1. September 4, 2015Data Mining: Concepts and Techniques1 Data Mining: Concepts and Techniques — Chapter 2 —

2. September 4, 2015Data Mining: Concepts and Techniques2 General data characteristics Basic data description and exploration Measuring data similarity What is about Data?

3. September 4, 2015Data Mining: Concepts and Techniques3 What is Data? Collection of data objects and their attributes An attribute is a property or characteristic of an object Examples: eye color of a person, temperature, etc. Attribute is also known as variable, field, characteristic, or feature A collection of attributes describe an object Object is also known as record, point, case, sample, entity, or instance Attributes Objects

4. September 4, 2015Data Mining: Concepts and Techniques4 Important Characteristics of Structured Data Dimensionality Curse of dimensionality Sparsity Only presence counts Resolution Patterns depend on the scale Similarity Distance measure

5. September 4, 2015Data Mining: Concepts and Techniques5 Attribute Values Attribute values are numbers or symbols assigned to an attribute Distinction between attributes and attribute values Same attribute can be mapped to different attribute values Example: height can be measured in feet or meters Different attributes can be mapped to the same set of values Example: Attribute values for ID and age are integers But properties of attribute values can be different ID has no limit but age has a maximum and minimum value

6. September 4, 2015Data Mining: Concepts and Techniques6 Types of Attribute Values Nominal E.g., profession, ID numbers, eye color, zip codes Ordinal E.g., rankings (e.g., army, professions), grades, height in {tall, medium, short} Binary E.g., medical test (positive vs. negative) Interval E.g., calendar dates, body temperatures Ratio E.g., temperature in Kelvin, length, time, counts

7. September 4, 2015Data Mining: Concepts and Techniques7 Properties of Attribute Values The type of an attribute depends on which of the following properties it possesses: Distinctness: =  Order: Addition: + - Multiplication: * / Nominal attribute: distinctness Ordinal attribute: distinctness & order Interval attribute: distinctness, order & addition Ratio attribute: all 4 properties

8. September 4, 2015Data Mining: Concepts and Techniques8 Attribute Type DescriptionExamplesOperations NominalThe values of a nominal attribute are just different names, i.e., nominal attributes provide only enough information to distinguish one object from another. (=,  ) zip codes, employee ID numbers, eye color, sex: {male, female} mode, entropy, contingency correlation,  2 test OrdinalThe 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 IntervalFor 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 RatioFor 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

9. September 4, 2015Data Mining: Concepts and Techniques9 Discrete vs. Continuous Attributes Discrete Attribute Has only a finite or countably infinite set of values E.g., zip codes, profession, or the set of words in a collection of documents Sometimes, represented as integer variables Note: Binary attributes are a special case of discrete attributes Continuous Attribute Has real numbers as attribute values Examples: temperature, height, or weight Practically, real values can only be measured and represented using a finite number of digits Continuous attributes are typically represented as floating-point variables

10. September 4, 2015Data Mining: Concepts and Techniques10 Types of data sets Record Data Matrix Document Data Transaction Data Graph World Wide Web Molecular Structures Ordered Spatial Data Temporal Data Sequential Data Genetic Sequence Data

11. September 4, 2015Data Mining: Concepts and Techniques11 Important Characteristics of Structured Data Dimensionality Curse of Dimensionality Sparsity Only presence counts Resolution Patterns depend on the scale

12. September 4, 2015Data Mining: Concepts and Techniques12 Record Data Data that consists of a collection of records, each of which consists of a fixed set of attributes

13. September 4, 2015Data Mining: Concepts and Techniques13 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

14. September 4, 2015Data Mining: Concepts and Techniques14 Document Data Each document becomes a `term' vector, each term is a component (attribute) of the vector, the value of each component is the number of times the corresponding term occurs in the document.

15. September 4, 2015Data Mining: Concepts and Techniques15 Transaction Data A special type of record data, where each record (transaction) involves a set of items. For example, consider a grocery store. The set of products purchased by a customer during one shopping trip constitute a transaction, while the individual products that were purchased are the items.

