Data Mining: Data Lecture Notes for Chapter 2 Introduction to Data Mining , 2nd Edition by Tan, Steinbach, Karpatne, Kumar 01/22/2018 Introduction to Data Mining, 2nd Edition 1 Outline Attributes and Objects Types of Data Data Quality Similarity and Distance Data Preprocessing 01/22/2018 Introduction to Data Mining, 2nd Edition 2 What is Data? Attributes Collection of data objects and their attributes – Examples: eye color of a person, temperature, etc. – Attribute is also known as variable, field, characteristic, dimension, or feature Objects An attribute is a property or characteristic of an object A collection of attributes describe an object – Object is also known as record, point, case, sample, entity, or instance 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 Single 90K Yes 10 No 10 60K A More Complete View of Data Data may have parts The different parts of the data may have relationships More generally, data may have structure Data can be incomplete We will discuss this in more detail later 01/22/2018 Introduction to Data Mining, 2nd Edition 4 Attribute Values Attribute values are numbers or symbols assigned to an attribute for a particular object 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 ◆ 01/22/2018 Introduction to Data Mining, 2nd Edition 5 Measurement of Length The way you measure an attribute may not match the attributes properties. 5 A 1 B 7 This scale preserves only the ordering property of length. 2 C 8 3 D 10 4 E 15 5 This scale preserves the ordering and additvity properties of length. 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 {tall, medium, short} – Interval ◆ Examples: calendar dates, temperatures in Celsius or Fahrenheit. – Ratio ◆ 01/22/2018 Examples: temperature in Kelvin, length, time, counts Introduction to Data Mining, 2nd Edition 7 Properties of Attribute Values The type of an attribute depends on which of the following properties/operations it possesses: – Distinctness: =  – Order: < > – Differences are + meaningful : – Ratios are meaningful * / – Nominal attribute: distinctness – Ordinal attribute: distinctness & order – Interval attribute: distinctness, order & meaningful differences – Ratio attribute: all 4 properties/operations 01/22/2018 Introduction to Data Mining, 2nd Edition 8 Difference Between Ratio and Interval Is it physically meaningful to say that a temperature of 10 ° is twice that of 5° on – the Celsius scale? – the Fahrenheit scale? – the Kelvin scale? Consider measuring the height above average – If Bill’s height is three inches above average and Bob’s height is six inches above average, then would we say that Bob is twice as tall as Bill? – Is this situation analogous to that of temperature? 01/22/2018 Introduction to Data Mining, 2nd Edition 9 Categorical Qualitative Attribute Description Type Nominal Nominal attribute values only distinguish. (=, ) zip codes, employee ID numbers, eye color, sex: {male, female} Ordinal Ordinal attribute values also order objects. () For interval attributes, differences between values are meaningful. (+, - ) For ratio variables, both differences and ratios are meaningful. (*, /) hardness of minerals, {good, better, best}, grades, street numbers calendar dates, temperature in Celsius or Fahrenheit Interval Numeric Quantitative Examples Ratio Operations mode, entropy, contingency correlation, 2 test median, percentiles, rank correlation, run tests, sign tests mean, standard deviation, Pearson's correlation, t and F tests temperature in Kelvin, geometric mean, monetary quantities, harmonic mean, counts, age, mass, percent variation length, current This categorization of attributes is due to S. S. Stevens Numeric Quantitative Categorical Qualitative Attribute Transformation Type Comments 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}. Interval new_value = a * old_value + b where a and b are constants Ratio new_value = a * old_value Thus, the Fahrenheit and Celsius temperature scales differ in terms of where their zero value is and the size of a unit (degree). Length can be measured in meters or feet. This categorization of attributes is due to S. S. Stevens 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 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 floatingpoint variables. 