1)For this week, you will be working through the steps of an affinity diagram. Choose one of the following problem statements:Power outages cause downtimeMalicious code causes systems to crash and production lossHardware failure causes data loss on the database serverOnce you pick a statement, generate ideas and brainstorm based on this article: https://asq.org/quality-resources/affinityFor your peer responses, pick 2 and group the ideas based on step 3.2)Assignment (Similarity and distance measures: )Review Chapter 2-recording, and Chapter-2 text and answer the following question.Submit the answer in a Word document. (Note: this is NOT a group assignment).Compute the Hamming distance and the Jaccard similarity between the following two binary vectors:x=0101010001y=0100011000
chap2_data.pptx
Unformatted Attachment Preview
Data Mining: Data
Lecture Notes for Chapter 2
Introduction to Data Mining , 2nd Edition
by
Tan, Steinbach, Karpatne, Kumar
01/22/2018
Introduction to Data Mining, 2nd Edition
1
Outline
Attributes and Objects
Types of Data
Data Quality
Similarity and Distance
Data Preprocessing
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What is Data?
Attributes
Collection of data objects
and their attributes
– Examples: eye color of a
person, temperature, etc.
– Attribute is also known as
variable, field, characteristic,
dimension, or feature
Objects
An attribute is a property
or characteristic of an
object
A collection of attributes
describe an object
– Object is also known as
record, point, case, sample,
entity, or instance
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
Single
90K
Yes
10 No
10
60K
A More Complete View of Data
Data may have parts
The different parts of the data may have
relationships
More generally, data may have structure
Data can be incomplete
We will discuss this in more detail later
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Attribute Values
Attribute values are numbers or symbols
assigned to an attribute for a particular object
Distinction between attributes and attribute values
– Same attribute can be mapped to different attribute
values
◆
Example: height can be measured in feet or meters
– Different attributes can be mapped to the same set of
values
Example: Attribute values for ID and age are integers
◆ But properties of attribute values can be different
◆
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Measurement of Length
The way you measure an attribute may not match the
attributes properties.
5
A
1
B
7
This scale
preserves
only the
ordering
property of
length.
2
C
8
3
D
10
4
E
15
5
This scale
preserves
the ordering
and additvity
properties of
length.
Types of Attributes
There are different types of attributes
– Nominal
◆
Examples: ID numbers, eye color, zip codes
– Ordinal
◆
Examples: rankings (e.g., taste of potato chips on a
scale from 1-10), grades, height {tall, medium, short}
– Interval
◆
Examples: calendar dates, temperatures in Celsius or
Fahrenheit.
– Ratio
◆
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Examples: temperature in Kelvin, length, time, counts
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Properties of Attribute Values
The type of an attribute depends on which of the
following properties/operations it possesses:
– Distinctness:
=
– Order:
< >
– Differences are
+ meaningful :
– Ratios are
meaningful
* /
– Nominal attribute: distinctness
– Ordinal attribute: distinctness & order
– Interval attribute: distinctness, order & meaningful
differences
– Ratio attribute: all 4 properties/operations
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Difference Between Ratio and Interval
Is it physically meaningful to say that a
temperature of 10 ° is twice that of 5° on
– the Celsius scale?
– the Fahrenheit scale?
– the Kelvin scale?
Consider measuring the height above average
– If Bill’s height is three inches above average and
Bob’s height is six inches above average, then would
we say that Bob is twice as tall as Bill?
– Is this situation analogous to that of temperature?
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Categorical
Qualitative
Attribute Description
Type
Nominal
Nominal attribute
values only
distinguish. (=, )
zip codes, employee
ID numbers, eye
color, sex: {male,
female}
Ordinal
Ordinal attribute
values also order
objects.
(<, >)
For interval
attributes,
differences between
values are
meaningful. (+, – )
For ratio variables,
both differences and
ratios are
meaningful. (*, /)
hardness of minerals,
{good, better, best},
grades, street
numbers
calendar dates,
temperature in
Celsius or Fahrenheit
Interval
Numeric
Quantitative
Examples
Ratio
Operations
mode, entropy,
contingency
correlation, 2
test
median,
percentiles, rank
correlation, run
tests, sign tests
mean, standard
deviation,
Pearson’s
correlation, t and
F tests
temperature in Kelvin, geometric mean,
monetary quantities,
harmonic mean,
counts, age, mass,
percent variation
length, current
This categorization of attributes is due to S. S. Stevens
Numeric
Quantitative
Categorical
Qualitative
Attribute Transformation
Type
Comments
Nominal
Any permutation of values
If all employee ID numbers
were reassigned, would it
make any difference?
