#### Transcript Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques — Chapter 2 — TUGAS 1 dikiumpulkan tanggal 10 April 2010 ( PRogramming ) 2orang 1 kelompok April 13, 2017 Data Mining: Concepts and Techniques 1 Chapter 2: Data Preprocessing Karakteristik data secara umum Diskripsi data dan eksplorasi Mengukur kesamaan data Data cleaning Integrasi data dan transformasi Reduksi data Kesimpulan April 13, 2017 Data Mining: Concepts and Techniques 2 Types of Attribute Values Nominal E.g., profession, ID numbers, eye color, zip codes Ordinal E.g., rankings (e.g., army, professions), grades, height in {tall, medium, short} Binary E.g., medical test (positive vs. negative) Interval E.g., calendar dates, body temperatures Ratio E.g., temperature in Kelvin, length, time, counts April 13, 2017 Data Mining: Concepts and Techniques 3 Discrete vs. Continuous Attributes Discrete Attribute Has only a finite or countably infinite set of values E.g., zip codes, profession, or the set of words in a collection of documents Sometimes, represented as integer variables Note: Binary attributes are a special case of discrete attributes Continuous Attribute Has real numbers as attribute values Examples: temperature, height, or weight Practically, real values can only be measured and represented using a finite number of digits Continuous attributes are typically represented as floating-point variables April 13, 2017 Data Mining: Concepts and Techniques 4 Chapter 2: Data Preprocessing General data characteristics Basic data description and exploration Measuring data similarity Data cleaning Data integration and transformation Data reduction Summary April 13, 2017 Data Mining: Concepts and Techniques 5 Mining Data Descriptive Characteristics Motivasi Karakteristik dari sebaran data Untuk memahami data: sebaran, kecenderungan terpusat, dan variasi median, max, min, quartiles, outliers, variance Dimensi numerik terkait dengan interval yang terurut April 13, 2017 Boxplot atau quantile analysis pada interval yang terurut Data Mining: Concepts and Techniques 6 Mengukur kecenderungan terpusat ( Central Tendency) Rata-rata (sample vs. population): 1 n x xi n i 1 Weighted arithmetic mean: x N n Trimmed mean: chopping extreme values x Median: A holistic measure w x i 1 n i i w i 1 i Middle value if odd number of values, or average of the middle two values otherwise Estimated by interpolation (for grouped data): median L1 ( Mode Value that occurs most frequently in the data Unimodal, bimodal, trimodal Empirical formula: April 13, 2017 N / 2 ( freq)l freqmedian ) width mean mode 3 (mean median) Data Mining: Concepts and Techniques 7 Symmetric vs. Skewed Data Median, mean and mode of symmetric, positively and negatively skewed data positively skewed April 13, 2017 symmetric negatively skewed Data Mining: Concepts and Techniques 8 Contoh : Upah Karyawan PT. Satria Semarang Upah Harian F 200 - 219 220 - 239 240 - 259 260 - 279 280 - 299 300 - 319 320 - 339 4 8 17 24 15 9 5 F.Kumulatif 4 12 29 53 68 77 82 F = 82 Me = 82 : 2= 41 Kelas : 260 - 279 82 259 260 TepiKelasBawah 259,5 2 279 280 TepiKelasA tas 279,5 2 F .sk Me TKB xi Fd 12 Me 259,5 x 20 24 240 Me 259,5 24 Me 259,5 10 Me 269,50 F .sl Me TKA xi Fd 12 Me 279,5 x 20 24 240 Me 279,5 24 Me 279,5 10 269,50 F .sk Me TKB xi Fd 14 Me 64,5 x10 23 140 Me 64,5 23 Me 64,5 6,1 Me 76 Measuring the Dispersion of Data Quartiles, outliers and boxplots Quartiles: Q1 (25th percentile), Q3 (75th percentile) Inter-quartile range: IQR = Q3 – Q1 Five number summary: min, Q1, M, Q3, max Boxplot: ends of the box are the quartiles, median is marked, whiskers, and plot outlier individually Outlier: usually, a value higher/lower than 1.5 x IQR Variance and standard deviation (sample: s, population: σ) Variance: (algebraic, scalable computation) 1 n 1 n 2 1 n 2 s ( xi x ) [ xi ( xi ) 2 ] n 1 i 1 n 1 i 1 n i 1 2 1 N 2 n 1 ( x ) i N i 1 2 n xi 2 2 i 1 Standard deviation s (or σ) is the square root of variance s2 (or σ2) April 13, 2017 Data Mining: Concepts and Techniques 12 Properties of Normal Distribution Curve The normal (distribution) curve From μ–σ to μ+σ: contains about 68% of the measurements (μ: mean, σ: standard deviation) From μ–2σ to μ+2σ: contains about 95% of it From μ–3σ to μ+3σ: contains about 99.7% of it April 13, 2017 Data Mining: Concepts and Techniques 13 Graphic Displays of Basic Statistical Descriptions Boxplot: graphic display of five-number summary Histogram: x-axis are values, y-axis repres. frequencies Scatter plot: each pair of values is a pair of coordinates and plotted as points in the plane Loess (local regression) curve: add a smooth curve to a scatter plot to provide better perception of the pattern of dependence April 13, 2017 Data Mining: Concepts and Techniques 15 Histogram Analysis Graph displays of basic statistical class descriptions Frequency histograms April 13, 2017 A univariate graphical method Consists of a set of rectangles that reflect the counts or frequencies of the classes present in the given data Data Mining: Concepts and Techniques 