examples.ipynb

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    "import urllib\n",
    "import requests\n",
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = requests.get(\"https://raw.githubusercontent.com/saadtony/NumericalMethods/master/styles/custom.css\")\n",
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   "source": [
    "# Examples of Using PyMaxEnt\n",
    "**Tony Saad and Giovanna R. Ruai** <br/>\n",
    "**Department of Chemical Engineering, University of Utah**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below are the necessary libraries used in the functions below. See examples of functions below their coded definition."
   ]
  },
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   "cell_type": "code",
   "execution_count": 2,
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   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rc(\"font\", size=11, family='serif')\n",
    "plt.rc('text', usetex=True)\n",
    "import numpy as np\n",
    "import scipy\n",
    "from scipy.special import gamma\n",
    "from scipy.stats import lognorm\n",
    "from src.pymaxent import *\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Single Moment Results in a Uniform Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667]\n"
     ]
    }
   ],
   "source": [
    "from src.pymaxent import reconstruct\n",
    "mu = np.array([1]) # set the moments\n",
    "x = np.array([1,2,3,4,5,6]) # set all possible x-values\n",
    "sol, lambdas = reconstruct(mu=mu,rndvar=x)\n",
    "print(sol)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unbiased Die"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667]\n"
     ]
    }
   ],
   "source": [
    "from src.pymaxent import reconstruct\n",
    "mu = [1,3.5] # set the moments\n",
    "x = [1,2,3,4,5,6] # set all possible x-values\n",
    "sol, lambdas = reconstruct(mu=mu,rndvar=x)\n",
    "print(sol)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Biased Die"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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     "name": "stdout",
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     "text": [
      "4.416666666666666\n",
      "μ1 for reconstructed probabilities:  4.416666666664865\n",
      "[0.0614559  0.08595796 0.12022884 0.16816329 0.23520889 0.32898512]\n"
     ]
    }
   ],
   "source": [
    "from src.pymaxent import reconstruct\n",
    "pi = np.array([1/12,1/12,1/12,2/12,3/12,4/12])\n",
    "x = [1,2,3,4,5,6] # set all possible x-values\n",
    "# compute μ1\n",
    "μ1 = np.sum(pi*x)\n",
    "print(μ1)\n",
    "mu = [1,μ1] # set the moments\n",
    "sol, lambdas = reconstruct(mu=mu,rndvar=x)\n",
    "print('μ1 for reconstructed probabilities: ', np.sum(sol*x))\n",
    "print(sol)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gaussian Reconstruction"
   ]
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   "cell_type": "code",
   "execution_count": 6,
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   "source": [
    "def gauss(x):\n",
    "    σ = 0.2\n",
    "    μ = 1.0\n",
    "    A = 1/(σ*np.sqrt(2*np.pi))\n",
    "    return A*np.exp(-(x-μ)**2/(2*σ**2))\n",
    "x = np.linspace(0,2)\n",
    "plt.plot(x,gauss(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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     "metadata": {},
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   "source": [
    "from src.pymaxent import reconstruct\n",
    "# Use the following function to generate a set of moments to test against\n",
