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data Preprocessing different techniques summarized
1. 1
1
Chapter 3: Data Preprocessing
Data Preprocessing: An Overview
Data Quality
Major Tasks in Data Preprocessing
Data Cleaning
Data Integration
Data Reduction
Data Transformation and Data Discretization
Summary
2. 2
Data Quality: Why Preprocess the Data?
Measures for data quality: A multidimensional view
Accuracy: correct or wrong, accurate or not
Completeness: not recorded, unavailable, …
Consistency: some modified but some not, dangling, …
Timeliness: timely update?
Believability: how trustable the data are correct?
Interpretability: how easily the data can be
understood?
3. 3
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 reduction
Dimensionality reduction
Numerosity reduction
Data compression
Data transformation and data discretization
Normalization
Concept hierarchy generation
4. 4
4
Chapter 3: Data Preprocessing
Data Preprocessing: An Overview
Data Quality
Major Tasks in Data Preprocessing
Data Cleaning
Data Integration
Data Reduction
Data Transformation and Data Discretization
Summary
5. 5
Data Cleaning
Data in the Real World Is Dirty: Lots of potentially incorrect data,
e.g., instrument faulty, human or computer error, transmission error
incomplete: lacking attribute values, lacking certain attributes of
interest, or containing only aggregate data
e.g., Occupation=“ ” (missing data)
noisy: containing noise, errors, or outliers
e.g., Salary=“−10” (an error)
inconsistent: containing discrepancies in codes or names, e.g.,
Age=“42”, Birthday=“03/07/2010”
Was rating “1, 2, 3”, now rating “A, B, C”
discrepancy between duplicate records
Intentional (e.g., disguised missing data)
Jan. 1 as everyone’s birthday?
6. 6
Incomplete (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
7. 7
How to Handle Missing Data?
Ignore the tuple: usually done when class label is missing
(when doing classification)—not effective when the % of
missing values per attribute varies considerably
Fill in the missing value manually: tedious + infeasible?
Fill in it automatically with
a global constant : e.g., “unknown”, a new class?!
the attribute mean
the attribute mean for all samples belonging to the
same class: smarter
the most probable value: inference-based such as
Bayesian formula or decision tree
8. 8
Noisy Data
Noise: random error or variance in a measured variable
Incorrect attribute values may be due to
faulty data collection instruments
data entry problems
data transmission problems
technology limitation
inconsistency in naming convention
Other data problems which require data cleaning
duplicate records
incomplete data
inconsistent data
9. 9
How to Handle Noisy Data?
Binning
first sort data and partition into (equal-frequency)
bins
then one can smooth by bin means, smooth by bin
median, smooth by bin boundaries, etc.
Regression
smooth by fitting the data into regression functions
Clustering
detect and remove outliers
Combined computer and human inspection
detect suspicious values and check by human (e.g.,
deal with possible outliers)
10. 10
Data Cleaning as a Process
Data discrepancy detection
Use metadata (e.g., domain, range, dependency, distribution)
Check field overloading
Check uniqueness rule, consecutive rule and null rule
Use commercial tools
Data scrubbing: use simple domain knowledge (e.g., postal
code, spell-check) to detect errors and make corrections
Data auditing: by analyzing data to discover rules and
relationship to detect violators (e.g., correlation and
clustering to find outliers)
Data migration and integration
Data migration tools: allow transformations to be specified
ETL (Extraction/Transformation/Loading) tools: allow users to
specify transformations through a graphical user interface
Integration of the two processes
Iterative and interactive (e.g., Potter’s Wheels)
11. 11
11
Chapter 3: Data Preprocessing
Data Preprocessing: An Overview
Data Quality
Major Tasks in Data Preprocessing
Data Cleaning
Data Integration
Data Reduction
Data Transformation and Data Discretization
Summary
12. 12
12
Data Integration
Data integration:
Combines data from multiple sources into a coherent store
Schema integration: e.g., A.cust-id B.cust-#
Integrate metadata from different sources
Entity identification problem:
Identify real world entities from multiple data sources, e.g., Bill
Clinton = William Clinton
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
13. 13
13
Handling Redundancy in Data Integration
Redundant data occur often when integration of
multiple databases
Object identification: The same attribute or object
may have different names in different databases
Derivable data: One attribute may be a “derived”
attribute in another table, e.g., annual revenue
Redundant attributes may be able to be detected by
correlation analysis and covariance analysis
Careful integration of the data from multiple sources
may help reduce/avoid redundancies and
inconsistencies and improve mining speed and quality
14. 14
Correlation Analysis (Numeric Data)
Correlation coefficient (also called Pearson’s product
moment coefficient)
where n is the number of tuples, and are the respective
means of A and B, σA and σB are the respective standard deviation
of A and B, and Σ(aibi) is the sum of the AB cross-product.