01/22/2018 Introduction to Data Mining, 2nd Edition 12 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.” We need two asymmetric binary attributes to represent one ordinary binary attribute – Association analysis uses asymmetric attributes Asymmetric attributes typically arise from objects that are sets 01/22/2018 Introduction to Data Mining, 2nd Edition 13 Some Extensions and Critiques Velleman, Paul F., and Leland Wilkinson. "Nominal, ordinal, interval, and ratio typologies are misleading." The American Statistician 47, no. 1 (1993): 65-72. Mosteller, Frederick, and John W. Tukey. "Data analysis and regression. A second course in statistics." AddisonWesley Series in Behavioral Science: Quantitative Methods, Reading, Mass.: Addison-Wesley, 1977. Chrisman, Nicholas R. "Rethinking levels of measurement for cartography."Cartography and Geographic Information Systems 25, no. 4 (1998): 231-242. 01/22/2018 Introduction to Data Mining, 2nd Edition 14 Critiques Incomplete – Asymmetric binary – Cyclical – Multivariate – Partially ordered – Partial membership – Relationships between the data Real data is approximate and noisy – This can complicate recognition of the proper attribute type – Treating one attribute type as another may be approximately correct 01/22/2018 Introduction to Data Mining, 2nd Edition 15 Critiques … Not a good guide for statistical analysis – May unnecessarily restrict operations and results ◆ Statistical analysis is often approximate ◆ Thus, for example, using interval analysis for ordinal values may be justified – Transformations are common but don’t preserve scales ◆ Can transform data to a new scale with better statistical properties ◆ Many statistical analyses depend only on the distribution 01/22/2018 Introduction to Data Mining, 2nd Edition 16 More Complicated Examples ID numbers – Nominal, ordinal, or interval? Number of cylinders in an automobile engine – Nominal, ordinal, or ratio? Biased Scale – Interval or Ratio 01/22/2018 Introduction to Data Mining, 2nd Edition 17 Key Messages for Attribute Types The types of operations you choose should be “meaningful” for the type of data you have – Distinctness, order, meaningful intervals, and meaningful ratios are only four properties of data – The data type you see – often numbers or strings – may not capture all the properties or may suggest properties that are not there – Analysis may depend on these other properties of the data ◆ Many statistical analyses depend only on the distribution – Many times what is meaningful is measured by statistical significance – But in the end, what is meaningful is measured by the domain 01/22/2018 Introduction to Data Mining, 2nd Edition 18 Types of data sets Record – Data Matrix – Document Data – Transaction Data Graph – World Wide Web – Molecular Structures Ordered – – – – Spatial Data Temporal Data Sequential Data Genetic Sequence Data 01/22/2018 Introduction to Data Mining, 2nd Edition 19 Important Characteristics of Data – Dimensionality (number of attributes) ◆ High dimensional data brings a number of challenges – Sparsity ◆ Only presence counts – Resolution ◆ Patterns depend on the scale – Size ◆ Type of analysis may depend on size of data 01/22/2018 Introduction to Data Mining, 2nd Edition 20 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 01/22/2018 Introduction to Data Mining, 2nd Edition 21 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 01/22/2018 Introduction to Data Mining, 2nd Edition 22 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 play ball score game win lost timeout season 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 01/22/2018 Introduction to Data Mining, 2nd Edition 23 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. 01/22/2018 TID Items 1 Bread, Coke, Milk 2 3 4 5 Beer, Bread Beer, Coke, Diaper, Milk Beer, Bread, Diaper, Milk Coke, Diaper, Milk Introduction to Data Mining, 2nd Edition 24 Graph Data Examples: Generic graph, a molecule, and webpages 2 1 5 2 5 Benzene Molecule: C6H6 01/22/2018 Introduction to Data Mining, 2nd Edition 25 Ordered Data Sequences of transactions Items/Events An element of the sequence 01/22/2018 Introduction to Data Mining, 2nd Edition 26 Ordered Data Genomic sequence data GGTTCCGCCTTCAGCCCCGCGCC CGCAGGGCCCGCCCCGCGCCGTC GAGAAGGGCCCGCCTGGCGGGCG GGGGGAGGCGGGGCCGCCCGAGC CCAACCGAGTCCGACCAGGTGCC CCCTCTGCTCGGCCTAGACCTGA GCTCATTAGGCGGCAGCGGACAG GCCAAGTAGAACACGCGAAGCGC TGGGCTGCCTGCTGCGACCAGGG 01/22/2018 Introduction to Data Mining, 2nd Edition 27 Ordered Data Spatio-Temporal Data Average Monthly Temperature of land and ocean 01/22/2018 Introduction to Data Mining, 2nd Edition 28 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 01/22/2018 Introduction to Data Mining, 2nd Edition 29 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 Wrong data 01/22/2018 Introduction to Data Mining, 2nd Edition 30 Noise For objects, noise is an extraneous object For attributes, 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 Two Sine Waves 01/22/2018 Two Sine Waves + Noise Introduction to Data Mining, 2nd Edition 31 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 Causes? 