多维数据的可视化知识发现与机器学习

机器学习

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"《Visual Knowledge Discovery and Machine Learning》是由Boris Kovalerchuk撰写的一本书，探讨了在数据科学背景下，如何结合视觉分析和机器学习技术来发现多维数据中的模式。书中强调了人类独特的感知和认知能力在知识发现过程中的重要性，并指出由于我们无法直接观察高维数据，需要借助增强的可视化工具（如“n-D眼镜”）来无损地将高维数据转换为二维表示，以便于理解和发现模式。尽管存在多种高维数据的可视化方法，但许多方法都是不可逆和有损的，限制了对高维数据模式的完整理解和发现。作者提出了一种新的可逆、无损的高维数据可视化表示方法——General Line Coordinate (GLC)及其具体规格，并通过多个实际应用案例，如挑战者号灾难数据、世界饥饿问题、语义变化分析、图像处理、医疗计算机辅助诊断、股票市场和货币汇率预测等，阐述了GLC的优势。这种可逆的可视化方法能够增强人类的认知能力，更有效地发现高维数据中的模式。" 这本书的核心知识点包括： 1. 数据科学的崛起：数据科学已成为当前计算机科学及众多领域研究和应用的重点，涵盖了知识发现、机器学习和数据挖掘。 2. 视觉分析与机器学习的结合：利用视觉和分析手段的机器学习技术是发现多维数据模式的有力工具，它允许人类的感知和认知能力直接参与模式发现过程。 3. 高维数据可视化挑战：人类的感知和认知能力局限于三维世界，无法直接观察高维数据，需要发展无损的可视化技术来完全呈现高维数据。 4. 可逆和无损的可视化方法：平行坐标和径向/星坐标是目前最强大的可逆、无损高维数据可视化方法，但存在遮挡问题。GLC作为一种新的可视化表示方法，旨在扩展这一类别，解决遮挡问题。 5. General Line Coordinate (GLC)：GLC是一种新的高维数据可视化表示形式，具有可逆性和无损性，能更好地支持知识发现。 6. 应用案例：书中通过多个实例展示了GLC在不同领域的应用，包括灾难数据分析、社会问题研究、文本语义分析、图像处理、医疗诊断、金融市场预测等，验证了GLC的有效性和优势。 7. 认知增强：可逆的高维数据可视化方法可以增强人类的认知能力，帮助更准确地发现和理解高维数据中的复杂模式。

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Since only 2-D and 3-D data can be directly visualized in the physical 3-D world,

visualization of n-D data becomes more difﬁcult with higher dimensions. Further

progress in data science require greater involvement of end users in constructing

machine learning models, along with more scalable, intuitive and efﬁcient visual

discovery methods and tools that we discuss in Chap. 12 .

In Data Mining (DM), Machine Learning (ML), and related ﬁelds one of these

challenges is ineffective heuristic initial selection of a class of models. Often we do

not have both (1) prior knowledg e to select a class of these models directly, and

(2) visualization tools to facilitate model selection losslessly and without occlusion.

In DM/ML often we are in essence guessing the class of models in advance, e.g.,

linear regression, decision trees, SVM, linear discrimination, linear programming, SOM

and so on. In contrast the success is evident in model selection in low-dimensional 2-D

or 3-D data that we can observe with a naked eye as we illustrate later. While iden-

tifying a class of ML models for a given data is rather an art than science, there is a

progress in automating this process. For instance, a method to learn a kernel function

for SVM automatically is proposed in (Nguyen et al. 2017).

In visualization of multi-dimensional data, the major challenges are (1) occlusion,

(2)lossofsigniﬁcant n-D information in 2-D visualization of n-D data, and (3) dif ﬁ-

culties of ﬁnding visual representation with clear and meaningful 2-D patterns.

