Data Warehousing and Data Mining Chapter 3 Data Preprocessing

1
Data Warehousing and Data Mining Chapter 3
Data Preprocessing

• Spring
• 2005

2
Chapter 3 Data Preprocessing

• Why preprocess the data?
• Data cleaning
• Data integration and transformation
• Data reduction
• Discretization and concept hierarchy generation
• Time-dependent data
• Summary

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Why Data Preprocessing?

• Data in the real world is dirty
• incomplete lacking attribute values, lacking
  certain attributes of interest, or containing
  only aggregate data
• e.g., occupation
• noisy containing errors or outliers
• e.g., Salary-10
• inconsistent containing discrepancies in codes
  or names
• e.g., Age42 Birthday03/07/1997
• e.g., Was rating 1,2,3, now rating A, B, C
• e.g., discrepancy between duplicate records

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Why Is Data Dirty?

• Incomplete data comes from
• n/a data value when collected
• different consideration between the time when the
  data was collected and when it is analyzed.
• human/hardware/software problems
• Noisy data comes from the process of data
• collection
• entry
• transmission
• Inconsistent data comes from
• different data sources
• functional dependency violation

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Why Is Data Preprocessing Important?

• No quality data, no quality mining results!
• Quality decisions must be based on quality data
• e.g., duplicate or missing data may cause
  incorrect or even misleading statistics.
• Data warehouse needs consistent integration of
  quality data
• Data extraction, cleaning, and transformation
  comprises the majority of the work of building a
  data warehouse

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Multi-Dimensional Measure of Data Quality

• A well-accepted multidimensional view
• Accuracy
• Completeness
• Consistency
• Timeliness
• Believability
• Value added
• Interpretability
• Accessibility
• Broad categories
• intrinsic, contextual, representational, and
  accessibility.

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Major Tasks in Data Preprocessing

• Data cleaning
• Fill in missing values, smooth noisy data,
  identify or remove outliers, and resolve
  inconsistencies
• Data integration
• Integration of multiple databases, data cubes, or
  files
• Data transformation
• Normalization and aggregation
• Data reduction
• Obtains reduced representation in volume but
  produces the same or similar analytical results
• Data discretization
• Part of data reduction but with particular
  importance, especially for numerical data

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Forms of data preprocessing
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Chapter 3 Data Preprocessing

• Why preprocess the data?
• Data cleaning
• Data integration and transformation
• Data reduction
• Discretization and concept hierarchy generation
• Time-dependent data
• Summary

10
Data Cleaning

• Importance
• Data cleaning is one of the three biggest
  problems in data warehousingRalph Kimball
• Data cleaning is the number one problem in data
  warehousingDCI survey
• Data cleaning tasks
• Fill in missing values
• Identify outliers and smooth out noisy data
• Correct inconsistent data
• Resolve redundancy caused by data integration

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Missing Data

• Data is not always available
• E.g., many tuples have no recorded value for
  several attributes, such as customer income in
  sales data
• Missing data may be due to
• equipment malfunction
• inconsistent with other recorded data and thus
  deleted
• data not entered due to misunderstanding
• certain data may not be considered important at
  the time of entry
• not register history or changes of the data
• Missing data may need to be inferred.

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How to Handle Missing Data?

• Ignore the tuple usually done when class label
  is missing (assuming the tasks in
  classificationnot effective when the percentage
  of missing values per attribute varies
  considerably.
• Fill in the missing value manually tedious
  infeasible?
• Use a global constant to fill in the missing
  value e.g., unknown, a new class?!
• Use the attribute mean,median or mode to fill in
  the missing value
• Use the attribute mean for all samples belonging
  to the same class to fill in the missing value
  smarter
• Use the most probable value to fill in the
  missing value inference-based such as Bayesian
  formula or decision tree

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Most Probable Value

• Use the most probable value to fill in the
  missing value inference-based such as Bayesian
  formula or decision tree
• develop a submodel to predict the category or
  numerical value of the missing case
• income f(education,sex, age,...)
• A neural network model
• K-NN model
• Decision tree
• Regression
• ...

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An Extreme Case

• Suppose all values of an attribute is missing
• This attribute may be considered as the unknown
  label in unsupervised learning
• Assigning an object into a cluster can be thought
  of filling the missing values of an attribute

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Time Series and Spacial Data

• Filling missing values in time series and
  spatial data needs a different tratement

Stock index
x
x data exisits o missing
x
x
o
x
o
x
x
o
x
Time in days
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Noisy Data

• Noise random error or variance in a measured
  variable
• Incorrect attribute values may due to
• faulty data collection instruments
• data entry problemshuman or computer error
• data transmission problems
• technology limitation
• inconsistency in naming convention
• duplicate records
• Some noice is inavitable and due to variables or
  factors that can not be measured

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How to Handle Noisy Data?

• Binning method
• first sort data and partition into (equi-depth)
  bins
• then one can smooth by bin means, smooth by bin
  median, smooth by bin boundaries, etc.
• Clustering
• detect and remove outliers
• Combined computer and human inspection
• detect suspicious values and check by human
• Regression
• smooth by fitting the data into regression
  functions

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Simple Discretization Methods Binning

• Equal-width (distance) partitioning
• It divides the range into N intervals of equal
  size uniform grid
• if A and B are the lowest and highest values of
  the attribute, the width of intervals will be W
  (B-A)/N.
• The most straightforward
• But outliers may dominate presentation
• Skewed data is not handled well.
• Equal-depth (frequency) partitioning
• It divides the range into N intervals, each
  containing approximately same number of samples
• Managing categorical attributes can be tricky.

