We are given a data set of items, which have certain features, and values for these features (like a vector). We are tasked with categorizing those items into groups. To achieve this, we will use the kMeans algorithm, an unsupervised learning algorithm. Language of implementation is Python.

### Overview

(It will help if you think of items as points in an n-dimensional space)

The algorithm will categorize the items into k groups of similarity. To calculate that similarity, we will use the euclidean distance as measurement.

The algorithm works as follows: First we initialize k points, called means, randomly. We categorize each item to its closest mean and we update the mean’s coordinates, which are the averages of the items categorized in that mean so far. We repeat the process for a given number of iterations and at the end, we have our clusters.

The “points” mentioned above are called means, because they hold the mean values of the items categorized in it. To initialize these means, we have a lot of options. An intuitive method is to initialize the means at random items in the data set. Another method is to initialize the means at random values between the boundaries of the data set (if for a feature x the items have values in [0,3], we will initialize the means with values for x at [0,3]).

The above algorithm in pseudocode:

```Initialize k means with random values

For a given number of iterations:
Iterate through items:
Find the mean closest to the item
Assign item to mean
Update mean
```

We receive input as a text file (‘data.txt’). Each line represents an item, and it contains numerical values (one for each feature) split by commas. You can find a sample data set here.

We will read the data from the file, saving it into a list. Each element of the list is another list containing the item values for the features. We do this with the following function:

```def ReadData(fileName):
#Read the file, splitting by lines
f = open(fileName,'r');
f.close();

items = [];

for i in range(1,len(lines)):
line = lines[i].split(',');
itemFeatures = [];

for j in range(len(line)-1):
v = float(line[j]); #Convert feature value to float
itemFeatures.append(v); #Add feature value to dict

items.append(itemFeatures);

shuffle(items);

return items;
```

### Initialize Means

We want to initialize each mean’s values in the range of the feature values of the items. For that, we need to find the min and max for each feature. We accomplish that with the following function:

```def FindColMinMax(items):
n = len(items[0]);
minima = [sys.maxint for i in range(n)];
maxima = [-sys.maxint -1 for i in range(n)];

for item in items:
for f in range(len(item)):
if(item[f]  maxima[f]):
maxima[f] = item[f];

return minima,maxima;
```

The variables minima, maxima are lists containing the min and max values of the items respectively. We initialize each mean’s feature values randomly between the corresponding minimum and maximum in those above two lists:

```def InitializeMeans(items,k,cMin,cMax):
#Initialize means to random numbers between
#the min and max of each column/feature

f = len(items[0]); #number of features
means = [[0 for i in range(f)] for j in range(k)];

for mean in means:
for i in range(len(mean)):
#Set value to a random float
#(adding +-1 to avoid a wide placement of a mean)
mean[i] = uniform(cMin[i]+1,cMax[i]-1);

return means;
```

### Euclidean Distance

We will be using the euclidean distance as a metric of similarity for our data set (note: depending on your items, you can use another similarity metric).

```def EuclideanDistance(x,y):
S = 0; #The sum of the squared differences of the elements
for i in range(len(x)):
S += math.pow(x[i]-y[i],2);

return math.sqrt(S); #The square root of the sum
```

### Update Means

To update a mean, we need to find the average value for its feature, for all the items in the mean/cluster. We can do this by adding all the values and then dividing by the number of items, or we can use a more elegant solution. We will calculate the new average without having to re-add all the values, by doing the following:

$m = \dfrac{m*(n-1)+x}{n}$

where m is the mean value for a feature, n is the number of items in the cluster and x is the feature value for the added item. We do the above for each feature to get the new mean.

You can read more on this technique on this post.

Code:

```def UpdateMean(n,mean,item):
for i in range(len(mean)):
m = mean[i];
m = (m*(n-1)+item[i])/float(n);
mean[i] = round(m,3);

return mean;
```

### Classify Item

Now we need to write a function to classify an item to a group/cluster. For the given item, we will find its similarity to each mean, and we will classify the item to the closest one.

```def Classify(means,item):
#Classify item to the mean with minimum distance

minimum = sys.maxint;
index = -1;

for i in range(len(means)):
#Find distance from item to mean
dis = EuclideanDistance(item,means[i]);

if(dis < minimum):
minimum = dis;
index = i;

return index;
```

### Find Means

To actually find the means, we will loop through all the items, classify them to their nearest cluster and update the cluster’s mean. We will repeat the process for some fixed number of iterations. If between two iterations no item changes classification, we stop the process as the algorithm has found the optimal solution.

The below function takes as input k (the number of desired clusters), the items and the number of maximum iterations, and returns the means and the clusters. The classification of an item is stored in the array belongsTo and the number of items in a cluster is stored in clusterSizes.

```def CalculateMeans(k,items,maxIterations=100000):
#Find the minima and maxima for columns
cMin, cMax = FindColMinMax(items);

#Initialize means at random points
means = InitializeMeans(items,k,cMin,cMax);

#Initialize clusters, the array to hold
#the number of items in a class
clusterSizes= [0 for i in range(len(means))];

#An array to hold the cluster an item is in
belongsTo = [0 for i in range(len(items))];

#Calculate means
for e in range(maxIterations):
#If no change of cluster occurs, halt
noChange = True;
for i in range(len(items)):
item = items[i];
#Classify item into a cluster and update the
#corresponding means.

index = Classify(means,item);

clusterSizes[index] += 1;
cSize = clusterSizes[index];
means[index] = UpdateMean(cSize,means[index],item);

#Item changed cluster
if(index != belongsTo[i]):
noChange = False;

belongsTo[i] = index;

#Nothing changed, return
if(noChange):
break;

return means;
```

### Find Clusters

Finally we want to find the clusters, given the means. We will iterate through all the items and we will classify each item to its closest cluster.

```def FindClusters(means,items):
clusters = [[] for i in range(len(means))]; #Init clusters

for item in items:
#Classify item into a cluster
index = Classify(means,item);