Data Mining Concepts and Techniques Chapter 2 Jiawei

Data Mining: Concepts and Techniques — Chapter 2 — Jiawei Han, Micheline Kamber, and Jian Pei University of Illinois at Urbana-Champaign Simon Fraser University © 2008 Jiawei Han, Micheline Kamber, and Jian Pei. All rights reserved. June 5, 2021 Data Mining: Concepts and Techniques 1

June 5, 2021 Data Mining: Concepts and Techniques 2

Chapter 2: Getting to Know Your Data n Data Objects and Attribute Types n Basic Statistical Descriptions of Data n Data Visualization n Measuring Data Similarity and Dissimilarity n Summary 3

Types of Data Sets n n Record n Relational records n Data matrix, e. g. , numerical matrix, crosstabs n Document data: text documents: termfrequency vector n Transaction data Graph and network n World Wide Web n Social or information networks n Molecular Structures Ordered n Video data: sequence of images n Temporal data: time-series n Sequential Data: transaction sequences n Genetic sequence data Spatial, image and multimedia: n Spatial data: maps n Image data: n Video data: 4

Important Characteristics of Structured Data n Dimensionality n n Sparsity n n Only presence counts Resolution n n Curse of dimensionality Patterns depend on the scale Distribution n Centrality and dispersion 5

Data Objects n Data sets are made up of data objects. n A data object represents an entity. n Examples: n n sales database: customers, store items, sales n medical database: patients, treatments n university database: students, professors, courses Also called samples , examples, instances, data points, objects, tuples. n Data objects are described by attributes. n Database rows -> data objects; columns ->attributes. 6

Attributes n Attribute (or dimensions, features, variables): a data field, representing a characteristic or feature of a data object. n n n E. g. , customer _ID, name, address A set of attributes used to describe an object is called an attribute vector or feature vector Types: n Nominal, Binary, ordinal, or numeric n Numeric: quantitative n Interval-scaled n Ratio-scaled 7

Attribute Types n n Nominal: categories, codes, states, or “names of things” n Hair_color = {auburn, black, blond, brown, grey, red, white} n marital status, occupation, ID numbers, zip codes n Also referred as categorical n Nominal attributes can be numeric but do not have any meaningful order and are not quantitative, e. g. customer_ID n No sense to find mean or median n Interesting feature: mode (commonly occurring value) Binary n Nominal attribute with only 2 states (0 and 1) n Symmetric binary: both outcomes equally important n e. g. , gender n Asymmetric binary: outcomes not equally important. n e. g. , medical test ( positive vs. negative) n Convention: assign 1 to most important outcome (e. g. , HIV positive) 8

Attribute Types n Ordinal n Values have a meaningful order (ranking) but magnitude between successive values is not known. n Size = {small, medium, large}, grades, army rankings n Measure subjective assessments of qualities that cannot be measured objectively n Used in surveys for ratings (0: very dissatisfied, 1: somewhat satisfied, 2: neutral, 3: satisfied, 4: very satisfied) n Central tendency represented by mode and median (middle value) 9

Numeric Attribute Types n n n Quantity (integer or real-valued) Interval n Measured on a scale of equal-sized units n Values have order, can be positive, 0, negative n E. g. , temperature in C˚or F˚, calendar dates n No true zero-point n Can compute mean, median and mode Ratio n Inherent zero-point n 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˚). n e. g. , temperature in Kelvin, length, counts, monetary quantities 10

Discrete vs. Continuous Attributes n n Discrete Attribute n Has only a finite or countably infinite set of values n E. g. , zip codes, profession, or the set of words in a collection of documents n Sometimes, represented as integer variables n Note: Binary attributes are a special case of discrete attributes Continuous Attribute n Has real numbers as attribute values n E. g. , temperature, height, or weight n Practically, real values can only be measured and represented using a finite number of digits n Continuous attributes are typically represented as floating-point variables 11

Chapter 2: Getting to Know Your Data n Data Objects and Attribute Types n Basic Statistical Descriptions of Data n Data Visualization n Measuring Data Similarity and Dissimilarity n Summary 12

Basic Statistical Descriptions of Data Motivation n To better understand the data: central tendency, variation and spread n Data dispersion characteristics n median, max, min, quantiles, outliers, variance, etc. n Numerical dimensions correspond to sorted intervals n Data dispersion: analyzed with multiple granularities of precision n Boxplot or quantile analysis on sorted intervals n Dispersion analysis on computed measures n Folding measures into numerical dimensions n Boxplot or quantile analysis on the transformed cube n 13

