Courtesy of Prof. Shixia University

Transcription

1 Courtesy of Prof. Shixia University

2 Outline Introduction Classification of Techniques Table Scatter Plot Matrices Projections Parallel Coordinates Summary

3 Motivation Real world data contain multiple dimensions

4 Multivariate/Multidimensional Data Visualization Multivariate data visualization is a specific type of information visualization that deals with multivariate/multidimensional data The data to be visualized are of high dimensionality in which the correlations between these many attributes are of interest

5 Dimensionality Refers to the number of attributes that presents in the data 1: one-dimensional 1D / univariate 2: two-dimensional 2D/ bivaraite 3: three-dimensional 3D / trivariate 3: multidimensional / hypervarite / multivariate Boundary between high and low dimensionality not clear, generally high dimensionality has >4 variables

6 Terminology Dimensions Variables Multidimensional Multivariate Dimensionality of the independent dimensions Dimensionality of the dependent variables

7 Outline Introduction Classification of Techniques Projections Parallel Coordinates Table Scatter Plot Matrices Summary

8 Classification of Techniques Projection Parallel Coordinates Plot Table Scatter Plot Matrix

9 What if we have too many dimensions? A intuitive way is to project to low dimension space Linear projections Nonlinear projections A projection (X -> Y) maps points {x1, x2,, xm} in an n-dimensional space into a p-dimensional space as {y1, y2,, ym} (p << n) while preserving distance measures of data items.

10 Classification Linear projection Example: PCA (principal component analysis) Non-linear projection Example: t-sne (t-distributed stochastic neighbor embedding)

11 PCA Seeks a space of lower dimensionality (magenta) Such that the orthogonal projection of the data points (red) onto this subspace maximizes the variance of the projected points (green)

12 Maximizes Variance To begin with, consider the projection onto a onedimensional space The direction of this space Variance Trick: How to maximize this?

13 Maximizes Variance (cont d) Eigenvalue

14 One Example

15 Extension to M-dimension Define additional principal components in an incremental fashion (details refer to Chapter 12 in Patter Recognition and Machine Learning) Conclusion of M dimension: The M eigenvectors u1,...,um of the data covariance matrix S corresponding to the M largest eigenvalues λ1,...,λm

16 Covariance Matrix Covariance

17 Fit an n-d Ellipsoid to the Data

18 T-SNE

19 T-SNE Particularly well-suited for embedding highdimensional data into a space of two or three dimensions, which can then be visualized in a scatter plot

20 Major Goal t-distributed stochastic neighbor embedding (t- SNE) minimizes the divergence between two distributions: a distribution that measures pairwise similarities of the input objects and a distribution that measures pairwise similarities of the corresponding low-dimensional points in the embedding.

21 Two Main Stages First, t-sne constructs a probability distribution over pairs of high-dimensional objects Similar objects have a high probability of being picked Dissimilar points have an extremely small probability of being picked

22 Example Step 1

23 Two Main Stages (cont d) Second, t-sne defines a probability distribution over the points in the low-dimensional map Similar to the one in high-dimensional space Minimizes the Kullback Leibler divergence between the two distributions with respect to the locations of the points in the map. Heavy-tailed student-t distribution

24 Example: Step Two

25 Example: Step Two Before optimization

26 Example: Final Result Student t-distribution Gaussian distribution

27 The t-student distribution The volume of the N-dimensional ball of radius r scales is When N is large, if we pick random points uniformly in the ball, most points will be close to the surface, and very few will be near the center.

28 The t-student distribution If the same Gaussian distribution is used for the low dimensional map points, not enough space is available in low dimensional space The crowding problem Use a t-student with one degree of freedom (or Cauchy) distribution instead for the map points. Has a much heavier tail than the Gaussian distribution, which compensates the original imbalance.

29 Comparison

30 The Distribution Model Probability model for high-dimensional data points Probability model for low-dimensional map points The different between two distributions

31 The Solution To minimize this score, we perform a gradient descent. The gradient can be computed analytically: Update y i iteratively

32 One Example

33 Example: MNIST Hand written digit (0-9)

34 Package Laurens van der Maate L.J.P. van der Maaten. Accelerating t-sne using Tree-Based Algorithms. Journal of Machine Learning Research 15(Oct): , L.J.P. van der Maaten and G.E. Hinton. Visualizing Non-Metric Similarities in Multiple Maps. Machine Learning 87(1):33-55, L.J.P. van der Maaten. Learning a Parametric Embedding by Preserving Local Structure. In Proceedings of the Twelfth International Conference on Artificial Intelligence & Statistics (AI- STATS), JMLR W&CP 5: , PDF L.J.P. van der Maaten and G.E. Hinton. Visualizing High- Dimensional Data Using t-sne. Journal of Machine Learning Research 9(Nov): , 2008.