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Binning Methods for Data Smoothing

• Sorted data for price (in dollars) 4, 8, 9,
  15, 21, 21, 24, 25, 26, 28, 29, 34
• Partition into (equi-depth) bins
• - Bin 1 4, 8, 9, 15
• - Bin 2 21, 21, 24, 25
• - Bin 3 26, 28, 29, 34
• Smoothing by bin means
• - Bin 1 9, 9, 9, 9
• - Bin 2 23, 23, 23, 23
• - Bin 3 29, 29, 29, 29
• Smoothing by bin boundaries
• - Bin 1 4, 4, 4, 15
• - Bin 2 21, 21, 25, 25
• - Bin 3 26, 26, 26, 34

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Cluster Analysis
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Regression
y
Y1
y x 1
Y1
x
X1
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Example use of regression to determine outliers

• Problem predict the yearly spending of AE
  customers from their income.
• Y yearly spending in YTL
• X average montly income in YTL

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Data set
X income, Y spending First data is probabliy an
outlier But examining only spending data that is
Distribution of Y 200 spending of the
first Customer can not be identified as an
outlier

• X Y
• 50 200
• 100 50
• 200 70
• 250 80
• 300 85
• 400 120
• 500 130
• 700 150
• 750 150
• 800 180
• 1000 200

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Regression
y
Y1200
y 0.15x 20
Y1
x
X150
X101000
Examine residuals Y1-Y1 is high But Y10-Y10 is
low So data point 1 is an outlier
inccome
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Notes

• Some methods are used for both smoothing and data
  reduction or discretization
• binning
• used in decision tress to reduce number of
  categories
• concept hierarchies
• Example price a numerical variable
• in to concepts as expensive, moderately prised,
  expensive

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Inconsistent Data

• Inconsistent data may due to
• faulty data collection instruments
• data entry problemshuman or computer error
• data transmission problems
• Chang in scale over time
• 1,2,3 to A, B. C
• inconsistency in naming convention
• Data integration
• Different units used for the same variable
• TL or dollar
• Value added tax included in one source not in
  other
• duplicate records

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Chapter 3 Data Preprocessing

• Why preprocess the data?
• Data cleaning
• Data integration and transformation
• Data reduction
• Discretization and concept hierarchy generation
• Time-dependent data
• Summary

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Data Integration

• Data integration
• combines data from multiple sources into a
  coherent store
• Schema integration
• integrate metadata from different sources
• Entity identification problem identify real
  world entities from multiple data sources, e.g.,
  A.cust-id ? B.cust-
• Detecting and resolving data value conflicts
• for the same real world entity, attribute values
  from different sources are different
• possible reasons different representations,
  different scales, e.g., metric vs. British units
• price may include value added tax in one source
  and not include in the other
• Name are entered by different convensions
• name lastname or lastname name or ...

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Handling Redundant Data in Data Integration

• Redundant data occur often when integration of
  multiple databases
• The same attribute may have different names in
  different databases
• One attribute may be a derived attribute in
  another table, e.g., annual revenue
• Redundant data may be able to be detected by
  correlational analysis
• rx,y ?ni1(xi-mean_x)(yi-mean_y)/(n-1)?x?y
• where ?x,?y are standard deviation of X and Y
• as r?1 positive correlation x? y? or x? y?, one
  is redundant
• r ? 0, X and X are independent
• r ? -1 negative correlation x? y? or x? y? one
  is redundant

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Positively and Negatively Correlated Data
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Not Correlated Data
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Exercise

• Construct a data set or a functional relationship
  of two variables X and Y where there is a perfect
  relation between X and Y
• knowing value of X, Y corresponding to that X can
  be predicted with no error
• but correlation coefficient between X and Y is
• ZERO

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Handling Redundant Data in Data Integration

• how to detect redundancy for categorical
  variables?
• Find a measure of association for categorical
  variables
• Careful integration of the data from multiple
  sources may help reduce/avoid redundancies and
  inconsistencies and improve mining speed and
  quality

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Correlation Analysis (Categorical Data)

• ?2 (chi-square) test
• The larger the ?2 value, the more likely the
  variables are related
• The cells that contribute the most to the ? 2
  value are those whose actual count is very
  different from the expected count

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Chi-Square Test

• Under the null hypothesis that attribute valuea
  are independent or not correlated
• ? 2 statistics is approximately distributed as
  chi-square distribution with (kA-1)(kB-1) degree
  of freedom
• KA and KB are number of distinct values of
  attribute A and B
• Given a confidence level ?
• Reject null hypothesis of no correlation if
• ? 2gt ? 2(df,?)

