perceptron.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook is focused on getting a feel for the Perceptron.  This is similar information to the very nice [webpage](http://sebastianraschka.com/Articles/2015_singlelayer_neurons.html) by Sebastian Raschka, which has very good graphics. He covers all this material much better there, although his code is Python instead of Julia.  As with the other notebooks in this series, I hope it helps people get a better feel for using the [Julia](http://julialang.org/) programming language.\n",
    "\n",
    "This notebook was originally designed as an adjunct to a [talk I gave](https://www.meetup.com/Monterey-Bay-Data-Science-Meetup/events/241814181/) on Aug 3rd, 2017 at [CSUMB](https://csumb.edu/) introducing neural networks.\n",
    "\n",
    "I've updated the Julia code to version 1.1 for giving a [talk on Nov 19, 2019](https://www.meetup.com/Santa-Cruz-Machine-Learning-Meetup-Group/events/265387888/) at ProductOps for the Santa Cruz Machine Learning Meetup."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Perceptron "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **Perceptron** is the basic building block of a neural network, first described by Frank Rosenblatt in his 1957 paper, *The Perceptron--a perceiving and recognizing automaton*. It was conceived as a system that could take input and learn patterns through supervision. A single layer perceptron is able to learn any linearly separable pattern.\n",
    "\n",
    "We will use it to linearly separate categories of flowers from the famous Iris dataset, based on the variables petal length and petal width.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Get the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Those with an R background are likely familiar with the Iris dataset.  Several popular R datasets are available in Julia through the `RDatasets` library.  To use dataframes in Julia, we have to import the library."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>SepalLength</th><th>SepalWidth</th><th>PetalLength</th><th>PetalWidth</th><th>Species</th></tr><tr><th></th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Categorical…</th></tr></thead><tbody><p>150 rows × 5 columns</p><tr><th>1</th><td>5.1</td><td>3.5</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>2</th><td>4.9</td><td>3.0</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>3</th><td>4.7</td><td>3.2</td><td>1.3</td><td>0.2</td><td>setosa</td></tr><tr><th>4</th><td>4.6</td><td>3.1</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>5</th><td>5.0</td><td>3.6</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>6</th><td>5.4</td><td>3.9</td><td>1.7</td><td>0.4</td><td>setosa</td></tr><tr><th>7</th><td>4.6</td><td>3.4</td><td>1.4</td><td>0.3</td><td>setosa</td></tr><tr><th>8</th><td>5.0</td><td>3.4</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>9</th><td>4.4</td><td>2.9</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>10</th><td>4.9</td><td>3.1</td><td>1.5</td><td>0.1</td><td>setosa</td></tr><tr><th>11</th><td>5.4</td><td>3.7</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>12</th><td>4.8</td><td>3.4</td><td>1.6</td><td>0.2</td><td>setosa</td></tr><tr><th>13</th><td>4.8</td><td>3.0</td><td>1.4</td><td>0.1</td><td>setosa</td></tr><tr><th>14</th><td>4.3</td><td>3.0</td><td>1.1</td><td>0.1</td><td>setosa</td></tr><tr><th>15</th><td>5.8</td><td>4.0</td><td>1.2</td><td>0.2</td><td>setosa</td></tr><tr><th>16</th><td>5.7</td><td>4.4</td><td>1.5</td><td>0.4</td><td>setosa</td></tr><tr><th>17</th><td>5.4</td><td>3.9</td><td>1.3</td><td>0.4</td><td>setosa</td></tr><tr><th>18</th><td>5.1</td><td>3.5</td><td>1.4</td><td>0.3</td><td>setosa</td></tr><tr><th>19</th><td>5.7</td><td>3.8</td><td>1.7</td><td>0.3</td><td>setosa</td></tr><tr><th>20</th><td>5.1</td><td>3.8</td><td>1.5</td><td>0.3</td><td>setosa</td></tr><tr><th>21</th><td>5.4</td><td>3.4</td><td>1.7</td><td>0.2</td><td>setosa</td></tr><tr><th>22</th><td>5.1</td><td>3.7</td><td>1.5</td><td>0.4</td><td>setosa</td></tr><tr><th>23</th><td>4.6</td><td>3.6</td><td>1.0</td><td>0.2</td><td>setosa</td></tr><tr><th>24</th><td>5.1</td><td>3.3</td><td>1.7</td><td>0.5</td><td>setosa</td></tr><tr><th>25</th><td>4.8</td><td>3.4</td><td>1.9</td><td>0.2</td><td>setosa</td></tr><tr><th>26</th><td>5.0</td><td>3.0</td><td>1.6</td><td>0.2</td><td>setosa</td></tr><tr><th>27</th><td>5.0</td><td>3.4</td><td>1.6</td><td>0.4</td><td>setosa</td></tr><tr><th>28</th><td>5.2</td><td>3.5</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>29</th><td>5.2</td><td>3.4</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>30</th><td>4.7</td><td>3.2</td><td>1.6</td><td>0.2</td><td>setosa</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccccc}\n",
