piecewise_linear_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Piecewise Linear Regression Demo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: Piecewise is mostly not used by itself. Most of regression technique assumes independent variables to have linear boundry. Piecewise can be used to break up non-linear features on breakpoint to create linear interpretation or used to segment key features that can further be used to fit individual models\n",
    "\n",
    "Importing required libraries. We will use pwlf python library to demonstrate simple piecewise linear regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pwlf\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us first generate a sample dataset and define ponts with non linear boundry. We will use the data to fit piecewise regression and also linear regression "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_train = np.array([1.70, 1.45, 1.50, 1.42, 1.39, 1.90, 2.49, 2.21, 2.57, 2.90, 3.55, 3.80, 4.27, 4.10, 4.60, 4.42, 4.30, 4.52, 4.40, 4.50, 4.15,4.30,4.35], order=None)\n",
    "x_train = np.array([478, 525, 580,  650,  700,  720,  780,  825,  850,  900,  930,  980, 1020, 1040, 1050, 1075, 1081, 1100, 1160, 1180, 1200,1220,1240], order=None)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualize generated dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the plot it is very evident that fitting an linear regression line might result in larger standard error. Option is to go to slightly complex algorithm like SVM and Tree that can fit non linear boundry. But for here let us see if we can use piecewise linear regression to better fit the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simpler models are better interpretable and explainable. Scenario like credit decisioning which are regulated by fair act lending law explainability is key. If we visualize credit approval based on FICO as independent variable and probability of default as dependent, decision will always tend to favor high FICO score customers when using linear regression models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One way of solving below decision boundry is to segment users manually (in above case FICO score) and fit linear models for individual segments\n",
    "\n",
    "Piecewise does something similar but uses a global optimization to find breakpoint locations that minimize the sum of squares error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYUAAAEKCAYAAAD9xUlFAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15933fcb668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_train, y_train, 'o', color='black')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialize piecewise linear fit with x_train and y_train data point"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pwlf = pwlf.PiecewiseLinFit(x_train, y_train)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the plot above it is evident we can fit the data using 3 regression lines"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pwise_model = pwlf.fit(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate evenly spaced test dataset to check model fit similar to training interval <br/>\n",
    "Predict on generated test data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_test = np.linspace(min(x_train), max(x_train), num=100)\n",
    "yHat_piecewise = pwlf.predict(x_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the results of piecewise fitted model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAD8CAYAAACMwORRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1593603ecf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(x_train, y_train, 'o')\n",
    "plt.plot(x_test, yHat_piecewise, '-')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Get details of piecewise breakpoint boundry and model co-efficients "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "fit_breaks extracts 3 regression breakpoint model was fitted on"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each break defines the segment of data individual linear regression line is fitted on <br/>\n",
    "First line is fitted on X data point 0 till 478 <br/>\n",
    "Second line is fitted on X data point between 479 to 681.67 <br/>\n",
    "Third line is fitted on X data point between 681.68 till 1056.53"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "478.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pwlf.fit_breaks[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "681.6752402750844"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pwlf.fit_breaks[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1056.5382951233303"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pwlf.fit_breaks[2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Co-efficients and Intercepts of every fitted line. In this case 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.00131881,  0.00821945, -0.00068192])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pwlf.slopes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.25382915, -4.24817018,  5.1564764 ])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pwlf.intercepts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's end of piecewise regression. You can follow below to see what single regression line would have done in this case"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Check for bonus material towards the end on how to deploy piecewise in production"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simple Linear regression Just to compare the fit with piecewise plot above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_train=x_train.reshape((x_train.size,1))\n",
    "y_train=y_train.reshape((y_train.size,1))\n",
    "x_test=x_test.reshape((x_test.size,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "reg_model = LinearRegression().fit(x_train, y_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "yHat_reg=reg_model.predict(x_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Based on plot below we can see single regression line has higher standard error "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAD8CAYAAACMwORRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15936648940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(x_train, y_train, 'o')\n",
    "plt.plot(x_test, yHat_reg, '-')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.00501184]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg_model.coef_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bonus  - Deploying piecewise in production"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## While the biggest bonus is this is not overfitting line. Piecewise generalizes very well. Try it out if you dont trust me"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Deploying is as simple as building an hirearchical tree of each piecewise fitted regression model <br/>\n",
    "Remember from top the 3 segment the mocel was fitted on <br/>\n",
    "First line is fitted on X data point 0 till 478 <br/>\n",
    "Second line is fitted on X data point between 479 to 681.67 <br/>\n",
    "Third line is fitted on X data point between 681.68 till 1056.53 <br/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the intercepts and co-eeficient for 3 regression line are <br/>\n",
    "Intercepts - array([ 2.25382915, -4.24817018,  5.1564764 ]) <br/>\n",
    "Co-efficients - array([-0.00131881,  0.00821945, -0.00068192])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In application just use the below to deploy it within your business process"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "if x < 478  <br/>\n",
    "    y_pred = 2.25382915 + x &#8727; -0.00131881 <br/>\n",
    "else if x > 478 and x <= 681.67 <br/>\n",
    "    y_pred = -4.24817018 + x &#8727; 0.00821945 <br/>\n",
    "else <br/>\n",
    "    y_pred = 5.1564764 + x &#8727; -0.00068192\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another way is to use below equation but in this case you are assuming you are calculation prediction for all 3 classes\n",
    "\n",
    "pf(x) = y1 &#8727; max(x − breakpoint1, 0) + y2 &#8727; max(x − breakpoint2, 0) + y3 &#8727; max(x − breakpoint3, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}