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Chi-Square Calculation An Example

• ? 2 (chi-square) calculation (numbers in
  parenthesis are expected counts calculated based
  on the data distribution in the two categories)
• ?2((2-1)(2-1),0.05)5.99lt507.93
• It shows that like_science_fiction and play_chess
  are correlated in the group

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Calculating expected frequencies

• Expected values are computed under the
  independence assumption
• E(i,j)pipjN
• pipjN probability of observing variable A at
  state i
• pj probability of observing variable B at state j
• N total number of observations
• E.g. E(like_scfic,play_chess)
  (450/1500)(300/1500)150090

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Exercises

• What is the limitation of chi-square measure of
  correlation between categorical variables
• Suggest a better way of measuring correlation
  between categorical variables
• Suggest a way of measuring association between a
  categorical and a numerical variable

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Correlation and Causation

• Correlation does not imply causality
• of hospitals and of car-theft in a city are
  correlated
• Both are causally linked to the third variable
  population

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Exercise

• Correlation coefficient show degree of linear
  association between two variables say X and Y
• Suppose in the data set there is a third variable
  Z potentially correlated with X and Y
• Then the association between X and Y is affected
  by the presence of the third variable Z
• E.g. three variables
• Y crop yield, Xrainfall, Z temperature
• Assume
• No association between Y and X
• Positive association between Y and Z and
• Negative association between X and Z

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Exercise cont.

• How do you interpret the associations
• Design such a dataset
• So the simple correlation coefficient between X
  and X is misleading
• Contradicts with common sense knowledge
• Develop an association measure showing the
  association between Y and X elliminating the
  effect of Z(third variable) on both X and Y

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Another attribute relevance measureEntropy and
Information Gain

• S contains si tuples of class Ci for i 1, ,
  m
• Information measures info required to classify
  any arbitrary tuple
• Entropy of attribute A with values a1,a2,,av
• Information gained by branching on attribute A

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Entropy and Information Gain

• Choose the attributes as relevant
• Whose information gains is greater than a
  critical value

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Exercise

• Construct a simple data set of two categorical
  attributes A and B such that
• Knowing the values of A provides perfect
  information to predict B but
• Knowing the values of B does not provide any
  information to precict A
• Entroy is not a symetric measure of association
  like correlation coefficient is

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Exercise

• Consider a data set of three attributes A, B, and
  C. A has two distinct values as positive p
  and negative n and can be considered as the
  class label. The unconditional probabilities of p
  and n are 0.5 each. B and C have four and three
  distinct values as b1 b2 b3 b4 and c1 c2 c3 each
  having unconditional probabilities ¼ and 1/3
  .respectively. Construct a data set such that B
  is perfectly relevant in predicting the class
  type A but C is perfectly irrelevant for the
  prediction of A.

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Data Transformation

• Smoothing remove noise from data
• Aggregation summarization, data cube
  construction
• From daily sales to monthly sales
• Generalization concept hierarchy climbing
• From original ages to categories
  young,middle,senior
• Normalization scaled to fall within a small,
  specified range
• min-max normalization
• z-score normalization
• normalization by decimal scaling
• Attribute/feature construction
• New attributes constructed from the given ones
• profit, financial ratios,

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Data Transformation Normalization

• min-max normalization

• When a score is outside the new range the
  algorithm may give an error or warning message
• in the presence of outliers regular observations
  are squized in to a small interval

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Data Transformation Normalization

• z-score normalization

Good in handling outliers as the new range is
between -?to ? no out of range error
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Data Transformation Normalization

• Decimal scaling

Where j is the smallest integer such that Max(
)lt1

• Ex max v984 then j 3 v0.984
• preserves the appearance of figures

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Data TransformationLogarithmic Transformation

• Logarithmic transformations
• Y logY
• Used in some distance based methods
• Clustering
• For ratio scaled variables
• E.g. weight, TL/dollar
• Distance between objects is related to percentage
  changes rather then actual differences

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Linear transformations

• Note that linear transformations preserve the
  shape of the distribution
• Whereas non linear transformations distors the
  distribution of data
• Example
• Logistic function f(x) 1/(1exp(-x))
• Transforms x between 0 and 1
• New data are always between 0-1

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Exercise

• Device a transformation such that new data are
  always between 0-1 interval
• but preserve the shape of the sample distribution
  as much as possible

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Attribute/feature construction

• Automatic attribute generation
• Using product or and opperations
• E.g. a regression model to explain spending of a
  customer shown by Y
• Using X1 age, X2 income, ...
• Y a b1X1 b2X2 a linear model
• a,b1,b2 are parameters to be estimated
• Y a b1X1 b2X2 cX1X2
• a nonlinear model a,b1,b2,c are parameters to be
  estimated but linear in parameters as
• X1X2 can be directly computed from income and
  age

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Ratios or differences

• Define new attributes
• E.g.
• Real variables in macroeconomics
• Real GNP
• Financial ratios
• Profit revenue cost

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Chapter 3 Data Preprocessing

• Why preprocess the data?
• Data cleaning
• Data integration and transformation
• Data reduction
• Discretization and concept hierarchy generation
• Time-dependent data
• Summary

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Data Reduction Strategies

• Warehouse may store terabytes of data Complex
  data analysis/mining may take a very long time to
  run on the complete data set
• Data reduction
• Obtains a reduced representation of the data set
  that is much smaller in volume but yet produces
  the same (or almost the same) analytical results
• Data reduction strategies
• Data cube aggregation
• Dimensionality reductionremove unimportant
  attributes
• Data Compression
• Numerosity reductionfit data into models
• Discretization and concept hierarchy generation

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Credit Card Promotion Database
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Data Reduction

• Eliminate redandent attributes
• correlation coefficients show that
• Watch and magazine are associated
• Eliminate one
• Combine variables to reduce number of independent
  variables
• Principle component analysis
• Sampling reduce number of cases (records) by
  sampling from secondary storage
• Discretization
• Age to categorical values

