week-7/7-non-linear-regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Non-linear Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7fab0bcee860>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1200x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, 10, 100)\n",
    "y = np.exp(x)\n",
    "\n",
    "plt.rcParams['figure.figsize'] = (12,8)\n",
    "plt.scatter(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "y^ = θ + θ1 * θ2^x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Exponantial\n",
    "- Quadratic\n",
    "- Cubic\n",
    "- ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Polynomial Regression\n",
    "\n",
    "y^ = θ0 + θ1 * x + θ2 * x^2 + θ3 * x^3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "x1 = x\n",
    "x2 = x^2\n",
    "x3 = x^3\n",
    "\n",
    "y^ = θ0 + θ1 * x1 + θ2 * x2 + θ3 * x3\n",
    "\n",
    "Polynomial regression is considered to be a special case of traditional multiple linear regression.\n",
    "    - Polynomial regression models can fit using the model of \"least squares\"\n",
    "    - Minimize the sum of the squares of the differences between y and y^ (y_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Non-linear regression\n",
    "- non-linear relationship between the dependet variable and a set of independent variables\n",
    "- For a model to be considered non-linear y^ must be a non-linear function of the paramaters θ, not necessarily the features of x.\n",
    "- in Non-linear regression, a model is non-linear by parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How can I know if a problem is linear or non-linear?\n",
    "\n",
    "- Visually figure out relation, use plots of output variables with each input variable.\n",
    "- Calculate the correlation coefficient between independet and dependent variables. \n",
    "    - if for all variables it is 0.7 or higher there is a linearity, and it's not appropriate to fit a non-linear regression."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How should I model my data, if it displays non-linear on a scatterplot?\n",
    "- Polynomial regression\n",
    "- Non-linear regression model\n",
    "- Transform your data"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.10.13"
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 },
 "nbformat": 4,
 "nbformat_minor": 2
}