       "\t& SepalLength & SepalWidth & PetalLength & PetalWidth & Species\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Categorical…\\\\\n",
       "\t\\hline\n",
       "\t1 & 5.1 & 3.5 & 1.4 & 0.2 & setosa \\\\\n",
       "\t2 & 4.9 & 3.0 & 1.4 & 0.2 & setosa \\\\\n",
       "\t3 & 4.7 & 3.2 & 1.3 & 0.2 & setosa \\\\\n",
       "\t4 & 4.6 & 3.1 & 1.5 & 0.2 & setosa \\\\\n",
       "\t5 & 5.0 & 3.6 & 1.4 & 0.2 & setosa \\\\\n",
       "\t6 & 5.4 & 3.9 & 1.7 & 0.4 & setosa \\\\\n",
       "\t7 & 4.6 & 3.4 & 1.4 & 0.3 & setosa \\\\\n",
       "\t8 & 5.0 & 3.4 & 1.5 & 0.2 & setosa \\\\\n",
       "\t9 & 4.4 & 2.9 & 1.4 & 0.2 & setosa \\\\\n",
       "\t10 & 4.9 & 3.1 & 1.5 & 0.1 & setosa \\\\\n",
       "\t11 & 5.4 & 3.7 & 1.5 & 0.2 & setosa \\\\\n",
       "\t12 & 4.8 & 3.4 & 1.6 & 0.2 & setosa \\\\\n",
       "\t13 & 4.8 & 3.0 & 1.4 & 0.1 & setosa \\\\\n",
       "\t14 & 4.3 & 3.0 & 1.1 & 0.1 & setosa \\\\\n",
       "\t15 & 5.8 & 4.0 & 1.2 & 0.2 & setosa \\\\\n",
       "\t16 & 5.7 & 4.4 & 1.5 & 0.4 & setosa \\\\\n",
       "\t17 & 5.4 & 3.9 & 1.3 & 0.4 & setosa \\\\\n",
       "\t18 & 5.1 & 3.5 & 1.4 & 0.3 & setosa \\\\\n",
       "\t19 & 5.7 & 3.8 & 1.7 & 0.3 & setosa \\\\\n",
       "\t20 & 5.1 & 3.8 & 1.5 & 0.3 & setosa \\\\\n",
       "\t21 & 5.4 & 3.4 & 1.7 & 0.2 & setosa \\\\\n",
       "\t22 & 5.1 & 3.7 & 1.5 & 0.4 & setosa \\\\\n",
       "\t23 & 4.6 & 3.6 & 1.0 & 0.2 & setosa \\\\\n",
       "\t24 & 5.1 & 3.3 & 1.7 & 0.5 & setosa \\\\\n",
       "\t25 & 4.8 & 3.4 & 1.9 & 0.2 & setosa \\\\\n",
       "\t26 & 5.0 & 3.0 & 1.6 & 0.2 & setosa \\\\\n",
       "\t27 & 5.0 & 3.4 & 1.6 & 0.4 & setosa \\\\\n",
       "\t28 & 5.2 & 3.5 & 1.5 & 0.2 & setosa \\\\\n",
       "\t29 & 5.2 & 3.4 & 1.4 & 0.2 & setosa \\\\\n",
       "\t30 & 4.7 & 3.2 & 1.6 & 0.2 & setosa \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "150×5 DataFrame\n",
       "│ Row │ SepalLength │ SepalWidth │ PetalLength │ PetalWidth │ Species      │\n",
       "│     │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mCategorical…\u001b[39m │\n",
       "├─────┼─────────────┼────────────┼─────────────┼────────────┼──────────────┤\n",
       "│ 1   │ 5.1         │ 3.5        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 2   │ 4.9         │ 3.0        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 3   │ 4.7         │ 3.2        │ 1.3         │ 0.2        │ setosa       │\n",
       "│ 4   │ 4.6         │ 3.1        │ 1.5         │ 0.2        │ setosa       │\n",
       "│ 5   │ 5.0         │ 3.6        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 6   │ 5.4         │ 3.9        │ 1.7         │ 0.4        │ setosa       │\n",
       "│ 7   │ 4.6         │ 3.4        │ 1.4         │ 0.3        │ setosa       │\n",
       "│ 8   │ 5.0         │ 3.4        │ 1.5         │ 0.2        │ setosa       │\n",
       "│ 9   │ 4.4         │ 2.9        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 10  │ 4.9         │ 3.1        │ 1.5         │ 0.1        │ setosa       │\n",
       "⋮\n",
       "│ 140 │ 6.9         │ 3.1        │ 5.4         │ 2.1        │ virginica    │\n",
       "│ 141 │ 6.7         │ 3.1        │ 5.6         │ 2.4        │ virginica    │\n",
       "│ 142 │ 6.9         │ 3.1        │ 5.1         │ 2.3        │ virginica    │\n",
       "│ 143 │ 5.8         │ 2.7        │ 5.1         │ 1.9        │ virginica    │\n",
       "│ 144 │ 6.8         │ 3.2        │ 5.9         │ 2.3        │ virginica    │\n",
       "│ 145 │ 6.7         │ 3.3        │ 5.7         │ 2.5        │ virginica    │\n",
       "│ 146 │ 6.7         │ 3.0        │ 5.2         │ 2.3        │ virginica    │\n",
       "│ 147 │ 6.3         │ 2.5        │ 5.0         │ 1.9        │ virginica    │\n",
       "│ 148 │ 6.5         │ 3.0        │ 5.2         │ 2.0        │ virginica    │\n",
       "│ 149 │ 6.2         │ 3.4        │ 5.4         │ 2.3        │ virginica    │\n",
       "│ 150 │ 5.9         │ 3.0        │ 5.1         │ 1.8        │ virginica    │"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using DataFrames;\n",
    "using RDatasets;\n",
    "iris = dataset(\"datasets\", \"iris\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Peek at data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(150, 5)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "size(iris)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>Species</th><th>x1</th></tr><tr><th></th><th>Categorical…</th><th>Int64</th></tr></thead><tbody><p>3 rows × 2 columns</p><tr><th>1</th><td>setosa</td><td>50</td></tr><tr><th>2</th><td>versicolor</td><td>50</td></tr><tr><th>3</th><td>virginica</td><td>50</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cc}\n",