01/22/2018 Introduction to Data Mining, 2nd Edition 32 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 or variables – Estimate missing values Example: time series of temperature ◆ Example: census results ◆ – Ignore the missing value during analysis 01/22/2018 Introduction to Data Mining, 2nd Edition 33 Missing Values … Missing completely at random (MCAR) – Missingness of a value is independent of attributes – Fill in values based on the attribute – Analysis may be unbiased overall Missing at Random (MAR) – Missingness is related to other variables – Fill in values based other values – Almost always produces a bias in the analysis Missing Not at Random (MNAR) – Missingness is related to unobserved measurements – Informative or non-ignorable missingness Not possible to know the situation from the data 01/22/2018 Introduction to Data Mining, 2nd Edition 34 Duplicate Data Data set may include data objects that are duplicates, or almost duplicates of one another – Major issue when merging data from heterogeneous sources Examples: – Same person with multiple email addresses Data cleaning – Process of dealing with duplicate data issues When should duplicate data not be removed? 01/22/2018 Introduction to Data Mining, 2nd Edition 35 Similarity and Dissimilarity Measures Similarity measure – Numerical measure of how alike two data objects are. – Is higher when objects are more alike. – Often falls in the range [0,1] Dissimilarity measure – 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 01/22/2018 Introduction to Data Mining, 2nd Edition 36 Similarity/Dissimilarity for Simple Attributes The following table shows the similarity and dissimilarity between two objects, x and y, with respect to a single, simple attribute. 01/22/2018 Introduction to Data Mining, 2nd Edition 37 Euclidean Distance Euclidean Distance where n is the number of dimensions (attributes) and xk and yk are, respectively, the kth attributes (components) or data objects x and y. Standardization is necessary, if scales differ. 01/22/2018 Introduction to Data Mining, 2nd Edition 38 Euclidean Distance 3 point p1 p2 p3 p4 p1 2 p3 p4 1 p2 0 0 1 2 3 4 5 p1 p1 p2 p3 p4 0 2.828 3.162 5.099 x 0 2 3 5 y 2 0 1 1 6 p2 2.828 0 1.414 3.162 p3 3.162 1.414 0 2 p4 5.099 3.162 2 0 Distance Matrix 01/22/2018 Introduction to Data Mining, 2nd Edition 39 Minkowski Distance Minkowski Distance is a generalization of Euclidean Distance Where r is a parameter, n is the number of dimensions (attributes) and xk and yk are, respectively, the kth attributes (components) or data objects x and y. 01/22/2018 Introduction to Data Mining, 2nd Edition 40 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 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. 01/22/2018 Introduction to Data Mining, 2nd Edition 41 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 01/22/2018 Introduction to Data Mining, 2nd Edition 42 Mahalanobis Distance 𝐦𝐚𝐡𝐚𝐥𝐚𝐧𝐨𝐛𝐢𝐬 𝐱, 𝐲 = (𝐱 − 𝐲)𝑇 Ʃ−1 (𝐱 − 𝐲)  is the covariance matrix For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6. 01/22/2018 Introduction to Data Mining, 2nd Edition 43 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 01/22/2018 Introduction to Data Mining, 2nd Edition 44 Common Properties of a Distance Distances, such as the Euclidean distance, have some well known properties. 1. d(x, y)  0 for all x and y and d(x, y) = 0 only if x = y. (Positive definiteness) 2. d(x, y) = d(y, x) for all x and y. (Symmetry) 3. d(x, z)  d(x, y) + d(y, z) for all points x, y, and z. (Triangle Inequality) where d(x, y) is the distance (dissimilarity) between points (data objects), x and y. A distance that satisfies these properties is a metric 01/22/2018 Introduction to Data Mining, 2nd Edition 45 Common Properties of a Similarity Similarities, also have some well known properties. 