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Data Cube Aggregation

• The lowest level of a data cube
• the aggregated data for an individual entity of
  interest
• e.g., a customer in a phone calling data
  warehouse.
• Multiple levels of aggregation in data cubes
• Further reduce the size of data to deal with
• Reference appropriate levels
• Use the smallest representation which is enough
  to solve the task
• Queries regarding aggregated information should
  be answered using data cube, when possible

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Dimensionality Reduction

• Feature selection (i.e., attribute subset
  selection)
• Select a minimum set of features such that the
  probability distribution of different classes
  given the values for those features is as close
  as possible to the original distribution given
  the values of all features
• reduce of patterns in the patterns, easier to
  understand
• Heuristic methods (due to exponential of
  choices)
• step-wise forward selection
• step-wise backward elimination
• combining forward selection and backward
  elimination
• decision-tree induction

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Example for irrelevent features
Average spending of a customer by income and age
Droping age Expain spending by Income only
Does not change results significantly
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Example of Decision Tree Induction
Initial attribute set A1, A2, A3, A4, A5, A6
A4 ?
A6?
A1?
Class 2
Class 2
Class 1
Class 1
Reduced attribute set A1, A4, A6
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Heuristic Feature Selection Methods

• There are 2d possible sub-features of d features
• Several heuristic feature selection methods
• Best single features under the feature
  independence assumption choose by significance
  tests.
• Best step-wise feature selection
• The best single-feature is picked first
• Then next best feature condition to the first,
  ...
• Step-wise feature elimination
• Repeatedly eliminate the worst feature
• Best combined feature selection and elimination
• Optimal branch and bound
• Use feature elimination and backtracking

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Example Economic indicator problem

• There are tens of macroeconomic variables
• say totally 45
• Which ones is the best predictor for inflation
  rate three months ahead?
• Develop a simple model to predict inflation by
  using only a couple of those 45 macro variables
• Best-stepwise feature selection
• The single macro variable predicting inflation
  among the 45 is seleced first try 45 models say
  /TL
• repeat
• The k th variable is entered among 45-(k-1)
  variables
• Stop at some point introducing new variables

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Example cont.

• Best feature elimination
• Develop a model including all 45 variables
• Remove just one of them try 45 models each
  excluding just one out of 45 variables
• repeat
• Continue eliminating a new variable at each step
• Unitl a stoping criteria
• Rearly used comared to feature selection

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Principal Component Analysis

• Given N data vectors from k-dimensions, find c lt
  k orthogonal vectors that can be best used to
  represent data
• The original data set is reduced to one
  consisting of N data vectors on c principal
  components (reduced dimensions)
• Each data vector is a linear combination of the c
  principal component vectors
• Works for numeric data only
• Used when the number of dimensions is large

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Principal Component Analysis
X2
Y1
Y2

X1

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• X Nk matrix of data in original form Xj is j th
  column of data matrix X where j 1,.,k
• xij is the j th variable of i th observation
• First normalize all variables X1..Xk to Z1.. Zk
• such that mean of Zj j1,..,k 0 E(Zi) 0
• If the scales of variables are differnt bring the
  scales of all variables to same units apply
• Zi (Xi-mean_X)/std_X

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• The first principle component a1 is a linear
  combination of the data Z
• y1 Za1 where y1 is N1 vector Z is NK and
  ak1
• y1,1 z1,1a1,1 z1,2a2,1... z1,kak,1
• y2,1 z2,1a1,1 z2,2a2,1... z2,kak,1

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Za1 y1
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• a1 is a unit vector that is norm of a1 1
• a1a1 1 or a1,12 a2,12... ak,121
• Find the principle comp. vector a1 so as to get
  the maximum variation in y1
• maximize the variance of y1
• var(y1) var(y1,1) var(y2,1)... var(yN,1)
• var(y1,1)i(y1,1-mean_y1)2
• note that mean_y10
• variation is y1 is y1,12 y2,12... yN,12
• or in matrix notation (Za1)Za1 a1ZZa1
• dimensions1k kN Nk k1 11 a scaler

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a1ZZa1 y21
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• Z 1, 1.5, -0.5, 1
• 2, 0.5, -1 ,-1.5
• Z 1 2
• 1.5 0.5
• -0.5 -.1
• 1 -1.5
• ZZ 4.5 1.75
• 4 7.5

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The maximization problem

• Max variation in y
• subject to the restriction that a1a1
• or max a1ZZa1
• st a1a1
• find a1 a1,1 a2,1 ak,1 so as to maximize the
  variation in the first transformed vector y1
• Form the lagranien
• L a1ZZa1 -?(a1a1-1)
• Variation in y 0 constrain

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• L a1ZZa1 -?(a1a1-1)
• dL/da0
• take derivative of Lagrenian with respect to a
  and
• dL/da 2ZZa1 -2?a10 or
• ZZa1 ?a1 so
• multiplying by a1
• a1ZZa1 ? a1a1 or
• a1ZZa1 ?
• a1 is the first eigen vector of ZZ ,? is the
  maximum variation is the first eigen value of ZZ

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Eigen Value problem

• Ax ?x
• Where A is an n by n matrix
• X is an n by 1 vector
• ? is a constant
• Ax is multiplication of a matrix by a vector
• A transformation from Rn to Rn
• Find such vectors whose transformations are
  multiples of themselves

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• Ax- ?x 0
• (A- ?I)x 0
• Two unkonwns x and ?
• But a homogenuous equation
• X0 is a trivial solution
• Rank A- ?I less then n
• Determinant A- ?I should be zero

