Data Mining and Analytics: AIM411 Introduction to Data Mining PDF

Data Mining and Analytics: AIM411 Introduction to Data Mining 1 Outline Attributes and Objects Types of Data Data Quality Similarity and Distance Data Preprocessing What is Data? Collection of data objects Att...

Data Mining and Analytics: AIM411 Introduction to Data Mining 1 Outline Attributes and Objects Types of Data Data Quality Similarity and Distance Data Preprocessing What is Data? Collection of data objects Attributes and their attributes An attribute is a property Tid Refund Marital Taxable or characteristic of an Status Income Cheat object 1 Yes Single 125K No – Examples: eye color of a 2 No Married 100K No person, temperature, etc. 3 No Single 70K No Objects – Attribute is also known as variable, field, characteristic, 4 Yes Married 120K No dimension, or feature 5 No Divorced 95K Yes A collection of attributes 6 No Married 60K No describe an object 7 Yes Divorced 220K No – Object is also known as 8 No Single 85K Yes record, point, case, sample, 9 No Married 75K No entity, or instance 10 No Single 90K Yes 10 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 can be different than the properties of the values used to represent the attribute  Measurement of Length The way you measure an attribute may not match the attributes properties. 5 A 1 B 7 2 C This scale This scale 8 3 preserves preserves only the the ordering ordering D and property of additivity length. 10 4 properties of length. E 15 5 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 ◆ Examples: temperature in Kelvin, length, counts, elapsed time (e.g., time to run a race) 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 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?  Attribute Description Examples Operations Type Nominal Nominal attribute zip codes, employee mode, entropy, values only ID numbers, eye contingency distinguish. (=, ) color, sex: {male, correlation, 2 Categorical Qualitative female} test Ordinal Ordinal attribute hardness of minerals, median, values also order {good, better, best}, percentiles, rank objects. grades, street correlation, run () numbers tests, sign tests Interval For interval calendar dates, mean, standard attributes, temperature in deviation, differences between Celsius or Fahrenheit Pearson's Quantitative Numeric values are correlation, t and meaningful. (+, - ) F tests Ratio For ratio variables, temperature in Kelvin, geometric mean, both differences and monetary quantities, harmonic mean, ratios are counts, age, mass, percent variation meaningful. (*, /) length, current This categorization of attributes is due to S. S. Stevens  Attribute Transformation Comments Type Nominal Any permutation of values If all employee ID numbers were reassigned, would it make any difference? Categorical Qualitative Ordinal An order preserving change of An attribute encompassing values, i.e., the notion of good, better best new_value = f(old_value) can be represented equally where f is a monotonic function well by the values {1, 2, 3} or by { 0.5, 1, 10}. Interval new_value = a * old_value + b Thus, the Fahrenheit and where a and b are constants Celsius temperature scales Quantitative Numeric differ in terms of where their zero value is and the size of a unit (degree). Ratio new_value = a * old_value 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 floating- point variables. 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.” Critiques of the attribute categorization Incomplete – Asymmetric binary – Cyclical – Multivariate – Partially ordered – Partial membership – Relationships between the data Real data is approximate and noisy – This can complicate the recognition of the proper attribute type – Treating one attribute type as another may be approximately correct 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 (among many possible) 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 present – Analysis may depend on these other properties of the data ◆ Many statistical analyses depend only on the distribution – In the end, what is meaningful can be specific to domain 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 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 Record Data Data that consists of a collection of records, each of which consists of a fixed set of attributes Tid Refund Marital Taxable Status 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 60K No 7 Yes Divorced 220K No 8 No Single 85K Yes 9 No Married 75K No 10 No Single 90K Yes 10 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 a 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 Projection Distance Load Thickness of x Load of y load 10.23 5.27 15.22 2.7 1.2 12.65 6.25 16.22 2.2 1.1 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. timeout season coach game score play team win ball lost 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 Transaction Data A special type of data, where – Each 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. – Can represent transaction data as record data TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk  Graph Data Examples: Generic graph, a molecule, and webpages 2 5 1 2 5 Benzene Molecule: C6H6 Ordered Data Sequences of transactions Items/Events An element of the sequence Ordered Data Genomic sequence data GGTTCCGCCTTCAGCCCCGCGCC CGCAGGGCCCGCCCCGCGCCGTC GAGAAGGGCCCGCCTGGCGGGCG GGGGGAGGCGGGGCCGCCCGAGC CCAACCGAGTCCGACCAGGTGCC CCCTCTGCTCGGCCTAGACCTGA GCTCATTAGGCGGCAGCGGACAG GCCAAGTAGAACACGCGAAGCGC TGGGCTGCCTGCTGCGACCAGGG Ordered Data Spatio-Temporal Data Average Monthly Temperature of land and ocean Data Quality Poor data quality negatively affects many data processing efforts 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 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 – Wrong data – Fake data – Missing values – Duplicate data 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 – The figures below show two sine waves of the same magnitude and different frequencies, the waves combined, and the two sine waves with random noise ◆ The magnitude and shape of the original signal is distorted 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? 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 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? 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  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. 