1. s(x, y) = 1 (or maximum similarity) only if x = y. 2. s(x, y) = s(y, x) for all x and y. (Symmetry) where s(x, y) is the similarity between points (data objects), x and y. 01/22/2018 Introduction to Data Mining, 2nd Edition 46 Similarity Between Binary Vectors Common situation is that objects, p and q, have only binary attributes Compute similarities using the following quantities f01 = the number of attributes where p was 0 and q was 1 f10 = the number of attributes where p was 1 and q was 0 f00 = the number of attributes where p was 0 and q was 0 f11 = the number of attributes where p was 1 and q was 1 Simple Matching and Jaccard Coefficients SMC = number of matches / number of attributes = (f11 + f00) / (f01 + f10 + f11 + f00) J = number of 11 matches / number of non-zero attributes = (f11) / (f01 + f10 + f11) 01/22/2018 Introduction to Data Mining, 2nd Edition 47 SMC versus Jaccard: Example x= 1000000000 y= 0000001001 f01 = 2 (the number of attributes where p was 0 and q was 1) f10 = 1 (the number of attributes where p was 1 and q was 0) f00 = 7 (the number of attributes where p was 0 and q was 0) f11 = 0 (the number of attributes where p was 1 and q was 1) SMC = (f11 + f00) / (f01 + f10 + f11 + f00) = (0+7) / (2+1+0+7) = 0.7 J = (f11) / (f01 + f10 + f11) = 0 / (2 + 1 + 0) = 0 01/22/2018 Introduction to Data Mining, 2nd Edition 48 Cosine Similarity If d1 and d2 are two document vectors, then cos( d1, d2 ) = / ||d1|| ||d2|| , where 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 = 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 01/22/2018 Introduction to Data Mining, 2nd Edition 49 Extended Jaccard Coefficient (Tanimoto) Variation of Jaccard for continuous or count attributes – Reduces to Jaccard for binary attributes 01/22/2018 Introduction to Data Mining, 2nd Edition 50 Correlation measures the linear relationship between objects 01/22/2018 Introduction to Data Mining, 2nd Edition 51 Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1. 01/22/2018 Introduction to Data Mining, 2nd Edition 52 Drawback of Correlation x = (-3, -2, -1, 0, 1, 2, 3) y = (9, 4, 1, 0, 1, 4, 9) yi = xi2 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 01/22/2018 Introduction to Data Mining, 2nd Edition 53 Comparison of Proximity Measures Domain of application – Similarity measures tend to be specific to the type of attribute and data – Record data, images, graphs, sequences, 3D-protein structure, etc. tend to have different measures However, one can talk about various properties that you would like a proximity measure to have – – – – Symmetry is a common one Tolerance to noise and outliers is another Ability to find more types of patterns? Many others possible The measure must be applicable to the data and produce results that agree with domain knowledge 01/22/2018 Introduction to Data Mining, 2nd Edition 54 Information Based Measures Information theory is a well-developed and fundamental disciple with broad applications Some similarity measures are based on information theory – Mutual information in various versions – Maximal Information Coefficient (MIC) and related measures – General and can handle non-linear relationships – Can be complicated and time intensive to compute 01/22/2018 Introduction to Data Mining, 2nd Edition 55 Information and Probability Information relates to possible outcomes of an event – transmission of a message, flip of a coin, or measurement of a piece of data The more certain an outcome, the less information that it contains and vice-versa – For example, if a coin has two heads, then an outcome of heads provides no information – More quantitatively, the information is related the probability of an outcome ◆ The smaller the probability of an outcome, the more information it provides and vice-versa – Entropy is the commonly used measure 01/22/2018 Introduction to Data Mining, 2nd Edition 56 Entropy For – – – – a variable (event), X, with n possible values (outcomes), x1, x2 …, xn each outcome having probability, p1, p2 …, pn the entropy of X , H(X), is given by 𝑛 𝐻 𝑋 = − ෍ 𝑝𝑖 log 2 𝑝𝑖 𝑖=1 Entropy is between 0 and log2n and is measured in bits – Thus, entropy is a measure of how many bits it takes to represent an observation of X on average 01/22/2018 Introduction to Data Mining, 2nd Edition 57 Entropy Examples For a coin with probability p of heads and probability q = 1 – p of tails 𝐻 = −𝑝 log 2 𝑝 − 𝑞 log 2 𝑞 – For p= 0.5, q = 0.5 (fair coin) H = 1 – For p = 1 or q = 1, H = 0 What is the entropy of a fair four-sided die? 01/22/2018 Introduction to Data Mining, 2nd Edition 58 Entropy for Sample Data: Example Hair Color Count p -plog2p Black 75 0.75 0.3113 Brown 15 0.15 0.4105 Blond 5 0.05 0.2161 Red 0 0.00 0 Other 5 0.05 0.2161 