From this question, I got the getUnique function.
Array.prototype.getUnique = function(){
   var u = {}, a = [];
   for(var i = 0, l = this.length; i < l; ++i){
      if(u.hasOwnProperty(this[i])) {
         continue;
      }
      a.push(this[i]);
      u[this[i]] = 1;
   }
   return a;
}
then you can add a function that counts ocurrences of a string inside an array of strings:
function getOcurrencesInStrings(targetString, arrayOfStrings){
  var ocurrencesCount = 0;
  for(var i = 0, arrayOfStrings.length; i++){
    if(arrayOfStrings[i].indexOf(targetString) > -1){
      ocurrencesCount++;
    }
  }
  return ocurrencesCount;
}
then you just:
getOcurrencesInStrings('dots', initialArray.getUnique())
This will return the number you want.
It's not the smallest piece of code, but It's highly reusable.