Example run.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Load the libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "library(pracma)\n",
    "library(igraph)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Load the functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_s=function(y,s){\n",
    "  a=mean(y^s)\n",
    "  a=a^(1/s)\n",
    "  return(a)\n",
    "}\n",
    "obj_power.k.means=function(X,theta,s){\n",
    "  s1=0\n",
    "  n=dim(X)[1]\n",
    "  k=dim(theta)[1]\n",
    "  y=numeric(k)\n",
    "  for(i in 1:n){\n",
    "    for(j in 1:k){\n",
    "      y[j]=sum((X[i,]-theta[j,])^2)\n",
    "    }\n",
    "    s1=s1+m_s(y,s)\n",
    "  }\n",
    "  return(s1)\n",
    "}\n",
    "\n",
    "power.k.means=function(X,s=-1,k,eta=1.04,\n",
    "                       tmax=200,tol=10^(-3)){\n",
    "  n=dim(X)[1]\n",
    "  p=dim(X)[2]\n",
    "  W=matrix(0,nrow=n,ncol=k)\n",
    "  m=matrix(0,nrow=n,ncol=k)\n",
    "  label=numeric(n)\n",
    "  dd=numeric(k)\n",
    "  #initialization\n",
    "  sam=sample(n,k)\n",
    "  theta=X[sam,]\n",
    "  #theta=rand(k,p)\n",
    "  #theta=rand(k,p)\n",
    "  for(i in 1:p){\n",
    "    theta[,i]=theta[,i]+rnorm(k,0,0.1)\n",
    "  }\n",
    "  val1=theta\n",
    "  coef = numeric(n)\n",
    "  for(t in 1:tmax){\n",
    "    if(t==tmax){\n",
    "      cat(coef)\n",
    "      cat(\"\\n\")\n",
    "      cat(W[1,])\n",
    "      cat(\"\\n\")\n",
    "    }\n",
    "    #update weights\n",
    "    for(i in 1:n){\n",
    "      for(l in 1:k){\n",
    "        m[i,l]=sum((X[i,]-theta[l,])^2)\n",
    "      }\n",
    "      coef[i] = (sum(m[i,]^s))^((1/s)-1)\n",
    "    }\n",
    "    for(i in 1:n){\n",
    "      for(l in 1:k){\n",
    "        W[i,l]=m[i,l]^(s-1)*coef[i]\n",
    "      }\n",
    "    }\n",
    "    #update centroids\n",
    "    for(l in 1:k){\n",
    "      for(d in 1:p){\n",
    "        theta[l,d]=(sum(W[,l]*X[,d]))/sum(W[,l])\n",
    "      }\n",
    "    }\n",
    "    #update s\n",
    "    if(t%%2==0){s=eta*s}\n",
    "    cat(t)\n",
    "    cat('\\n')\n",
    "    t=t+1\n",
    "    val2=theta\n",
    "    if(norm(val1-val2)<tol){\n",
    "      break\n",
    "    }else{\n",
    "      val1=val2\n",
    "    }\n",
    "  }\n",
    "  for(i in 1:n){\n",
    "    for(j in 1:k){\n",
    "      dd[j]=sum((X[i,]-theta[j,])^2)\n",
    "    }\n",
    "    label[i]=which.min(dd)\n",
    "  }\n",
    "  list1=list(theta,label)\n",
    "  names(list1)=c('theta','label')\n",
    "  return(list1)\n",
    "}\n",
    "\n",
    "entropy_obj_weighted.power.k.means=function(X,theta,s,W,lambda){\n",
    "  s1=0\n",
    "  n=dim(X)[1]\n",
    "  k=dim(theta)[1]\n",
    "  y=numeric(k)\n",
    "  for(i in 1:n){\n",
    "    for(j in 1:k){\n",
    "      y[j]=sum((X[i,]-theta[j,])^2*W)\n",
    "    }\n",
    "    s1=s1+m_s(y,s)\n",
    "  }\n",
    "  s1=s1+lambda*sum(W*log(W))\n",
    "  return(s1)\n",
    "}\n",
    "\n",
    "w.norm=function(y,w,beta){\n",
    "  return(sum(y^2*w^(beta)))\n",
    "}\n",
    "entropy_weight.power.k.means=function(X,s=-1,k,lambda,eta=1.04,beta=1,tmax=200,tol=10^(-5)){\n",
    "  n=dim(X)[1]\n",
    "  p=dim(X)[2]\n",
    "  phi=matrix(0,nrow=n,ncol=k)\n",
    "  m=matrix(0,nrow=n,ncol=k)\n",
    "  label=numeric(n)\n",
    "  dd=numeric(k)\n",
    "  #initialization\n",
    "  sam=sample(n,k)\n",
    "  theta=X[sam,]\n",
    "  #theta=rand(k,p)\n",
    "  #theta=rand(k,p)\n",
