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Data Mining: Data Lecture Notes for Chapter 2 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› What is Data?  Collection of data objects and their attributes  An attribute is a property or characteristic of an object A...

Data Mining: Data Lecture Notes for Chapter 2 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› What is Data?  Collection of data objects and their attributes  An attribute is a property or characteristic of an object Attributes – Examples: eye color of a person, temperature, etc. – Attribute is also known as variable, field, characteristic, or feature Objects  A collection of attributes describe an object – Object is also known as record, point, case, sample, entity, or instance © Tan,Steinbach, Kumar Introduction to Data Mining 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 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Measurement of Length  The way you measure an attribute is somewhat may not match the attributes properties. A 5 1 B 7 This scale preserves only the 2 C 8 3 ordering property of length. D 10 4 E 15 This scale preserves the ordering and additivity properties of length. 5 A mapping to lengths to numbers that captures only the order properties of length A mapping to lengths to numbers that captures both order and additivity properties of length Thus, an attribute can be measured in a way that does not capture all the properties of the attribute. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Types of Attributes  There are different types of attributes – Nominal  Examples: ID numbers, eye color, zip codes – Ordinal  Examples: rankings (e.g., taste of potato chips on a scale from 1-10), grades, height in {tall, medium, short} – Interval  Examples: calendar dates, temperatures in Celsius or Fahrenheit. – Ratio  Examples: temperature in Kelvin, length, time, counts © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Properties of Attribute Values  The type of an attribute depends on which of the following properties it has: =  < > + */ – – – – Distinctness: Order: Addition: Multiplication: – – – – Nominal attribute: distinctness Ordinal attribute: distinctness & order Interval attribute: distinctness, order & addition Ratio attribute: all 4 properties © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 Operations This categorization of attributes is due to S. S. Stevens Comments Categorical (or qualitative) attribute Transformation 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}. Numeric (Quantitative) attributes Attribute Level 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). new_value = a * old_value Length can be measured in meters or feet. Ratio The types of attributes can also be described in terms of transformations that do not change the meaning of an attribute. Discrete and Continuous Attributes  Discrete Attribute – Has only a finite or countably infinite set of values – Examples: 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 assume  only two values, e.g., true/false, yes/no, male/female, or 0/1. 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Asymmetric Attributes  Only presence (a non-zero attribute value) is regarded as important      Words present in documents Items present in customer transactions If we met a friend in the grocery store would we ever say the following? “I see our purchases are very similar since we didn’t buy most of the same things.” It is more meaningful and more efficient to focus on the nonzero values. Binary attributes where only non-zero values are important are called asymmetric binary attributes. – Association analysis uses asymmetric attributes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Types of data sets    Record – Data Matrix – Document Data – Transaction Data Graph-based – World Wide Web – Molecular Structures Ordered – Spatial Data – Temporal Data – Sequential Data – Genetic Sequence Data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Important Characteristics of Data – Dimensionality (number of attributes)  Curse of Dimensionality – Sparsity  Only presence counts – Resolution  Patterns depend on the scale – Size  Type of analysis may depend on size of data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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. TID Items 1 Bread, Coke, Milk 2 3 4 5 Beer, Bread Beer, Coke, Diaper, Milk Beer, Bread, Diaper, Milk Coke, Diaper, Milk © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Graph Data  Examples: Generic graph, a molecule, and webpages 2 1 5 2 5 Benzene Molecule: C6H6 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Ordered Data  Sequences of transactions Items/Events An element of the sequence © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Ordered Data  Genomic sequence data GGTTCCGCCTTCAGCCCCGCGCC CGCAGGGCCCGCCCCGCGCCGTC GAGAAGGGCCCGCCTGGCGGGCG GGGGGAGGCGGGGCCGCCCGAGC CCAACCGAGTCCGACCAGGTGCC CCCTCTGCTCGGCCTAGACCTGA GCTCATTAGGCGGCAGCGGACAG GCCAAGTAGAACACGCGAAGCGC TGGGCTGCCTGCTGCGACCAGGG © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Ordered Data  Spatio-Temporal Data Average Monthly Temperature of land and ocean © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Data Quality Poor data quality negatively affects many data processing efforts “The most important point is that poor data quality is an unfolding disaster. – Poor data quality costs the typical company at least ten percent (10%) of revenue; twenty percent (20%) is probably a better estimate.” Thomas C. Redman, DM Review, August 2004   Data mining example: a classification model for detecting people who are loan risks is built using poor data – Some credit-worthy candidates are denied loans – More loans are given to individuals that default © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Data Quality  What kinds of data quality problems? How can we detect problems with the data? What can we do about these problems?  