       "\t& Species & x1\\\\\n",
       "\t\\hline\n",
       "\t& Categorical… & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & setosa & 50 \\\\\n",
       "\t2 & versicolor & 50 \\\\\n",
       "\t3 & virginica & 50 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "3×2 DataFrame\n",
       "│ Row │ Species      │ x1    │\n",
       "│     │ \u001b[90mCategorical…\u001b[39m │ \u001b[90mInt64\u001b[39m │\n",
       "├─────┼──────────────┼───────┤\n",
       "│ 1   │ setosa       │ 50    │\n",
       "│ 2   │ versicolor   │ 50    │\n",
       "│ 3   │ virginica    │ 50    │"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the `by` function allows examining a dataframe by a category variable\n",
    "by(iris, :Species, nrow)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>variable</th><th>mean</th><th>min</th><th>median</th><th>max</th><th>nunique</th><th>nmissing</th><th>eltype</th></tr><tr><th></th><th>Symbol</th><th>Union…</th><th>Any</th><th>Union…</th><th>Any</th><th>Union…</th><th>Nothing</th><th>DataType</th></tr></thead><tbody><p>5 rows × 8 columns</p><tr><th>1</th><td>SepalLength</td><td>5.84333</td><td>4.3</td><td>5.8</td><td>7.9</td><td></td><td></td><td>Float64</td></tr><tr><th>2</th><td>SepalWidth</td><td>3.05733</td><td>2.0</td><td>3.0</td><td>4.4</td><td></td><td></td><td>Float64</td></tr><tr><th>3</th><td>PetalLength</td><td>3.758</td><td>1.0</td><td>4.35</td><td>6.9</td><td></td><td></td><td>Float64</td></tr><tr><th>4</th><td>PetalWidth</td><td>1.19933</td><td>0.1</td><td>1.3</td><td>2.5</td><td></td><td></td><td>Float64</td></tr><tr><th>5</th><td>Species</td><td></td><td>setosa</td><td></td><td>virginica</td><td>3</td><td></td><td>CategoricalString{UInt8}</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccccccc}\n",
       "\t& variable & mean & min & median & max & nunique & nmissing & eltype\\\\\n",
       "\t\\hline\n",
       "\t& Symbol & Union… & Any & Union… & Any & Union… & Nothing & DataType\\\\\n",
       "\t\\hline\n",
       "\t1 & SepalLength & 5.84333 & 4.3 & 5.8 & 7.9 &  &  & Float64 \\\\\n",
       "\t2 & SepalWidth & 3.05733 & 2.0 & 3.0 & 4.4 &  &  & Float64 \\\\\n",
       "\t3 & PetalLength & 3.758 & 1.0 & 4.35 & 6.9 &  &  & Float64 \\\\\n",
       "\t4 & PetalWidth & 1.19933 & 0.1 & 1.3 & 2.5 &  &  & Float64 \\\\\n",
       "\t5 & Species &  & setosa &  & virginica & 3 &  & CategoricalString\\{UInt8\\} \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "5×8 DataFrame. Omitted printing of 2 columns\n",
       "│ Row │ variable    │ mean    │ min    │ median │ max       │ nunique │\n",
       "│     │ \u001b[90mSymbol\u001b[39m      │ \u001b[90mUnion…\u001b[39m  │ \u001b[90mAny\u001b[39m    │ \u001b[90mUnion…\u001b[39m │ \u001b[90mAny\u001b[39m       │ \u001b[90mUnion…\u001b[39m  │\n",
       "├─────┼─────────────┼─────────┼────────┼────────┼───────────┼─────────┤\n",
       "│ 1   │ SepalLength │ 5.84333 │ 4.3    │ 5.8    │ 7.9       │         │\n",
       "│ 2   │ SepalWidth  │ 3.05733 │ 2.0    │ 3.0    │ 4.4       │         │\n",
       "│ 3   │ PetalLength │ 3.758   │ 1.0    │ 4.35   │ 6.9       │         │\n",
       "│ 4   │ PetalWidth  │ 1.19933 │ 0.1    │ 1.3    │ 2.5       │         │\n",
       "│ 5   │ Species     │         │ setosa │        │ virginica │ 3       │"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Julia uses `describe` to get a summary of basic statistics\n",
    "describe(iris)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# load plotting libraries\n",
    "using Plots\n",
    "using StatsPlots\n",
    "gr(size=(800,600))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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   "source": [
    "@df iris scatter(:SepalLength, :SepalWidth, group=:Species, title = \"sepal length vs. width\", \n",
    "xlabel = \"sepal length\", ylabel = \"sepal width\", marker=([:circle :rect :cross], 5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, there are 150 data points, with 50 of each species.  Using a perceptron, we are going to try to find the linear separation between setosa and non-setosa, using sepal length and sepal width.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Perceptron](http://sebastianraschka.com/images/blog/2015/singlelayer_neural_networks_files/perceptron_schematic.png) Image by Sebastian Raschka from his [article](http://sebastianraschka.com/Articles/2015_singlelayer_neurons.html) as mentioned above.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### implementation\n",
    "We will implement the algorithm step by step to see what the data looks like, then implement it in a loop to see how many iterations it takes to converge."