While n-D data visualization is a well-studied area, none of the current solutions

fully address these long-standing challenges (Agrawal et al. 2015; Bertini, et al. 2011;

Ward et al. 2010;Inselberg2009;Simovetal.2008; Tergan and Keller 2005;Keim

et al. 2002; Wong and Bergeron 1997;HeerandPerer2014;Wangetal.2015). In this

book, we consider the problem of the loss of information in visualization as a problem

of developing reversible lossless visual representation of multidimensional (n-D) data

in 2-D and 3-D. This challenging task is addressed by generalizing Parallel and Radial

coordinates with a new concept of General Line Coordinates (GLC).

1.2 Visualization: From n-D Points to 2-D Points

The simplest method to represent n-D data in 2-D is splitting n-D space

 X

 …  X

into all 2-D projections X

 X

, i, j =1,…, n and showing

them to the user. It produces a large number of fragmented visual representations of

n-D data and destroys the integrity of n-D data. In each projection X

 X

, this

method maps each n-D point to a single 2-D point. We will call such mapping as

n-D point to 2-D-point mapping and denote is as P-to- P representation for short.

Multidimensional scaling (MDS) and other similar nonreversible lossy methods are

such point-to-point representations. These methods aim preserving the proximity of n-D

points in 2-D using speciﬁc metrics (Jäckle et al. 2016; Kruskal and Wish 1978;Mead

1992). It means that n-D information beyond proximity can be lost in 2-D in general,

because its preservation is not controlled. Next, the proximity captured by these

methods may or may not be relevant to the user’s task, such as classiﬁcation of n-D

points, when the proximity measure is imposed on the task externally not derived from

2 1 Motivation, Problems and Approach

it. As a result, such methods can drastically distort initial data structures (Duch et al.

2000) that were relevant to the user’s task. For instance, a formal proximity measure

such as the Euclidean metric can contradict meaningful similarity of n-D points known

in the given domain. Domain experts can know that n-D points a and b are closer to

each other than n-D points c and d,|a, b|<|c, d|, but the formal externally imposed

metric F may set up an opposite relation, F(a, b)>F(c, d). In contrast, lossless data

displays presented in this book provide opportunity to improve interpretability of

visualization result and its understanding by subject matter experts (SME).

The common expectation of metric approaches is that they will produce relatively

simple clouds of 2-Dpointson the plane with distinct lengths, widths, orientations,

crossings, and densities. Otherwise, if patterns differ from such clouds, these methods

do not help much to use other unique human visual perception and shape recognition

capabilities in visualization (Grishin 1982; Grishin et al. 2003). Together all these

deﬁc iencies lead to a shallow understanding of complex n-D data.

To cope with abilities of the vision system to observe directly only 2-D/3-D spaces,

many other common approaches such as Principal Components Analysis (PCA) also

project every n-D data point into a single 2-D or 3-D point. In PCA and similar

dimension reduction methods, it is done by plotting the two main components of these

n-D points (e.g., Jeong et al. 2009). These two components show only a fraction of all

information contained in these n-D points. There is no way to restore completely n-D

points from these two components in general beyond some very special datasets. In

other words, these methods do not provide an isomorphic (bijective, lossless, rever-

sible) mapping between an n-D dataset and a 2-D dataset. These methods provide only

a one-way irreversible mapping from an n-D dataset to a 2-D data set.

Such lossy visualization algorithms may not ﬁnd complex relations even after

multiple time-consuming adjustments of parameters of the visualization algorithms,

because they cut out needed information from entering the visualization channel. As

a result, decisions based on such truncated visual information can be incorrect

Thus, we have two major types of 2-D visualizations of n-D data available to be

combined in the hybrid approach:

(1) each n-D point is mapped to a 2-D point (P-to-P mapping), and

(2) each n-D point is mapped to a 2-D structure such as a graph (we denote this

mapping as P-to-G), which is the focus of this book.

Both types of mapping have their own advantages and disadvantages.