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The Second principle component a2

• The second principle component a2 can be found in
  a similar manner y2 Za2 where
• y2,1 z1,1a1,2 z1,2a2,2... z1,kak,2
• y2,2 z2,1a2,1 z2,2a2,2... z2,kak,2
• and so on
• a2 is a unit vector so a2a21 and orthogonal to
  a1

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• Max variation in y
• subject to the restriction that a2a21 and
  a1a20
• max a2ZZa2
• st a2a21 and a1a20 the first and
  second PC unit vectors are orthogonal to each
  other
• find a2 a2,1 a2,2 a2,k so as to maximize the
  variation in the transformed compnents y2
• L a2ZZa2 -?2(a2a2-1) - ?a1a2

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The Second principle component a2

• L a2ZZa2 -?2(a2a2-1) - ?a1a2
• dL/da20
• take derivative of lagrenian with respect to a2
  and
• dL/da 2ZZa2 -2?a2- ?a10 or
• ZZa2 ?a2 ?a1
• multiplying by a2
• a2ZZa2 ? a2a2 ?a2 a1
• a2ZZa2 ?
• Variantion in second direction

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• Multiply by a1
• a1ZZa2 ? a1a2 ?a1 a1
• a1ZZa2 ? 0 as
• a1ZZ ?a1 so ?a1a2 0 ?
• Remember that
• ZZa2 ?a2 ?a1
• ZZa2 ?a2
• the second transformation vector is the second
  eigen vector variation of the second principle
  component is the second eigen value ?2

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C th principle component

• Notice that ?2lt ?1
• the first principle component is found in the
  direction of maximum variation
• the second has the second maximum variation whose
  direction is orthogonal to the first PC
• continue the optimization procedure until
  finining c lt k components
• R(c) ?ci1 ?i/ ?ki1 ?i
• Variation in c components/total variation
• R(c) gt specified limit

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Principal Component Analysis
X2
Y1
Y2

X1

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Principal Component Analysis
X2
Y1
Y2

• Coordinate of Z1 is
• Z1,Z2 in
• when using principle
• components
• Z y1,y2 expressed in
• unit vectors a1 a2

• a1

• a2

Z1
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Exercise

• Z
• (-0.5,0.5),(1,1.5), (2.5,1), (-1,-1.5),
  (-1.5,-1.5)
• N5 k2
• plot the point in Z1 Z2 axis
• find the first principle component
• without a matrix formulation
• with a matrix formulation
• express the data in terms of first PC
• find the second principle component
• Draw on a figure the principle components

86
Histograms

• A popular data reduction technique
• Divide data into buckets and store average (sum)
  for each bucket
• Can be constructed optimally in one dimension
  using dynamic programming
• Related to quantization problems.

87

• Equiwidht the width of each bucket range is
  uniform
• Equidepth (equiheight)each bucket contains
  roughly the same number of continuous data
  samples
• V-Optimalleast variance
• histogram variance is the is a weighted sum of
  the original values that each bucket represents
• bucket weight values in bucket

88
MaxDiff

• MaxDiff make a bucket boundry between adjacent
  values if the difference is one of the largest k
  differences
• xi1 - xi gt max_k-1(x1,..xN)

89
V-optimal design

• Sort the values
• assign equal number of values in each bucket
• compute variance
• repeat
• change buckets of boundary values
• compute new variance
• until no reduction in variance
• variance (n1Var1n2Var2...nkVark)/N
• N n1n2..nk,
• Vari ?nij1(xj-x_meani)2/ni,
• Note that V-Optimal in one dimension is
  equivalent to K-means clustering in one dimension

90
Clustering

• Partition data set into clusters, and one can
  store cluster representation only
• Can be very effective if data is clustered but
  not if data is smeared
• Can have hierarchical clustering and be stored in
  multi-dimensional index tree structures
• There are many choices of clustering definitions
  and clustering algorithms, further detailed in
  Chapter 8

91
Sampling

• Allow a mining algorithm to run in complexity
  that is potentially sub-linear to the size of the
  data
• Choose a representative subset of the data
• Simple random sampling may have very poor
  performance in the presence of skew
• Develop adaptive sampling methods
• Stratified sampling
• Approximate the percentage of each class (or
  subpopulation of interest) in the overall
  database
• Used in conjunction with skewed data
• Sampling may not reduce database I/Os (page at a
  time).

92
Sampling methods

• Simple random sample without replacement (SRSWOR)
  of size n
• n of N tuples from D nltN
• P(drawing any tuple)1/N all are equally likely
• Simple random sample with replacement (SRSWR) of
  size n
• each time a tuple is drawn from D, it is recorded
  and then replaced
• it may be drawn again

93
Sampling methods cont.

• Cluster Sample if tuples in D are grouped into M
  mutually disjoint clusters then an SRS of m
  clusters can be obtained where m lt M
• Tuples in a database are retrieved a page at a
  time
• Each pages can be considered as a cluster
• Stratified Sample if D is divided into mutually
  disjoint parts called strata.
• Obtain a SRS at each stratum
• a representative sample when data are skewed
• Ex customer data a stratum for each age group
• The age group having the smallest number of
  customers will be sure to be presented

94
Confidence intervals and sample size

• Cost of obtaining a sample is propotional to the
  size of the sample, n
• Specifying a confidence interval and
• you should be able to determine n number of
  samples required so that the sample mean will be
  within the confidence interval with (1-p)
  confident
• n is very small compared to the size of the
  database N
• nltltN

