Preprocessing and Visualization. Jonathan Diehl
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1 RWTH Aachen University Chair of Computer Science VI Prof. Dr.-Ing. Hermann Ney Seminar Data Mining WS 2003/2004 Preprocessing and Visualization Jonathan Diehl January 19, 2004 onathan Diehl Preprocessing and Visualization January 19,
2 Structure Preprocessing and Visualization Introduction Theoretical Fundamentals Visualization Preprocessing onathan Diehl Preprocessing and Visualization January 19,
3 Introduction Literature D. Hand, H. Manila, P. Smyth: Principles of Data Mining. MIT Press, Cambridge, MA, 2001, Chapters 2 and 3 J. Han, M. Kamber: Data Mining: Concepts and Techniques. Academic Press, San Diego, CA, 2001, Chapter 3 S. Roweis, L.Saul: Locally Linear Embedding Homepage. roweis/lle/ onathan Diehl Preprocessing and Visualization January 19,
4 Introduction Overview ata Preprocessing: data manipulation prior to mining improves quality or speed of actual mining ata Exploration: combined human and computer analysis utilizes human s natural abilities ata Visualization: methods to display data to the human data is plotted graphically onathan Diehl Preprocessing and Visualization January 19,
5 Structure Preprocessing and Visualization Introduction and Motivation Theoretical Fundamentals Visualization Preprocessing onathan Diehl Preprocessing and Visualization January 19,
6 Theoretical Fundamentals Data Representation data consists of N samples (measures, etc.) and M attributes (variables, dimensions, etc.) for each sample n and each attribute m there exists one data value x n,m onathan Diehl Preprocessing and Visualization January 19,
7 Theoretical Fundamentals Data Representation ata Set: X := ( X(1),...,X(N)) T with X(n) =(x n,1,...,x n,m ) ata Matrix: x 1,1... x 1,M X :=.. x N,1... x N,M ith x n,m value of n-th sample and m-th attribute onathan Diehl Preprocessing and Visualization January 19,
8 Theoretical Fundamentals Measures of Location ample Mean: ˆµ = 1 N edian: N n=1 X(n) with X set of N data values 50% of values below median ode: data point which occurs most often onathan Diehl Preprocessing and Visualization January 19,
9 Theoretical Fundamentals Measures of Dispersion ariance: ˆσ 2 = 1 N N ( ) 2 X(n) µ with X set of N data values and mean µ n=1 uartiles: 25%/75% of values below quartile ange: difference between lowest and highest data value onathan Diehl Preprocessing and Visualization January 19,
10 Theoretical Fundamentals Measures of Correlation ovariance (of attributes A and B): Cov(A, B) = 1 N N n=1 ( A(n) µa ) ( B(n) µb ) ovariance Matrix: V = 1 N XT X with X zero-centered data matrix orrelation Coefficient: Cov(A, B) ρ A,B = σ A σ B onathan Diehl Preprocessing and Visualization January 19,
11 Theoretical Fundamentals Modelling Data inear Regression (for two-dimensional data set): Y = a + b X with a, b coefficients and X, Y attributes ultiple Linear Regression: Y = a + N n=1 b n X n onathan Diehl Preprocessing and Visualization January 19,
12 Structure Preprocessing and Visualization Introduction and Motivation Theoretical Fundamentals Visualization Preprocessing onathan Diehl Preprocessing and Visualization January 19,
13 Visualization Histogram onathan Diehl Preprocessing and Visualization January 19,
14 Visualization Kernel Method stimated density of kernel function K with bandwidth h: f(x) = 1 N ( ) x X(n) K for given data set X of N values N h n=1 here K(t)dt =1 onathan Diehl Preprocessing and Visualization January 19,
15 Visualization Kernel Method onathan Diehl Preprocessing and Visualization January 19,
16 Visualization Boxplot onathan Diehl Preprocessing and Visualization January 19,
17 Visualization Scatterplot onathan Diehl Preprocessing and Visualization January 19,
18 Visualization Contour Plot onathan Diehl Preprocessing and Visualization January 19,
19 Visualization Trellis Plot onathan Diehl Preprocessing and Visualization January 19,
20 Structure Preprocessing and Visualization Introduction and Motivation Theoretical Fundamentals Visualization Preprocessing onathan Diehl Preprocessing and Visualization January 19,
21 Data Cleaning roblems: incomplete data (missing values) erroneous (noisy) data inconsistent data ( data integration) onathan Diehl Preprocessing and Visualization January 19,
22 Missing Values olutions: ignore samples loose important information determine most likely value: 1. make use of statistical measures (mean/median, class mean/median) 2. construct model of attribute relations (regression) and calculate value 3. construct decision tree and derive value... onathan Diehl Preprocessing and Visualization January 19,
