![Discretization Without Using Class Labels
Data
Equal interval width
Equal frequency
K-means
Attribute TransformationA function that maps the entire set of
values of a given attribute to a new set of replacement values
such that each old value can be identified with one of the new
valuesSimple functions: xk, log(x), ex, |x|Standardization and
Normalization
Similarity and DissimilaritySimilarityNumerical measure of
how alike two data objects are.Is higher when objects are more
alike.Often falls in the range [0,1]DissimilarityNumerical
measure of how different are two data objectsLower when
objects are more alikeMinimum dissimilarity is often 0Upper
limit variesProximity refers to a similarity or dissimilarity
Similarity/Dissimilarity for Simple Attributes
p and q are the attribute values for two data objects.
Euclidean DistanceEuclidean Distance](https://image.slidesharecdn.com/dataminingdatalecturenotesforchapter2introduc-230108151730-37993377/85/Data-Mining-DataLecture-Notes-for-Chapter-2Introduc-docx-21-320.jpg)
![12. Provide a brief description of what Principal Components
Analysis (PCA) does. [Hint: See Appendix A and your lecture
notes.] State what's the input and what the output of PCA is.
13. What's the difference between dimensionality reduction and
feature selection?
14. Describe in detail 2 different techniques for feature
selection.
15. Given a sample dataset (represented by a set of attributes, a
correlation matrix, a co-variance matrix, ...), apply feature
selection techniques to select the best attributes to keep (or
equivalently, the best attributes to remove).
16. What's the difference between feature selection and feature
extraction?
17. Give two examples of data in which feature extraction
would be useful.
18. Given a sample dataset, apply feature extraction.
19. What's data discretization and when is it needed?](https://image.slidesharecdn.com/dataminingdatalecturenotesforchapter2introduc-230108151730-37993377/85/Data-Mining-DataLecture-Notes-for-Chapter-2Introduc-docx-47-320.jpg)
Data Mining DataLecture Notes for Chapter 2Introduc.docx
- 1. 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 objectExamples: eye color of a person, temperature, etc.Attribute is also known as variable, field, characteristic, or featureA collection of attributes describe an objectObject is also known as record, point, case, sample, entity, or instance Attributes Objects
- 2. Attribute ValuesAttribute values are numbers or symbols assigned to an attribute Distinction between attributes and attribute valuesSame 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 Measurement of Length The way you measure an attribute is somewhat may not match the attributes properties. Types of Attributes There are different types of attributesNominalExamples: ID numbers, eye color, zip codesOrdinalExamples: rankings (e.g., taste of potato chips on a scale from 1-10), grades, height in {tall, medium, short}IntervalExamples: calendar dates, temperatures in Celsius or Fahrenheit.RatioExamples: temperature in Kelvin, length, time, counts
- 3. Properties of Attribute Values The type of an attribute depends on which of the following properties it possesses:Distinctness: - Multiplication: * / Nominal attribute: distinctnessOrdinal attribute: distinctness & orderInterval attribute: distinctness, order & additionRatio 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 zip codes, employee ID numbers, eye color, sex: {male, female} Ordinal
- 4. 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
- 5. 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
- 6. new_value = a * old_value Length can be measured in meters or feet. Discrete and Continuous Attributes Discrete AttributeHas only a finite or countably infinite set of valuesExamples: 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 AttributeHas real numbers as attribute valuesExamples: 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 RecordData MatrixDocument DataTransaction DataGraphWorld Wide WebMolecular StructuresOrderedSpatial DataTemporal DataSequential DataGenetic Sequence Data Important Characteristics of Structured DataDimensionality
- 7. 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 NoMarried 100K
- 9. 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 DataEach 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�
- 11. 2� 0� 3� 0� Transaction DataA 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. TIDItems 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk Graph Data Examples: Generic graph and HTML Links
- 12. Chemical Data Benzene Molecule: C6H6 Ordered Data Sequences of transactions An element of the sequence Items/Events Ordered Data Genomic sequence data Ordered DataSpatio-Temporal Data Average Monthly Temperature of land and ocean Data Quality What kinds of data quality problems?How can we
- 13. detect problems with the data? What can we do about these problems? Examples of data quality problems: Noise and outliers missing values duplicate data NoiseNoise refers to modification of original valuesExamples: distortion of a person’s voice when talking on a poor phone and “snow” on television screen Two Sine Waves Two Sine Waves + Noise OutliersOutliers are data objects with characteristics that are considerably different than most of the other data objects in the data set Missing ValuesReasons for missing valuesInformation is not collected (e.g., people decline to give their age and weight)Attributes