95
Sampling
SRSWOR (simple random sample without
replacement)
SRSWR
96
Sampling
Cluster/Stratified Sample
Raw Data
97
Hierarchical Reduction

• Use multi-resolution structure with different
  degrees of reduction
• Hierarchical clustering is often performed but
  tends to define partitions of data sets rather
  than clusters
• Parametric methods are usually not amenable to
  hierarchical representation
• Hierarchical aggregation
• An index tree hierarchically divides a data set
  into partitions by value range of some attributes
• Each partition can be considered as a bucket
• Thus an index tree with aggregates stored at each
  node is a hierarchical histogram

98
Chapter 3 Data Preprocessing

• Why preprocess the data?
• Data cleaning
• Data integration and transformation
• Data reduction
• Discretization and concept hierarchy generation
• Time-dependent data
• Summary

99
Discretization

• Three types of attributes
• Nominal values from an unordered set
• Ordinal values from an ordered set
• Continuous real numbers
• Discretization
• divide the range of a continuous attribute into
  intervals
• Some classification algorithms only accept
  categorical attributes.
• Reduce data size by discretization
• Prepare for further analysis

100
Discretization and Concept hierachy

• 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.
• Types of discretization
• Supervised (class information used)
• unsupervised (not used or not available)
• dynamic - static
• local - global
• top down - bottom up

101
Concept hierarchies

• reduce the data by collecting and replacing low
  level concepts (such as numeric values for the
  attribute age) by higher level concepts (such as
  young, middle-aged, or senior).
• detail is lost
• generalized data
• more meaningful
• easier to interpret,
• require less space
• mining on a reduced data set require
• fewer input/output opperations
• be more efficient

102
Discretization Process

• Univariate - one feature at a time
• Steps of a typical discritization process
• sorting the continuous values of the feature
• evaluating a cut-point for splitting or adjacent
  intervals for merging
• splitting or merging intervals of continuous
  value
• top-down approach intervals are split
• chose the best cut point to split the values
  into two partitions
• until a stopping criterion is satisfied

103
Discretization Process cont.

• bottom-up approach intervals are merged
• find best pair of intervals to merge
• until stopping criterion is satisfied
• stopping at some point
• lower better understanding less accuracy
• high poorer understanding higher accuracy
• Stopping criteria
• number of intervals reach a critical value
• number of observations in each interval exceeds a
  value
• all observations in an interval are of the same
  class
• a minimum information gain is not satisfied

104
Generalized Splitting Algorithm

• S sorted values of feature f
• splitting(S)
• if stopping criterion satisfied
• return
• T GetBestSplitPoint(S)
• S1GetLeftPart(S,T)
• S2GetRightPart(S,T)
• Splitting(S1)
• Splitting(S2)

105
Discretization and concept hierarchy generation
for numeric data

• Binning (see sections before)
• Histogram analysis (see sections before)
• Clustering analysis (see sections before)
• Entropy-based discretization
• Segmentation by natural partitioning

106
1R a supervised discretization method

• After sorting the continuous values,
• divides the range of continuous values into a
  number of disjoint intervals and adjusts the
  boundaries based on the class labels
• Example
• 11 14 15 18 19 20 21 ? 22 23 25 30 31 33 35 ? 36
• R C C R C R C ? R C C R C R
  C ? R
• C ? C
  ? R
• interval of class C from 11-21
• another interval of C from 22-35
• last of class R including just 36
• the two leftmost intervals are merged as they
  predict the same class
• stoping each interval should contain a
  prespecified minimum number of instances

107
Entropy

• A measure of (im)purity of an sample variable S
  is defined as
• Ent(S) -?spslog2ps
• s is a value of S
• Ps is its estimated probability
• average amount of information per event
• Information of an event is
• I(s) -log2ps
• information is high for lower probable events and
  low otherwise

108
Entropy-Based Discretization

• Given a set of samples S, if S is partitioned
  into two intervals S1 and S2 using boundary T,
  the entropy after partitioning is
• The boundary that minimizes the entropy function
  over all possible boundaries is selected as a
  binary discretization.
• The process is recursively applied to partitions
  obtained until some stopping criterion is met,
  e.g.,
• Experiments show that it may reduce data size and
  improve classification accuracy

109
Age Splitting Example

• A sample of ages for 15 customers
• 19 27 29 35 38 39 40 41 42 43 43 43 45 55 55
• y n y y y y y y n y y n n
  n n
• classes are y or n response to life insurance
• Responding to life promotion (y or n) v.s. Age of
  customer
• y or n yes or no
• entropy of the whole sample without partitioning
• ent(S)(-6/15)log2(6/15)(-9/15)log2(9/15)
• -0.4(-1.322)(-0.6)(-0.734)0.969

110
Age Splitting Example

• Splitting value of age41.5
• S1 agelt41.5 7 y, 1 n
• S2 Tgt 41.5 2 y, 5 n
• I(S,T)S1/Sent(S1)S2/Sent(S2)
• (8/15)ent(S1)(7/15)ent(S2)
• ent(S1)(-1/8)log2(1/8)(-7/8)log2(7/8)
• -0.125(-3.000)(-0.875(-0.193)0.544
  ent(S2)(-2/7)log2(2/7)(-5/7)log2(5/7)
• -0.286(-1.806)(-0.714(-0.486)0.863
  
• ent(S1,S2) (8/15)0.544(7/15)0.8630.692

111
Age Splitting Example

• Information gain
• 0.969-0.6920.277
• try all possible partitions
• chose the partition providing maximum information
  gain
• then reccursively partition each interval
• until the stoping criteria is satisfied
• when information gain lt a treshold value

112
Exercise

• Consider a data set of two attributes A and B. A
  is continuous, whereas B is categorical, having
  two values as y and n, which can be
  considered as class of each observation. When
  attribute A is discretized into two equiwidth
  intervals no information is provided by the class
  attribute B but when discretized into three
  equiwidth intervals there is perfect information
  provided by B. Construct a simple dataset obeying
  these characteristics.

