Important Characteristics of Structured Data

• Dimensionality
• Curse of dimensionality
• Sparsity
• Only presence counts
• Resolution
• Patterns depend on the scale
• Distribution
• Centrality and dispersion

Data Objects

• Data sets are made up of data objects.
• A data object represents an entity.
• Examples:
• sales database: customers, store items, sales
• medical database: patients, treatments
• university database: students, professors, courses
• Also called samples , examples, instances, data points, objects, tuples.
• Data objects are described by attributes.
• Database rows -> data objects; columns ->attributes.

Attributes

• Attribute (or dimensions, features, variables): a data field, representing a characteristic or feature of a data object.
• E.g., customer _ID, name, address
• Types:
• Nominal
• Binary
• Numeric: quantitative
• Interval-scaled
• Ratio-scaled

Attribute Types

• Nominal: categories, states, or “names of things”
• Hair_color = {auburn, black, blond, brown, grey, red, white}
• marital status, occupation, ID numbers, zip codes
• Binary
• Nominal attribute with only 2 states (0 and 1)
• Symmetric binary: both outcomes equally important
• e.g., gender
• Asymmetric binary: outcomes not equally important.
• e.g., medical test (positive vs. negative)
• Convention: assign 1 to most important outcome (e.g., HIV positive)
• Ordinal
• Values have a meaningful order (ranking) but magnitude between successive values is not known.
• Size = {small, medium, large}, grades, army rankings

Numeric Attribute Types

• Quantity (integer or real-valued)
• Interval
• Measured on a scale of equal-sized units
• Values have order
• E.g., temperature in C˚or F˚, calendar dates
• No true zero-point
• Ratio
• Inherent zero-point
• We can speak of values as being an order of magnitude larger than the unit of measurement (10 K˚ is twice as high as 5 K˚).
• e.g., temperature in Kelvin, length, counts, monetary quantities

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
• E.g., 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

Basic Statistical Descriptions of Data

• Motivation
• To better understand the data: central tendency, variation and spread
• Data dispersion characteristics
• median, max, min, quantiles, outliers, variance, etc.
• Numerical dimensions correspond to sorted intervals
• Data dispersion: analyzed with multiple granularities of precision
• Boxplot or quantile analysis on sorted intervals
• Dispersion analysis on computed measures
• Folding measures into numerical dimensions
• Boxplot or quantile analysis on the transformed cube

• Median, mean and mode of symmetric, positively and negatively skewed data 

• Quartiles, outliers and boxplots
• Quartiles: Q1 (25th percentile), Q3 (75th percentile)
• Inter-quartile range: IQR = Q3 – Q1
• Five number summary: min, Q1, median, Q3, max
• Boxplot: ends of the box are the quartiles; median is marked; add whiskers, and plot outliers individually
• Outlier: usually, a value higher/lower than 1.5 x IQR
• Variance and standard deviation (sample: s, population: σ)
• Variance: (algebraic, scalable computation)

• Standard deviation s (or σ) is the square root of variance s2 (or σ2)  

• Five-number summary of a distribution
• Minimum, Q1, Median, Q3, Maximum

• Boxplot
• Data is represented with a box
• The ends of the box are at the first and third quartiles, i.e., the height of the box is IQR
• The median is marked by a line within the box
• Whiskers: two lines outside the box extended to Minimum and Maximum
• Outliers: points beyond a specified outlier threshold, plotted individually

• 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

• Boxplot: graphic display of five-number summary
• Histogram: x-axis are values, y-axis repres. frequencies
• Quantile plot: each value xi is paired with fi indicating that approximately 100 fi % of data are ≤ xi
• Quantile-quantile (q-q) plot: graphs the quantiles of one univariant distribution against the corresponding quantiles of another
• Scatter plot: each pair of values is a pair of coordinates and plotted as points in the plane

• Histogram: Graph display of tabulated frequencies, shown as bars
• It shows what proportion of cases fall into each of several categories
• Differs from a bar chart in that it is the area of the bar that denotes the value, not the height as in bar charts, a crucial distinction when the categories are not of uniform width
• The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent

Quantile Plot

• Displays all of the data (allowing the user to assess both the overall behavior and unusual occurrences)
• Plots quantile information
• For a data xi data sorted in increasing order, fi indicates that approximately 100 fi% of the data are below or equal to the value xi

Quantile-Quantile (Q-Q) Plot

• Graphs the quantiles of one univariate distribution against the corresponding quantiles of another
• View: Is there is a shift in going from one distribution to another?
• Example shows unit price of items sold at Branch 1 vs. Branch 2 for each quantile. Unit prices of items sold at Branch 1 tend to be lower than those at Branch 2.