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. Euclidean Distance 3 point x y 2 p1 p1 0 2 p3 p4 1 p2 2 0 p2 p3 3 1 0 p4 5 1 0 1 2 3 4 5 6 p1 p2 p3 p4 p1 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 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. Minkowski Distance: Examples r = 1. City block (Manhattan, taxicab, L1 norm) distance. – A common example of this for binary vectors 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. Minkowski Distance 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 point x y p1 0 2 L2 p1 p2 p3 p4 p2 2 0 p1 0 2.828 3.162 5.099 p3 3 1 p2 2.828 0 1.414 3.162 p4 5 1 p3 3.162 1.414 0 2 p4 5.099 3.162 2 0 L p1 p2 p3 p4 p1 0 2 3 5 p2 2 0 1 3 p3 3 1 0 2 p4 5 3 2 0 Distance Matrix  Mahalanobis Distance 𝐦𝐚𝐡𝐚𝐥𝐚𝐧𝐨𝐛𝐢𝐬 𝐱, 𝐲 = ((𝐱 − 𝐲)𝑇 Ʃ−1 (𝐱 − 𝐲))-0.5  is the covariance matrix For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6. Mahalanobis Distance Covariance Matrix:  0.3 0.2 =  C  0.2 0.3 B A: (0.5, 0.5) B: (0, 1) A C: (1.5, 1.5) Mahal(A,B) = 5 Mahal(A,C) = 4 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 if and only if x = y. 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 Common Properties of a Similarity Similarities, also have some well known properties. 1. s(x, y) = 1 (or maximum similarity) only if x = y. (does not always hold, e.g., cosine) 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. Similarity Between Binary Vectors Common situation is that objects, x and y, have only binary attributes Compute similarities using the following quantities f01 = the number of attributes where x was 0 and y was 1 f10 = the number of attributes where x was 1 and y was 0 f00 = the number of attributes where x was 0 and y was 0 f11 = the number of attributes where x was 1 and y 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) SMC versus Jaccard: Example x= 1000000000 y= 0000001001 f01 = 2 (the number of attributes where x was 0 and y was 1) f10 = 1 (the number of attributes where x was 1 and y was 0) f00 = 7 (the number of attributes where x was 0 and y was 0) f11 = 0 (the number of attributes where x was 1 and y 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 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 Correlation measures the linear relationship between objects Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1.  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 Correlation vs Cosine vs Euclidean Distance Compare the three proximity measures according to their behavior under variable transformation – scaling: multiplication by a value – translation: adding a constant Property Cosine Correlation Euclidean Distance Invariant to scaling Yes Yes No (multiplication) Invariant to translation No Yes No (addition) Consider the example – x = (1, 2, 4, 3, 0, 0, 0), y = (1, 2, 3, 4, 0, 0, 0) – ys = y * 2 (scaled version of y), yt = y + 5 (translated version) Measure (x , y) (x , ys) (x , yt) Cosine 0.9667 0.9667 0.7940 Correlation 0.9429 0.9429 0.9429 Euclidean Distance 1.4142 5.8310 14.2127 Correlation vs cosine vs Euclidean distance Choice of the right proximity measure depends on the domain What is the correct choice of proximity measure for the following situations? – Comparing documents using the frequencies of words ◆ Documents are considered similar if the word frequencies are similar – Comparing the temperature in Celsius of two locations ◆ Two locations are considered similar if the temperatures are similar in magnitude – Comparing two time series of temperature measured in Celsius ◆ Two time series are considered similar if their “shape” is similar, i.e., they vary in the same way over time, achieving minimums and maximums at similar times, etc. 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 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 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 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 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? 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 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 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)  Mutual Information Example Student Count p -plog2p Student Grade Count p -plog2p Status Status Undergrad 45 0.45 0.5184 Undergrad A 5 0.05 0.2161 Grad 55 0.55 0.4744 Undergrad B 30 0.30 0.5211 Total 100 1.00 0.9928 Undergrad C 10 0.10 0.3322 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 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 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 Using Weights to Combine Similarities May not want to treat all attributes the same. – Use non-negative weights 𝜔𝑘 σ𝑛 𝑘=1 𝜔𝑘 𝛿𝑘 𝑠𝑘 (𝐱,𝐲) – 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝐱, 𝐲 = σ𝑛 𝑘=1 𝜔𝑘 𝛿𝑘 Can also define a weighted form of distance Data Preprocessing Aggregation Sampling Discretization and Binarization Attribute Transformation Dimensionality Reduction Feature subset selection Feature creation 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 - aggregated data tends to have less variability 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. Example: Precipitation in Australia … Variation of Precipitation in Australia Standard Deviation of Average Standard Deviation of Monthly Precipitation Average Yearly Precipitation 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. 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 Sample Size 8000 points 2000 Points 500 Points 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 Sample Size What sample size is necessary to get at least one object from each of 10 equal-sized groups. 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 used in both unsupervised and supervised settings Unsupervised Discretization Data consists of four groups of points and two outliers. Data is one- dimensional, but a random y component is added to reduce overlap. Unsupervised Discretization Equal interval width approach used to obtain 4 values. Unsupervised Discretization Equal frequency approach used to obtain 4 values. Unsupervised Discretization K-means approach to obtain 4 values. Discretization in Supervised Settings – 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 following example. Binarization Binarization maps a continuous or categorical attribute into one or more binary variables 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 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 Atlanta Sao Paolo Minneapolis 1.0000 0.7591 -0.7581 Atlanta 0.7591 1.0000 -0.5739 Sao Paolo -0.7581 -0.5739 1.0000 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 Atlanta Sao Paolo Minneapolis 1.0000 0.0492 0.0906 Atlanta 0.0492 1.0000 -0.0154 Sao Paolo 0.0906 -0.0154 1.0000 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 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 Dimensionality Reduction: PCA Goal is to find a projection that captures the largest amount of variation in data x2 e x1 Dimensionality Reduction: PCA 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 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 Mapping Data to a New Space Fourier and wavelet transform Frequency Two Sine Waves + Noise Frequency