16. September 4, 2015Data Mining: Concepts and Techniques16 Graph Data Examples: Generic graph and HTML Links

17. September 4, 2015Data Mining: Concepts and Techniques17 Chemical Data Benzene Molecule: C 6 H 6

18. September 4, 2015Data Mining: Concepts and Techniques18 Ordered Data Sequences of transactions An element of the sequence Items/Events

19. September 4, 2015Data Mining: Concepts and Techniques19 Ordered Data Genomic sequence data

20. September 4, 2015Data Mining: Concepts and Techniques20 Ordered Data Spatio-Temporal Data Average Monthly Temperature of land and ocean

21. September 4, 2015Data Mining: Concepts and Techniques21 General data characteristics Basic data description and exploration Measuring data similarity

22. September 4, 2015Data Mining: Concepts and Techniques22 Mining Data Descriptive Characteristics Motivation To better understand the data: central tendency, variation and spread Data dispersion characteristics median, max, min, quantiles, outliers, variance, etc. Numerical dimensions correspond to sorted intervals Data dispersion: analyzed with multiple granularities of precision Boxplot or quantile analysis on sorted intervals Dispersion analysis on computed measures Folding measures into numerical dimensions Boxplot or quantile analysis on the transformed cube

23. September 4, 2015Data Mining: Concepts and Techniques23 Measuring the Central Tendency Mean (algebraic measure) (sample vs. population): Weighted arithmetic mean: Trimmed mean: chopping extreme values Median: A holistic measure Middle value if odd number of values, or average of the middle two values otherwise Estimated by interpolation (for grouped data): Mode Value that occurs most frequently in the data Unimodal, bimodal, trimodal Empirical formula:

24. September 4, 2015Data Mining: Concepts and Techniques24 Symmetric vs. Skewed Data Median, mean and mode of symmetric, positively and negatively skewed data positively skewednegatively skewed symmetric

25. September 4, 2015Data Mining: Concepts and Techniques25 Measuring the Dispersion of Data Quartiles, outliers and boxplots Quartiles: Q 1 (25 th percentile), Q 3 (75 th percentile) Inter-quartile range: IQR = Q 3 – Q 1 Five number summary: min, Q 1, M, Q 3, max Boxplot: ends of the box are the quartiles, median is marked, whiskers, and plot outlier individually Outlier: usually, a value higher/lower than 1.5 x IQR Variance and standard deviation (sample: s, population: σ) Variance: (algebraic, scalable computation) Standard deviation s (or σ) is the square root of variance s 2 ( or σ 2)

26. September 4, 2015Data Mining: Concepts and Techniques26 Boxplot Analysis Five-number summary of a distribution: Minimum, Q1, M, Q3, Maximum Boxplot Data is represented with a box The ends of the box are at the first and third quartiles, i.e., the height of the box is IQR The median is marked by a line within the box Whiskers: two lines outside the box extend to Minimum and Maximum

27. September 4, 2015Data Mining: Concepts and Techniques27 Histogram Analysis Graph displays of basic statistical class descriptions Frequency histograms A univariate graphical method Consists of a set of rectangles that reflect the counts or frequencies of the classes present in the given data

28. September 4, 2015Data Mining: Concepts and Techniques28 Histograms Often Tells More than Boxplots The two histograms shown in the left may have the same boxplot representation The same values for: min, Q1, median, Q3, max But they have rather different data distributions

29. September 4, 2015Data Mining: Concepts and Techniques29 Quantile Plot Displays all of the data (allowing the user to assess both the overall behavior and unusual occurrences) Plots quantile information For a data x i data sorted in increasing order, f i indicates that approximately 100 f i % of the data are below or equal to the value x i

30. September 4, 2015Data Mining: Concepts and Techniques30 Quantile-Quantile (Q-Q) Plot Graphs the quantiles of one univariate distribution against the corresponding quantiles of another Allows the user to view whether there is a shift in going from one distribution to another

31. September 4, 2015Data Mining: Concepts and Techniques31 Scatter plot Provides a first look at bivariate data to see clusters of points, outliers, etc Each pair of values is treated as a pair of coordinates and plotted as points in the plane

32. September 4, 2015Data Mining: Concepts and Techniques32 Loess Curve Adds a smooth curve to a scatter plot in order to provide better perception of the pattern of dependence Loess curve is fitted by setting two parameters: a smoothing parameter, and the degree of the polynomials that are fitted by the regression

33. September 4, 2015Data Mining: Concepts and Techniques33 Positively and Negatively Correlated Data The left half fragment is positively correlated The right half is negative correlated