Ordinal
An order preserving change of
values, i.e.,
new_value = f(old_value)
where f is a monotonic function
An attribute encompassing
the notion of good, better best
can be represented equally
well by the values {1, 2, 3} or
by { 0.5, 1, 10}.
Interval
new_value = a * old_value + b
where a and b are constants
Ratio
new_value = a * old_value
Thus, the Fahrenheit and
Celsius temperature scales
differ in terms of where their
zero value is and the size of a
unit (degree).
Length can be measured in
meters or feet.
This categorization of attributes is due to S. S. Stevens
Discrete and Continuous Attributes
Discrete Attribute
– Has only a finite or countably infinite set of values
– Examples: zip codes, counts, or the set of words in a
collection of documents
– Often represented as integer variables.
– Note: binary attributes are a special case of discrete
attributes
Continuous Attribute
– Has real numbers as attribute values
– Examples: temperature, height, or weight.
– Practically, real values can only be measured and
represented using a finite number of digits.
– Continuous attributes are typically represented as floatingpoint variables.
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Asymmetric Attributes
Only presence (a non-zero attribute value) is regarded as
important
◆
◆
Words present in documents
Items present in customer transactions
If we met a friend in the grocery store would we ever say the
following?
“I see our purchases are very similar since we didn’t buy most of the
same things.”
We need two asymmetric binary attributes to represent one
ordinary binary attribute
– Association analysis uses asymmetric attributes
Asymmetric attributes typically arise from objects that are
sets
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Some Extensions and Critiques
Velleman, Paul F., and Leland Wilkinson. “Nominal,
ordinal, interval, and ratio typologies are misleading.” The
American Statistician 47, no. 1 (1993): 65-72.
Mosteller, Frederick, and John W. Tukey. “Data analysis
and regression. A second course in statistics.” AddisonWesley Series in Behavioral Science: Quantitative
Methods, Reading, Mass.: Addison-Wesley, 1977.
Chrisman, Nicholas R. “Rethinking levels of measurement
for cartography.”Cartography and Geographic Information
Systems 25, no. 4 (1998): 231-242.
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Critiques
Incomplete
– Asymmetric binary
– Cyclical
– Multivariate
– Partially ordered
– Partial membership
– Relationships between the data
Real data is approximate and noisy
– This can complicate recognition of the proper attribute type
– Treating one attribute type as another may be approximately
correct
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Critiques …
Not a good guide for statistical analysis
– May unnecessarily restrict operations and results
◆
Statistical analysis is often approximate
◆
Thus, for example, using interval analysis for ordinal values
may be justified
– Transformations are common but don’t preserve
scales
◆
Can transform data to a new scale with better statistical
properties
◆
Many statistical analyses depend only on the distribution
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More Complicated Examples
ID numbers
– Nominal, ordinal, or interval?
Number of cylinders in an automobile engine
– Nominal, ordinal, or ratio?
Biased Scale
– Interval or Ratio
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Key Messages for Attribute Types
The types of operations you choose should be
“meaningful” for the type of data you have
– Distinctness, order, meaningful intervals, and meaningful ratios
are only four properties of data
– The data type you see – often numbers or strings – may not
capture all the properties or may suggest properties that are not
there
– Analysis may depend on these other properties of the data
◆
Many statistical analyses depend only on the distribution
– Many times what is meaningful is measured by statistical
significance
– But in the end, what is meaningful is measured by the domain
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Types of data sets
Record
– Data Matrix
– Document Data
– Transaction Data
Graph
– World Wide Web
– Molecular Structures
Ordered
–
–
–
–
Spatial Data
Temporal Data
Sequential Data
Genetic Sequence Data
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Important Characteristics of Data
– Dimensionality (number of attributes)
◆
High dimensional data brings a number of challenges
– Sparsity
◆
Only presence counts
– Resolution
◆
Patterns depend on the scale
– Size
◆
Type of analysis may depend on size of data
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Record Data
Data that consists of a collection of records, each
of which consists of a fixed set of attributes
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
10
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Data Matrix
If data objects have the same fixed set of numeric
attributes, then the data objects can be thought of as
points in a multi-dimensional space, where each
dimension represents a distinct attribute
Such data set can be represented by an m by n matrix,
where there are m rows, one for each object, and n
columns, one for each attribute
Projection
of x Load
Projection
of y load
Distance
Load
Thickness
10.23
5.27
15.22
2.7
1.2
12.65
6.25
16.22
2.2
1.1
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Document Data
Each document becomes a ‘term’ vector
– Each term is a component (attribute) of the vector
– The value of each component is the number of times
the corresponding term occurs in the document.
