discussion- Chapter 2 q&a- chapter 3 q&a

Data Mining: Exploring Data

Lecture Notes for Chapter 3
Slides by Tan, Steinbach, Kumar adapted by Michael Hahsler

Look for accompanying R
code on the course web site.

© Tan,Steinbach, Kumar Introduction to Data Mining 8/05/2005 ‹#›

Techniques Used In Data Exploration

 In EDA, as originally defined by Tukey
– The focus was on visualization
– Clustering and anomaly detection were viewed as

exploratory techniques
– In data mining, clustering and anomaly detection are

major areas of interest, and not thought of as just
exploratory

 In our discussion of data exploration, we focus on
– Summary statistics
– Visualization
– Online Analytical Processing (OLAP)

What is data exploration?

• Key motivations of data exploration include
– Helping to select the right tool for preprocessing or analysis
– Making use of humans’ abilities to recognize patterns

• People can recognize patterns not captured by data analysis
tools

• Related to the area of Exploratory Data Analysis (EDA)
– Created by statistician John Tukey
– Seminal book is “Exploratory Data Analysis” by Tukey
– A nice online introduction can be found in Chapter 1 of the NIST

Engineering Statistics Handbook
http://www.itl.nist.gov/div898/handbook/index.htm

A preliminary exploration of the data to
better understand its characteristics.

Iris Sample Data Set

• Many of the exploratory data techniques are illustrated
with the Iris Plant data set.
– Can be obtained from the UCI Machine Learning Repository

http://www.ics.uci.edu/~mlearn/MLRepository.html

– From the statistician R.A. Fisher
– Three flower types (classes):

• Setosa
• Virginica
• Versicolour

– Four (non-class) attributes
• Sepal width and length
• Petal width and length Virginica. Robert H. Mohlenbrock. USDA

NRCS. 1995. Northeast wetland flora: Field
office guide to plant species. Northeast
National Technical Center, Chester, PA.
Courtesy of USDA NRCS Wetland Science
Institute.

Summary Statistics

• Summary statistics are numbers that summarize
properties of the data

– Summarized properties include location and
spread for continuous data

• Examples: location – mean
spread – standard deviation

-Most summary statistics can be calculated in a
single pass through the data

Frequency and Mode

• The frequency of an attribute value is the
percentage of time the value occurs in the
data set
– For example, given the attribute ‘gender’ and a

representative population of people, the gender
‘female’ occurs about 50% of the time.

• The mode of an attribute is the most frequent attribute
value

• The notions of frequency and mode are typically used
with categorical data

Measures of Location: Mean and Median

• The mean is the most common measure of the
location of a set of points.

• However, the mean is very sensitive to outliers.
• Thus, the median or a trimmed mean is also

commonly used.

Measures of Spread: Range and Variance

• Range is the difference between the max and min
• The variance or standard deviation is the most

common measure of the spread of a set of points.

• However, this is also sensitive to outliers, so that
other measures are often used.

Percentiles

x p
x p

Median – 50% of the
cases has a smaller value & 50%

are larger

Multivariate Summary Statistics
Object x

1
x
2

1 12 15
2 2 4
… … …
m 18 4

• Covariance between
features i and j

• Correlation

si is the variance of
feature i

sij=
1

m−1∑k=1
m

( xki− x̄i)(xkj− x̄ j)

r ij=
sij
si s j

Topics

• Exploratory Data Analysis
• Summary Statistics
• Visualization

Visualization

Visualization is the conversion of data into a visual
or tabular format so that the characteristics of the
data and the relationships among data items or
attributes can be analyzed or reported.

• Visualization of data is one of the most powerful and
appealing techniques for data exploration.
– Humans have a well developed ability to analyze

large amounts of information that is presented
visually
– Can detect general patterns and trends
– Can detect outliers and unusual patterns

Example: Sea Surface Temperature
• The following shows the Sea Surface Temperature

(SST) for July 1982
– Tens of thousands of data points are

summarized in a single figure

Representation

• Is the mapping of information to a visual format
• Data objects, their attributes, and the relationships

among data objects are translated into graphical
elements such as points, lines, shapes, and colors.