If rA,B > 0, A and B are positively correlated (A’s values
increase as B’s).
rA,B = 0: independent; rAB < 0: negatively correlated
B
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17. 17
Correlation (viewed as linear relationship)
Correlation measures the linear relationship
between objects
To compute correlation, we standardize data
objects, A and B, and then take their dot
product
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18. 18
Covariance (Numeric Data)
Covariance is similar to correlation
where n is the number of tuples, and are the respective mean or
expected values of A and B, σA and σB are the respective standard
deviation of A and B.
Positive covariance: If CovA,B > 0, then A and B both tend to be larger than
their expected values.
Negative covariance: If CovA,B < 0 then if A is larger than its expected value,
B is likely to be smaller than its expected value.
Independence: CovA,B = 0 but the converse is not true:
Some pairs of random variables may have a covariance of 0 but are not
independent. Only under some additional assumptions (e.g., the data follow
multivariate normal distributions) does a covariance of 0 imply independence
A B
Correlation coefficient:
19. Co-Variance: An Example
It can be simplified in computation as
Suppose two stocks A and B have the following values in one week:
(2, 5), (3, 8), (5, 10), (4, 11), (6, 14).
Question: If the stocks are affected by the same industry trends,
will their prices rise or fall together?
E(A) = (2 + 3 + 5 + 4 + 6)/ 5 = 20/5 = 4
E(B) = (5 + 8 + 10 + 11 + 14) /5 = 48/5 = 9.6
Cov(A,B) = (2×5+3×8+5×10+4×11+6×14)/5 4 × 9.6 = 4
−
Thus, A and B rise together since Cov(A, B) > 0.
20. 20
20
Chapter 3: Data Preprocessing
Data Preprocessing: An Overview
Data Quality
Major Tasks in Data Preprocessing
Data Cleaning
Data Integration
Data Reduction
Data Transformation and Data Discretization
Summary
21. 21
Data Reduction Strategies
Data reduction: Obtain a reduced representation of the data set
that is much smaller in volume but yet produces the same (or
almost the same) analytical results
Why data reduction? — A database/data warehouse may store
terabytes of data. Complex data analysis may take a very long time
to run on the complete data set.
