Concepts and Techniques Chapter 2 Jiawei Han, Micheline Kamber, and Jian Pei University of Illinois at Urbana-Champaign Simon Fraser University 2011 Han, Kamber, and Pei. All rights reserved. 1 Chapter 2: Getting to Know Your Data Data Objects and Attribute Types Basic Statistical Descriptions of Data Data Visualization Measuring Data Similarity and Dissimilarity Summary 2

Types of Data Sets Record Relational records Data matrix, e.g., numerical matrix, crosstabs Document data: text documents: termfrequency vector Transaction data Graph and network World Wide Web Social or information networks

Molecular Structures Document 1 3 0 5 0 2 6 0 2 0 2 Document 2 0 7

0 2 1 0 0 3 0 0 Document 3 0 1 0 0 1 2 2

0 3 0 Ordered Video data: sequence of images Temporal data: time-series Sequential Data: transaction sequences Genetic sequence data Spatial, image and multimedia: Spatial data: maps Image data:

Video data: TID Items 1 2 3 4 5 Bread, Coke, Milk Beer, Bread Beer, Coke, Diaper, Milk Beer, Bread, Diaper, Milk Coke, Diaper, Milk 3 Important Characteristics of Structured Data Dimensionality

Sparsity Only presence counts Resolution Curse of dimensionality Patterns depend on the scale Distribution Centrality and dispersion 4 Data Objects Data sets are made up of data objects. A data object represents an entity.

Examples: sales database: customers, store items, sales medical database: patients, treatments university database: students, professors, courses Also called samples , examples, instances, data points, objects, tuples. Data objects are described by attributes. Database rows -> data objects; columns ->attributes. 5 Attributes

Attribute (or dimensions, features, variables): a data field, representing a characteristic or feature of a data object. E.g., customer _ID, name, address Types: Nominal Binary Numeric: quantitative Interval-scaled Ratio-scaled 6 Attribute Types Nominal: categories, states, or names of things Hair_color = {auburn, black, blond, brown, grey, red, white} marital status, occupation, ID numbers, zip codes Binary Nominal attribute with only 2 states (0 and 1)

Symmetric binary: both outcomes equally important e.g., gender Asymmetric binary: outcomes not equally important. e.g., medical test (positive vs. negative) Convention: assign 1 to most important outcome (e.g., HIV positive) Ordinal Values have a meaningful order (ranking) but magnitude between successive values is not known. Size = {small, medium, large}, grades, army rankings 7 Numeric Attribute Types Quantity (integer or real-valued) Interval Measured on a scale of equal-sized units Values have order E.g., temperature in Cor F, calendar dates

No true zero-point Ratio Inherent zero-point We can speak of values as being an order of magnitude larger than the unit of measurement (10 K is twice as high as 5 K). e.g., temperature in Kelvin, length, counts, monetary quantities 8 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 E.g., 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 9 Chapter 2: Getting to Know Your Data Data Objects and Attribute Types Basic Statistical Descriptions of Data Data Visualization Measuring Data Similarity and Dissimilarity Summary 10 Basic Statistical Descriptions of Data

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 11 Measuring the Central Tendency Note: n is sample size and N is population size. 1 n x xi

n i 1 Mean (algebraic measure) (sample vs. population): N n Weighted arithmetic mean: Trimmed mean: chopping extreme values Median: x wx i i x i 1n w i 1 Middle value if odd number of values, or average of i

the middle two values otherwise Estimated by interpolation (for grouped data): Mode median L1 ( n / 2 ( freq )l freq median ) width Value that occurs most frequently in the data Unimodal, bimodal, trimodal Empirical formula: mean mode 3 (mean median) 12 Symmetric vs.

