#### Transcript What is Data?

Data Mining: Concepts and Techniques — Chapter 2 — March 21, 2017 Data Mining: Concepts and Techniques 1 What is about Data? General data characteristics Basic data description and exploration Measuring data similarity March 21, 2017 Data Mining: Concepts and Techniques 2 What is Data? Attributes 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 Objects feature A collection of attributes describe an object Object is also known as record, point, case, sample, entity, or instance March 21, 2017 Tid Refund Marital Status Taxable Income Cheat 1 Yes Single 125K No 2 No Married 100K No 3 No Single 70K No 4 Yes Married 120K No 5 No Divorced 95K Yes 6 No Married No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 60K 10 Data Mining: Concepts and Techniques 3 Important Characteristics of Structured Data Dimensionality Curse of dimensionality Sparsity Only presence counts Resolution Patterns depend on the scale Similarity Distance measure March 21, 2017 Data Mining: Concepts and Techniques 4 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 March 21, 2017 Data Mining: Concepts and Techniques 5 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 March 21, 2017 Data Mining: Concepts and Techniques 6 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 March 21, 2017 Data Mining: Concepts and Techniques 7 Attribute Type Description Examples Nominal The 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 Ordinal 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 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 Ratio March 21, 2017 Data Mining: Concepts and Techniques Operations 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 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 March 21, 2017 Data Mining: Concepts and Techniques 9 Types of data sets Record Graph Data Matrix Document Data Transaction Data World Wide Web Molecular Structures Ordered Spatial Data Temporal Data Sequential Data Genetic Sequence Data March 21, 2017 Data Mining: Concepts and Techniques 10 Important Characteristics of Structured Data Dimensionality Sparsity Only presence counts Resolution March 21, 2017 Curse of Dimensionality Patterns depend on the scale Data Mining: Concepts and Techniques 11 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 No Married 100K No 3 No Single 70K No 4 Yes Married 120K No 5 No Divorced 95K Yes 6 No Married No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 60K 10 March 21, 2017 Data Mining: Concepts and Techniques 12 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 Projection of x Load Projection of y load Distance Load Thickness 10.23 5.27 15.22 2.7 1.2 12.65 6.25 16.22 2.2 1.1 March 21, 2017 Data Mining: Concepts and Techniques 13 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. team coach pla y ball score game wi n lost timeout season March 21, 2017 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 Data Mining: Concepts and Techniques 14 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. March 21, 2017 TID Items 1 Bread, Coke, Milk 2 3 4 5 Beer, Bread Beer, Coke, Diaper, Milk Beer, Bread, Diaper, Milk Coke, Diaper, Milk Data Mining: Concepts and Techniques 15 Graph Data Examples: Generic graph and HTML Links 2 1 5 2 <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 5 March 21, 2017 Data Mining: Concepts and Techniques 16 Chemical Data Benzene Molecule: C6H6 March 21, 2017 Data Mining: Concepts and Techniques 17 Ordered Data Sequences of transactions Items/Events An element of the sequence March 21, 2017 Data Mining: Concepts and Techniques 18 Ordered Data Genomic sequence data GGTTCCGCCTTCAGCCCCGCGCC CGCAGGGCCCGCCCCGCGCCGTC GAGAAGGGCCCGCCTGGCGGGCG GGGGGAGGCGGGGCCGCCCGAGC CCAACCGAGTCCGACCAGGTGCC CCCTCTGCTCGGCCTAGACCTGA GCTCATTAGGCGGCAGCGGACAG GCCAAGTAGAACACGCGAAGCGC TGGGCTGCCTGCTGCGACCAGGG March 21, 2017 Data Mining: Concepts and Techniques 19 Ordered Data Spatio-Temporal Data Average Monthly Temperature of land and ocean March 21, 2017 Data Mining: Concepts and Techniques 20 General data characteristics Basic data description and exploration Measuring data similarity March 21, 2017 Data Mining: Concepts and Techniques 21 Mining Data Descriptive Characteristics Motivation Data dispersion characteristics To better understand the data: central tendency, variation and spread 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 March 21, 2017 Data Mining: Concepts and Techniques 22 Measuring the Central Tendency 1 n Mean (algebraic measure) (sample