   ]
  },
  {
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   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.051062 seconds (78.20 k allocations: 3.795 MiB)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "150-element Array{Int64,1}:\n",
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     "execution_count": 12,
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     "output_type": "execute_result"
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   ],
   "source": [
    "# create targets\n",
    "@time target = [s==\"setosa\" ? 1 : -1 for s in iris[!, :Species]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For our data, we have two inputs corresponding to $x_1$ and $x_2$ in the equations above (sepal length and sepal width).  To create our data input array, we add a third column of 1's for the bias corresponding to $x_0$ above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
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    "jupyter": {
     "outputs_hidden": false
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000035 seconds (9 allocations: 5.266 KiB)\n"
     ]
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     "data": {
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       " ⋮            \n",
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     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# to create input data array X, append a third column of 1's for the bias\n",
    "@time X = hcat(iris[!, :SepalLength], iris[!, :SepalWidth], ones(150))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.027893 seconds (98.36 k allocations: 4.813 MiB)\n"
     ]
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     "execution_count": 14,
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   ],
   "source": [
    "# initialize weights w as small random numbers\n",
    "@time w = rand(3).-0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.146601 seconds (309.35 k allocations: 15.500 MiB, 5.65% gc time)\n"
     ]
    },
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     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# calculate output (predicted y) from the product X*w, and squash\n",
    "@time output = [o >= 0 ? 1 : -1 for o in X*w]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you look at the equations, $\\Delta w_0$ is simply multiplied by each of the members of our input X to get the weight changes.\n",
    "\n",
    "$$\\quad \\Delta w_0 = \\eta \\,(target^{(0)} - output^{(0)})$$\n",
    "$$\\quad \\Delta w_1 = \\eta \\,(target^{(1)} - output^{(1)})\\,x_1^{(1)}$$\n",
    "$$\\quad \\Delta w_2 = \\eta \\,(target^{(2)} - output^{(2)})\\,x_2^{(2)}$$\n",
    "\n",
    "means\n",
    "\n",
    "$$\\quad \\Delta w_0 = \\eta \\,(target^{(0)} - output^{(0)})$$\n",
    "$$\\quad \\Delta w_1 = \\eta \\,(\\Delta w_0)\\,x_1^{(1)}$$\n",
    "$$\\quad \\Delta w_2 = \\eta \\,(\\Delta w_0)\\,x_2^{(2)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.149563 seconds (426.96 k allocations: 21.413 MiB, 5.72% gc time)\n"
     ]
    },
    {
     "data": {
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       "150-element Array{Float64,1}:\n",
       "  0.0\n",
       "  0.0\n",
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     "execution_count": 16,
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   "source": [
    "# calculate delta Δw0\n",
    "η = 0.1\n",
    "@time delta = η*(target.-output)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
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    "error = count(x->x!=0, delta)/150"
   ]
  },
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   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.236802 seconds (451.43 k allocations: 22.203 MiB, 4.39% gc time)\n"
     ]
    },
    {
     "data": {
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       "  -57.440000000000005\n",
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     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "@time [sum((X.*delta)[:,i]) for i in 1:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.173654 seconds (318.11 k allocations: 16.164 MiB, 9.29% gc time)\n"
     ]
    },
    {
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     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# Julia has a cumsum function which is a bit more efficient\n",
    "@time cumsum(X.*delta, dims=1)[end,:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### loop with learning rate η = 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   "outputs": [
    {
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   "source": [
    "# put into loop and save weights and errors for plotting, η = 0.1\n",
    "epochs = 30\n",
    "w = rand(3).-0.5\n",
    "η = 0.1\n",
    "errors = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for i in 1:epochs\n",
    "    output = [o >= 0 ? 1 : -1 for o in X*w]\n",
    "    delta = η*(target.-output)\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[i,:] = w\n",
    "    push!(errors, count(x->x!=0, delta)/150)\n",
    "end\n",
    "errors"
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    "weights"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we plot the errors we saved above, we see that it jumps up and down between 1/3 and 2/3 until epoch 14 when it drops below 0.01."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
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     "execution_count": 38,
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     "output_type": "execute_result"
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   ],
   "source": [
    "plot(errors, xlabel=\"iteration\", ylabel=\"% error\", legend=:none, \n",
    "title=\"Perceptron learning (learning rate = $η)\")"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### loop with learning rate η = 0.001\n",
    "Let's see what happens when we make the learning rate smaller."