- 14. may not be applicable to all cases (e.g., annual income is not applicable to children) Handling missing valuesEliminate Data ObjectsEstimate Missing ValuesIgnore the Missing Value During AnalysisReplace with all possible values (weighted by their probabilities) Duplicate DataData set may include data objects that are duplicates, or almost duplicates of one anotherMajor issue when merging data from heterogeous sources Examples:Same person with multiple email addresses Data cleaningProcess of dealing with duplicate data issues Data PreprocessingAggregationSamplingDimensionality ReductionFeature subset selectionFeature creationDiscretization and BinarizationAttribute Transformation AggregationCombining two or more attributes (or objects) into a single attribute (or object) PurposeData reduction Reduce the number of attributes or objectsChange of scale Cities aggregated into regions, states, countries, etcMore “stable” data Aggregated data tends to have less variability
- 15. Aggregation Standard Deviation of Average Monthly Precipitation Standard Deviation of Average Yearly Precipitation Variation of Precipitation in Australia 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
- 16. Types of SamplingSimple Random SamplingThere is an equal probability of selecting any particular item Sampling without replacementAs each item is selected, it is removed from the population Sampling with replacementObjects 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 samplingSplit the data into several partitions; then draw random samples from each partition Sample Size 8000 points 2000 Points500 Points Sample SizeWhat sample size is necessary to get at least one object from each of 10 groups. Curse of DimensionalityWhen dimensionality increases, data becomes increasingly sparse in the space that it occupies
- 17. Definitions of density and distance between points, which is critical for clustering and outlier detection, become less meaningful Randomly generate 500 pointsCompute difference between max and min distance between any pair of points Dimensionality ReductionPurpose:Avoid curse of dimensionalityReduce amount of time and memory required by data mining algorithmsAllow data to be more easily visualizedMay help to eliminate irrelevant features or reduce noise TechniquesPrinciple Component AnalysisSingular Value DecompositionOthers: supervised and non-linear techniques Dimensionality Reduction: PCAGoal is to find a projection that captures the largest amount of variation in data
- 18. x2 x1 e Dimensionality Reduction: PCAFind the eigenvectors of the covariance matrixThe eigenvectors define the new space x2 x1 e
- 19. Dimensionality Reduction: ISOMAPConstruct a neighbourhood graphFor each pair of points in the graph, compute the shortest path distances – geodesic distances By: Tenenbaum, de Silva, Langford (2000) Dimensionality Reduction: PCA Feature Subset SelectionAnother way to reduce dimensionality of data Redundant features duplicate much or all of the information contained in one or more other attributesExample: purchase price of a product and the amount of sales tax paid Irrelevant featurescontain no information that is useful for the data mining task at handExample: students' ID is often irrelevant to the task of predicting students' GPA Feature Subset SelectionTechniques:Brute-force approch:Try all possible feature subsets as input to data mining algorithmEmbedded approaches: Feature selection occurs naturally as part of the data mining algorithmFilter approaches: Features are selected before data mining algorithm is runWrapper approaches: Use the data mining algorithm as a black box to find best subset of attributes
- 20. Feature CreationCreate 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- specificMapping Data to New SpaceFeature Construction combining features Mapping Data to a New Space Two Sine Waves Two Sine Waves + Noise FrequencyFourier transformWavelet transform Discretization Using Class LabelsEntropy based approach 3 categories for both x and y 5 categories for both x and y
- 21. Discretization Without Using Class Labels Data Equal interval width Equal frequency K-means Attribute TransformationA function that maps the entire set of values of a given attribute to a new set of replacement values such that each old value can be identified with one of the new valuesSimple functions: xk, log(x), ex, |x|Standardization and Normalization Similarity and DissimilaritySimilarityNumerical measure of how alike two data objects are.Is higher when objects are more alike.Often falls in the range [0,1]DissimilarityNumerical measure of how different are two data objectsLower when objects are more alikeMinimum dissimilarity is often 0Upper limit variesProximity refers to a similarity or dissimilarity Similarity/Dissimilarity for Simple Attributes p and q are the attribute values for two data objects. Euclidean DistanceEuclidean Distance
- 22. 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. Euclidean Distance Distance Matrix Sheet1pointxy02p220p331p451pointxyp102p220p331p451 p1 Sheet2 Sheet3 Sheet1pointxy02p220p331p451pointxyp102p220p331p451p1p2p 3p4p102.8283.1625.099p22.82801.4143.162p33.1621.41402p45. 0993.16220 p1 Sheet2 Sheet3 Minkowski DistanceMinkowski Distance is a generalization of Euclidean Distance
- 23. Where r is a parameter, n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q. Minkowski Distance: Examplesr = 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 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. Minkowski Distance Distance Matrix Sheet1pointxy02p220p331p451pointxyp102p220p331p451p1p2p 3p4p102.8283.1625.099p22.82801.4143.162p33.1621.41402p45. 0993.16220pointxyp102p220p331p451L1p1p2p3p4p10446p2402 4p34202p46420L2p1p2p3p4p102.8283.1625.099p22.82801.4143 .162p33.1621.41402p45.0993.16220p1p2p3p4p10235p22013p33 102p45320 p1 Sheet2 Sheet3 Sheet1pointxy02p220p331p451pointxyp102p220p331p451p1p2p
- 25. distance is 6. Mahalanobis Distance Covariance Matrix: B A C A: (0.5, 0.5) B: (0, 1) C: (1.5, 1.5) Mahal(A,B) = 5 Mahal(A,C) = 4 Common Properties of a DistanceDistances, such as the Euclidean distance, have some well known properties. p = q. (Positive definiteness) d(p, q) = d(q, p) for all p and q. (Symmetry) d(p (Triangle Inequality) where d(p, q) is the distance (dissimilarity) between points (data objects), p and q.