Examples of data quality problems:   – Noise and outliers – missing values – duplicate data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Noise  Noise refers to modification of original values – Examples: distortion of a person’s voice when talking on a poor phone and “snow” on television screen © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Outliers  Outliers are data objects with characteristics that are considerably different than most of the other data objects in the data set – Case 1: Outliers are noise that interferes with data analysis – Case 2: Outliers are the goal of our analysis  Credit card fraud  Intrusion detection © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Missing Values  Reasons for missing values – Information is not collected (e.g., people decline to give their age and weight) – Attributes may not be applicable to all cases (e.g., annual income is not applicable to children)  Handling missing values – – – – Eliminate Data Objects Estimate Missing Values ?: missing value Ignore the Missing Value During Analysis Replace with all possible values (weighted by their probabilities) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Duplicate Data  Data set may include data objects that are duplicates, or almost duplicates of one another – Major issue when merging data from heterogeous sources  Examples: – Same person with multiple email addresses  Data cleaning – Process of dealing with duplicate data issues © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Data Preprocessing        Aggregation Sampling Dimensionality Reduction Feature Subset Selection Feature Creation Discretization and Binarization Attribute Transformation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Aggregation  Combining two or more attributes (or objects) into a single attribute (or object)  Purpose – Data reduction  Reduce the number of attributes or objects – Change of scale  Cities aggregated into regions, states, countries, etc. – More “stable” data  Aggregated data tends to have less variability © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Aggregation Variation of Precipitation in Australia apparent decrease in the amount of std. dev. owing to aggregation Standard Deviation of Average Monthly Precipitation © Tan,Steinbach, Kumar Introduction to Data Mining Standard Deviation of Average Yearly Precipitation 4/18/2004 ‹#› 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Types of Sampling  Simple Random Sampling – There is an equal probability of selecting any particular item  Sampling without replacement – As each item is selected, it is removed from the population  Sampling with replacement – Objects 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 sampling – Split the data into several partitions; then draw random samples from each partition © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Sample Size 8000 points © Tan,Steinbach, Kumar 2000 Points Introduction to Data Mining 500 Points 4/18/2004 ‹#› Sample Size  What sample size is necessary to get at least one object from each of 10 equal-sized groups. The figure showing an idealized set of clusters (groups) from which these points might be drawn © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Curse of Dimensionality  When dimensionality increases, data becomes increasingly sparse in the space that it occupies  Definitions of density and distance between points, which is critical for clustering and outlier detection, become less meaningful • Randomly generate 500 points • Compute difference between max and min distance between any pair of points © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Dimensionality Reduction  Purpose: – Avoid curse of dimensionality – Reduce amount of time and memory required by data mining algorithms – Allow data to be more easily visualized – May help to eliminate irrelevant features or reduce noise  Techniques – Principle Component Analysis (PCA) – Singular Value Decomposition – Others: supervised and non-linear techniques © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Dimensionality Reduction: PCA  Goal is to find a projection that captures the largest amount of variation in data x2 e x1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Dimensionality Reduction: PCA   Find the eigenvectors of the covariance matrix The eigenvectors define the new space x2 e x1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Dimensionality Reduction: PCA  Principal Components Analysis (PCA) is a linear algebra technique for continuous attributes that finds new attributes (principal components) that – (1) are linear combinations of the original attributes, – (2) are orthogonal (perpendicular) to each other, and – (3) capture the maximum amount of variation in the data. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Dimensionality Reduction: PCA © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Feature Subset Selection  Another way to reduce dimensionality of data  Redundant features – duplicate much or all of the information contained in one or more other attributes – Example: purchase price of a product and the amount of sales tax paid  Irrelevant features – contain no information that is useful for the data mining task at hand – Example: students' ID is often irrelevant to the task of predicting students' GPA © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Feature Creation  Create new attributes that can capture the important information in a data set much more efficiently than the original attributes  Three general methodologies: – Feature extraction  Example: extracting edges from images – Feature construction  Example: dividing mass by volume to get density – Mapping data to new space  Example: Fourier and wavelet analysis © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Mapping Data to a New Space  Fourier transform  Wavelet transform Two Sine Waves © Tan,Steinbach, Kumar Two Sine Waves + Noise Introduction to Data Mining Frequency 4/18/2004 ‹#› Discretization  Discretization is the process of converting a continuous attribute into an ordinal attribute – A potentially infinite number of values are mapped into a small number of categories – Discretization is commonly used in classification – Many classification algorithms work best if both the independent and dependent variables have only a few values – We give an illustration of the usefulness of discretization using the Iris data set © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Iris Sample Data Set  Iris Plant data set. – Can be obtained from the UCI Machine Learning Repository http://www.ics.uci.edu/~mlearn/MLRepository.html – From the statistician Douglas Fisher – Three flower types (classes):  Setosa  Versicolour  Virginica – Four (non-class) attributes  Sepal width and length Virginica. Robert H. Mohlenbrock. USDA  Petal width and length NRCS. 1995. Northeast wetland flora: Field office guide to plant species. Northeast National Technical Center, Chester, PA. Courtesy of USDA NRCS Wetland Science Institute. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Discretization: Iris Example Petal width low or petal length low implies Setosa. Petal width medium or petal length medium implies Versicolour. Petal width high or petal length high implies Virginica. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Discretization: Iris Example …  How can we tell what the best discretization is? – Unsupervised discretization: find breaks in the data values 50  Example: 40 Counts Petal Length 30 20 10 0 0 2 4 6 Petal Length 8 – Supervised discretization: Use class labels to find breaks © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Discretization Without Using Class Labels Data consists of four groups of points and two outliers. Data is onedimensional, but a random y component is added to reduce overlap. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Discretization Without Using Class Labels Equal interval width approach used to obtain 4 values. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Discretization Without Using Class Labels Equal frequency approach used to obtain 4 values. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Discretization Without Using Class Labels K-means approach to obtain 4 values. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Discretization Without Using Class Labels Data Equal interval width Equal frequency © Tan,Steinbach, Kumar K-means Introduction to Data Mining 4/18/2004 ‹#› Discretization Using Class Labels  Entropy based approach 3 categories for both x and y 5 categories for both x and y Discretizing x and y attributes for four groups (classes) of points. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Binarization  Binarization maps a continuous or categorical attribute into one or more binary variables  Typically used for association analysis  Often convert a continuous attribute to a categorical attribute and then convert a categorical attribute to a set of binary attributes – Association analysis needs asymmetric binary attributes – Examples: eye color and height measured as {low, medium, high} © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Binarization © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Attribute Transformation  A 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 values – Simple functions: xk, log(x), ex, |x| – Standardization and Normalization © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Attribute Transformation  An attribute transform is a 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 values – Simple functions: xk, log(x), ex, |x| – Normalization Refers to various techniques to adjust to differences among attributes in terms of frequency of occurrence, mean, variance, range  Take out unwanted, common signal, e.g., seasonality – In statistics, standardization refers to subtracting off the means and dividing by the standard deviation  © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Example: Sample Time Series of Plant Growth Minneapolis Net Primary Production (NPP) is a measure of plant growth used by ecosystem scientists. Correlations between time series Correlations between time series Minneapolis Minneapolis 1.0000 Atlanta 0.7591 Sao Paolo -0.7581 © Tan,Steinbach, Kumar Atlanta 0.7591 1.0000 -0.5739 Introduction to Data Mining Sao Paolo -0.7581 -0.5739 1.0000 4/18/2004 ‹#› Seasonality Accounts for Much Correlation Minneapolis Normalized using monthly Z Score: Subtract off monthly mean and divide by monthly standard deviation Correlations between time series Correlations between time series Minneapolis Minneapolis 1.0000 Atlanta 0.0492 Sao Paolo 0.0906 © Tan,Steinbach, Kumar Atlanta 0.0492 1.0000 -0.0154 Introduction to Data Mining Sao Paolo 0.0906 -0.0154 1.0000 4/18/2004 ‹#› Similarity and Dissimilarity  Similarity – Numerical measure of how alike two data objects are. – Is higher when objects are more alike. – Often falls in the range [0,1]  Dissimilarity – 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Similarity/Dissimilarity for Simple Attributes p and q are the attribute values for two data objects. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Similarity/Dissimilarity transformation examples For the dissimilaritiy values of 0, 1, 10, 100; transformation equation results in similaritiy values of 1, 0.5, 0.09, 0.01, respectively. transformation equation results in similaritiy values of 1.00, 0.99, 0.00, 0.00, respectively. transformation equation results in similaritiy values of 1.00, 0.37, 0.00, 0.00, respectively. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Euclidean Distance  Euclidean Distance dist  n  ( pk k 1  qk ) 2 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Euclidean Distance 3 point p1 p2 p3 p4 p1 2 p3 p4 1 p2 0 0 1 2 3 4 5 y 2 0 1 1 6 p1 p1 p2 p3 p4 x 0 2 3 5 0 2.828 3.162 5.099 p2 2.828 0 1.414 3.162 p3 3.162 1.414 0 2 p4 5.099 3.162 2 0 Distance Matrix © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Minkowski Distance  Minkowski Distance is a generalization of Euclidean Distance n dist  (  | pk  qk k 1 1 r r |) 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Minkowski Distance: Examples  r = 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 (L2 norm)  r  . “supremum” (Lmax norm, L norm) distance. – This is 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Mahalanobis Distance  is the covariance matrix For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Mahalanobis Distance Covariance Matrix: C  0 .3 0 .2    0.2 0.3 A: (0.5, 0.5) B B: (0, 1) A C: (1.5, 1.5) Mahal(A,B) = 5 Mahal(A,C) = 4 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Common Properties of a Distance (Dissimilarity)  Distances, such as the Euclidean distance, have some well known properties. 1. d(p, q)  0 for all p and q and d(p, q) = 0 only if p = q. (Positive definiteness) 2. d(p, q) = d(q, p) for all p and q. (Symmetry) 3. d(p, r)  d(p, q) + d(q, r) for all points p, q, and r. (Triangle Inequality) where d(p, q) is the distance (dissimilarity) between points (data objects), p and q.  A distance that satisfies these properties is a metric © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Common Properties of a Similarity  Similarities, also have some well known properties. 1. s(p, q) = 1 (or maximum similarity) only if p = q. 2. 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Similarity Between Binary Vectors  Common 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) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› SMC versus Jaccard: Example p= 1000000000 q= 0000001001 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Cosine Similarity If d1 and d2 are two document vectors, then cos( d1, d2 ) = <d1,d2> / ||d1|| ||d2|| , where <d1,d2> indicates inner product or vector dot product of vectors, d1 and d2, and || d || is 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 <d1, d2> = 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 = 2.449 cos(d1, d2 ) = 0.3150 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Extended Jaccard Coefficient (Tanimoto)  Variation of Jaccard for continuous or count attributes – Reduces to Jaccard for binary attributes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Correlation   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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Correlation measures the linear relationship between objects © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1. If the correlation is 0, then there is no linear relationship between the attributes. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Drawback of Correlation   x = (-3, -2, -1, 0, 1, 2, 3) y = (9, 4, 1, 0, 1, 4, 9) yi = x i 2  mean(x) = 0, mean(y) = 4 std(x) = 2.16, std(y) = 3.74  corr = (-3)(5)+(-2)(0)+(-1)(-3)+(0)(-4)+(1)(-3)+(2)(0)+3(5) / ( 6 * 2.16 * 3.74 )  =0 If the correlation is 0, then there is no linear relationship between the attributes of the two data objects. However, non-linear relationships may still exist as in this example. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› General Approach for Combining Similarities  Sometimes attributes are of many different types, but an overall similarity is needed. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Using Weights to Combine Similarities  May not want to treat all attributes the same. – Use weights wk which are between 0 and 1 and sum to 1. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Density     Measures the degree to which data objects are close to each other in a specified area. The notion of density is closely related to that of proximity. Concept of density is typically used for clustering and anomaly detection. Examples: – Euclidean density  Euclidean density = number of points per unit volume – Probability density  Estimate what the distribution of the data looks like – Graph-based density  Connectivity © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#› Euclidean Density: Grid-based Approach  Simplest approach is to divide region into a number of rectangular cells of equal volume and define density as # of points the cell contains Grid-based density. © Tan,Steinbach, Kumar Introduction to Data Mining Counts for each cell. 4/18/2004 ‹#› Euclidean Density: Center-Based  Euclidean density is the number of points within a specified radius of the point Illustration of center-based density. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›