   ]
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   "execution_count": 28,
   "metadata": {
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   "source": [
    "# loop η = .001\n",
    "epochs = 60\n",
    "w = rand(3).-0.5\n",
    "η = 0.001\n",
    "errors = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for i in 1:epochs\n",
    "    output = [o >= 0 ? 1 : -1 for o in X*w]\n",
    "    delta = η*(target.-output)\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[i,:] = w\n",
    "    push!(errors, count(x->x!=0, delta)/150)\n",
    "end\n",
    "errors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we expect, it takes longer to drop to find a set of solutions with lower learning rate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
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      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(errors, xlabel=\"iteration\", ylabel=\"% error\", legend=:none, \n",
    "title=\"Perceptron learning (learning rate = $η)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Need to actually go out to 50 epochs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false,
    "jupyter": {
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   ],
   "source": [
    "# loop η = .001\n",
    "epochs = 50\n",
    "w = rand(3).-0.5\n",
    "η = 0.001\n",
    "errors = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for i in 1:epochs\n",
    "    output = [o >= 0 ? 1 : -1 for o in X*w]\n",
    "    delta = η*(target.-output)\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[i,:] = w\n",
    "    push!(errors, count(x->x!=0, delta)/150)\n",
    "end\n",
    "errors"
   ]
  },
  {
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   "execution_count": 36,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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       "  2000.68,465.153 2056.96,1178.24 2113.24,1178.24 2169.53,465.153 2225.81,1178.24 2282.09,1257.47 2338.37,524.577 2394.65,1178.24 2450.93,1376.32 2507.21,999.969 \n",
       "  2563.49,1178.24 2619.77,2129.03 2676.06,2129.03 2732.34,2129.03 2788.62,2129.03 2844.9,2129.03 2901.18,2129.03 2957.46,2129.03 3013.74,2148.83 3070.02,2129.03 \n",
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       "</svg>\n"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(errors, xlabel=\"iteration\", ylabel=\"% error\", legend=:none, \n",
    "title=\"Perceptron learning (learning rate = $η)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### loop with learning rate η = 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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       " 0.3333333333333333 \n",
       " 0.3333333333333333 \n",
       " 0.46               \n",
       " 0.3333333333333333 \n",
       " 0.31333333333333335\n",
       " 0.6666666666666666 \n",
       " 0.3333333333333333 \n",
       " 0.3333333333333333 \n",
       " 0.5533333333333333 \n",
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    "# loop η = .01\n",
    "epochs = 30\n",
    "w = rand(3).-0.5\n",
    "η = 0.01\n",
    "errors = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for i in 1:epochs\n",
    "    output = [o >= 0 ? 1 : -1 for o in X*w]\n",
    "    delta = η*(target.-output)\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[i,:] = w\n",
    "    push!(errors, count(x->x!=0, delta)/150)\n",
    "end\n",
    "errors"
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    "plot(errors, xlabel=\"iteration\", ylabel=\"% error\", legend=:none, \n",
    "title=\"Perceptron learning (learning rate = $η)\")"
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    "### Identify decision boundary\n",
    "\n",
    "To visualize the solutions that we come up with, let's plot the decision boundary.  If we are plotting sepal length vs sepal width, we can find out what line would be generated by the weights we calculated using the formula:\n",
    "\n",
    "$ x_2 = \\frac{-w_1}{w_2} x_1 - \\frac{w_0}{w_2}$\n",
    "\n",
    "since we are looking for the place where \n",
    "\n",
    "$ w_2 x_2 + w_1 x_1 + w_0 = 0$"
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   "source": [
    "# in Julia, plot! modifies a previous plot\n",
    "@df iris scatter(:SepalLength, :SepalWidth, group=:Species, title = \"Linear separation with Perceptron\", \n",
    "xlabel = \"sepal length\", ylabel = \"sepal width\", marker=([:circle :rect :cross], 5))\n",
    "x = LinRange(4.0, 8.0, 50)\n",
    "y = (-w[1]*x.-w[3])/w[2]\n",
    "plot!(x,y, linecolor=:black, label=\"decision boundary\")"
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   "source": [
    "The line above was calculated with the weights we got at the end of all iterations.  If we plot a few other lines from our saved weights (iterations =5, =10, =15 and =20), we can get a feel for how the solutions evolve in the learning process.  Note that when iterations = 5, the line doesn't show up in the plot below because it is not even in the chosen x and y limits."
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  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
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     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# draw boundaries at different numbers of iterations (note #5 isn't even in the plot frame)\n",
    "xmin=4\n",
    "xmax=8\n",
    "ymin=1.5\n",
    "ymax=5\n",
    "@df iris scatter(:SepalLength, :SepalWidth, group=:Species, title = \"Linear separation with Perceptron\", \n",
    "xlabel = \"sepal length\", ylabel = \"sepal width\", marker=([:circle :rect :cross], 5))\n",
    "x = LinRange(4.0, 8.0, 50)\n",
    "i=5\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)], \n",
    "label=\"boundary $i\", xlims=(xmin,xmax), ylims=(ymin,ymax))\n",
    "i=10\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)], \n",
    "label=\"boundary $i\", xlims=(xmin,xmax), ylims=(ymin,ymax))\n",