34. September 4, 2015Data Mining: Concepts and Techniques34 Not Correlated Data

35. September 4, 2015Data Mining: Concepts and Techniques35 Data Visualization and Its Methods Why data visualization? Gain insight into an information space by mapping data onto graphical primitives Provide qualitative overview of large data sets Search for patterns, trends, structure, irregularities, relationships among data Help find interesting regions and suitable parameters for further quantitative analysis Provide a visual proof of computer representations derived Typical visualization methods: Geometric techniques Icon-based techniques Hierarchical techniques

36. September 4, 2015Data Mining: Concepts and Techniques36 Geometric Techniques Visualization of geometric transformations and projections of the data Methods Landscapes Projection pursuit technique Finding meaningful projections of multidimensional data Scatterplot matrices Prosection views Hyperslice Parallel coordinates

37. September 4, 2015Data Mining: Concepts and Techniques37 Scatterplot Matrices Matrix of scatterplots (x-y-diagrams) of the k-dim. data Used by ermission of M. Ward, Worcester Polytechnic Institute

38. September 4, 2015Data Mining: Concepts and Techniques38 news articles visualized as a landscape Used by permission of B. Wright, Visible Decisions Inc. Landscapes Visualization of the data as perspective landscape The data needs to be transformed into a (possibly artificial) 2D spatial representation which preserves the characteristics of the data

39. September 4, 2015Data Mining: Concepts and Techniques39 Parallel Coordinates n equidistant axes which are parallel to one of the screen axes and correspond to the attributes The axes are scaled to the [minimum, maximum]: range of the corresponding attribute Every data item corresponds to a polygonal line which intersects each of the axes at the point which corresponds to the value for the attribute

40. September 4, 2015Data Mining: Concepts and Techniques40 Parallel Coordinates of a Data Set

41. September 4, 2015Data Mining: Concepts and Techniques41 Icon-based Techniques Visualization of the data values as features of icons Methods: Chernoff Faces Stick Figures Shape Coding: Color Icons: TileBars: The use of small icons representing the relevance feature vectors in document retrieval

42. September 4, 2015Data Mining: Concepts and Techniques42 Chernoff Faces A way to display variables on a two-dimensional surface, e.g., let x be eyebrow slant, y be eye size, z be nose length, etc. The figure shows faces produced using 10 characteristics--head eccentricity, eye size, eye spacing, eye eccentricity, pupil size, eyebrow slant, nose size, mouth shape, mouth size, and mouth opening): Each assigned one of 10 possible values, generated using Mathematica (S. Dickson) Mathematica REFERENCE: Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, p. 212, 1993The Cartoon Guide to Statistics. Weisstein, Eric W. "Chernoff Face." From MathWorld--A Wolfram Web Resource. mathworld.wolfram.com/ChernoffFace.html mathworld.wolfram.com/ChernoffFace.html

43. September 4, 2015Data Mining: Concepts and Techniques43 Hierarchical Techniques Visualization of the data using a hierarchical partitioning into subspaces. Methods Dimensional Stacking Worlds-within-Worlds Treemap Cone Trees InfoCube

44. September 4, 2015Data Mining: Concepts and Techniques44 Tree-Map Screen-filling method which uses a hierarchical partitioning of the screen into regions depending on the attribute values The x- and y-dimension of the screen are partitioned alternately according to the attribute values (classes) MSR Netscan Image

45. September 4, 2015Data Mining: Concepts and Techniques45 Tree-Map of a File System (Schneiderman)

46. September 4, 2015Data Mining: Concepts and Techniques46 General data characteristics Basic data description and exploration Measuring data similarity (Sec. 7.2)

47. September 4, 2015Data Mining: Concepts and Techniques47 Similarity and Dissimilarity Similarity Numerical measure of how alike two data objects are Value is higher when objects are more alike Often falls in the range [0,1] Dissimilarity (i.e., distance) Numerical measure of how different are two data objects Lower when objects are more alike Minimum dissimilarity is often 0 Upper limit varies Proximity refers to a similarity or dissimilarity

48. September 4, 2015Data Mining: Concepts and Techniques48 Data Matrix and Dissimilarity Matrix Data matrix n data points with p dimensions Two modes Dissimilarity matrix n data points, but registers only the distance A triangular matrix Single mode

49. September 4, 2015Data Mining: Concepts and Techniques49 Example: Data Matrix and Distance Matrix Distance Matrix (i.e., Dissimilarity Matrix) for Euclidean Distance Data Matrix

