15 Sep What is an attribute and note the importance? What are the different types of attributes? What is the differenc
After completing the reading this week answer the following questions:
Chapter 2:
- What is an attribute and note the importance?
- What are the different types of attributes?
- What is the difference between discrete and continuous data?
- Why is data quality important?
- What occurs in data preprocessing?
- In section 2.4, review the measures of similarity and dissimilarity, select one topic and note the key factors.
Data Mining: Data
Lecture Notes for Chapter 2
Introduction to Data Mining
by
Tan, Steinbach, Kumar
What is Data?
- Collection of data objects and their attributes
- An attribute is a property or characteristic of an object
- Examples: eye color of a person, temperature, etc.
- Attribute is also known as variable, field, characteristic, or feature
- A collection of attributes describe an object
- Object is also known as record, point, case, sample, entity, or instance
Attributes
Objects
Attribute Values
- Attribute values are numbers or symbols assigned to an attribute
- 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
ID has no limit but age has a maximum and minimum value
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 in {tall, medium, short}
- Interval
- Examples: calendar dates, temperatures in Celsius or Fahrenheit.
- Ratio
- Examples: temperature in Kelvin, length, time, counts
Properties of Attribute Values
- The type of an attribute depends on which of the following properties it possesses:
- Distinctness: =
- Order: < >
- Addition: + –
- Multiplication: * /
- Nominal attribute: distinctness
- Ordinal attribute: distinctness & order
- Interval attribute: distinctness, order & addition
- Ratio attribute: all 4 properties
Attribute Type
Description
Examples
Operations
Nominal
The values of a nominal attribute are just different names, i.e., nominal attributes provide only enough information to distinguish one object from another. (=, )
zip codes, employee ID numbers, eye color, sex: {male, female}
mode, entropy, contingency correlation, 2 test
Ordinal
The values of an ordinal attribute provide enough information to order objects. (<, >)
hardness of minerals, {good, better, best},
grades, street numbers
median, percentiles, rank correlation, run tests, sign tests
Interval
For interval attributes, the differences between values are meaningful, i.e., a unit of measurement exists.
(+, – )
calendar dates, temperature in Celsius or Fahrenheit
mean, standard deviation, Pearson's correlation, t and F tests
Ratio
For ratio variables, both differences and ratios are meaningful. (*, /)
temperature in Kelvin, monetary quantities, counts, age, mass, length, electrical current
geometric mean, harmonic mean, percent variation
Attribute Level
Transformation
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
Thus, the Fahrenheit and Celsius temperature scales differ in terms of where their zero value is and the size of a unit (degree).
Ratio
new_value = a * old_value
Length can be measured in meters or feet.
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 floating-point variables.
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
Important Characteristics of Structured Data
- Dimensionality
- Curse of Dimensionality
- Sparsity
- Only presence counts
- Resolution
- Patterns depend on the scale
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 |
60K |
No |
|
7 |
Yes |
Divorced |
220K |
No |
|
8 |
No |
Single |
85K |
Yes |
|
9 |
No |
Married |
75K |
No |
|
10 |
No |
Single |
90K |
Yes |
10
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
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.
Document 1�
season�
timeout�
lost�
win�
game�
score�
ball�
play�
coach�
team�
Document 2�
Document 3�
3�
0�
5�
0�
2�
6�
0�
2�
0�
2�
0�
0�
7�
0�
2�
1�
0�
0�
3�
0�
0�
1�
0�
0�
1�
2�
2�
0�
3�
0�
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.
Graph Data
- Examples: Generic graph and HTML Links
Ordered Data
- Sequences of transactions
An element of the sequence
Items/Events
Ordered Data
- Genomic sequence data
Ordered Data
- Spatio-Temporal Data
Average Monthly Temperature of land and ocean
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
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
- Estimate Missing Values
- Ignore the Missing Value During Analysis
- Replace with all possible values (weighted by their probabilities)
Duplicate Data
- Data set may include data objects that are duplicates, or almost duplicates of one another
- Major issue when merging data from heterogeous sources
- Examples:
- Same person with multiple email addresses
- Data cleaning
- Process of dealing with duplicate data issues
Data Preprocessing
- Aggregation
- Sampling
- Dimensionality Reduction
- Feature subset selection
- Feature creation
- Discretization and Binarization
- Attribute Transformation
Aggregation
- Combining two or more attributes (or objects) into a single attribute (or object)
- Purpose
- Data reduction
- Reduce the number of attributes or objects
- Change of scale
- Cities aggregated into regions, states, countries, etc
- More “stable” data
- Aggregated data tends to have less variability
Sampling
- Sampling is the main technique employed for data selection.
- It is often used for both the preliminary investigation of the data and the final data analysis.
- Statisticians sample because obtaining the entire set of data of interest is too expensive or time consuming.
- Sampling is used in data mining because processing the entire set of data of interest is too expensive or time consuming.
Sampling …
- The key principle for effective sampling is the following:
- using a sample will work almost as well as using the entire data sets, if the sample is representative
- A sample is representative if it has approximately the same property (of interest) as the original set of data
Types of Sampling
- Simple Random Sampling
- There is an equal probability of selecting any particular item
- Sampling without replacement
- As each item is selected, it is removed from the population
- Sampling with replacement
- Objects are not removed from the population as they are selected for the sample.