16 Histograms Often Tells More than Boxplots The two histograms shown in the left may have the same boxplot representation April 13, 2017 The same values for: min, Q1, median, Q3, max But they have rather different data distributions Data Mining: Concepts and Techniques 17 Scatter plot Provides a first look at bivariate data to see clusters of points, outliers, etc Each pair of values is treated as a pair of coordinates and plotted as points in the plane April 13, 2017 Data Mining: Concepts and Techniques 18 Loess Curve Adds a smooth curve to a scatter plot in order to provide better perception of the pattern of dependence Loess curve is fitted by setting two parameters: a smoothing parameter, and the degree of the polynomials that are fitted by the regression April 13, 2017 Data Mining: Concepts and Techniques 19 Positively and Negatively Correlated Data The left half fragment is positively correlated April 13, 2017 The right half is negative correlated Data Mining: Concepts and Techniques 20 Not Correlated Data April 13, 2017 Data Mining: Concepts and Techniques 21 Used by permission of M. Ward, Worcester Polytechnic Institute Scatterplot Matrices Matrix of scatterplots (x-y-diagrams) of the k-dim. data [total of C(k, 2) = (k2 ̶ k)/2 scatterplots] April 13, 2017 Data Mining: Concepts and Techniques 22 Chapter 2: Data Preprocessing General data characteristics Basic data description and exploration Measuring data similarity (Sec. 7.2) Data cleaning Data integration and transformation Data reduction Summary April 13, 2017 Data Mining: Concepts and Techniques 23 Similarity and Dissimilarity Similarity Numerical measure of how alike two data objects are Value is higher when objects are more alike Often falls in the range [0,1] Dissimilarity (i.e., distance) Numerical measure of how different are two data objects Lower when objects are more alike Minimum dissimilarity is often 0 Upper limit varies Proximity refers to a similarity or dissimilarity April 13, 2017 Data Mining: Concepts and Techniques 24 Data Matrix and Dissimilarity Matrix Data matrix n data points with p dimensions Two modes Dissimilarity matrix n data points, but registers only the distance A triangular matrix Single mode April 13, 2017 x11 ... x i1 ... x n1 ... x1f ... ... ... ... xif ... ... ... ... ... xnf ... ... 0 d(2,1) 0 d(3,1) d ( 3,2) 0 : : : d ( n,1) d ( n,2) ... Data Mining: Concepts and Techniques x1p ... xip ... xnp ... 0 25 Example: Data Matrix and Distance Matrix 3 point p1 p2 p3 p4 p1 2 p3 p4 1 p2 0 0 1 2 3 4 5 p1 p2 p3 p4 0 2.828 3.162 5.099 y 2 0 1 1 Data Matrix 6 p1 x 0 2 3 5 p2 2.828 0 1.414 3.162 p3 3.162 1.414 0 2 p4 5.099 3.162 2 0 Distance Matrix (i.e., Dissimilarity Matrix) for Euclidean Distance April 13, 2017 Data Mining: Concepts and Techniques 26 Minkowski Distance Minkowski distance: A popular distance measure d (i, j) q (| x x |q | x x |q ... | x x |q ) i1 j1 i2 j2 ip jp where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is the order Properties d(i, j) > 0 if i ≠ j, and d(i, i) = 0 (Positive definiteness) d(i, j) = d(j, i) (Symmetry) d(i, j) d(i, k) + d(k, j) (Triangle Inequality) A distance that satisfies these properties is a metric April 13, 2017 Data Mining: Concepts and Techniques 27 Special Cases of Minkowski Distance q = 1: Manhattan (city block, L1 norm) distance E.g., the Hamming distance: the number of bits that are different between two binary vectors d (i, j) | x x | | x x | ... | x x | i1 j1 i2 j 2 ip jp q= 2: (L2 norm) Euclidean distance d (i, j) (| x x |2 | x x |2 ... | x x |2 ) i1 j1 i2 j2 ip jp q . “supremum” (Lmax norm, L norm) distance. This is the maximum difference between any component of the vectors Do not confuse q with n, i.e., all these distances are defined for all numbers of dimensions. Also, one can use weighted distance, parametric Pearson product moment correlation, or other dissimilarity measures April 13, 2017 Data Mining: Concepts and Techniques 28 Example: Minkowski Distance point p1 p2 p3 p4 x 0 2 3 5 y 2 0 1 1 L1 p1 p2 p3 p4 p1 0 4 4 6 p2 4 0 2 4 p3 4 2 0 2 p4 6 4 2 0 L2 p1 p2 p3 p4 p1 p2 2.828 0 1.414 3.162 p3 3.162 1.414 0 2 p4 5.099 3.162 2 0 L p1 p2 p3 p4 p1 p2 p3 p4 0 2.828 3.162 5.099 0 2 3 5 2 0 1 3 3 1 0 2 5 3 2 0 Distance Matrix April 13, 2017 Data Mining: Concepts and Techniques 29 Interval-valued variables Standardize data Calculate the mean absolute deviation: sf 1 n (| x1 f m f | | x2 f m f | ... | xnf m f |) where m f 1n (x1 f x2 f ... xnf ) . Calculate the standardized measurement (z-score) xif m f zif sf Using mean absolute deviation is more robust than using standard deviation Then calculate the Enclidean distance of other Minkowski distance April 13, 2017 Data Mining: Concepts and Techniques 30 Binary Variables 1 0 a b A contingency table for binary data Object i 1 0 c d sum a c b d Distance measure for symmetric d (i, j) binary variables: Distance measure for asymmetric binary variables: Jaccard coefficient (similarity measure for asymmetric binary variables): Object j d (i, j) sum a b cd p bc a bc d bc a