1. Elias Raftopoulos HY539 29/3/06 Prof. Maria Papadopouli
Clustering Tutorial
Elias Raftopoulos
HY539 29/3/06
Prof. Maria Papadopouli

2. Roadmap
Math Reminder
Principle Components Analysis
Clustering
ANOVA

3. Standard Deviation
Statistics – analyzing data sets in terms of the relationships between the individual points
Standard Deviation is a measure of the spread of the data
Calculation: average distance from the mean of the data

4. Variance Another measure of the spread of the data in a data set
Calculation:
Var( X ) = E(( x – μ )^2)
Why have both variance and SD to calculate the spread of data?
Variance is claimed to be the original statistical measure of spread of data. However it’s unit would be expressed as a square e.g. cm^2, which is unrealistic to express heights or other measures. Hence SD as the square root of variance was born.

5. Covariance
Variance – measure of the deviation from the mean for points in one dimension e.g. heights
Covariance as a measure of how much each of the dimensions vary from the mean with respect to each other.
Covariance is measured between 2 dimensions to see if there is a relationship between the 2 dimensions e.g. number of hours studied & marks obtained
The covariance between one dimension and itself is the variance

6. Covariance Matrix
Representing Covariance between dimensions as a matrix e.g. for 3 dimensions:
cov(x,x) cov(x,y) cov(x,z)
C = cov(y,x) cov(y,y) cov(y,z)
cov(z,x) cov(z,y) cov(z,z)
Diagonal is the variances of x, y and z
cov(x,y) = cov(y,x) hence matrix is symmetrical about the diagonal
N-dimensional data will result in nxn covariance matrix

7. Covariance Exact value is not as important as it’s sign.
A positive value of covariance indicates both dimensions increase or decrease together e.g. as the number of hours studied increases, the marks in that subject increase.
A negative value indicates while one increases the other decreases, or vice-versa e.g. active social life at RIT vs performance in CS dept.
If covariance is zero: the two dimensions are independent of each other e.g. heights of students vs the marks obtained in a subject

8. Transformation matrices
Consider:
Square transformation matrix transforms (3,2) from its original location. Now if we were to take a multiple of (3,2)
3 6
2 4 x = = 4 x 2 x = x = = 4 x

9. Transformation matrices
Scale vector (3,2) by a value 2 to get (6,4)
Multiply by the square transformation matrix
We see the result is still a multiple of 4.
WHY?
A vector consists of both length and direction. Scaling a vector only changes its length and not its direction. This is an important observation in the transformation of matrices leading to formation of eigenvectors and eigenvalues.
Irrespective of how much we scale (3,2) by, the solution is always a multiple of 4.

10. eigenvalue problem
The eigenvalue problem is any problem having the following form:
A . v = λ . v
A: n x n matrix
v: n x 1 non-zero vector
λ: scalar
Any value of λ for which this equation has a solution is called the eigenvalue of A and vector v which corresponds to this value is called the eigenvector of A.

11. eigenvalue problem 2 3 3 12 3 2 1 2 8 2 A . v = λ . v
A . v = λ . v
Therefore, (3,2) is an eigenvector of the square matrix A and 4 is an eigenvalue of A
Given matrix A, how can we calculate the eigenvector and eigenvalues for A? x = = 4 x

12. Calculating eigenvectors & eigenvalues
Given A . v = λ . v
A . v - λ . I . v = 0
(A - λ . I ). v = 0
Finding the roots of |A - λ . I| will give the eigenvalues and for each of these eigenvalues there will be an eigenvector
Example …

13. Calculating eigenvectors & eigenvalues
If A =
-2 -3
Then |A - λ . I| = λ = 0 λ
-λ = λ2 + 3λ + 2 = 0
-2 -3-λ
This gives us 2 eigenvalues:
λ1 = -1 and λ2 = -2

14. Properties of eigenvectors and eigenvalues
Note that Irrespective of how much we scale (3,2) by, the solution is always a multiple of 4.
Eigenvectors can only be found for square matrices and not every square matrix has eigenvectors.
Given an n x n matrix, we can find n eigenvectors

15. Roadmap
Principle Components Analysis
Clustering
ANOVA

16. PCA
principal components analysis (PCA) is a technique that can be used to simplify a dataset
It is a linear transformation that chooses a new coordinate system for the data set such that
greatest variance by any projection of the data set comes to lie on the first axis (then called the first principal component),
the second greatest variance on the second axis, and so on.
PCA can be used for reducing dimensionality by eliminating the later principal components.