Data reduction strategies
Dimensionality reduction, e.g., remove unimportant attributes
Principal Components Analysis (PCA)
Feature subset selection, feature creation
Numerosity reduction (some simply call it: Data Reduction)
Regression and Log-Linear Models
Histograms, clustering, sampling
Data cube aggregation
Data compression
22. 22
Data Reduction 1: Dimensionality Reduction
Curse of dimensionality
When dimensionality increases, data becomes increasingly sparse
Density and distance between points, which is critical to clustering,
outlier analysis, becomes less meaningful
The possible combinations of subspaces will grow exponentially
Dimensionality reduction
Avoid the curse of dimensionality
Help eliminate irrelevant features and reduce noise
Reduce time and space required in data mining
Allow easier visualization
Dimensionality reduction techniques
Principal Component Analysis (Feature extraction)
Supervised and nonlinear techniques (e.g., feature selection)
23. 23
x2
x1
e
Principal Component Analysis (PCA)
Find a projection that captures the largest amount of variation in data
The original data are projected onto a much smaller space, resulting in
dimensionality reduction. We find the eigenvectors of the covariance
matrix, and these eigenvectors define the new space
Benefits:
1.Faster execution of Algo
2.Easier visualization
24. 24
Given N data vectors from n-dimensions, find k ≤ n orthogonal
vectors (principal components) that can be best used to represent data
Normalize input data: Each attribute falls within the same range
Compute k orthonormal (unit) vectors, i.e., principal components
Each input data (vector) is a linear combination of the k principal
component vectors
The principal components are sorted in order of decreasing
“significance” or strength
Since the components are sorted, the size of the data can be
reduced by eliminating the weak components, i.e., those with low
variance (i.e., using the strongest principal components, it is
possible to reconstruct a good approximation of the original data)
Works for numeric data only
Principal Component Analysis (Steps)
25. 25
Attribute Subset Selection
Another way to reduce dimensionality of data
Redundant attributes
Duplicate much or all of the information contained
in one or more other attributes
E.g., purchase price of a product and the amount of
sales tax paid
Irrelevant attributes
Contain no information that is useful for the data
mining task at hand
E.g., students' ID is often irrelevant to the task of
predicting students' GPA
26. 26
Heuristic Search in Attribute Selection
There are 2d
possible attribute combinations of d attributes
Typical heuristic attribute selection methods:
Best single attribute under the attribute independence
assumption: choose by significance tests
Best step-wise feature selection:
The best single-attribute is picked first
Then next best attribute condition to the first, ...
Step-wise attribute elimination:
Repeatedly eliminate the worst attribute
Best combined attribute selection and elimination
Optimal branch and bound:
Use attribute elimination and backtracking
27. 27
Attribute Creation (Feature Generation)
Create new attributes (features) that can capture the
important information in a data set more effectively than
the original ones
Three general methodologies
Attribute extraction
Domain-specific
Mapping data to new space (see: data reduction)
E.g., Fourier transformation, wavelet transformation,
manifold approaches (not covered)
Attribute construction
Combining features (see: discriminative frequent
patterns in Chapter 7)
Data discretization
28. 28
Data Reduction 2: Numerosity Reduction
Reduce data volume by choosing alternative, smaller
forms of data representation
Parametric methods (e.g., regression)
Assume the data fits some model, estimate model
parameters, store only the parameters, and
discard the data (except possible outliers)
Ex.: Log-linear models—obtain value at a point in
m-D space as the product on appropriate marginal
subspaces
Non-parametric methods
Do not assume models
Major families: histograms, clustering, sampling, …
29. 29
Parametric Data Reduction: Regression
and Log-Linear Models
Linear regression
Data modeled to fit a straight line
Often uses the least-square method to fit the line
Multiple regression
Allows a response variable Y to be modeled as a
linear function of multidimensional feature vector
Log-linear model
Approximates discrete multidimensional
probability distributions
30. 30
Regression Analysis
Regression analysis: A collective name for
techniques for the modeling and analysis
of numerical data consisting of values of
a dependent variable (also called
response variable or measurement) and of
one or more independent variables (aka.
explanatory variables or predictors)
The parameters are estimated so as to
give a "best fit" of the data
Most commonly the best fit is evaluated
by using the least squares method, but
other criteria have also been used
Used for prediction
(including forecasting of
time-series data),
inference, hypothesis
testing, and modeling of
causal relationships
y
x
y = x + 1
X1
Y1
Y1’
31. 31
Linear regression: Y = w X + b
Two regression coefficients, w and b, specify the line and are to be
estimated by using the data at hand
Using the least squares criterion to the known values of Y1, Y2, …, X1,
X2, ….