bc simJaccard (i, j) a a b c Note: Jaccard coefficient is the same as “coherence”: coherence(i, j) April 13, 2017 sup(i, j) a sup(i) sup( j) sup(i, j) (a b) (a c) a Data Mining: Concepts and Techniques 31 Dissimilarity between Binary Variables Example Name Jack Mary Jim Gender M F M Fever Y Y Y Cough N N P Test-1 P P N Test-2 N N N Test-3 N P N Test-4 N N N gender is a symmetric attribute the remaining attributes are asymmetric binary let the values Y and P be set to 1, and the value N be set to 0 01 0.33 2 01 11 d ( jack , jim ) 0.67 111 1 2 d ( jim , mary ) 0.75 11 2 d ( jack , mary ) April 13, 2017 Data Mining: Concepts and Techniques 32 Nominal Variables A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green Method 1: Simple matching m: # of matches, p: total # of variables m d (i, j) p p Method 2: Use a large number of binary variables April 13, 2017 creating a new binary variable for each of the M nominal states Data Mining: Concepts and Techniques 33 Ordinal Variables An ordinal variable can be discrete or continuous Order is important, e.g., rank Can be treated like interval-scaled replace xif by their rank map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by zif rif {1,...,M f } rif 1 M f 1 compute the dissimilarity using methods for intervalscaled variables April 13, 2017 Data Mining: Concepts and Techniques 34 Ratio-Scaled Variables Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt Methods: treat them like interval-scaled variables—not a good choice! (why?—the scale can be distorted) apply logarithmic transformation yif = log(xif) April 13, 2017 treat them as continuous ordinal data treat their rank as interval-scaled Data Mining: Concepts and Techniques 35 Variables of Mixed Types A database may contain all the six types of variables symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio One may use a weighted formula to combine their effects pf 1 ij( f ) dij( f ) d (i, j) pf 1 ij( f ) f is binary or nominal: dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise f is interval-based: use the normalized distance f is ordinal or ratio-scaled Compute ranks rif and r 1 zif M 1 Treat zif as interval-scaled if f April 13, 2017 Data Mining: Concepts and Techniques 36 Vector Objects: Cosine Similarity Vector objects: keywords in documents, gene features in micro-arrays, … Applications: information retrieval, biologic taxonomy, ... Cosine measure: If d1 and d2 are two vectors, then cos(d1, d2) = (d1 d2) /||d1|| ||d2|| , where indicates vector dot product, ||d||: the length of vector d Example: d1 = 3 2 0 5 0 0 0 2 0 0 d2 = 1 0 0 0 0 0 0 1 0 2 d1d2 = 3*1+2*0+0*0+5*0+0*0+0*0+0*0+2*1+0*0+0*2 = 5 ||d1||= (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5=(42)0.5 = 6.481 ||d2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2)0.5=(6) 0.5 = 2.245 cos( d1, d2 ) = .3150 April 13, 2017 Data Mining: Concepts and Techniques 37 Chapter 2: Data Preprocessing General data characteristics Basic data description and exploration Measuring data similarity Data cleaning Data integration and transformation Data reduction Summary April 13, 2017 Data Mining: Concepts and Techniques 38 Tugas Pokok dalam Pemrosesan awal data Data cleaning Mengisi nilai yang hilang, memperhalus data noise, mengidentifikasi atau menghilangkan outlier dan memecahkan ketidak konsistenanan Integrasi data Mengintegrasikan berbagai database, data cube atau file-file Transformasi data Data transformation Normalisasi dan aggregation Reduksi data Mendapatkan representasi dalam volume data yung sudah terkurangi tetapi menghasilkan hasil analitis yang sama atau serupa Diskritisasi data : bagian dari reduksi data, bagian penting untuk data numerik April 13, 2017 Data Mining: Concepts and Techniques 39 Data Cleaning Data yang tidak berkualitas , hasil data mining yang tidak berkualitas! Keputusan yang berkualitas harus didasarkan pada data yang berkualitas e.g., data ganda atau data yang hilang mungkin menyebabkan ketidakbenaran atau bahkan menyesatkan Ekstaksi data, pembersihan, dan transformasi data merupakan tugas utama dalam data warehouse Tugas-tugas data cleaning Mengisi nilai-nilai yang hilang Mengidentifikasi outliers dan memperhalus data noise Memperbaiki ketidakkonsitenan data Memecahkan redudansi yang disebabkan oleh integrasi data April 13, 2017 Data Mining: Concepts and Techniques 40 Data in the Real World Is Dirty incomplete: lacking attribute values, lacking certain attributes of interest, or containing only aggregate data e.g., children=“ ” (missing data) noisy: containing noise, errors, or outliers e.g., Salary=“−10” (an error) inconsistent: containing discrepancies in codes or names, e.g., Age=“42” Birthday=“03/07/1997” Was rating “1,2,3”, now rating “A, B, C” discrepancy between duplicate records April 13, 2017 Data Mining: Concepts and Techniques 41 Why Is Data Dirty? Data yang tidak lengkap mungkin diperoleh dari Noisy data (incorrect values) may come from Faulty data collection instruments Human or computer error at data entry Errors in data transmission Inconsistent