14-Chapter 14. Adventures in Covariance.ipynb

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   "id": "1743ed37",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Turing\n",
    "using DataFrames\n",
    "using CSV\n",
    "using Random\n",
    "using Distributions\n",
    "using StatisticalRethinking\n",
    "using StatisticalRethinking: link\n",
    "using StatisticalRethinkingPlots\n",
    "using StatsPlots\n",
    "using StatsBase\n",
    "using Logging\n",
    "using LinearAlgebra\n",
    "\n",
    "default(label=false);\n",
    "Logging.disable_logging(Logging.Warn);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82af9c69",
   "metadata": {},
   "source": [
    "# 14.1 Varying slopes by construction"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "274bb549",
   "metadata": {},
   "source": [
    "Code 14.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f01dcee2",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 3.5    # average morning wait time\n",
    "b = -1     # average difference afternoon wait time\n",
    "σ_a = 1    # std dev in intercepts\n",
    "σ_b = 0.5  # std dev in slopes\n",
    "ρ = -0.7;  # correlation between intercepts and slopes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58b50f15",
   "metadata": {},
   "source": [
    "Code 14.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "12db9e64",
   "metadata": {},
   "outputs": [],
   "source": [
    "μ = [a, b];"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea532213",
   "metadata": {},
   "source": [
    "Code 14.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ca55f875",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Matrix{Float64}:\n",
       "  1.0   -0.35\n",
       " -0.35   0.25"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cov_ab = σ_a * σ_b * ρ\n",
    "Σ = [[σ_a^2, cov_ab] [cov_ab, σ_b^2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75c27a70",
   "metadata": {},
   "source": [
    "Code 14.4\n",
    "\n",
    "Julia has similar \"column-first\" matix order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "89890d9c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 reshape(::UnitRange{Int64}, 2, 2) with eltype Int64:\n",
       " 1  3\n",
       " 2  4"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reshape(1:4, (2,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ccd45231",
   "metadata": {},
   "source": [
    "Code 14.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "7de5925d",
   "metadata": {},
   "outputs": [],
   "source": [
    "sigmas = [σ_a, σ_b]\n",
    "Ρ = [[1, ρ] [ρ, 1]]\n",
    "Σ = Diagonal(sigmas) * Ρ * Diagonal(sigmas);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2afc04ef",
   "metadata": {},
   "source": [
    "Code 14.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9ef22aac",
   "metadata": {},
   "outputs": [],
   "source": [
    "N_cafes = 20;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34aa4fa0",
   "metadata": {},
   "source": [
    "Code 14.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "c570b27d",
   "metadata": {},
   "outputs": [],
   "source": [
    "Random.seed!(5)\n",
    "vary_effect = rand(MvNormal(μ, Σ), N_cafes);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc58fae3",
   "metadata": {},
   "source": [
    "Code 14.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "77442dc0",
   "metadata": {},
   "outputs": [],
   "source": [
    "a_cafe = vary_effect[1,:]\n",
    "b_cafe = vary_effect[2,:];"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e194afbc",
   "metadata": {},
   "source": [
    "Code 14.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "e22aca56",
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       "  2228.69,748.894 2217.11,734.226 2204.62,719.575 2191.24,704.957 2176.97,690.385 2161.83,675.875 2145.84,661.441 2129.01,647.097 2111.36,632.857 2092.91,618.737 \n",
       "  2073.67,604.749 2053.66,590.908 2032.91,577.228 2011.43,563.722 1989.25,550.403 1966.39,537.286 1942.86,524.383 1918.7,511.706 1893.93,499.269 1868.58,487.084 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = scatter(a_cafe, b_cafe, xlab=\"intercepts (a_cafe)\", ylab=\"slopes (b_cafe)\")\n",
    "\n",
    "d = acos(Σ[1,2])\n",
    "chi = Chisq(2)\n",
    "\n",
    "for l ∈ (0.1, 0.3, 0.5, 0.8, 0.99)\n",
    "    scale = sqrt(quantile(chi, l))\n",
    "    xₜ(t) = scale*Σ[1,1]*cos(t + d/2) + μ[1]\n",
    "    yₜ(t) = scale*Σ[2,2]*cos(t - d/2) + μ[2]\n",
    "\n",
    "    plot!(xₜ, yₜ, 0, 2π, c=:black, alpha=0.3)\n",
    "end\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af309f36",
   "metadata": {},
   "source": [
    "Code 14.10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "7a02c335",
   "metadata": {},
   "outputs": [],
   "source": [
    "Random.seed!(22)\n",
    "N_visits = 10\n",
    "\n",
    "afternoon = repeat(0:1, N_visits*N_cafes ÷ 2)\n",
    "cafe_id = repeat(1:N_cafes, inner=N_visits)\n",
    "μ = a_cafe[cafe_id] + b_cafe[cafe_id] .* afternoon\n",
    "σ = 0.5\n",
    "wait = rand.(Normal.(μ, σ))\n",
    "d = DataFrame(cafe=cafe_id, afternoon=afternoon, wait=wait);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "620b27b3",
   "metadata": {},
   "source": [
    "Code 14.11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e607c8e5",
   "metadata": {},
   "outputs": [
    {
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       "  2196.9,1382.61 2207.31,1383.28 2217.72,1383.7 2228.13,1383.94 2238.54,1384.08 2248.96,1384.16 2259.37,1384.2 2269.78,1384.22 2280.19,1384.23 2290.6,1384.24 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = rand(LKJ(2, 2), 10^4);\n",
    "density(getindex.(R, 2), xlab=\"Correlation\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0708f5f1",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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