Skewed Data Median, mean and mode of symmetric, positively and negatively skewed data positively skewed January 21, 2020 symmetric negatively skewed Data Mining: Concepts and Techniques 13 Measuring the Dispersion of Data Quartiles, outliers and boxplots Quartiles: Q1 (25th percentile), Q3 (75th percentile)

Inter-quartile range: IQR = Q3 Q1 Five number summary: min, Q1, median, Q3, max Boxplot: ends of the box are the quartiles; median is marked; add whiskers, and plot outliers individually Outlier: usually, a value higher/lower than 1.5 x IQR Variance and standard deviation (sample: s, population: ) Variance: (algebraic, scalable computation) 1 n 1 n 2 1 n 2 s ( xi x ) [ xi ( xi ) 2 ] n 1 i 1 n 1 i 1 n i 1

2 1 N 2 n 1 ( x ) i N i 1 2 n 2 xi 2 i 1 Standard deviation s (or ) is the square root of variance s2 (or 2)

14 Boxplot Analysis Five-number summary of a distribution Minimum, Q1, Median, 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 extended to Minimum and Maximum Outliers: points beyond a specified outlier threshold, plotted individually

15 Visualization of Data Dispersion: 3-D Boxplots January 21, 2020 Data Mining: Concepts and Techniques 16 Properties of Normal Distribution Curve The normal (distribution) curve From to +: contains about 68% of the measurements (: mean, : standard deviation) From 2 to +2: contains about 95% of it From 3 to +3: contains about 99.7% of it 17 Graphic Displays of Basic Statistical Descriptions Boxplot: graphic display of five-number summary

Histogram: x-axis are values, y-axis repres. frequencies Quantile plot: each value xi is paired with fi indicating that approximately 100 fi % of data are xi Quantile-quantile (q-q) plot: graphs the quantiles of one univariant distribution against the corresponding quantiles of another Scatter plot: each pair of values is a pair of coordinates and plotted as points in the plane 18 Histogram Analysis Histogram: Graph display of tabulated frequencies, shown as bars

35 It shows what proportion of cases 30 fall into each of several categories Differs from a bar chart in that it is 25 the area of the bar that denotes the value, not the height as in bar charts, a crucial distinction when the categories are not of uniform width 40 20 15 10 5 The categories are usually 0 specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent 10000 30000 50000

70000 90000 19 Histograms Often Tell 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 20 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 xi data sorted in increasing order, fi indicates that approximately 100 fi% of the data are below or equal to the value xi Data Mining: Concepts and Techniques 21 Quantile-Quantile (Q-Q) Plot Graphs the quantiles of one univariate distribution against the corresponding quantiles of another View: Is there is a shift in going from one distribution to another? Example shows unit price of items sold at Branch 1 vs. Branch 2 for each quantile. Unit prices of items sold at Branch 1 tend to be lower than those at Branch 2. 22 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 23 Positively and Negatively Correlated Data The left half fragment is positively correlated The right half is negative correlated 24 Uncorrelated Data 25 Chapter 2: Getting to Know Your Data

Data Objects and Attribute Types Basic Statistical Descriptions of Data Data Visualization Measuring Data Similarity and Dissimilarity Summary 26 Data Visualization Why data visualization? 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 Gain insight into an information space by mapping data onto graphical primitives Provide a visual proof of computer representations derived Categorization of visualization methods: Pixel-oriented visualization techniques Geometric projection visualization techniques Icon-based visualization techniques Hierarchical visualization techniques

Visualizing complex data and relations 27 Pixel-Oriented Visualization Techniques For a data set of m dimensions, create m windows on the screen, one for each dimension The m dimension values of a record are mapped to m pixels at the corresponding positions in the windows The colors of the pixels reflect the corresponding values (a) Income (b) Credit Limit (c) transaction volume (d) age 28

Laying Out Pixels in Circle Segments To save space and show the connections among multiple dimensions, space filling is often done in a circle segment (a) Representing a data record in circle segment (b) Laying out pixels in circle segment 29 Geometric Projection Visualization Techniques Visualization of geometric transformations and projections of the data Methods Direct visualization