data may come from Different considerations between the time when the data was collected and when it is analyzed. Human/hardware/software problems Different data sources Duplicate records also need data cleaning April 13, 2017 Data Mining: Concepts and Techniques 42 Missing Data Data is not always available E.g., many tuples have no recorded value for several attributes, such as customer income in sales data Missing data may be due to equipment malfunction inconsistent with other recorded data and thus deleted data not entered due to misunderstanding certain data may not be considered important at the time of entry not register history or changes of the data Missing data may need to be inferred April 13, 2017 Data Mining: Concepts and Techniques 43 Bagaimana mengatasi Missing Value ( data yang hilang ) Mengabaikan record-record: biasanya dilakukan bila label class hilang (tidak efektif bila % dari nilai yang hilang per atribut sangat diperhatikan Mengisi nilai yang hilang secara manual Mengisi secara otomatis dengan Global konstant : e.g., “unknown”, a new class?! Rata-rata dari atribut Rata-rata atribut untuk seluruh sample dengan kelas yang sama : smarter nilai yang lebih memungkinkan: yaitu dengan menggunakan metode Bayesian April 13, 2017 Data Mining: Concepts and Techniques 44 Noisy Data Noise: random error or variance in a measured variable Incorrect attribute values may due to faulty data collection instruments data entry problems data transmission problems technology limitation Other data problems which requires data cleaning duplicate records incomplete data inconsistent data April 13, 2017 Data Mining: Concepts and Techniques 45 How to Handle Noisy Data? Binning first sort data and partition into (equal-frequency) bins then one can smooth by bin means, smooth by bin median, smooth by bin boundaries, etc. Regression smooth by fitting the data into regression functions Clustering detect and remove outliers Combined computer and human inspection detect suspicious values and check by human (e.g., deal with possible outliers) April 13, 2017 Data Mining: Concepts and Techniques 46 Simple Discretization Methods: Binning Equal-width (distance) partitioning Divides the range into N intervals of equal size: uniform grid if A and B are the lowest and highest values of the attribute, the width of intervals will be: W = (B –A)/N. The most straightforward, but outliers may dominate presentation Skewed data is not handled well Equal-depth (frequency) partitioning Divides the range into N intervals, each containing approximately same number of samples Good data scaling Managing categorical attributes can be tricky April 13, 2017 Data Mining: Concepts and Techniques 47 Binning Methods for Data Smoothing Sorted data for price (in dollars): 4, 8, 9, 15, 21, 21, 24, 25, 26, 28, 29, 34 * Partition into equal-frequency (equi-depth) bins: - Bin 1: 4, 8, 9, 15 - Bin 2: 21, 21, 24, 25 - Bin 3: 26, 28, 29, 34 * Smoothing by bin means: - Bin 1: 9, 9, 9, 9 - Bin 2: 23, 23, 23, 23 - Bin 3: 29, 29, 29, 29 * Smoothing by bin boundaries: - Bin 1: 4, 4, 4, 15 - Bin 2: 21, 21, 25, 25 - Bin 3: 26, 26, 26, 34 April 13, 2017 Data Mining: Concepts and Techniques 48 Regression y Y1 Y1’ y=x+1 X1 April 13, 2017 Data Mining: Concepts and Techniques x 49 Cluster Analysis April 13, 2017 Data Mining: Concepts and Techniques 50 Data Cleaning as a Process Data discrepancy detection Use metadata (e.g., domain, range, dependency, distribution) Check field overloading Check uniqueness rule, consecutive rule and null rule Use commercial tools Data scrubbing: use simple domain knowledge (e.g., postal code, spell-check) to detect errors and make corrections Data auditing: by analyzing data to discover rules and relationship to detect violators (e.g., correlation and clustering to find outliers) Data migration and integration Data migration tools: allow transformations to be specified ETL (Extraction/Transformation/Loading) tools: allow users to specify transformations through a graphical user interface Integration of the two processes Iterative and interactive (e.g., Potter’s Wheels) April 13, 2017 Data Mining: Concepts and Techniques 51 Chapter 2: Data Preprocessing General data characteristics Basic data description and exploration Measuring data similarity Data cleaning Data integration and transformation Data reduction Summary April 13, 2017 Data Mining: Concepts and Techniques 52 Data Integration Data integration: Combines data from multiple sources into a coherent store Schema integration: e.g., A.cust-id B.cust-# Integrate metadata from different sources Entity identification problem: Identify real world entities from multiple data sources, e.g., Bill Clinton = William Clinton Detecting and resolving data value conflicts For the same real world entity, attribute values from different sources are different Possible reasons: different representations, different scales, e.g., metric vs. British units April 13, 2017 Data Mining: Concepts and Techniques 53 Handling Redundancy in Data Integration Redundant