113
Segmentation by natural partitioning

• 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 (equiwith
  intervals for 3,6,9)
• 3 intervals in grouping of 2-3-2 for 7
• 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

114
Example of 3-4-5 rule
(-4000 -5,000)
Step 4
115
Example 3.5

• profits at different branches of AE
• Min-351,976.00 Max4,700,896.50
• 5th percentile -159,876
• 95th percentile 1,838,761
• low-159,876, high1,838,761
• msd1,000,000
• rounding low to milliondigit low-1,000,000
• rounding high to milliondigit high2,000,000
• (2,000,000-(-1,000,000))/1,000,0003
• partitioned into 3 equiwidth subsegments
• (-1,000,000...0,(0...1,000,000,(1,000,000...2,00
  0,000

116
Example 3.5

• First interval
• low(-1,000,000)ltmin
• adjust the left boundary
• msd of min 100,000
• rounding min to -400,000 min
• first interval(-400,000...0
• last interval (1,000,000...2,000,000 not cover
  max
• max lthigh
• a new interval (2,000,000...5,000,000
• rounding max to 5,000,000

117
Example 3.5

• Recursively each interval can be further
  partitioned according to 3-4-5 rule
• first interval (-400,00...0 partitioned into 4
  subintervals
• (-400,000...-300,000,(-300,000...-200,000,
• (-200,000...-100,000,(-100,000...0
• second interval (0...1,000,000 partitioned into
  5 subintervals
• (0...200,000,(200,000...400,000,
• (400,000...600,000,(600,000...800,000
• (800,000...1,000,000
• last interval into 3 subintervals
• (2,000,000..3,000,000(3,000,000..4,000,000,(4,00
  0,000..5,000,000

118
Concept hierarchy generation for categorical data

• finite but possibly large distinct values with no
  ordering
• geographic location,job categories, item type
• Specification of a partial ordering of attributes
  explicitly at the schema level by users or
  experts
• Specification of a portion of a hierarchy by
  explicit data grouping
• Specification of a set of attributes, but not of
  their partial ordering
• Specification of only a partial set of attributes

119
Specification of a set of attributes

• Concept hierarchy can be automatically generated
  based on the number of distinct values per
  attribute in the given attribute set. The
  attribute with the most distinct values is placed
  at the lowest level of the hierarchy.

15 distinct values
country
65 distinct values
province_or_ state
3567 distinct values
city
674,339 distinct values
street
120
Chapter 3 Data Preprocessing

• Why preprocess the data?
• Data cleaning
• Data integration and transformation
• Data reduction
• Discretization and concept hierarchy generation
• Time-dependent data
• Summary

121
Time Series Data

• A time series database consists of sequences of
  values or events changing with time.
• The values are typically measured at equal time
  intervals
• Examples
• daily closing values of stock market index
• sales of products
• economic variables GNP, exchange rates...

122
Mining Time-Series and Sequence Data
Time-series plot
123
Objective (1)

• forecasting future unknown values based on
  historical observed data
• one step ahead
• predict Yt1 from YtYt-1, Yt-2...
• multiple steps ahead
• predict Yt1Yt2 Yt3 .. from YtYt-1, Yt-2...
• univariate just one variable
• forecast of TL/ based on its own values

124
Objective (2)

• multivariate more then one variable is
  forecasted simultaneously
• inflation, stock index, exchange rate, interest
  rates influence each other
• model based forecasts

125
Data preprosessing

• Handling missing values
• methods for cross-sectional data are not
  applicable
• fill missing values by smoothing
• Aggregation disaggregation problems
• frequency of raw data and desired forecast
  frequency may be different
• aggregation form low level to high level
• sum,average, minimum,maximum, last,first
• which function is used depends on the type of data

126

• disaggregation from high to low frequency
• different concept from OLAPs drill down where low
  level data is already available
• here low level data is not available
• GNP is available at the quarterly level
• but may be needed in monthly forecast of GNP
• Methods
• constant value is used
• equally partitioned
• linear smoothing
• cubic smoothing

127
Components of time series data

• long-term or trend
• indicate the general direction in which a time
  series variable is moving over a long interval of
  time
• a line or curve
• Cyclic movements or cyclic variations
• long term oscillations about a trend line or
  curve may or may not be periodic

128
Components of time series data

• Seasonal movements or variations
• due to events that recur annually
• sales of beverages varies from season to season
• fuel oil sales in winter are higher than in
  summer
• identical or nearly identical patterns during
  corresponding months or seasons of successive
  years
• Irregular or random movements
• sporadic motion of time series due to random or
  chance events

129
Decomposition of time series

• product of the four variables
• Yi TiCiSiIi,
• or sum of the four variables
• Yi Ti Ci Si Ii,
• Identify each components and try to forecast each
  seperately
• The I component can not be forecasted or
  predicted as it is irregular

