Leave-One-Out Cross Validation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2 [Leave-One-Out Cross Validation, 15 Marks] \n",
    "\n",
    "    I. A special case of K-Fold cross-validation is Leave-One-Out cross-validation where K (i.e., the number of folds/subsets) is equal to the size of the training dataset. In each iteration, one training data point is used as the validation set. Implement a Leave-One-Out cross-validation (CV) function for your KNN regressor: \n",
    "    *cv(train.data, train.label, K, numFold=?)* \n",
    "    which takes the training data and their labels (continuous values), K value, the number of folds, and returns errors for different folds of the training data.\n",
    "    \n",
    "    II. Using the training data in Question 1, run your Leave-One-Out CV. Change the value of K=1,..,15 in your KNN regressor, and for each K compute tthe average of error values you have got for folds. Plot the average of error values versus 1/K for K=1,..,15 in your KNN regressor. Save the plot in your Jupyter Notebook file for Question 2.\n",
    "    \n",
    "#### Importing Libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(reshape2) # Reshape\n",
    "library(ggplot2) # Plots\n",
    "library(corrplot) # Analyse correlations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Importing the Dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "test <- read.csv(\"data\\\\Sintetic1A_test.csv\")\n",
    "train <- read.csv(\"data\\\\Sintetic1A_train.csv\")\n",
    "\n",
    "# Create a column for index in the dataframes for support\n",
    "row.names(train) <- 1:nrow(train)\n",
    "\n",
    "# Create a column for index in the dataframes for support\n",
    "train.data <- train[1]\n",
    "train.label <- train[2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### KNN Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# CV function\n",
    "cv <- function(train.data, train.label, K, numFold = nrow(train.data)){\n",
    "    \n",
    "    error.vector <- c()\n",
    "    random.sample <- sample(1:nrow(train.data), size = numFold, replace = FALSE)\n",
    "        \n",
    "    for (i in random.sample){\n",
    "        \n",
    "        ## Define new data\n",
    "        validation.data <- train.data[i,]\n",
    "        trainning.data <- train.data[-i,]\n",
    "\n",
    "        ## Define validation features\n",
    "        validation.label <- train.label[i,]\n",
    "        trainning.label <- train.label[-i,]\n",
    "       \n",
    "        ## calculate distances between samples\n",
    "        dist <- as.matrix(dist(c(validation.data, trainning.data), method= 'manhattan'))[1,-1]\n",
    "    \n",
    "        ### ...find its K nearest neighbours from training sampels...\n",
    "        nn <- as.data.frame(sort(dist, index.return = TRUE))[1:K,2]\n",
    "        \n",
    "        ###... and calculate the predicted label according to the majority vote\n",
    "        predicted.label <- sum(train.label[nn,])/K\n",
    "        \n",
    "        ###... Find error and store on vector\n",
    "        error.vector <- c(error.vector, (validation.label - predicted.label)^2)\n",
    "        \n",
    "    }\n",
    "    \n",
    "    ## return the class labels as output\n",
    "    return (round(error.vector,3))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "    II. Using the training data in Question 1, run your Leave-One-Out CV. Change the value of K=1,..,15 in your KNN regressor, and for each K compute the average of error values you have got for folds. Plot the average of error values versus 1/K for K=1,..,15 in your KNN regressor. Save the plot in your Jupyter Notebook file for Question 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_error <- data.frame('K'=1:15, 'error'=rep(0,15)) #support table with errors rates\n",
    "\n",
    "for (k in 1:15){\n",
    "    df_error[k,'error'] <- mean(cv(train.data, train.label, K = k)) # train error\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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e2006qqqoJZKgAAAFBMNcU+wR133HHS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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 300,
       "width": 720
      },
      "text/plain": {
       "height": 300,
       "width": 720
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=12, repr.plot.height=5) # area of display\n",
    "\n",
    "ggplot(data=df_error, aes(x=1/K, y=error)) + geom_line() +\n",
    "       scale_color_discrete(guide = guide_legend(title = NULL)) + theme_minimal() +\n",
    "       ggtitle(\"Missclassification Error\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "    III. Report (in your Jupyter Notebook file) the optimum value for K based on your plot for this Leave-One-Out cross validation in the previous part (Part II)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimal Parameter:  2"
     ]
    }
   ],
   "source": [
    "min.error <- min(df_error$error)\n",
    "min.K <- df_error[df_error$error == min.error,\"K\"]\n",
    "cat(\"Optimal Parameter: \", min.K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After importing the KNN function and performed the Leave-One-Out Cross Validation simulations we have found the optimal `K = 2`.  "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}