17. PCA
By finding the eigenvalues and eigenvectors of the covariance matrix, we find that the eigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the dataset.
This is the principal component.
PCA is a useful statistical technique that has found application in:
fields such as face recognition and image compression
finding patterns in data of high dimension

18. PCA process –STEP 1 Subtract the mean
from each of the data dimensions. All the x values have x subtracted and y values have y subtracted from them. This produces a data set whose mean is zero.
Subtracting the mean makes variance and covariance calculation easier by simplifying their equations. The variance and co-variance values are not affected by the mean value.

19. PCA process –STEP 1 DATA: x y 2.5 2.4 0.5 0.7 2.2 2.9 1.9 2.2 3.1 3.0
ZERO MEAN DATA:
x y

20. PCA process –STEP 1

21. PCA process –STEP 2 Calculate the covariance matrix
since the non-diagonal elements in this covariance matrix are positive, we should expect that both the x and y variable increase together.

22. PCA process –STEP 3
Calculate the eigenvectors and eigenvalues of the covariance matrix
eigenvalues =
eigenvectors =

23. PCA process –STEP 3
eigenvectors are plotted as diagonal dotted lines on the plot.
Note they are perpendicular to each other.
Note one of the eigenvectors goes through the middle of the points, like drawing a line of best fit.
The second eigenvector gives us the other, less important, pattern in the data, that all the points follow the main line, but are off to the side of the main line by some amount.

24. PCA process –STEP 4
Reduce dimensionality and form feature vector the eigenvector with the highest eigenvalue is the principle component of the data set.
In our example, the eigenvector with the larges eigenvalue was the one that pointed down the middle of the data.
Once eigenvectors are found from the covariance matrix, the next step is to order them by eigenvalue, highest to lowest. This gives you the components in order of significance.

25. PCA process –STEP 4
Now, if you like, you can decide to ignore the components of lesser significance
You do lose some information, but if the eigenvalues are small, you don’t lose much
n dimensions in your data
calculate n eigenvectors and eigenvalues
choose only the first p eigenvectors
final data set has only p dimensions.

26. PCA process –STEP 4 Feature Vector
FeatureVector = (eig1 eig2 eig3 … eign)
We can either form a feature vector with both of the eigenvectors:
or, we can choose to leave out the smaller, less significant component and only have a single column:

27. PCA process –STEP 5 Deriving the new data
FinalData = RowFeatureVector x RowZeroMeanData
RowFeatureVector is the matrix with the eigenvectors in the columns transposed so that the eigenvectors are now in the rows, with the most significant eigenvector at the top
RowZeroMeanData is the mean-adjusted data transposed, ie. the data items are in each column, with each row holding a separate dimension.

28. PCA process –STEP 5 S R = U VT factors variables factors variables
sig.
significant
noise
noise
significant
noise
factors
factors
samples
samples

29. PCA process –STEP 5
FinalData is the final data set, with data items in columns, and dimensions along rows.
What will this give us?
It will give us the original data solely in terms of the vectors we chose.
We have changed our data from being in terms of the axes x and y , and now they are in terms of our 2 eigenvectors.

30. PCA process –STEP 5 FinalData transpose: dimensions along columns x y

31. PCA process –STEP 5

32. Reconstruction of original Data
If we reduced the dimensionality, obviously, when reconstructing the data we would lose those dimensions we chose to discard. In our example let us assume that we considered only the x dimension…

33. Reconstruction of original Datax

34. Roadmap
Principle Components Analysis
Clustering
ANOVA

35. What is Cluster Analysis?
Cluster: a collection of data objects
Similar to the objects in the same cluster (Intraclass similarity)
Dissimilar to the objects in other clusters (Interclass dissimilarity)
Cluster analysis
Statistical method for grouping a set of data objects into clusters
A good clustering method produces high quality clusters with high intraclass similarity and low interclass similarity
Clustering is unsupervised classification
Can be a stand-alone tool or as a preprocessing step for other algorithms

36. Group objects according to their similarity
Cluster:
a set of objects
that are similar
to each other
and separated
from the other
objects.
Example: green/
red data points
were generated
from two different
normal distributions

37. Clustering data object expression data matrix
Experiments/samples are given as the row and column vectors of an expression data matrix
Clustering may be applied either to objects experiments (regarded as vectors in Ro or Rn).
n experiments
o objects

38. Pattern matrix  Proximity matrix
Pattern matrix (nxp)
p=attributes
n=# of objects
Proximity matrix (nxn)
d(i,j)=difference/
dissimilarity between i and j