130
How to determine trend

• A moving average of order m
• (y1y2...ym)/m
• a moving average tends to reduce the amount of
  variation present in the data set
• eliminates unwanted fluctuations
• smoothing of time series
• Weighted moving average of order m

131
Example

• original data
• 3 7 2 0 4 5 9 7 2
• MA(3) 4 3 2 3 6 7 6
• weighed
• MA(3) 5.5 2.5 1 3.5 5.5 8 6.5
• 1 4 1
• the first WMA value
• (134712)/(141)5.5
• MA loses the data at the beginning and end of a
  series may sometimes

132
Detecting trend

• By moving averages
• cyclic, seasonal and irregular patterns in the
  data can be eliminated
• resulting only the trend movement
• Free-hand method
• approximate curve or line is drawn to fit a set
  of data based on the users own judgement
• costly and not reliable

133
least square method

• Fit a line by regression
• Yt a bt a linear trend
• Yt a bt ct2 a quadratic trend
• Yt atb,convert into linear by logarithmic
  transformation
• lnYt lnablnt
• Yt aebt, exponential
• lnYt lna bt

134
Seasonal variations

• identify and remove seasonal variations
• deseasonalize the data for trend and cyclic
  analysis
• A seasonal index set of number showing the
  relative values of a variable during the months
  of a year
• yt yt/st,
• where yt is deseasonalized variable
• st is seasonal index value at time t

135

• Estimation of seasonal variations
• Seasonal index
• Set of numbers showing the relative values of a
  variable during the months of the year
• E.g., if the sales during October, November, and
  December are 80, 120, and 140 of the average
  monthly sales for the whole year, respectively,
  then 80, 120, and 140 are seasonal index numbers
  for these months
• Deseasonalized data
• Data adjusted for seasonal variations
• E.g., divide the original monthly data by the
  seasonal index numbers for the corresponding
  months

136
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137

• Visually inspect the data
• to identify trend cyclic or seasonal components

138
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139
Mining Time-Series and Sequence Data

• Time-series database
• Consists of sequences of values or events
  changing with time
• Data is recorded at regular intervals
• Characteristic time-series components
• Trend, cycle, seasonal, irregular
• Applications
• Financial stock price, inflation
• Biomedical blood pressure
• Meteorological precipitation

140
Mining Time-Series and Sequence Data Trend
analysis

• A time series can be illustrated as a time-series
  graph which describes a point moving with the
  passage of time
• Categories of Time-Series Movements
• Long-term or trend movements (trend curve)
• Cyclic movements or cycle variations, e.g.,
  business cycles
• Seasonal movements or seasonal variations
• i.e, almost identical patterns that a time series
  appears to follow during corresponding months of
  successive years.
• Irregular or random movements

141
Estimation of Trend Curve

• The freehand method
• Fit the curve by looking at the graph
• Costly and barely reliable for large-scaled data
  mining
• The least-square method
• Find the curve minimizing the sum of the squares
  of the deviation of points on the curve from the
  corresponding data points
• The moving-average method
• Eliminate cyclic, seasonal and irregular patterns
• Loss of end data
• Sensitive to outliers

142
Discovery of Trend in Time-Series (1)

• Estimation of seasonal variations
• Seasonal index
• Set of numbers showing the relative values of a
  variable during the months of the year
• E.g., if the sales during October, November, and
  December are 80, 120, and 140 of the average
  monthly sales for the whole year, respectively,
  then 80, 120, and 140 are seasonal index numbers
  for these months
• Deseasonalized data
• Data adjusted for seasonal variations
• E.g., divide the original monthly data by the
  seasonal index numbers for the corresponding
  months

143
Discovery of Trend in Time-Series (2)

• Estimation of cyclic variations
• If (approximate) periodicity of cycles occurs,
  cyclic index can be constructed in much the same
  manner as seasonal indexes
• Estimation of irregular variations
• By adjusting the data for trend, seasonal and
  cyclic variations
• With the systematic analysis of the trend,
  cyclic, seasonal, and irregular components, it is
  possible to make long- or short-term predictions
  with reasonable quality

144
Chapter 3 Data Preprocessing

• Why preprocess the data?
• Data cleaning
• Data integration and transformation
• Data reduction
• Discretization and concept hierarchy generation
• Summary

145
Summary

• Data preparation is a big issue for both
  warehousing and mining
• Data preparation includes
• Data cleaning and data integration
• Data reduction and feature selection
• Discretization
• A lot a methods have been developed but still an
  active area of research

146
References

• D. P. Ballou and G. K. Tayi. Enhancing data
  quality in data warehouse environments.
  Communications of ACM, 4273-78, 1999.
• Jagadish et al., Special Issue on Data Reduction
  Techniques. Bulletin of the Technical Committee
  on Data Engineering, 20(4), December 1997.
• D. Pyle. Data Preparation for Data Mining. Morgan
  Kaufmann, 1999.
• T. Redman. Data Quality Management and
  Technology. Bantam Books, New York, 1992.
• Y. Wand and R. Wang. Anchoring data quality
  dimensions ontological foundations.
  Communications of ACM, 3986-95, 1996.
• R. Wang, V. Storey, and C. Firth. A framework for
  analysis of data quality research. IEEE Trans.
  Knowledge and Data Engineering, 7623-640, 1995.

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