    "mu = moments(gauss,3,bnds=[-5,5])\n",
    "sol, lambdas = reconstruct(mu=mu,bnds=[0,2])\n",
    "print(lambdas)\n",
    "x = np.linspace(0,2)\n",
    "plt.plot(x,gauss(x),x,sol(x),'o')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bimodal Gaussian Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "def gauss2(x):\n",
    "    σ0 = 1.0/14.0\n",
    "    μ0 = 1.0/4.0\n",
    "    A0 = 1.0/(2.0*σ0*np.sqrt(2*np.pi))\n",
    "    σ1 = 1.0/20.0\n",
    "    μ1 = 2.0/4.0\n",
    "    A1 = 1/(2*σ1*np.sqrt(2*np.pi))\n",
    "    return A0*np.exp(-(x-μ0)**2/(2*σ0**2)) + A1*np.exp(-(x-μ1)**2/(2*σ1**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "bnd=[0,1]\n",
    "f = gauss2\n",
    "xaxis= np.linspace(bnd[0], bnd[1], 200)\n",
    "μin5 = moments(f,5,bnds=bnd)\n",
    "μin10 = moments(f,10,bnds=bnd)\n",
    "sol5,lambdas5 = reconstruct(μin5,bnds=bnd)\n",
    "sol10,lambdas10 = reconstruct(μin10,bnds=bnd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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   "source": [
    "fig = plt.figure(figsize=(4,2.5), dpi=400)\n",
    "plt.plot(xaxis,gauss2(xaxis),'k-',label='Exact')\n",
    "plt.plot(xaxis,sol5(xaxis), 'b*-',linewidth=1,label='5 moments',markevery=4,markersize=4)\n",
    "plt.plot(xaxis,sol10(xaxis), 'r.-',linewidth=1,label='10 moments',markevery=4,markersize=4)\n",
    "# plt.title('Bimodal Gaussian Reconstruction')\n",
    "plt.ylabel('$f$')\n",
    "plt.xlabel('$x$')\n",
    "plt.legend()\n",
    "plt.show()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Trimodal Gaussian Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "def gauss3(x):\n",
    "    σ0 = 1.0/14.0\n",
    "    μ0 = 1.0/4.0\n",
    "    A0 = 1.0/(2.0*σ0*np.sqrt(2*np.pi))\n",
    "    σ1 = 1.0/20.0\n",
    "    μ1 = 2.0/4.0\n",
    "    A1 = 1/(2*σ1*np.sqrt(2*np.pi))\n",
    "    σ2 = 1.0/20.0\n",
    "    μ2 = 3.0/4.0\n",
    "    A2 = 1/(2*σ1*np.sqrt(2*np.pi))\n",
    "    return A0*np.exp(-(x-μ0)**2/(2*σ0**2)) + A1*np.exp(-(x-μ1)**2/(2*σ1**2)) + A2*np.exp(-(x-μ2)**2/(2*σ2**2))"
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  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "bnd=[0,1]\n",
    "f = gauss3\n",
    "xaxis= np.linspace(bnd[0], bnd[1], 200)\n",
    "μin5 = moments(f,5,bnds=bnd)\n",
    "μin13 = moments(f,13,bnds=bnd)\n",
    "sol5,lambdas5 = reconstruct(μin5,bnds=bnd)\n",
    "sol13,lambdas13 = reconstruct(μin13,bnds=bnd)"
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   "execution_count": 13,
   "metadata": {},
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      "text/plain": [
       "<Figure size 1600x1000 with 1 Axes>"
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   "source": [
    "fig = plt.figure(figsize=(4,2.5), dpi=400)\n",
    "plt.plot(xaxis,f(xaxis),'k-',label='Exact')\n",
    "plt.plot(xaxis,sol5(xaxis), 'b*-',linewidth=1,label='5 moments',markevery=4,markersize=4)\n",
    "plt.plot(xaxis,sol13(xaxis), 'r.-',linewidth=1,label='13 moments',markevery=4,markersize=4)\n",
    "# plt.title('Bimodal Gaussian Reconstruction')\n",
    "plt.ylabel('$f$')\n",
    "plt.xlabel('$x$')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Beta Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def beta(x):\n",
    "    β = 9\n",
    "    α = 3\n",
    "    B = lambda X: X**(α-1)*(1-X)**(β-1)\n",
    "    btm = quad(B,0,1)[0]\n",
    "    return x**(α-1)*(1-x)**(β-1)/btm"
   ]
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  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "bnd=[0,1]\n",
    "f = beta\n",
