1.8 discretization

1
Discretization and Concept
Hierarchy Generation

2
Discretization
 Types of attributes:
 Nominal — values from an unordered set, e.g., color, profession
 Ordinal — values from an ordered set, e.g., military or academic
rank
 Continuous — real numbers, e.g., integer or real numbers
 Discretization:
 Divide the range of a continuous attribute into intervals
 Reduce data size by discretization

3
Discretization and Concept Hierarchy
 Discretization
 Reduce the number of values for a given continuous attribute
by dividing the range of the attribute into intervals
 Interval labels can then be used to replace actual data values
 Supervised vs. unsupervised
 Split (top-down) vs. merge (bottom-up)
 Discretization can be performed recursively on an attribute

4
Concept hierarchy
 Concept hierarchy formation
 Recursively reduce the data by collecting and replacing low level
concepts (such as numeric values for age) by higher level
concepts (such as young, middle-aged, or senior)
 Detail lost
 More meaningful
 Easier to interpret
 Mining becomes easier
 Several concept hierarchies can be defined for the same
attribute
 Manual / Implicit

5
Discretization and Concept Hierarchy
Generation for Numeric Data
 Typical methods:
 Binning
 Histogram analysis
 Clustering analysis
 Entropy-based discretization
 χ2
merging
 Segmentation by natural partitioning
All the methods can be applied recursively

6
Techniques
 Binning
 Distribute values into bins
 Replace by bin mean / median
 Recursive application – leads to concept hierarchies
 Unsupervised technique
 Histogram Analysis
 Data Distribution – Partition
 Equiwidth – (0-100], (100-200], …
 Equidepth
 Recursive
 Minimum Interval size
 Unsupervised

7
Techniques
 Cluster Analysis
 Clusters form nodes of concept hierarchy
 Can decompose / combine
 Lower level / higher level of hierarchy

8
Entropy-Based Discretization
 Given a set of samples S, if S is partitioned into two intervals S1 and S2
using boundary T, the expected information requirement after partitioning is
 Entropy is calculated based on class distribution of the samples in the set.
Given m classes, the entropy of S1 is
where pi is the probability of class i in S1
 The boundary that minimizes the expected information requirement over all
possible boundaries is selected as a binary discretization
 The process is recursively applied to partitions obtained until some stopping
criterion is met
)(
||
||
)(
||
||
),( 2
2
1
1
SEntropy
S
S
SEntropy
S
S
TSI +=
∑=
−=
m
i
ii ppSEntropy
1
21 )(log)(

9
 Reduces data size
 Class information is considered
 Improves accuracy
Entropy-Based Discretization

10
Interval Merging by χ2
Analysis
 ChiMerge
 Bottom-up approach
 find the best neighbouring intervals and merges them to form larger intervals
 Supervised
 If two adjacent intervals have similar distribution of classes – they can be
merged
 Initially each value is in a separate interval
 χ2
tests are performed for adjacent intervals. Those with least
values are merged
 Can be repeated
 Stopping condition (Threshold, Number of intervals)

11
Segmentation by Natural Partitioning
 A simply 3-4-5 rule can be used to segment numeric data into
relatively uniform, “natural” intervals.
 If an interval covers 3, 6, 7 or 9 distinct values at the most
significant digit, partition the range into 3 equi-width intervals
 If it covers 2, 4, or 8 distinct values at the most significant digit,
partition the range into 4 intervals
 If it covers 1, 5, or 10 distinct values at the most significant digit,
partition the range into 5 intervals

12
 Outliers could be present
 Consider only the majority values
 5th
percentile – 95th
percentile
Segmentation by Natural Partitioning

13
Example of 3-4-5 Rule
(-$400 -$5,000)
(-$400 - 0)
(-$400 -
-$300)
(-$300 -
-$200)
(-$200 -
-$100)
(-$100 -
0)
(0 - $1,000)
(0 -
$200)
($200 -
$400)
($400 -
$600)
($600 -
$800) ($800 -
$1,000)
($2,000 - $5, 000)
($2,000 -
$3,000)
($3,000 -
$4,000)
($4,000 -
$5,000)
($1,000 - $2, 000)
($1,000 -
$1,200)
($1,200 -
$1,400)
($1,400 -
$1,600)
($1,600 -
$1,800)
($1,800 -
$2,000)
msd=1,000 Low=-$1,000 High=$2,000Step 2:
Step 4:
Step 1: -$351 -$159 profit $1,838 $4,700
Min Low (i.e, 5%-tile) High(i.e, 95%-tile) Max
count
(-$1,000 - $2,000)
(-$1,000 - 0) (0 -$ 1,000)
Step 3:
($1,000 - $2,000)

14
Concept Hierarchy Generation for
Categorical Data
 Specification of a partial ordering of attributes explicitly at
the schema level by users or experts
 User / Expert defines hierarchy
 Street < city < state < country
 Specification of a portion of a hierarchy by explicit data
grouping
 Manual
 Intermediate level information specified
 Industrial, Agricultural..

15
Concept Hierarchy Generation for
Categorical Data
 Specification of a set of attributes but not their partial
ordering
 Automatically inferring the hierarchy
 Heuristic rule
 High level concepts contain a smaller number of values
 Specification of only a partial set of attributes
 Embedding data semantics
 Attributes with tight semantic connections are pinned together

1.8 discretization

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1.8 discretization