Total 100 1.0 1.1540 Maximum entropy is log25 = 2.3219 01/22/2018 Introduction to Data Mining, 2nd Edition 59 Entropy for Sample Data Suppose we have – a number of observations (m) of some attribute, X, e.g., the hair color of students in the class, – where there are n different possible values – And the number of observation in the ith category is mi – Then, for this sample 𝑛 𝑚𝑖 𝑚𝑖 𝐻 𝑋 = − ෍ log 2 𝑚 𝑚 𝑖=1 For continuous data, the calculation is harder 01/22/2018 Introduction to Data Mining, 2nd Edition 60 Mutual Information Information one variable provides about another Formally, 𝐼 𝑋, 𝑌 = 𝐻 𝑋 + 𝐻 𝑌 − 𝐻(𝑋, 𝑌), where H(X,Y) is the joint entropy of X and Y, 𝐻 𝑋, 𝑌 = − ෍ ෍ 𝑝𝑖𝑗log 2 𝑝𝑖𝑗 𝑖 𝑗 Where pij is the probability that the ith value of X and the jth value of Y occur together For discrete variables, this is easy to compute Maximum mutual information for discrete variables is log2(min( nX, nY ), where nX (nY) is the number of values of X (Y) 01/22/2018 Introduction to Data Mining, 2nd Edition 61 Mutual Information Example Student Count Status p -plog2p Undergrad 45 0.45 0.5184 Grad 55 0.55 0.4744 Student Grade Status Count p -plog2p Undergrad A 5 0.05 0.2161 Undergrad B 30 0.30 0.5211 Undergrad C 10 0.10 0.3322 Total 100 1.00 0.9928 Grade Count p -plog2p Grad A 30 0.30 0.5211 A 35 0.35 0.5301 Grad B 20 0.20 0.4644 B 50 0.50 0.5000 Grad C 5 0.05 0.2161 C 15 0.15 0.4105 Total 100 1.00 2.2710 Total 100 1.00 1.4406 Mutual information of Student Status and Grade = 0.9928 + 1.4406 - 2.2710 = 0.1624 01/22/2018 Introduction to Data Mining, 2nd Edition 62 Maximal Information Coefficient Reshef, David N., Yakir A. Reshef, Hilary K. Finucane, Sharon R. Grossman, Gilean McVean, Peter J. Turnbaugh, Eric S. Lander, Michael Mitzenmacher, and Pardis C. Sabeti. "Detecting novel associations in large data sets." science 334, no. 6062 (2011): 1518-1524. Applies mutual information to two continuous variables Consider the possible binnings of the variables into discrete categories – nX × nY ≤ N0.6 where ◆ ◆ ◆ nX is the number of values of X nY is the number of values of Y N is the number of samples (observations, data objects) Compute the mutual information – Normalized by log2(min( nX, nY ) Take the highest value 01/22/2018 Introduction to Data Mining, 2nd Edition 63 General Approach for Combining Similarities Sometimes attributes are of many different types, but an overall similarity is needed. 1: For the kth attribute, compute a similarity, sk(x, y), in the range [0, 1]. 2: Define an indicator variable, k, for the kth attribute as follows: k = 0 if the kth attribute is an asymmetric attribute and both objects have a value of 0, or if one of the objects has a missing value for the kth attribute k = 1 otherwise 3. Compute 01/22/2018 Introduction to Data Mining, 2nd Edition 64 Using Weights to Combine Similarities May not want to treat all attributes the same. – Use non-negative weights 𝜔𝑘 – 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝐱, 𝐲 = σ𝑛 𝑘=1 𝜔𝑘 𝛿𝑘 𝑠𝑘 (𝐱,𝐲) σ𝑛 𝑘=1 𝜔𝑘 𝛿𝑘 Can also define a weighted form of distance 01/22/2018 Introduction to Data Mining, 2nd Edition 65 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 01/22/2018 Introduction to Data Mining, 2nd Edition 66 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. 01/22/2018 Counts for each cell. Introduction to Data Mining, 2nd Edition 67 Euclidean Density: Center-Based Euclidean density is the number of points within a specified radius of the point Illustration of center-based density. 01/22/2018 Introduction to Data Mining, 2nd Edition 68 Data Preprocessing Aggregation Sampling Dimensionality Reduction Feature subset selection Feature creation Discretization and Binarization Attribute Transformation 01/22/2018 Introduction to Data Mining, 2nd Edition 69 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. ◆ Days aggregated into weeks, months, or years – More “stable” data ◆ 01/22/2018 Aggregated data tends to have less variability Introduction to Data Mining, 2nd Edition 70 Example: Precipitation in Australia This example is based on precipitation in Australia from the period 1982 to 1993. The next slide shows – A histogram for the standard deviation of average monthly precipitation for 3,030 0.5◦ by 0.5◦ grid cells in Australia, and – A histogram for the standard deviation of the average yearly precipitation for the same locations. The average yearly precipitation has less variability than the average monthly precipitation. All precipitation measurements (and their standard deviations) are in centimeters. 