Measuring the Central Tendency n Mean (algebraic measure) (sample vs. population): Note: n is sample size and N is population size. n n Weighted arithmetic mean: n Trimmed mean: chopping extreme values Median: n Middle value if odd number of values, or average of the middle two values otherwise n Estimated by interpolation (for grouped data): n L 1 – lower boundary of the median interval n N – number of values in the entire data set n (∑freq)l – sum of the frequencies of all of the intervals that are lower than the median interval n Freq median – frequency of the median interval 14

Measuring the Central Tendency n Mode n Value that occurs most frequently in the data n Unimodal, bimodal, trimodal – data sets with one, two, or three modes n Unimodal data that are moderately skewed (asymmertical), we have the following empirical relation: n This implies that the mode for unimodal frequency curves that are moderately skewed can easily be approximated if the mean and median values are known n Midrange n Average of the largest and smallest values in the set n Easy to compute using SQL aggregate function max() and min() 15

Symmetric vs. Skewed Data n Median, mean and mode of symmetric, positively and negatively skewed data positively skewed June 5, 2021 symmetric negatively skewed Data Mining: Concepts and Techniques 16

Measuring the Dispersion of Data n Quartiles, outliers and boxplots n Quartiles: Q 1 (25 th percentile), Q 3 (75 th percentile) n Inter-quartile range: IQR = Q 3 – Q 1 n Five number summary: min, Q 1, median, Q 3, max n Boxplot: ends of the box are the quartiles; median is marked by a line within the box; add whiskers, and plot outliers individually n n Outlier: usually, a value higher/lower than 1. 5 x IQR Variance and standard deviation (sample: s, population: σ) n Variance: (algebraic, scalable computation) n Standard deviation s (or σ) is the square root of variance s 2 (or σ2) 17

Boxplot Analysis n Five-number summary of a distribution n n Minimum, Q 1, Median, Q 3, Maximum Boxplot n n n 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 (1. 5 x IQR), plotted individually Boxplots can be computed in O(nlogn) time 18

Boxplot Analysis 19

Visualization of Data Dispersion: 3 -D Boxplots June 5, 2021 Data Mining: Concepts and Techniques 20

Properties of Normal Distribution Curve n The normal (distribution) curve n From μ–σ to μ+σ: contains about 68% of the measurements (μ: mean, σ: standard deviation) n From μ– 2σ to μ+2σ: contains about 95% of it n From μ– 3σ to μ+3σ: contains about 99. 7% of it 21

Graphic Displays of Basic Statistical Descriptions n Boxplot: graphic display of five-number summary n Histogram: x-axis are values, y-axis repres. frequencies n Quantile plot: each value xi is paired with fi indicating that approximately fi x 100% of data are xi n Quantile-quantile (q-q) plot: graphs the quantiles of one univariant distribution against the corresponding quantiles of another n Scatter plot: each pair of values is a pair of coordinates and plotted as points in the plane 22

Histogram Analysis n n Histogram: Graph display of tabulated frequencies, shown as bars It shows what proportion of cases fall into each of several categories Differs from a bar chart in that it is 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 The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent 40 35 30 25 20 15 10 5 0 10000 30000 50000 70000 90000 23

Histograms Often Tell More than Boxplots n The two histograms shown in the left may have the same boxplot representation n n The same values for: min, Q 1, median, Q 3, max But they have rather different data distributions 24

Quantile Plot n n Displays all of the data (allowing the user to assess both the overall behavior and unusual occurrences) Plots quantile information n For a data xi data sorted in increasing order, fi indicates that approximately fi x 100% of the data are below or equal to the value xi n Let Data Mining: Concepts and Techniques 25

Quantile-Quantile (Q-Q) Plot n n n Graphs the quantiles of one univariate distribution against the corresponding quantiles of another View: If 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. 26

Scatter plot n n Provides a first look at bivariate data to see clusters of points, outliers, or correlationships Each pair of values is treated as a pair of coordinates and plotted as points in the plane 27

Positively and Negatively Correlated Data n X and Y are correlated if one attribute implies the other n The left half fragment is positively correlated n The right half is negative correlated 28

Uncorrelated Data 29

Chapter 2: Getting to Know Your Data n Data Objects and Attribute Types n Basic Statistical Descriptions of Data n Data Visualization n Measuring Data Similarity and Dissimilarity n Summary 30

Data Visualization n Why data visualization? n n n 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 Categorization of visualization methods: n Pixel-oriented visualization techniques n Geometric projection visualization techniques n Icon-based visualization techniques n Hierarchical visualization techniques n Visualizing complex data and relations 31

Pixel-Oriented Visualization Techniques n n n 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 32