Principal Component Analysis (PCA) (Jolliffe 1986;Yin2002), Multidimensional

Scaling (MDS) (Kruskal and Wish 1978), Self-Organized maps (SOM) (Kohonen

1984), RadVis (Sharko et al. 2008) are examples of (1), and Parallel Coordinates

(PC) (Inselberg 2009), and General Line Coordinates (GLC) presented in this book are

examples of (2). The P-to-P representations (1) are not reversible (lossy), i.e., in general

there is no way to restore the n-D point from its 2-D representation. In contrast, PC and

GLC graphs are reversible as we discuss in depth later.

The next issue is preserving n-D distance in 2-D. W hile such P-to-P repre-

sentations as MDS and SOM are speciﬁcally designed to meet this goal, in fact,

they only minimize the mean difference in distance between the points in n-D and

1.2 Visualization: From n-D Points to 2-D Points 3

their representations in 2-D. PCA minimizes the mean- square difference between

the original points and the projected ones (Yin 2002). For individual points, the

difference can be quite large. For a 4-D hypercube SOM and MDS have Kruskal’s

stress values S

SOM

= 0.327 and S

MDS

= 0.312, respectively, i.e., on average the

distances in 2-D differ from distances in n-D over 30% (Duch et al. 2000).

Such high distortion of n-D distances (loss of the actual distance information)

can lead to misclassiﬁcation, when such corrupted 2-D distances are used for the

classiﬁcation in 2-D. This problem is well known and several attempts have been

made to address by controlling and decreasing it, e.g., for SOM in (Yin 2002). It

can lead to disasters and loss of life in tasks with high cost of error that are common

in medical, engineering and defense applications.

In curren t machine learning practice, 2-D representation is commonly used for

illustration and explanation of the ideas of the algorithms such as SVM or LDA, but

much less for actual discovery of n-D rules due to the difﬁculties to adequately

represent the n-D data in 2-D, which we discussed above. In the hybrid approach

that co mbined analytical and visual machi ne learning presented in this book the

visualization guides both:

• Getting the information about the structure of data, and pattern discovery,

• Finding most informative splits of data into the training–validation pairs for

evaluation of machine learning model s. This includes worst, best and median

split of data.

1.3 Visualization: From n-D Points to 2-D Structures

While mapping n-D points to 2-D points provides an intuitive and simple visual

metaphor for n-D data in 2-D, it is also a major source of the loss of information in

2-D visualization. For visualization methods discussed in the previous section, this

mapping is a self-in ﬂicted limitation. In fact, it is not mandatory for visualization of

n-D data to represent each n-D point as a single 2-D point.

Each n-D point can be represented as a 2-D structure or a glyph . Some of them

can be reversible and lossless. Several such representations are already well-known

for a long time, such as radial coordinates (star glyphs), parallel coordinates (PC),

bar- and pie-graphs, and heat maps. However, these methods have different lim-

itations on the size and dimension of data that are illustrated below.

Figure 1.1 shows two 7-D points A and B inBar(column)-graphchartandinParallel

Coordinates. In a bar-graph each value of coordinates of an n-D point is represented by

the height of a rectangle instead of a point on the axis in the Parallel Coordinates.

The PC lines in Fig. 1.1b can be obtained by connecting tops of the bars (columns)

7-D points A and B. The backward process allows getting Fig. 1.1afromFig.1.1b.

The major difference between these visualizations is in scalability. The length of

the Bar-graph will be 100 times wider than in Fig. 1.1a if we put 100 7-D points to

the Bar graph with the same width of the bars. It will not ﬁt the page. If we try to

keep the same size of the graph as in Fig. 1.1, then the width of bars will be 100

times smaller, making bars invisible.

4 1 Motivation, Problems and Approach

In contrast, PC and Radial coordinates (see Fig. 1.2a) can accommodate 100 lines

without increasing the size of the chart, but with signiﬁcant occlusion. An alternative

Bar-graph with bars for point B drawn on the same location as A (onthetopof

A without shifting to the right) will keep the size of the chart, but with severe occlusion.