data occur often when integration of multiple databases Object identification: The same attribute or object may have different names in different databases Derivable data: One attribute may be a “derived” attribute in another table, e.g., annual revenue Redundant attributes may be able to be detected by correlation analysis Careful integration of the data from multiple sources may help reduce/avoid redundancies and inconsistencies and improve mining speed and quality April 13, 2017 Data Mining: Concepts and Techniques 54 Correlation Analysis (Numerical Data) Correlation coefficient (also called Pearson’s product moment coefficient) rp ,q ( p p)( q q) ( pq) n p q (n 1) p q (n 1) p q where n is the number of baris ( record) , q and are the p respective means of p and q, σp and σq are the respective standard deviation of p and q, and Σ(pq) is the sum of the pq cross-product. If rp,q > 0, p and q are positively correlated (p’s values increase as q’s). The higher, the stronger correlation. rp,q = 0: independent; rpq < 0: negatively correlated April 13, 2017 Data Mining: Concepts and Techniques 55 Correlation (viewed as linear relationship) Correlation measures the linear relationship between objects To compute correlation, we standardize data objects, p and q, and then take their dot product pk ( pk mean( p)) / std ( p) qk (qk mean(q)) / std (q) correlation( p, q) p q April 13, 2017 Data Mining: Concepts and Techniques 56 Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1. April 13, 2017 Data Mining: Concepts and Techniques 57 Correlation Analysis (Categorical Data) Χ2 (chi-square) test (Observed Expected) Expected 2 2 The larger the Χ2 value, the more likely the variables are related The cells that contribute the most to the Χ2 value are those whose actual count is very different from the expected count Correlation does not imply causality # of hospitals and # of car-theft in a city are correlated Both are causally linked to the third variable: population April 13, 2017 Data Mining: Concepts and Techniques 58 Chi-Square Calculation: An Example Play chess Not play chess Sum (row) Like science fiction 250(90) 200(360) 450 Not like science fiction 50(210) 1000(840) 1050 Sum(col.) 300 1200 1500 Χ2 (chi-square) calculation (numbers in parenthesis are expected counts calculated based on the data distribution in the two categories) (250 90) 2 (50 210) 2 (200 360) 2 (1000 840) 2 507.93 90 210 360 840 2 It shows that like_science_fiction and play_chess are correlated in the group April 13, 2017 Data Mining: Concepts and Techniques 59 Data Transformation A function that maps the entire set of values of a given attribute to a new set of replacement values s.t. each old value can be identified with one of the new values Methods Smoothing: Remove noise from data Aggregation: Summarization, data cube construction Generalization: Concept hierarchy climbing Normalization: Scaled to fall within a small, specified range min-max normalization z-score normalization normalization by decimal scaling Attribute/feature construction New attributes constructed from the given ones April 13, 2017 Data Mining: Concepts and Techniques 60 Data Transformation: Normalization Min-max normalization: to [new_minA, new_maxA] v' v minA (new _ maxA new _ minA) new _ minA maxA minA Ex. Let income range $12,000 to $98,000 normalized to [0.0, 73,600 12,000 1.0]. Then $73,000 is mapped to 98,000 12,000 (1.0 0) 0 0.716 Z-score normalization (μ: mean, σ: standard deviation): v' v A A Ex. Let μ = 54,000, σ = 16,000. Then 73,600 54,000 1.225 16,000 Normalization by decimal scaling v v' j 10 April 13, 2017 Where j is the smallest integer such that Max(|ν’|) < 1 Data Mining: Concepts and Techniques 61 Chapter 2: Data Preprocessing General data characteristics Basic data description and exploration Measuring data similarity Data cleaning Data integration and transformation Data reduction Summary April 13, 2017 Data Mining: Concepts and Techniques 62 Data Reduction Strategies Why data reduction? A database/data warehouse may store terabytes of data Complex data analysis/mining may take a very long time to run on the complete data set Data reduction: Obtain a reduced representation of the data set that is much smaller in volume but yet produce the same (or almost the same) analytical results Data reduction strategies Dimensionality reduction — e.g., remove unimportant attributes Numerosity reduction (some simply call it: Data Reduction) Data cub aggregation Data compression Regression Discretization (and concept hierarchy generation) April 13, 2017 Data Mining: Concepts and Techniques 63 Dimensionality Reduction Curse of dimensionality When dimensionality increases, data becomes increasingly sparse Density and distance between points, which is critical to clustering, outlier analysis, becomes less meaningful The possible combinations of subspaces will grow exponentially Dimensionality reduction Avoid the curse of dimensionality Help eliminate irrelevant features and reduce noise Reduce time and space required in data