    "  for(i in 1:p){\n",
    "    theta[,i]=theta[,i]+rnorm(k,0,0.1)\n",
    "  }\n",
    "  W=rep(1/p,p)\n",
    "  D=rep(0,p)\n",
    "  val1=theta\n",
    "  \n",
    "  for(t in 1:tmax){\n",
    "    #update membership\n",
    "    for(i in 1:n){\n",
    "      for(l in 1:k){\n",
    "        m[i,l]=w.norm((X[i,]-theta[l,]),W,beta)\n",
    "      }\n",
    "    }\n",
    "    for(i in 1:n){\n",
    "      for(l in 1:k){\n",
    "        phi[i,l]=(m[i,l]+0.0001)^(s-1)*(sum(m[i,]^s)+0.0001)^((1/s)-1)\n",
    "      }\n",
    "    }\n",
    "    #update centroids\n",
    "    for(l in 1:k){\n",
    "      for(d in 1:p){\n",
    "        theta[l,d]=(sum(phi[,l]*X[,d]))/sum(phi[,l])\n",
    "      }\n",
    "    }\n",
    "    #update s\n",
    "    if(t%%2==0){\n",
    "    s=eta*s\n",
    "    }\n",
    "    #update feature weights\n",
    "    for(l in 1:p){\n",
    "      s1=0\n",
    "      for(i in 1:n){\n",
    "        for(j in 1:k){\n",
    "          s1=s1+phi[i,j]*(X[i,l]-theta[j,l])^2\n",
    "        }\n",
    "      }\n",
    "      D[l]=exp(-s1/lambda)\n",
    "    }\n",
    "    W=D/sum(D)\n",
    "    cat(t)\n",
    "    cat('\\n')\n",
    "    t=t+1\n",
    "    val2=theta\n",
    "    if(norm(val1-val2)<tol){\n",
    "      break\n",
    "    }else{\n",
    "      val1=val2\n",
    "    }\n",
    "  }\n",
    "  for(i in 1:n){\n",
    "    for(j in 1:k){\n",
    "      dd[j]=sum((X[i,]-theta[j,])^2*W)\n",
    "    }\n",
    "    label[i]=which.min(dd)\n",
    "  }\n",
    "  list2=list(theta,label,W)\n",
    "  names(list2)=c('theta','label','weight')\n",
    "  return(list2)\n",
    "}\n",
    "data_generate=function(n,M,prob1,sigma,sigma2){\n",
    "  p=dim(M)[2]\n",
    "  X=matrix(0,n,p)\n",
    "  k=dim(M)[1]\n",
    "  label=numeric(n)\n",
    "  for(i in 1:n){\n",
    "    s=sample(1:k,size=1,prob=prob1)\n",
    "    for(l in 1:p){\n",
    "      if(M[s,l]==0){\n",
    "        X[i,l]=rnorm(1,M[s,l],sigma2)\n",
    "      }else{\n",
    "        X[i,l]=rnorm(1,M[s,l],sigma)\n",
    "      }\n",
    "\n",
    "    }\n",
    "    label[i]=s\n",
    "  }\n",
    "  ls=list(X,label)\n",
    "  names(ls)=c('data','label')\n",
    "  return(ls)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generate the Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "k=5\n",
    "p=20\n",
    "M=rand(k,p)\n",
    "\n",
    "prop=0.95\n",
    "s=sample(p,floor(p*prop))\n",
    "s=6:20\n",
    "M[,s]=0\n",
    "X=data_generate(1000,M,rep(1/k,k),0.02,1)\n",
    "toss=X$label\n",
    "X=X$data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Normalise the data (optional)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "p=dim(X)[2]\n",
    "for(i in 1:p){\n",
    "  X[,i]=(X[,i]-mean(X[,i]))/sd(X[,i])\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "2\n",
      "3\n",
      "4\n",
      "5\n",
      "6\n",
      "7\n",
      "8\n",
      "9\n",
      "10\n",
      "11\n",
      "12\n",
      "13\n",
      "14\n"
     ]
    }
   ],
   "source": [
    "l=entropy_weight.power.k.means(X,k=k,s=-1,lambda=10,tmax=200,tol=0.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "1"
      ],
      "text/latex": [
       "1"
      ],
      "text/markdown": [
       "1"
      ],
      "text/plain": [
       "[1] 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "1"
      ],
      "text/latex": [
       "1"
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      "text/markdown": [