• Example:
– Objects are often represented as points
– Their attribute values can be represented as the

position of the points or the characteristics of the
points, e.g., color, size, and shape
– If position is used, then the relationships of points,

i.e., whether they form groups or a point is an
outlier, is easily perceived.

Arrangement

• Is the placement of visual elements within a
display

• Can make a large difference in how easy it is to
understand the data

• Example:

Selection

• Is the elimination or the deemphasis of certain objects and
attributes

• Selection may involve the choosing a subset of attributes
– Dimensionality reduction is often used to reduce the

number of dimensions to two or three
– Alternatively, pairs of attributes can be considered

• Selection may also involve choosing a subset of objects
– A region of the screen can only show so many points
– Can sample, but want to preserve points in sparse areas

Histograms

• Usually shows the distribution of values of a single variable
• Divide the values into bins and show a bar plot of the number of objects in each bin.
• The height of each bar indicates the number of objects
• Shape of histogram depends on the number of bins

• Example: Petal Width (10 and 20 bins, respectively)

Empirical Cumulative Distribution
Function (ECDF)

• Probability Density
Function (PDF): describes
the relative likelihood for
this random variable to take
on a given value

• Cumulative Distribution
Function (CDF): Shows
the distribution of data as
the fraction of points that
are less than this value.

CDF

PDF

Example: ECDF

Two-Dimensional Histograms
• Show the joint distribution of the values of two attributes
• Example: petal width and petal length

– What does this tell us?

Box Plots
• Invented by J. Tukey
• Another way of displaying the distribution of data
• Following figure shows the basic part of a box plot

outlier

25th percentile
– 1.5 IQR

25th percentile

75th percentile

50th percentile

75th percentile
+ 1.5 IQR

IQR

Q1 Q3

IQR

Median

Q3 + 1.5 × IQRQ1 − 1.5 × IQR

−0.6745 σ 0.6745 σ 2.698 σ−2.698 σ

50%24.65% 24.65%

68.27% 15.73%15.73%

−4 σ −3 σ −2 σ −1 σ 0 σ 1 σ 3 σ2 σ 4 σ

−4 σ −3 σ −2 σ −1 σ 0 σ 1 σ 3 σ2 σ 4 σ

−4 σ −3 σ −2 σ −1 σ 0 σ 1 σ 3 σ2 σ 4 σ

Example of Box Plots
• Box plots can be used to compare attributes

Scatter Plots

– Attributes values determine the position
– Two-dimensional scatter plots most common,

but can have three-dimensional scatter plots
-Often additional attributes can be displayed by

using the size, shape, and color of the markers
that represent the objects
– It is useful to have arrays of scatter plots can

compactly summarize the relationships of
several pairs of attributes

• See example on the next slide

Scatter Plot Array of Iris Attributes

Contour Plots

– Useful when a continuous attribute is measured

on a spatial grid
– They partition the plane into regions of similar

values
– The contour lines that form the boundaries of

these regions connect points with equal values
– The most common example is contour maps of

elevation
– Can also display temperature, rainfall, air

pressure, etc.
 An example for Sea Surface Temperature (SST) is

provided on the next slide

Contour Plot Example: SST Dec, 1998

Celsius

Matrix Plots

– Can plot a data matrix
– Can be useful when objects are sorted

according to class
– Typically, the attributes are normalized to

prevent one attribute from dominating the plot
– Plots of similarity or distance matrices can also

be useful for visualizing the relationships
between objects

Visualization of the Iris Data Matrix

standard
deviation

Deviation form feature mean

Visualization of the Iris Correlation Matrix

Parallel Coordinates

– Used to plot the attribute values of high-

dimensional data
– Instead of using perpendicular axes, use a set of

parallel axes
– The attribute values of each object are plotted as a

point on each corresponding coordinate axis and
the points are connected by a line

