1. Data Mining: Concepts and Techniques — Chapter 2 —
Jiawei Han, Micheline Kamber, and Jian Pei
University of Illinois at Urbana-Champaign
Simon Fraser University
©2013 Han, Kamber, and Pei. All rights reserved.

2. Chapter 2: Getting to Know Your Data
Data Objects and Attribute Types
Basic Statistical Descriptions of Data
Data Visualization
Measuring Data Similarity and Dissimilarity
Summary

3. Types of Data Sets Record Relational records
Data matrix, e.g., numerical matrix, crosstabs
Document data: text documents: term-frequency vector
Transaction data
Graph and network
World Wide Web
Social or information networks
Molecular Structures
Ordered
Video data: sequence of images
Temporal data: time-series
Sequential Data: transaction sequences
Genetic sequence data
Spatial, image and multimedia:
Spatial data: maps
Image data:
Video data:

4. Data Objects Data sets are made up of data objects.
A data object represents an entity.
Examples:
sales database: customers, store items, sales
medical database: patients, treatments
university database: students, professors, courses
Also called samples , examples, instances, data points, objects, tuples.
Data objects are described by attributes.
Database rows -> data objects; columns ->attributes.

5. Attributes
Attribute (or dimensions, features, variables): a data field, representing a characteristic or feature of a data object.
E.g., customer _ID, name, address
Types:
Nominal
Binary
Ordinal
Numeric
Interval-scaled
Ratio-scaled

6. Attribute Types
Nominal: categories, states, or “names of things”, order not meaningful
Hair_color = {auburn, black, blond, brown, grey, red, white}
Binary
Nominal attribute with only 2 states (0 and 1)
Symmetric binary: both outcomes equally important
e.g., gender
Asymmetric binary: outcomes not equally important.
e.g., medical test (positive vs. negative)
Convention: assign 1 to most important outcome (e.g., HIV positive)
Ordinal
Values have a meaningful order (ranking) but magnitude between successive values is not known.
Size = {small, medium, large}, grades, army rankings

7. Numeric Attribute Types
Quantity (integer or real-valued)
Interval
Measured on a scale of equal-sized units
Values have order
E.g., temperature in C˚or F˚, calendar dates
No true zero-point
Ratio
Inherent zero-point
We can speak of values as being an order of magnitude larger than the unit of measurement (10 K˚ is twice as high as 5 K˚).
e.g., temperature in Kelvin, length, counts, monetary quantities

8. Discrete vs. Continuous Attributes
Discrete Attribute
Has only a finite or countably infinite set of values
E.g., zip codes, profession, or the set of words in a collection of documents
Continuous Attribute
Has real numbers as attribute values
E.g., temperature, height, or weight
Continuous attributes are typically represented as floating-point variables

9. Chapter 2: Getting to Know Your Data
Data Objects and Attribute Types
Basic Statistical Descriptions of Data
Data Visualization
Measuring Data Similarity and Dissimilarity
Summary

10. Basic Statistical Descriptions of Data
Measures of central tendency
Mean, median, mode
Dispersion of data
range, quartiles and interquartile range, five-number summary and boxplots, variance and standard deviation

11. Measuring the Central Tendency
Mean:
Note: n is sample size and N is population size.
Weighted arithmetic mean:
Trimmed mean: chopping extreme values
Median:
Middle value if odd number of values, or average of the middle two values otherwise
Estimated by interpolation (for grouped data):
Mode
Value that occurs most frequently in the data
Unimodal, bimodal, trimodal
Empirical formula:
Median interval

12. Symmetric vs. Skewed Data
Median, mean and mode of symmetric, positively and negatively skewed data
symmetric
positively skewed
negatively skewed
April 17, 2017
Data Mining: Concepts and Techniques

13. Measuring the Dispersion of Data
Quartiles: Q1 (25th percentile), Q3 (75th percentile)
Inter-quartile range: IQR = Q3 – Q1
Five number summary: min, Q1, median, Q3, max
Boxplot: A simple graphical device to display the overall shape of a distribution, including the outliers. Ends of the box are the quartiles; median is marked within the box; add whiskers to mark min and max, and plot outliers individually
Outlier: Values less than Q1-1.5*IQR and greater than Q3+1.5*IQR are outliers

14. Boxplot Analysis Draw a box plot for the following dataset.
10.2, 14.1, , 14.4, 14.5, 14.5, 14.6, 14.7, 14.7, 14.7, 14.9, 15.1, 15.9,
Here,
Q2(median) = 14.6
Q1 = 14.4
Q3 = 14.9
IQR = Q3 – Q1 = = 0.5
Outliers will be any points below Q1 – 1.5×IQR = 14.4 – 0.75 = or above Q ×IQR = =
So, the outliers are at 10.2, 15.9, and 16.4.
The ends of the box are at 14.4 and 14.9.
The median 14.6 is marked within the box.
The whiskers extend to 14.1 and 15.1.
The outliers 10.2, 15.9, and 16.4 are plotted individually.