    "xaxis= np.linspace(bnd[0], bnd[1], 200)\n",
    "μin3 = moments(f,3,bnds=bnd)\n",
    "μin5 = moments(f,5,bnds=bnd)\n",
    "sol3,lambdas3 = reconstruct(μin3,bnds=bnd)\n",
    "sol5,lambdas5 = reconstruct(μin5,bnds=bnd)"
   ]
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   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
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      "text/plain": [
       "<Figure size 1600x1000 with 1 Axes>"
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     "output_type": "display_data"
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   ],
   "source": [
    "fig = plt.figure(figsize=(4,2.5), dpi=400)\n",
    "plt.plot(xaxis,f(xaxis),'k-',label='Exact')\n",
    "plt.plot(xaxis,sol3(xaxis), 'b*-',linewidth=1,label='3 moments',markevery=4,markersize=4)\n",
    "plt.plot(xaxis,sol5(xaxis), 'r.-',linewidth=1,label='5 moments',markevery=4,markersize=4)\n",
    "# plt.title('Bimodal Gaussian Reconstruction')\n",
    "plt.ylabel('$f$')\n",
    "plt.xlabel('$x$')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Beta Distribution 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def beta2(x):\n",
    "    α = β = 0.5\n",
    "    B = lambda X: X**(α-1)*(1-X)**(β-1)\n",
    "    btm = quad(B,0,1)[0]\n",
    "    return x**(α-1)*(1-x)**(β-1)/btm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "bnd=[0,1]\n",
    "f = beta2\n",
    "xaxis= np.linspace(bnd[0], bnd[1], 200)\n",
    "μin3 = moments(f,3,bnds=bnd)\n",
    "μin5 = moments(f,5,bnds=bnd)\n",
    "sol3,lambdas3 = reconstruct(μin3,bnds=bnd)\n",
    "sol5,lambdas5 = reconstruct(μin5,bnds=bnd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 1600x1000 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "fig = plt.figure(figsize=(4,2.5), dpi=400)\n",
    "plt.plot(xaxis,f(xaxis),'k-',label='Exact')\n",
    "plt.plot(xaxis,sol3(xaxis), 'b*-',linewidth=1,label='3 moments',markevery=4,markersize=4)\n",
    "plt.plot(xaxis,sol5(xaxis), 'r.-',linewidth=1,label='5 moments',markevery=4,markersize=4)\n",
    "# plt.title('Bimodal Gaussian Reconstruction')\n",
    "plt.ylabel('$f$')\n",
    "plt.xlabel('$x$')\n",
    "plt.legend()\n",
    "# plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# LogNormal Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a20141a58>]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
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   "source": [
    "x = np.linspace(0,3,200)\n",
    "stddev = 1/4\n",
    "mean = 0.2\n",
    "dist=lognorm([stddev],loc=mean)\n",
    "plt.plot(x,dist.pdf(x))"
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  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "bnd=[0,5]\n",
    "f = dist.pdf\n",
    "xaxis= np.linspace(bnd[0], bnd[1], 200)\n",
    "μin3 = moments(f,3,bnds=bnd)\n",
    "μin5 = moments(f,5,bnds=bnd)\n",
    "sol3,lambdas3 = reconstruct(μin3,bnds=bnd)\n",
    "sol5,lambdas5 = reconstruct(μin5,bnds=bnd)"
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    }
   ],
   "source": [
    "fig = plt.figure(figsize=(4,2.5), dpi=400)\n",
    "plt.plot(xaxis,f(xaxis),'k-',label='Exact')\n",
    "plt.plot(xaxis,sol3(xaxis), 'b*-',linewidth=1,label='3 moments',markevery=4,markersize=4)\n",
    "plt.plot(xaxis,sol5(xaxis), 'r.-',linewidth=1,label='5 moments',markevery=4,markersize=4)\n",
    "# plt.title('Bimodal Gaussian Reconstruction')\n",
    "plt.ylabel('$f$')\n",
    "plt.xlabel('$x$')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  }
 ],
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