Scatterplot and scatterplot matrices Landscapes Projection pursuit technique: Help users find meaningful projections of multidimensional data Prosection views Hyperslice Parallel coordinates 30 Direct Data Visualization Ribbons with Twists Based on Vorticity Data Mining: Concepts and Techniques 31 Used by ermission of M. Ward, Worcester Polytechnic Institute

Scatterplot Matrices Matrix of scatterplots (x-y-diagrams) of the k-dim. data [total of (k2/2-k) scatterplots] 32 Used by permission of B. Wright, Visible Decisions Inc. Landscapes news articles visualized as a landscape 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 33 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 Attr. 1 Attr. 2 Attr. 3 Attr. k 34 Parallel Coordinates of a Data Set 35 Icon-Based Visualization Techniques Visualization of the data values as features of icons Typical visualization methods

Chernoff Faces Stick Figures General techniques Shape coding: Use shape to represent certain information encoding Color icons: Use color icons to encode more information Tile bars: Use small icons to represent the relevant feature vectors in document retrieval 36 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) REFERENCE: Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, p. 212, 1993 Weisstein, Eric W. "Chernoff Face." From MathWorld--A Wolfram Web Resource. mathworld.wolfram.com/ChernoffFace.html 37 used by permission of G. Grinstein, University of Massachusettes at Lowell Stick Figure A census data figure showing age, income, gender, education, etc. A 5-piece stick figure (1 body and 4 limbs w. different angle/length)

Two attributes mapped to axes, remaining attributes mapped to angle or length of limbs. Look at 38 Hierarchical Visualization Techniques Visualization of the data using a hierarchical partitioning into subspaces Methods Dimensional Stacking Worlds-within-Worlds Tree-Map Cone Trees

InfoCube 39 Dimensional Stacking attribute 4 attribute 2 attribute 3 attribute 1 Partitioning of the n-dimensional attribute space in 2-D subspaces, which are stacked into each other Partitioning of the attribute value ranges into classes. The important attributes should be used on the outer levels. Adequate for data with ordinal attributes of low cardinality But, difficult to display more than nine dimensions Important to map dimensions appropriately 40 Dimensional Stacking Used by permission of M. Ward, Worcester Polytechnic Institute Visualization of oil mining data with longitude and latitude mapped to

the outer x-, y-axes and ore grade and depth mapped to the inner x-, yaxes 41 Worlds-within-Worlds Assign the function and two most important parameters to innermost world Fix all other parameters at constant values - draw other (1 or 2 or 3 dimensional worlds choosing these as the axes) Software that uses this paradigm Nvision: Dynamic interaction through data glove and stereo displays, including rotation, scaling (inner) and translation (inner/outer) Auto Visual: Static interaction by means of queries

42 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 Ack.: 43 Tree-Map of a File System (Schneiderman) 44 InfoCube A 3-D visualization technique where hierarchical information is displayed as nested semi-transparent cubes The outermost cubes correspond to the top

level data, while the subnodes or the lower level data are represented as smaller cubes inside the outermost cubes, and so on 45 Three-D Cone Trees 3D cone tree visualization technique works well for up to a thousand nodes or so First build a 2D circle tree that arranges its nodes in concentric circles centered on the root node Cannot avoid overlaps when projected to 2D G. Robertson, J. Mackinlay, S. Card. Cone Trees: Animated 3D Visualizations of Hierarchical Information, ACM SIGCHI'91

Graph from Nadeau Software Consulting website: Visualize a social network data set that models the way an infection spreads from one person to the next Ack.: http://nadeausoftware.com/articles/visualization 46 Visualizing Complex Data and Relations Visualizing non-numerical data: text and social networks Tag cloud: visualizing user-generated tags The importance of tag is represented by font size/color Besides text data, there are also methods to visualize relationships, such as visualizing social networks Newsmap: Google News Stories in

Chapter 2: Getting to Know Your Data Data Objects and Attribute Types Basic Statistical Descriptions of Data Data Visualization Measuring Data Similarity and Dissimilarity Summary 48 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 (e.g., distance) Numerical measure of how different two data objects are Lower when objects are more alike Minimum dissimilarity is often 0 Upper limit varies Proximity refers to a similarity or dissimilarity 49 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 x11

... x i1 ... x n1 ... x1f ... ... ... ... ... xif ... ... ... ... xnf ... ...