15. Measuring the Dispersion of Data
Variance and standard deviation
Variance:
Standard deviation σ is the square root of variance σ2

16. Example of Standard Deviation
Find out the Mean, the Variance, and the Standard Deviation of the following dataset.
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Here, the mean = 7
Using the formula for variance,
σ2 = 8.9
Therefore, the standard deviation is σ = 2.98

17. Graphic Displays of Basic Statistical Descriptions
Boxplot: graphic display of five-number summary
Histogram: x-axis are values, y-axis represents frequencies
Quantile plot: each value xi is paired with fi indicating that approximately fi * 100% of data are below the value xi
Quantile-quantile (q-q) plot: graphs the quantiles of one univariant distribution against the corresponding quantiles of another
Scatter plot: each pair of values is a pair of coordinates and plotted as points in the plane

18. Histogram Example

19. Quantile Plot Used to check whether your data is normal
To make a quantile plot:
If the data distribution is close to normal, the plotted points will lie close to a slopped straight line

20. Quantile Plot Example
Create a quantile plot for the following dataset.
3, 5, 1, 4, 10
First sort the data: 1, 3, 4, 5, 10
Calculate the sample quantiles: 0.1, 0.3, 0.5, 0.7, 0.9
Plot the graph

21. Scatter Plot

22. Positively and Negatively Correlated Data

23. Uncorrelated Data

24. Chapter 2: Getting to Know Your Data
Data Objects and Attribute Types
Basic Statistical Descriptions of Data
Data Visualization
Measuring Data Similarity and Dissimilarity
Summary

25. Data Visualization Why data visualization? Gain insight into the data
Search for patterns, trends, structure, irregularities, relationships among data
Categorization of visualization methods:
Pixel-oriented visualization techniques
Geometric projection visualization techniques
Icon-based visualization techniques
Hierarchical visualization techniques
Visualizing complex data and relations

26. Pixel-Oriented Visualization Techniques
For a data set of m dimensions, create m windows on the screen, one for each dimension
The m dimension values of a record are mapped to m pixels at the corresponding positions in the windows
The colors of the pixels reflect the corresponding values
Income
(b) Credit Limit
(c) transaction volume
(d) age 26 26

27. Geometric Projection Visualization Techniques
Visualization of geometric transformations and projections of the data
Helps users find interesting projections of multidimensional data
Methods
Scatterplot and scatterplot matrices

28. Icon-Based Visualization Techniques
Visualization of the data values as features of icons
Typical visualization methods
Chernoff Faces
Stick Figures

29. Chernoff Faces
A way to display multidimensional data of up to 18 dimensions as a cartoon human face
Components of the faces such as eyes, ears, mouth, and nose represent values of the dimensions by their shape, size, placement and orientation

30. Data Mining: Concepts and Techniques
Stick Figure
A 5-piece stick figure (1 body and 4 limbs)
Two attributes mapped to the two axes, remaining attributes mapped to angle or length of limbs
A census data figure showing age, income, gender, education, etc.
used by permission of G. Grinstein, University of Massachusettes at Lowell
Data Mining: Concepts and Techniques

31. Hierarchical Visualization Techniques
Visualization of the data using a hierarchical partitioning into subspaces instead of visualizing all dimensions at the same time
Methods
Dimensional Stacking
Worlds-within-Worlds
Tree-Map
Cone Trees
InfoCube

32. Dimensional Stacking
Used by permission of M. Ward, Worcester Polytechnic Institute
Visualization of oil mining data with longitude and latitude mapped to the outer x-, y-axes and ore grade and depth mapped to the inner x-, y-axes

33. Visualizing Complex Data and Relations
Visualizing non-numerical data: text and social networks
Tag cloud: visualizing user-generated tags
The importance of tag is represented by font size/color
Newsmap: Google News Stories in 2005

34. Chapter 2: Getting to Know Your Data
Data Objects and Attribute Types
Basic Statistical Descriptions of Data
Data Visualization
Measuring Data Similarity and Dissimilarity
Summary

35. Similarity and Dissimilarity
Numerical measure of how alike two data objects are
Value is higher when objects are more alike
Often falls in the range [0,1]
Dissimilarity
Numerical measure of how different two data objects are
Lower when objects are more alike
Two data structures commonly used to measure the above
Data matrix
Dissimilarity matrix

36. Data Matrix and Dissimilarity Matrix
Stores n data points with p dimensions
Two modes – stores both objects and attributes
Dissimilarity matrix
Stores n data points, but registers only the dissimilarity between objects i and j
A triangular matrix
Single mode as it only stores dissimilarity values

37. Proximity Measure for Nominal Attributes
Can take 2 or more states, e.g., map_color may have attributes red, yellow, blue and green
Dissimilarity can be computed based on the ratio of mismatches
m: # of matches, p: total # of attributes

38. Dissimilarity between Nominal Attributes
Here, we have one nominal attribute, test-1, so p=1.
The dissimilarity matrix is as shown below:

39. Proximity Measure for Binary Attributes
Object j
A contingency table for binary data
Dissimilarity for symmetric binary attributes:
Dissimilarity for asymmetric binary attributes :
Jaccard coefficient (similarity measure for asymmetric binary attributes):
Object i