team
coach
play
ball
score
game
win
lost
timeout
season
Document 1
3
0
5
0
2
6
0
2
0
2
Document 2
0
7
0
2
1
0
0
3
0
0
Document 3
0
1
0
0
1
2
2
0
3
0
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Transaction Data
A special type of record data, where
– Each record (transaction) involves a set of items.
– For example, consider a grocery store. The set of
products purchased by a customer during one
shopping trip constitute a transaction, while the
individual products that were purchased are the items.
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TID
Items
1
Bread, Coke, Milk
2
3
4
5
Beer, Bread
Beer, Coke, Diaper, Milk
Beer, Bread, Diaper, Milk
Coke, Diaper, Milk
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Graph Data
Examples: Generic graph, a molecule, and webpages
2
1
5
2
5
Benzene Molecule: C6H6
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Ordered Data
Sequences of transactions
Items/Events
An element of
the sequence
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Ordered Data
Genomic sequence data
GGTTCCGCCTTCAGCCCCGCGCC
CGCAGGGCCCGCCCCGCGCCGTC
GAGAAGGGCCCGCCTGGCGGGCG
GGGGGAGGCGGGGCCGCCCGAGC
CCAACCGAGTCCGACCAGGTGCC
CCCTCTGCTCGGCCTAGACCTGA
GCTCATTAGGCGGCAGCGGACAG
GCCAAGTAGAACACGCGAAGCGC
TGGGCTGCCTGCTGCGACCAGGG
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Ordered Data
Spatio-Temporal Data
Average Monthly
Temperature of
land and ocean
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Data Quality
Poor data quality negatively affects many data processing
efforts
“The most important point is that poor data quality is an unfolding
disaster.
– Poor data quality costs the typical company at least ten
percent (10%) of revenue; twenty percent (20%) is
probably a better estimate.”
Thomas C. Redman, DM Review, August 2004
Data mining example: a classification model for detecting
people who are loan risks is built using poor data
– Some credit-worthy candidates are denied loans
– More loans are given to individuals that default
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Data Quality …
What kinds of data quality problems?
How can we detect problems with the data?
What can we do about these problems?
Examples of data quality problems:
–
–
–
–
Noise and outliers
Missing values
Duplicate data
Wrong data
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Noise
For objects, noise is an extraneous object
For attributes, noise refers to modification of original values
– Examples: distortion of a person’s voice when talking on a poor
phone and “snow” on television screen
Two Sine Waves
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Two Sine Waves + Noise
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Outliers
Outliers are data objects with characteristics that
are considerably different than most of the other
data objects in the data set
– Case 1: Outliers are
noise that interferes
with data analysis
– Case 2: Outliers are
the goal of our analysis
◆
Credit card fraud
◆
Intrusion detection
Causes?
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Missing Values
Reasons for missing values
– Information is not collected
(e.g., people decline to give their age and weight)
– Attributes may not be applicable to all cases
(e.g., annual income is not applicable to children)
Handling missing values
– Eliminate data objects or variables
– Estimate missing values
Example: time series of temperature
◆ Example: census results
◆
– Ignore the missing value during analysis
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Missing Values …
Missing completely at random (MCAR)
– Missingness of a value is independent of attributes
– Fill in values based on the attribute
– Analysis may be unbiased overall
Missing at Random (MAR)
– Missingness is related to other variables
– Fill in values based other values
– Almost always produces a bias in the analysis
Missing Not at Random (MNAR)
– Missingness is related to unobserved measurements
– Informative or non-ignorable missingness
Not possible to know the situation from the data
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Duplicate Data
Data set may include data objects that are
duplicates, or almost duplicates of one another
– Major issue when merging data from heterogeneous
sources
Examples:
– Same person with multiple email addresses
Data cleaning
– Process of dealing with duplicate data issues
When should duplicate data not be removed?
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Similarity and Dissimilarity Measures
Similarity measure
– Numerical measure of how alike two data objects are.
– Is higher when objects are more alike.