Space-filling curves n n Space-filling curve n Curve whose range covers the entire ndimensional unit hypercube Filling a window by laying out the data records in a linear way may not work well for a window Some frequently used 2 -dimensional space-filling curves 33

Laying Out Pixels in Circle Segments n To save space and show the connections among multiple dimensions, the windows are located side by side in the shape of segments in a circle (a) Representing a data record in circle segment (b) Laying out pixels in circle segment 34

Geometric Projection Visualization Techniques n n n Visualization of geometric transformations help find interesting projections of multidimensional data sets Help visualize high domensional space on a 2 -dimensional display Methods n Direct visualization n Scatterplot and scatterplot matrices n Landscapes n Projection pursuit technique: Help users find meaningful projections of multidimensional data n Prosection views n Hyperslice n Parallel coordinates 35

Direct Data Visualization Ribbons with Twists Based on Vorticity Data Mining: Concepts and Techniques 36

Visualization using Scatter Plots X, Y - spacial attributes, shapes represent third dimension 37

Scatter Plot showing 4 -dimensional data 38

• IRIS Data set • 450 samples from each of three species of iris flowers • 5 dimensions in data set – length, width of sepal and petal and the species • Less effective with higher dimensionality Used by ermission of M. Ward, Worcester Polytechnic Institute Scatterplot Matrices – dimension > 4 Matrix of scatterplots (x-y-diagrams) of the k-dim. data [total of (k 2/2 -k) scatterplots] 39

Used by permission of B. Wright, Visible Decisions Inc. Landscapes n n news articles visualized as a landscape Visualization of the data as perspective landscape The data needs to be transformed into a (possibly artificial) 2 D spatial representation which preserves the characteristics of the data 40

Parallel Coordinates n n 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 Limitation – cannot effectively show a data set of many records 41

Parallel Coordinates of a Data Set 42

Icon-Based Visualization Techniques n Visualization of the data values as features of icons n Typical visualization methods n n Chernoff Faces n Stick Figures General techniques n n n 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 43

Chernoff Faces n n 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 Math. World--A Wolfram Web Resource. mathworld. wolfram. com/Chernoff. Face. html 44

Chernoff Faces n n Display multidimensional data of up to eighteen variables (or dimensions) Limitations n Relating multiple relationships n Specific data values are not shown n Features of the face vary in perceived importance n The similarity of two faces can vary depending on the order in which dimensions are assigned to facial characteristics 45

Stick Figure used by permission of G. Grinstein, University of Massachusettes at Lowell A census data figure showing age, income, gender, education, etc. A 5 -piece stick figure (1 body and 4 limbs w. different angle/length) Data Mining: Concepts and Techniques 46

Hierarchical Visualization Techniques n n Visualization of the data using a hierarchical partitioning into subspaces Methods n Dimensional Stacking n Worlds-within-Worlds n Tree-Map n Cone Trees n Info. Cube 47

Dimensional Stacking n n n 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 48

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-, y-axes 49

Worlds-within-Worlds (n-Vision) Assign the function and two most important parameters to innermost world n Fix all other parameters at constant values - draw other (1 or 2 or 3 dimensional worlds choosing these as the axes) n Software that uses this paradigm n n n N–vision: Dynamic interaction through data glove and stereo displays, including rotation, scaling (inner) and translation (inner/outer) Auto Visual: Static interaction by means of queries 50

Tree-Map n n 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) Newsmap: Google News Stories in 2005 MSR Netscan Image 51

Tree-Map of a File System (Schneiderman) 52

Info. Cube n n 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 53

Three-D Cone Trees n 3 D cone tree visualization technique works well for up to a thousand nodes or so n n First build a 2 D circle tree that arranges its nodes in concentric circles centered on the root node Cannot avoid overlaps when projected to 2 D G. Robertson, J. Mackinlay, S. Card. “Cone Trees: Animated 3 D 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 54

Visualizing Complex Data and Relations n n n Visualizing non-numerical data: text and social networks Tag cloud: visualizing statistic of user-generated tags n The importance of tag is represented by font size/color Tags used in two ways n n For single item, size of tag represents number of times it is applied For multiple items, size represents number of items tag is applied

Visualizing Complex Data and Relations n Visualize complex relations among data entries n Disease influence graph to visualize correlation between diseases n Nodes – diseases, size – proportional to prevalence of the disease n Link – shows strong correlation between diseases 56

Chapter 2: Getting to Know Your Data n Data Objects and Attribute Types n Basic Statistical Descriptions of Data n Data Visualization n Measuring Data Similarity and Dissimilarity n Summary 57