    "i=15\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)], \n",
    "label=\"boundary $i\", xlims=(xmin,xmax), ylims=(ymin,ymax))\n",
    "i=20\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)], \n",
    "label=\"boundary $i\", xlims=(xmin,xmax), ylims=(ymin,ymax))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Adaline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **ADA**pative **LI**near **NE**uron, developed by Widrow and Hoff (1960), uses a **linear activation function** to provide a graded feedback compared to Rosenblatt's original perceptron.  The weights are changed using the rule:\n",
    "\n",
    "update weights ($w_j$) for data at $j$,\n",
    "$$\\quad \\Delta w_j = \\eta \\,(target^{(j)} - output^{(j)}) \\,x_i^{(j)}$$\n",
    "\n",
    "Unlike the perceptron, this output is the weighted sum *without* squashing.  It is derived from taking the partial derviative of the cost function:\n",
    "\n",
    "$$\\quad J(\\mathbf{w}) = \\frac{1}{2} \\sum\\limits_{j}( target^{(j)} - output^{(j)} )^2$$\n",
    "\n",
    "so that the update is in the opposite direction of the gradient:\n",
    "\n",
    "$$\\quad \\Delta \\mathbf{w} = - \\eta\\,\\nabla \\, J(\\mathbf{w})$$\n",
    "\n",
    "Thus, the term *gradient descent* for minimizing the cost function.\n"
   ]
  },
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   "execution_count": 27,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
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   "outputs": [
    {
     "data": {
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       " ⋮                   \n",
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     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "# here is the code from the perceptron for comparison\n",
    "epochs = 30\n",
    "w = rand(3).-0.5\n",
    "η = 0.1\n",
    "errors = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for i in 1:epochs\n",
    "    output = [o >= 0 ? 1 : -1 for o in X*w]\n",
    "    delta = η*(target.-output)\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[i,:] = w\n",
    "    push!(errors, count(x->x!=0, delta)/150)\n",
    "end\n",
    "errors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## loop with learning rate η = 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   },
   "outputs": [
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       "      8.238267025263883e41  \n",
       "      3.64671980280301e45   \n",
       "      1.6142430537118523e49 \n",
       "      ⋮                     \n",
       "      1.0132082760393039e144\n",
       "      4.48502903980352e147  \n",
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       "      8.78816740340711e154  \n",
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       "      6.611462001370576e176 \n",
       "      2.9266045069843327e180\n",
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     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# loop η = .01\n",
    "# gradient doesn't descend if use an η like above (it explodes!)\n",
    "epochs = 50\n",
    "w = rand(3).-0.5\n",
    "η = 0.01\n",
    "costs = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for j in 1:epochs\n",
    "    delta = η*(target.-(X*w))\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[j,:] = w\n",
    "    cost = (sum(abs2, target.-(X*w)))/2\n",
    "    push!(costs, cost)\n",
    "end\n",
    "costs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   "outputs": [
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   "source": [
    "plot(costs, xlabel=\"iteration\", ylabel=\"sum squared error\", legend=:none, \n",
    "title=\"Adaline learning (learning rate = $η)\")"
   ]
  },
  {
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   "execution_count": 41,
   "metadata": {
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       "  2581.53,548.023 2636.09,508.003 2690.65,467.982 2745.21,427.962 2799.76,387.942 2854.32,347.921 2908.88,307.901 2963.44,267.881 3018,227.861 3072.56,187.84 \n",
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       "</svg>\n"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(costs, yscale=:log10, xlabel=\"iteration\", ylabel=\"sum squared error (log)\", legend=:none, \n",
    "title=\"Adaline learning (learning rate = $η)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## loop with learning rate η = 0.0001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
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       " 146.03417341021103 \n",
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       "  80.39367100898389 \n",
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       "  76.54072309524054 \n",
       "  75.92193135912208 \n",
       "  75.30927937917706 \n",
       "   ⋮                \n",
       "  61.3476132036392  \n",
       "  60.87956960437343 \n",
       "  60.41616998741546 \n",
       "  59.957368273869044\n",
       "  59.50311884204593 \n",
       "  59.053376522929184\n",
       "  58.608096595681786\n",
       "  58.1672347831996  \n",
       "  57.73074724770852 \n",
       "  57.29859058640533 \n",
       "  56.87072182714181 \n",
       "  56.447098424151626"
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     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# loop η = .0001\n",
    "epochs = 50\n",
    "w = rand(3).-0.5\n",
    "η = 0.0001\n",
    "costs = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for j in 1:epochs\n",
    "    delta = η*(target.-(X*w))\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[j,:] = w\n",
    "    cost = (sum(abs2, target.-(X*w)))/2\n",
    "    push!(costs, cost)\n",
    "end\n",
    "costs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
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       "  1445.72,1838.56 1501.75,1850.82 1557.77,1862.95 1613.79,1874.97 1669.82,1886.87 1725.84,1898.65 1781.86,1910.31 1837.89,1921.86 1893.91,1933.29 1949.93,1944.61 \n",
