{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "46ea4f25",
   "metadata": {},
   "source": [
    "# FORM and correlations\n",
    "\n",
    "### Reliability calculations with FORM\n",
    "\n",
    "In this example, we will demonstrate how to perform reliability calculations using the First Order Reliability Method (FORM).\n",
    "\n",
    "### Define model\n",
    "\n",
    "First, let's import the necessary classes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c0d851fb",
   "metadata": {},
   "outputs": [],
   "source": [
    "from probabilistic_library import ReliabilityProject, DistributionType, ReliabilityMethod, StartMethod"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b00bceae",
   "metadata": {},
   "source": [
    "Next, we define a simple limit state function: \n",
    "\n",
    "$Z = 1.9 - (a+b)$\n",
    "\n",
    "This is a linear model involving two variables, $a$ and $b$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ea8b7c51",
   "metadata": {},
   "outputs": [],
   "source": [
    "from utils.models import linear_a_b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96d6a96d",
   "metadata": {},
   "source": [
    "To perform a reliability analysis, we create a reliability project and specify the limit state function (model):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "42d26675",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model linear_a_b:\n",
      "Input parameters:\n",
      "  a\n",
      "  b\n",
      "Output parameters:\n",
      "  Z\n"
     ]
    }
   ],
   "source": [
    "project = ReliabilityProject()\n",
    "project.model = linear_a_b\n",
    "\n",
    "project.model.print()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac0bb5dc",
   "metadata": {},
   "source": [
    "We assume that variables $a$ and $b$ are uniformly distributed over the interval $[-1, 1]$. This is defined as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b93101e2",
   "metadata": {},
   "outputs": [],
   "source": [
    "project.variables[\"a\"].distribution = DistributionType.uniform\n",
    "project.variables[\"a\"].minimum = -1\n",
    "project.variables[\"a\"].maximum = 1\n",
    "\n",
    "project.variables[\"b\"].distribution = DistributionType.uniform\n",
    "project.variables[\"b\"].minimum = -1\n",
    "project.variables[\"b\"].maximum = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c42f2630",
   "metadata": {},
   "source": [
    "### Define reliability method\n",
    "\n",
    "We use the reliability method `form`. We choose the calculation settings: `relaxation_factor`, `maximum_iterations` and `epsilon_beta`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "56af65d5",
   "metadata": {},
   "outputs": [],
   "source": [
    "project.settings.reliability_method = ReliabilityMethod.form\n",
    "project.settings.relaxation_factor = 0.15\n",
    "project.settings.maximum_iterations = 50\n",
    "project.settings.epsilon_beta = 0.01"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0073e13",
   "metadata": {},
   "source": [
    "We can also define the start method, with the following options available: `ray_search`, `one`, `fixed_value`, `sensitivity_search`, and `sphere_search`. If the fixed_value option is selected, a start value must be specified for each variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8f02c6ee",
   "metadata": {},
   "outputs": [],
   "source": [
    "# fixed_value\n",
    "project.settings.start_method = StartMethod.fixed_value\n",
    "project.settings.stochast_settings[\"a\"].start_value = 1.2\n",
    "project.settings.stochast_settings[\"b\"].start_value = 2.4\n",
    "\n",
    "# in this example we choose the ray_search method\n",
    "project.settings.start_method = StartMethod.ray_search"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8ef2993",
   "metadata": {},
   "source": [
    "### Perform calculations\n",
    "\n",
    "We use `project.run()` to execute the reliability analysis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "0572047d",
   "metadata": {},
   "outputs": [],
   "source": [
    "project.run()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "866f87f3",
   "metadata": {},
   "source": [
    "The results are written to `project.design_point` and consist of:\n",
    "* reliability index $\\beta$\n",
    "* failure probability $P_f$\n",
    "* influence coefficients $\\alpha$-values\n",
    "* design point $x$-values\n",
    "* information about the convergence of FORM\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "25d5a13e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Reliability (FORM)\n",
      " Reliability index = 2.773\n",
      " Probability of failure = 0.0028\n",
      " Convergence = 0.0088 (converged)\n",
      " Model runs = 54\n",
      "Alpha values:\n",
      " a: alpha = -0.7071, x = 0.9501\n",
      " b: alpha = -0.7071, x = 0.9501\n",
      "\n",
      "Contributing design points:\n",
      "  Reliability (Start point)\n",
      "   Reliability index = 3.6937\n",
      "   Probability of failure = 1.104871e-04\n",
      "   Model runs = 11\n",
      "  Alpha values:\n",
      "   a: alpha = -0.4472, x = 0.9014\n",
      "   b: alpha = -0.8944, x = 0.999\n",
      "\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "project.design_point.print()\n",
    "project.design_point.plot_alphas()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5e7e964",
   "metadata": {},
   "source": [
    "### Correlated variables\n",
    "\n",
    "In the above example, variables $a$ and $b$ are independent. However, it is also possible to introduce a correlation between $a$ and $b$. The example below demonstrates this with a partial correlation coefficient of $0.8$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "cb9faec5",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "gallery",
     "reliability",
     "1"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Reliability (FORM)\n",
      " Reliability index = 2.0646\n",
      " Probability of failure = 0.0195\n",
      " Convergence = 0.0084 (converged)\n",
      " Model runs = 39\n",
      "Alpha values:\n",
      " a: alpha = -0.9565, x = 0.9517\n",
      " b: alpha = -0.2916, x = 0.9478\n",
      "\n",
      "Contributing design points:\n",
      "  Reliability (Start point)\n",
      "   Reliability index = 3.6937\n",
      "   Probability of failure = 1.104871e-04\n",
      "   Model runs = 11\n",
      "  Alpha values:\n",
      "   a: alpha = -0.4472, x = 0.9014\n",
      "   b: alpha = -0.8944, x = 0.999\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "project.correlation_matrix[\"a\", \"b\"] = 0.8\n",
    "project.run()\n",
    "\n",
    "project.design_point.print()\n",
    "project.design_point.plot_alphas()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "base",
   "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": "3.12.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}