mining Allow easier visualization Dimensionality reduction techniques Principal component analysis Singular value decomposition Supervised and nonlinear techniques (e.g., feature selection) April 13, 2017 Data Mining: Concepts and Techniques 64 Dimensionality Reduction: Principal Component Analysis (PCA) Find a projection that captures the largest amount of variation in data Find the eigenvectors of the covariance matrix, and these eigenvectors define the new space x2 e x1 April 13, 2017 Data Mining: Concepts and Techniques 65 Principal Component Analysis (Steps) Given N data vectors from n-dimensions, find k ≤ n orthogonal vectors (principal components) that can be best used to represent data Normalize input data: Each attribute falls within the same range Compute k orthonormal (unit) vectors, i.e., principal components Each input data (vector) is a linear combination of the k principal component vectors The principal components are sorted in order of decreasing “significance” or strength Since the components are sorted, the size of the data can be reduced by eliminating the weak components, i.e., those with low variance (i.e., using the strongest principal components, it is possible to reconstruct a good approximation of the original data) Works for numeric data only April 13, 2017 Data Mining: Concepts and Techniques 66 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 E.g., 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 E.g., students' ID is often irrelevant to the task of predicting students' GPA April 13, 2017 Data Mining: Concepts and Techniques 67 Heuristic Search in Feature Selection There are 2d possible feature combinations of d features Typical heuristic feature selection methods: Best single features under the feature independence assumption: choose by significance tests Best step-wise feature selection: The best single-feature is picked first Then next best feature condition to the first, ... Step-wise feature elimination: Repeatedly eliminate the worst feature Best combined feature selection and elimination Optimal branch and bound: Use feature elimination and backtracking April 13, 2017 Data Mining: Concepts and Techniques 68 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 domain-specific Mapping data to new space (see: data reduction) E.g., Fourier transformation, wavelet transformation Feature construction Combining features Data discretization April 13, 2017 Data Mining: Concepts and Techniques 69 Mapping Data to a New Space Fourier transform Wavelet transform Two Sine Waves April 13, 2017 Two Sine Waves + Noise Data Mining: Concepts and Techniques Frequency 70 Numerosity (Data) Reduction Reduce data volume by choosing alternative, smaller forms of data representation Parametric methods (e.g., regression) Assume the data fits some model, estimate model parameters, store only the parameters, and discard the data (except possible outliers) Example: Log-linear models—obtain value at a point in m-D space as the product on appropriate marginal subspaces Non-parametric methods Do not assume models Major families: histograms, clustering, sampling April 13, 2017 Data Mining: Concepts and Techniques 71 Parametric Data Reduction: Regression and Log-Linear Models Linear regression: Data are modeled to fit a straight line Often uses the least-square method to fit the line Multiple regression: allows a response variable Y to be modeled as a linear function of multidimensional feature vector Log-linear model: approximates discrete multidimensional probability distributions April 13, 2017 Data Mining: Concepts and Techniques 72 Regress Analysis and Log-Linear Models Linear regression: Y = w X + b Two regression coefficients, w and b, specify the line and are to be estimated by using the data at hand Using the least squares criterion to the known values of Y1, Y2, …, X1, X2, …. Multiple regression: Y = b0 + b1 X1 + b2 X2. Many nonlinear functions can be transformed into the above Log-linear models: The multi-way table of joint probabilities is approximated by a product of lower-order tables Probability: p(a, b, c, d) = ab acad bcd Data Cube Aggregation The lowest level of a data cube (base cuboid) The aggregated data for an individual entity of interest E.g., a customer in a phone calling data warehouse Multiple levels of aggregation in data cubes Reference appropriate levels Further reduce the size of data to deal with Use the smallest representation which is enough to solve the task Queries regarding aggregated information should be answered using data cube, when possible April 13, 2017 Data Mining: Concepts and Techniques 74 Data Compression String compression There are extensive theories and well-tuned algorithms Typically lossless But only limited manipulation is possible without expansion Audio/video compression Typically lossy compression, with progressive refinement Sometimes small fragments of signal can be reconstructed without reconstructing the whole Time sequence is not audio Typically short and vary slowly with time April 13, 2017 