準備"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "潜在変数の初期値は事前分布からサンプリングする"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Gauss,1}:\n",
       " Gauss([-5.45594, 10.6532], [1.08619 0.572529; 0.572529 0.308124]) \n",
       " Gauss([-1.11151, 1.82516], [0.789222 0.198393; 0.198393 0.640528])\n",
       " Gauss([-0.98439, 2.21395], [1.54854 0.207806; 0.207806 0.297267]) "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mixing_coeff = rand(Dirichlet(α0))\n",
    "\n",
    "μk = zeros(K, D)\n",
    "Λk = zeros(K, D, D)\n",
    "\n",
    "for k in 1:K\n",
    "    Λk[k, :, :] = rand(Wishart(ν0, W0))\n",
    "    μk[k, :] = rand(MvNormal(μ0, PDMat(Symmetric(inv(β0 * Λk[k, :, :])))))\n",
    "end\n",
    "\n",
    "gaussians = Gauss[Gauss(μk[k, :], Λk[k, :, :]) for k in 1:K]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "gmm = GMM(D, K, mixing_coeff, gaussians)\n",
    "hidden_state = rand(Categorical(gmm.mixing_coeff), N);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "データとクラスタの割当をプロット"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plt = plot()\n",
    "plot_contours(gmm)\n",
    "scatter!(x_train[1, :], x_train[2, :], markercolor=hidden_state)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 潜在変数のサンプリングの実装"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resample Hidden States"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sample_hidden_state (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sample_hidden_state(gmm::GMM, x_train::Array{Float64})\n",
    "    hidden_sttate = zeros(Int32, N)\n",
    "    \n",
    "    mvnormals = MvNormal[\n",
    "        MvNormal(\n",
    "            gmm.components[k].μ, \n",
    "            PDMat(Symmetric(inv(gmm.components[k].Λ)))\n",
    "        ) for k in 1:K]\n",
    "    \n",
    "    # temporary variable\n",
    "    p_xi = zeros(N, K)\n",
    "\n",
    "    for k in 1:K\n",
    "        p_xi[:, k] = pdf(mvnormals[k], x_train)\n",
    "    end\n",
    "\n",
    "    for n in 1:N\n",
    "        p_xi[n, :] .*= gmm.mixing_coeff\n",
    "    end\n",
    "\n",
    "    for n in 1:N\n",
    "        mixing_coeff_post = p_xi[n, :] / sum(p_xi[n, :])\n",
    "        hidden_state[n] = rand(Categorical(mixing_coeff_post))\n",
    "    end\n",
    "    \n",
    "    return hidden_state\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resampling mixing coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sample_mixing_coeff (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sample_mixing_coeff(gmm::GMM, hidden_state::Array{Int64,1})\n",
    "    state_ids = Array{Int64,1}(1:gmm.K)\n",
    "    m = [count(x -> x == i, hidden_state) for i in state_ids]\n",
    "    \n",
    "    α_post = α0 + m\n",
    "    new_mixing_coeff = rand(Dirichlet(α_post))\n",
    "    \n",
    "    return GMM(gmm.D, gmm.K, new_mixing_coeff, gmm.components)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resampling $\\mu_k$ and $\\Lambda_k$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sample_gauss (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sample_gauss(gmm::GMM, x_train::Array{Float64}, hidden_state::Array{Int64,1})\n",
    "    new_μk = zeros(K, D)\n",
    "    new_Λk = zeros(K, D, D)\n",
    "\n",
    "    # temporary variables\n",
    "    state_ids = Array{Int64,1}(1:gmm.K)\n",
    "    m = [count(x -> x == i, hidden_state) for i in state_ids]\n",
    "\n",
    "    for k in 1:K\n",
    "        # Get data in the k-th class\n",
    "        x_k = x_train[:, hidden_state .== k]\n",
    "        # Compute the covariance matrix of the k-th class\n",
    "        S_k = x_k * x_k'\n",
    "\n",
    "        # Compute parameters of posterior distributions\n",
    "        β_post = β0 + m[k]\n",
    "        μ_post = (sum(x_k, dims=2) + β0 * μ0) / β_post\n",
    "        μ_post = μ_post[:, 1]\n",
    "\n",
    "        ν_post = ν0 + m[k]\n",
    "        W_post_inv = S_k + β0 * μ0 * μ0' - β_post * μ_post * μ_post' + inv(W0)\n",
    "        W_post = PDMat(Symmetric(inv(W_post_inv)))\n",
    "        \n",
    "        # Sample parameters from posterior distributions\n",
    "        new_Λk[k, :, :] = rand(Wishart(ν_post, W_post))\n",
    "        new_μk[k, :] = rand(MvNormal(μ_post, PDMat(Symmetric(inv(β_post * new_Λk[k, :, :])))))\n",
    "    end\n",
    "    \n",
    "    new_components = Gauss[Gauss(new_μk[k, :], new_Λk[k, :, :]) for k in 1:K]\n",
    "    \n",
    "    return GMM(gmm.D, gmm.K, gmm.mixing_coeff, new_components)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Gibbs Samplingの実行とアニメーションの生成"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "アニメーション生成の参考\n",
    "\n",
    "https://qiita.com/Lirimy/items/8976bb4dcf5febad178e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "anim = Animation()\n",
    "\n",
    "plt = scatter(x_train[1, :], x_train[2, :], markercolor=hidden_state, legend=:none)\n",
    "plot_contours(gmm)\n",
    "title!(\"Iteration: 0\")\n",
    "frame(anim, plt)\n",
    "\n",
    "for iter in tqdm(1:120)\n",
    "    hidden_state = sample_hidden_state(gmm, x_train)\n",
    "    gmm = sample_mixing_coeff(gmm, hidden_state)\n",
    "    gmm = sample_gauss(gmm, x_train, hidden_state)\n",
    "    \n",
    "    plt = plot()\n",
    "    scatter!(x_train[1, :], x_train[2, :], markercolor=hidden_state, legend=:none)\n",
    "    plot_contours(gmm)\n",
    "    title!(\"Iteration: $iter\")\n",
    "    frame(anim, plt)\n",
    "end\n",
    "\n",
    "gif(anim, \"movie.gif\", fps=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最終結果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plt = plot()\n",
    "plot_contours(gmm)\n",
    "scatter!(x_train[1, :], x_train[2, :], markercolor=hidden_state)"
   ]
  }
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