vs. population): x xi n i 1 Weighted arithmetic mean: x N n Trimmed mean: chopping extreme values x Median: A holistic measure w x i 1 n i i w i 1 i Middle value if odd number of values, or average of the middle two values otherwise Estimated by interpolation (for grouped data): median L1 ( Mode Value that occurs most frequently in the data Unimodal, bimodal, trimodal Empirical formula: March 21, 2017 N / 2 ( freq)l freqmedian ) width mean mode 3 (mean median) Data Mining: Concepts and Techniques 23 Symmetric vs. Skewed Data Median, mean and mode of symmetric, positively and negatively skewed data positively skewed March 21, 2017 symmetric negatively skewed Data Mining: Concepts and Techniques 24 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, M, Q3, 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) 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 xi 2 2 i 1 Standard deviation s (or σ) is the square root of variance s2 (or σ2) March 21, 2017 Data Mining: Concepts and Techniques 25 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 March 21, 2017 Data Mining: Concepts and Techniques 26 Histogram Analysis Graph displays of basic statistical class descriptions Frequency histograms March 21, 2017 A univariate graphical method Consists of a set of rectangles that reflect the counts or frequencies of the classes present in the given data Data Mining: Concepts and Techniques 27 Histograms Often Tells More than Boxplots The two histograms shown in the left may have the same boxplot representation March 21, 2017 The same values for: min, Q1, median, Q3, max But they have rather different data distributions Data Mining: Concepts and Techniques 28 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 March 21, 2017 Data Mining: Concepts and Techniques 29 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 March 21, 2017 Data Mining: Concepts and Techniques 30 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 March 21, 2017 Data Mining: Concepts and Techniques 31 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 March 21, 2017 Data Mining: Concepts and Techniques 32 Positively and Negatively Correlated Data The left half fragment is positively correlated March 21, 2017 The right half is negative correlated Data Mining: Concepts and Techniques 33 Not Correlated Data March 21, 2017 Data Mining: Concepts and Techniques 34 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 March 21, 2017 Data Mining: Concepts and Techniques 35 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 March 21, 2017 Data Mining: Concepts and Techniques 36 Used by ermission of M. Ward, Worcester Polytechnic Institute Scatterplot Matrices Matrix of scatterplots (x-y-diagrams) of the k-dim. data March 21, 2017 Data Mining: Concepts and Techniques 37 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 March 21, 2017 Data Mining: Concepts and Techniques 38 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 March 21, 2017 Attr. 2 Attr. 3 Data Mining: Concepts and Techniques Attr. k 39 Parallel Coordinates of a Data Set March 21, 2017 Data Mining: Concepts and Techniques 40 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 March 21, 2017 Data Mining: Concepts and Techniques 41 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 March 21, 2017 Data Mining: Concepts and Techniques 42 Hierarchical Techniques Visualization of the data using a hierarchical partitioning into subspaces. Methods Dimensional Stacking Worlds-within-Worlds Treemap Cone Trees InfoCube March 21, 2017 Data Mining: Concepts and Techniques 43 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 March 21, 2017 Data Mining: Concepts and Techniques 44 Tree-Map of a File System (Schneiderman) March 21, 2017 Data Mining: Concepts and Techniques 45 General data characteristics Basic data description and exploration Measuring data similarity (Sec. 7.2) March 21, 2017 Data Mining: Concepts and Techniques 46 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 March 21, 2017 Data Mining: Concepts and Techniques 47 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 March 21, 2017 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) ... Data Mining: Concepts and Techniques x1p ... xip ... xnp ... 