Data Mining: Concepts and Techniques 75 Data Compression Compressed Data Original Data lossless Original Data Approximated April 13, 2017 Data Mining: Concepts and Techniques 76 Data Reduction Method: Clustering Partition data set into clusters based on similarity, and store cluster representation (e.g., centroid and diameter) only Can be very effective if data is clustered but not if data is “smeared” Can have hierarchical clustering and be stored in multidimensional index tree structures There are many choices of clustering definitions and clustering algorithms Cluster analysis will be studied in depth in Chapter 7 April 13, 2017 Data Mining: Concepts and Techniques 77 Data Reduction Method: Sampling Sampling: obtaining a small sample s to represent the whole data set N Allow a mining algorithm to run in complexity that is potentially sub-linear to the size of the data Key principle: Choose a representative subset of the data Simple random sampling may have very poor performance in the presence of skew Develop adaptive sampling methods, e.g., stratified sampling: Note: Sampling may not reduce database I/Os (page at a time) April 13, 2017 Data Mining: Concepts and Techniques 78 Types of Sampling Simple random sampling There is an equal probability of selecting any particular item Sampling without replacement Once an object is selected, it is removed from the population Sampling with replacement A selected object is not removed from the population Stratified sampling: Partition the data set, and draw samples from each partition (proportionally, i.e., approximately the same percentage of the data) Used in conjunction with skewed data April 13, 2017 Data Mining: Concepts and Techniques 79 Sampling: Cluster or Stratified Sampling Raw Data April 13, 2017 Cluster/Stratified Sample Data Mining: Concepts and Techniques 80 Data Reduction: Discretization Three types of attributes: Nominal — values from an unordered set, e.g., color, profession Ordinal — values from an ordered set, e.g., military or academic rank Continuous — real numbers, e.g., integer or real numbers Discretization: Divide the range of a continuous attribute into intervals Some classification algorithms only accept categorical attributes. Reduce data size by discretization Prepare for further analysis April 13, 2017 Data Mining: Concepts and Techniques 81 Discretization and Concept Hierarchy Discretization Reduce the number of values for a given continuous attribute by dividing the range of the attribute into intervals Interval labels can then be used to replace actual data values Supervised vs. unsupervised Split (top-down) vs. merge (bottom-up) Discretization can be performed recursively on an attribute Concept hierarchy formation Recursively reduce the data by collecting and replacing low level concepts (such as numeric values for age) by higher level concepts (such as young, middle-aged, or senior) April 13, 2017 Data Mining: Concepts and Techniques 82 Discretization and Concept Hierarchy Generation for Numeric Data Typical methods: All the methods can be applied recursively Binning (covered above) Histogram analysis (covered above) Top-down split, unsupervised, Top-down split, unsupervised Clustering analysis (covered above) Either top-down split or bottom-up merge, unsupervised Entropy-based discretization: supervised, top-down split Interval merging by 2 Analysis: unsupervised, bottom-up merge Segmentation by natural partitioning: top-down split, unsupervised April 13, 2017 Data Mining: Concepts and Techniques 83 Discretization Using Class Labels Entropy based approach 3 categories for both x and y April 13, 2017 5 categories for both x and y Data Mining: Concepts and Techniques 84 Entropy-Based Discretization Given a set of samples S, if S is partitioned into two intervals S1 and S2 using boundary T, the information gain after partitioning is I (S , T ) | S1 | |S | Entropy( S1) 2 Entropy( S 2) |S| |S| Entropy is calculated based on class distribution of the samples in the set. Given m classes, the entropy of S1 is m Entropy( S1 ) pi log 2 ( pi ) i 1 where pi is the probability of class i in S1 The boundary that minimizes the entropy function over all possible boundaries is selected as a binary discretization The process is recursively applied to partitions obtained until some stopping criterion is met Such a boundary may reduce data size and improve classification accuracy April 13, 2017 Data Mining: Concepts and Techniques 85 Discretization Without Using Class Labels Data Equal frequency April 13, 2017 Equal interval width K-means Data Mining: Concepts and Techniques 86 Interval Merge by 2 Analysis Merging-based (bottom-up) vs. splitting-based methods Merge: Find the best neighboring intervals and merge them to form larger intervals recursively ChiMerge [Kerber AAAI 1992, See also Liu et al. DMKD 2002] Initially, each distinct value of a numerical attr. A is considered to be one interval 2 tests are performed for every pair of adjacent intervals Adjacent intervals with the least 2 values are merged