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

Data Visualization

• Why data visualization?
• Gain insight into an information space by mapping data onto graphical primitives
• Provide qualitative overview of large data sets
• Search for patterns, trends, structure, irregularities, relationships among data
• Help find interesting regions and suitable parameters for further quantitative analysis
• Provide a visual proof of computer representations derived
• Categorization of visualization methods:
• Pixel-oriented visualization techniques
• Geometric projection visualization techniques
• Icon-based visualization techniques
• Hierarchical visualization techniques
• Visualizing complex data and relations

• For a data set of m dimensions, create m windows on the screen, one for each dimension
• The m dimension values of a record are mapped to m pixels at the corresponding positions in the windows
• The colors of the pixels reflect the corresponding values

Geometric Projection Visualization Techniques

• Visualization of geometric transformations and projections of the data
• Methods
• Direct visualization
• Scatterplot and scatterplot matrices
• Landscapes
• Projection pursuit technique: Help users find meaningful projections of multidimensional data
• Prosection views
• Hyperslice
• Parallel coordinates

Direct Data Visualization

• Ribbons with Twists Based on Vorticity

• Matrix of scatterplots (x-y-diagrams) of the k-dim. data [total of (k2/2-k) scatterplots]

• Visualization of the data as perspective landscape
• The data needs to be transformed into a (possibly artificial) 2D spatial representation which preserves the characteristics of the data 

• n equidistant axes which are parallel to one of the screen axes and correspond to the attributes
• The axes are scaled to the [minimum, maximum]: range of the corresponding attribute
• Every data item corresponds to a polygonal line which intersects each of the axes at the point which corresponds to the value for the attribute

Icon-Based Visualization Techniques

• Visualization of the data values as features of icons
• Typical visualization methods
• Chernoff Faces
• Stick Figures
• General techniques
• Shape coding: Use shape to represent certain information encoding
• Color icons: Use color icons to encode more information
• Tile bars: Use small icons to represent the relevant feature vectors in document retrieval

• A way to display variables on a two-dimensional surface, e.g., let x be eyebrow slant, y be eye size, z be nose length, etc.
• The figure shows faces produced using 10 characteristics--head eccentricity, eye size, eye spacing, eye eccentricity, pupil size, eyebrow slant, nose size, mouth shape, mouth size, and mouth opening): Each assigned one of 10 possible values, generated using Mathematica (S. Dickson)

• REFERENCE: Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, p. 212, 1993
• Weisstein, Eric W. "Chernoff Face." From MathWorld--A Wolfram Web Resource. mathworld.wolfram.com/ChernoffFace.html 

Hierarchical Visualization Techniques

• Visualization of the data using a hierarchical partitioning into subspaces
• Methods
• Dimensional Stacking
• Worlds-within-Worlds
• Tree-Map
• Cone Trees
• InfoCube

• Partitioning of the n-dimensional attribute space in 2-D subspaces, which are ‘stacked’ into each other
• Partitioning of the attribute value ranges into classes. The important attributes should be used on the outer levels.
• Adequate for data with ordinal attributes of low cardinality
• But, difficult to display more than nine dimensions
• Important to map dimensions appropriately

Used by permission of M. Ward, Worcester Polytechnic Institute

• Visualization of oil mining data with longitude and latitude mapped to the outer x-, y-axes and ore grade and depth mapped to the inner x-, y-axes

• Assign the function and two most important parameters to innermost world
• Fix all other parameters at constant values - draw other (1 or 2 or 3 dimensional worlds choosing these as the axes)
• Software that uses this paradigm
  • N–vision: Dynamic interaction through data glove and stereo displays, including rotation, scaling (inner) and translation (inner/outer) 
  • Auto Visual: Static interaction by means of queries

• Screen-filling method which uses a hierarchical partitioning of the screen into regions depending on the attribute values
• The x- and y-dimension of the screen are partitioned alternately according to the attribute values (classes)

InfoCube

• A 3-D visualization technique where hierarchical information is displayed as nested semi-transparent cubes
• The outermost cubes correspond to the top level data, while the subnodes or the lower level data are represented as smaller cubes inside the outermost cubes, and so on

• Visualizing non-numerical data: text and social networks
• Tag cloud: visualizing user-generated tags
  • The importance of tag is represented by font size/color
• Besides text data, there are also methods to visualize relationships, such as visualizing social networks

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 (e.g., distance)
• 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

• Can take 2 or more states, e.g., red, yellow, blue, green (generalization of a binary attribute)
• Method 1: Simple matching
• m : # of matches, p : total # of variables

\[ d(i,j)=\frac{p-m}{p} \]