50. September 4, 2015Data Mining: Concepts and Techniques50 Minkowski Distance Minkowski distance: A popular distance measure where i = (x i1, x i2, …, x ip ) and j = (x j1, x j2, …, x jp ) are two p-dimensional data objects, and q is the order Properties d(i, j) > 0 if i ≠ j, and d(i, i) = 0 (Positive definiteness) d(i, j) = d(j, i) (Symmetry) d(i, j)  d(i, k) + d(k, j) (Triangle Inequality) A distance that satisfies these properties is a metric

51. September 4, 2015Data Mining: Concepts and Techniques51 Special Cases of Minkowski Distance q = 1: Manhattan (city block, L 1 norm) distance E.g., the Hamming distance: the number of bits that are different between two binary vectors q= 2: (L 2 norm) Euclidean distance q  . “supremum” (L max norm, L  norm) distance. This is the maximum difference between any component of the vectors Do not confuse q with n, i.e., all these distances are defined for all numbers of dimensions. Also, one can use weighted distance, parametric Pearson product moment correlation, or other dissimilarity measures

52. September 4, 2015Data Mining: Concepts and Techniques52 Example: Minkowski Distance Distance Matrix

53. September 4, 2015Data Mining: Concepts and Techniques53 Binary Variables A contingency table for binary data Distance measure for symmetric binary variables: Distance measure for asymmetric binary variables: Jaccard coefficient (similarity measure for asymmetric binary variables): A binary variable is symmetric if both of its states are equally valuable and carry the same weight. Object i Object j

54. September 4, 2015Data Mining: Concepts and Techniques54 Dissimilarity between Binary Variables Example gender is a symmetric attribute the remaining attributes are asymmetric binary let the values Y and P be set to 1, and the value N be set to 0

55. September 4, 2015Data Mining: Concepts and Techniques55 Nominal Variables A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green Method 1: Simple matching m: # of matches, p: total # of variables Method 2: Use a large number of binary variables creating a new binary variable for each of the M nominal states

56. September 4, 2015Data Mining: Concepts and Techniques56 Ordinal Variables An ordinal variable can be discrete or continuous Order is important, e.g., rank Can be treated like interval-scaled replace x if by their rank map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by compute the dissimilarity using methods for interval- scaled variables

57. September 4, 2015Data Mining: Concepts and Techniques57 Ratio-Scaled Variables Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as Ae Bt or Ae -Bt Methods: treat them like interval-scaled variables—not a good choice! (why?—the scale can be distorted) apply logarithmic transformation y if = log(x if ) treat them as continuous ordinal data treat their rank as interval-scaled

58. September 4, 2015Data Mining: Concepts and Techniques58 Variables of Mixed Types A database may contain all the six types of variables symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio One may use a weighted formula to combine their effects f is binary or nominal: d ij (f) = 0 if x if = x jf, or d ij (f) = 1 otherwise f is interval-based: use the normalized distance f is ordinal or ratio-scaled Compute ranks r if and Treat z if as interval-scaled

59. September 4, 2015Data Mining: Concepts and Techniques59 Vector Objects: Cosine Similarity Vector objects: keywords in documents, gene features in micro-arrays, … Applications: information retrieval, biologic taxonomy,... Cosine measure: If d 1 and d 2 are two vectors, then cos(d 1, d 2 ) = (d 1  d 2 ) /||d 1 || ||d 2 ||, where  indicates vector dot product, ||d||: the length of vector d Example: d 1 = 3 2 0 5 0 0 0 2 0 0 d 2 = 1 0 0 0 0 0 0 1 0 2 d 1  d 2 = 3*1+2*0+0*0+5*0+0*0+0*0+0*0+2*1+0*0+0*2 = 5 ||d 1 ||= (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 ||d 2 || = (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( d 1, d 2 ) =.3150

60. September 4, 2015Data Mining: Concepts and Techniques60 Correlation Analysis (Numerical Data) Correlation coefficient (also called Pearson’s product moment coefficient) where n is the number of tuples, and are the respective means of p and q, σ p and σ q are the respective standard deviation of p and q, and Σ(pq) is the sum of the pq cross-product. If r p,q > 0, p and q are positively correlated (p’s values increase as q’s). The higher, the stronger correlation. r p,q = 0: independent; r pq < 0: negatively correlated

61. September 4, 2015Data Mining: Concepts and Techniques61 Correlation (viewed as linear relationship) Correlation measures the linear relationship between objects To compute correlation, we standardize data objects, p and q, and then take their dot product

62. September 4, 2015Data Mining: Concepts and Techniques62 Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1.