Multiple regression: Y = b0 + b1 X1 + b2 X2
Many nonlinear functions can be transformed into the above
Log-linear models:
Approximate discrete multidimensional probability distributions
Estimate the probability of each point (tuple) in a multi-dimensional
space for a set of discretized attributes, based on a smaller subset
of dimensional combinations
Useful for dimensionality reduction and data smoothing
Regress Analysis and Log-Linear Models
33. 33
Clustering
Partition data set into clusters based on similarity,
and store cluster representation (e.g., centroid and
diameter) 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
Cluster analysis will be studied in depth in Chapter 10
34. 34
Sampling
Sampling: obtaining a small sample s to represent the
whole data set N
Allow a mining algorithm to run in complexity that is
potentially sub-linear to the size of the data
Key principle: Choose a representative subset of the data
Simple random sampling may have very poor
performance in the presence of skew
Develop adaptive sampling methods, e.g., stratified
sampling:
Note: Sampling may not reduce database I/Os (page at a
time)
35. 35
Types of Sampling
Simple random sampling
There is an equal probability of selecting any
particular item
Sampling without replacement
Once an object is selected, it is removed from the
population
Sampling with replacement
A selected object is not removed from the population
Stratified sampling:
Partition the data set, and draw samples from each
partition (proportionally, i.e., approximately the
same percentage of the data)
Used in conjunction with skewed data
36. 36
Sampling: With or without Replacement
SRSWOR
(simple random
sample without
replacement)
SRSWR
Raw Data
38. 38
Data Cube Aggregation
The lowest level of a data cube (base cuboid)
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
39. 39
Data Reduction 3: Data Compression
String compression
There are extensive theories and well-tuned algorithms
Typically lossless, but only limited manipulation is
possible without expansion
Audio/video compression
Typically lossy compression, with progressive refinement
Sometimes small fragments of signal can be
reconstructed without reconstructing the whole
Time sequence is not audio
Typically short and vary slowly with time
Dimensionality and numerosity reduction may also be
considered as forms of data compression
41. 41
Chapter 3: Data Preprocessing
Data Preprocessing: An Overview
Data Quality
Major Tasks in Data Preprocessing
Data Cleaning
Data Integration
Data Reduction
Data Transformation and Data Discretization
Summary
42. 42
Data Transformation
A function that maps the entire set of values of a given attribute to a
new set of replacement values s.t. each old value can be identified
with one of the new values
Methods
Smoothing: Remove noise from data
Attribute/feature construction
New attributes constructed from the given ones
Aggregation: Summarization, data cube construction
Normalization: Scaled to fall within a smaller, specified range
min-max normalization
z-score normalization
normalization by decimal scaling
Discretization: Concept hierarchy climbing
43. 43
Normalization
Min-max normalization: to [new_minA, new_maxA]
Ex. Let income range $12,000 to $98,000 normalized to [0.0,
1.0]. Then $73,000 is mapped to
Z-score normalization (μ: mean, σ: standard deviation):
Ex. Let μ = 54,000, σ = 16,000. Then
Normalization by decimal scaling
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44. 44
Discretization
Three 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
Numeric—real numbers, e.g., integer or real numbers
Discretization: Divide the range of a continuous attribute into
intervals
Interval labels can then be used to replace actual data values
Reduce data size by discretization
Supervised vs. unsupervised
Split (top-down) vs. merge (bottom-up)
Discretization can be performed recursively on an attribute
Prepare for further analysis, e.g., classification
45. 45
Data Discretization Methods
Typical methods: All the methods can be applied recursively
Binning
Top-down split, unsupervised
Histogram analysis
Top-down split, unsupervised
Clustering analysis (unsupervised, top-down split or
bottom-up merge)
Decision-tree analysis (supervised, top-down split)
Correlation (e.g., 2
) analysis (unsupervised, bottom-up
merge)
46. 46
Simple Discretization: Binning
Equal-width (distance) partitioning
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
Divides the range into N intervals, each containing approximately
same number of samples
Good data scaling
Managing categorical attributes can be tricky
47. 47
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 equal-frequency (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
48. 48
Discretization Without Using Class Labels
(Binning vs. Clustering)
Data Equal interval width
(binning)
Equal frequency (binning) K-means clustering leads to better
results
49. 49
Discretization by Classification &
Correlation Analysis
Classification (e.g., decision tree analysis)
Supervised: Given class labels, e.g., cancerous vs. benign
Using entropy to determine split point (discretization point)
Top-down, recursive split
Details to be covered in Chapter 7
Correlation analysis (e.g., Chi-merge: χ2
-based discretization)
Supervised: use class information
Bottom-up merge: find the best neighboring intervals (those having
similar distributions of classes, i.e., low χ2
values) to merge
Merge performed recursively, until a predefined stopping condition
50. 50
Concept Hierarchy Generation
Concept hierarchy organizes concepts (i.e., attribute values)
hierarchically and is usually associated with each dimension in a
data warehouse
Concept hierarchies facilitate drilling and rolling in data
warehouses to view data in multiple granularity
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 youth, adult, or senior)
Concept hierarchies can be explicitly specified by domain experts
and/or data warehouse designers
Concept hierarchy can be automatically formed for both numeric
and nominal data. For numeric data, use discretization methods
shown.