– Thus, each object is represented as a line
– Often, the lines representing a distinct class of

objects group together, at least for some attributes
– Ordering of attributes is important in seeing such

groupings

Parallel Coordinates Plots for Iris Data

Reordered features

Other Visualization Techniques

Star Plots
– Similar approach to parallel coordinates, but axes radiate

from a central point
– The line connecting the values of an object is a polygon

Chernoff Faces
– Approach created by Herman Chernoff
– This approach associates each attribute with a characteristic

of a face
– The values of each attribute determine the appearance of the

corresponding facial characteristic
– Each object becomes a separate face
– Relies on human’s ability to distinguish faces

Star Plots for Iris Data

Setosa

Versicolor

Virginica

Chernoff Faces for Iris Data
Setosa

Versicolor

Virginica

OLAP
• On-Line Analytical Processing (OLAP) was

proposed by E. F. Codd, the father of the
relational database.

• Relational databases put data into tables, while
OLAP uses a multidimensional array
representation.
– Such representations of data previously existed in

statistics and other fields

• There are a number of data analysis and data
exploration operations that are easier with such
a data representation.

Creating a Multidimensional Array
• Two key steps in converting tabular data into a

multidimensional array.
– First, identify which attributes are to be the dimensions

and which attribute is to be the target attribute whose
values appear as entries in the multidimensional array.
• The attributes used as dimensions must have discrete values
• The target value is typically a count or continuous value, e.g.,

the cost of an item
• Can have no target variable at all except the count of objects

that have the same set of attribute values

– Second, find the value of each entry in the
multidimensional array by summing the values (of the
target attribute) or count of all objects that have the
attribute values corresponding to that entry.

Example: Iris data
• We show how the attributes, petal length, petal

width, and species type can be converted to a
multidimensional array
– First, we discretized the petal width and length to

have categorical values: low, medium, and high

– We get the following table – note the count
attribute

Example: Iris data (continued)
• Each unique tuple of petal width, petal length,

and species type identifies one element of the
array.

• This element is assigned the corresponding
count value.

• The figure illustrates
the result.

• All non-specified
tuples are 0.

Example: Iris data (continued)
• Slices of the multidimensional array are shown

by the following cross-tabulations

• What do these tables tell us?

OLAP Operations: Data Cube
• The key operation of a OLAP is the formation of a

data cube
• A data cube is a multidimensional representation

of data, together with all possible aggregates.
• By all possible aggregates, we mean the aggregates

that result by selecting a proper subset of the
dimensions and summing over all remaining
dimensions.

• For example, if we choose the species type
dimension of the Iris data and sum over all other
dimensions, the result will be a one-dimensional
entry with three entries, each of which gives the
number of flowers of each type.

Data Cube Example
• Consider a data set that records the sales of

products at a number of company stores at
various dates.

• This data can be represented
as a 3 dimensional array

• There are 3 two-dimensional
aggregates (3 choose 2 ),
3 one-dimensional aggregates,
and 1 zero-dimensional
aggregate (the overall total)

Data Cube Example (continued)
• The following figure table shows one of the two

dimensional aggregates, along with two of the
one-dimensional aggregates, and the overall
total

OLAP Operations: Slicing and Dicing
• Slicing is selecting a group of cells from the

entire multidimensional array by specifying a
specific value for one or more dimensions.

• Dicing involves selecting a subset of cells by
specifying a range of attribute values.

– This is equivalent to defining a subarray from the
complete array.

• In practice, both operations can also be
accompanied by aggregation over some
dimensions.

OLAP Operations: Roll-up and Drill-down

• Attribute values often have a hierarchical
structure.
– Each date is associated with a year, month, and week.

– A location is associated with a continent, country, state
(province, etc.), and city.

– Products can be divided into various categories, such as
clothing, electronics, and furniture.

• Note that these categories often nest and form a
tree or lattice
– A year contains months which contains day

– A country contains a state which contains a city

OLAP Operations: Roll-up and Drill-down

• This hierarchical structure gives rise to the roll-
up and drill-down operations.

– For sales data, we can aggregate (roll up) the sales
across all the dates in a month.

– Conversely, given a view of the data where the time
dimension is broken into months, we could split the
monthly sales totals (drill down) into daily sales
totals.

– Likewise, we can drill down or roll up on the
location or product ID attributes.