Cluster Validation

CLUSTER ANALYSIS
CASE STUDY OF TIBETIAN SKULLS
A.W.D. UDAYA SHALIKA
158888L

Overview
• What is cluster analysis?
• Why Cluster Analysis
• Clustering Methods
• Analysis
Background of Data
Objectives
Hierarchical Clustering
K mean Clustering
Validation

What is cluster analysis?
• What this means?
When plotted geometrically,
objects within clusters should be
very close together and clusters
will be far apart.
• Clusters should exhibit high
internal homogeneity and high
external heterogeneity
Cluster analysis is a multivariate data mining technique whose goal is to
groups objects based on a set of user selected characteristics

Why Cluster Analysis
Cluster analysis is a major tool in a number of applications in many fields
of Business, Engineering & etc.(The odoridis and Koutroubas, 1999):
• Data reduction.
• Hypothesis generation.
• Hypothesis testing.
• Prediction based on groups.

Cluster Analysis – Classification
Clustering
Hierarchical clustering
Divisive
Agglomerative
Partitional clustering
K-means
Fuzzy K-means
Isodata
Density based
clustering
Denclust
CLUPOT
SVC
Parzen-
Watershed
Grid based clustering
STING
CLIQUE

Background Story
• These data, collected by Colonel L.A. Waddel, were
first reported in Morant (1923) . The data consist of
five measurements on each of 32 skulls found in the
southwestern and eastern districts ofTibet.
• The first comprises skulls 1 to 17 found in graves in
Sikkim and the neighboring area of Tibet (Type A
skulls). The remaining 15 skulls (Type B skulls) were
picked up on a battlefield in the Lhasa district and
are believed to be those of native soldiers from the
eastern province of Khams.

Objective
Hypothesis test:
Tibetans from Khams might be survivors of a particular human type,
unrelated to the Mongolian and Indian types that surrounded them.
• Greatest length of skull (Length)
• Greatest horizontal breadth of skull (Breadth)
• Height of skull (Height)
• Upper face length (Fheight)
• Face breadth between outermost points of cheekbones (Fbreadth)

Matrix Plot
Preliminary graphical display of the data
might be useful and here we will display
them as a scatter plot matrix in which
group membership is indicated. While this
diagram only allows us to asses the group
separation in two dimensions, it seems to
suggest that face breadth between outer-
most points of cheek bones (Fbreadth),
greatest length of skull (Length), and
upper face length (Fheight) provide the
greatest discrimination between the two
skull types

Descriptive / Correlations
Highest mean is 179.94 in Length of skull which
varying in between 200 (max) and 162.5(min).
lowest mean is 72.94 in Fheight which varying in
between 82.5 to 62.
As per the correlation matrix there are high
correlation among Length and
Fheight(0.755,p=0.00). Other than that Fheight
and Fbreadth (0.617,0.00), Length and
Fbreadth(0.567,0.01), Breadth and
Fbreadth(0.549,0.01) are significant

Number of clusters
The correct choice of number of clusters is often ambiguous, with interpretations
depending on the shape and scale of the distribution of points in a data set and the
desired clustering resolution of the user
• The rules of thumb is k=(n/2)^1/2 : where k is the number of clusters
• Plotting % of variance vs number of clusters
• In our case number of clusters is 2

Hierarchical Clustering
• Agglomerative (Bottom-up):
Principle: compute the Distance-Matrix between all objects (initially
one object = one cluster). Find the two clusters with the closest
distance and put those two clusters into one. Compute the new
Distance-Matrix

Hierarchical Clustering contd.
• data were analyzed under different linkage measurements vs squared
Euclidian distance methods
SUMMARYTABLE
Linkage Distance Cluster 1 Cluster 2 Cluster 1 Cluster 2
A B A B
Average Squared Euclidian A-6 A-11, B-15 35% 0% 65% 100%
Centroid Squared Euclidian A-6 A-11, B-15 35% 0% 65% 100%
Complete Squared Euclidian A-14, B-5 A-3, B-10 82% 33% 18% 67%
Single Squared Euclidian A-17, B-14 B-1 100% 93% 0% 7%
Ward Squared Euclidian A-16, B-5 A-1, B-10 94% 33% 6% 67%