– Often falls in the range [0,1]
Dissimilarity measure
– Numerical measure of how different two data objects
are
– Lower when objects are more alike
– Minimum dissimilarity is often 0
– Upper limit varies
Proximity refers to a similarity or dissimilarity
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Similarity/Dissimilarity for Simple Attributes
The following table shows the similarity and dissimilarity
between two objects, x and y, with respect to a single, simple
attribute.
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Euclidean Distance
Euclidean Distance
where n is the number of dimensions (attributes) and
xk and yk are, respectively, the kth attributes
(components) or data objects x and y.
Standardization is necessary, if scales differ.
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Euclidean Distance
3
point
p1
p2
p3
p4
p1
2
p3
p4
1
p2
0
0
1
2
3
4
5
p1
p1
p2
p3
p4
0
2.828
3.162
5.099
x
0
2
3
5
y
2
0
1
1
6
p2
2.828
0
1.414
3.162
p3
3.162
1.414
0
2
p4
5.099
3.162
2
0
Distance Matrix
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Minkowski Distance
Minkowski Distance is a generalization of Euclidean
Distance
Where r is a parameter, n is the number of dimensions
(attributes) and xk and yk are, respectively, the kth
attributes (components) or data objects x and y.
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Minkowski Distance: Examples
r = 1. City block (Manhattan, taxicab, L1 norm) distance.
– A common example of this is the Hamming distance, which
is just the number of bits that are different between two
binary vectors
r = 2. Euclidean distance
r → . “supremum” (Lmax norm, L norm) distance.
– This is the maximum difference between any component of
the vectors
Do not confuse r with n, i.e., all these distances are
defined for all numbers of dimensions.
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Minkowski Distance
point
p1
p2
p3
p4
x
0
2
3
5
y
2
0
1
1
L1
p1
p2
p3
p4
p1
0
4
4
6
p2
4
0
2
4
p3
4
2
0
2
p4
6
4
2
0
L2
p1
p2
p3
p4
p1
p2
2.828
0
1.414
3.162
p3
3.162
1.414
0
2
p4
5.099
3.162
2
0
L
p1
p2
p3
p4
p1
p2
p3
p4
0
2.828
3.162
5.099
0
2
3
5
2
0
1
3
3
1
0
2
5
3
2
0
Distance Matrix
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Mahalanobis Distance
𝐦𝐚𝐡𝐚𝐥𝐚𝐧𝐨𝐛𝐢𝐬 𝐱, 𝐲 = (𝐱 − 𝐲)𝑇 Ʃ−1 (𝐱 − 𝐲)
is the covariance matrix
For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6.
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Mahalanobis Distance
Covariance
Matrix:
C
0.3 0.2
=
0
.
2
0
.
3
A: (0.5, 0.5)
B
B: (0, 1)
A
C: (1.5, 1.5)
Mahal(A,B) = 5
Mahal(A,C) = 4
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Common Properties of a Distance
Distances, such as the Euclidean distance,
have some well known properties.
1. d(x, y) 0 for all x and y and d(x, y) = 0 only if
x = y. (Positive definiteness)
2. d(x, y) = d(y, x) for all x and y. (Symmetry)
3. d(x, z) d(x, y) + d(y, z) for all points x, y, and z.
(Triangle Inequality)
where d(x, y) is the distance (dissimilarity) between
points (data objects), x and y.
A distance that satisfies these properties is a
metric
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Common Properties of a Similarity
Similarities, also have some well known
properties.
1.
s(x, y) = 1 (or maximum similarity) only if x = y.
2.
s(x, y) = s(y, x) for all x and y. (Symmetry)
where s(x, y) is the similarity between points (data
objects), x and y.
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Similarity Between Binary Vectors
Common situation is that objects, p and q, have only
binary attributes
Compute similarities using the following quantities
f01 = the number of attributes where p was 0 and q was 1
f10 = the number of attributes where p was 1 and q was 0
f00 = the number of attributes where p was 0 and q was 0
f11 = the number of attributes where p was 1 and q was 1
Simple Matching and Jaccard Coefficients
SMC = number of matches / number of attributes
= (f11 + f00) / (f01 + f10 + f11 + f00)
J = number of 11 matches / number of non-zero attributes
= (f11) / (f01 + f10 + f11)
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SMC versus Jaccard: Example
x= 1000000000
y= 0000001001
f01 = 2 (the number of attributes w …
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