01/22/2018 Introduction to Data Mining, 2nd Edition 71 Example: Precipitation in Australia … Variation of Precipitation in Australia Standard Deviation of Average Monthly Precipitation 01/22/2018 Standard Deviation of Average Yearly Precipitation Introduction to Data Mining, 2nd Edition 72 Sampling Sampling is the main technique employed for data reduction. – It is often used for both the preliminary investigation of the data and the final data analysis. Statisticians often sample because obtaining the entire set of data of interest is too expensive or time consuming. Sampling is typically used in data mining because processing the entire set of data of interest is too expensive or time consuming. 01/22/2018 Introduction to Data Mining, 2nd Edition 73 Sampling … The key principle for effective sampling is the following: – Using a sample will work almost as well as using the entire data set, if the sample is representative – A sample is representative if it has approximately the same properties (of interest) as the original set of data 01/22/2018 Introduction to Data Mining, 2nd Edition 74 Sample Size 8000 points 01/22/2018 2000 Points Introduction to Data Mining, 2nd Edition 500 Points 75 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 01/22/2018 Introduction to Data Mining, 2nd Edition 76 Sample Size What sample size is necessary to get at least one object from each of 10 equal-sized groups. 01/22/2018 Introduction to Data Mining, 2nd Edition 77 Curse of Dimensionality When dimensionality increases, data becomes increasingly sparse in the space that it occupies Definitions of density and distance between points, which are 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 01/22/2018 Introduction to Data Mining, 2nd Edition 78 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 – Principal Components Analysis (PCA) – Singular Value Decomposition – Others: supervised and non-linear techniques 01/22/2018 Introduction to Data Mining, 2nd Edition 79 Dimensionality Reduction: PCA Goal is to find a projection that captures the largest amount of variation in data x2 e x1 01/22/2018 Introduction to Data Mining, 2nd Edition 80 Dimensionality Reduction: PCA 01/22/2018 Introduction to Data Mining, 2nd Edition 81 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 Many techniques developed, especially for classification 01/22/2018 Introduction to Data Mining, 2nd Edition 82 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 ◆ 01/22/2018 Example: Fourier and wavelet analysis Introduction to Data Mining, 2nd Edition 83 Mapping Data to a New Space Fourier and wavelet transform Frequency Two Sine Waves + Noise 01/22/2018 Frequency Introduction to Data Mining, 2nd Edition 84 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 01/22/2018 Introduction to Data Mining, 2nd Edition 85 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 01/22/2018 NRCS. 1995. Northeast wetland flora: Field office guide to plant species. Northeast National Technical Center, Chester, PA. Courtesy of USDA NRCS Wetland Science Institute. Introduction to Data Mining, 2nd Edition 86 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. 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 01/22/2018 Introduction to Data Mining, 2nd Edition 88 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. 01/22/2018 Introduction to Data Mining, 2nd Edition 89 Discretization Without Using Class Labels Equal interval width approach used to obtain 4 values. 01/22/2018 Introduction to Data Mining, 2nd Edition 90 Discretization Without Using Class Labels Equal frequency approach used to obtain 4 values. 01/22/2018 Introduction to Data Mining, 2nd Edition 91 Discretization Without Using Class Labels K-means approach to obtain 4 values. 01/22/2018 Introduction to Data Mining, 2nd Edition 92 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} 01/22/2018 Introduction to Data Mining, 2nd Edition 93 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 ◆ 01/22/2018 Introduction to Data Mining, 2nd Edition 94 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 01/22/2018 Atlanta 0.7591 1.0000 -0.5739 Sao Paolo -0.7581 -0.5739 1.0000 Introduction to Data Mining, 2nd Edition 95 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 01/22/2018 Atlanta 0.0492 1.0000 -0.0154 Sao Paolo 0.0906 -0.0154 1.0000 Introduction to Data Mining, 2nd Edition 96
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