0 d(2,1) 0 d(3,1) d ( 3,2) 0 : : : d ( n,1) d ( n,2) ... x1p ... xip ... xnp ... 0 50 Proximity Measure for Nominal Attributes

Can take 2 or more states, e.g., red, yellow, blue, green (generalization of a binary attribute) Method 1: Simple matching m: # of matches, p: total # of variables d (i, j) p p m Method 2: Use a large number of binary attributes creating a new binary attribute for each of the M nominal states 51 Proximity Measure for Binary Attributes Object j A contingency table for binary data Object i

Distance measure for symmetric binary variables: Distance measure for asymmetric binary variables: Jaccard coefficient (similarity measure for asymmetric binary variables): Note: Jaccard coefficient is the same as coherence: 52 Dissimilarity between Binary Variables Example Name Jack Mary Jim

Gender M F M Fever Y Y Y Cough N N P Test-1 P P N Test-2 N N N Test-3 N P

N Test-4 N N N Gender is a symmetric attribute The remaining attributes are asymmetric binary Let the values Y and P be 1, and the value N 0 0 1 0.33 2 0 1 11 d ( jack , jim ) 0.67 111 1 2 d ( jim , mary ) 0.75 11 2 d ( jack , mary ) 53 Standardizing Numeric Data x Z-score: z

X: raw score to be standardized, : mean of the population, : standard deviation the distance between the raw score and the population mean in units of the standard deviation negative when the raw score is below the mean, + when above An alternative way: Calculate the mean absolute deviation 1 where s f n (| x1 f m f | | x2 f m f | ... | xnf m f |) m f 1n (x1 f x2 f ... xnf ) . standardized measure (z-score): xif m f zif s f

Using mean absolute deviation is more robust than using standard deviation 54 Example: Data Matrix and Dissimilarity Matrix x2 Data Matrix x4 point x1 x2 x3 x4 4 2 attribute1 attribute2 1 2 3 5 2 0 4 5

x1 Dissimilarity Matrix (with Euclidean Distance) x3 0 2 x1 4 x1 x2 x3 x4 x2 0 3.61 5.1 4.24 x3 0 5.1 1 x4

0 5.39 0 55 Distance on Numeric Data: Minkowski Distance Minkowski distance: A popular distance measure where i = (xi1, xi2, , xip) and j = (xj1, xj2, , xjp) are two p-dimensional data objects, and h is the order (the distance so defined is also called L-h norm) 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 56 Special Cases of Minkowski Distance h = 1: Manhattan (city block, L1 norm) distance E.g., the Hamming distance: the number of bits that are different between two binary vectors d (i, j) | x x | | x x | ... | x x | i1 j1 i2 j 2 ip jp h = 2: (L2 norm) Euclidean distance d (i, j) (| x x |2 | x x |2 ... | x x |2 ) i1 j1 i2 j 2 ip jp h . supremumsupremum (Lmax norm, L norm) distance. This is the maximum difference between any component

(attribute) of the vectors 57 Example: Minkowski Distance Dissimilarity Matrices point x1 x2 x3 x4 attribute 1 attribute 2 1 2 3 5 2 0 4 5 x2 x4 4 2 x1

x2 x3 x4 0 5 3 6 x2 x3 x4 0 6 1 0 7 0 x2 x3 x4 Euclidean (L2) L2

x1 x2 x3 x4 x1 0 3.61 2.24 4.24 0 5.1 1 0 5.39 0 Supremum x1 x3 0 Manhattan (L1)L x1 2