0 48 Example: Data Matrix and Distance Matrix 3 point p1 p2 p3 p4 p1 2 p3 p4 1 p2 0 0 1 2 3 4 5 p1 p2 p3 p4 0 2.828 3.162 5.099 y 2 0 1 1 Data Matrix 6 p1 x 0 2 3 5 p2 2.828 0 1.414 3.162 p3 3.162 1.414 0 2 p4 5.099 3.162 2 0 Distance Matrix (i.e., Dissimilarity Matrix) for Euclidean Distance March 21, 2017 Data Mining: Concepts and Techniques 49 Minkowski Distance Minkowski distance: A popular distance measure d (i, j) q (| x x |q | x x |q ... | x x |q ) i1 j1 i2 j2 ip jp where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) 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 March 21, 2017 Data Mining: Concepts and Techniques 50 Special Cases of Minkowski Distance q = 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 q= 2: (L2 norm) Euclidean distance d (i, j) (| x x |2 | x x |2 ... | x x |2 ) i1 j1 i2 j2 ip jp q . “supremum” (Lmax 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 March 21, 2017 Data Mining: Concepts and Techniques 51 Example: Minkowski Distance point p1 p2 p3 p4 x 0 2 3 5 y 2 0 1 1 L1 p1 p2 p3 p4 p1 0 4 4 6 p2 4 0 2 4 p3 4 2 0 2 p4 6 4 2 0 L2 p1 p2 p3 p4 p1 p2 2.828 0 1.414 3.162 p3 3.162 1.414 0 2 p4 5.099 3.162 2 0 L p1 p2 p3 p4 p1 p2 p3 p4 0 2.828 3.162 5.099 0 2 3 5 2 0 1 3 3 1 0 2 5 3 2 0 Distance Matrix March 21, 2017 Data Mining: Concepts and Techniques 52 Binary Variables 1 0 a b A contingency table for binary data Object i 1 0 c d sum a c b d Distance measure for symmetric d (i, j) binary variables: Distance measure for asymmetric binary variables: Jaccard coefficient (similarity measure for asymmetric binary variables): Object j d (i, j) sum a b cd p bc a bc d bc a bc simJaccard (i, j) a a b c A binary variable is symmetric if both of its states are equally valuable and carry the same weight. March 21, 2017 Data Mining: Concepts and Techniques 53 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 set to 1, and the value N be set to 0 01 0.33 2 01 11 d ( jack , jim ) 0.67 111 1 2 d ( jim , mary ) 0.75 11 2 d ( jack , mary ) March 21, 2017 Data Mining: Concepts and Techniques 54 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 m d (i, j) p p Method 2: Use a large number of binary variables creating a new binary variable for each of the M nominal states March 21, 2017 Data Mining: Concepts and Techniques 55 Ordinal Variables An ordinal variable can be discrete or continuous Order is important, e.g., rank Can be treated like interval-scaled replace xif by their rank map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by zif rif {1,...,M f } rif 1 M f 1 compute the dissimilarity using methods for intervalscaled variables March 21, 2017 Data Mining: Concepts and Techniques 56 Ratio-Scaled Variables Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt Methods: treat them like interval-scaled variables—not a good choice! (why?—the scale can be distorted) apply logarithmic transformation yif = log(xif) March 21, 2017 treat them as continuous ordinal data treat their rank as interval-scaled Data Mining: Concepts and Techniques 57 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 pf 1 ij( f ) dij( f ) d (i, j) pf 1 ij( f ) f is binary or nominal: dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise f is interval-based: use the normalized distance f is ordinal or ratio-scaled Compute ranks rif and r 1 zif M 1 Treat zif as interval-scaled if f March 21, 2017 Data Mining: Concepts and Techniques 58 Vector Objects: Cosine Similarity Vector objects: keywords in documents, gene features in micro-arrays, … Applications: information retrieval, biologic taxonomy, ... Cosine measure: If d1 and d2 are two vectors, then cos(d1, d2) = (d1 d2) /||d1|| ||d2|| , where indicates vector dot product, ||d||: the length 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 d1d2 = 3*1+2*0+0*0+5*0+0*0+0*0+0*0+2*1+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 March 21, 2017 Data Mining: Concepts and Techniques 59 Correlation Analysis (Numerical Data) Correlation coefficient (also called Pearson’s product moment coefficient) rp ,q ( p p)( q q) ( pq) n p q (n 1) p q (n 1) p q where n is the number of tuples, p and q are the respective means of p and q, σp and σq are the respective standard deviation of p and q, and Σ(pq) is the sum of the pq cross-product. If rp,q > 0, p and q are positively correlated (p’s values increase as q’s). The higher, the stronger correlation. rp,q = 0: independent; rpq < 0: negatively correlated March 21, 2017 Data Mining: Concepts and Techniques 60 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 pk ( pk mean( p)) / std ( p) qk (qk mean(q)) / std (q) correlation( p, q) p q March 21, 2017 Data Mining: Concepts and Techniques 61 Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1. March 21, 2017 Data Mining: Concepts and Techniques 62