39. Proximity matrix
Clustering methods require that a index of proximity, or alikeness, or affinity or association be established between pairs of patterns
A proximity index is either a similarity or a dissimilarity
The crucial problem in identifying clusters in data is to specify what proximity is and how to measure it

40. Proximity indices
A proximity index between the ith and kth patterns is denoted d(i,k) and must satisfy the following three properties:
1. (a) for a dissimilarity: d(i,i) = 0, all i
(b) for a similarity: d(i,i) ≥ max d(i,k), all I
2. d(i,k) = d(k,i), all (i,k)
3. d(i,k) ≥ 0, all (i,k)

41. Different proximity measures
r = 2(Euclidean distance)
[ ]1/2 = 4.472
r = 1(Manhattan distance)
4 + 2 = 6
r → ∞ (“sup” distance)
max{4,2} = 4

42. K-Means Clustering The meaning of ‘K-means’
Why it is called ‘K-means’ clustering: K points are used to represent the clustering result; each point corresponds to the centre (mean) of a cluster
Each point is assigned to the cluster with the closest center point
The number K, must be specified
Basic algorithm

43. The K-Means Clustering Method
Given k, the k-means algorithm is implemented in 4 steps:
Partition objects into k non-empty subsets
Arbitrarily choose k points as initial centers
Assign each object to the cluster with the nearest seed point (center)
Calculate the mean of the cluster and update the seed point
Go back to Step 3, stop when no more new assignment

44. The K-Means Clustering Method (cntd)
The basic step of k-means clustering is simple:
Iterate until stable (= no object move group):
Determine the centroid coordinate
Determine the distance of each object to the centroids
Group the object based on minimum distance

45. The K-Means Clustering Method (cntd)

46. The K-Means Clustering Results
Example 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5
Update the cluster means 4
Assign each objects to most similar center 3 2 1 1 2 3 4 5 6 7 8 9 10
reassign
reassign
K=2
Arbitrarily choose K object as initial cluster center
Update the cluster means

47. Weaknesses of the K-Means Method
Unable to handle noisy data and outliers
Very large or very small values could skew the mean
Not suitable to discover clusters with non-convex shapes

48. Hierarchical Clustering
Start with every data point in a separate cluster
Keep merging the most similar pairs of data points/clusters until we have one big cluster left
This is called a bottom-up or agglomerative method

49. Hierarchical Clustering (cont.)
This produces a binary tree or dendrogram
The final cluster is the root and each data item is a leaf
The height of the bars indicate how close the items are

50. Hierarchical Clustering Demo

51. Levels of Clustering

52. Linkage in Hierarchical Clustering
We already know about distance measures between data items, but what about between a data item and a cluster or between two clusters?
We just treat a data point as a cluster with a single item, so our only problem is to define a linkage method between clusters
As usual, there are lots of choices…

53. Average Linkage Definition
Each cluster ci is associated with a mean vector i which is the mean of all the data items in the cluster
The distance between two clusters ci and cj is then just d(i , j )
This is somewhat non-standard – this method is usually referred to as centroid linkage and average linkage is defined as the average of all pairwise distances between points in the two clusters

54. Single Linkage
The minimum of all pairwise distances between points in the two clusters
Tends to produce long, “loose” clusters

55. Complete Linkage
The maximum of all pairwise distances between points in the two clusters
Tends to produce very tight clusters

56. Distances between clusters (summary)
Calculation of the distance between two clusters is based on the pairwise distances between members of the clusters.
Complete linkage: largest distance between points
Average linkage: average distance between points
Single linkage: smallest distance between points
Centroid: distance between centroids
Complete linkage gives preference to compact/spherical
clusters. Single linkage can produce long stretched clusters.

57. A B C D E 1 2 3 4 5
EXAMPLE

58. More on Hierarchical Clustering Methods
Major advantage
Conceptually very simple
Easy to implement  most commonly used technique
Major weakness of agglomerative clustering methods
do not scale well: time complexity of at least O(n2), where n is the number of total objects
can never undo what was done previously  high likelihood of getting stuck in local minima

59. Roadmap
Principle Components Analysis
Clustering
ANOVA

60. (M)ANOVA
The analysis of variance technique in One-Way Analysis of Variance (ANOVA) takes a set of grouped data and determine whether the mean of a variable differs significantly between groups
Often there are multiple variables and you are interested in determining whether the entire set of means is different from one group to the next
There is a multivariate version of analysis of variance that can address that problem (MANOVA)