- 26. A distance that satisfies these properties is a metric Common Properties of a SimilaritySimilarities, 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 VectorsCommon 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
- 27. 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 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 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 Extended Jaccard Coefficient (Tanimoto)Variation of Jaccard for continuous or count attributesReduces to Jaccard for binary attributes
- 28. CorrelationCorrelation measures the linear relationship between objectsTo compute correlation, we standardize data objects, p and q, and then take their dot product Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1. General Approach for Combining SimilaritiesSometimes attributes are of many different types, but an overall similarity is needed. Using Weights to Combine SimilaritiesMay not want to treat all attributes the same.Use weights wk which are between 0 and 1 and sum to 1. DensityDensity-based clustering require a notion of density Examples:Euclidean density Euclidean density = number of
- 29. points per unit volume Probability density Graph-based density Euclidean Density – Cell-basedSimplest approach is to divide region into a number of rectangular cells of equal volume and define density as # of points the cell contains Euclidean Density – Center-basedEuclidean density is the number of points within a specified radius of the point Tid Refund Marital Status Taxable Income Cheat 1 Yes Single 125K
- 33. C D E 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
- 34. of x Load Document 1 s e a s o n t i m e o u t l o s t w i n g a m e s c o r e b a l l p
- 35. l a y c o a c h t e a m Document 2 Document 3 3050260202 0 0 702100300 100122030 TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4
- 36. Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk 5 2 1 2 5 <a href="papers/papers.html#bbbb"> Data Mining </a> <li> <a href="papers/papers.html#aaaa"> Graph Partitioning </a> <li> <a href="papers/papers.html#aaaa"> Parallel Solution of Sparse Linear System of Equations </a> <li> <a href="papers/papers.html#ffff"> N-Body Computation and Dense Linear System Solvers GGTTCCGCCTTCAGCCCCGCGCC CGCAGGGCCCGCCCCGCGCCGTC GAGAAGGGCCCGCCTGGCGGGCG
- 37. GGGGGAGGCGGGGCCGCCCGAGC CCAACCGAGTCCGACCAGGTGCC CCCTCTGCTCGGCCTAGACCTGA GCTCATTAGGCGGCAGCGGACAG GCCAAGTAGAACACGCGAAGCGC TGGGCTGCCTGCTGCGACCAGGG Dimensions = 10 Dimensions = 40 Dimensions = 80 Dimensions = 120 Dimensions = 160 Dimensions = 206 å = - = n k k k q p dist 1 2
- 45. n correlatio ¢ · ¢ = ) , ( ITS-632 Intro to Data Mining Dr. Patrick Haney Dept. of Information Technology & School of Computer and Information Sciences University of the Cumberlands Chapter 2 Assignment 1. What's an attribute? What's a data instance? 2. What's noise? How can noise be reduced in a dataset? 3. Define outlier. Describe 2 different approaches to detect
- 46. outliers in a dataset. 4. Describe 3 different techniques to deal with missing values in a dataset. Explain when each of these techniques would be most appropriate. 5. Given a sample dataset with missing values, apply an appropriate technique to deal with them. 6. Give 2 examples in which aggregation is useful. 7. Given a sample dataset, apply aggregation of data values. 8. What's sampling? 9. What's simple random sampling? Is it possible to sample data instances using a distribution different from the uniform distribution? If so, give an example of a probability distribution of the data instances that is different from uniform (i.e., equal probability). 10. What's stratified sampling? 11. What's "the curse of dimensionality"?
- 47. 12. Provide a brief description of what Principal Components Analysis (PCA) does. [Hint: See Appendix A and your lecture notes.] State what's the input and what the output of PCA is. 13. What's the difference between dimensionality reduction and feature selection? 14. Describe in detail 2 different techniques for feature selection. 15. Given a sample dataset (represented by a set of attributes, a correlation matrix, a co-variance matrix, ...), apply feature selection techniques to select the best attributes to keep (or equivalently, the best attributes to remove). 16. What's the difference between feature selection and feature extraction? 17. Give two examples of data in which feature extraction would be useful. 18. Given a sample dataset, apply feature extraction. 19. What's data discretization and when is it needed?
- 48. 20. What's the difference between supervised and unsupervised discretization? 21. Given a sample dataset, apply unsupervised (e.g., equal width, equal frequency) discretization, or supervised discretization (e.g., using entropy). 22. Describe 2 approaches to handle nominal attributes with too many values. 23. Given a dataset, apply variable transformation: Either a simple given function, normalization, or standardization. 24. Definition of Correlation and Covariance, and how to use them in data pre-processing (see pp. 76-78).