       "  2005.96,1955.82 2061.98,1966.92 2118,1977.9 2174.03,1988.78 2230.05,1999.55 2286.07,2010.21 2342.1,2020.77 2398.12,2031.22 2454.14,2041.57 2510.17,2051.81 \n",
       "  2566.19,2061.95 2622.21,2072 2678.24,2081.94 2734.26,2091.78 2790.28,2101.53 2846.31,2111.18 2902.33,2120.74 2958.35,2130.2 3014.38,2139.56 3070.4,2148.83 \n",
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       "</svg>\n"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(costs, xlabel=\"iteration\", ylabel=\"sum squared error\", legend=:none, \n",
    "title=\"Adaline learning (learning rate = $η)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## loop with learning rate η = 0.00001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       " 167.76317274937225 \n",
       " 150.38766885497213 \n",
       " 135.2770849894618  \n",
       " 122.13584222564045 \n",
       " 110.70693592408749 \n",
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       "  67.42023416364711 \n",
       "  63.11346193260241 \n",
       "  59.36611170094554 \n",
       "   ⋮                \n",
       "  34.66838447041562 \n",
       "  34.56321035159927 \n",
       "  34.469204298247966\n",
       "  34.384911374333306\n",
       "  34.30906651655501 \n",
       "  34.24056975515559 \n",
       "  34.178464668522714\n",
       "  34.1219196495564  \n",
       "  34.07021161685383 \n",
       "  34.022711851652964\n",
       "  33.97887368311444 \n",
       "  33.93822178072583 "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# loop η = .00001\n",
    "epochs = 50\n",
    "w = rand(3).-0.5\n",
    "η = 0.00001\n",
    "costs = Float64[]\n",
    "weights = Array{Float64}(undef,epochs,3)\n",
    "for j in 1:epochs\n",
    "    delta = η*(target.-(X*w))\n",
    "    w += cumsum(X.*delta, dims=1)[end,:]\n",
    "    weights[j,:] = w\n",
    "    cost = (sum(abs2, target.-(X*w)))/2\n",
    "    push!(costs, cost)\n",
    "end\n",
    "costs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   "outputs": [
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       "</g>\n",
       "<g clip-path=\"url(#clip9700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 210.899, 1198.3)\" x=\"210.899\" y=\"1198.3\">100</text>\n",
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       "<g clip-path=\"url(#clip9700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 210.899, 465.632)\" x=\"210.899\" y=\"465.632\">150</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 1697.83, 2349.53)\" x=\"1697.83\" y=\"2349.53\">iteration</text>\n",
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       "  325.253,187.84 381.277,442.451 437.3,663.873 493.324,856.437 549.347,1023.91 605.37,1169.56 661.394,1296.25 717.417,1406.44 773.441,1502.29 829.464,1585.67 \n",
       "  885.488,1658.21 941.511,1721.32 997.534,1776.23 1053.56,1824.01 1109.58,1865.6 1165.6,1901.8 1221.63,1933.31 1277.65,1960.75 1333.68,1984.64 1389.7,2005.46 \n",
       "  1445.72,2023.59 1501.75,2039.4 1557.77,2053.18 1613.79,2065.21 1669.82,2075.7 1725.84,2084.86 1781.86,2092.86 1837.89,2099.86 1893.91,2105.98 1949.93,2111.34 \n",
       "  2005.96,2116.04 2061.98,2120.16 2118,2123.79 2174.03,2126.97 2230.05,2129.78 2286.07,2132.26 2342.1,2134.46 2398.12,2136.4 2454.14,2138.13 2510.17,2139.68 \n",
       "  2566.19,2141.05 2622.21,2142.29 2678.24,2143.4 2734.26,2144.4 2790.28,2145.31 2846.31,2146.14 2902.33,2146.9 2958.35,2147.6 3014.38,2148.24 3070.4,2148.83 \n",
       "  \n",
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       "</svg>\n"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(costs, xlabel=\"iteration\", ylabel=\"sum squared error\", legend=:none, \n",
    "title=\"Adaline learning (learning rate = $η)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "50×3 Array{Float64,2}:\n",
       " -0.135519  0.45496   0.358836\n",
       " -0.147507  0.449271  0.356899\n",
       " -0.158699  0.443994  0.355097\n",
       " -0.16915   0.439099  0.353419\n",
       " -0.17891   0.434562  0.351858\n",
       " -0.188025  0.430357  0.350405\n",
       " -0.196538  0.426463  0.349054\n",
       " -0.204491  0.422858  0.347797\n",
       " -0.211921  0.419524  0.346629\n",
       " -0.218863  0.416441  0.345542\n",
       " -0.225351  0.413593  0.344532\n",
       " -0.231415  0.410963  0.343594\n",
       " -0.237084  0.408538  0.342722\n",
       "  ⋮                           \n",
       " -0.30567   0.386111  0.333331\n",
       " -0.306705  0.386073  0.33324 \n",
       " -0.307685  0.386064  0.333157\n",
       " -0.308612  0.386082  0.333084\n",
       " -0.309491  0.386124  0.333019\n",
       " -0.310324  0.38619   0.332961\n",
       " -0.311115  0.386277  0.332911\n",
       " -0.311867  0.386385  0.332867\n",
       " -0.312582  0.386511  0.332829\n",
       " -0.313262  0.386654  0.332797\n",
       " -0.313911  0.386814  0.33277 \n",
       " -0.314529  0.386989  0.332749"
      ]
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     "execution_count": 35,
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    }
   ],
   "source": [
    "weights"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scaling\n",
    "We can start the process in a better range if we normalize the input data around zero.  This will help the process zero in faster."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "150×3 Array{Float64,2}:\n",
       " -0.897674    1.0156     -0.33527  \n",
       " -1.1392     -0.131539    0.150102 \n",
       " -1.38073     0.327318   -0.0737293\n",
       " -1.50149     0.0978893  -0.418584 \n",
       " -1.01844     1.24503     0.485188 \n",
       " -0.535384    1.93331     0.225613 \n",
       " -1.50149     0.786174   -0.212607 \n",
       " -1.01844     0.786174    0.193534 \n",
       " -1.74302    -0.360967   -0.238101 \n",
       " -1.1392      0.0978893  -0.350043 \n",
       " -0.535384    1.47446     0.132263 \n",