35 Comparison PCA, MDS Linear technique Keep the low-dimensional representations of dissimilar data points far apart t-sne Non-linear technique Capture much of the local structure of the highdimensional data very well, while also revealing global structure such as the presence of clusters at multiple scales.

36 Comparison

37 Inselberg, "Multidimensional detective" (parallel coordiantes), 1997

38 Parallel Coordinates: Visual Design Max: 1 Min: dim1 dim2 dim3 dimn 0.25 Dimensions as parallel axes Data items as line segments Intersections on the axes indicates the values of the corresponding attributes

39 Parallel Coordinates: Pros and Cons!"Correlations among attributes studied by spotting the locations of the intersection points!"effective for revealing data distributions and functional dependencies #"Visual clutter due to limited space available for each parallel axis #"Axes packed very closely when dimensionality is high

40 Clustering and filtering approaches Dimension reordering approaches Visual enhancement approaches Out5d dataset (5 dimensions, data items)

41 Star Coordinates Scatterplots for higher dimensions: attribute as axis on a circle, data item as point Change the length of axis $ alters contribution of attribute Change the direction of axis $ angles not equal, adjusts correlations between attributes!"useful for gaining insight into hierarchically clustered datasets and for multi-factor analysis for decision-making

42 Table Lens Represents rows as data items and columns as attributes Each column viewed as histogram or plot Information along rows or columns interrelated!"uses the familiar concept table The table lens: merging graphical and symbolic representations in an interactive focus+ context visualization for tabular information

43 Scatterplot Matrix Scatterplot: 2 attributes projected along the x- and y-axis Collection of scatterplots is organized in a matrix!"straightforward% #"Important patterns in higher dimensions barely recognized #"Chaotic when number of data items too large

44 Outline Introduction Classification of Techniques Table Scatter Plot Matrices Projections Parallel Coordinates Pixel-Oriented Techniques Iconography Summary

45 Visualizations Advantages Disadvantages Clear visual patterns Clear visual patterns 1. Obscured semantics 2. Loss of information 3. Visual Clutter Visual Clutter Uses the familiar concept table Simple Support limited numbers of dimensions 1. Visual clutter 2. Unclear patterns

46 Further Reading Survey Dos Santos, Selan, and Ken Brodlie. "Gaining understanding of multivariate and multidimensional data through visualization." Computers & Graphics28.3 (2004): Website website/

47 Further Reading Evaluation Rubio-Sánchez, Manuel, et al. "A comparative study between RadViz and Star Coordinates." IEEE transactions on visualization and computer graphics 22.1 (2016):

48 References Rao, Ramana, and Stuart K. Card. "The table lens: merging graphical and symbolic representations in an interactive focus+ context visualization for tabular information." Proceedings of the SIGCHI conference on Human factors in computing systems. ACM, Gratzl, Samuel, et al. "Lineup: Visual analysis of multi-attribute rankings."ieee transactions on visualization and computer graphics (2013): van Wijk, Jarke J., and Robert van Liere. "HyperSlice: visualization of scalar functions of many variables." Proceedings of the 4th conference on Visualization'93. IEEE Computer Society, Kim, Hannah, et al. "InterAxis: Steering Scatterplot Axes via Observation-Level Interaction." IEEE transactions on visualization and computer graphics22.1 (2016):

49 References Maaten, Laurens van der, and Geoffrey Hinton. "Visualizing data using t-sne." Journal of Machine Learning Research 9.Nov (2008): Zhou, Hong, et al. "Visual clustering in parallel coordinates." Computer Graphics Forum. Vol. 27. No. 3., Ferdosi, Bilkis J., and Jos BTM Roerdink. "Visualizing High Dimensional Structures by Dimension Ordering and Filtering using Subspace Analysis."Computer Graphics Forum. Vol. 30. No. 3, Novotny, Matej, and Helwig Hauser. "Outlier-preserving focus+ context visualization in parallel coordinates." IEEE Transactions on Visualization and Computer Graphics 12.5 (2006):

50 References Keim, Daniel A., and H-P. Kriegel. "Visualization techniques for mining large databases: A comparison." IEEE Transactions on knowledge and data engineering 8.6 (1996):