• Method 2: Use a large number of binary attributes
• creating a new binary attribute for each of the M nominal states

• A contingency table for binary data

• Distance measure for symmetric binary variables: 

• Distance measure for asymmetric binary variables: 

\[ d(i,j)=\frac{r+s}{q+r+s} \]

• Jaccard coefficient (similarity measure for asymmetric binary variables):

\[ sim_{Jaccard}(i,j)=\frac{q}{q+r+s} \]

• Note: Jaccard coefficient is the same as “coherence”:

• Example

• Gender is a symmetric attribute
• The remaining attributes are asymmetric binary
• Let the values Y and P be 1, and the value N 0

• Z-score: 

\[ z=\frac{x-\mu}{\sigma } \]

• X: raw score to be standardized, μ: mean of the population, σ: standard deviation
• the distance between the raw score and the population mean in units of the standard deviation
• negative when the raw score is below the mean, “+” when above
• An alternative way: Calculate the mean absolute deviation, where

\[ m_{f}= \frac{1}{n}(x_{1f}+x_{2f}+...+x_{nf}) \]

• standardized measure (z-score):

\[ z_{if}=\frac{(x_{if}-m_{f})}{S_{f}} \]

• Using mean absolute deviation is more robust than using standard deviation 

• Minkowski distance: A popular distance measure

where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and h is the order (the distance so defined is also called L-h norm)

• 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

• h = 1: Manhattan (city block, L1 norm) distance
• E.g., the Hamming distance: the number of bits that are different between two binary vectors

• h = 2: (L2 norm) Euclidean distance

\[ d(i,j)=\sqrt{(|x_{i1}-x_{j1}|^2+|x_{i2}-x_{j2}|^2+...+|x_{ip}-x_{jp}|^2)} \]

• h →≈ . “supremum” (Lmax norm, L≈ norm) distance.
• This is the maximum difference between any component (attribute) of the vectors

• 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 

\[ r_{if} \epsilon \left \{ 1,...,M_{f} \right \} \]

• map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by
• compute the dissimilarity using methods for interval-scaled variables

• A database may contain all attribute types
• Nominal, symmetric binary, asymmetric binary, numeric, ordinal
• One may use a weighted formula to combine their effects

\[ d(i,j) = \frac{\sum_{f=1}^{p} \delta _{ij}^{(f)} d_{ij}^{(f)}}{\sum_{f=1}^{p} \delta _{ij}^{(f)}} \]

• f is binary or nominal:
• dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise
• f is numeric: use the normalized distance
• f is ordinal
• Compute ranks rif and
• Treat zif as interval-scaled

• A document can be represented by thousands of attributes, each recording the frequency of a particular word (such as keywords) or phrase in the document.

• Other vector objects: gene features in micro-arrays, …
• Applications: information retrieval, biologic taxonomy, gene feature mapping, ...
  
• Cosine measure: If d1 and d2 are two vectors (e.g., term-frequency vectors), then
               cos(d1, d2) =  (d1 ⋅ d2) /||d1|| ||d2|| ,
     where ⋅ indicates vector dot product, ||d||: the length of vector d
  
  

Summary

• Data attribute types: nominal, binary, ordinal, interval-scaled, ratio-scaled
• Many types of data sets, e.g., numerical, text, graph, Web, image.
• Gain insight into the data by:
• Basic statistical data description: central tendency, dispersion, graphical displays
• Data visualization: map data onto graphical primitives
• Measure data similarity
• Above steps are the beginning of data preprocessing.
• Many methods have been developed but still an active area of research.

References

• W. Cleveland, Visualizing Data, Hobart Press, 1993
• T. Dasu and T. Johnson. Exploratory Data Mining and Data Cleaning. John Wiley, 2003
• U. Fayyad, G. Grinstein, and A. Wierse. Information Visualization in Data Mining and Knowledge Discovery, Morgan Kaufmann, 2001
• L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990.
• H. V. Jagadish et al., Special Issue on Data Reduction Techniques. Bulletin of the Tech. Committee on Data Eng., 20(4), Dec. 1997
• D. A. Keim. Information visualization and visual data mining, IEEE trans. on Visualization and Computer Graphics, 8(1), 2002
• D. Pyle. Data Preparation for Data Mining. Morgan Kaufmann, 1999
• S.  Santini and R. Jain,” Similarity measures”, IEEE Trans. on Pattern Analysis and Machine Intelligence, 21(9), 1999
• E. R. Tufte. The Visual Display of Quantitative Information, 2nd ed., Graphics Press, 2001
• C. Yu et al., Visual data mining of multimedia data for social and behavioral studies, Information Visualization, 8(1), 2009