- In sampling with replacement, the same object can be picked up more than once
- Stratified sampling
- Split the data into several partitions; then draw random samples from each partition
Dimensionality Reduction
- Purpose:
- Avoid curse of dimensionality
- Reduce amount of time and memory required by data mining algorithms
- Allow data to be more easily visualized
- May help to eliminate irrelevant features or reduce noise
- Techniques
- Principle Component Analysis
- Singular Value Decomposition
- Others: supervised and non-linear techniques
Feature Subset Selection
- Another way to reduce dimensionality of data
- Redundant features
- duplicate much or all of the information contained in one or more other attributes
- Example: purchase price of a product and the amount of sales tax paid
- Irrelevant features
- contain no information that is useful for the data mining task at hand
- Example: students' ID is often irrelevant to the task of predicting students' GPA
Feature Subset Selection
- Techniques:
- Brute-force approch:
- Try all possible feature subsets as input to data mining algorithm
- Embedded approaches:
- Feature selection occurs naturally as part of the data mining algorithm
- Filter approaches:
- Features are selected before data mining algorithm is run
- Wrapper approaches:
- Use the data mining algorithm as a black box to find best subset of attributes
Feature Creation
- Create new attributes that can capture the important information in a data set much more efficiently than the original attributes
- Three general methodologies:
- Feature Extraction
- domain-specific
- Mapping Data to New Space
- Feature Construction
- combining features
Similarity and Dissimilarity
- Similarity
- Numerical measure of how alike two data objects are.
- Is higher when objects are more alike.
- Often falls in the range [0,1]
- Dissimilarity
- Numerical measure of how different are two data objects
- Lower when objects are more alike
- Minimum dissimilarity is often 0
- Upper limit varies
- Proximity refers to a similarity or dissimilarity
Similarity/Dissimilarity for Simple Attributes
p and q are the attribute values for two data objects.
Euclidean Distance
- Euclidean Distance
Where n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q.
- Standardization is necessary, if scales differ.
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.
Common Properties of a Distance
- Distances, such as the Euclidean distance, have some well known properties.
d(p, q) 0 for all p and q and d(p, q) = 0 only if
p = q. (Positive definiteness)
d(p, q) = d(q, p) for all p and q. (Symmetry)
d(p, r) d(p, q) + d(q, r) for all points p, q, and r.
(Triangle Inequality)
where d(p, q) is the distance (dissimilarity) between points (data objects), p and q.
- A distance that satisfies these properties is a metric
Common Properties of a Similarity
- Similarities, also have some well known properties.
s(p, q) = 1 (or maximum similarity) only if p = q.
s(p, q) = s(q, p) for all p and q. (Symmetry)
where s(p, q) is the similarity between points (data objects), p and q.
Similarity Between Binary Vectors
- Common situation is that objects, p and q, have only binary attributes
- Compute similarities using the following quantities
M01 = the number of attributes where p was 0 and q was 1
M10 = the number of attributes where p was 1 and q was 0
M00 = the number of attributes where p was 0 and q was 0
M11 = the number of attributes where p was 1 and q was 1
- Simple Matching and Jaccard Coefficients
SMC = number of matches / number of attributes
= (M11 + M00) / (M01 + M10 + M11 + M00)
J = number of 11 matches / number of not-both-zero attributes values
= (M11) / (M01 + M10 + M11)
SMC versus Jaccard: Example
p = 1 0 0 0 0 0 0 0 0 0
q = 0 0 0 0 0 0 1 0 0 1
M01 = 2 (the number of attributes where p was 0 and q was 1)
M10 = 1 (the number of attributes where p was 1 and q was 0)
M00 = 7 (the number of attributes where p was 0 and q was 0)
M11 = 0 (the number of attributes where p was 1 and q was 1)
SMC = (M11 + M00)/(M01 + M10 + M11 + M00) = (0+7) / (2+1+0+7) = 0.7
J = (M11) / (M01 + M10 + M11) = 0 / (2 + 1 + 0) = 0
Cosine Similarity
- If d1 and d2 are two document vectors, then
cos( d1, d2 ) = (d1 d2) / ||d1|| ||d2|| ,
where indicates vector dot product and || d || is the length of vector d.
- Example:
d1 = 3 2 0 5 0 0 0 2 0 0
d2 = 1 0 0 0 0 0 0 1 0 2
d1 d2= 3*1 + 2*0 + 0*0 + 5*0 + 0*0 + 0*0 + 0*0 + 2*1 + 0*0 + 0*2 = 5
||d1|| = (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5 = (42) 0.5 = 6.481
||d2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2) 0.5 = (6) 0.5 = 2.245
cos( d1, d2 ) = .3150
Correlation
- Correlation measures the linear relationship between objects
- To compute correlation, we standardize data objects, p and q, and then take their dot product
General Approach for Combining Similarities
- Sometimes attributes are of many different types, but an overall similarity is needed.
Density
- Density-based clustering require a notion of density
- Examples:
- Euclidean density
- Euclidean density = number of points per unit volume
- Probability density
- Graph-based density
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
60K
No
7
Yes
Divorced
220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Singl
e
90K
Yes
10
1.1
2.2
16.22
6.25
12.65
1.2
2.7
15.22
5.27
10.23
Thickness
Load
Distance
Projection
of y load
Projection
of x Load
1.1
2.2
16.22
6.25
12.65
1.2
2.7
15.22
5.27
10.23
Thickness
Load
Distance
Projection
of y load
Projection
of x Load
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