Hierarchical Clustering Contd.
Complete Linkage Squared Euclidean Ward Linkage Squared Euclidean

Complete Linkage Squared Euclidean Ward Linkage Squared Euclidean
The Dendrogram displays the information in the amalgamation table in the form of a tree
diagram. The first table summarizes each cluster by the number of observations, the within
cluster sum of squares, In general, a cluster with a small sum of squares is more compact than
one with a large sum of squares. The centroid is the vector of variable means for the
observations in that cluster and is used as a cluster midpoint. The second table displays the
centroids for the individual clusters while the third table gives distances between cluster
centroids.

Partitional Clustering
• K-mean

Case Number Type Cluster Case Number Type Cluster
1 A 1 17 A 2
2 A 2 18 B 1
3 A 2 19 B 2
4 A 2 20 B 1
5 A 2 21 B 1
6 A 2 22 B 1
7 A 2 23 B 2
8 A 2 24 B 1
9 A 2 25 B 1
10 A 2 26 B 1
11 A 2 27 B 1
12 A 2 28 B 1
13 A 2 29 B 2
14 A 1 30 B 1
15 A 2 31 B 1
16 A 2 32 B 2
K Mean Contd. Cluster Membership

K Mean Clustering Contd.
Out of 17 of A-type skulls 15 skulls present in one cluster(88.2%).
Out of 15 skulls 11 B-type skulls present in another cluster (77.3%).
The output of K-mean clustering quite better than the out put of
ward and complete in hierarchical clustering.

Cluster Validation
• Fisher’s Linear Discriminant Analysis
• Use discriminant analysis to classify observations into two or more
groups if you have a sample with known groups. Discriminant
analysis can also used to investigate how variables contribute to
group separation.
• For two groups, the null hypothesis is that the means of the two
groups on the discriminant function-the centroids, are equal.
• Centroids are the mean discriminant score for each group. Wilk’s
lambda is used to test for significant differences between groups

Cluster Validation contd.
The canonical relation is a correlation between
the discriminant scores and the levels of the
dependent variable. A high correlation (0.825)
indicates a function that discriminates well.
Wilks’ Lambda is the ratio of within-groups sums
of squares to the total sums of squares. Wilks'
lambda is a measure of how well each function
separates cases into groups. Smaller values of
Wilks' lambda indicate greater discriminatory
ability of the function. The associated
significance value indicate whether the
difference is significant. Here, the Lambda of
0.319 and significant p= 0.00)thus, the group
means appear to significantly different from
each other

Cluster Validation contd.
This table is used to assess how well the
discriminant function works, and if it works equally
well for each group of the dependent variable.
Here cross validated accuracy is about 87.5% and
this is quite good result

Summary
• Out of 17 of A-type skulls 16 skulls present in one cluster(94%). Out of
15 skulls 10 B-type skulls present in another cluster (67%) according to
Ward linkage hierarchical clustering
• Out of 17 of A-type skulls 15 skulls present in one cluster(88.2%). Out
of 15 skulls 11 B-type skulls present in another cluster (77.3%)
according to K-mean clustering
• Group means appear to significantly different from each other
according to discriminant analysis
• Thus Tibetans from Khams district significantly different from
generalTibetans according to the skull measurements

References
• Joseph F. Hair Jr.,Willim C. Black, Barry J. Babin, Rolph E. Anderson –
“Multivariate Data Analysis – a global perspective”
• Brian S. Everitt ,Sabine Landau “Cluster Analysis” 5th edition
• Mo'oamin M. R. El-Hanjouri , Bashar S. Hamad “Using Cluster Analysis
and Discriminant Analysis Methods in Classification with Application on
Standard of Living Family in Palestinian Areas” International Journal of
Statistics and Applications,2015

THANKYOU
19/3/2017
A.W.D Udaya Shalika
158888L

Cluster Validation

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Editor's Notes