4 L x1 x2 x3 x4 x1 x2 0 3 2 3 x3 0 5 1 x4 0 5 0 58 Ordinal Variables

An ordinal variable can be discrete or continuous Order is important, e.g., rank Can be treated like interval-scaled replace x by their rank rif {1,..., M f } if map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by rif 1 zif Mf 1 compute the dissimilarity using methods for interval-scaled variables 59 Attributes of Mixed Type

A database may contain all attribute types Nominal, symmetric binary, asymmetric binary, numeric, ordinal One may use a weighted formula to combine their effects pf 1 ij( f ) dij( f ) d (i, j) p f 1 ij( f ) f is binary or nominal: dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise f is numeric: use the normalized distance f is ordinal Compute ranks rif and r if 1 zif Treat zif as interval-scaled Mf 1 60 Cosine Similarity

A document can be represented by thousands of attributes, each recording the frequency of a particular word (such as keywords) or phrase in the document. Other vector objects: gene features in micro-arrays, Applications: information retrieval, biologic taxonomy, gene feature mapping, ... Cosine measure: If d1 and d2 are two vectors (e.g., term-frequency vectors), then cos(d1, d2) = (d1 d2) /||d1|| ||d2|| , where indicates vector dot product, ||d||: the length of vector d 61 Example: Cosine Similarity cos(d1, d2) = (d1 d2) /||d1|| ||d2|| , where indicates vector dot product, ||d|: the length of vector d Ex: Find the similarity between documents 1 and 2. d1 = (5, 0, 3, 0, 2, 0, 0, 2, 0, 0) d2 = (3, 0, 2, 0, 1, 1, 0, 1, 0, 1) d1d2 = 5*3+0*0+3*2+0*0+2*1+0*1+0*1+2*1+0*0+0*1 = 25

||d1||= (5*5+0*0+3*3+0*0+2*2+0*0+0*0+2*2+0*0+0*0) 0.5=(42)0.5 = 6.481 ||d2||= (3*3+0*0+2*2+0*0+1*1+1*1+0*0+1*1+0*0+1*1) 0.5=(17)0.5 = 4.12 cos(d1, d2 ) = 0.94 62 Chapter 2: Getting to Know Your Data Data Objects and Attribute Types Basic Statistical Descriptions of Data Data Visualization Measuring Data Similarity and Dissimilarity Summary 63 Summary

Data attribute types: nominal, binary, ordinal, interval-scaled, ratio-scaled Many types of data sets, e.g., numerical, text, graph, Web, image. Gain insight into the data by: Basic statistical data description: central tendency, dispersion, graphical displays Data visualization: map data onto graphical primitives Measure data similarity Above steps are the beginning of data preprocessing.

Many methods have been developed but still an active area of research. 64 References W. Cleveland, Visualizing Data, Hobart Press, 1993 T. Dasu and T. Johnson. Exploratory Data Mining and Data Cleaning. John Wiley, 2003 U. Fayyad, G. Grinstein, and A. Wierse. Information Visualization in Data Mining and Knowledge Discovery, Morgan Kaufmann, 2001 L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990. H. V. Jagadish, et al., Special Issue on Data Reduction Techniques. Bulletin of the Tech. Committee on Data Eng., 20(4), Dec. 1997 D. A. Keim. Information visualization and visual data mining, IEEE trans. on Visualization and Computer Graphics, 8(1), 2002

D. Pyle. Data Preparation for Data Mining. Morgan Kaufmann, 1999 S. Santini and R. Jain, Similarity measures, IEEE Trans. on Pattern Analysis and Machine Intelligence, 21(9), 1999 E. R. Tufte. The Visual Display of Quantitative Information, 2nd ed., Graphics Press, 2001 C. Yu , et al., Visual data mining of multimedia data for social and behavioral studies, Information Visualization, 8(1), 2009 65