       " -1.25996     0.786174    0.445958 \n",
       " -1.25996    -0.131539   -0.450483 \n",
       "  ⋮                                \n",
       "  0.189196   -0.131539   -0.291508 \n",
       "  1.27607     0.0978893  -0.335894 \n",
       "  1.03454     0.0978893   0.391401 \n",
       "  1.27607     0.0978893  -0.216794 \n",
       " -0.0523308  -0.819823   -0.262303 \n",
       "  1.1553      0.327318   -0.449911 \n",
       "  1.03454     0.556746   -0.252071 \n",
       "  1.03454    -0.131539   -0.305214 \n",
       "  0.551486   -1.27868     0.120528 \n",
       "  0.793012   -0.131539   -0.489577 \n",
       "  0.430722    0.786174    0.265956 \n",
       "  0.0684325  -0.131539   -0.0328858"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# scale data\n",
    "Xs = Array{Float64}(undef, 150,3)\n",
    "for i in 1:3\n",
    "    Xs[:,i] = convert(Array, X[:,i].-Statistics.mean(X[:,i]))./std(X[:,i]);\n",
    "end\n",
    "Xs[:,3] = rand(150).-0.5\n",
    "Xs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## loop with learning rate η = 0.001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20-element Array{Float64,1}:\n",
       " 45.137198160504624\n",
       " 38.64462236448352 \n",
       " 34.02529861971254 \n",
       " 30.73393396047286 \n",
       " 28.38518103632474 \n",
       " 26.70641086845401 \n",
       " 25.50451016984793 \n",
       " 24.642518247749155\n",
       " 24.023172438064353\n",
       " 23.57730795527149 \n",
       " 23.255671645066112\n",
       " 23.023138715940163\n",
       " 22.854622465328493\n",
       " 22.73217796796357 \n",
       " 22.64294866802153 \n",
       " 22.577708689758825\n",
       " 22.529826653540756\n",
       " 22.494528090250586\n",
       " 22.46836965136598 \n",
       " 22.448863742174453"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# loop with scaling data η = .001\n",
    "epochs = 20\n",
    "w = rand(3).-0.5\n",
    "η = 0.001\n",
    "costs = Float64[]\n",
    "weights = Array{Float64}(undef, epochs,3)\n",
    "for i in 1:epochs\n",
    "    delta = η*(target.-(Xs*w))\n",
    "    w += cumsum(Xs.*delta, dims=1)[end,:]\n",
    "    weights[i,:] = w\n",
    "    cost = (sum(abs2, target.-(Xs*w)))/2\n",
    "    push!(costs, cost)\n",
    "end\n",
    "costs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
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     "execution_count": 52,
     "metadata": {},
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   "source": [
    "plot(costs, xlabel=\"iteration\", ylabel=\"sum squared error\", legend=:none, \n",
    "title=\"Adaline learning (learning rate = $η)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## loop with learning rate η = 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
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       " 44.439172908691475\n",
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       " 24.332973096292005\n",
       " 23.29882303586227 \n",
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       " 22.492363970010963\n",
       " 22.484607925886916\n",
       " 22.48117616292677 \n",
       " 22.479657734556532\n",
       " 22.478985886074895\n",
       " 22.47868861793111 \n",
       " 22.47855708775464 \n",
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       " 22.478461746893473\n",
       " 22.478456705698097"
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     "execution_count": 39,
     "metadata": {},
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   "source": [
    "# loop with scaling data η = .01\n",
    "epochs = 20\n",
    "w = rand(3).-0.5\n",
    "η = 0.01\n",
    "costs = Float64[]\n",
    "weights = Array{Float64}(undef, epochs,3)\n",
    "for i in 1:epochs\n",
    "    delta = η*(target.-(Xs*w))\n",
    "    w += cumsum(Xs.*delta, dims=1)[end,:]\n",
    "    weights[i,:] = w\n",
    "    cost = (sum(abs2, target.-(Xs*w)))/2\n",
    "    push!(costs, cost)\n",
    "end\n",
    "costs"
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  },
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   "execution_count": 40,
   "metadata": {
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     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "plot(costs, xlabel=\"iteration\", ylabel=\"sum squared error\", legend=:none, \n",
    "title=\"Adaline learning (learning rate = $η)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot decision boundary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(Xs[:,1],Xs[:,2], group=iris[!, :Species], title = \"Linear separation with Adaline\", \n",
    "xlabel = \"Length\", ylabel = \"Width\", marker=([:circle :rect :cross], 5))\n",
    "\n",
    "x = LinRange(minimum(Xs[:,1]), maximum(Xs[:,1]), 50)\n",
    "i=3\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)+1], \n",
    "label=\"boundary $i\")\n",
    "i=6\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)+1], \n",
    "label=\"boundary $i\")\n",
    "i=9\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)+1], \n",
    "label=\"boundary $i\")\n",
    "i=12\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)+1], \n",
    "label=\"boundary $i\")\n",
    "i=15\n",
    "y= (-weights[i,1]*x.-weights[i,3])/weights[i,2]\n",
    "plot!(x,y, linecolor=[:violet, :blue, :green, :orange, :red][fld(i,5)+1], \n",
    "label=\"boundary $i\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multi Layer Perceptrons"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{\\partial E^n}{\\partial w_{ji}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "[<img style=\"float: left;\" src=\"https://i.creativecommons.org/l/by-sa/4.0/88x31.png\">](http://creativecommons.org/licenses/by-sa/4.0/)  \n",
    "\n",
    "Licensed under a [Creative Commons Attribution-ShareAlike 4.0 International License](http://creativecommons.org/licenses/by-sa/4.0/)."
   ]
  }
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