40. Dissimilarity between Binary Variables
Example
Gender is a symmetric attribute, others are asymmetric binary
Let the values Y and P be 1, and the value N 0
Suppose the distance between patients is computed based only on the asymmetric attributes

41. Distance on Numeric Data: Minkowski Distance
Minkowski distance: A popular distance measure
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p- dimensional data objects, and h is the order (the distance so defined is also called L-h norm)
Properties
d(i, j) > 0 if i ≠ j, and d(i, i) = 0 (Positive definiteness)
d(i, j) = d(j, i) (Symmetry)
d(i, j)  d(i, k) + d(k, j) (Triangle Inequality)
A distance that satisfies these properties is a metric

42. Special Cases of Minkowski Distance
h = 1: Manhattan distance
h = 2: Euclidean distance
h  . supremum distance
This is the maximum difference between any component (attribute) of the vectors

43. Example: Minkowski Distance
Dissimilarity Matrices
Manhattan (L1)
Euclidean (L2)
Supremum

44. Proximity Measures for Ordinal Variables
Let M represent the number of possible values that an ordinal attribute can have
replace each xif by its corresponding rank
map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by
compute the dissimilarity using any of the distance measures for numeric attributes, e.g., Euclidean distance

45. Proximity Measures for Ordinal Variables
Consider the data in the adjacent table:
Here, the attribute Test has three states: fair, good and excellent, so Mf=3
For step 1, the four attribute values are assigned the ranks 3,1,2 and 3 respectively.
Step 2 normalizes the ranking by mapping rank 1 to 0.0, rank 2 to 0.5 and rank 3 to 1.0
For step 3, using Euclidean distance, a dissimilarity matrix is obtained as shown
Therefore, students 1 and 2 are most dissimilar, as are students 2 and 4
Student
Test 1
Excellent 2
Fair 3
Good 4

46. Attributes of Mixed Type
A database may contain all attribute types : Nominal, symmetric binary, asymmetric binary, numeric, ordinal
One may use a weighted formula to combine their effects
For nominal and ordinal attributes, use the technique mentioned earlier to compute dissimilarity matrix
For numeric attributes use the following formula to calculate dissimilarity

47. Attributes of Mixed Type - Example
Consider the data in the table:
The dissimilarity matrices for the nominal and ordinal data are shown to the right computed using the methods discussed before
To compute dissimilarity matrix for the numeric attribute, maxhxh=64, minhxh=22. Using the formula from previous slide, the dissimilarity matrix is obtained as shown below:

48. Attributes of Mixed Type - Example
The three dissimilarity matrices can now be used to compute the overall dissimilarity between two objects using the equation
The resulting dissimilarity matrix is

49. Cosine Similarity
Cosine similarity is a measure of similarity that can be used to compare documents.
A document can be represented by thousands of attributes, each recording the frequency of a particular word (such as keywords) or phrase in the document called term-frequency vector.
Cosine measure: If d1 and d2 are two term-frequency vectors, then
cos(x, y) = (x  y) /||x|| ||y|| ,
where  indicates dot product, ||x|| is the length of vector x defined as
. Similarly, ||y|| is the length of vector y.
A cosine value of 0 means that the two vectors are at 90 degrees to each other and there is no match.
The closer the cosine value to 1, the greater the match between the vectors

50. Example: Cosine Similarity
Ex: Find the similarity between documents 1 and 2 from previous slide.
d1 = (5, 0, 3, 0, 2, 0, 0, 2, 0, 0)
d2 = (3, 0, 2, 0, 1, 1, 0, 1, 0, 1)
Using the formula, cos(d1, d2) = (d1  d2) /||d1|| ||d2|| ,
d1d2 = 5*3+0*0+3*2+0*0+2*1+0*1+0*1+2*1+0*0+0*1 = 25
||d1||= (5*5+0*0+3*3+0*0+2*2+0*0+0*0+2*2+0*0+0*0)0.5=(42)0.5 = 6.481
||d2||= (3*3+0*0+2*2+0*0+1*1+1*1+0*0+1*1+0*0+1*1)0.5=(17) = 4.12
cos(d1, d2 ) = 0.94
The cosine similarity shows that the two documents are quite similar.

51. Chapter 2: Getting to Know Your Data
Data Objects and Attribute Types
Basic Statistical Descriptions of Data
Data Visualization
Measuring Data Similarity and Dissimilarity
Summary

52. Summary
Data attribute types: nominal, binary, ordinal, interval-scaled, ratio-scaled
Many types of data sets, e.g., numerical, text, graph, image, etc.
Gain insight into the data by:
Basic statistical data description: central tendency, dispersion, graphical displays
Data visualization:
Measure data similarity
Above steps are the beginning of data preprocessing

53. Exercise 1 of 2
Find the dissimilarity value between Alyssa and Chris and between Alyssa and Diane.
Student
Gender
HairColor
Test1
Test2
Alyssa F
Black
Excellent 80
Chris M
Good 85
Jessica
Brown
Fair 55
Diane
Blonde

54. Exercise 2 of 2
Given two objects represented by the tuples (22, 1, 42, 10) and (20, 0, 36, 8), compute the:
Euclidean distance
Manhattan distance
Supremum distance
Cosine similarity