51. 51
Concept Hierarchy Generation
for Nominal Data
Specification of a partial/total ordering of attributes
explicitly at the schema level by users or experts
street < city < state < country
Specification of a hierarchy for a set of values by
explicit data grouping
{Urbana, Champaign, Chicago} < Illinois
Specification of only a partial set of attributes
E.g., only street < city, not others
Automatic generation of hierarchies (or attribute
levels) by the analysis of the number of distinct values
E.g., for a set of attributes: {street, city, state, country}
52. 52
Automatic Concept Hierarchy Generation
Some hierarchies can be automatically generated based on
the analysis of the number of distinct values per attribute in
the data set
The attribute with the most distinct values is placed at
the lowest level of the hierarchy
Exceptions, e.g., weekday, month, quarter, year
country
province_or_ state
city
street
15 distinct values
365 distinct values
3567 distinct values
674,339 distinct values
53. 53
Chapter 3: Data Preprocessing
Data Preprocessing: An Overview
Data Quality
Major Tasks in Data Preprocessing
Data Cleaning
Data Integration
Data Reduction
Data Transformation and Data Discretization
Summary
54. 54
Summary
Data quality: accuracy, completeness, consistency, timeliness,
believability, interpretability
Data cleaning: e.g. missing/noisy values, outliers
Data integration from multiple sources:
Entity identification problem
Remove redundancies
Detect inconsistencies
Data reduction
Dimensionality reduction
Numerosity reduction
Data compression
Data transformation and data discretization
Normalization
Concept hierarchy generation
55. 55
References
D. P. Ballou and G. K. Tayi. Enhancing data quality in data warehouse environments. Comm. of
ACM, 42:73-78, 1999
A. Bruce, D. Donoho, and H.-Y. Gao. Wavelet analysis. IEEE Spectrum, Oct 1996
T. Dasu and T. Johnson. Exploratory Data Mining and Data Cleaning. John Wiley, 2003
J. Devore and R. Peck. Statistics: The Exploration and Analysis of Data. Duxbury Press, 1997.
H. Galhardas, D. Florescu, D. Shasha, E. Simon, and C.-A. Saita. Declarative data cleaning:
Language, model, and algorithms. VLDB'01
M. Hua and J. Pei. Cleaning disguised missing data: A heuristic approach. KDD'07
H. V. Jagadish, et al., Special Issue on Data Reduction Techniques. Bulletin of the Technical
Committee on Data Engineering, 20(4), Dec. 1997
H. Liu and H. Motoda (eds.). Feature Extraction, Construction, and Selection: A Data Mining
Perspective. Kluwer Academic, 1998
J. E. Olson. Data Quality: The Accuracy Dimension. Morgan Kaufmann, 2003
D. Pyle. Data Preparation for Data Mining. Morgan Kaufmann, 1999
V. Raman and J. Hellerstein. Potters Wheel: An Interactive Framework for Data Cleaning and
Transformation, VLDB’2001
T. Redman. Data Quality: The Field Guide. Digital Press (Elsevier), 2001
R. Wang, V. Storey, and C. Firth. A framework for analysis of data quality research. IEEE Trans.
Knowledge and Data Engineering, 7:623-640, 1995
56. 56
Correlation Analysis (Nominal Data)
Χ2
(chi-square) test
The larger the Χ2
value, the more likely the variables
are related
The cells that contribute the most to the Χ2
value are
those whose actual count is very different from the
expected count
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
Expected
Expected
Observed 2
2 )
(
57. 57
Chi-Square Calculation: An Example
Χ2
(chi-square) calculation (numbers in parenthesis are
expected counts calculated based on the data
distribution in the two categories)
It shows that like_science_fiction and play_chess are
correlated in the group
93
.
507
840
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840
1000
(
360
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360
200
(
210
)
210
50
(
90
)
90
250
( 2
2
2
2
2
Play
chess
Not play chess Sum (row)
Like science fiction 250(90) 200(360) 450
Not like science
fiction
50(210) 1000(840) 1050
Sum(col.) 300 1200 1500
58. 58
Mapping Data to a New Space
Two Sine Waves Two Sine Waves + Noise Frequency
Fourier transform
Wavelet transform
59. 59
What Is Wavelet Transform?
Decomposes a signal into
different frequency subbands
Applicable to n-dimensional
signals
Data are transformed to
preserve relative distance
between objects at different
levels of resolution
Allow natural clusters to
become more distinguishable
Used for image compression
60. 60
Wavelet Transformation
Discrete wavelet transform (DWT) for linear signal
processing, multi-resolution analysis
Compressed approximation: store only a small fraction of
the strongest of the wavelet coefficients
Similar to discrete Fourier transform (DFT), but better
lossy compression, localized in space
Method:
Length, L, must be an integer power of 2 (padding with 0’s, when
necessary)
Each transform has 2 functions: smoothing, difference
Applies to pairs of data, resulting in two set of data of length L/2
Applies two functions recursively, until reaches the desired length
Haar2 Daubechie4
61. 61
Wavelet Decomposition
Wavelets: A math tool for space-efficient hierarchical
decomposition of functions
S = [2, 2, 0, 2, 3, 5, 4, 4] can be transformed to S^ = [23
/4,
-11
/4, 1
/2, 0, 0, -1, -1, 0]
Compression: many small detail coefficients can be
replaced by 0’s, and only the significant coefficients
are retained
63. 63
Why Wavelet Transform?
Use hat-shape filters
Emphasize region where points cluster
Suppress weaker information in their boundaries
Effective removal of outliers
Insensitive to noise, insensitive to input order
Multi-resolution
Detect arbitrary shaped clusters at different scales
Efficient
Complexity O(N)
Only applicable to low dimensional data
![43
Normalization
Min-max normalization: to [new_minA, new_maxA]
Ex. Let income range $12,000 to $98,000 normalized to [0.0,
1.0]. Then $73,000 is mapped to
Z-score normalization (μ: mean, σ: standard deviation):
Ex. Let μ = 54,000, σ = 16,000. Then
Normalization by decimal scaling
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,
73
](https://image.slidesharecdn.com/03preprocessing1-250401050835-50723f66/85/data-Preprocessing-different-techniques-summarized-43-320.jpg)
![61
Wavelet Decomposition
Wavelets: A math tool for space-efficient hierarchical
decomposition of functions
S = [2, 2, 0, 2, 3, 5, 4, 4] can be transformed to S^ = [23
/4,
-11
/4, 1
/2, 0, 0, -1, -1, 0]
Compression: many small detail coefficients can be
replaced by 0’s, and only the significant coefficients
are retained](https://image.slidesharecdn.com/03preprocessing1-250401050835-50723f66/85/data-Preprocessing-different-techniques-summarized-61-320.jpg)