The last three bars of point A will be completely covered by bars from point

B. The same will happen if lines in PC will be represented as ﬁlled areas. See

Fig. 1.2b. Thus, when we visualize only a singl e n-D point a bar-graph is equivalent

to the lines in PC. Both methods are lossless in this situation. For more n-D points,

these methods are not equivalent in general beyond some speciﬁc data.

Figure 1.2a shows points A and B in Radial (star) Coordinates and Fig. 1.3

shows 6-D point C = (2, 4, 6, 2, 5, 4) in the Area (pie ) chart and Radial (star)

Coordinates. The p ie-chart uses the height of sectors (or length of the sectors)

instead of the length of radii in the radial coordinates.

Tops of the pieces of the pie in Fig. 1.3a can be connected to get visualization of point

C in Radial Coordinates. The backward process allows getting Fig. 1.3afromFig.1.3b.

Thus, such pie-graph is equivalent to its representation in the Radial Coordinates.

As was pointed out above, more n-D points in the same plot occlude each other

very signiﬁcantly, making quickly these visual representations inefﬁcient. To avoid

X1 X2 X3 X4 X5 X6 X7 X1 X2 X3 X4 X5 X6 X7

(b) 7-D points A and B in Parallel Coordinates.(a) 7-D points A and B in Bar-

raph

Fig. 1.1 7D points A = (7, 9, 4, 10, 8, 3, 6) in red and B = (6, 8, 3, 9, 10, 4, 6) in blue in a

Bar-graph chart (a) and in Parallel coordinates (b)

(a) 7-D points A and B in Radial

Coordinates.

(b) 7-D points A and B in Area chart based on

PC.

X4X5

X1 X2 X3 X4 X5 X6 X7

Fig. 1.2 7D points A = (7, 9, 4, 10, 8, 3, 6) in red and B = (6, 8, 3, 9, 10, 4, 6) in Area-Graph

based on PC (b) and in Radial Coordinates (a)

1.3 Visualization: From n-D Points to 2-D Structures 5

occlusion, n-D points can be shown side -by-side in multiple plots not in a single

plot. In this case, we are limited by the number of the plots that can be shown

side-by-side on the screen and by perceptual abilities of humans to analyze multiple

plots at the same time.

Parallel and radial coordinates have fundamental advantage over bar- and pie-charts

allowing the visualization of larger n-D datasets with less occlusion. However parallel

and radial coordinates suffer from occlusion just for larger datasets.

To visualize each n-D data point x =(x

, x

, … x

) the heat map uses a line of n-

bars (cells) of the same size with values of color intensity of the bar (cell) matched

to the value of x

. While the heat map does not suffer from the occlusion, it is

limited in the number of n-D points and dimension n that can be presented to the

user on a single screen. It is also unable to show all n-D points that are close to the

given n-D point next to that n-D point. Only two n-D points can be shown on the

adjacent rows.

The discussed visualization approaches can be interpreted as speciﬁc glyph-

based approaches where each glyph is a sequence of bars (cells), segments, or

connected points speciﬁcally located on the plane in the parallel or radial way.

These visual representations provide homomorphism or isomorphism of each n-D

data point into visual features of some ﬁgures, e.g., a “star”.

Homomorphic mapping is a source of one of the difﬁculty of these visualiza-

tions, because it maps two or more equal n-D points to a single visual representation

(e.g., to a single polyline in the parallel coordinates). As a result, the information

about frequencies of n-D points in the dataset is lost in 2-D visualization.

Commonly it is addressed by drawing wider lines to represent more often n-D

points, but with higher occlusion. In the heat map all equal points can be preserved

at the cost of less numbe r of different n-D points shown.

(a) 6-D point C in Pie-chart.

(b) 6-D point C=(2,4,6,2,5,4) in Radial

Coordinates.

Fig. 1.3 6-D point C = (2, 4, 6, 2, 5, 4) in Pie-chart ( a) and Radial Coordinates (b)

6 1 Motivation, Problems and Approach

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多维数据的可视化知识发现与机器学习

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