Similarity and Dissimilarity n n n Similarity n Numerical measure of how alike two data objects are n Value is higher when objects are more alike n Often falls in the range [0, 1] Dissimilarity (e. g. , distance) n Numerical measure of how different two data objects are n Lower when objects are more alike n Minimum dissimilarity is often 0 n Upper limit varies Proximity refers to a similarity or dissimilarity 58

Data Matrix and Dissimilarity Matrix n n n Data matrix n n data points with p dimensions/attributes n Two modes Similarity in terms of dissimilarity n Sim(i, j) = 1 – d(i, j) Dissimilarity matrix n n data points, but registers only the distance n A triangular matrix n Single mode 59

Proximity Measure for Nominal Attributes n n Can take 2 or more states, e. g. , red, yellow, blue, green (generalization of a binary attribute) Method 1: Simple matching n n m: # of matches (# of attributes for which i and j are in the same state), p: total # of variables Method 2: Use a large number of binary attributes n Nominal attributes can be encoded by asymmetric binary attributes by creating a new binary attribute for each of the M nominal states 60

Proximity Measure for Nominal Attributes Take only test-1, which is nominal. p = 1 Dissimilarity matrix = = 61

Proximity Measure for Binary Attributes Object j n A contingency table for binary data Object i n Distance measure for symmetric binary variables: n Distance measure for asymmetric binary variables: n Jaccard coefficient (similarity measure for asymmetric binary variables): n Note: Jaccard coefficient is the same as “coherence”: 62

Dissimilarity between Binary Variables n Example 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 be 0 63

Standardizing Numeric Data n Z-score: n n X: raw score to be standardized, μ: mean of the population, σ: 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 where n n standardized measure (z-score): Using mean absolute deviation is more robust than using standard deviation 64

Example: Data Matrix and Dissimilarity Matrix Data Matrix Dissimilarity Matrix (with Euclidean Distance) 65

Distance on Numeric Data: Minkowski Distance n Minkowski distance: A popular distance measure where i = (xi 1, xi 2, …, xip) and j = (xj 1, xj 2, …, xjp) are two p-dimensional data objects, and h is the order (the distance so defined is also called Lh norm) n n Properties n d(i, j) > 0 if i ≠ j, and d(i, i) = 0 (Positive definiteness) n d(i, j) = d(j, i) (Symmetry) n d(i, j) d(i, k) + d(k, j) (Triangle Inequality) A distance that satisfies these properties is a metric 66

Special Cases of Minkowski Distance n n n h = 1: Manhattan (city block, L 1 norm) distance n E. g. , the Hamming distance: the number of bits that are different between two binary vectors h = 2: (L 2 norm) Euclidean distance h . “supremum” (Lmax norm, L norm and Chebyshev) distance. n This is the maximum difference between any component (attribute) of the vectors 67

Special Cases of Minkowski Distance n Weighted Euclidean distance 68

Example: Minkowski Distance Dissimilarity Matrices Manhattan (L 1) Euclidean (L 2) Supremum 69

Ordinal Variables n An ordinal variable can be discrete or continuous n Order is important, e. g. , rank n Can be treated like interval-scaled n n n replace xif by their rank map the range of each attribute onto [0, 1] by replacing i -th object in the f -th attribute by compute the dissimilarity using methods for intervalscaled attributes 70

Ordinal Variables n n n Example: consider object identifier and test-2 States – fair, good, excellent Mf = 3 Step 1 - Replace the states by their ranks Step 2 – normalize ranks by mapping 1 to 0, 2 to 0. 5, 3 to 1 Step 3 – Euclidean distance 71

Attributes of Mixed Type n 72

Attributes of Mixed Type n 73

Attributes of Mixed Type n 74

Attributes of Mixed Type n 75

Cosine Similarity n 76

Example: Cosine Similarity n 77

Cosine Similarity for Binary Attributes n 78

Chapter 2: Getting to Know Your Data n Data Objects and Attribute Types n Basic Statistical Descriptions of Data n Data Visualization n Measuring Data Similarity and Dissimilarity n Summary 79

Summary n Data attribute types: nominal, binary, ordinal, interval-scaled, ratioscaled n Many types of data sets, e. g. , numerical, text, graph, Web, image. n Gain insight into the data by: n Basic statistical data description: central tendency, dispersion, graphical displays n Data visualization: map data onto graphical primitives n Measure data similarity n Above steps are the beginning of data preprocessing. n Many methods have been developed but still an active area of research.

References n n n n n 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, 2 nd ed. , Graphics Press, 2001 C. Yu , et al, Visual data mining of multimedia data for social and behavioral studies, Information Visualization, 8(1), 2009

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