together, since low 2 values for a pair indicate similar class distributions This merge process proceeds recursively until a predefined stopping criterion is met (such as significance level, max-interval, max inconsistency, etc.) April 13, 2017 Data Mining: Concepts and Techniques 87 Segmentation by Natural Partitioning A simply 3-4-5 rule can be used to segment numeric data into relatively uniform, “natural” intervals. If an interval covers 3, 6, 7 or 9 distinct values at the most significant digit, partition the range into 3 equiwidth intervals If it covers 2, 4, or 8 distinct values at the most significant digit, partition the range into 4 intervals If it covers 1, 5, or 10 distinct values at the most significant digit, partition the range into 5 intervals April 13, 2017 Data Mining: Concepts and Techniques 88 Example of 3-4-5 Rule count Step 1: Step 2: -$351 -$159 Min Low (i.e, 5%-tile) msd=1,000 profit High(i.e, 95%-0 tile) Low=-$1,000 (-$1,000 - 0) (-$400 - 0) (-$200 -$100) (-$100 0) April 13, 2017 Max High=$2,000 ($1,000 - $2,000) (0 -$ 1,000) (-$400 -$5,000) Step 4: (-$300 -$200) $4,700 (-$1,000 - $2,000) Step 3: (-$400 -$300) $1,838 ($1,000 - $2, 000) (0 - $1,000) (0 $200) ($1,000 $1,200) ($200 $400) ($1,200 $1,400) ($1,400 $1,600) ($400 $600) ($600 $800) ($800 $1,000) ($1,600 ($1,800 $1,800) $2,000) Data Mining: Concepts and Techniques ($2,000 - $5, 000) ($2,000 $3,000) ($3,000 $4,000) ($4,000 $5,000) 89 Concept Hierarchy Generation for Categorical Data Specification of a partial/total ordering of attributes explicitly at the schema level by users or experts Specification of a hierarchy for a set of values by explicit data grouping {Urbana, Champaign, Chicago} < Illinois Specification of only a partial set of attributes street < city < state < country E.g., only street < city, not others Automatic generation of hierarchies (or attribute levels) by the analysis of the number of distinct values E.g., for a set of attributes: {street, city, state, country} April 13, 2017 Data Mining: Concepts and Techniques 90 Automatic Concept Hierarchy Generation Some hierarchies can be automatically generated based on the analysis of the number of distinct values per attribute in the data set The attribute with the most distinct values is placed at the lowest level of the hierarchy Exceptions, e.g., weekday, month, quarter, year 15 distinct values country province_or_ state 365 distinct values city 3567 distinct values street April 13, 2017 674,339 distinct values Data Mining: Concepts and Techniques 91 Chapter 2: Data Preprocessing General data characteristics Basic data description and exploration Measuring data similarity Data cleaning Data integration and transformation Data reduction Summary April 13, 2017 Data Mining: Concepts and Techniques 92 Summary Data preparation/preprocessing: A big issue for data mining Data description, data exploration, and measure data similarity set the base for quality data preprocessing Data preparation includes Data cleaning Data integration and data transformation Data reduction (dimensionality and numerosity reduction) A lot a methods have been developed but data preprocessing still an active area of research April 13, 2017 Data Mining: Concepts and Techniques 93 References D. P. Ballou and G. K. Tayi. Enhancing data quality in data warehouse environments. Communications of ACM, 42:73-78, 1999 W. Cleveland, Visualizing Data, Hobart Press, 1993 T. Dasu and T. Johnson. Exploratory Data Mining and Data Cleaning. John Wiley, 2003 T. Dasu, T. Johnson, S. Muthukrishnan, V. Shkapenyuk. Mining Database Structure; Or, How to Build a Data Quality Browser. SIGMOD’02 U. Fayyad, G. Grinstein, and A. Wierse. Information Visualization in Data Mining and Knowledge Discovery, Morgan Kaufmann, 2001 H. V. Jagadish et al., Special Issue on Data Reduction Techniques. Bulletin of the Technical Committee on Data Engineering, 20(4), Dec. 1997 D. Pyle. Data Preparation for Data Mining. Morgan Kaufmann, 1999 E. Rahm and H. H. Do. Data Cleaning: Problems and Current Approaches. IEEE Bulletin of the Technical Committee on Data Engineering. Vol.23, No.4 V. Raman and J. Hellerstein. Potters Wheel: An Interactive Framework for Data Cleaning and Transformation, VLDB’2001 T. Redman. Data Quality: Management and Technology. Bantam Books, 1992 E. R. Tufte. The Visual Display of Quantitative Information, 2nd ed., Graphics Press, 2001 R. Wang, V. Storey, and C. Firth. A framework for analysis of data quality research. IEEE Trans. Knowledge and Data Engineering, 7:623-640, 1995 April 13, 2017 Data Mining: Concepts and Techniques 94 Feature Subset Selection Techniques Brute-force approach: Try all possible feature subsets as input to data mining algorithm Embedded approaches: Feature selection occurs naturally as part of the data mining algorithm Filter approaches: Features are selected before data mining algorithm is run Wrapper approaches: Use the data mining algorithm as a black box to find best subset of attributes April 13, 2017 Data Mining: Concepts and Techniques 95