23 Noisy Data: Regression. construct model of entire data set: linear Regression multiple Regression (regression over weighted linear combination of attributes). calculate data points from regression equation global smoothing, strong data reduction onathan Diehl Preprocessing and Visualization January 19,
24 Noisy Data: Regression 40 original data regression onathan Diehl Preprocessing and Visualization January 19,
25 Data Integration roblems: entity identification problem ( metadata) data redundancy ( correlation analysis) value conflicts ( data transformation) onathan Diehl Preprocessing and Visualization January 19,
26 Data Transformation roblem: data has to meet certain criteria before further processing.g. attribute values must be weighted equally for many analysis methods onathan Diehl Preprocessing and Visualization January 19,
27 Normalization it attribute A of data X into predefined range (e.g. [0.0, 1.0]) min-max normalization: f(i) = A(i) min A max A min A (newmax A newmin A )+newmin A z-score normalization: g(i) = A(i) µ A σ A onathan Diehl Preprocessing and Visualization January 19,
28 Attribute Construction onstruct new (redundant) attributes summarizing stored information sums, products of samples/data classes Fast online access to summarized data onathan Diehl Preprocessing and Visualization January 19,
29 Data Reduction roblem: huge databases with many attributes very slow mining process onathan Diehl Preprocessing and Visualization January 19,
30 Data Cube Aggregation multidimensional arrangement of aggregated information dimensions (axes) represent attributes multiple levels of abstraction possible (concept hierarchy) Efficient organization of summarized data onathan Diehl Preprocessing and Visualization January 19,
31 Data Cube Aggregation onathan Diehl Preprocessing and Visualization January 19,
32 Attribute Subset Selection etermine significance of attributes: correlation coefficient information gain ake selection: stepwise selection/elimination decision tree induction onathan Diehl Preprocessing and Visualization January 19,
33 Attribute Subset Selection A? Y N B? C? Y N Y N Class 1 Class 2 Class 2 Class 1 onathan Diehl Preprocessing and Visualization January 19,
34 Principal Component Analysis im: Maximize variability project data onto principal components (linear combination of attributes) normalize principal components zero-centered data set (subtract mean from all values) onathan Diehl Preprocessing and Visualization January 19,
35 Principal Component Analysis ariability of projection vector a (S := X T X scatter matrix): (Xa) T ( ) s 2 a = Xa = a T X T Xa = a T Sa pose normalization constraint and maximize: u = a T Sa λ(a T a 1) δu = 2Sa 2λa =0 δa }{{} eigenvalue form max eigenvector a with largest eigenvalue is first principal component onathan Diehl Preprocessing and Visualization January 19,
36 Principal Component Analysis data is projected onto the first K eigenvectors small values of K sufficient because late principal components are insignificant (eigenvalue approaches zero) for K =2principal component analysis can be used to project data into a plane for visualization onathan Diehl Preprocessing and Visualization January 19,
37 Multidimensional Scaling im: Preserve Distances fit multidimensional data into plane minimize squared distance error zero-centered data set (subtract mean from all values) onathan Diehl Preprocessing and Visualization January 19,
38 Multidimensional Scaling iven B = XX T solve (euclidian distance) d 2 ij = x i x j 2 = x T i X i 2x T i x j + x T j x j = b ii + b jj 2b ij hen minimize N N (δ ij d ij ) 2 i=1 j=1 ith δ ij multidimensional distance between data point i and j onathan Diehl Preprocessing and Visualization January 19,
39 Locally Linear Embedding im: Preserve Neighborhoods reduce data to lower dimensional embedding find locally linear patches of the data reconstruct data points from its neighbors invariant to rotations, rescalings and translations onathan Diehl Preprocessing and Visualization January 19,
40 Locally Linear Embedding onathan Diehl Preprocessing and Visualization January 19,
41 Locally Linear Embedding. Find neighbors. Reconstruct data point as linear combination of neighbors Weight matrix. Calculate low-dimensional representation of weight matrix using eigenvector analysis onathan Diehl Preprocessing and Visualization January 19,
42 onathan Diehl Preprocessing and Visualization January 19,
43 and Visualization Summary and Discussion onathan Diehl Preprocessing and Visualization January 19,
Preprocessing and Visualization. Jonathan Diehl
RWTH Aachen University Chair of Computer Science VI Prof.Dr.-Ing.HermannNey Seminar Data Mining in WS 2003 / 2004 Preprocessing and Visualization Jonathan Diehl Matrikelnummer 235087 January 19, 2004 Tutor:
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