       "1"
      ],
      "text/plain": [
       "[1] 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "compare(toss,l$label,'nmi')\n",
    "compare(toss,l$label,'adjusted.rand')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.138012191855092</li>\n",
       "\t<li>0.202263769425908</li>\n",
       "\t<li>0.162645620606395</li>\n",
       "\t<li>0.264927870768136</li>\n",
       "\t<li>0.232150547344469</li>\n",
       "\t<li>7.9823547293465e-42</li>\n",
       "\t<li>5.27088497960263e-42</li>\n",
       "\t<li>8.87092522890206e-42</li>\n",
       "\t<li>8.4403357561847e-42</li>\n",
       "\t<li>6.96277933412724e-42</li>\n",
       "\t<li>8.15735490355873e-42</li>\n",
       "\t<li>9.44206815122039e-42</li>\n",
       "\t<li>7.42735435007309e-42</li>\n",
       "\t<li>7.56201840761902e-42</li>\n",
       "\t<li>1.28412555913383e-41</li>\n",
       "\t<li>6.36157271640332e-42</li>\n",
       "\t<li>1.16510746566077e-41</li>\n",
       "\t<li>2.47749040168835e-41</li>\n",
       "\t<li>8.28784081512283e-42</li>\n",
       "\t<li>9.01574389692695e-42</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.138012191855092\n",
       "\\item 0.202263769425908\n",
       "\\item 0.162645620606395\n",
       "\\item 0.264927870768136\n",
       "\\item 0.232150547344469\n",
       "\\item 7.9823547293465e-42\n",
       "\\item 5.27088497960263e-42\n",
       "\\item 8.87092522890206e-42\n",
       "\\item 8.4403357561847e-42\n",
       "\\item 6.96277933412724e-42\n",
       "\\item 8.15735490355873e-42\n",
       "\\item 9.44206815122039e-42\n",
       "\\item 7.42735435007309e-42\n",
       "\\item 7.56201840761902e-42\n",
       "\\item 1.28412555913383e-41\n",
       "\\item 6.36157271640332e-42\n",
       "\\item 1.16510746566077e-41\n",
       "\\item 2.47749040168835e-41\n",
       "\\item 8.28784081512283e-42\n",
       "\\item 9.01574389692695e-42\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.138012191855092\n",
       "2. 0.202263769425908\n",
       "3. 0.162645620606395\n",
       "4. 0.264927870768136\n",
       "5. 0.232150547344469\n",
       "6. 7.9823547293465e-42\n",
       "7. 5.27088497960263e-42\n",
       "8. 8.87092522890206e-42\n",
       "9. 8.4403357561847e-42\n",
       "10. 6.96277933412724e-42\n",
       "11. 8.15735490355873e-42\n",
       "12. 9.44206815122039e-42\n",
       "13. 7.42735435007309e-42\n",
       "14. 7.56201840761902e-42\n",
       "15. 1.28412555913383e-41\n",
       "16. 6.36157271640332e-42\n",
       "17. 1.16510746566077e-41\n",
       "18. 2.47749040168835e-41\n",
       "19. 8.28784081512283e-42\n",
       "20. 9.01574389692695e-42\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 1.380122e-01 2.022638e-01 1.626456e-01 2.649279e-01 2.321505e-01\n",
       " [6] 7.982355e-42 5.270885e-42 8.870925e-42 8.440336e-42 6.962779e-42\n",
       "[11] 8.157355e-42 9.442068e-42 7.427354e-42 7.562018e-42 1.284126e-41\n",
       "[16] 6.361573e-42 1.165107e-41 2.477490e-41 8.287841e-42 9.015744e-42"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "l$weight"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+S3UPrHw0ofEQleg6DMJrM70S3kFXLT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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(X,col=l$label+1)\n",
    "points(l$theta,pch=18,col=1)"
   ]
  }
 ],
 "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": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}